1,1,68,114,0.0752448,"\int e^{\coth ^{-1}(a x)} x^3 \, dx","Integrate[E^ArcCoth[a*x]*x^3,x]","\frac{9 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+a x \sqrt{1-\frac{1}{a^2 x^2}} \left(6 a^3 x^3+8 a^2 x^2+9 a x+16\right)}{24 a^4}","\frac{3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a^2}+\frac{1}{4} x^4 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{x^3 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}+\frac{3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{8 a^4}+\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3}",1,"(a*Sqrt[1 - 1/(a^2*x^2)]*x*(16 + 9*a*x + 8*a^2*x^2 + 6*a^3*x^3) + 9*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(24*a^4)","A",0
2,1,60,90,0.0495397,"\int e^{\coth ^{-1}(a x)} x^2 \, dx","Integrate[E^ArcCoth[a*x]*x^2,x]","\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2+3 a x+4\right)+3 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{6 a^3}","\frac{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a}+\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^2}+\frac{1}{3} x^3 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a^3}",1,"(a*Sqrt[1 - 1/(a^2*x^2)]*x*(4 + 3*a*x + 2*a^2*x^2) + 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(6*a^3)","A",0
3,1,49,63,0.0374328,"\int e^{\coth ^{-1}(a x)} x \, dx","Integrate[E^ArcCoth[a*x]*x,x]","\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} (a x+2)+\log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{2 a^2}","\frac{1}{2} x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a^2}",1,"(a*Sqrt[1 - 1/(a^2*x^2)]*x*(2 + a*x) + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(2*a^2)","A",0
4,1,41,36,0.0240965,"\int e^{\coth ^{-1}(a x)} \, dx","Integrate[E^ArcCoth[a*x],x]","x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{\log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a}","x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}",1,"Sqrt[1 - 1/(a^2*x^2)]*x + Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]/a","A",0
5,1,36,22,0.0137802,"\int \frac{e^{\coth ^{-1}(a x)}}{x} \, dx","Integrate[E^ArcCoth[a*x]/x,x]","\log \left(x \left(\sqrt{\frac{a^2 x^2-1}{a^2 x^2}}+1\right)\right)-\sin ^{-1}\left(\frac{1}{a x}\right)","\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)-\csc ^{-1}(a x)",1,"-ArcSin[1/(a*x)] + Log[x*(1 + Sqrt[(-1 + a^2*x^2)/(a^2*x^2)])]","A",0
6,1,27,24,0.0198312,"\int \frac{e^{\coth ^{-1}(a x)}}{x^2} \, dx","Integrate[E^ArcCoth[a*x]/x^2,x]","a \left(\sqrt{1-\frac{1}{a^2 x^2}}-\sin ^{-1}\left(\frac{1}{a x}\right)\right)","a \sqrt{1-\frac{1}{a^2 x^2}}-a \csc ^{-1}(a x)",1,"a*(Sqrt[1 - 1/(a^2*x^2)] - ArcSin[1/(a*x)])","A",0
7,1,42,38,0.045028,"\int \frac{e^{\coth ^{-1}(a x)}}{x^3} \, dx","Integrate[E^ArcCoth[a*x]/x^3,x]","\frac{a \left(\sqrt{1-\frac{1}{a^2 x^2}} (2 a x+1)-a x \sin ^{-1}\left(\frac{1}{a x}\right)\right)}{2 x}","\frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a+\frac{1}{x}\right)-\frac{1}{2} a^2 \csc ^{-1}(a x)",1,"(a*(Sqrt[1 - 1/(a^2*x^2)]*(1 + 2*a*x) - a*x*ArcSin[1/(a*x)]))/(2*x)","A",0
8,1,51,75,0.0888502,"\int \frac{e^{\coth ^{-1}(a x)}}{x^4} \, dx","Integrate[E^ArcCoth[a*x]/x^4,x]","\frac{1}{6} a \left(\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left(4 a^2 x^2+3 a x+2\right)}{x^2}-3 a^2 \sin ^{-1}\left(\frac{1}{a x}\right)\right)","-\frac{1}{2} a^3 \csc ^{-1}(a x)+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 x}-\frac{1}{3} a^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}+a^3 \sqrt{1-\frac{1}{a^2 x^2}}",1,"(a*((Sqrt[1 - 1/(a^2*x^2)]*(2 + 3*a*x + 4*a^2*x^2))/x^2 - 3*a^2*ArcSin[1/(a*x)]))/6","A",0
9,1,59,88,0.1008056,"\int \frac{e^{\coth ^{-1}(a x)}}{x^5} \, dx","Integrate[E^ArcCoth[a*x]/x^5,x]","\frac{1}{24} a \left(\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left(16 a^3 x^3+9 a^2 x^2+8 a x+6\right)}{x^3}-9 a^3 \sin ^{-1}\left(\frac{1}{a x}\right)\right)","-\frac{3}{8} a^4 \csc ^{-1}(a x)+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 x^2}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}+\frac{1}{24} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left(16 a+\frac{9}{x}\right)",1,"(a*((Sqrt[1 - 1/(a^2*x^2)]*(6 + 8*a*x + 9*a^2*x^2 + 16*a^3*x^3))/x^3 - 9*a^3*ArcSin[1/(a*x)]))/24","A",0
10,1,43,43,0.0203527,"\int e^{2 \coth ^{-1}(a x)} x^3 \, dx","Integrate[E^(2*ArcCoth[a*x])*x^3,x]","\frac{2 \log (1-a x)}{a^4}+\frac{2 x}{a^3}+\frac{x^2}{a^2}+\frac{2 x^3}{3 a}+\frac{x^4}{4}","\frac{2 \log (1-a x)}{a^4}+\frac{2 x}{a^3}+\frac{x^2}{a^2}+\frac{2 x^3}{3 a}+\frac{x^4}{4}",1,"(2*x)/a^3 + x^2/a^2 + (2*x^3)/(3*a) + x^4/4 + (2*Log[1 - a*x])/a^4","A",1
11,1,33,33,0.0149638,"\int e^{2 \coth ^{-1}(a x)} x^2 \, dx","Integrate[E^(2*ArcCoth[a*x])*x^2,x]","\frac{2 \log (1-a x)}{a^3}+\frac{2 x}{a^2}+\frac{x^2}{a}+\frac{x^3}{3}","\frac{2 \log (1-a x)}{a^3}+\frac{2 x}{a^2}+\frac{x^2}{a}+\frac{x^3}{3}",1,"(2*x)/a^2 + x^2/a + x^3/3 + (2*Log[1 - a*x])/a^3","A",1
12,1,26,26,0.0132756,"\int e^{2 \coth ^{-1}(a x)} x \, dx","Integrate[E^(2*ArcCoth[a*x])*x,x]","\frac{2 \log (1-a x)}{a^2}+\frac{2 x}{a}+\frac{x^2}{2}","\frac{2 \log (1-a x)}{a^2}+\frac{2 x}{a}+\frac{x^2}{2}",1,"(2*x)/a + x^2/2 + (2*Log[1 - a*x])/a^2","A",1
13,1,14,14,0.0114635,"\int e^{2 \coth ^{-1}(a x)} \, dx","Integrate[E^(2*ArcCoth[a*x]),x]","\frac{2 \log (1-a x)}{a}+x","\frac{2 \log (1-a x)}{a}+x",1,"x + (2*Log[1 - a*x])/a","A",1
14,1,14,14,0.0084806,"\int \frac{e^{2 \coth ^{-1}(a x)}}{x} \, dx","Integrate[E^(2*ArcCoth[a*x])/x,x]","2 \log (1-a x)-\log (x)","2 \log (1-a x)-\log (x)",1,"-Log[x] + 2*Log[1 - a*x]","A",1
15,1,19,19,0.0107751,"\int \frac{e^{2 \coth ^{-1}(a x)}}{x^2} \, dx","Integrate[E^(2*ArcCoth[a*x])/x^2,x]","-2 a \log (x)+2 a \log (1-a x)+\frac{1}{x}","-2 a \log (x)+2 a \log (1-a x)+\frac{1}{x}",1,"x^(-1) - 2*a*Log[x] + 2*a*Log[1 - a*x]","A",1
16,1,33,33,0.0136216,"\int \frac{e^{2 \coth ^{-1}(a x)}}{x^3} \, dx","Integrate[E^(2*ArcCoth[a*x])/x^3,x]","-2 a^2 \log (x)+2 a^2 \log (1-a x)+\frac{2 a}{x}+\frac{1}{2 x^2}","-2 a^2 \log (x)+2 a^2 \log (1-a x)+\frac{2 a}{x}+\frac{1}{2 x^2}",1,"1/(2*x^2) + (2*a)/x - 2*a^2*Log[x] + 2*a^2*Log[1 - a*x]","A",1
17,1,40,40,0.0158526,"\int \frac{e^{2 \coth ^{-1}(a x)}}{x^4} \, dx","Integrate[E^(2*ArcCoth[a*x])/x^4,x]","-2 a^3 \log (x)+2 a^3 \log (1-a x)+\frac{2 a^2}{x}+\frac{a}{x^2}+\frac{1}{3 x^3}","-2 a^3 \log (x)+2 a^3 \log (1-a x)+\frac{2 a^2}{x}+\frac{a}{x^2}+\frac{1}{3 x^3}",1,"1/(3*x^3) + a/x^2 + (2*a^2)/x - 2*a^3*Log[x] + 2*a^3*Log[1 - a*x]","A",1
18,1,75,118,0.0876663,"\int e^{3 \coth ^{-1}(a x)} x^2 \, dx","Integrate[E^(3*ArcCoth[a*x])*x^2,x]","\frac{33 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^3 x^3+7 a^2 x^2+19 a x-52\right)}{a x-1}}{6 a^3}","\frac{3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a}+\frac{14 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^2}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left(a-\frac{1}{x}\right)}+\frac{1}{3} x^3 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{11 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a^3}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-52 + 19*a*x + 7*a^2*x^2 + 2*a^3*x^3))/(-1 + a*x) + 33*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(6*a^3)","A",0
19,1,66,92,0.0677071,"\int e^{3 \coth ^{-1}(a x)} x \, dx","Integrate[E^(3*ArcCoth[a*x])*x,x]","\frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(a^2 x^2+5 a x-14\right)}{a x-1}+9 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{2 a^2}","\frac{1}{2} x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{3 x \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a \left(a-\frac{1}{x}\right)}+\frac{9 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a^2}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-14 + 5*a*x + a^2*x^2))/(-1 + a*x) + 9*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(2*a^2)","A",0
20,1,54,62,0.0496871,"\int e^{3 \coth ^{-1}(a x)} \, dx","Integrate[E^(3*ArcCoth[a*x]),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x-5)}{a x-1}+\frac{3 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a}","x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a-\frac{1}{x}}+\frac{3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(-5 + a*x))/(-1 + a*x) + (3*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a","A",0
21,1,53,46,0.0625075,"\int \frac{e^{3 \coth ^{-1}(a x)}}{x} \, dx","Integrate[E^(3*ArcCoth[a*x])/x,x]","-\frac{4 a x \sqrt{1-\frac{1}{a^2 x^2}}}{a x-1}+\log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\sin ^{-1}\left(\frac{1}{a x}\right)","-\frac{4 a \sqrt{1-\frac{1}{a^2 x^2}}}{a-\frac{1}{x}}+\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+\csc ^{-1}(a x)",1,"(-4*a*Sqrt[1 - 1/(a^2*x^2)]*x)/(-1 + a*x) + ArcSin[1/(a*x)] + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]","A",0
22,1,41,51,0.0909998,"\int \frac{e^{3 \coth ^{-1}(a x)}}{x^2} \, dx","Integrate[E^(3*ArcCoth[a*x])/x^2,x]","\frac{a \sqrt{1-\frac{1}{a^2 x^2}} (1-5 a x)}{a x-1}+3 a \sin ^{-1}\left(\frac{1}{a x}\right)","-\frac{2 \left(a+\frac{1}{x}\right)^2}{a \sqrt{1-\frac{1}{a^2 x^2}}}-3 a \sqrt{1-\frac{1}{a^2 x^2}}+3 a \csc ^{-1}(a x)",1,"(a*Sqrt[1 - 1/(a^2*x^2)]*(1 - 5*a*x))/(-1 + a*x) + 3*a*ArcSin[1/(a*x)]","A",0
23,1,56,91,0.1020343,"\int \frac{e^{3 \coth ^{-1}(a x)}}{x^3} \, dx","Integrate[E^(3*ArcCoth[a*x])/x^3,x]","\frac{1}{2} a \left(\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left(-14 a^2 x^2+5 a x+1\right)}{x (a x-1)}+9 a \sin ^{-1}\left(\frac{1}{a x}\right)\right)","-\frac{9}{2} a^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{9}{2} a^2 \csc ^{-1}(a x)-\frac{a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{\left(a-\frac{1}{x}\right)^3}-\frac{3 a^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 \left(a-\frac{1}{x}\right)}",1,"(a*((Sqrt[1 - 1/(a^2*x^2)]*(1 + 5*a*x - 14*a^2*x^2))/(x*(-1 + a*x)) + 9*a*ArcSin[1/(a*x)]))/2","A",0
24,1,66,93,0.123883,"\int \frac{e^{3 \coth ^{-1}(a x)}}{x^4} \, dx","Integrate[E^(3*ArcCoth[a*x])/x^4,x]","\frac{1}{6} a \left(33 a^2 \sin ^{-1}\left(\frac{1}{a x}\right)+\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left(-52 a^3 x^3+19 a^2 x^2+7 a x+2\right)}{x^2 (a x-1)}\right)","\frac{11}{2} a^3 \csc ^{-1}(a x)-\frac{1}{6} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(28 a+\frac{3}{x}\right)-\frac{1}{3} a \sqrt{1-\frac{1}{a^2 x^2}} \left(3 a+\frac{1}{x}\right)^2-\frac{\left(a+\frac{1}{x}\right)^3}{\sqrt{1-\frac{1}{a^2 x^2}}}",1,"(a*((Sqrt[1 - 1/(a^2*x^2)]*(2 + 7*a*x + 19*a^2*x^2 - 52*a^3*x^3))/(x^2*(-1 + a*x)) + 33*a^2*ArcSin[1/(a*x)]))/6","A",0
25,1,57,57,0.0503101,"\int e^{4 \coth ^{-1}(a x)} x^3 \, dx","Integrate[E^(4*ArcCoth[a*x])*x^3,x]","\frac{4}{a^4 (1-a x)}+\frac{16 \log (1-a x)}{a^4}+\frac{12 x}{a^3}+\frac{4 x^2}{a^2}+\frac{4 x^3}{3 a}+\frac{x^4}{4}","\frac{4}{a^4 (1-a x)}+\frac{16 \log (1-a x)}{a^4}+\frac{12 x}{a^3}+\frac{4 x^2}{a^2}+\frac{4 x^3}{3 a}+\frac{x^4}{4}",1,"(12*x)/a^3 + (4*x^2)/a^2 + (4*x^3)/(3*a) + x^4/4 + 4/(a^4*(1 - a*x)) + (16*Log[1 - a*x])/a^4","A",1
26,1,47,47,0.03741,"\int e^{4 \coth ^{-1}(a x)} x^2 \, dx","Integrate[E^(4*ArcCoth[a*x])*x^2,x]","\frac{4}{a^3 (1-a x)}+\frac{12 \log (1-a x)}{a^3}+\frac{8 x}{a^2}+\frac{2 x^2}{a}+\frac{x^3}{3}","\frac{4}{a^3 (1-a x)}+\frac{12 \log (1-a x)}{a^3}+\frac{8 x}{a^2}+\frac{2 x^2}{a}+\frac{x^3}{3}",1,"(8*x)/a^2 + (2*x^2)/a + x^3/3 + 4/(a^3*(1 - a*x)) + (12*Log[1 - a*x])/a^3","A",1
27,1,39,39,0.0308899,"\int e^{4 \coth ^{-1}(a x)} x \, dx","Integrate[E^(4*ArcCoth[a*x])*x,x]","\frac{4}{a^2 (1-a x)}+\frac{8 \log (1-a x)}{a^2}+\frac{4 x}{a}+\frac{x^2}{2}","\frac{4}{a^2 (1-a x)}+\frac{8 \log (1-a x)}{a^2}+\frac{4 x}{a}+\frac{x^2}{2}",1,"(4*x)/a + x^2/2 + 4/(a^2*(1 - a*x)) + (8*Log[1 - a*x])/a^2","A",1
28,1,26,27,0.0194275,"\int e^{4 \coth ^{-1}(a x)} \, dx","Integrate[E^(4*ArcCoth[a*x]),x]","-\frac{4}{a (a x-1)}+\frac{4 \log (1-a x)}{a}+x","\frac{4}{a (1-a x)}+\frac{4 \log (1-a x)}{a}+x",1,"x - 4/(a*(-1 + a*x)) + (4*Log[1 - a*x])/a","A",1
29,1,13,13,0.0105809,"\int \frac{e^{4 \coth ^{-1}(a x)}}{x} \, dx","Integrate[E^(4*ArcCoth[a*x])/x,x]","\frac{4}{1-a x}+\log (x)","\frac{4}{1-a x}+\log (x)",1,"4/(1 - a*x) + Log[x]","A",1
30,1,32,32,0.0273423,"\int \frac{e^{4 \coth ^{-1}(a x)}}{x^2} \, dx","Integrate[E^(4*ArcCoth[a*x])/x^2,x]","\frac{4 a}{1-a x}+4 a \log (x)-4 a \log (1-a x)-\frac{1}{x}","\frac{4 a}{1-a x}+4 a \log (x)-4 a \log (1-a x)-\frac{1}{x}",1,"-x^(-1) + (4*a)/(1 - a*x) + 4*a*Log[x] - 4*a*Log[1 - a*x]","A",1
31,1,46,46,0.0322117,"\int \frac{e^{4 \coth ^{-1}(a x)}}{x^3} \, dx","Integrate[E^(4*ArcCoth[a*x])/x^3,x]","\frac{4 a^2}{1-a x}+8 a^2 \log (x)-8 a^2 \log (1-a x)-\frac{4 a}{x}-\frac{1}{2 x^2}","\frac{4 a^2}{1-a x}+8 a^2 \log (x)-8 a^2 \log (1-a x)-\frac{4 a}{x}-\frac{1}{2 x^2}",1,"-1/2*1/x^2 - (4*a)/x + (4*a^2)/(1 - a*x) + 8*a^2*Log[x] - 8*a^2*Log[1 - a*x]","A",1
32,1,54,54,0.0469307,"\int \frac{e^{4 \coth ^{-1}(a x)}}{x^4} \, dx","Integrate[E^(4*ArcCoth[a*x])/x^4,x]","\frac{4 a^3}{1-a x}+12 a^3 \log (x)-12 a^3 \log (1-a x)-\frac{8 a^2}{x}-\frac{2 a}{x^2}-\frac{1}{3 x^3}","\frac{4 a^3}{1-a x}+12 a^3 \log (x)-12 a^3 \log (1-a x)-\frac{8 a^2}{x}-\frac{2 a}{x^2}-\frac{1}{3 x^3}",1,"-1/3*1/x^3 - (2*a)/x^2 - (8*a^2)/x + (4*a^3)/(1 - a*x) + 12*a^3*Log[x] - 12*a^3*Log[1 - a*x]","A",1
33,1,68,114,0.0717015,"\int e^{-\coth ^{-1}(a x)} x^3 \, dx","Integrate[x^3/E^ArcCoth[a*x],x]","\frac{9 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+a x \sqrt{1-\frac{1}{a^2 x^2}} \left(6 a^3 x^3-8 a^2 x^2+9 a x-16\right)}{24 a^4}","\frac{3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a^2}+\frac{1}{4} x^4 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{x^3 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}+\frac{3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{8 a^4}-\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3}",1,"(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-16 + 9*a*x - 8*a^2*x^2 + 6*a^3*x^3) + 9*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(24*a^4)","A",0
34,1,60,90,0.0528811,"\int e^{-\coth ^{-1}(a x)} x^2 \, dx","Integrate[x^2/E^ArcCoth[a*x],x]","\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2-3 a x+4\right)-3 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{6 a^3}","-\frac{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a}+\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^2}+\frac{1}{3} x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a^3}",1,"(a*Sqrt[1 - 1/(a^2*x^2)]*x*(4 - 3*a*x + 2*a^2*x^2) - 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(6*a^3)","A",0
35,1,49,64,0.0495896,"\int e^{-\coth ^{-1}(a x)} x \, dx","Integrate[x/E^ArcCoth[a*x],x]","\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-2)+\log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{2 a^2}","\frac{1}{2} x^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a^2}",1,"(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-2 + a*x) + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(2*a^2)","A",0
36,1,42,37,0.0227142,"\int e^{-\coth ^{-1}(a x)} \, dx","Integrate[E^(-ArcCoth[a*x]),x]","x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{\log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a}","x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}",1,"Sqrt[1 - 1/(a^2*x^2)]*x - Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]/a","A",0
37,1,34,20,0.0155107,"\int \frac{e^{-\coth ^{-1}(a x)}}{x} \, dx","Integrate[1/(E^ArcCoth[a*x]*x),x]","\log \left(x \left(\sqrt{\frac{a^2 x^2-1}{a^2 x^2}}+1\right)\right)+\sin ^{-1}\left(\frac{1}{a x}\right)","\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+\csc ^{-1}(a x)",1,"ArcSin[1/(a*x)] + Log[x*(1 + Sqrt[(-1 + a^2*x^2)/(a^2*x^2)])]","A",0
38,1,26,25,0.0216263,"\int \frac{e^{-\coth ^{-1}(a x)}}{x^2} \, dx","Integrate[1/(E^ArcCoth[a*x]*x^2),x]","-a \left(\sqrt{1-\frac{1}{a^2 x^2}}+\sin ^{-1}\left(\frac{1}{a x}\right)\right)","a \left(-\csc ^{-1}(a x)\right)-a \sqrt{1-\frac{1}{a^2 x^2}}",1,"-(a*(Sqrt[1 - 1/(a^2*x^2)] + ArcSin[1/(a*x)]))","A",0
39,1,41,40,0.0502746,"\int \frac{e^{-\coth ^{-1}(a x)}}{x^3} \, dx","Integrate[1/(E^ArcCoth[a*x]*x^3),x]","\frac{a \left(\sqrt{1-\frac{1}{a^2 x^2}} (2 a x-1)+a x \sin ^{-1}\left(\frac{1}{a x}\right)\right)}{2 x}","\frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a-\frac{1}{x}\right)+\frac{1}{2} a^2 \csc ^{-1}(a x)",1,"(a*(Sqrt[1 - 1/(a^2*x^2)]*(-1 + 2*a*x) + a*x*ArcSin[1/(a*x)]))/(2*x)","A",0
40,1,52,76,0.084496,"\int \frac{e^{-\coth ^{-1}(a x)}}{x^4} \, dx","Integrate[1/(E^ArcCoth[a*x]*x^4),x]","-\frac{1}{2} a^3 \sin ^{-1}\left(\frac{1}{a x}\right)-\frac{a \sqrt{1-\frac{1}{a^2 x^2}} \left(4 a^2 x^2-3 a x+2\right)}{6 x^2}","-\frac{1}{2} a^3 \csc ^{-1}(a x)+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 x}+\frac{1}{3} a^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}-a^3 \sqrt{1-\frac{1}{a^2 x^2}}",1,"-1/6*(a*Sqrt[1 - 1/(a^2*x^2)]*(2 - 3*a*x + 4*a^2*x^2))/x^2 - (a^3*ArcSin[1/(a*x)])/2","A",0
41,1,59,88,0.1029605,"\int \frac{e^{-\coth ^{-1}(a x)}}{x^5} \, dx","Integrate[1/(E^ArcCoth[a*x]*x^5),x]","\frac{1}{24} a \left(9 a^3 \sin ^{-1}\left(\frac{1}{a x}\right)+\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left(16 a^3 x^3-9 a^2 x^2+8 a x-6\right)}{x^3}\right)","\frac{3}{8} a^4 \csc ^{-1}(a x)+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 x^2}-\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}+\frac{1}{24} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left(16 a-\frac{9}{x}\right)",1,"(a*((Sqrt[1 - 1/(a^2*x^2)]*(-6 + 8*a*x - 9*a^2*x^2 + 16*a^3*x^3))/x^3 + 9*a^3*ArcSin[1/(a*x)]))/24","A",0
42,1,42,42,0.0212692,"\int e^{-2 \coth ^{-1}(a x)} x^3 \, dx","Integrate[x^3/E^(2*ArcCoth[a*x]),x]","\frac{2 \log (a x+1)}{a^4}-\frac{2 x}{a^3}+\frac{x^2}{a^2}-\frac{2 x^3}{3 a}+\frac{x^4}{4}","\frac{2 \log (a x+1)}{a^4}-\frac{2 x}{a^3}+\frac{x^2}{a^2}-\frac{2 x^3}{3 a}+\frac{x^4}{4}",1,"(-2*x)/a^3 + x^2/a^2 - (2*x^3)/(3*a) + x^4/4 + (2*Log[1 + a*x])/a^4","A",1
43,1,33,33,0.015188,"\int e^{-2 \coth ^{-1}(a x)} x^2 \, dx","Integrate[x^2/E^(2*ArcCoth[a*x]),x]","-\frac{2 \log (a x+1)}{a^3}+\frac{2 x}{a^2}-\frac{x^2}{a}+\frac{x^3}{3}","-\frac{2 \log (a x+1)}{a^3}+\frac{2 x}{a^2}-\frac{x^2}{a}+\frac{x^3}{3}",1,"(2*x)/a^2 - x^2/a + x^3/3 - (2*Log[1 + a*x])/a^3","A",1
44,1,25,25,0.0134093,"\int e^{-2 \coth ^{-1}(a x)} x \, dx","Integrate[x/E^(2*ArcCoth[a*x]),x]","\frac{2 \log (a x+1)}{a^2}-\frac{2 x}{a}+\frac{x^2}{2}","\frac{2 \log (a x+1)}{a^2}-\frac{2 x}{a}+\frac{x^2}{2}",1,"(-2*x)/a + x^2/2 + (2*Log[1 + a*x])/a^2","A",1
45,1,13,13,0.0111607,"\int e^{-2 \coth ^{-1}(a x)} \, dx","Integrate[E^(-2*ArcCoth[a*x]),x]","x-\frac{2 \log (a x+1)}{a}","x-\frac{2 \log (a x+1)}{a}",1,"x - (2*Log[1 + a*x])/a","A",1
46,1,13,13,0.0074231,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{x} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*x),x]","2 \log (a x+1)-\log (x)","2 \log (a x+1)-\log (x)",1,"-Log[x] + 2*Log[1 + a*x]","A",1
47,1,18,18,0.0099781,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{x^2} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*x^2),x]","2 a \log (x)-2 a \log (a x+1)+\frac{1}{x}","2 a \log (x)-2 a \log (a x+1)+\frac{1}{x}",1,"x^(-1) + 2*a*Log[x] - 2*a*Log[1 + a*x]","A",1
48,1,32,32,0.0127827,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{x^3} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*x^3),x]","-2 a^2 \log (x)+2 a^2 \log (a x+1)-\frac{2 a}{x}+\frac{1}{2 x^2}","-2 a^2 \log (x)+2 a^2 \log (a x+1)-\frac{2 a}{x}+\frac{1}{2 x^2}",1,"1/(2*x^2) - (2*a)/x - 2*a^2*Log[x] + 2*a^2*Log[1 + a*x]","A",1
49,1,40,40,0.0150693,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{x^4} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*x^4),x]","2 a^3 \log (x)-2 a^3 \log (a x+1)+\frac{2 a^2}{x}-\frac{a}{x^2}+\frac{1}{3 x^3}","2 a^3 \log (x)-2 a^3 \log (a x+1)+\frac{2 a^2}{x}-\frac{a}{x^2}+\frac{1}{3 x^3}",1,"1/(3*x^3) - a/x^2 + (2*a^2)/x + 2*a^3*Log[x] - 2*a^3*Log[1 + a*x]","A",1
50,1,83,136,0.0948678,"\int e^{-3 \coth ^{-1}(a x)} x^3 \, dx","Integrate[x^3/E^(3*ArcCoth[a*x]),x]","\frac{51 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^4 x^4-6 a^3 x^3+11 a^2 x^2-29 a x-80\right)}{a x+1}}{8 a^4}","\frac{19 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a^2}+\frac{1}{4} x^4 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{x^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{51 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{8 a^4}-\frac{6 x \sqrt{1-\frac{1}{a^2 x^2}}}{a^3}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^3 \left(a+\frac{1}{x}\right)}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-80 - 29*a*x + 11*a^2*x^2 - 6*a^3*x^3 + 2*a^4*x^4))/(1 + a*x) + 51*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(8*a^4)","A",0
51,1,75,116,0.0834126,"\int e^{-3 \coth ^{-1}(a x)} x^2 \, dx","Integrate[x^2/E^(3*ArcCoth[a*x]),x]","\frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^3 x^3-7 a^2 x^2+19 a x+52\right)}{a x+1}-33 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{6 a^3}","-\frac{3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a}+\frac{14 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^2}+\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a^2 \left(a+\frac{1}{x}\right)}+\frac{1}{3} x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{11 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a^3}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(52 + 19*a*x - 7*a^2*x^2 + 2*a^3*x^3))/(1 + a*x) - 33*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(6*a^3)","A",0
52,1,66,90,0.0681707,"\int e^{-3 \coth ^{-1}(a x)} x \, dx","Integrate[x/E^(3*ArcCoth[a*x]),x]","\frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(a^2 x^2-5 a x-14\right)}{a x+1}+9 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{2 a^2}","\frac{1}{2} x^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 x \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a \left(a+\frac{1}{x}\right)}+\frac{9 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a^2}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-14 - 5*a*x + a^2*x^2))/(1 + a*x) + 9*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(2*a^2)","A",0
53,1,54,60,0.0512454,"\int e^{-3 \coth ^{-1}(a x)} \, dx","Integrate[E^(-3*ArcCoth[a*x]),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x+5)}{a x+1}-\frac{3 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a}","x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a+\frac{1}{x}}-\frac{3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(5 + a*x))/(1 + a*x) - (3*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a","A",0
54,1,55,46,0.0568116,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{x} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*x),x]","-\frac{4 a x \sqrt{1-\frac{1}{a^2 x^2}}}{a x+1}+\log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)-\sin ^{-1}\left(\frac{1}{a x}\right)","-\frac{4 a \sqrt{1-\frac{1}{a^2 x^2}}}{a+\frac{1}{x}}+\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)-\csc ^{-1}(a x)",1,"(-4*a*Sqrt[1 - 1/(a^2*x^2)]*x)/(1 + a*x) - ArcSin[1/(a*x)] + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]","A",0
55,1,41,53,0.0725368,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*x^2),x]","\frac{a \sqrt{1-\frac{1}{a^2 x^2}} (5 a x+1)}{a x+1}+3 a \sin ^{-1}\left(\frac{1}{a x}\right)","\frac{2 \left(a-\frac{1}{x}\right)^2}{a \sqrt{1-\frac{1}{a^2 x^2}}}+3 a \sqrt{1-\frac{1}{a^2 x^2}}+3 a \csc ^{-1}(a x)",1,"(a*Sqrt[1 - 1/(a^2*x^2)]*(1 + 5*a*x))/(1 + a*x) + 3*a*ArcSin[1/(a*x)]","A",0
56,1,56,87,0.1453633,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{x^3} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*x^3),x]","\frac{1}{2} a \left(\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left(-14 a^2 x^2-5 a x+1\right)}{x (a x+1)}-9 a \sin ^{-1}\left(\frac{1}{a x}\right)\right)","-\frac{9}{2} a^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{9}{2} a^2 \csc ^{-1}(a x)-\frac{a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{\left(a+\frac{1}{x}\right)^3}-\frac{3 a^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 \left(a+\frac{1}{x}\right)}",1,"(a*((Sqrt[1 - 1/(a^2*x^2)]*(1 - 5*a*x - 14*a^2*x^2))/(x*(1 + a*x)) - 9*a*ArcSin[1/(a*x)]))/2","A",0
57,1,66,96,0.113087,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{x^4} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*x^4),x]","\frac{1}{6} a \left(33 a^2 \sin ^{-1}\left(\frac{1}{a x}\right)+\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left(52 a^3 x^3+19 a^2 x^2-7 a x+2\right)}{x^2 (a x+1)}\right)","\frac{11}{2} a^3 \csc ^{-1}(a x)+\frac{1}{6} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(28 a-\frac{3}{x}\right)+\frac{1}{3} a \sqrt{1-\frac{1}{a^2 x^2}} \left(3 a-\frac{1}{x}\right)^2+\frac{\left(a-\frac{1}{x}\right)^3}{\sqrt{1-\frac{1}{a^2 x^2}}}",1,"(a*((Sqrt[1 - 1/(a^2*x^2)]*(2 - 7*a*x + 19*a^2*x^2 + 52*a^3*x^3))/(x^2*(1 + a*x)) + 33*a^2*ArcSin[1/(a*x)]))/6","A",0
58,1,75,133,0.0568092,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*x^5),x]","-\frac{51}{8} a^4 \sin ^{-1}\left(\frac{1}{a x}\right)-\frac{a \sqrt{1-\frac{1}{a^2 x^2}} \left(80 a^4 x^4+29 a^3 x^3-11 a^2 x^2+6 a x-2\right)}{8 x^3 (a x+1)}","-\frac{51}{8} a^4 \csc ^{-1}(a x)-\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{x^2}-\frac{a \left(a-\frac{1}{x}\right)^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}-\frac{27}{4} a^4 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{9}{8} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a-\frac{3}{x}\right)",1,"-1/8*(a*Sqrt[1 - 1/(a^2*x^2)]*(-2 + 6*a*x - 11*a^2*x^2 + 29*a^3*x^3 + 80*a^4*x^4))/(x^3*(1 + a*x)) - (51*a^4*ArcSin[1/(a*x)])/8","A",0
59,1,173,253,5.2243967,"\int e^{\frac{1}{2} \coth ^{-1}(a x)} x^4 \, dx","Integrate[E^(ArcCoth[a*x]/2)*x^4,x]","\frac{\frac{9620 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{34000 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}+\frac{64640 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^3}+\frac{62976 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^4}+\frac{24576 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^5}-465 \log \left(1-e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+465 \log \left(e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)+930 \tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{3840 a^5}","\frac{31 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{128 a^5}+\frac{31 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{128 a^5}+\frac{611 x \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{1920 a^4}+\frac{269 x^2 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{960 a^3}+\frac{11 x^3 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{48 a^2}+\frac{1}{5} x^5 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}+\frac{9 x^4 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{40 a}",1,"((24576*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^5 + (62976*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^4 + (64640*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^3 + (34000*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^2 + (9620*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x])) + 930*ArcTan[E^(ArcCoth[a*x]/2)] - 465*Log[1 - E^(ArcCoth[a*x]/2)] + 465*Log[1 + E^(ArcCoth[a*x]/2)])/(3840*a^5)","A",0
60,1,149,216,5.1917425,"\int e^{\frac{1}{2} \coth ^{-1}(a x)} x^3 \, dx","Integrate[E^(ArcCoth[a*x]/2)*x^3,x]","\frac{\frac{980 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{2512 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}+\frac{3200 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^3}+\frac{1536 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^4}-33 \log \left(1-e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+33 \log \left(e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)+66 \tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{384 a^4}","\frac{11 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{64 a^4}+\frac{11 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{64 a^4}+\frac{83 x \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{192 a^3}+\frac{29 x^2 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{96 a^2}+\frac{1}{4} x^4 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}+\frac{7 x^3 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{24 a}",1,"((1536*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^4 + (3200*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^3 + (2512*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^2 + (980*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x])) + 66*ArcTan[E^(ArcCoth[a*x]/2)] - 33*Log[1 - E^(ArcCoth[a*x]/2)] + 33*Log[1 + E^(ArcCoth[a*x]/2)])/(384*a^4)","A",0
61,1,399,179,9.064432,"\int e^{\frac{1}{2} \coth ^{-1}(a x)} x^2 \, dx","Integrate[E^(ArcCoth[a*x]/2)*x^2,x]","-\frac{e^{-\frac{7}{2} \coth ^{-1}(a x)} \left(1280 e^{8 \coth ^{-1}(a x)} \left(1346 e^{2 \coth ^{-1}(a x)}+557 e^{4 \coth ^{-1}(a x)}+821\right) \, _4F_3\left(2,2,2,\frac{9}{4};1,1,\frac{21}{4};e^{2 \coth ^{-1}(a x)}\right)+10240 e^{8 \coth ^{-1}(a x)} \left(42 e^{2 \coth ^{-1}(a x)}+19 e^{4 \coth ^{-1}(a x)}+23\right) \, _5F_4\left(2,2,2,2,\frac{9}{4};1,1,1,\frac{21}{4};e^{2 \coth ^{-1}(a x)}\right)+20480 e^{8 \coth ^{-1}(a x)} \, _6F_5\left(2,2,2,2,2,\frac{9}{4};1,1,1,1,\frac{21}{4};e^{2 \coth ^{-1}(a x)}\right)+40960 e^{10 \coth ^{-1}(a x)} \, _6F_5\left(2,2,2,2,2,\frac{9}{4};1,1,1,1,\frac{21}{4};e^{2 \coth ^{-1}(a x)}\right)+20480 e^{12 \coth ^{-1}(a x)} \, _6F_5\left(2,2,2,2,2,\frac{9}{4};1,1,1,1,\frac{21}{4};e^{2 \coth ^{-1}(a x)}\right)+732349800 e^{2 \coth ^{-1}(a x)} \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};e^{2 \coth ^{-1}(a x)}\right)-635067810 e^{4 \coth ^{-1}(a x)} \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};e^{2 \coth ^{-1}(a x)}\right)-384831720 e^{6 \coth ^{-1}(a x)} \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};e^{2 \coth ^{-1}(a x)}\right)+60913125 e^{8 \coth ^{-1}(a x)} \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};e^{2 \coth ^{-1}(a x)}\right)+1070609085 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};e^{2 \coth ^{-1}(a x)}\right)-946471617 e^{2 \coth ^{-1}(a x)}+369641285 e^{4 \coth ^{-1}(a x)}+351173641 e^{6 \coth ^{-1}(a x)}-23818496 e^{8 \coth ^{-1}(a x)}-1070609085\right)}{1909440 a^3}","\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{8 a^3}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{8 a^3}+\frac{11 x \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{24 a^2}+\frac{1}{3} x^3 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}+\frac{5 x^2 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{12 a}",1,"-1/1909440*(-1070609085 - 946471617*E^(2*ArcCoth[a*x]) + 369641285*E^(4*ArcCoth[a*x]) + 351173641*E^(6*ArcCoth[a*x]) - 23818496*E^(8*ArcCoth[a*x]) + 1070609085*Hypergeometric2F1[1/4, 1, 5/4, E^(2*ArcCoth[a*x])] + 732349800*E^(2*ArcCoth[a*x])*Hypergeometric2F1[1/4, 1, 5/4, E^(2*ArcCoth[a*x])] - 635067810*E^(4*ArcCoth[a*x])*Hypergeometric2F1[1/4, 1, 5/4, E^(2*ArcCoth[a*x])] - 384831720*E^(6*ArcCoth[a*x])*Hypergeometric2F1[1/4, 1, 5/4, E^(2*ArcCoth[a*x])] + 60913125*E^(8*ArcCoth[a*x])*Hypergeometric2F1[1/4, 1, 5/4, E^(2*ArcCoth[a*x])] + 1280*E^(8*ArcCoth[a*x])*(821 + 1346*E^(2*ArcCoth[a*x]) + 557*E^(4*ArcCoth[a*x]))*HypergeometricPFQ[{2, 2, 2, 9/4}, {1, 1, 21/4}, E^(2*ArcCoth[a*x])] + 10240*E^(8*ArcCoth[a*x])*(23 + 42*E^(2*ArcCoth[a*x]) + 19*E^(4*ArcCoth[a*x]))*HypergeometricPFQ[{2, 2, 2, 2, 9/4}, {1, 1, 1, 21/4}, E^(2*ArcCoth[a*x])] + 20480*E^(8*ArcCoth[a*x])*HypergeometricPFQ[{2, 2, 2, 2, 2, 9/4}, {1, 1, 1, 1, 21/4}, E^(2*ArcCoth[a*x])] + 40960*E^(10*ArcCoth[a*x])*HypergeometricPFQ[{2, 2, 2, 2, 2, 9/4}, {1, 1, 1, 1, 21/4}, E^(2*ArcCoth[a*x])] + 20480*E^(12*ArcCoth[a*x])*HypergeometricPFQ[{2, 2, 2, 2, 2, 9/4}, {1, 1, 1, 1, 21/4}, E^(2*ArcCoth[a*x])])/(a^3*E^((7*ArcCoth[a*x])/2))","C",0
62,1,66,142,0.1829681,"\int e^{\frac{1}{2} \coth ^{-1}(a x)} x \, dx","Integrate[E^(ArcCoth[a*x]/2)*x,x]","\frac{\frac{2 e^{\frac{1}{2} \coth ^{-1}(a x)} \left(5 e^{2 \coth ^{-1}(a x)}-1\right)}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}+\tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+\tanh ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{4 a^2}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{4 a^2}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{4 a^2}+\frac{1}{2} x^2 \left(1-\frac{1}{a x}\right)^{3/4} \left(\frac{1}{a x}+1\right)^{5/4}+\frac{x \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{4 a}",1,"((2*E^(ArcCoth[a*x]/2)*(-1 + 5*E^(2*ArcCoth[a*x])))/(-1 + E^(2*ArcCoth[a*x]))^2 + ArcTan[E^(ArcCoth[a*x]/2)] + ArcTanh[E^(ArcCoth[a*x]/2)])/(4*a^2)","A",0
63,1,51,96,0.0982498,"\int e^{\frac{1}{2} \coth ^{-1}(a x)} \, dx","Integrate[E^(ArcCoth[a*x]/2),x]","\frac{\frac{2 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+\tanh ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{a}","x \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{a}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{a}",1,"((2*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x])) + ArcTan[E^(ArcCoth[a*x]/2)] + ArcTanh[E^(ArcCoth[a*x]/2)])/a","A",0
64,1,30,291,0.0412091,"\int \frac{e^{\frac{1}{2} \coth ^{-1}(a x)}}{x} \, dx","Integrate[E^(ArcCoth[a*x]/2)/x,x]","\frac{8}{5} e^{\frac{5}{2} \coth ^{-1}(a x)} \, _2F_1\left(\frac{5}{8},1;\frac{13}{8};e^{4 \coth ^{-1}(a x)}\right)","\frac{\log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}-\frac{\log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}-\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)+2 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)+2 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)",1,"(8*E^((5*ArcCoth[a*x])/2)*Hypergeometric2F1[5/8, 1, 13/8, E^(4*ArcCoth[a*x])])/5","C",0
65,1,148,267,0.2576063,"\int \frac{e^{\frac{1}{2} \coth ^{-1}(a x)}}{x^2} \, dx","Integrate[E^(ArcCoth[a*x]/2)/x^2,x]","a \left(\frac{2 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}+1}+\frac{\log \left(-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right)}{2 \sqrt{2}}-\frac{\log \left(\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right)}{2 \sqrt{2}}+\frac{\tan ^{-1}\left(1-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)}{\sqrt{2}}\right)","a \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}+\frac{a \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{2 \sqrt{2}}-\frac{a \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{2 \sqrt{2}}-\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{\sqrt{2}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}",1,"a*((2*E^(ArcCoth[a*x]/2))/(1 + E^(2*ArcCoth[a*x])) + ArcTan[1 - Sqrt[2]*E^(ArcCoth[a*x]/2)]/Sqrt[2] - ArcTan[1 + Sqrt[2]*E^(ArcCoth[a*x]/2)]/Sqrt[2] + Log[1 - Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]]/(2*Sqrt[2]))","A",0
66,1,173,319,0.2314502,"\int \frac{e^{\frac{1}{2} \coth ^{-1}(a x)}}{x^3} \, dx","Integrate[E^(ArcCoth[a*x]/2)/x^3,x]","\frac{1}{16} a^2 \left(\frac{40 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}+1}-\frac{32 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}+1\right)^2}+\sqrt{2} \log \left(-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right)-\sqrt{2} \log \left(\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right)+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)\right)","\frac{1}{2} a^2 \left(1-\frac{1}{a x}\right)^{3/4} \left(\frac{1}{a x}+1\right)^{5/4}+\frac{1}{4} a^2 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}+\frac{a^2 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}-\frac{a^2 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}-\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{4 \sqrt{2}}+\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{4 \sqrt{2}}",1,"(a^2*((-32*E^(ArcCoth[a*x]/2))/(1 + E^(2*ArcCoth[a*x]))^2 + (40*E^(ArcCoth[a*x]/2))/(1 + E^(2*ArcCoth[a*x])) + 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*E^(ArcCoth[a*x]/2)] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*E^(ArcCoth[a*x]/2)] + Sqrt[2]*Log[1 - Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]] - Sqrt[2]*Log[1 + Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]]))/16","A",0
67,1,93,356,0.1142825,"\int \frac{e^{\frac{1}{2} \coth ^{-1}(a x)}}{x^4} \, dx","Integrate[E^(ArcCoth[a*x]/2)/x^4,x]","\frac{1}{96} a^3 \left(9 \text{RootSum}\left[\text{$\#$1}^4+1\&,\frac{\coth ^{-1}(a x)-2 \log \left(e^{\frac{1}{2} \coth ^{-1}(a x)}-\text{$\#$1}\right)}{\text{$\#$1}^3}\&\right]+\frac{8 e^{\frac{1}{2} \coth ^{-1}(a x)} \left(6 e^{2 \coth ^{-1}(a x)}+29 e^{4 \coth ^{-1}(a x)}+9\right)}{\left(e^{2 \coth ^{-1}(a x)}+1\right)^3}\right)","\frac{1}{12} a^3 \left(1-\frac{1}{a x}\right)^{3/4} \left(\frac{1}{a x}+1\right)^{5/4}+\frac{3}{8} a^3 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}+\frac{3 a^3 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{16 \sqrt{2}}-\frac{3 a^3 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{16 \sqrt{2}}-\frac{3 a^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{8 \sqrt{2}}+\frac{3 a^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}+\frac{a^2 \left(1-\frac{1}{a x}\right)^{3/4} \left(\frac{1}{a x}+1\right)^{5/4}}{3 x}",1,"(a^3*((8*E^(ArcCoth[a*x]/2)*(9 + 6*E^(2*ArcCoth[a*x]) + 29*E^(4*ArcCoth[a*x])))/(1 + E^(2*ArcCoth[a*x]))^3 + 9*RootSum[1 + #1^4 & , (ArcCoth[a*x] - 2*Log[E^(ArcCoth[a*x]/2) - #1])/#1^3 & ]))/96","C",0
68,1,173,253,5.2550956,"\int e^{\frac{3}{2} \coth ^{-1}(a x)} x^4 \, dx","Integrate[E^((3*ArcCoth[a*x])/2)*x^4,x]","\frac{\frac{5500 e^{\frac{3}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{14032 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}+\frac{23936 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^3}+\frac{22016 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^4}+\frac{8192 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^5}-1185 \log \left(1-e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+1185 \log \left(e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)-2370 \tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{1280 a^5}","-\frac{237 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{128 a^5}+\frac{237 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{128 a^5}+\frac{557 x \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{640 a^4}+\frac{157 x^2 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{320 a^3}+\frac{5 x^3 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{16 a^2}+\frac{1}{5} x^5 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}+\frac{11 x^4 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{40 a}",1,"((8192*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^5 + (22016*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^4 + (23936*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^3 + (14032*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^2 + (5500*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x])) - 2370*ArcTan[E^(ArcCoth[a*x]/2)] - 1185*Log[1 - E^(ArcCoth[a*x]/2)] + 1185*Log[1 + E^(ArcCoth[a*x]/2)])/(1280*a^5)","A",0
69,1,149,216,5.2043274,"\int e^{\frac{3}{2} \coth ^{-1}(a x)} x^3 \, dx","Integrate[E^((3*ArcCoth[a*x])/2)*x^3,x]","\frac{\frac{532 e^{\frac{3}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{1008 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}+\frac{1152 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^3}+\frac{512 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^4}-123 \log \left(1-e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+123 \log \left(e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)-246 \tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{128 a^4}","-\frac{123 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{64 a^4}+\frac{123 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{64 a^4}+\frac{63 x \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{64 a^3}+\frac{15 x^2 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{32 a^2}+\frac{1}{4} x^4 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}+\frac{3 x^3 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{8 a}",1,"((512*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^4 + (1152*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^3 + (1008*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^2 + (532*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x])) - 246*ArcTan[E^(ArcCoth[a*x]/2)] - 123*Log[1 - E^(ArcCoth[a*x]/2)] + 123*Log[1 + E^(ArcCoth[a*x]/2)])/(128*a^4)","A",0
70,1,125,179,5.1754936,"\int e^{\frac{3}{2} \coth ^{-1}(a x)} x^2 \, dx","Integrate[E^((3*ArcCoth[a*x])/2)*x^2,x]","\frac{\frac{180 e^{\frac{3}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{240 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}+\frac{128 e^{\frac{3}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^3}-51 \log \left(1-e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+51 \log \left(e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)-102 \tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{48 a^3}","-\frac{17 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{8 a^3}+\frac{17 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{8 a^3}+\frac{23 x \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{24 a^2}+\frac{1}{3} x^3 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}+\frac{7 x^2 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{12 a}",1,"((128*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^3 + (240*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^2 + (180*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x])) - 102*ArcTan[E^(ArcCoth[a*x]/2)] - 51*Log[1 - E^(ArcCoth[a*x]/2)] + 51*Log[1 + E^(ArcCoth[a*x]/2)])/(48*a^3)","A",0
71,1,70,142,0.1865027,"\int e^{\frac{3}{2} \coth ^{-1}(a x)} x \, dx","Integrate[E^((3*ArcCoth[a*x])/2)*x,x]","\frac{\frac{2 e^{\frac{3}{2} \coth ^{-1}(a x)} \left(7 e^{2 \coth ^{-1}(a x)}-3\right)}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}-9 \tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+9 \tanh ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{4 a^2}","-\frac{9 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{4 a^2}+\frac{9 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{4 a^2}+\frac{1}{2} x^2 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/4}+\frac{3 x \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{4 a}",1,"((2*E^((3*ArcCoth[a*x])/2)*(-3 + 7*E^(2*ArcCoth[a*x])))/(-1 + E^(2*ArcCoth[a*x]))^2 - 9*ArcTan[E^(ArcCoth[a*x]/2)] + 9*ArcTanh[E^(ArcCoth[a*x]/2)])/(4*a^2)","A",0
72,1,56,98,0.0665329,"\int e^{\frac{3}{2} \coth ^{-1}(a x)} \, dx","Integrate[E^((3*ArcCoth[a*x])/2),x]","\frac{8 e^{\frac{3}{2} \coth ^{-1}(a x)} \left(\left(e^{2 \coth ^{-1}(a x)}-1\right) \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};e^{2 \coth ^{-1}(a x)}\right)+1\right)}{a \left(e^{2 \coth ^{-1}(a x)}-1\right)}","x \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}-\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{a}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{a}",1,"(8*E^((3*ArcCoth[a*x])/2)*(1 + (-1 + E^(2*ArcCoth[a*x]))*Hypergeometric2F1[3/4, 2, 7/4, E^(2*ArcCoth[a*x])]))/(a*(-1 + E^(2*ArcCoth[a*x])))","C",0
73,1,30,291,0.0498743,"\int \frac{e^{\frac{3}{2} \coth ^{-1}(a x)}}{x} \, dx","Integrate[E^((3*ArcCoth[a*x])/2)/x,x]","\frac{8}{7} e^{\frac{7}{2} \coth ^{-1}(a x)} \, _2F_1\left(\frac{7}{8},1;\frac{15}{8};e^{4 \coth ^{-1}(a x)}\right)","-\frac{\log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}+\frac{\log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}-\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)-2 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)+2 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)",1,"(8*E^((7*ArcCoth[a*x])/2)*Hypergeometric2F1[7/8, 1, 15/8, E^(4*ArcCoth[a*x])])/7","C",0
74,1,46,268,0.0826137,"\int \frac{e^{\frac{3}{2} \coth ^{-1}(a x)}}{x^2} \, dx","Integrate[E^((3*ArcCoth[a*x])/2)/x^2,x]","-8 a e^{\frac{3}{2} \coth ^{-1}(a x)} \left(\, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-e^{2 \coth ^{-1}(a x)}\right)-\frac{1}{e^{2 \coth ^{-1}(a x)}+1}\right)","a \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}-\frac{3 a \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{2 \sqrt{2}}+\frac{3 a \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{2 \sqrt{2}}-\frac{3 a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{\sqrt{2}}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}",1,"-8*a*E^((3*ArcCoth[a*x])/2)*(-(1 + E^(2*ArcCoth[a*x]))^(-1) + Hypergeometric2F1[3/4, 2, 7/4, -E^(2*ArcCoth[a*x])])","C",0
75,1,76,319,0.0856466,"\int \frac{e^{\frac{3}{2} \coth ^{-1}(a x)}}{x^3} \, dx","Integrate[E^((3*ArcCoth[a*x])/2)/x^3,x]","-\frac{8}{3} a^2 e^{\frac{3}{2} \coth ^{-1}(a x)} \left(\, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-e^{2 \coth ^{-1}(a x)}\right)-3 \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-e^{2 \coth ^{-1}(a x)}\right)+2 \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-e^{2 \coth ^{-1}(a x)}\right)\right)","\frac{1}{2} a^2 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/4}+\frac{3}{4} a^2 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}-\frac{9 a^2 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}+\frac{9 a^2 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}-\frac{9 a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{4 \sqrt{2}}+\frac{9 a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{4 \sqrt{2}}",1,"(-8*a^2*E^((3*ArcCoth[a*x])/2)*(Hypergeometric2F1[3/4, 1, 7/4, -E^(2*ArcCoth[a*x])] - 3*Hypergeometric2F1[3/4, 2, 7/4, -E^(2*ArcCoth[a*x])] + 2*Hypergeometric2F1[3/4, 3, 7/4, -E^(2*ArcCoth[a*x])]))/3","C",0
76,1,93,356,0.132174,"\int \frac{e^{\frac{3}{2} \coth ^{-1}(a x)}}{x^4} \, dx","Integrate[E^((3*ArcCoth[a*x])/2)/x^4,x]","\frac{1}{96} a^3 \left(51 \text{RootSum}\left[\text{$\#$1}^4+1\&,\frac{\coth ^{-1}(a x)-2 \log \left(e^{\frac{1}{2} \coth ^{-1}(a x)}-\text{$\#$1}\right)}{\text{$\#$1}}\&\right]+\frac{8 e^{\frac{3}{2} \coth ^{-1}(a x)} \left(30 e^{2 \coth ^{-1}(a x)}+45 e^{4 \coth ^{-1}(a x)}+17\right)}{\left(e^{2 \coth ^{-1}(a x)}+1\right)^3}\right)","\frac{1}{4} a^3 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/4}+\frac{17}{24} a^3 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}-\frac{17 a^3 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{16 \sqrt{2}}+\frac{17 a^3 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{16 \sqrt{2}}-\frac{17 a^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{8 \sqrt{2}}+\frac{17 a^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}+\frac{a^2 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/4}}{3 x}",1,"(a^3*((8*E^((3*ArcCoth[a*x])/2)*(17 + 30*E^(2*ArcCoth[a*x]) + 45*E^(4*ArcCoth[a*x])))/(1 + E^(2*ArcCoth[a*x]))^3 + 51*RootSum[1 + #1^4 & , (ArcCoth[a*x] - 2*Log[E^(ArcCoth[a*x]/2) - #1])/#1 & ]))/96","C",0
77,1,198,287,5.2831738,"\int e^{\frac{5}{2} \coth ^{-1}(a x)} x^4 \, dx","Integrate[E^((5*ArcCoth[a*x])/2)*x^4,x]","\frac{-8 e^{\frac{1}{2} \coth ^{-1}(a x)}+\frac{4117 e^{\frac{1}{2} \coth ^{-1}(a x)}}{192 \left(e^{2 \coth ^{-1}(a x)}-1\right)}+\frac{1661 e^{\frac{1}{2} \coth ^{-1}(a x)}}{48 \left(e^{2 \coth ^{-1}(a x)}-1\right)^2}+\frac{233 e^{\frac{1}{2} \coth ^{-1}(a x)}}{6 \left(e^{2 \coth ^{-1}(a x)}-1\right)^3}+\frac{122 e^{\frac{1}{2} \coth ^{-1}(a x)}}{5 \left(e^{2 \coth ^{-1}(a x)}-1\right)^4}+\frac{32 e^{\frac{1}{2} \coth ^{-1}(a x)}}{5 \left(e^{2 \coth ^{-1}(a x)}-1\right)^5}-\frac{1003}{256} \log \left(1-e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+\frac{1003}{256} \log \left(e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)+\frac{1003}{128} \tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{a^5}","-\frac{26111 \sqrt[4]{\frac{1}{a x}+1}}{1920 a^5 \sqrt[4]{1-\frac{1}{a x}}}+\frac{1003 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{128 a^5}+\frac{1003 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{128 a^5}+\frac{5533 x \sqrt[4]{\frac{1}{a x}+1}}{1920 a^4 \sqrt[4]{1-\frac{1}{a x}}}+\frac{1189 x^2 \sqrt[4]{\frac{1}{a x}+1}}{960 a^3 \sqrt[4]{1-\frac{1}{a x}}}+\frac{181 x^3 \sqrt[4]{\frac{1}{a x}+1}}{240 a^2 \sqrt[4]{1-\frac{1}{a x}}}+\frac{x^5 \sqrt[4]{\frac{1}{a x}+1}}{5 \sqrt[4]{1-\frac{1}{a x}}}+\frac{21 x^4 \sqrt[4]{\frac{1}{a x}+1}}{40 a \sqrt[4]{1-\frac{1}{a x}}}",1,"(-8*E^(ArcCoth[a*x]/2) + (32*E^(ArcCoth[a*x]/2))/(5*(-1 + E^(2*ArcCoth[a*x]))^5) + (122*E^(ArcCoth[a*x]/2))/(5*(-1 + E^(2*ArcCoth[a*x]))^4) + (233*E^(ArcCoth[a*x]/2))/(6*(-1 + E^(2*ArcCoth[a*x]))^3) + (1661*E^(ArcCoth[a*x]/2))/(48*(-1 + E^(2*ArcCoth[a*x]))^2) + (4117*E^(ArcCoth[a*x]/2))/(192*(-1 + E^(2*ArcCoth[a*x]))) + (1003*ArcTan[E^(ArcCoth[a*x]/2)])/128 - (1003*Log[1 - E^(ArcCoth[a*x]/2)])/256 + (1003*Log[1 + E^(ArcCoth[a*x]/2)])/256)/a^5","A",0
78,1,161,250,5.2458224,"\int e^{\frac{5}{2} \coth ^{-1}(a x)} x^3 \, dx","Integrate[E^((5*ArcCoth[a*x])/2)*x^3,x]","\frac{-3072 e^{\frac{1}{2} \coth ^{-1}(a x)}+\frac{6292 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{7376 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}+\frac{5248 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^3}+\frac{1536 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^4}-1425 \log \left(1-e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+1425 \log \left(e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)+2850 \tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{384 a^4}","-\frac{2467 \sqrt[4]{\frac{1}{a x}+1}}{192 a^4 \sqrt[4]{1-\frac{1}{a x}}}+\frac{475 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{64 a^4}+\frac{475 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{64 a^4}+\frac{521 x \sqrt[4]{\frac{1}{a x}+1}}{192 a^3 \sqrt[4]{1-\frac{1}{a x}}}+\frac{113 x^2 \sqrt[4]{\frac{1}{a x}+1}}{96 a^2 \sqrt[4]{1-\frac{1}{a x}}}+\frac{x^4 \sqrt[4]{\frac{1}{a x}+1}}{4 \sqrt[4]{1-\frac{1}{a x}}}+\frac{17 x^3 \sqrt[4]{\frac{1}{a x}+1}}{24 a \sqrt[4]{1-\frac{1}{a x}}}",1,"(-3072*E^(ArcCoth[a*x]/2) + (1536*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^4 + (5248*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^3 + (7376*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^2 + (6292*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x])) + 2850*ArcTan[E^(ArcCoth[a*x]/2)] - 1425*Log[1 - E^(ArcCoth[a*x]/2)] + 1425*Log[1 + E^(ArcCoth[a*x]/2)])/(384*a^4)","A",0
79,1,137,213,5.2129733,"\int e^{\frac{5}{2} \coth ^{-1}(a x)} x^2 \, dx","Integrate[E^((5*ArcCoth[a*x])/2)*x^2,x]","\frac{-384 e^{\frac{1}{2} \coth ^{-1}(a x)}+\frac{548 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{400 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}+\frac{128 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^3}-165 \log \left(1-e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+165 \log \left(e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)+330 \tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{48 a^3}","-\frac{287 \sqrt[4]{\frac{1}{a x}+1}}{24 a^3 \sqrt[4]{1-\frac{1}{a x}}}+\frac{55 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{8 a^3}+\frac{55 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{8 a^3}+\frac{61 x \sqrt[4]{\frac{1}{a x}+1}}{24 a^2 \sqrt[4]{1-\frac{1}{a x}}}+\frac{x^3 \sqrt[4]{\frac{1}{a x}+1}}{3 \sqrt[4]{1-\frac{1}{a x}}}+\frac{13 x^2 \sqrt[4]{\frac{1}{a x}+1}}{12 a \sqrt[4]{1-\frac{1}{a x}}}",1,"(-384*E^(ArcCoth[a*x]/2) + (128*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^3 + (400*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^2 + (548*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x])) + 330*ArcTan[E^(ArcCoth[a*x]/2)] - 165*Log[1 - E^(ArcCoth[a*x]/2)] + 165*Log[1 + E^(ArcCoth[a*x]/2)])/(48*a^3)","A",0
80,1,80,176,0.2258893,"\int e^{\frac{5}{2} \coth ^{-1}(a x)} x \, dx","Integrate[E^((5*ArcCoth[a*x])/2)*x,x]","\frac{-\frac{2 e^{\frac{1}{2} \coth ^{-1}(a x)} \left(-45 e^{2 \coth ^{-1}(a x)}+16 e^{4 \coth ^{-1}(a x)}+25\right)}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}+25 \tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+25 \tanh ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{4 a^2}","-\frac{25 \sqrt[4]{\frac{1}{a x}+1}}{2 a^2 \sqrt[4]{1-\frac{1}{a x}}}+\frac{25 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{4 a^2}+\frac{25 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{4 a^2}+\frac{x^2 \left(\frac{1}{a x}+1\right)^{9/4}}{2 \sqrt[4]{1-\frac{1}{a x}}}+\frac{5 x \left(\frac{1}{a x}+1\right)^{5/4}}{4 a \sqrt[4]{1-\frac{1}{a x}}}",1,"((-2*E^(ArcCoth[a*x]/2)*(25 - 45*E^(2*ArcCoth[a*x]) + 16*E^(4*ArcCoth[a*x])))/(-1 + E^(2*ArcCoth[a*x]))^2 + 25*ArcTan[E^(ArcCoth[a*x]/2)] + 25*ArcTanh[E^(ArcCoth[a*x]/2)])/(4*a^2)","A",0
81,1,67,130,0.1432746,"\int e^{\frac{5}{2} \coth ^{-1}(a x)} \, dx","Integrate[E^((5*ArcCoth[a*x])/2),x]","\frac{-\frac{2 e^{\frac{1}{2} \coth ^{-1}(a x)} \left(4 e^{2 \coth ^{-1}(a x)}-5\right)}{e^{2 \coth ^{-1}(a x)}-1}+5 \tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+5 \tanh ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{a}","\frac{x \left(\frac{1}{a x}+1\right)^{5/4}}{\sqrt[4]{1-\frac{1}{a x}}}-\frac{10 \sqrt[4]{\frac{1}{a x}+1}}{a \sqrt[4]{1-\frac{1}{a x}}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{a}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{a}",1,"((-2*E^(ArcCoth[a*x]/2)*(-5 + 4*E^(2*ArcCoth[a*x])))/(-1 + E^(2*ArcCoth[a*x])) + 5*ArcTan[E^(ArcCoth[a*x]/2)] + 5*ArcTanh[E^(ArcCoth[a*x]/2)])/a","A",0
82,1,30,320,0.0746153,"\int \frac{e^{\frac{5}{2} \coth ^{-1}(a x)}}{x} \, dx","Integrate[E^((5*ArcCoth[a*x])/2)/x,x]","8 e^{\frac{1}{2} \coth ^{-1}(a x)} \left(\, _2F_1\left(\frac{1}{8},1;\frac{9}{8};e^{4 \coth ^{-1}(a x)}\right)-1\right)","-\frac{8 \sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}-\frac{\log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}+\frac{\log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}+\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)+2 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)+2 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)",1,"8*E^(ArcCoth[a*x]/2)*(-1 + Hypergeometric2F1[1/8, 1, 9/8, E^(4*ArcCoth[a*x])])","C",0
83,1,173,299,0.3959473,"\int \frac{e^{\frac{5}{2} \coth ^{-1}(a x)}}{x^2} \, dx","Integrate[E^((5*ArcCoth[a*x])/2)/x^2,x]","a \left(-\frac{10 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}+1}-\frac{8 e^{\frac{5}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}+1}-\frac{5 \log \left(-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right)}{2 \sqrt{2}}+\frac{5 \log \left(\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right)}{2 \sqrt{2}}-\frac{5 \tan ^{-1}\left(1-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{\sqrt{2}}+\frac{5 \tan ^{-1}\left(\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)}{\sqrt{2}}\right)","-\frac{4 a \left(\frac{1}{a x}+1\right)^{5/4}}{\sqrt[4]{1-\frac{1}{a x}}}-5 a \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}-\frac{5 a \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{2 \sqrt{2}}+\frac{5 a \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{2 \sqrt{2}}+\frac{5 a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{\sqrt{2}}-\frac{5 a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}",1,"a*((-10*E^(ArcCoth[a*x]/2))/(1 + E^(2*ArcCoth[a*x])) - (8*E^((5*ArcCoth[a*x])/2))/(1 + E^(2*ArcCoth[a*x])) - (5*ArcTan[1 - Sqrt[2]*E^(ArcCoth[a*x]/2)])/Sqrt[2] + (5*ArcTan[1 + Sqrt[2]*E^(ArcCoth[a*x]/2)])/Sqrt[2] - (5*Log[1 - Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]])/(2*Sqrt[2]) + (5*Log[1 + Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]])/(2*Sqrt[2]))","A",0
84,1,186,351,0.2853997,"\int \frac{e^{\frac{5}{2} \coth ^{-1}(a x)}}{x^3} \, dx","Integrate[E^((5*ArcCoth[a*x])/2)/x^3,x]","\frac{1}{16} a^2 \left(-128 e^{\frac{1}{2} \coth ^{-1}(a x)}-\frac{104 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}+1}+\frac{32 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}+1\right)^2}-25 \sqrt{2} \log \left(-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right)+25 \sqrt{2} \log \left(\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right)-50 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+50 \sqrt{2} \tan ^{-1}\left(\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)\right)","-\frac{2 a^2 \left(\frac{1}{a x}+1\right)^{9/4}}{\sqrt[4]{1-\frac{1}{a x}}}-\frac{5}{2} a^2 \left(1-\frac{1}{a x}\right)^{3/4} \left(\frac{1}{a x}+1\right)^{5/4}-\frac{25}{4} a^2 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}-\frac{25 a^2 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}+\frac{25 a^2 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}+\frac{25 a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{4 \sqrt{2}}-\frac{25 a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{4 \sqrt{2}}",1,"(a^2*(-128*E^(ArcCoth[a*x]/2) + (32*E^(ArcCoth[a*x]/2))/(1 + E^(2*ArcCoth[a*x]))^2 - (104*E^(ArcCoth[a*x]/2))/(1 + E^(2*ArcCoth[a*x])) - 50*Sqrt[2]*ArcTan[1 - Sqrt[2]*E^(ArcCoth[a*x]/2)] + 50*Sqrt[2]*ArcTan[1 + Sqrt[2]*E^(ArcCoth[a*x]/2)] - 25*Sqrt[2]*Log[1 - Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]] + 25*Sqrt[2]*Log[1 + Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]]))/16","A",0
85,1,104,385,0.154426,"\int \frac{e^{\frac{5}{2} \coth ^{-1}(a x)}}{x^4} \, dx","Integrate[E^((5*ArcCoth[a*x])/2)/x^4,x]","a^3 \left(-\frac{55}{32} \text{RootSum}\left[\text{$\#$1}^4+1\&,\frac{\coth ^{-1}(a x)-2 \log \left(e^{\frac{1}{2} \coth ^{-1}(a x)}-\text{$\#$1}\right)}{\text{$\#$1}^3}\&\right]-\frac{e^{\frac{1}{2} \coth ^{-1}(a x)} \left(462 e^{2 \coth ^{-1}(a x)}+425 e^{4 \coth ^{-1}(a x)}+96 e^{6 \coth ^{-1}(a x)}+165\right)}{12 \left(e^{2 \coth ^{-1}(a x)}+1\right)^3}\right)","-\frac{1}{3} a^3 \left(1-\frac{1}{a x}\right)^{3/4} \left(\frac{1}{a x}+1\right)^{9/4}-\frac{2 a^3 \left(\frac{1}{a x}+1\right)^{9/4}}{\sqrt[4]{1-\frac{1}{a x}}}-\frac{11}{4} a^3 \left(1-\frac{1}{a x}\right)^{3/4} \left(\frac{1}{a x}+1\right)^{5/4}-\frac{55}{8} a^3 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}-\frac{55 a^3 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{16 \sqrt{2}}+\frac{55 a^3 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{16 \sqrt{2}}+\frac{55 a^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{8 \sqrt{2}}-\frac{55 a^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}",1,"a^3*(-1/12*(E^(ArcCoth[a*x]/2)*(165 + 462*E^(2*ArcCoth[a*x]) + 425*E^(4*ArcCoth[a*x]) + 96*E^(6*ArcCoth[a*x])))/(1 + E^(2*ArcCoth[a*x]))^3 - (55*RootSum[1 + #1^4 & , (ArcCoth[a*x] - 2*Log[E^(ArcCoth[a*x]/2) - #1])/#1^3 & ])/32)","C",0
86,1,173,253,5.3019134,"\int e^{-\frac{1}{2} \coth ^{-1}(a x)} x^4 \, dx","Integrate[x^4/E^(ArcCoth[a*x]/2),x]","\frac{\frac{9620 e^{\frac{3}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}-\frac{34000 e^{\frac{7}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}+\frac{64640 e^{\frac{11}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^3}-\frac{62976 e^{\frac{15}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^4}+\frac{24576 e^{\frac{19}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^5}+465 \log \left(1-e^{-\frac{1}{2} \coth ^{-1}(a x)}\right)-465 \log \left(e^{-\frac{1}{2} \coth ^{-1}(a x)}+1\right)-930 \tan ^{-1}\left(e^{-\frac{1}{2} \coth ^{-1}(a x)}\right)}{3840 a^5}","\frac{31 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{128 a^5}-\frac{31 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{128 a^5}+\frac{611 x \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{1920 a^4}-\frac{269 x^2 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{960 a^3}+\frac{11 x^3 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{48 a^2}+\frac{1}{5} x^5 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}-\frac{9 x^4 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{40 a}",1,"((24576*E^((19*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^5 - (62976*E^((15*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^4 + (64640*E^((11*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^3 - (34000*E^((7*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^2 + (9620*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x])) - 930*ArcTan[E^(-1/2*ArcCoth[a*x])] + 465*Log[1 - E^(-1/2*ArcCoth[a*x])] - 465*Log[1 + E^(-1/2*ArcCoth[a*x])])/(3840*a^5)","A",0
87,1,149,216,5.2558704,"\int e^{-\frac{1}{2} \coth ^{-1}(a x)} x^3 \, dx","Integrate[x^3/E^(ArcCoth[a*x]/2),x]","\frac{-\frac{980 e^{\frac{3}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{2512 e^{\frac{7}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}-\frac{3200 e^{\frac{11}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^3}+\frac{1536 e^{\frac{15}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^4}-33 \log \left(1-e^{-\frac{1}{2} \coth ^{-1}(a x)}\right)+33 \log \left(e^{-\frac{1}{2} \coth ^{-1}(a x)}+1\right)+66 \tan ^{-1}\left(e^{-\frac{1}{2} \coth ^{-1}(a x)}\right)}{384 a^4}","-\frac{11 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{64 a^4}+\frac{11 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{64 a^4}-\frac{83 x \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{192 a^3}+\frac{29 x^2 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{96 a^2}+\frac{1}{4} x^4 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}-\frac{7 x^3 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{24 a}",1,"((1536*E^((15*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^4 - (3200*E^((11*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^3 + (2512*E^((7*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^2 - (980*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x])) + 66*ArcTan[E^(-1/2*ArcCoth[a*x])] - 33*Log[1 - E^(-1/2*ArcCoth[a*x])] + 33*Log[1 + E^(-1/2*ArcCoth[a*x])])/(384*a^4)","A",0
88,1,389,179,9.0292932,"\int e^{-\frac{1}{2} \coth ^{-1}(a x)} x^2 \, dx","Integrate[x^2/E^(ArcCoth[a*x]/2),x]","-\frac{e^{-\frac{5}{2} \coth ^{-1}(a x)} \left(256 e^{6 \coth ^{-1}(a x)} \left(1090 e^{2 \coth ^{-1}(a x)}+437 e^{4 \coth ^{-1}(a x)}+685\right) \, _4F_3\left(\frac{7}{4},2,2,2;1,1,\frac{19}{4};e^{2 \coth ^{-1}(a x)}\right)+2048 e^{6 \coth ^{-1}(a x)} \left(38 e^{2 \coth ^{-1}(a x)}+17 e^{4 \coth ^{-1}(a x)}+21\right) \, _5F_4\left(\frac{7}{4},2,2,2,2;1,1,1,\frac{19}{4};e^{2 \coth ^{-1}(a x)}\right)+4096 e^{6 \coth ^{-1}(a x)} \, _6F_5\left(\frac{7}{4},2,2,2,2,2;1,1,1,1,\frac{19}{4};e^{2 \coth ^{-1}(a x)}\right)+8192 e^{8 \coth ^{-1}(a x)} \, _6F_5\left(\frac{7}{4},2,2,2,2,2;1,1,1,1,\frac{19}{4};e^{2 \coth ^{-1}(a x)}\right)+4096 e^{10 \coth ^{-1}(a x)} \, _6F_5\left(\frac{7}{4},2,2,2,2,2;1,1,1,1,\frac{19}{4};e^{2 \coth ^{-1}(a x)}\right)+17244920 e^{2 \coth ^{-1}(a x)} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};e^{2 \coth ^{-1}(a x)}\right)-9077530 e^{4 \coth ^{-1}(a x)} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};e^{2 \coth ^{-1}(a x)}\right)-7043960 e^{6 \coth ^{-1}(a x)} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};e^{2 \coth ^{-1}(a x)}\right)+446985 e^{8 \coth ^{-1}(a x)} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};e^{2 \coth ^{-1}(a x)}\right)+22034705 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};e^{2 \coth ^{-1}(a x)}\right)-26688365 e^{2 \coth ^{-1}(a x)}-3731255 e^{4 \coth ^{-1}(a x)}+3122405 e^{6 \coth ^{-1}(a x)}-22034705\right)}{221760 a^3}","\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{8 a^3}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{8 a^3}+\frac{11 x \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{24 a^2}+\frac{1}{3} x^3 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}-\frac{5 x^2 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{12 a}",1,"-1/221760*(-22034705 - 26688365*E^(2*ArcCoth[a*x]) - 3731255*E^(4*ArcCoth[a*x]) + 3122405*E^(6*ArcCoth[a*x]) + 22034705*Hypergeometric2F1[3/4, 1, 7/4, E^(2*ArcCoth[a*x])] + 17244920*E^(2*ArcCoth[a*x])*Hypergeometric2F1[3/4, 1, 7/4, E^(2*ArcCoth[a*x])] - 9077530*E^(4*ArcCoth[a*x])*Hypergeometric2F1[3/4, 1, 7/4, E^(2*ArcCoth[a*x])] - 7043960*E^(6*ArcCoth[a*x])*Hypergeometric2F1[3/4, 1, 7/4, E^(2*ArcCoth[a*x])] + 446985*E^(8*ArcCoth[a*x])*Hypergeometric2F1[3/4, 1, 7/4, E^(2*ArcCoth[a*x])] + 256*E^(6*ArcCoth[a*x])*(685 + 1090*E^(2*ArcCoth[a*x]) + 437*E^(4*ArcCoth[a*x]))*HypergeometricPFQ[{7/4, 2, 2, 2}, {1, 1, 19/4}, E^(2*ArcCoth[a*x])] + 2048*E^(6*ArcCoth[a*x])*(21 + 38*E^(2*ArcCoth[a*x]) + 17*E^(4*ArcCoth[a*x]))*HypergeometricPFQ[{7/4, 2, 2, 2, 2}, {1, 1, 1, 19/4}, E^(2*ArcCoth[a*x])] + 4096*E^(6*ArcCoth[a*x])*HypergeometricPFQ[{7/4, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 19/4}, E^(2*ArcCoth[a*x])] + 8192*E^(8*ArcCoth[a*x])*HypergeometricPFQ[{7/4, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 19/4}, E^(2*ArcCoth[a*x])] + 4096*E^(10*ArcCoth[a*x])*HypergeometricPFQ[{7/4, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 19/4}, E^(2*ArcCoth[a*x])])/(a^3*E^((5*ArcCoth[a*x])/2))","C",0
89,1,66,142,0.1694776,"\int e^{-\frac{1}{2} \coth ^{-1}(a x)} x \, dx","Integrate[x/E^(ArcCoth[a*x]/2),x]","\frac{-\frac{2 e^{\frac{3}{2} \coth ^{-1}(a x)} \left(e^{2 \coth ^{-1}(a x)}-5\right)}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}-\tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+\tanh ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{4 a^2}","-\frac{\tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{4 a^2}+\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{4 a^2}+\frac{1}{2} x^2 \left(1-\frac{1}{a x}\right)^{5/4} \left(\frac{1}{a x}+1\right)^{3/4}-\frac{x \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}}{4 a}",1,"((-2*E^((3*ArcCoth[a*x])/2)*(-5 + E^(2*ArcCoth[a*x])))/(-1 + E^(2*ArcCoth[a*x]))^2 - ArcTan[E^(ArcCoth[a*x]/2)] + ArcTanh[E^(ArcCoth[a*x]/2)])/(4*a^2)","A",0
90,1,33,97,0.0464847,"\int e^{-\frac{1}{2} \coth ^{-1}(a x)} \, dx","Integrate[E^(-1/2*ArcCoth[a*x]),x]","-\frac{8 e^{\frac{3}{2} \coth ^{-1}(a x)} \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};e^{2 \coth ^{-1}(a x)}\right)}{3 a}","x \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}+\frac{\tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{a}-\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{a}",1,"(-8*E^((3*ArcCoth[a*x])/2)*Hypergeometric2F1[3/4, 2, 7/4, E^(2*ArcCoth[a*x])])/(3*a)","C",0
91,1,30,291,0.0525135,"\int \frac{e^{-\frac{1}{2} \coth ^{-1}(a x)}}{x} \, dx","Integrate[1/(E^(ArcCoth[a*x]/2)*x),x]","\frac{8}{3} e^{\frac{3}{2} \coth ^{-1}(a x)} \, _2F_1\left(\frac{3}{8},1;\frac{11}{8};e^{4 \coth ^{-1}(a x)}\right)","\frac{\log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}-\frac{\log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}+\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)-2 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)+2 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)",1,"(8*E^((3*ArcCoth[a*x])/2)*Hypergeometric2F1[3/8, 1, 11/8, E^(4*ArcCoth[a*x])])/3","C",0
92,1,33,268,0.0515156,"\int \frac{e^{-\frac{1}{2} \coth ^{-1}(a x)}}{x^2} \, dx","Integrate[1/(E^(ArcCoth[a*x]/2)*x^2),x]","-\frac{8}{3} a e^{\frac{3}{2} \coth ^{-1}(a x)} \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-e^{2 \coth ^{-1}(a x)}\right)","-a \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}-\frac{a \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{2 \sqrt{2}}+\frac{a \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{2 \sqrt{2}}-\frac{a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{\sqrt{2}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}",1,"(-8*a*E^((3*ArcCoth[a*x])/2)*Hypergeometric2F1[3/4, 2, 7/4, -E^(2*ArcCoth[a*x])])/3","C",0
93,1,56,319,0.0777881,"\int \frac{e^{-\frac{1}{2} \coth ^{-1}(a x)}}{x^3} \, dx","Integrate[1/(E^(ArcCoth[a*x]/2)*x^3),x]","-\frac{8}{3} a^2 e^{\frac{3}{2} \coth ^{-1}(a x)} \left(\, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-e^{2 \coth ^{-1}(a x)}\right)-2 \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-e^{2 \coth ^{-1}(a x)}\right)\right)","\frac{1}{2} a^2 \left(1-\frac{1}{a x}\right)^{5/4} \left(\frac{1}{a x}+1\right)^{3/4}+\frac{1}{4} a^2 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}+\frac{a^2 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}-\frac{a^2 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}+\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{4 \sqrt{2}}-\frac{a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{4 \sqrt{2}}",1,"(-8*a^2*E^((3*ArcCoth[a*x])/2)*(Hypergeometric2F1[3/4, 2, 7/4, -E^(2*ArcCoth[a*x])] - 2*Hypergeometric2F1[3/4, 3, 7/4, -E^(2*ArcCoth[a*x])]))/3","C",0
94,1,93,356,0.1338039,"\int \frac{e^{-\frac{1}{2} \coth ^{-1}(a x)}}{x^4} \, dx","Integrate[1/(E^(ArcCoth[a*x]/2)*x^4),x]","\frac{1}{96} a^3 \left(9 \text{RootSum}\left[\text{$\#$1}^4+1\&,\frac{2 \log \left(e^{-\frac{1}{2} \coth ^{-1}(a x)}-\text{$\#$1}\right)+\coth ^{-1}(a x)}{\text{$\#$1}^3}\&\right]-\frac{8 e^{\frac{3}{2} \coth ^{-1}(a x)} \left(6 e^{2 \coth ^{-1}(a x)}+9 e^{4 \coth ^{-1}(a x)}+29\right)}{\left(e^{2 \coth ^{-1}(a x)}+1\right)^3}\right)","-\frac{1}{12} a^3 \left(1-\frac{1}{a x}\right)^{5/4} \left(\frac{1}{a x}+1\right)^{3/4}-\frac{3}{8} a^3 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}-\frac{3 a^3 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{16 \sqrt{2}}+\frac{3 a^3 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{16 \sqrt{2}}-\frac{3 a^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{8 \sqrt{2}}+\frac{3 a^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}+\frac{a^2 \left(1-\frac{1}{a x}\right)^{5/4} \left(\frac{1}{a x}+1\right)^{3/4}}{3 x}",1,"(a^3*((-8*E^((3*ArcCoth[a*x])/2)*(29 + 6*E^(2*ArcCoth[a*x]) + 9*E^(4*ArcCoth[a*x])))/(1 + E^(2*ArcCoth[a*x]))^3 + 9*RootSum[1 + #1^4 & , (ArcCoth[a*x] + 2*Log[E^(-1/2*ArcCoth[a*x]) - #1])/#1^3 & ]))/96","C",0
95,1,173,253,5.3388992,"\int e^{-\frac{3}{2} \coth ^{-1}(a x)} x^4 \, dx","Integrate[x^4/E^((3*ArcCoth[a*x])/2),x]","\frac{\frac{5500 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}-\frac{14032 e^{\frac{5}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}+\frac{23936 e^{\frac{9}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^3}-\frac{22016 e^{\frac{13}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^4}+\frac{8192 e^{\frac{17}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^5}+1185 \log \left(1-e^{-\frac{1}{2} \coth ^{-1}(a x)}\right)-1185 \log \left(e^{-\frac{1}{2} \coth ^{-1}(a x)}+1\right)+2370 \tan ^{-1}\left(e^{-\frac{1}{2} \coth ^{-1}(a x)}\right)}{1280 a^5}","-\frac{237 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{128 a^5}-\frac{237 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{128 a^5}+\frac{557 x \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{640 a^4}-\frac{157 x^2 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{320 a^3}+\frac{5 x^3 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{16 a^2}+\frac{1}{5} x^5 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}-\frac{11 x^4 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{40 a}",1,"((8192*E^((17*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^5 - (22016*E^((13*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^4 + (23936*E^((9*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^3 - (14032*E^((5*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^2 + (5500*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x])) + 2370*ArcTan[E^(-1/2*ArcCoth[a*x])] + 1185*Log[1 - E^(-1/2*ArcCoth[a*x])] - 1185*Log[1 + E^(-1/2*ArcCoth[a*x])])/(1280*a^5)","A",0
96,1,149,216,5.2709062,"\int e^{-\frac{3}{2} \coth ^{-1}(a x)} x^3 \, dx","Integrate[x^3/E^((3*ArcCoth[a*x])/2),x]","\frac{-\frac{532 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{1008 e^{\frac{5}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}-\frac{1152 e^{\frac{9}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^3}+\frac{512 e^{\frac{13}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^4}-123 \log \left(1-e^{-\frac{1}{2} \coth ^{-1}(a x)}\right)+123 \log \left(e^{-\frac{1}{2} \coth ^{-1}(a x)}+1\right)-246 \tan ^{-1}\left(e^{-\frac{1}{2} \coth ^{-1}(a x)}\right)}{128 a^4}","\frac{123 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{64 a^4}+\frac{123 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{64 a^4}-\frac{63 x \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{64 a^3}+\frac{15 x^2 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{32 a^2}+\frac{1}{4} x^4 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}-\frac{3 x^3 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{8 a}",1,"((512*E^((13*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^4 - (1152*E^((9*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^3 + (1008*E^((5*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^2 - (532*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x])) - 246*ArcTan[E^(-1/2*ArcCoth[a*x])] - 123*Log[1 - E^(-1/2*ArcCoth[a*x])] + 123*Log[1 + E^(-1/2*ArcCoth[a*x])])/(128*a^4)","A",0
97,1,389,179,8.6520001,"\int e^{-\frac{3}{2} \coth ^{-1}(a x)} x^2 \, dx","Integrate[x^2/E^((3*ArcCoth[a*x])/2),x]","-\frac{e^{-\frac{7}{2} \coth ^{-1}(a x)} \left(256 e^{6 \coth ^{-1}(a x)} \left(850 e^{2 \coth ^{-1}(a x)}+325 e^{4 \coth ^{-1}(a x)}+557\right) \, _4F_3\left(\frac{5}{4},2,2,2;1,1,\frac{17}{4};e^{2 \coth ^{-1}(a x)}\right)+2048 e^{6 \coth ^{-1}(a x)} \left(34 e^{2 \coth ^{-1}(a x)}+15 e^{4 \coth ^{-1}(a x)}+19\right) \, _5F_4\left(\frac{5}{4},2,2,2,2;1,1,1,\frac{17}{4};e^{2 \coth ^{-1}(a x)}\right)+4096 e^{6 \coth ^{-1}(a x)} \, _6F_5\left(\frac{5}{4},2,2,2,2,2;1,1,1,1,\frac{17}{4};e^{2 \coth ^{-1}(a x)}\right)+8192 e^{8 \coth ^{-1}(a x)} \, _6F_5\left(\frac{5}{4},2,2,2,2,2;1,1,1,1,\frac{17}{4};e^{2 \coth ^{-1}(a x)}\right)+4096 e^{10 \coth ^{-1}(a x)} \, _6F_5\left(\frac{5}{4},2,2,2,2,2;1,1,1,1,\frac{17}{4};e^{2 \coth ^{-1}(a x)}\right)+14157000 e^{2 \coth ^{-1}(a x)} \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};e^{2 \coth ^{-1}(a x)}\right)-2472210 e^{4 \coth ^{-1}(a x)} \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};e^{2 \coth ^{-1}(a x)}\right)-3598920 e^{6 \coth ^{-1}(a x)} \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};e^{2 \coth ^{-1}(a x)}\right)+21645 e^{8 \coth ^{-1}(a x)} \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};e^{2 \coth ^{-1}(a x)}\right)+15779205 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};e^{2 \coth ^{-1}(a x)}\right)-17312841 e^{2 \coth ^{-1}(a x)}-1213875 e^{4 \coth ^{-1}(a x)}+2199249 e^{6 \coth ^{-1}(a x)}-15779205\right)}{112320 a^3}","-\frac{17 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{8 a^3}-\frac{17 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{8 a^3}+\frac{23 x \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{24 a^2}+\frac{1}{3} x^3 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}-\frac{7 x^2 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{12 a}",1,"-1/112320*(-15779205 - 17312841*E^(2*ArcCoth[a*x]) - 1213875*E^(4*ArcCoth[a*x]) + 2199249*E^(6*ArcCoth[a*x]) + 15779205*Hypergeometric2F1[1/4, 1, 5/4, E^(2*ArcCoth[a*x])] + 14157000*E^(2*ArcCoth[a*x])*Hypergeometric2F1[1/4, 1, 5/4, E^(2*ArcCoth[a*x])] - 2472210*E^(4*ArcCoth[a*x])*Hypergeometric2F1[1/4, 1, 5/4, E^(2*ArcCoth[a*x])] - 3598920*E^(6*ArcCoth[a*x])*Hypergeometric2F1[1/4, 1, 5/4, E^(2*ArcCoth[a*x])] + 21645*E^(8*ArcCoth[a*x])*Hypergeometric2F1[1/4, 1, 5/4, E^(2*ArcCoth[a*x])] + 256*E^(6*ArcCoth[a*x])*(557 + 850*E^(2*ArcCoth[a*x]) + 325*E^(4*ArcCoth[a*x]))*HypergeometricPFQ[{5/4, 2, 2, 2}, {1, 1, 17/4}, E^(2*ArcCoth[a*x])] + 2048*E^(6*ArcCoth[a*x])*(19 + 34*E^(2*ArcCoth[a*x]) + 15*E^(4*ArcCoth[a*x]))*HypergeometricPFQ[{5/4, 2, 2, 2, 2}, {1, 1, 1, 17/4}, E^(2*ArcCoth[a*x])] + 4096*E^(6*ArcCoth[a*x])*HypergeometricPFQ[{5/4, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 17/4}, E^(2*ArcCoth[a*x])] + 8192*E^(8*ArcCoth[a*x])*HypergeometricPFQ[{5/4, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 17/4}, E^(2*ArcCoth[a*x])] + 4096*E^(10*ArcCoth[a*x])*HypergeometricPFQ[{5/4, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 17/4}, E^(2*ArcCoth[a*x])])/(a^3*E^((7*ArcCoth[a*x])/2))","C",0
98,1,70,142,0.2040858,"\int e^{-\frac{3}{2} \coth ^{-1}(a x)} x \, dx","Integrate[x/E^((3*ArcCoth[a*x])/2),x]","\frac{-\frac{2 e^{\frac{1}{2} \coth ^{-1}(a x)} \left(3 e^{2 \coth ^{-1}(a x)}-7\right)}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}+9 \tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+9 \tanh ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{4 a^2}","\frac{9 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{4 a^2}+\frac{9 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{4 a^2}+\frac{1}{2} x^2 \left(1-\frac{1}{a x}\right)^{7/4} \sqrt[4]{\frac{1}{a x}+1}-\frac{3 x \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}}{4 a}",1,"((-2*E^(ArcCoth[a*x]/2)*(-7 + 3*E^(2*ArcCoth[a*x])))/(-1 + E^(2*ArcCoth[a*x]))^2 + 9*ArcTan[E^(ArcCoth[a*x]/2)] + 9*ArcTanh[E^(ArcCoth[a*x]/2)])/(4*a^2)","A",0
99,1,55,98,0.1226318,"\int e^{-\frac{3}{2} \coth ^{-1}(a x)} \, dx","Integrate[E^((-3*ArcCoth[a*x])/2),x]","\frac{\frac{2 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}-3 \tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)-3 \tanh ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{a}","x \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}-\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{a}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{a}",1,"((2*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x])) - 3*ArcTan[E^(ArcCoth[a*x]/2)] - 3*ArcTanh[E^(ArcCoth[a*x]/2)])/a","A",0
100,1,28,291,0.0748981,"\int \frac{e^{-\frac{3}{2} \coth ^{-1}(a x)}}{x} \, dx","Integrate[1/(E^((3*ArcCoth[a*x])/2)*x),x]","8 e^{\frac{1}{2} \coth ^{-1}(a x)} \, _2F_1\left(\frac{1}{8},1;\frac{9}{8};e^{4 \coth ^{-1}(a x)}\right)","-\frac{\log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}+\frac{\log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}+\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)+2 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)+2 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)",1,"8*E^(ArcCoth[a*x]/2)*Hypergeometric2F1[1/8, 1, 9/8, E^(4*ArcCoth[a*x])]","C",0
101,1,149,269,0.3011188,"\int \frac{e^{-\frac{3}{2} \coth ^{-1}(a x)}}{x^2} \, dx","Integrate[1/(E^((3*ArcCoth[a*x])/2)*x^2),x]","a \left(-\frac{2 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}+1}+\frac{3 \log \left(-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right)}{2 \sqrt{2}}-\frac{3 \log \left(\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right)}{2 \sqrt{2}}+\frac{3 \tan ^{-1}\left(1-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}\right)}{\sqrt{2}}-\frac{3 \tan ^{-1}\left(\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)}{\sqrt{2}}\right)","-a \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}+\frac{3 a \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{2 \sqrt{2}}-\frac{3 a \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{2 \sqrt{2}}-\frac{3 a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{\sqrt{2}}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}",1,"a*((-2*E^(ArcCoth[a*x]/2))/(1 + E^(2*ArcCoth[a*x])) + (3*ArcTan[1 - Sqrt[2]*E^(ArcCoth[a*x]/2)])/Sqrt[2] - (3*ArcTan[1 + Sqrt[2]*E^(ArcCoth[a*x]/2)])/Sqrt[2] + (3*Log[1 - Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]])/(2*Sqrt[2]) - (3*Log[1 + Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]])/(2*Sqrt[2]))","A",0
102,1,174,319,0.2240353,"\int \frac{e^{-\frac{3}{2} \coth ^{-1}(a x)}}{x^3} \, dx","Integrate[1/(E^((3*ArcCoth[a*x])/2)*x^3),x]","\frac{1}{16} a^2 \left(\frac{24 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}+1}+\frac{32 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}+1\right)^2}-9 \sqrt{2} \log \left(-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right)+9 \sqrt{2} \log \left(\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right)-18 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+18 \sqrt{2} \tan ^{-1}\left(\sqrt{2} e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)\right)","\frac{1}{2} a^2 \left(1-\frac{1}{a x}\right)^{7/4} \sqrt[4]{\frac{1}{a x}+1}+\frac{3}{4} a^2 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}-\frac{9 a^2 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}+\frac{9 a^2 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}+\frac{9 a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{4 \sqrt{2}}-\frac{9 a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{4 \sqrt{2}}",1,"(a^2*((32*E^(ArcCoth[a*x]/2))/(1 + E^(2*ArcCoth[a*x]))^2 + (24*E^(ArcCoth[a*x]/2))/(1 + E^(2*ArcCoth[a*x])) - 18*Sqrt[2]*ArcTan[1 - Sqrt[2]*E^(ArcCoth[a*x]/2)] + 18*Sqrt[2]*ArcTan[1 + Sqrt[2]*E^(ArcCoth[a*x]/2)] - 9*Sqrt[2]*Log[1 - Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]] + 9*Sqrt[2]*Log[1 + Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]]))/16","A",0
103,1,93,356,0.1506574,"\int \frac{e^{-\frac{3}{2} \coth ^{-1}(a x)}}{x^4} \, dx","Integrate[1/(E^((3*ArcCoth[a*x])/2)*x^4),x]","\frac{1}{96} a^3 \left(51 \text{RootSum}\left[\text{$\#$1}^4+1\&,\frac{2 \log \left(e^{-\frac{1}{2} \coth ^{-1}(a x)}-\text{$\#$1}\right)+\coth ^{-1}(a x)}{\text{$\#$1}}\&\right]-\frac{8 e^{\frac{1}{2} \coth ^{-1}(a x)} \left(30 e^{2 \coth ^{-1}(a x)}+17 e^{4 \coth ^{-1}(a x)}+45\right)}{\left(e^{2 \coth ^{-1}(a x)}+1\right)^3}\right)","-\frac{1}{4} a^3 \left(1-\frac{1}{a x}\right)^{7/4} \sqrt[4]{\frac{1}{a x}+1}-\frac{17}{24} a^3 \left(1-\frac{1}{a x}\right)^{3/4} \sqrt[4]{\frac{1}{a x}+1}+\frac{17 a^3 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{16 \sqrt{2}}-\frac{17 a^3 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{16 \sqrt{2}}-\frac{17 a^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{8 \sqrt{2}}+\frac{17 a^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}+\frac{a^2 \left(1-\frac{1}{a x}\right)^{7/4} \sqrt[4]{\frac{1}{a x}+1}}{3 x}",1,"(a^3*((-8*E^(ArcCoth[a*x]/2)*(45 + 30*E^(2*ArcCoth[a*x]) + 17*E^(4*ArcCoth[a*x])))/(1 + E^(2*ArcCoth[a*x]))^3 + 51*RootSum[1 + #1^4 & , (ArcCoth[a*x] + 2*Log[E^(-1/2*ArcCoth[a*x]) - #1])/#1 & ]))/96","C",0
104,1,198,287,5.4282888,"\int e^{-\frac{5}{2} \coth ^{-1}(a x)} x^4 \, dx","Integrate[x^4/E^((5*ArcCoth[a*x])/2),x]","\frac{8 e^{-\frac{1}{2} \coth ^{-1}(a x)}-\frac{4117 e^{-\frac{1}{2} \coth ^{-1}(a x)}}{192 \left(e^{-2 \coth ^{-1}(a x)}-1\right)}-\frac{1661 e^{-\frac{1}{2} \coth ^{-1}(a x)}}{48 \left(e^{-2 \coth ^{-1}(a x)}-1\right)^2}-\frac{233 e^{-\frac{1}{2} \coth ^{-1}(a x)}}{6 \left(e^{-2 \coth ^{-1}(a x)}-1\right)^3}-\frac{122 e^{-\frac{1}{2} \coth ^{-1}(a x)}}{5 \left(e^{-2 \coth ^{-1}(a x)}-1\right)^4}-\frac{32 e^{-\frac{1}{2} \coth ^{-1}(a x)}}{5 \left(e^{-2 \coth ^{-1}(a x)}-1\right)^5}+\frac{1003}{256} \log \left(1-e^{-\frac{1}{2} \coth ^{-1}(a x)}\right)-\frac{1003}{256} \log \left(e^{-\frac{1}{2} \coth ^{-1}(a x)}+1\right)-\frac{1003}{128} \tan ^{-1}\left(e^{-\frac{1}{2} \coth ^{-1}(a x)}\right)}{a^5}","\frac{26111 \sqrt[4]{1-\frac{1}{a x}}}{1920 a^5 \sqrt[4]{\frac{1}{a x}+1}}+\frac{1003 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{128 a^5}-\frac{1003 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{128 a^5}+\frac{5533 x \sqrt[4]{1-\frac{1}{a x}}}{1920 a^4 \sqrt[4]{\frac{1}{a x}+1}}-\frac{1189 x^2 \sqrt[4]{1-\frac{1}{a x}}}{960 a^3 \sqrt[4]{\frac{1}{a x}+1}}+\frac{181 x^3 \sqrt[4]{1-\frac{1}{a x}}}{240 a^2 \sqrt[4]{\frac{1}{a x}+1}}+\frac{x^5 \sqrt[4]{1-\frac{1}{a x}}}{5 \sqrt[4]{\frac{1}{a x}+1}}-\frac{21 x^4 \sqrt[4]{1-\frac{1}{a x}}}{40 a \sqrt[4]{\frac{1}{a x}+1}}",1,"(8/E^(ArcCoth[a*x]/2) - 32/(5*E^(ArcCoth[a*x]/2)*(-1 + E^(-2*ArcCoth[a*x]))^5) - 122/(5*E^(ArcCoth[a*x]/2)*(-1 + E^(-2*ArcCoth[a*x]))^4) - 233/(6*E^(ArcCoth[a*x]/2)*(-1 + E^(-2*ArcCoth[a*x]))^3) - 1661/(48*E^(ArcCoth[a*x]/2)*(-1 + E^(-2*ArcCoth[a*x]))^2) - 4117/(192*E^(ArcCoth[a*x]/2)*(-1 + E^(-2*ArcCoth[a*x]))) - (1003*ArcTan[E^(-1/2*ArcCoth[a*x])])/128 + (1003*Log[1 - E^(-1/2*ArcCoth[a*x])])/256 - (1003*Log[1 + E^(-1/2*ArcCoth[a*x])])/256)/a^5","A",0
105,1,161,250,5.3225796,"\int e^{-\frac{5}{2} \coth ^{-1}(a x)} x^3 \, dx","Integrate[x^3/E^((5*ArcCoth[a*x])/2),x]","\frac{-3072 e^{-\frac{1}{2} \coth ^{-1}(a x)}-\frac{6292 e^{\frac{3}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{7376 e^{\frac{7}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}-\frac{5248 e^{\frac{11}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^3}+\frac{1536 e^{\frac{15}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^4}-1425 \log \left(1-e^{-\frac{1}{2} \coth ^{-1}(a x)}\right)+1425 \log \left(e^{-\frac{1}{2} \coth ^{-1}(a x)}+1\right)+2850 \tan ^{-1}\left(e^{-\frac{1}{2} \coth ^{-1}(a x)}\right)}{384 a^4}","-\frac{2467 \sqrt[4]{1-\frac{1}{a x}}}{192 a^4 \sqrt[4]{\frac{1}{a x}+1}}-\frac{475 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{64 a^4}+\frac{475 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{64 a^4}-\frac{521 x \sqrt[4]{1-\frac{1}{a x}}}{192 a^3 \sqrt[4]{\frac{1}{a x}+1}}+\frac{113 x^2 \sqrt[4]{1-\frac{1}{a x}}}{96 a^2 \sqrt[4]{\frac{1}{a x}+1}}+\frac{x^4 \sqrt[4]{1-\frac{1}{a x}}}{4 \sqrt[4]{\frac{1}{a x}+1}}-\frac{17 x^3 \sqrt[4]{1-\frac{1}{a x}}}{24 a \sqrt[4]{\frac{1}{a x}+1}}",1,"(-3072/E^(ArcCoth[a*x]/2) + (1536*E^((15*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^4 - (5248*E^((11*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^3 + (7376*E^((7*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^2 - (6292*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x])) + 2850*ArcTan[E^(-1/2*ArcCoth[a*x])] - 1425*Log[1 - E^(-1/2*ArcCoth[a*x])] + 1425*Log[1 + E^(-1/2*ArcCoth[a*x])])/(384*a^4)","A",0
106,1,137,213,5.285122,"\int e^{-\frac{5}{2} \coth ^{-1}(a x)} x^2 \, dx","Integrate[x^2/E^((5*ArcCoth[a*x])/2),x]","\frac{384 e^{-\frac{1}{2} \coth ^{-1}(a x)}+\frac{548 e^{\frac{3}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}-\frac{400 e^{\frac{7}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}+\frac{128 e^{\frac{11}{2} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^3}+165 \log \left(1-e^{-\frac{1}{2} \coth ^{-1}(a x)}\right)-165 \log \left(e^{-\frac{1}{2} \coth ^{-1}(a x)}+1\right)-330 \tan ^{-1}\left(e^{-\frac{1}{2} \coth ^{-1}(a x)}\right)}{48 a^3}","\frac{287 \sqrt[4]{1-\frac{1}{a x}}}{24 a^3 \sqrt[4]{\frac{1}{a x}+1}}+\frac{55 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{8 a^3}-\frac{55 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{8 a^3}+\frac{61 x \sqrt[4]{1-\frac{1}{a x}}}{24 a^2 \sqrt[4]{\frac{1}{a x}+1}}+\frac{x^3 \sqrt[4]{1-\frac{1}{a x}}}{3 \sqrt[4]{\frac{1}{a x}+1}}-\frac{13 x^2 \sqrt[4]{1-\frac{1}{a x}}}{12 a \sqrt[4]{\frac{1}{a x}+1}}",1,"(384/E^(ArcCoth[a*x]/2) + (128*E^((11*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^3 - (400*E^((7*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x]))^2 + (548*E^((3*ArcCoth[a*x])/2))/(-1 + E^(2*ArcCoth[a*x])) - 330*ArcTan[E^(-1/2*ArcCoth[a*x])] + 165*Log[1 - E^(-1/2*ArcCoth[a*x])] - 165*Log[1 + E^(-1/2*ArcCoth[a*x])])/(48*a^3)","A",0
107,1,121,176,0.2672755,"\int e^{-\frac{5}{2} \coth ^{-1}(a x)} x \, dx","Integrate[x/E^((5*ArcCoth[a*x])/2),x]","-\frac{e^{-\frac{1}{2} \coth ^{-1}(a x)} \left(-90 e^{2 \coth ^{-1}(a x)}+50 e^{4 \coth ^{-1}(a x)}+25 e^{\frac{1}{2} \coth ^{-1}(a x)} \left(e^{2 \coth ^{-1}(a x)}-1\right)^2 \tan ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)-25 e^{\frac{1}{2} \coth ^{-1}(a x)} \left(e^{2 \coth ^{-1}(a x)}-1\right)^2 \tanh ^{-1}\left(e^{\frac{1}{2} \coth ^{-1}(a x)}\right)+32\right)}{4 a^2 \left(e^{2 \coth ^{-1}(a x)}-1\right)^2}","-\frac{25 \sqrt[4]{1-\frac{1}{a x}}}{2 a^2 \sqrt[4]{\frac{1}{a x}+1}}-\frac{25 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{4 a^2}+\frac{25 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{4 a^2}+\frac{x^2 \left(1-\frac{1}{a x}\right)^{9/4}}{2 \sqrt[4]{\frac{1}{a x}+1}}-\frac{5 x \left(1-\frac{1}{a x}\right)^{5/4}}{4 a \sqrt[4]{\frac{1}{a x}+1}}",1,"-1/4*(32 - 90*E^(2*ArcCoth[a*x]) + 50*E^(4*ArcCoth[a*x]) + 25*E^(ArcCoth[a*x]/2)*(-1 + E^(2*ArcCoth[a*x]))^2*ArcTan[E^(ArcCoth[a*x]/2)] - 25*E^(ArcCoth[a*x]/2)*(-1 + E^(2*ArcCoth[a*x]))^2*ArcTanh[E^(ArcCoth[a*x]/2)])/(a^2*E^(ArcCoth[a*x]/2)*(-1 + E^(2*ArcCoth[a*x]))^2)","A",0
108,1,31,130,0.0760173,"\int e^{-\frac{5}{2} \coth ^{-1}(a x)} \, dx","Integrate[E^((-5*ArcCoth[a*x])/2),x]","\frac{8 e^{-\frac{1}{2} \coth ^{-1}(a x)} \, _2F_1\left(-\frac{1}{4},2;\frac{3}{4};e^{2 \coth ^{-1}(a x)}\right)}{a}","\frac{x \left(1-\frac{1}{a x}\right)^{5/4}}{\sqrt[4]{\frac{1}{a x}+1}}+\frac{10 \sqrt[4]{1-\frac{1}{a x}}}{a \sqrt[4]{\frac{1}{a x}+1}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{a}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)}{a}",1,"(8*Hypergeometric2F1[-1/4, 2, 3/4, E^(2*ArcCoth[a*x])])/(a*E^(ArcCoth[a*x]/2))","C",0
109,1,28,320,0.1030852,"\int \frac{e^{-\frac{5}{2} \coth ^{-1}(a x)}}{x} \, dx","Integrate[1/(E^((5*ArcCoth[a*x])/2)*x),x]","-8 e^{-\frac{1}{2} \coth ^{-1}(a x)} \, _2F_1\left(-\frac{1}{8},1;\frac{7}{8};e^{4 \coth ^{-1}(a x)}\right)","-\frac{8 \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}-\frac{\log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}+\frac{\log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}-\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)-2 \tan ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)+2 \tanh ^{-1}\left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}\right)",1,"(-8*Hypergeometric2F1[-1/8, 1, 7/8, E^(4*ArcCoth[a*x])])/E^(ArcCoth[a*x]/2)","C",0
110,1,31,299,0.0838441,"\int \frac{e^{-\frac{5}{2} \coth ^{-1}(a x)}}{x^2} \, dx","Integrate[1/(E^((5*ArcCoth[a*x])/2)*x^2),x]","8 a e^{-\frac{1}{2} \coth ^{-1}(a x)} \, _2F_1\left(-\frac{1}{4},2;\frac{3}{4};-e^{2 \coth ^{-1}(a x)}\right)","\frac{4 a \left(1-\frac{1}{a x}\right)^{5/4}}{\sqrt[4]{\frac{1}{a x}+1}}+5 a \left(\frac{1}{a x}+1\right)^{3/4} \sqrt[4]{1-\frac{1}{a x}}+\frac{5 a \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{2 \sqrt{2}}-\frac{5 a \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{2 \sqrt{2}}+\frac{5 a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{\sqrt{2}}-\frac{5 a \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{\sqrt{2}}",1,"(8*a*Hypergeometric2F1[-1/4, 2, 3/4, -E^(2*ArcCoth[a*x])])/E^(ArcCoth[a*x]/2)","C",0
111,1,101,351,0.1518048,"\int \frac{e^{-\frac{5}{2} \coth ^{-1}(a x)}}{x^3} \, dx","Integrate[1/(E^((5*ArcCoth[a*x])/2)*x^3),x]","-\frac{8}{3} a^2 e^{-\frac{1}{2} \coth ^{-1}(a x)} \left(e^{2 \coth ^{-1}(a x)} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-e^{2 \coth ^{-1}(a x)}\right)+e^{2 \coth ^{-1}(a x)} \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-e^{2 \coth ^{-1}(a x)}\right)+2 e^{2 \coth ^{-1}(a x)} \, _2F_1\left(\frac{3}{4},3;\frac{7}{4};-e^{2 \coth ^{-1}(a x)}\right)+3\right)","-\frac{2 a^2 \left(1-\frac{1}{a x}\right)^{9/4}}{\sqrt[4]{\frac{1}{a x}+1}}-\frac{5}{2} a^2 \left(\frac{1}{a x}+1\right)^{3/4} \left(1-\frac{1}{a x}\right)^{5/4}-\frac{25}{4} a^2 \left(\frac{1}{a x}+1\right)^{3/4} \sqrt[4]{1-\frac{1}{a x}}-\frac{25 a^2 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}+\frac{25 a^2 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}-\frac{25 a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{4 \sqrt{2}}+\frac{25 a^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{4 \sqrt{2}}",1,"(-8*a^2*(3 + E^(2*ArcCoth[a*x])*Hypergeometric2F1[3/4, 1, 7/4, -E^(2*ArcCoth[a*x])] + E^(2*ArcCoth[a*x])*Hypergeometric2F1[3/4, 2, 7/4, -E^(2*ArcCoth[a*x])] + 2*E^(2*ArcCoth[a*x])*Hypergeometric2F1[3/4, 3, 7/4, -E^(2*ArcCoth[a*x])]))/(3*E^(ArcCoth[a*x]/2))","C",0
112,1,104,385,0.205939,"\int \frac{e^{-\frac{5}{2} \coth ^{-1}(a x)}}{x^4} \, dx","Integrate[1/(E^((5*ArcCoth[a*x])/2)*x^4),x]","a^3 \left(\frac{e^{-\frac{1}{2} \coth ^{-1}(a x)} \left(425 e^{2 \coth ^{-1}(a x)}+462 e^{4 \coth ^{-1}(a x)}+165 e^{6 \coth ^{-1}(a x)}+96\right)}{12 \left(e^{2 \coth ^{-1}(a x)}+1\right)^3}-\frac{55}{32} \text{RootSum}\left[\text{$\#$1}^4+1\&,\frac{2 \log \left(e^{-\frac{1}{2} \coth ^{-1}(a x)}-\text{$\#$1}\right)+\coth ^{-1}(a x)}{\text{$\#$1}^3}\&\right]\right)","\frac{1}{3} a^3 \left(1-\frac{1}{a x}\right)^{9/4} \left(\frac{1}{a x}+1\right)^{3/4}+\frac{11}{4} a^3 \left(1-\frac{1}{a x}\right)^{5/4} \left(\frac{1}{a x}+1\right)^{3/4}+\frac{55}{8} a^3 \sqrt[4]{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/4}+\frac{2 a^3 \left(1-\frac{1}{a x}\right)^{9/4}}{\sqrt[4]{\frac{1}{a x}+1}}+\frac{55 a^3 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{16 \sqrt{2}}-\frac{55 a^3 \log \left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{16 \sqrt{2}}+\frac{55 a^3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}\right)}{8 \sqrt{2}}-\frac{55 a^3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+1\right)}{8 \sqrt{2}}",1,"a^3*((96 + 425*E^(2*ArcCoth[a*x]) + 462*E^(4*ArcCoth[a*x]) + 165*E^(6*ArcCoth[a*x]))/(12*E^(ArcCoth[a*x]/2)*(1 + E^(2*ArcCoth[a*x]))^3) - (55*RootSum[1 + #1^4 & , (ArcCoth[a*x] + 2*Log[E^(-1/2*ArcCoth[a*x]) - #1])/#1^3 & ])/32)","C",0
113,1,189,285,5.287956,"\int e^{\frac{1}{3} \coth ^{-1}(x)} x^2 \, dx","Integrate[E^(ArcCoth[x]/3)*x^2,x]","\frac{1}{324} \left(\frac{732 e^{\frac{1}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}-1}+\frac{1368 e^{\frac{1}{3} \coth ^{-1}(x)}}{\left(e^{2 \coth ^{-1}(x)}-1\right)^2}+\frac{864 e^{\frac{1}{3} \coth ^{-1}(x)}}{\left(e^{2 \coth ^{-1}(x)}-1\right)^3}-38 \log \left(1-e^{\frac{1}{3} \coth ^{-1}(x)}\right)+38 \log \left(e^{\frac{1}{3} \coth ^{-1}(x)}+1\right)-19 \log \left(-e^{\frac{1}{3} \coth ^{-1}(x)}+e^{\frac{2}{3} \coth ^{-1}(x)}+1\right)+19 \log \left(e^{\frac{1}{3} \coth ^{-1}(x)}+e^{\frac{2}{3} \coth ^{-1}(x)}+1\right)+38 \sqrt{3} \tan ^{-1}\left(\frac{2 e^{\frac{1}{3} \coth ^{-1}(x)}-1}{\sqrt{3}}\right)+38 \sqrt{3} \tan ^{-1}\left(\frac{2 e^{\frac{1}{3} \coth ^{-1}(x)}+1}{\sqrt{3}}\right)\right)","\frac{1}{3} \sqrt[6]{\frac{1}{x}+1} \left(\frac{x-1}{x}\right)^{5/6} x^3+\frac{7}{18} \sqrt[6]{\frac{1}{x}+1} \left(\frac{x-1}{x}\right)^{5/6} x^2+\frac{11}{27} \sqrt[6]{\frac{1}{x}+1} \left(\frac{x-1}{x}\right)^{5/6} x-\frac{19}{324} \log \left(\frac{\sqrt[3]{\frac{1}{x}+1}}{\sqrt[3]{\frac{x-1}{x}}}-\frac{\sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}+1\right)+\frac{19}{324} \log \left(\frac{\sqrt[3]{\frac{1}{x}+1}}{\sqrt[3]{\frac{x-1}{x}}}+\frac{\sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}+1\right)-\frac{19 \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}}{\sqrt{3}}\right)}{54 \sqrt{3}}+\frac{19 \tan ^{-1}\left(\frac{\frac{2 \sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}+1}{\sqrt{3}}\right)}{54 \sqrt{3}}+\frac{19}{81} \tanh ^{-1}\left(\frac{\sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}\right)",1,"((864*E^(ArcCoth[x]/3))/(-1 + E^(2*ArcCoth[x]))^3 + (1368*E^(ArcCoth[x]/3))/(-1 + E^(2*ArcCoth[x]))^2 + (732*E^(ArcCoth[x]/3))/(-1 + E^(2*ArcCoth[x])) + 38*Sqrt[3]*ArcTan[(-1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] + 38*Sqrt[3]*ArcTan[(1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 38*Log[1 - E^(ArcCoth[x]/3)] + 38*Log[1 + E^(ArcCoth[x]/3)] - 19*Log[1 - E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)] + 19*Log[1 + E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)])/324","A",0
114,1,167,258,0.4057842,"\int e^{\frac{1}{3} \coth ^{-1}(x)} x \, dx","Integrate[E^(ArcCoth[x]/3)*x,x]","\frac{1}{36} \left(\frac{84 e^{\frac{1}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}-1}+\frac{72 e^{\frac{1}{3} \coth ^{-1}(x)}}{\left(e^{2 \coth ^{-1}(x)}-1\right)^2}-2 \log \left(1-e^{\frac{1}{3} \coth ^{-1}(x)}\right)+2 \log \left(e^{\frac{1}{3} \coth ^{-1}(x)}+1\right)-\log \left(-e^{\frac{1}{3} \coth ^{-1}(x)}+e^{\frac{2}{3} \coth ^{-1}(x)}+1\right)+\log \left(e^{\frac{1}{3} \coth ^{-1}(x)}+e^{\frac{2}{3} \coth ^{-1}(x)}+1\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 e^{\frac{1}{3} \coth ^{-1}(x)}-1}{\sqrt{3}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 e^{\frac{1}{3} \coth ^{-1}(x)}+1}{\sqrt{3}}\right)\right)","\frac{1}{2} \left(\frac{1}{x}+1\right)^{7/6} \left(\frac{x-1}{x}\right)^{5/6} x^2+\frac{1}{6} \sqrt[6]{\frac{1}{x}+1} \left(\frac{x-1}{x}\right)^{5/6} x-\frac{1}{36} \log \left(\frac{\sqrt[3]{\frac{1}{x}+1}}{\sqrt[3]{\frac{x-1}{x}}}-\frac{\sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}+1\right)+\frac{1}{36} \log \left(\frac{\sqrt[3]{\frac{1}{x}+1}}{\sqrt[3]{\frac{x-1}{x}}}+\frac{\sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}+1\right)-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}}{\sqrt{3}}\right)}{6 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\frac{2 \sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}+1}{\sqrt{3}}\right)}{6 \sqrt{3}}+\frac{1}{9} \tanh ^{-1}\left(\frac{\sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}\right)",1,"((72*E^(ArcCoth[x]/3))/(-1 + E^(2*ArcCoth[x]))^2 + (84*E^(ArcCoth[x]/3))/(-1 + E^(2*ArcCoth[x])) + 2*Sqrt[3]*ArcTan[(-1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] + 2*Sqrt[3]*ArcTan[(1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 2*Log[1 - E^(ArcCoth[x]/3)] + 2*Log[1 + E^(ArcCoth[x]/3)] - Log[1 - E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)] + Log[1 + E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)])/36","A",0
115,1,35,223,0.050415,"\int e^{\frac{1}{3} \coth ^{-1}(x)} \, dx","Integrate[E^(ArcCoth[x]/3),x]","2 e^{\frac{1}{3} \coth ^{-1}(x)} \left(\, _2F_1\left(\frac{1}{6},1;\frac{7}{6};e^{2 \coth ^{-1}(x)}\right)+\frac{1}{e^{2 \coth ^{-1}(x)}-1}\right)","\sqrt[6]{\frac{1}{x}+1} \left(\frac{x-1}{x}\right)^{5/6} x-\frac{1}{6} \log \left(\frac{\sqrt[3]{\frac{1}{x}+1}}{\sqrt[3]{\frac{x-1}{x}}}-\frac{\sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}+1\right)+\frac{1}{6} \log \left(\frac{\sqrt[3]{\frac{1}{x}+1}}{\sqrt[3]{\frac{x-1}{x}}}+\frac{\sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}+1\right)-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{\tan ^{-1}\left(\frac{\frac{2 \sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}+1}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{2}{3} \tanh ^{-1}\left(\frac{\sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}\right)",1,"2*E^(ArcCoth[x]/3)*((-1 + E^(2*ArcCoth[x]))^(-1) + Hypergeometric2F1[1/6, 1, 7/6, E^(2*ArcCoth[x])])","C",0
116,1,26,402,0.0447328,"\int \frac{e^{\frac{1}{3} \coth ^{-1}(x)}}{x} \, dx","Integrate[E^(ArcCoth[x]/3)/x,x]","\frac{12}{7} e^{\frac{7}{3} \coth ^{-1}(x)} \, _2F_1\left(\frac{7}{12},1;\frac{19}{12};e^{4 \coth ^{-1}(x)}\right)","-\frac{1}{2} \log \left(\frac{\sqrt[3]{\frac{1}{x}+1}}{\sqrt[3]{\frac{x-1}{x}}}-\frac{\sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}+1\right)+\frac{1}{2} \log \left(\frac{\sqrt[3]{\frac{1}{x}+1}}{\sqrt[3]{\frac{x-1}{x}}}+\frac{\sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}+1\right)+\frac{1}{2} \sqrt{3} \log \left(\frac{\sqrt[3]{\frac{x-1}{x}}}{\sqrt[3]{\frac{1}{x}+1}}-\frac{\sqrt{3} \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}+1\right)-\frac{1}{2} \sqrt{3} \log \left(\frac{\sqrt[3]{\frac{x-1}{x}}}{\sqrt[3]{\frac{1}{x}+1}}+\frac{\sqrt{3} \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}+1\right)-\sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}}{\sqrt{3}}\right)+\sqrt{3} \tan ^{-1}\left(\frac{\frac{2 \sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}+1}{\sqrt{3}}\right)-\tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}\right)+\tan ^{-1}\left(\frac{2 \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}+\sqrt{3}\right)+2 \tan ^{-1}\left(\frac{\sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}\right)+2 \tanh ^{-1}\left(\frac{\sqrt[6]{\frac{1}{x}+1}}{\sqrt[6]{\frac{x-1}{x}}}\right)",1,"(12*E^((7*ArcCoth[x])/3)*Hypergeometric2F1[7/12, 1, 19/12, E^(4*ArcCoth[x])])/7","C",0
117,1,39,233,0.0597257,"\int \frac{e^{\frac{1}{3} \coth ^{-1}(x)}}{x^2} \, dx","Integrate[E^(ArcCoth[x]/3)/x^2,x]","-2 e^{\frac{1}{3} \coth ^{-1}(x)} \left(\, _2F_1\left(\frac{1}{6},1;\frac{7}{6};-e^{2 \coth ^{-1}(x)}\right)-\frac{1}{e^{2 \coth ^{-1}(x)}+1}\right)","\sqrt[6]{\frac{1}{x}+1} \left(\frac{x-1}{x}\right)^{5/6}+\frac{\log \left(\frac{\sqrt[3]{\frac{x-1}{x}}}{\sqrt[3]{\frac{1}{x}+1}}-\frac{\sqrt{3} \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}+1\right)}{2 \sqrt{3}}-\frac{\log \left(\frac{\sqrt[3]{\frac{x-1}{x}}}{\sqrt[3]{\frac{1}{x}+1}}+\frac{\sqrt{3} \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}+1\right)}{2 \sqrt{3}}-\frac{1}{3} \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}\right)+\frac{1}{3} \tan ^{-1}\left(\frac{2 \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}+\sqrt{3}\right)+\frac{2}{3} \tan ^{-1}\left(\frac{\sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}\right)",1,"-2*E^(ArcCoth[x]/3)*(-(1 + E^(2*ArcCoth[x]))^(-1) + Hypergeometric2F1[1/6, 1, 7/6, -E^(2*ArcCoth[x])])","C",0
118,1,124,260,0.7982004,"\int \frac{e^{\frac{1}{3} \coth ^{-1}(x)}}{x^3} \, dx","Integrate[E^(ArcCoth[x]/3)/x^3,x]","\frac{1}{54} \left(\text{RootSum}\left[\text{$\#$1}^4-\text{$\#$1}^2+1\&,\frac{\text{$\#$1}^2 \left(-\coth ^{-1}(x)\right)+3 \text{$\#$1}^2 \log \left(e^{\frac{1}{3} \coth ^{-1}(x)}-\text{$\#$1}\right)-6 \log \left(e^{\frac{1}{3} \coth ^{-1}(x)}-\text{$\#$1}\right)+2 \coth ^{-1}(x)}{2 \text{$\#$1}^3-\text{$\#$1}}\&\right]+\frac{18 e^{\frac{1}{3} \coth ^{-1}(x)} \left(7 e^{2 \coth ^{-1}(x)}+1\right)}{\left(e^{2 \coth ^{-1}(x)}+1\right)^2}-6 \tan ^{-1}\left(e^{\frac{1}{3} \coth ^{-1}(x)}\right)\right)","\frac{1}{2} \left(\frac{x-1}{x}\right)^{5/6} \left(\frac{1}{x}+1\right)^{7/6}+\frac{1}{6} \left(\frac{x-1}{x}\right)^{5/6} \sqrt[6]{\frac{1}{x}+1}+\frac{\log \left(\frac{\sqrt[3]{\frac{x-1}{x}}}{\sqrt[3]{\frac{1}{x}+1}}-\frac{\sqrt{3} \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}+1\right)}{12 \sqrt{3}}-\frac{\log \left(\frac{\sqrt[3]{\frac{x-1}{x}}}{\sqrt[3]{\frac{1}{x}+1}}+\frac{\sqrt{3} \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}+1\right)}{12 \sqrt{3}}-\frac{1}{18} \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}\right)+\frac{1}{18} \tan ^{-1}\left(\frac{2 \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}+\sqrt{3}\right)+\frac{1}{9} \tan ^{-1}\left(\frac{\sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}\right)",1,"((18*E^(ArcCoth[x]/3)*(1 + 7*E^(2*ArcCoth[x])))/(1 + E^(2*ArcCoth[x]))^2 - 6*ArcTan[E^(ArcCoth[x]/3)] + RootSum[1 - #1^2 + #1^4 & , (2*ArcCoth[x] - 6*Log[E^(ArcCoth[x]/3) - #1] - ArcCoth[x]*#1^2 + 3*Log[E^(ArcCoth[x]/3) - #1]*#1^2)/(-#1 + 2*#1^3) & ])/54","C",0
119,1,133,287,0.2249266,"\int \frac{e^{\frac{1}{3} \coth ^{-1}(x)}}{x^4} \, dx","Integrate[E^(ArcCoth[x]/3)/x^4,x]","\frac{1}{486} \left(-19 \text{RootSum}\left[\text{$\#$1}^4-\text{$\#$1}^2+1\&,\frac{\text{$\#$1}^2 \coth ^{-1}(x)-3 \text{$\#$1}^2 \log \left(e^{\frac{1}{3} \coth ^{-1}(x)}-\text{$\#$1}\right)+6 \log \left(e^{\frac{1}{3} \coth ^{-1}(x)}-\text{$\#$1}\right)-2 \coth ^{-1}(x)}{2 \text{$\#$1}^3-\text{$\#$1}}\&\right]+\frac{18 e^{\frac{1}{3} \coth ^{-1}(x)} \left(8 e^{2 \coth ^{-1}(x)}+61 e^{4 \coth ^{-1}(x)}+19\right)}{\left(e^{2 \coth ^{-1}(x)}+1\right)^3}-114 \tan ^{-1}\left(e^{\frac{1}{3} \coth ^{-1}(x)}\right)\right)","\frac{1}{18} \left(\frac{x-1}{x}\right)^{5/6} \left(\frac{1}{x}+1\right)^{7/6}+\frac{\left(\frac{x-1}{x}\right)^{5/6} \left(\frac{1}{x}+1\right)^{7/6}}{3 x}+\frac{19}{54} \left(\frac{x-1}{x}\right)^{5/6} \sqrt[6]{\frac{1}{x}+1}+\frac{19 \log \left(\frac{\sqrt[3]{\frac{x-1}{x}}}{\sqrt[3]{\frac{1}{x}+1}}-\frac{\sqrt{3} \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}+1\right)}{108 \sqrt{3}}-\frac{19 \log \left(\frac{\sqrt[3]{\frac{x-1}{x}}}{\sqrt[3]{\frac{1}{x}+1}}+\frac{\sqrt{3} \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}+1\right)}{108 \sqrt{3}}-\frac{19}{162} \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}\right)+\frac{19}{162} \tan ^{-1}\left(\frac{2 \sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}+\sqrt{3}\right)+\frac{19}{81} \tan ^{-1}\left(\frac{\sqrt[6]{\frac{x-1}{x}}}{\sqrt[6]{\frac{1}{x}+1}}\right)",1,"((18*E^(ArcCoth[x]/3)*(19 + 8*E^(2*ArcCoth[x]) + 61*E^(4*ArcCoth[x])))/(1 + E^(2*ArcCoth[x]))^3 - 114*ArcTan[E^(ArcCoth[x]/3)] - 19*RootSum[1 - #1^2 + #1^4 & , (-2*ArcCoth[x] + 6*Log[E^(ArcCoth[x]/3) - #1] + ArcCoth[x]*#1^2 - 3*Log[E^(ArcCoth[x]/3) - #1]*#1^2)/(-#1 + 2*#1^3) & ])/486","C",0
120,1,340,157,7.7724111,"\int e^{\frac{2}{3} \coth ^{-1}(x)} x^2 \, dx","Integrate[E^((2*ArcCoth[x])/3)*x^2,x]","-\frac{e^{-\frac{10}{3} \coth ^{-1}(x)} \left(54 e^{8 \coth ^{-1}(x)} \left(782 e^{2 \coth ^{-1}(x)}+325 e^{4 \coth ^{-1}(x)}+475\right) \, _4F_3\left(2,2,2,\frac{7}{3};1,1,\frac{16}{3};e^{2 \coth ^{-1}(x)}\right)+162 e^{8 \coth ^{-1}(x)} \left(64 e^{2 \coth ^{-1}(x)}+29 e^{4 \coth ^{-1}(x)}+35\right) \, _5F_4\left(2,2,2,2,\frac{7}{3};1,1,1,\frac{16}{3};e^{2 \coth ^{-1}(x)}\right)+486 e^{8 \coth ^{-1}(x)} \, _6F_5\left(2,2,2,2,2,\frac{7}{3};1,1,1,1,\frac{16}{3};e^{2 \coth ^{-1}(x)}\right)+972 e^{10 \coth ^{-1}(x)} \, _6F_5\left(2,2,2,2,2,\frac{7}{3};1,1,1,1,\frac{16}{3};e^{2 \coth ^{-1}(x)}\right)+486 e^{12 \coth ^{-1}(x)} \, _6F_5\left(2,2,2,2,2,\frac{7}{3};1,1,1,1,\frac{16}{3};e^{2 \coth ^{-1}(x)}\right)+15227940 e^{2 \coth ^{-1}(x)} \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};e^{2 \coth ^{-1}(x)}\right)-14083160 e^{4 \coth ^{-1}(x)} \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};e^{2 \coth ^{-1}(x)}\right)-8250060 e^{6 \coth ^{-1}(x)} \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};e^{2 \coth ^{-1}(x)}\right)+1456000 e^{8 \coth ^{-1}(x)} \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};e^{2 \coth ^{-1}(x)}\right)+22750000 \, _2F_1\left(\frac{1}{3},1;\frac{4}{3};e^{2 \coth ^{-1}(x)}\right)-20915440 e^{2 \coth ^{-1}(x)}+7026175 e^{4 \coth ^{-1}(x)}+7394140 e^{6 \coth ^{-1}(x)}-433485 e^{8 \coth ^{-1}(x)}-22750000\right)}{49140}","\frac{1}{3} \sqrt[3]{\frac{1}{x}+1} \left(\frac{x-1}{x}\right)^{2/3} x^3+\frac{4}{9} \sqrt[3]{\frac{1}{x}+1} \left(\frac{x-1}{x}\right)^{2/3} x^2+\frac{14}{27} \sqrt[3]{\frac{1}{x}+1} \left(\frac{x-1}{x}\right)^{2/3} x-\frac{11}{27} \log \left(\sqrt[3]{\frac{1}{x}+1}-\sqrt[3]{\frac{x-1}{x}}\right)-\frac{11 \log (x)}{81}-\frac{22 \tan ^{-1}\left(\frac{2 \sqrt[3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt[3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right)}{27 \sqrt{3}}",1,"-1/49140*(-22750000 - 20915440*E^(2*ArcCoth[x]) + 7026175*E^(4*ArcCoth[x]) + 7394140*E^(6*ArcCoth[x]) - 433485*E^(8*ArcCoth[x]) + 22750000*Hypergeometric2F1[1/3, 1, 4/3, E^(2*ArcCoth[x])] + 15227940*E^(2*ArcCoth[x])*Hypergeometric2F1[1/3, 1, 4/3, E^(2*ArcCoth[x])] - 14083160*E^(4*ArcCoth[x])*Hypergeometric2F1[1/3, 1, 4/3, E^(2*ArcCoth[x])] - 8250060*E^(6*ArcCoth[x])*Hypergeometric2F1[1/3, 1, 4/3, E^(2*ArcCoth[x])] + 1456000*E^(8*ArcCoth[x])*Hypergeometric2F1[1/3, 1, 4/3, E^(2*ArcCoth[x])] + 54*E^(8*ArcCoth[x])*(475 + 782*E^(2*ArcCoth[x]) + 325*E^(4*ArcCoth[x]))*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 16/3}, E^(2*ArcCoth[x])] + 162*E^(8*ArcCoth[x])*(35 + 64*E^(2*ArcCoth[x]) + 29*E^(4*ArcCoth[x]))*HypergeometricPFQ[{2, 2, 2, 2, 7/3}, {1, 1, 1, 16/3}, E^(2*ArcCoth[x])] + 486*E^(8*ArcCoth[x])*HypergeometricPFQ[{2, 2, 2, 2, 2, 7/3}, {1, 1, 1, 1, 16/3}, E^(2*ArcCoth[x])] + 972*E^(10*ArcCoth[x])*HypergeometricPFQ[{2, 2, 2, 2, 2, 7/3}, {1, 1, 1, 1, 16/3}, E^(2*ArcCoth[x])] + 486*E^(12*ArcCoth[x])*HypergeometricPFQ[{2, 2, 2, 2, 2, 7/3}, {1, 1, 1, 1, 16/3}, E^(2*ArcCoth[x])])/E^((10*ArcCoth[x])/3)","C",0
121,1,165,130,0.4855727,"\int e^{\frac{2}{3} \coth ^{-1}(x)} x \, dx","Integrate[E^((2*ArcCoth[x])/3)*x,x]","\frac{1}{9} \left(\frac{24 e^{\frac{2}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}-1}+\frac{18 e^{\frac{2}{3} \coth ^{-1}(x)}}{\left(e^{2 \coth ^{-1}(x)}-1\right)^2}-2 \log \left(1-e^{\frac{1}{3} \coth ^{-1}(x)}\right)-2 \log \left(e^{\frac{1}{3} \coth ^{-1}(x)}+1\right)+\log \left(-e^{\frac{1}{3} \coth ^{-1}(x)}+e^{\frac{2}{3} \coth ^{-1}(x)}+1\right)+\log \left(e^{\frac{1}{3} \coth ^{-1}(x)}+e^{\frac{2}{3} \coth ^{-1}(x)}+1\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 e^{\frac{1}{3} \coth ^{-1}(x)}-1}{\sqrt{3}}\right)-2 \sqrt{3} \tan ^{-1}\left(\frac{2 e^{\frac{1}{3} \coth ^{-1}(x)}+1}{\sqrt{3}}\right)\right)","\frac{1}{2} \left(\frac{1}{x}+1\right)^{4/3} \left(\frac{x-1}{x}\right)^{2/3} x^2+\frac{1}{3} \sqrt[3]{\frac{1}{x}+1} \left(\frac{x-1}{x}\right)^{2/3} x-\frac{1}{3} \log \left(\sqrt[3]{\frac{1}{x}+1}-\sqrt[3]{\frac{x-1}{x}}\right)-\frac{\log (x)}{9}-\frac{2 \tan ^{-1}\left(\frac{2 \sqrt[3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt[3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right)}{3 \sqrt{3}}",1,"((18*E^((2*ArcCoth[x])/3))/(-1 + E^(2*ArcCoth[x]))^2 + (24*E^((2*ArcCoth[x])/3))/(-1 + E^(2*ArcCoth[x])) + 2*Sqrt[3]*ArcTan[(-1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 2*Sqrt[3]*ArcTan[(1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 2*Log[1 - E^(ArcCoth[x]/3)] - 2*Log[1 + E^(ArcCoth[x]/3)] + Log[1 - E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)] + Log[1 + E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)])/9","A",0
122,1,85,96,0.1771308,"\int e^{\frac{2}{3} \coth ^{-1}(x)} \, dx","Integrate[E^((2*ArcCoth[x])/3),x]","\frac{1}{3} \left(\frac{6 e^{\frac{2}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}-1}-2 \log \left(1-e^{\frac{2}{3} \coth ^{-1}(x)}\right)+\log \left(e^{\frac{2}{3} \coth ^{-1}(x)}+e^{\frac{4}{3} \coth ^{-1}(x)}+1\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 e^{\frac{2}{3} \coth ^{-1}(x)}+1}{\sqrt{3}}\right)\right)","\sqrt[3]{\frac{1}{x}+1} \left(\frac{x-1}{x}\right)^{2/3} x-\log \left(\sqrt[3]{\frac{1}{x}+1}-\sqrt[3]{\frac{x-1}{x}}\right)-\frac{\log (x)}{3}-\frac{2 \tan ^{-1}\left(\frac{2 \sqrt[3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt[3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right)}{\sqrt{3}}",1,"((6*E^((2*ArcCoth[x])/3))/(-1 + E^(2*ArcCoth[x])) + 2*Sqrt[3]*ArcTan[(1 + 2*E^((2*ArcCoth[x])/3))/Sqrt[3]] - 2*Log[1 - E^((2*ArcCoth[x])/3)] + Log[1 + E^((2*ArcCoth[x])/3) + E^((4*ArcCoth[x])/3)])/3","A",0
123,1,26,155,0.0473273,"\int \frac{e^{\frac{2}{3} \coth ^{-1}(x)}}{x} \, dx","Integrate[E^((2*ArcCoth[x])/3)/x,x]","\frac{3}{2} e^{\frac{8}{3} \coth ^{-1}(x)} \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};e^{4 \coth ^{-1}(x)}\right)","-\frac{3}{2} \log \left(\sqrt[3]{\frac{1}{x}+1}-\sqrt[3]{\frac{x-1}{x}}\right)-\frac{3}{2} \log \left(\frac{\sqrt[3]{\frac{x-1}{x}}}{\sqrt[3]{\frac{1}{x}+1}}+1\right)-\frac{1}{2} \log \left(\frac{1}{x}+1\right)-\frac{\log (x)}{2}-\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt[3]{\frac{1}{x}+1}}\right)-\sqrt{3} \tan ^{-1}\left(\frac{2 \sqrt[3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt[3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right)",1,"(3*E^((8*ArcCoth[x])/3)*Hypergeometric2F1[2/3, 1, 5/3, E^(4*ArcCoth[x])])/2","C",0
124,1,87,99,0.1697734,"\int \frac{e^{\frac{2}{3} \coth ^{-1}(x)}}{x^2} \, dx","Integrate[E^((2*ArcCoth[x])/3)/x^2,x]","\frac{2 e^{\frac{2}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}+1}-\frac{2}{3} \log \left(e^{\frac{2}{3} \coth ^{-1}(x)}+1\right)+\frac{1}{3} \log \left(-e^{\frac{2}{3} \coth ^{-1}(x)}+e^{\frac{4}{3} \coth ^{-1}(x)}+1\right)-\frac{2 \tan ^{-1}\left(\frac{2 e^{\frac{2}{3} \coth ^{-1}(x)}-1}{\sqrt{3}}\right)}{\sqrt{3}}","\sqrt[3]{\frac{1}{x}+1} \left(\frac{x-1}{x}\right)^{2/3}-\log \left(\frac{\sqrt[3]{\frac{x-1}{x}}}{\sqrt[3]{\frac{1}{x}+1}}+1\right)-\frac{1}{3} \log \left(\frac{1}{x}+1\right)-\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt[3]{\frac{1}{x}+1}}\right)}{\sqrt{3}}",1,"(2*E^((2*ArcCoth[x])/3))/(1 + E^(2*ArcCoth[x])) - (2*ArcTan[(-1 + 2*E^((2*ArcCoth[x])/3))/Sqrt[3]])/Sqrt[3] - (2*Log[1 + E^((2*ArcCoth[x])/3)])/3 + Log[1 - E^((2*ArcCoth[x])/3) + E^((4*ArcCoth[x])/3)]/3","A",0
125,1,134,130,0.3150587,"\int \frac{e^{\frac{2}{3} \coth ^{-1}(x)}}{x^3} \, dx","Integrate[E^((2*ArcCoth[x])/3)/x^3,x]","-\frac{2}{27} \left(-\text{RootSum}\left[\text{$\#$1}^4-\text{$\#$1}^2+1\&,\frac{\text{$\#$1}^2 \coth ^{-1}(x)-3 \text{$\#$1}^2 \log \left(e^{\frac{1}{3} \coth ^{-1}(x)}-\text{$\#$1}\right)-3 \log \left(e^{\frac{1}{3} \coth ^{-1}(x)}-\text{$\#$1}\right)+\coth ^{-1}(x)}{\text{$\#$1}^2-2}\&\right]-2 \coth ^{-1}(x)-\frac{36 e^{\frac{2}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}+1}+\frac{27 e^{\frac{2}{3} \coth ^{-1}(x)}}{\left(e^{2 \coth ^{-1}(x)}+1\right)^2}+3 \log \left(e^{\frac{2}{3} \coth ^{-1}(x)}+1\right)\right)","\frac{1}{2} \left(\frac{x-1}{x}\right)^{2/3} \left(\frac{1}{x}+1\right)^{4/3}+\frac{1}{3} \left(\frac{x-1}{x}\right)^{2/3} \sqrt[3]{\frac{1}{x}+1}-\frac{1}{3} \log \left(\frac{\sqrt[3]{\frac{x-1}{x}}}{\sqrt[3]{\frac{1}{x}+1}}+1\right)-\frac{1}{9} \log \left(\frac{1}{x}+1\right)-\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt[3]{\frac{1}{x}+1}}\right)}{3 \sqrt{3}}",1,"(-2*((27*E^((2*ArcCoth[x])/3))/(1 + E^(2*ArcCoth[x]))^2 - (36*E^((2*ArcCoth[x])/3))/(1 + E^(2*ArcCoth[x])) - 2*ArcCoth[x] + 3*Log[1 + E^((2*ArcCoth[x])/3)] - RootSum[1 - #1^2 + #1^4 & , (ArcCoth[x] - 3*Log[E^(ArcCoth[x]/3) - #1] + ArcCoth[x]*#1^2 - 3*Log[E^(ArcCoth[x]/3) - #1]*#1^2)/(-2 + #1^2) & ]))/27","C",0
126,1,167,429,5.3415278,"\int e^{\frac{1}{4} \coth ^{-1}(a x)} x^2 \, dx","Integrate[E^(ArcCoth[a*x]/4)*x^2,x]","\frac{-33 \text{RootSum}\left[\text{$\#$1}^4+1\&,\frac{\coth ^{-1}(a x)-4 \log \left(e^{\frac{1}{4} \coth ^{-1}(a x)}-\text{$\#$1}\right)}{\text{$\#$1}^3}\&\right]-4 \left(-\frac{840 e^{\frac{1}{4} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}-\frac{1600 e^{\frac{1}{4} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^2}-\frac{1024 e^{\frac{1}{4} \coth ^{-1}(a x)}}{\left(e^{2 \coth ^{-1}(a x)}-1\right)^3}+33 \log \left(1-e^{\frac{1}{4} \coth ^{-1}(a x)}\right)-33 \log \left(e^{\frac{1}{4} \coth ^{-1}(a x)}+1\right)-66 \tan ^{-1}\left(e^{\frac{1}{4} \coth ^{-1}(a x)}\right)\right)}{1536 a^3}","-\frac{11 \log \left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}+1\right)}{128 \sqrt{2} a^3}+\frac{11 \log \left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}+1\right)}{128 \sqrt{2} a^3}-\frac{11 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}\right)}{64 \sqrt{2} a^3}+\frac{11 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}+1\right)}{64 \sqrt{2} a^3}+\frac{11 \tan ^{-1}\left(\frac{\sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}\right)}{64 a^3}+\frac{11 \tanh ^{-1}\left(\frac{\sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}\right)}{64 a^3}+\frac{37 x \left(1-\frac{1}{a x}\right)^{7/8} \sqrt[8]{\frac{1}{a x}+1}}{96 a^2}+\frac{1}{3} x^3 \left(1-\frac{1}{a x}\right)^{7/8} \sqrt[8]{\frac{1}{a x}+1}+\frac{3 x^2 \left(1-\frac{1}{a x}\right)^{7/8} \sqrt[8]{\frac{1}{a x}+1}}{8 a}",1,"(-4*((-1024*E^(ArcCoth[a*x]/4))/(-1 + E^(2*ArcCoth[a*x]))^3 - (1600*E^(ArcCoth[a*x]/4))/(-1 + E^(2*ArcCoth[a*x]))^2 - (840*E^(ArcCoth[a*x]/4))/(-1 + E^(2*ArcCoth[a*x])) - 66*ArcTan[E^(ArcCoth[a*x]/4)] + 33*Log[1 - E^(ArcCoth[a*x]/4)] - 33*Log[1 + E^(ArcCoth[a*x]/4)]) - 33*RootSum[1 + #1^4 & , (ArcCoth[a*x] - 4*Log[E^(ArcCoth[a*x]/4) - #1])/#1^3 & ])/(1536*a^3)","C",0
127,1,319,392,0.8034441,"\int e^{\frac{1}{4} \coth ^{-1}(a x)} x \, dx","Integrate[E^(ArcCoth[a*x]/4)*x,x]","\frac{\frac{6}{e^{\frac{1}{4} \coth ^{-1}(a x)}-1}+\frac{6}{e^{\frac{1}{4} \coth ^{-1}(a x)}+1}-\frac{12 e^{\frac{1}{4} \coth ^{-1}(a x)}}{e^{\frac{1}{2} \coth ^{-1}(a x)}+1}-\frac{40 e^{\frac{1}{4} \coth ^{-1}(a x)}}{e^{\coth ^{-1}(a x)}+1}+\frac{2}{\left(e^{\frac{1}{4} \coth ^{-1}(a x)}-1\right)^2}-\frac{2}{\left(e^{\frac{1}{4} \coth ^{-1}(a x)}+1\right)^2}+\frac{8 e^{\frac{1}{4} \coth ^{-1}(a x)}}{\left(e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)^2}+\frac{32 e^{\frac{1}{4} \coth ^{-1}(a x)}}{\left(e^{\coth ^{-1}(a x)}+1\right)^2}-\sqrt{2} \log \left(-\sqrt{2} e^{\frac{1}{4} \coth ^{-1}(a x)}+e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)+\sqrt{2} \log \left(\sqrt{2} e^{\frac{1}{4} \coth ^{-1}(a x)}+e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right)+4 \tan ^{-1}\left(e^{\frac{1}{4} \coth ^{-1}(a x)}\right)-2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} e^{\frac{1}{4} \coth ^{-1}(a x)}\right)+2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} e^{\frac{1}{4} \coth ^{-1}(a x)}+1\right)+4 \tanh ^{-1}\left(e^{\frac{1}{4} \coth ^{-1}(a x)}\right)}{64 a^2}","-\frac{\log \left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}+1\right)}{32 \sqrt{2} a^2}+\frac{\log \left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}+1\right)}{32 \sqrt{2} a^2}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}\right)}{16 \sqrt{2} a^2}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}+1\right)}{16 \sqrt{2} a^2}+\frac{\tan ^{-1}\left(\frac{\sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}\right)}{16 a^2}+\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}\right)}{16 a^2}+\frac{1}{2} x^2 \left(1-\frac{1}{a x}\right)^{7/8} \left(\frac{1}{a x}+1\right)^{9/8}+\frac{x \left(1-\frac{1}{a x}\right)^{7/8} \sqrt[8]{\frac{1}{a x}+1}}{8 a}",1,"(2/(-1 + E^(ArcCoth[a*x]/4))^2 + 6/(-1 + E^(ArcCoth[a*x]/4)) - 2/(1 + E^(ArcCoth[a*x]/4))^2 + 6/(1 + E^(ArcCoth[a*x]/4)) + (8*E^(ArcCoth[a*x]/4))/(1 + E^(ArcCoth[a*x]/2))^2 - (12*E^(ArcCoth[a*x]/4))/(1 + E^(ArcCoth[a*x]/2)) + (32*E^(ArcCoth[a*x]/4))/(1 + E^ArcCoth[a*x])^2 - (40*E^(ArcCoth[a*x]/4))/(1 + E^ArcCoth[a*x]) + 4*ArcTan[E^(ArcCoth[a*x]/4)] - 2*Sqrt[2]*ArcTan[1 - Sqrt[2]*E^(ArcCoth[a*x]/4)] + 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*E^(ArcCoth[a*x]/4)] + 4*ArcTanh[E^(ArcCoth[a*x]/4)] - Sqrt[2]*Log[1 - Sqrt[2]*E^(ArcCoth[a*x]/4) + E^(ArcCoth[a*x]/2)] + Sqrt[2]*Log[1 + Sqrt[2]*E^(ArcCoth[a*x]/4) + E^(ArcCoth[a*x]/2)])/(64*a^2)","A",0
128,1,56,352,0.0554437,"\int e^{\frac{1}{4} \coth ^{-1}(a x)} \, dx","Integrate[E^(ArcCoth[a*x]/4),x]","\frac{2 e^{\frac{1}{4} \coth ^{-1}(a x)} \left(\left(e^{2 \coth ^{-1}(a x)}-1\right) \, _2F_1\left(\frac{1}{8},1;\frac{9}{8};e^{2 \coth ^{-1}(a x)}\right)+1\right)}{a \left(e^{2 \coth ^{-1}(a x)}-1\right)}","x \left(1-\frac{1}{a x}\right)^{7/8} \sqrt[8]{\frac{1}{a x}+1}-\frac{\log \left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}+1\right)}{4 \sqrt{2} a}+\frac{\log \left(\frac{\sqrt[4]{\frac{1}{a x}+1}}{\sqrt[4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}+1\right)}{4 \sqrt{2} a}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}\right)}{2 \sqrt{2} a}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}+1\right)}{2 \sqrt{2} a}+\frac{\tan ^{-1}\left(\frac{\sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}\right)}{2 a}+\frac{\tanh ^{-1}\left(\frac{\sqrt[8]{\frac{1}{a x}+1}}{\sqrt[8]{1-\frac{1}{a x}}}\right)}{2 a}",1,"(2*E^(ArcCoth[a*x]/4)*(1 + (-1 + E^(2*ArcCoth[a*x]))*Hypergeometric2F1[1/8, 1, 9/8, E^(2*ArcCoth[a*x])]))/(a*(-1 + E^(2*ArcCoth[a*x])))","C",0
129,1,30,919,0.0468065,"\int \frac{e^{\frac{1}{4} \coth ^{-1}(a x)}}{x} \, dx","Integrate[E^(ArcCoth[a*x]/4)/x,x]","\frac{16}{9} e^{\frac{9}{4} \coth ^{-1}(a x)} \, _2F_1\left(\frac{9}{16},1;\frac{25}{16};e^{4 \coth ^{-1}(a x)}\right)","-\sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{1+\frac{1}{a x}}}}{\sqrt{2+\sqrt{2}}}\right)-\sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{1+\frac{1}{a x}}}}{\sqrt{2-\sqrt{2}}}\right)+\sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{1+\frac{1}{a x}}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right)+\sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{1+\frac{1}{a x}}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right)-\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[8]{1+\frac{1}{a x}}}{\sqrt[8]{1-\frac{1}{a x}}}\right)+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[8]{1+\frac{1}{a x}}}{\sqrt[8]{1-\frac{1}{a x}}}+1\right)+2 \tan ^{-1}\left(\frac{\sqrt[8]{1+\frac{1}{a x}}}{\sqrt[8]{1-\frac{1}{a x}}}\right)+2 \tanh ^{-1}\left(\frac{\sqrt[8]{1+\frac{1}{a x}}}{\sqrt[8]{1-\frac{1}{a x}}}\right)+\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left(\frac{\sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{1+\frac{1}{a x}}}-\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{1+\frac{1}{a x}}}+1\right)-\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left(\frac{\sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{1+\frac{1}{a x}}}+\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{1+\frac{1}{a x}}}+1\right)+\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left(\frac{\sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{1+\frac{1}{a x}}}-\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{1+\frac{1}{a x}}}+1\right)-\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left(\frac{\sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{1+\frac{1}{a x}}}+\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{1+\frac{1}{a x}}}+1\right)-\frac{\log \left(\frac{\sqrt[4]{1+\frac{1}{a x}}}{\sqrt[4]{1-\frac{1}{a x}}}-\frac{\sqrt{2} \sqrt[8]{1+\frac{1}{a x}}}{\sqrt[8]{1-\frac{1}{a x}}}+1\right)}{\sqrt{2}}+\frac{\log \left(\frac{\sqrt[4]{1+\frac{1}{a x}}}{\sqrt[4]{1-\frac{1}{a x}}}+\frac{\sqrt{2} \sqrt[8]{1+\frac{1}{a x}}}{\sqrt[8]{1-\frac{1}{a x}}}+1\right)}{\sqrt{2}}",1,"(16*E^((9*ArcCoth[a*x])/4)*Hypergeometric2F1[9/16, 1, 25/16, E^(4*ArcCoth[a*x])])/9","C",0
130,1,46,676,0.069878,"\int \frac{e^{\frac{1}{4} \coth ^{-1}(a x)}}{x^2} \, dx","Integrate[E^(ArcCoth[a*x]/4)/x^2,x]","-2 a e^{\frac{1}{4} \coth ^{-1}(a x)} \left(\, _2F_1\left(\frac{1}{8},1;\frac{9}{8};-e^{2 \coth ^{-1}(a x)}\right)-\frac{1}{e^{2 \coth ^{-1}(a x)}+1}\right)","a \left(1-\frac{1}{a x}\right)^{7/8} \sqrt[8]{\frac{1}{a x}+1}+\frac{1}{8} \sqrt{2-\sqrt{2}} a \log \left(\frac{\sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}-\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}+1\right)-\frac{1}{8} \sqrt{2-\sqrt{2}} a \log \left(\frac{\sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}+1\right)+\frac{1}{8} \sqrt{2+\sqrt{2}} a \log \left(\frac{\sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}-\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}+1\right)-\frac{1}{8} \sqrt{2+\sqrt{2}} a \log \left(\frac{\sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}+1\right)-\frac{1}{4} \sqrt{2+\sqrt{2}} a \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}}{\sqrt{2+\sqrt{2}}}\right)-\frac{1}{4} \sqrt{2-\sqrt{2}} a \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}}{\sqrt{2-\sqrt{2}}}\right)+\frac{1}{4} \sqrt{2+\sqrt{2}} a \tan ^{-1}\left(\frac{\frac{2 \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right)+\frac{1}{4} \sqrt{2-\sqrt{2}} a \tan ^{-1}\left(\frac{\frac{2 \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right)",1,"-2*a*E^(ArcCoth[a*x]/4)*(-(1 + E^(2*ArcCoth[a*x]))^(-1) + Hypergeometric2F1[1/8, 1, 9/8, -E^(2*ArcCoth[a*x])])","C",0
131,1,72,731,0.0993098,"\int \frac{e^{\frac{1}{4} \coth ^{-1}(a x)}}{x^3} \, dx","Integrate[E^(ArcCoth[a*x]/4)/x^3,x]","-\frac{a^2 e^{\frac{1}{4} \coth ^{-1}(a x)} \left(\left(e^{2 \coth ^{-1}(a x)}+1\right)^2 \, _2F_1\left(\frac{1}{8},1;\frac{9}{8};-e^{2 \coth ^{-1}(a x)}\right)-9 e^{2 \coth ^{-1}(a x)}-1\right)}{4 \left(e^{2 \coth ^{-1}(a x)}+1\right)^2}","\frac{1}{2} a^2 \left(1-\frac{1}{a x}\right)^{7/8} \left(\frac{1}{a x}+1\right)^{9/8}+\frac{1}{8} a^2 \left(1-\frac{1}{a x}\right)^{7/8} \sqrt[8]{\frac{1}{a x}+1}+\frac{1}{64} \sqrt{2-\sqrt{2}} a^2 \log \left(\frac{\sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}-\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}+1\right)-\frac{1}{64} \sqrt{2-\sqrt{2}} a^2 \log \left(\frac{\sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}+1\right)+\frac{1}{64} \sqrt{2+\sqrt{2}} a^2 \log \left(\frac{\sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}-\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}+1\right)-\frac{1}{64} \sqrt{2+\sqrt{2}} a^2 \log \left(\frac{\sqrt[4]{1-\frac{1}{a x}}}{\sqrt[4]{\frac{1}{a x}+1}}+\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}+1\right)-\frac{1}{32} \sqrt{2+\sqrt{2}} a^2 \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}}{\sqrt{2+\sqrt{2}}}\right)-\frac{1}{32} \sqrt{2-\sqrt{2}} a^2 \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}}{\sqrt{2-\sqrt{2}}}\right)+\frac{1}{32} \sqrt{2+\sqrt{2}} a^2 \tan ^{-1}\left(\frac{\frac{2 \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right)+\frac{1}{32} \sqrt{2-\sqrt{2}} a^2 \tan ^{-1}\left(\frac{\frac{2 \sqrt[8]{1-\frac{1}{a x}}}{\sqrt[8]{\frac{1}{a x}+1}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right)",1,"-1/4*(a^2*E^(ArcCoth[a*x]/4)*(-1 - 9*E^(2*ArcCoth[a*x]) + (1 + E^(2*ArcCoth[a*x]))^2*Hypergeometric2F1[1/8, 1, 9/8, -E^(2*ArcCoth[a*x])]))/(1 + E^(2*ArcCoth[a*x]))^2","C",0
132,1,47,45,0.026471,"\int e^{4 \coth ^{-1}(a x)} x^m \, dx","Integrate[E^(4*ArcCoth[a*x])*x^m,x]","\frac{x^{m+1} (-4 (m+1) (a x-1) \, _2F_1(1,m+1;m+2;a x)+a x-4 m-5)}{(m+1) (a x-1)}","-4 x^{m+1} \, _2F_1(1,m+1;m+2;a x)+\frac{4 x^{m+1}}{1-a x}+\frac{x^{m+1}}{m+1}",1,"(x^(1 + m)*(-5 - 4*m + a*x - 4*(1 + m)*(-1 + a*x)*Hypergeometric2F1[1, 1 + m, 2 + m, a*x]))/((1 + m)*(-1 + a*x))","A",1
133,1,228,151,0.3553007,"\int e^{3 \coth ^{-1}(a x)} x^m \, dx","Integrate[E^(3*ArcCoth[a*x])*x^m,x]","\frac{x^{m+1} \left(3 (m+1) \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{1-a x} \sqrt{\frac{a x+1}{a^2}} F_1\left(m;-\frac{1}{2},\frac{1}{2};m+1;-a x,a x\right)-2 (m+1) \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{1-a x} \sqrt{\frac{a x+1}{a^2}} F_1\left(m;-\frac{1}{2},\frac{3}{2};m+1;-a x,a x\right)+m \sqrt{a x-1} \sqrt{a x+1} \sqrt{x^2-\frac{1}{a^2}} \, _2F_1\left(-\frac{1}{2},-\frac{m}{2}-\frac{1}{2};\frac{1}{2}-\frac{m}{2};\frac{1}{a^2 x^2}\right)\right)}{m (m+1) \sqrt{a x-1} \sqrt{a x+1} \sqrt{x^2-\frac{1}{a^2}}}","-\frac{3 x^{m+1} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};\frac{1}{a^2 x^2}\right)}{m+1}+\frac{4 x^{m+1} \, _2F_1\left(\frac{3}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};\frac{1}{a^2 x^2}\right)}{m+1}-\frac{x^m \, _2F_1\left(\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right)}{a m}+\frac{4 x^m \, _2F_1\left(\frac{3}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right)}{a m}",1,"(x^(1 + m)*(3*(1 + m)*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[1 - a*x]*Sqrt[(1 + a*x)/a^2]*AppellF1[m, -1/2, 1/2, 1 + m, -(a*x), a*x] - 2*(1 + m)*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[1 - a*x]*Sqrt[(1 + a*x)/a^2]*AppellF1[m, -1/2, 3/2, 1 + m, -(a*x), a*x] + m*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[-a^(-2) + x^2]*Hypergeometric2F1[-1/2, -1/2 - m/2, 1/2 - m/2, 1/(a^2*x^2)]))/(m*(1 + m)*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[-a^(-2) + x^2])","C",0
134,1,26,35,0.0097514,"\int e^{2 \coth ^{-1}(a x)} x^m \, dx","Integrate[E^(2*ArcCoth[a*x])*x^m,x]","\frac{x^{m+1} (1-2 \, _2F_1(1,m+1;m+2;a x))}{m+1}","\frac{x^{m+1}}{m+1}-\frac{2 x^{m+1} \, _2F_1(1,m+1;m+2;a x)}{m+1}",1,"(x^(1 + m)*(1 - 2*Hypergeometric2F1[1, 1 + m, 2 + m, a*x]))/(1 + m)","A",1
135,1,128,74,0.4228694,"\int e^{\coth ^{-1}(a x)} x^m \, dx","Integrate[E^ArcCoth[a*x]*x^m,x]","x^{m+1} \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{m}{2}-\frac{1}{2};\frac{1}{2}-\frac{m}{2};\frac{1}{a^2 x^2}\right)}{m+1}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} \sqrt{x^2-\frac{1}{a^2}} F_1\left(m;-\frac{1}{2},\frac{1}{2};m+1;-a x,a x\right)}{m \sqrt{a x-1} \sqrt{\frac{a x+1}{a^2}} \sqrt{1-a^2 x^2}}\right)","\frac{x^{m+1} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};\frac{1}{a^2 x^2}\right)}{m+1}+\frac{x^m \, _2F_1\left(\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right)}{a m}",1,"x^(1 + m)*(-((Sqrt[1 - 1/(a^2*x^2)]*Sqrt[-a^(-2) + x^2]*AppellF1[m, -1/2, 1/2, 1 + m, -(a*x), a*x])/(m*Sqrt[-1 + a*x]*Sqrt[(1 + a*x)/a^2]*Sqrt[1 - a^2*x^2])) + Hypergeometric2F1[-1/2, -1/2 - m/2, 1/2 - m/2, 1/(a^2*x^2)]/(1 + m))","C",0
136,1,115,75,0.257068,"\int e^{-\coth ^{-1}(a x)} x^m \, dx","Integrate[x^m/E^ArcCoth[a*x],x]","x^{m+1} \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{m}{2}-\frac{1}{2};\frac{1}{2}-\frac{m}{2};\frac{1}{a^2 x^2}\right)}{m+1}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{a x-1}{a^2}} F_1\left(m;-\frac{1}{2},\frac{1}{2};m+1;a x,-a x\right)}{m \sqrt{1-a x} \sqrt{x^2-\frac{1}{a^2}}}\right)","\frac{x^{m+1} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};\frac{1}{a^2 x^2}\right)}{m+1}-\frac{x^m \, _2F_1\left(\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right)}{a m}",1,"x^(1 + m)*(-((Sqrt[1 - 1/(a^2*x^2)]*Sqrt[(-1 + a*x)/a^2]*AppellF1[m, -1/2, 1/2, 1 + m, a*x, -(a*x)])/(m*Sqrt[1 - a*x]*Sqrt[-a^(-2) + x^2])) + Hypergeometric2F1[-1/2, -1/2 - m/2, 1/2 - m/2, 1/(a^2*x^2)]/(1 + m))","C",0
137,1,27,36,0.0125067,"\int e^{-2 \coth ^{-1}(a x)} x^m \, dx","Integrate[x^m/E^(2*ArcCoth[a*x]),x]","\frac{x^{m+1} (1-2 \, _2F_1(1,m+1;m+2;-a x))}{m+1}","\frac{x^{m+1}}{m+1}-\frac{2 x^{m+1} \, _2F_1(1,m+1;m+2;-a x)}{m+1}",1,"(x^(1 + m)*(1 - 2*Hypergeometric2F1[1, 1 + m, 2 + m, -(a*x)]))/(1 + m)","A",1
138,1,192,150,0.2653334,"\int e^{-3 \coth ^{-1}(a x)} x^m \, dx","Integrate[x^m/E^(3*ArcCoth[a*x]),x]","\frac{x^{m+1} \left(-3 (m+1) \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{a x-1}{a^2}} F_1\left(m;-\frac{1}{2},\frac{1}{2};m+1;a x,-a x\right)+2 (m+1) \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{a x-1}{a^2}} F_1\left(m;-\frac{1}{2},\frac{3}{2};m+1;a x,-a x\right)+m \sqrt{1-a x} \sqrt{x^2-\frac{1}{a^2}} \, _2F_1\left(-\frac{1}{2},-\frac{m}{2}-\frac{1}{2};\frac{1}{2}-\frac{m}{2};\frac{1}{a^2 x^2}\right)\right)}{m (m+1) \sqrt{1-a x} \sqrt{x^2-\frac{1}{a^2}}}","-\frac{3 x^{m+1} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};\frac{1}{a^2 x^2}\right)}{m+1}+\frac{4 x^{m+1} \, _2F_1\left(\frac{3}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};\frac{1}{a^2 x^2}\right)}{m+1}+\frac{x^m \, _2F_1\left(\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right)}{a m}-\frac{4 x^m \, _2F_1\left(\frac{3}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{a^2 x^2}\right)}{a m}",1,"(x^(1 + m)*(-3*(1 + m)*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[(-1 + a*x)/a^2]*AppellF1[m, -1/2, 1/2, 1 + m, a*x, -(a*x)] + 2*(1 + m)*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[(-1 + a*x)/a^2]*AppellF1[m, -1/2, 3/2, 1 + m, a*x, -(a*x)] + m*Sqrt[1 - a*x]*Sqrt[-a^(-2) + x^2]*Hypergeometric2F1[-1/2, -1/2 - m/2, 1/2 - m/2, 1/(a^2*x^2)]))/(m*(1 + m)*Sqrt[1 - a*x]*Sqrt[-a^(-2) + x^2])","C",0
139,0,0,41,0.6471832,"\int e^{\frac{5}{2} \coth ^{-1}(a x)} x^m \, dx","Integrate[E^((5*ArcCoth[a*x])/2)*x^m,x]","\int e^{\frac{5}{2} \coth ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(-m-1;\frac{5}{4},-\frac{5}{4};-m;\frac{1}{a x},-\frac{1}{a x}\right)}{m+1}",1,"Integrate[E^((5*ArcCoth[a*x])/2)*x^m, x]","F",-1
140,0,0,41,0.6087067,"\int e^{\frac{3}{2} \coth ^{-1}(a x)} x^m \, dx","Integrate[E^((3*ArcCoth[a*x])/2)*x^m,x]","\int e^{\frac{3}{2} \coth ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(-m-1;\frac{3}{4},-\frac{3}{4};-m;\frac{1}{a x},-\frac{1}{a x}\right)}{m+1}",1,"Integrate[E^((3*ArcCoth[a*x])/2)*x^m, x]","F",-1
141,0,0,41,0.5412227,"\int e^{\frac{1}{2} \coth ^{-1}(a x)} x^m \, dx","Integrate[E^(ArcCoth[a*x]/2)*x^m,x]","\int e^{\frac{1}{2} \coth ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(-m-1;\frac{1}{4},-\frac{1}{4};-m;\frac{1}{a x},-\frac{1}{a x}\right)}{m+1}",1,"Integrate[E^(ArcCoth[a*x]/2)*x^m, x]","F",-1
142,0,0,41,0.6335249,"\int e^{-\frac{1}{2} \coth ^{-1}(a x)} x^m \, dx","Integrate[x^m/E^(ArcCoth[a*x]/2),x]","\int e^{-\frac{1}{2} \coth ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(-m-1;-\frac{1}{4},\frac{1}{4};-m;\frac{1}{a x},-\frac{1}{a x}\right)}{m+1}",1,"Integrate[x^m/E^(ArcCoth[a*x]/2), x]","F",-1
143,0,0,41,0.6094834,"\int e^{-\frac{3}{2} \coth ^{-1}(a x)} x^m \, dx","Integrate[x^m/E^((3*ArcCoth[a*x])/2),x]","\int e^{-\frac{3}{2} \coth ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(-m-1;-\frac{3}{4},\frac{3}{4};-m;\frac{1}{a x},-\frac{1}{a x}\right)}{m+1}",1,"Integrate[x^m/E^((3*ArcCoth[a*x])/2), x]","F",-1
144,0,0,41,0.6859692,"\int e^{-\frac{5}{2} \coth ^{-1}(a x)} x^m \, dx","Integrate[x^m/E^((5*ArcCoth[a*x])/2),x]","\int e^{-\frac{5}{2} \coth ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(-m-1;-\frac{5}{4},\frac{5}{4};-m;\frac{1}{a x},-\frac{1}{a x}\right)}{m+1}",1,"Integrate[x^m/E^((5*ArcCoth[a*x])/2), x]","F",-1
145,0,0,34,0.8519309,"\int e^{\frac{2}{3} \coth ^{-1}(x)} x^m \, dx","Integrate[E^((2*ArcCoth[x])/3)*x^m,x]","\int e^{\frac{2}{3} \coth ^{-1}(x)} x^m \, dx","\frac{x^{m+1} F_1\left(-m-1;\frac{1}{3},-\frac{1}{3};-m;\frac{1}{x},-\frac{1}{x}\right)}{m+1}",1,"Integrate[E^((2*ArcCoth[x])/3)*x^m, x]","F",-1
146,0,0,34,0.8335571,"\int e^{\frac{1}{3} \coth ^{-1}(x)} x^m \, dx","Integrate[E^(ArcCoth[x]/3)*x^m,x]","\int e^{\frac{1}{3} \coth ^{-1}(x)} x^m \, dx","\frac{x^{m+1} F_1\left(-m-1;\frac{1}{6},-\frac{1}{6};-m;\frac{1}{x},-\frac{1}{x}\right)}{m+1}",1,"Integrate[E^(ArcCoth[x]/3)*x^m, x]","F",-1
147,0,0,41,0.689531,"\int e^{\frac{1}{4} \coth ^{-1}(a x)} x^m \, dx","Integrate[E^(ArcCoth[a*x]/4)*x^m,x]","\int e^{\frac{1}{4} \coth ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(-m-1;\frac{1}{8},-\frac{1}{8};-m;\frac{1}{a x},-\frac{1}{a x}\right)}{m+1}",1,"Integrate[E^(ArcCoth[a*x]/4)*x^m, x]","F",-1
148,0,0,45,0.3616305,"\int e^{n \coth ^{-1}(a x)} x^m \, dx","Integrate[E^(n*ArcCoth[a*x])*x^m,x]","\int e^{n \coth ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(-m-1;\frac{n}{2},-\frac{n}{2};-m;\frac{1}{a x},-\frac{1}{a x}\right)}{m+1}",1,"Integrate[E^(n*ArcCoth[a*x])*x^m, x]","F",-1
149,1,118,174,0.6256314,"\int e^{n \coth ^{-1}(a x)} x^2 \, dx","Integrate[E^(n*ArcCoth[a*x])*x^2,x]","\frac{e^{n \coth ^{-1}(a x)} \left((n+2) \left(2 a^3 x^3+n \left(a^2 x^2-1\right)+\left(n^2+2\right) \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)+a n^2 x\right)+n \left(n^2+2\right) e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)\right)}{6 a^3 (n+2)}","\frac{2 \left(n^2+2\right) \left(\frac{1}{a x}+1\right)^{\frac{n-2}{2}} \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}} \, _2F_1\left(2,1-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{3 a^3 (2-n)}+\frac{1}{3} x^3 \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}}+\frac{n x^2 \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}}}{6 a}",1,"(E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n*(2 + n^2)*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(a*n^2*x + 2*a^3*x^3 + n*(-1 + a^2*x^2) + (2 + n^2)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(6*a^3*(2 + n))","A",0
150,1,98,122,0.3241878,"\int e^{n \coth ^{-1}(a x)} x \, dx","Integrate[E^(n*ArcCoth[a*x])*x,x]","\frac{e^{n \coth ^{-1}(a x)} \left((n+2) \left(a^2 x^2+n \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)+a n x-1\right)+n^2 e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)\right)}{2 a^2 (n+2)}","\frac{2 n \left(\frac{1}{a x}+1\right)^{\frac{n-2}{2}} \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}} \, _2F_1\left(2,1-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a^2 (2-n)}+\frac{1}{2} x^2 \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}}",1,"(E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n^2*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(-1 + a*n*x + a^2*x^2 + n*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(2*a^2*(2 + n))","A",0
151,1,82,78,0.1955887,"\int e^{n \coth ^{-1}(a x)} \, dx","Integrate[E^(n*ArcCoth[a*x]),x]","\frac{e^{n \coth ^{-1}(a x)} \left(n e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)+(n+2) \left(\, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)+a x\right)\right)}{a (n+2)}","\frac{4 \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n-2}{2}} \, _2F_1\left(2,1-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a (2-n)}",1,"(E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(a*x + Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(a*(2 + n))","A",0
152,1,142,127,0.1592126,"\int \frac{e^{n \coth ^{-1}(a x)}}{x} \, dx","Integrate[E^(n*ArcCoth[a*x])/x,x]","\frac{e^{n \coth ^{-1}(a x)} \left(n e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;-e^{2 \coth ^{-1}(a x)}\right)+n e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)-(n+2) \left(\, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;-e^{2 \coth ^{-1}(a x)}\right)-\, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)\right)\right)}{n (n+2)}","\frac{2^{\frac{n}{2}+1} \left(1-\frac{1}{a x}\right)^{-n/2} \, _2F_1\left(-\frac{n}{2},-\frac{n}{2};1-\frac{n}{2};\frac{a-\frac{1}{x}}{2 a}\right)}{n}-\frac{2 \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} \, _2F_1\left(1,-\frac{n}{2};1-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{n}",1,"(E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, -E^(2*ArcCoth[a*x])] + E^(2*ArcCoth[a*x])*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] - (2 + n)*(Hypergeometric2F1[1, n/2, 1 + n/2, -E^(2*ArcCoth[a*x])] - Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(n*(2 + n))","A",0
153,1,44,70,0.0391515,"\int \frac{e^{n \coth ^{-1}(a x)}}{x^2} \, dx","Integrate[E^(n*ArcCoth[a*x])/x^2,x]","-\frac{4 a e^{(n+2) \coth ^{-1}(a x)} \, _2F_1\left(2,\frac{n}{2}+1;\frac{n}{2}+2;-e^{2 \coth ^{-1}(a x)}\right)}{n+2}","\frac{a 2^{\frac{n}{2}+1} \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}} \, _2F_1\left(1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{2 a}\right)}{2-n}",1,"(-4*a*E^((2 + n)*ArcCoth[a*x])*Hypergeometric2F1[2, 1 + n/2, 2 + n/2, -E^(2*ArcCoth[a*x])])/(2 + n)","A",0
154,1,107,114,0.3888387,"\int \frac{e^{n \coth ^{-1}(a x)}}{x^3} \, dx","Integrate[E^(n*ArcCoth[a*x])/x^3,x]","-\frac{a^2 e^{n \coth ^{-1}(a x)} \left((n+2) \left(\frac{1}{a^2 x^2}+n \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;-e^{2 \coth ^{-1}(a x)}\right)+\frac{n}{a x}-1\right)-n^2 e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;-e^{2 \coth ^{-1}(a x)}\right)\right)}{2 (n+2)}","\frac{a^2 2^{n/2} n \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}} \, _2F_1\left(1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{2 a}\right)}{2-n}+\frac{1}{2} a^2 \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}}",1,"-1/2*(a^2*E^(n*ArcCoth[a*x])*(-(E^(2*ArcCoth[a*x])*n^2*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, -E^(2*ArcCoth[a*x])]) + (2 + n)*(-1 + 1/(a^2*x^2) + n/(a*x) + n*Hypergeometric2F1[1, n/2, 1 + n/2, -E^(2*ArcCoth[a*x])])))/(2 + n)","A",0
155,1,132,167,0.6656048,"\int \frac{e^{n \coth ^{-1}(a x)}}{x^4} \, dx","Integrate[E^(n*ArcCoth[a*x])/x^4,x]","-\frac{a^3 e^{n \coth ^{-1}(a x)} \left((n+2) \left(-\left(1-\frac{1}{a^2 x^2}\right) \left(\frac{2}{a x}+n\right)+\left(n^2+2\right) \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;-e^{2 \coth ^{-1}(a x)}\right)+\frac{n^2+2}{a x}\right)-n \left(n^2+2\right) e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;-e^{2 \coth ^{-1}(a x)}\right)\right)}{6 (n+2)}","\frac{a^3 2^{n/2} \left(n^2+2\right) \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}} \, _2F_1\left(1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{2 a}\right)}{3 (2-n)}+\frac{1}{6} a^3 n \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}}+\frac{a^2 \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}}}{3 x}",1,"-1/6*(a^3*E^(n*ArcCoth[a*x])*(-(E^(2*ArcCoth[a*x])*n*(2 + n^2)*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, -E^(2*ArcCoth[a*x])]) + (2 + n)*(-((1 - 1/(a^2*x^2))*(n + 2/(a*x))) + (2 + n^2)/(a*x) + (2 + n^2)*Hypergeometric2F1[1, n/2, 1 + n/2, -E^(2*ArcCoth[a*x])])))/(2 + n)","A",0
156,1,148,183,0.5451499,"\int \frac{e^{n \coth ^{-1}(a x)}}{x^5} \, dx","Integrate[E^(n*ArcCoth[a*x])/x^5,x]","-\frac{1}{24} a^4 e^{n \coth ^{-1}(a x)} \left(\frac{6}{a^4 x^4}+\frac{2 n}{a^3 x^3}+\frac{n^2}{a^2 x^2}-\frac{\left(n^2+8\right) n^2 e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;-e^{2 \coth ^{-1}(a x)}\right)}{n+2}+\left(n^2+8\right) n \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;-e^{2 \coth ^{-1}(a x)}\right)+\frac{n^3}{a x}+\frac{6 n}{a x}-n^2-6\right)","\frac{a^4 2^{\frac{n}{2}-2} n \left(n^2+8\right) \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}} \, _2F_1\left(1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{2 a}\right)}{3 (2-n)}+\frac{1}{24} a^3 \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(a \left(n^2+6\right)+\frac{2 n}{x}\right) \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}}+\frac{a^2 \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}}}{4 x^2}",1,"-1/24*(a^4*E^(n*ArcCoth[a*x])*(-6 - n^2 + 6/(a^4*x^4) + (2*n)/(a^3*x^3) + n^2/(a^2*x^2) + (6*n)/(a*x) + n^3/(a*x) - (E^(2*ArcCoth[a*x])*n^2*(8 + n^2)*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, -E^(2*ArcCoth[a*x])])/(2 + n) + n*(8 + n^2)*Hypergeometric2F1[1, n/2, 1 + n/2, -E^(2*ArcCoth[a*x])]))","A",0
157,1,131,143,0.0953703,"\int e^{\coth ^{-1}(a x)} (c-a c x)^p \, dx","Integrate[E^ArcCoth[a*x]*(c - a*c*x)^p,x]","\frac{\sqrt{\frac{1}{a x}+1} \left(\frac{a x-1}{a x+1}\right)^{-p} (c-a c x)^p \left(\sqrt{\frac{a x-1}{a x+1}} \, _2F_1\left(\frac{1}{2}-p,-p;1-p;\frac{2}{a x+1}\right)+p (a x-1) \left(\frac{a x-1}{a x+1}\right)^p\right)}{a p (p+1) \sqrt{1-\frac{1}{a x}}}","\frac{\sqrt{\frac{1}{a x}+1} \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{\frac{1}{2}-p} (c-a c x)^p \, _2F_1\left(\frac{1}{2}-p,-p;1-p;\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{a p (p+1) \sqrt{1-\frac{1}{a x}}}+\frac{x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1} (c-a c x)^p}{p+1}",1,"(Sqrt[1 + 1/(a*x)]*(c - a*c*x)^p*(p*(-1 + a*x)*((-1 + a*x)/(1 + a*x))^p + Sqrt[(-1 + a*x)/(1 + a*x)]*Hypergeometric2F1[1/2 - p, -p, 1 - p, 2/(1 + a*x)]))/(a*p*(1 + p)*Sqrt[1 - 1/(a*x)]*((-1 + a*x)/(1 + a*x))^p)","A",1
158,1,80,132,0.1997601,"\int e^{\coth ^{-1}(a x)} (c-a c x)^4 \, dx","Integrate[E^ArcCoth[a*x]*(c - a*c*x)^4,x]","\frac{c^4 \left(105 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+a x \sqrt{1-\frac{1}{a^2 x^2}} \left(24 a^4 x^4-90 a^3 x^3+112 a^2 x^2-15 a x-136\right)\right)}{120 a}","-\frac{7}{8} a c^4 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{7 c^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{8 a}+\frac{17}{15} a^2 c^4 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}+\frac{1}{5} a^4 c^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}-\frac{3}{4} a^3 c^4 x^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}",1,"(c^4*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-136 - 15*a*x + 112*a^2*x^2 - 90*a^3*x^3 + 24*a^4*x^4) + 105*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(120*a)","A",0
159,1,73,105,0.1692617,"\int e^{\coth ^{-1}(a x)} (c-a c x)^3 \, dx","Integrate[E^ArcCoth[a*x]*(c - a*c*x)^3,x]","\frac{c^3 \left(15 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)-a x \sqrt{1-\frac{1}{a^2 x^2}} \left(6 a^3 x^3-16 a^2 x^2+9 a x+16\right)\right)}{24 a}","-\frac{5}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{5 c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{8 a}+\frac{2}{3} a^2 c^3 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}-\frac{1}{4} a^3 c^3 x^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}",1,"(c^3*(-(a*Sqrt[1 - 1/(a^2*x^2)]*x*(16 + 9*a*x - 16*a^2*x^2 + 6*a^3*x^3)) + 15*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(24*a)","A",0
160,1,64,78,0.107729,"\int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx","Integrate[E^ArcCoth[a*x]*(c - a*c*x)^2,x]","\frac{c^2 \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2-3 a x-2\right)+3 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{6 a}","-\frac{1}{2} a c^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a}+\frac{1}{3} a^2 c^2 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}",1,"(c^2*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-2 - 3*a*x + 2*a^2*x^2) + 3*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(6*a)","A",0
161,1,51,47,0.0562271,"\int e^{\coth ^{-1}(a x)} (c-a c x) \, dx","Integrate[E^ArcCoth[a*x]*(c - a*c*x),x]","\frac{c \left(\log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)-a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a}","\frac{c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a}-\frac{1}{2} a c x^2 \sqrt{1-\frac{1}{a^2 x^2}}",1,"(c*(-(a^2*Sqrt[1 - 1/(a^2*x^2)]*x^2) + Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(2*a)","A",0
162,1,60,51,0.0596806,"\int \frac{e^{\coth ^{-1}(a x)}}{c-a c x} \, dx","Integrate[E^ArcCoth[a*x]/(c - a*c*x),x]","\frac{2 a x \sqrt{1-\frac{1}{a^2 x^2}}+(1-a x) \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a c (a x-1)}","\frac{2 \left(a+\frac{1}{x}\right)}{a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c}",1,"(2*a*Sqrt[1 - 1/(a^2*x^2)]*x + (1 - a*x)*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c*(-1 + a*x))","A",0
163,1,34,33,0.0552798,"\int \frac{e^{\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx","Integrate[E^ArcCoth[a*x]/(c - a*c*x)^2,x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x+1)}{3 c^2 (a x-1)^2}","-\frac{a^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{3 c^2 \left(a-\frac{1}{x}\right)^3}",1,"-1/3*(Sqrt[1 - 1/(a^2*x^2)]*x*(1 + a*x))/(c^2*(-1 + a*x)^2)","A",1
164,1,42,67,0.0598505,"\int \frac{e^{\coth ^{-1}(a x)}}{(c-a c x)^3} \, dx","Integrate[E^ArcCoth[a*x]/(c - a*c*x)^3,x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(a^2 x^2-3 a x-4\right)}{15 c^3 (a x-1)^3}","\frac{a^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{5 c^3 \left(a-\frac{1}{x}\right)^4}-\frac{4 a^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{15 c^3 \left(a-\frac{1}{x}\right)^3}",1,"-1/15*(Sqrt[1 - 1/(a^2*x^2)]*x*(-4 - 3*a*x + a^2*x^2))/(c^3*(-1 + a*x)^3)","A",1
165,1,51,100,0.0656204,"\int \frac{e^{\coth ^{-1}(a x)}}{(c-a c x)^4} \, dx","Integrate[E^ArcCoth[a*x]/(c - a*c*x)^4,x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^3 x^3-8 a^2 x^2+13 a x+23\right)}{105 c^4 (a x-1)^4}","-\frac{23 a^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{105 c^4 \left(a-\frac{1}{x}\right)^3}-\frac{a^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{7 c^4 \left(a-\frac{1}{x}\right)^5}+\frac{12 a^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{35 c^4 \left(a-\frac{1}{x}\right)^4}",1,"-1/105*(Sqrt[1 - 1/(a^2*x^2)]*x*(23 + 13*a*x - 8*a^2*x^2 + 2*a^3*x^3))/(c^4*(-1 + a*x)^4)","A",1
166,1,59,133,0.0696155,"\int \frac{e^{\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx","Integrate[E^ArcCoth[a*x]/(c - a*c*x)^5,x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^4 x^4-10 a^3 x^3+21 a^2 x^2-25 a x-58\right)}{315 c^5 (a x-1)^5}","-\frac{58 a^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{315 c^5 \left(a-\frac{1}{x}\right)^3}+\frac{a^5 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{9 c^5 \left(a-\frac{1}{x}\right)^6}-\frac{8 a^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{21 c^5 \left(a-\frac{1}{x}\right)^5}+\frac{47 a^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{105 c^5 \left(a-\frac{1}{x}\right)^4}",1,"-1/315*(Sqrt[1 - 1/(a^2*x^2)]*x*(-58 - 25*a*x + 21*a^2*x^2 - 10*a^3*x^3 + 2*a^4*x^4))/(c^5*(-1 + a*x)^5)","A",1
167,1,28,42,0.0248342,"\int e^{2 \coth ^{-1}(a x)} (c-a c x)^p \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a*c*x)^p,x]","\frac{(a p x+p+2) (c-a c x)^p}{a p (p+1)}","\frac{2 (c-a c x)^p}{a p}-\frac{(c-a c x)^{p+1}}{a c (p+1)}",1,"((c - a*c*x)^p*(2 + p + a*p*x))/(a*p*(1 + p))","A",1
168,1,23,37,0.0219279,"\int e^{2 \coth ^{-1}(a x)} (c-a c x)^5 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a*c*x)^5,x]","-\frac{c^5 (a x-1)^5 (5 a x+7)}{30 a}","\frac{2 c^5 (1-a x)^5}{5 a}-\frac{c^5 (1-a x)^6}{6 a}",1,"-1/30*(c^5*(-1 + a*x)^5*(7 + 5*a*x))/a","A",1
169,1,30,37,0.0181834,"\int e^{2 \coth ^{-1}(a x)} (c-a c x)^4 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a*c*x)^4,x]","\frac{1}{10} c^4 x \left(2 a^4 x^4-5 a^3 x^3+10 a x-10\right)","\frac{c^4 (1-a x)^4}{2 a}-\frac{c^4 (1-a x)^5}{5 a}",1,"(c^4*x*(-10 + 10*a*x - 5*a^3*x^3 + 2*a^4*x^4))/10","A",1
170,1,30,37,0.01608,"\int e^{2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a*c*x)^3,x]","-\frac{1}{12} c^3 x \left(3 a^3 x^3-4 a^2 x^2-6 a x+12\right)","\frac{2 c^3 (1-a x)^3}{3 a}-\frac{c^3 (1-a x)^4}{4 a}",1,"-1/12*(c^3*x*(12 - 6*a*x - 4*a^2*x^2 + 3*a^3*x^3))","A",1
171,1,17,20,0.0091846,"\int e^{2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a*c*x)^2,x]","-c^2 \left(x-\frac{a^2 x^3}{3}\right)","\frac{1}{3} a^2 c^2 x^3-c^2 x",1,"-(c^2*(x - (a^2*x^3)/3))","A",1
172,1,26,14,0.0083433,"\int e^{2 \coth ^{-1}(a x)} (c-a c x) \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a*c*x),x]","\frac{c \left(1-a^2 x^2\right) e^{2 \coth ^{-1}(a x)}}{2 a}","-\frac{1}{2} a c x^2-c x",1,"(c*E^(2*ArcCoth[a*x])*(1 - a^2*x^2))/(2*a)","C",1
173,1,30,32,0.016836,"\int \frac{e^{2 \coth ^{-1}(a x)}}{c-a c x} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a*c*x),x]","-\frac{\frac{2}{a (1-a x)}+\frac{\log (1-a x)}{a}}{c}","-\frac{2}{a c (1-a x)}-\frac{\log (1-a x)}{a c}",1,"-((2/(a*(1 - a*x)) + Log[1 - a*x]/a)/c)","A",1
174,1,25,14,0.0087786,"\int \frac{e^{2 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a*c*x)^2,x]","-\frac{(a x+1)^2}{4 a c^2 (1-a x)^2}","-\frac{x}{c^2 (1-a x)^2}",1,"-1/4*(1 + a*x)^2/(a*c^2*(1 - a*x)^2)","A",1
175,1,23,37,0.0167245,"\int \frac{e^{2 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a*c*x)^3,x]","\frac{3 a x+1}{6 a c^3 (a x-1)^3}","\frac{1}{2 a c^3 (1-a x)^2}-\frac{2}{3 a c^3 (1-a x)^3}",1,"(1 + 3*a*x)/(6*a*c^3*(-1 + a*x)^3)","A",1
176,1,23,37,0.0178529,"\int \frac{e^{2 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a*c*x)^4,x]","-\frac{2 a x+1}{6 a c^4 (a x-1)^4}","\frac{1}{3 a c^4 (1-a x)^3}-\frac{1}{2 a c^4 (1-a x)^4}",1,"-1/6*(1 + 2*a*x)/(a*c^4*(-1 + a*x)^4)","A",1
177,1,155,202,0.1481565,"\int e^{3 \coth ^{-1}(a x)} (c-a c x)^p \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a*c*x)^p,x]","\frac{\sqrt{\frac{1}{a x}+1} \left(\frac{a x-1}{a x+1}\right)^{-p} (c-a c x)^p \left(3 \sqrt{\frac{a x-1}{a x+1}} \, _2F_1\left(1-p,\frac{3}{2}-p;2-p;\frac{2}{a x+1}\right)+(p-1) (a x+1) (a p x+p+3) \left(\frac{a x-1}{a x+1}\right)^p\right)}{a (p-1) p (p+1) \sqrt{1-\frac{1}{a x}} (a x+1)}","-\frac{3 \sqrt{\frac{1}{a x}+1} \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{\frac{3}{2}-p} (c-a c x)^p \, _2F_1\left(1-p,\frac{3}{2}-p;2-p;\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{a^2 p \left(1-p^2\right) x \left(1-\frac{1}{a x}\right)^{3/2}}+\frac{x \left(\frac{1}{a x}+1\right)^{3/2} (c-a c x)^p}{(p+1) \sqrt{1-\frac{1}{a x}}}+\frac{3 \sqrt{\frac{1}{a x}+1} (c-a c x)^p}{a p (p+1) \sqrt{1-\frac{1}{a x}}}",1,"(Sqrt[1 + 1/(a*x)]*(c - a*c*x)^p*((-1 + p)*((-1 + a*x)/(1 + a*x))^p*(1 + a*x)*(3 + p + a*p*x) + 3*Sqrt[(-1 + a*x)/(1 + a*x)]*Hypergeometric2F1[1 - p, 3/2 - p, 2 - p, 2/(1 + a*x)]))/(a*(-1 + p)*p*(1 + p)*Sqrt[1 - 1/(a*x)]*((-1 + a*x)/(1 + a*x))^p*(1 + a*x))","A",1
178,1,80,105,0.2116304,"\int e^{3 \coth ^{-1}(a x)} (c-a c x)^4 \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a*c*x)^4,x]","\frac{c^4 \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(8 a^4 x^4-10 a^3 x^3-16 a^2 x^2+25 a x+8\right)-15 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{40 a}","\frac{3}{8} a c^4 x^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 c^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{8 a}+\frac{1}{5} a^4 c^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}-\frac{1}{4} a^3 c^4 x^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}",1,"(c^4*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(8 + 25*a*x - 16*a^2*x^2 - 10*a^3*x^3 + 8*a^4*x^4) - 15*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(40*a)","A",0
179,1,64,78,0.1529547,"\int e^{3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a*c*x)^3,x]","\frac{c^3 \left(a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(5-2 a^2 x^2\right)-3 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{8 a}","\frac{3}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{8 a}-\frac{1}{4} a^3 c^3 x^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}",1,"(c^3*(a^2*Sqrt[1 - 1/(a^2*x^2)]*x^2*(5 - 2*a^2*x^2) - 3*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(8*a)","A",0
180,1,64,78,0.1169677,"\int e^{3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a*c*x)^2,x]","\frac{c^2 \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2+3 a x-2\right)-3 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{6 a}","\frac{1}{2} a c^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a}+\frac{1}{3} a^2 c^2 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}",1,"(c^2*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-2 + 3*a*x + 2*a^2*x^2) - 3*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(6*a)","A",0
181,1,53,65,0.0851835,"\int e^{3 \coth ^{-1}(a x)} (c-a c x) \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a*c*x),x]","-\frac{c \left(a x \sqrt{1-\frac{1}{a^2 x^2}} (a x+4)+3 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{2 a}","-\frac{1}{2} a c x^2 \sqrt{1-\frac{1}{a^2 x^2}}-2 c x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a}",1,"-1/2*(c*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(4 + a*x) + 3*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/a","A",0
182,1,63,80,0.0955057,"\int \frac{e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a*c*x),x]","\frac{\frac{4 x \sqrt{1-\frac{1}{a^2 x^2}} (2 a x-1)}{(a x-1)^2}-\frac{3 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a}}{3 c}","\frac{8 \left(a+\frac{1}{x}\right)}{3 a^2 c \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}+\frac{4}{3 a^2 c x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c}",1,"((4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + 2*a*x))/(-1 + a*x)^2 - (3*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a)/(3*c)","A",0
183,1,36,33,0.0627753,"\int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a*c*x)^2,x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x+1)^2}{5 c^2 (a x-1)^3}","-\frac{a^4 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{5 c^2 \left(a-\frac{1}{x}\right)^5}",1,"-1/5*(Sqrt[1 - 1/(a^2*x^2)]*x*(1 + a*x)^2)/(c^2*(-1 + a*x)^3)","A",1
184,1,41,67,0.0691497,"\int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a*c*x)^3,x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x-6) (a x+1)^2}{35 c^3 (a x-1)^4}","\frac{a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{7 c^3 \left(a-\frac{1}{x}\right)^6}-\frac{6 a^4 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{35 c^3 \left(a-\frac{1}{x}\right)^5}",1,"-1/35*(Sqrt[1 - 1/(a^2*x^2)]*x*(-6 + a*x)*(1 + a*x)^2)/(c^3*(-1 + a*x)^4)","A",1
185,1,50,94,0.0688744,"\int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a*c*x)^4,x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x+1)^2 \left(2 a^2 x^2-14 a x+47\right)}{315 c^4 (a x-1)^5}","-\frac{\left(a+\frac{1}{x}\right)^7}{9 a^8 c^4 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}+\frac{16 \left(a+\frac{1}{x}\right)^6}{63 a^7 c^4 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}-\frac{47 \left(a+\frac{1}{x}\right)^5}{315 a^6 c^4 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}",1,"-1/315*(Sqrt[1 - 1/(a^2*x^2)]*x*(1 + a*x)^2*(47 - 14*a*x + 2*a^2*x^2))/(c^4*(-1 + a*x)^5)","A",1
186,1,58,125,0.0771783,"\int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a*c*x)^5,x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x+1)^2 \left(2 a^3 x^3-16 a^2 x^2+61 a x-152\right)}{1155 c^5 (a x-1)^6}","\frac{\left(a+\frac{1}{x}\right)^8}{11 a^9 c^5 \left(1-\frac{1}{a^2 x^2}\right)^{11/2}}-\frac{10 \left(a+\frac{1}{x}\right)^7}{33 a^8 c^5 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}+\frac{79 \left(a+\frac{1}{x}\right)^6}{231 a^7 c^5 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}-\frac{152 \left(a+\frac{1}{x}\right)^5}{1155 a^6 c^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}",1,"-1/1155*(Sqrt[1 - 1/(a^2*x^2)]*x*(1 + a*x)^2*(-152 + 61*a*x - 16*a^2*x^2 + 2*a^3*x^3))/(c^5*(-1 + a*x)^6)","A",1
187,1,50,66,0.0964632,"\int e^{4 \coth ^{-1}(a x)} (c-a c x)^p \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - a*c*x)^p,x]","\frac{\left(\frac{a x}{p+1}+\frac{4}{(p-1) (a x-1)}+\frac{3 p+4}{p (p+1)}\right) (c-a c x)^p}{a}","\frac{4 c (c-a c x)^{p-1}}{a (1-p)}+\frac{4 (c-a c x)^p}{a p}-\frac{(c-a c x)^{p+1}}{a c (p+1)}",1,"((c - a*c*x)^p*((4 + 3*p)/(p*(1 + p)) + (a*x)/(1 + p) + 4/((-1 + p)*(-1 + a*x))))/a","A",1
188,1,31,53,0.0252798,"\int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - a*c*x)^5,x]","-\frac{c^5 (a x-1)^4 \left(5 a^2 x^2+14 a x+11\right)}{30 a}","-\frac{c^5 (1-a x)^6}{6 a}+\frac{4 c^5 (1-a x)^5}{5 a}-\frac{c^5 (1-a x)^4}{a}",1,"-1/30*(c^5*(-1 + a*x)^4*(11 + 14*a*x + 5*a^2*x^2))/a","A",1
189,1,26,32,0.0170138,"\int e^{4 \coth ^{-1}(a x)} (c-a c x)^4 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - a*c*x)^4,x]","c^4 \left(\frac{a^4 x^5}{5}-\frac{2 a^2 x^3}{3}+x\right)","\frac{1}{5} a^4 c^4 x^5-\frac{2}{3} a^2 c^4 x^3+c^4 x",1,"c^4*(x - (2*a^2*x^3)/3 + (a^4*x^5)/5)","A",1
190,1,30,35,0.0160274,"\int e^{4 \coth ^{-1}(a x)} (c-a c x)^3 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - a*c*x)^3,x]","-\frac{1}{12} c^3 x \left(3 a^3 x^3+4 a^2 x^2-6 a x-12\right)","\frac{2 c^3 (a x+1)^3}{3 a}-\frac{c^3 (a x+1)^4}{4 a}",1,"-1/12*(c^3*x*(-12 - 6*a*x + 4*a^2*x^2 + 3*a^3*x^3))","A",1
191,1,21,17,0.0137862,"\int e^{4 \coth ^{-1}(a x)} (c-a c x)^2 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - a*c*x)^2,x]","c^2 \left(\frac{a^2 x^3}{3}+a x^2+x\right)","\frac{c^2 (a x+1)^3}{3 a}",1,"c^2*(x + a*x^2 + (a^2*x^3)/3)","A",1
192,1,26,27,0.0132752,"\int e^{4 \coth ^{-1}(a x)} (c-a c x) \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - a*c*x),x]","c \left(-\frac{a x^2}{2}-\frac{4 \log (1-a x)}{a}-3 x\right)","-\frac{1}{2} a c x^2-\frac{4 c \log (1-a x)}{a}-3 c x",1,"c*(-3*x - (a*x^2)/2 - (4*Log[1 - a*x])/a)","A",1
193,1,36,48,0.0235715,"\int \frac{e^{4 \coth ^{-1}(a x)}}{c-a c x} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - a*c*x),x]","\frac{4 a x+(a x-1)^2 (-\log (1-a x))-2}{a c (a x-1)^2}","-\frac{4}{a c (1-a x)}+\frac{2}{a c (1-a x)^2}-\frac{\log (1-a x)}{a c}",1,"(-2 + 4*a*x - (-1 + a*x)^2*Log[1 - a*x])/(a*c*(-1 + a*x)^2)","A",1
194,1,25,25,0.0097056,"\int \frac{e^{4 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - a*c*x)^2,x]","\frac{(a x+1)^3}{6 a c^2 (1-a x)^3}","\frac{(a x+1)^3}{6 a c^2 (1-a x)^3}",1,"(1 + a*x)^3/(6*a*c^2*(1 - a*x)^3)","A",1
195,1,31,52,0.0201971,"\int \frac{e^{4 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - a*c*x)^3,x]","\frac{3 a^2 x^2+2 a x+1}{6 a c^3 (a x-1)^4}","\frac{1}{2 a c^3 (1-a x)^2}-\frac{4}{3 a c^3 (1-a x)^3}+\frac{1}{a c^3 (1-a x)^4}",1,"(1 + 2*a*x + 3*a^2*x^2)/(6*a*c^3*(-1 + a*x)^4)","A",1
196,1,31,53,0.0211693,"\int \frac{e^{4 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - a*c*x)^4,x]","-\frac{5 a^2 x^2+5 a x+2}{15 a c^4 (a x-1)^5}","\frac{1}{3 a c^4 (1-a x)^3}-\frac{1}{a c^4 (1-a x)^4}+\frac{4}{5 a c^4 (1-a x)^5}",1,"-1/15*(2 + 5*a*x + 5*a^2*x^2)/(a*c^4*(-1 + a*x)^5)","A",1
197,1,76,94,0.0491651,"\int e^{-\coth ^{-1}(a x)} (c-a c x)^p \, dx","Integrate[(c - a*c*x)^p/E^ArcCoth[a*x],x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(\frac{a x-1}{a x+1}\right)^{-p-\frac{1}{2}} (c-a c x)^p \, _2F_1\left(-p-1,-p-\frac{1}{2};-p;\frac{2}{a x+1}\right)}{p+1}","\frac{x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1} \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{-p-\frac{1}{2}} (c-a c x)^p \, _2F_1\left(-p-1,-p-\frac{1}{2};-p;\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{p+1}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*((-1 + a*x)/(1 + a*x))^(-1/2 - p)*(c - a*c*x)^p*Hypergeometric2F1[-1 - p, -1/2 - p, -p, 2/(1 + a*x)])/(1 + p)","A",1
198,1,72,127,0.2066725,"\int e^{-\coth ^{-1}(a x)} (c-a c x)^3 \, dx","Integrate[(c - a*c*x)^3/E^ArcCoth[a*x],x]","\frac{c^3 \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(-6 a^3 x^3+32 a^2 x^2-81 a x+160\right)-105 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{24 a}","-\frac{27}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{20}{3} c^3 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{35 c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{8 a}+\frac{4}{3} a^2 c^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{1}{4} a^3 c^3 x^4 \sqrt{1-\frac{1}{a^2 x^2}}",1,"(c^3*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(160 - 81*a*x + 32*a^2*x^2 - 6*a^3*x^3) - 105*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(24*a)","A",0
199,1,64,100,0.1258691,"\int e^{-\coth ^{-1}(a x)} (c-a c x)^2 \, dx","Integrate[(c - a*c*x)^2/E^ArcCoth[a*x],x]","\frac{c^2 \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2-9 a x+22\right)-15 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{6 a}","-\frac{3}{2} a c^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{11}{3} c^2 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{5 c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a}+\frac{1}{3} a^2 c^2 x^3 \sqrt{1-\frac{1}{a^2 x^2}}",1,"(c^2*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(22 - 9*a*x + 2*a^2*x^2) - 15*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(6*a)","A",0
200,1,53,65,0.0829531,"\int e^{-\coth ^{-1}(a x)} (c-a c x) \, dx","Integrate[(c - a*c*x)/E^ArcCoth[a*x],x]","-\frac{c \left(a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-4)+3 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{2 a}","-\frac{1}{2} a c x^2 \sqrt{1-\frac{1}{a^2 x^2}}+2 c x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a}",1,"-1/2*(c*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-4 + a*x) + 3*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/a","A",0
201,1,34,23,0.042893,"\int \frac{e^{-\coth ^{-1}(a x)}}{c-a c x} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)),x]","-\frac{\log \left(a x \left(\sqrt{\frac{a^2 x^2-1}{a^2 x^2}}+1\right)\right)}{a c}","-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c}",1,"-(Log[a*x*(1 + Sqrt[(-1 + a^2*x^2)/(a^2*x^2)])]/(a*c))","A",0
202,1,27,28,0.0621008,"\int \frac{e^{-\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^2),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 (a x-1)}","-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \left(a-\frac{1}{x}\right)}",1,"-((Sqrt[1 - 1/(a^2*x^2)]*x)/(c^2*(-1 + a*x)))","A",1
203,1,34,62,0.0620606,"\int \frac{e^{-\coth ^{-1}(a x)}}{(c-a c x)^3} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^3),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x-2)}{3 c^3 (a x-1)^2}","\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left(a-\frac{1}{x}\right)^2}-\frac{2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^3 \left(a-\frac{1}{x}\right)}",1,"-1/3*(Sqrt[1 - 1/(a^2*x^2)]*x*(-2 + a*x))/(c^3*(-1 + a*x)^2)","A",1
204,1,43,95,0.0667445,"\int \frac{e^{-\coth ^{-1}(a x)}}{(c-a c x)^4} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^4),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2-6 a x+7\right)}{15 c^4 (a x-1)^3}","-\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{5 c^4 \left(a-\frac{1}{x}\right)^3}+\frac{8 a \sqrt{1-\frac{1}{a^2 x^2}}}{15 c^4 \left(a-\frac{1}{x}\right)^2}-\frac{7 \sqrt{1-\frac{1}{a^2 x^2}}}{15 c^4 \left(a-\frac{1}{x}\right)}",1,"-1/15*(Sqrt[1 - 1/(a^2*x^2)]*x*(7 - 6*a*x + 2*a^2*x^2))/(c^4*(-1 + a*x)^3)","A",1
205,1,51,128,0.0744829,"\int \frac{e^{-\coth ^{-1}(a x)}}{(c-a c x)^5} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^5),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^3 x^3-8 a^2 x^2+13 a x-12\right)}{35 c^5 (a x-1)^4}","-\frac{18 a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left(a-\frac{1}{x}\right)^3}+\frac{23 a \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left(a-\frac{1}{x}\right)^2}-\frac{12 \sqrt{1-\frac{1}{a^2 x^2}}}{35 c^5 \left(a-\frac{1}{x}\right)}+\frac{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}{7 c^5 \left(a-\frac{1}{x}\right)^4}",1,"-1/35*(Sqrt[1 - 1/(a^2*x^2)]*x*(-12 + 13*a*x - 8*a^2*x^2 + 2*a^3*x^3))/(c^5*(-1 + a*x)^4)","A",1
206,1,44,44,0.0171839,"\int e^{-2 \coth ^{-1}(a x)} (c-a c x)^p \, dx","Integrate[(c - a*c*x)^p/E^(2*ArcCoth[a*x]),x]","-\frac{(a x-1) (c-a c x)^p \left(\, _2F_1\left(1,p+1;p+2;\frac{1}{2} (1-a x)\right)-1\right)}{a (p+1)}","\frac{(c-a c x)^{p+2} \, _2F_1\left(1,p+2;p+3;\frac{1}{2} (1-a x)\right)}{2 a c^2 (p+2)}",1,"-(((-1 + a*x)*(c - a*c*x)^p*(-1 + Hypergeometric2F1[1, 1 + p, 2 + p, (1 - a*x)/2]))/(a*(1 + p)))","A",0
207,1,56,91,0.0231503,"\int e^{-2 \coth ^{-1}(a x)} (c-a c x)^4 \, dx","Integrate[(c - a*c*x)^4/E^(2*ArcCoth[a*x]),x]","\frac{c^4 \left(6 a^5 x^5-45 a^4 x^4+160 a^3 x^3-390 a^2 x^2+930 a x-960 \log (a x+1)-181\right)}{30 a}","-\frac{c^4 (1-a x)^5}{5 a}-\frac{c^4 (1-a x)^4}{2 a}-\frac{4 c^4 (1-a x)^3}{3 a}-\frac{4 c^4 (1-a x)^2}{a}-\frac{32 c^4 \log (a x+1)}{a}+16 c^4 x",1,"(c^4*(-181 + 930*a*x - 390*a^2*x^2 + 160*a^3*x^3 - 45*a^4*x^4 + 6*a^5*x^5 - 960*Log[1 + a*x]))/(30*a)","A",1
208,1,48,73,0.0207334,"\int e^{-2 \coth ^{-1}(a x)} (c-a c x)^3 \, dx","Integrate[(c - a*c*x)^3/E^(2*ArcCoth[a*x]),x]","-\frac{c^3 \left(3 a^4 x^4-20 a^3 x^3+66 a^2 x^2-180 a x+192 \log (a x+1)+35\right)}{12 a}","-\frac{c^3 (1-a x)^4}{4 a}-\frac{2 c^3 (1-a x)^3}{3 a}-\frac{2 c^3 (1-a x)^2}{a}-\frac{16 c^3 \log (a x+1)}{a}+8 c^3 x",1,"-1/12*(c^3*(35 - 180*a*x + 66*a^2*x^2 - 20*a^3*x^3 + 3*a^4*x^4 + 192*Log[1 + a*x]))/a","A",1
209,1,39,55,0.0162331,"\int e^{-2 \coth ^{-1}(a x)} (c-a c x)^2 \, dx","Integrate[(c - a*c*x)^2/E^(2*ArcCoth[a*x]),x]","\frac{c^2 \left(a^3 x^3-6 a^2 x^2+21 a x-24 \log (a x+1)-4\right)}{3 a}","-\frac{c^2 (1-a x)^3}{3 a}-\frac{c^2 (1-a x)^2}{a}-\frac{8 c^2 \log (a x+1)}{a}+4 c^2 x",1,"(c^2*(-4 + 21*a*x - 6*a^2*x^2 + a^3*x^3 - 24*Log[1 + a*x]))/(3*a)","A",1
210,1,26,26,0.0112562,"\int e^{-2 \coth ^{-1}(a x)} (c-a c x) \, dx","Integrate[(c - a*c*x)/E^(2*ArcCoth[a*x]),x]","-\frac{1}{2} a c x^2-\frac{4 c \log (a x+1)}{a}+3 c x","-\frac{1}{2} a c x^2-\frac{4 c \log (a x+1)}{a}+3 c x",1,"3*c*x - (a*c*x^2)/2 - (4*c*Log[1 + a*x])/a","A",1
211,1,14,14,0.0102215,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{c-a c x} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)),x]","-\frac{\log (a x+1)}{a c}","-\frac{\log (a x+1)}{a c}",1,"-(Log[1 + a*x]/(a*c))","A",1
212,1,12,12,0.0116675,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^2),x]","-\frac{\tanh ^{-1}(a x)}{a c^2}","-\frac{\tanh ^{-1}(a x)}{a c^2}",1,"-(ArcTanh[a*x]/(a*c^2))","A",1
213,1,32,33,0.0219442,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^3),x]","-\frac{\frac{1}{2 a (1-a x)}+\frac{\tanh ^{-1}(a x)}{2 a}}{c^3}","-\frac{1}{2 a c^3 (1-a x)}-\frac{\tanh ^{-1}(a x)}{2 a c^3}",1,"-((1/(2*a*(1 - a*x)) + ArcTanh[a*x]/(2*a))/c^3)","A",1
214,1,35,51,0.0282825,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^4),x]","\frac{a x+(a x-1)^2 \left(-\tanh ^{-1}(a x)\right)-2}{4 a c^4 (a x-1)^2}","-\frac{1}{4 a c^4 (1-a x)}-\frac{1}{4 a c^4 (1-a x)^2}-\frac{\tanh ^{-1}(a x)}{4 a c^4}",1,"(-2 + a*x - (-1 + a*x)^2*ArcTanh[a*x])/(4*a*c^4*(-1 + a*x)^2)","A",1
215,1,44,69,0.0339369,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^5),x]","\frac{3 a^2 x^2-9 a x-3 (a x-1)^3 \tanh ^{-1}(a x)+10}{24 a c^5 (a x-1)^3}","-\frac{1}{8 a c^5 (1-a x)}-\frac{1}{8 a c^5 (1-a x)^2}-\frac{1}{6 a c^5 (1-a x)^3}-\frac{\tanh ^{-1}(a x)}{8 a c^5}",1,"(10 - 9*a*x + 3*a^2*x^2 - 3*(-1 + a*x)^3*ArcTanh[a*x])/(24*a*c^5*(-1 + a*x)^3)","A",1
216,1,96,94,0.0694125,"\int e^{-3 \coth ^{-1}(a x)} (c-a c x)^p \, dx","Integrate[(c - a*c*x)^p/E^(3*ArcCoth[a*x]),x]","\frac{\sqrt{1-\frac{1}{a x}} (a x+1) \left(\frac{a x-1}{a x+1}\right)^{-p-\frac{1}{2}} (c-a c x)^p \, _2F_1\left(-p-\frac{3}{2},-p-1;-p;\frac{2}{a x+1}\right)}{a (p+1) \sqrt{\frac{1}{a x}+1}}","\frac{x \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{-p-\frac{3}{2}} (c-a c x)^p \, _2F_1\left(-p-\frac{3}{2},-p-1;-p;\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{(p+1) \sqrt{\frac{1}{a x}+1}}",1,"(Sqrt[1 - 1/(a*x)]*((-1 + a*x)/(1 + a*x))^(-1/2 - p)*(1 + a*x)*(c - a*c*x)^p*Hypergeometric2F1[-3/2 - p, -1 - p, -p, 2/(1 + a*x)])/(a*(1 + p)*Sqrt[1 + 1/(a*x)])","A",1
217,1,86,152,0.245231,"\int e^{-3 \coth ^{-1}(a x)} (c-a c x)^3 \, dx","Integrate[(c - a*c*x)^3/E^(3*ArcCoth[a*x]),x]","\frac{1}{8} c^3 \left(\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(-2 a^4 x^4+14 a^3 x^3-51 a^2 x^2+173 a x+496\right)}{a x+1}-\frac{315 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a}\right)","-\frac{67}{8} a c^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+30 c^3 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{32 c^3 \left(a-\frac{1}{x}\right)}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{315 c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{8 a}+2 a^2 c^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{1}{4} a^3 c^3 x^4 \sqrt{1-\frac{1}{a^2 x^2}}",1,"(c^3*((Sqrt[1 - 1/(a^2*x^2)]*x*(496 + 173*a*x - 51*a^2*x^2 + 14*a^3*x^3 - 2*a^4*x^4))/(1 + a*x) - (315*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a))/8","A",0
218,1,78,129,0.1701605,"\int e^{-3 \coth ^{-1}(a x)} (c-a c x)^2 \, dx","Integrate[(c - a*c*x)^2/E^(3*ArcCoth[a*x]),x]","\frac{1}{6} c^2 \left(\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^3 x^3-13 a^2 x^2+55 a x+166\right)}{a x+1}-\frac{105 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a}\right)","-\frac{5}{2} a c^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{35}{3} c^2 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{16 c^2 \left(a-\frac{1}{x}\right)}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{35 c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a}+\frac{1}{3} a^2 c^2 x^3 \sqrt{1-\frac{1}{a^2 x^2}}",1,"(c^2*((Sqrt[1 - 1/(a^2*x^2)]*x*(166 + 55*a*x - 13*a^2*x^2 + 2*a^3*x^3))/(1 + a*x) - (105*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a))/6","A",0
219,1,68,92,0.1502922,"\int e^{-3 \coth ^{-1}(a x)} (c-a c x) \, dx","Integrate[(c - a*c*x)/E^(3*ArcCoth[a*x]),x]","\frac{1}{2} c \left(\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(-a^2 x^2+7 a x+24\right)}{a x+1}-\frac{15 \log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a}\right)","-\frac{1}{2} a c x^2 \sqrt{1-\frac{1}{a^2 x^2}}+4 c x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{8 c \left(a-\frac{1}{x}\right)}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{15 c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{2 a}",1,"(c*((Sqrt[1 - 1/(a^2*x^2)]*x*(24 + 7*a*x - a^2*x^2))/(1 + a*x) - (15*Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a))/2","A",0
220,1,54,53,0.0691969,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{c-a c x} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)),x]","\frac{\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{a x+1}-\frac{\log \left(a x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a}}{c}","\frac{2 \left(a-\frac{1}{x}\right)}{a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c}",1,"((2*Sqrt[1 - 1/(a^2*x^2)]*x)/(1 + a*x) - Log[a*(1 + Sqrt[1 - 1/(a^2*x^2)])*x]/a)/c","A",0
221,1,26,28,0.0552608,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^2),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 (a x+1)}","\frac{a-\frac{1}{x}}{a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x)/(c^2*(1 + a*x))","A",1
222,1,33,21,0.0609787,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^3),x]","\frac{a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{c^3 \left(a^2 x^2-1\right)}","\frac{1}{a c^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(a*Sqrt[1 - 1/(a^2*x^2)]*x^2)/(c^3*(-1 + a^2*x^2))","A",1
223,1,50,61,0.0684749,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^4),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2-2 a x-1\right)}{3 c^4 (a x-1)^2 (a x+1)}","\frac{2}{3 a c^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{1}{3 a^2 c^4 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(a-\frac{1}{x}\right)}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(-1 - 2*a*x + 2*a^2*x^2))/(3*c^4*(-1 + a*x)^2*(1 + a*x))","A",1
224,1,57,94,0.0712691,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^5),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^3 x^3-4 a^2 x^2+a x+2\right)}{5 c^5 (a x-1)^3 (a x+1)}","-\frac{4 \left(a+\frac{1}{x}\right)}{5 a^2 c^5 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}+\frac{5 a+\frac{2}{x}}{5 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\left(a+\frac{1}{x}\right)^2}{5 a^3 c^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(2 + a*x - 4*a^2*x^2 + 2*a^3*x^3))/(5*c^5*(-1 + a*x)^3*(1 + a*x))","A",1
225,1,66,125,0.0797447,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^6} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^6),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(8 a^4 x^4-24 a^3 x^3+20 a^2 x^2+4 a x-13\right)}{35 c^6 (a x-1)^4 (a x+1)}","-\frac{46 \left(a+\frac{1}{x}\right)}{35 a^2 c^6 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}+\frac{35 a+\frac{13}{x}}{35 a^2 c^6 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\left(a+\frac{1}{x}\right)^3}{7 a^4 c^6 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}+\frac{24 \left(a+\frac{1}{x}\right)^2}{35 a^3 c^6 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(-13 + 4*a*x + 20*a^2*x^2 - 24*a^3*x^3 + 8*a^4*x^4))/(35*c^6*(-1 + a*x)^4*(1 + a*x))","A",1
226,1,83,254,0.0602723,"\int e^{\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - a*c*x)^(9/2),x]","\frac{2 c^4 \sqrt{\frac{1}{a x}+1} (a x+1) \left(315 a^4 x^4-1820 a^3 x^3+4530 a^2 x^2-6396 a x+5419\right) \sqrt{c-a c x}}{3465 a \sqrt{1-\frac{1}{a x}}}","\frac{9088 \left(\frac{1}{a x}+1\right)^{3/2} (c-a c x)^{9/2}}{3465 a^4 x^3 \left(1-\frac{1}{a x}\right)^{9/2}}-\frac{32 \left(a-\frac{1}{x}\right)^3 \left(\frac{1}{a x}+1\right)^{3/2} (c-a c x)^{9/2}}{99 a^4 \left(1-\frac{1}{a x}\right)^{9/2}}+\frac{2 x \left(a-\frac{1}{x}\right)^4 \left(\frac{1}{a x}+1\right)^{3/2} (c-a c x)^{9/2}}{11 a^4 \left(1-\frac{1}{a x}\right)^{9/2}}-\frac{768 \left(\frac{1}{a x}+1\right)^{3/2} (c-a c x)^{9/2}}{385 a^3 x^2 \left(1-\frac{1}{a x}\right)^{9/2}}+\frac{128 \left(\frac{1}{a x}+1\right)^{3/2} (c-a c x)^{9/2}}{231 a^2 x \left(1-\frac{1}{a x}\right)^{9/2}}",1,"(2*c^4*Sqrt[1 + 1/(a*x)]*(1 + a*x)*Sqrt[c - a*c*x]*(5419 - 6396*a*x + 4530*a^2*x^2 - 1820*a^3*x^3 + 315*a^4*x^4))/(3465*a*Sqrt[1 - 1/(a*x)])","A",1
227,1,75,197,0.0481654,"\int e^{\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - a*c*x)^(7/2),x]","-\frac{2 c^3 \sqrt{\frac{1}{a x}+1} (a x+1) \left(35 a^3 x^3-165 a^2 x^2+321 a x-319\right) \sqrt{c-a c x}}{315 a \sqrt{1-\frac{1}{a x}}}","-\frac{568 \left(\frac{1}{a x}+1\right)^{3/2} (c-a c x)^{7/2}}{315 a^3 x^2 \left(1-\frac{1}{a x}\right)^{7/2}}+\frac{2 x \left(a-\frac{1}{x}\right)^3 \left(\frac{1}{a x}+1\right)^{3/2} (c-a c x)^{7/2}}{9 a^3 \left(1-\frac{1}{a x}\right)^{7/2}}+\frac{48 \left(\frac{1}{a x}+1\right)^{3/2} (c-a c x)^{7/2}}{35 a^2 x \left(1-\frac{1}{a x}\right)^{7/2}}-\frac{8 \left(\frac{1}{a x}+1\right)^{3/2} (c-a c x)^{7/2}}{21 a \left(1-\frac{1}{a x}\right)^{7/2}}",1,"(-2*c^3*Sqrt[1 + 1/(a*x)]*(1 + a*x)*Sqrt[c - a*c*x]*(-319 + 321*a*x - 165*a^2*x^2 + 35*a^3*x^3))/(315*a*Sqrt[1 - 1/(a*x)])","A",1
228,1,67,115,0.0406341,"\int e^{\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - a*c*x)^(5/2),x]","\frac{2 c^2 \sqrt{\frac{1}{a x}+1} (a x+1) \left(15 a^2 x^2-54 a x+71\right) \sqrt{c-a c x}}{105 a \sqrt{1-\frac{1}{a x}}}","\frac{64 a^2 c^4 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{105 (c-a c x)^{3/2}}+\frac{16 a^2 c^3 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{35 \sqrt{c-a c x}}+\frac{2}{7} a^2 c^2 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \sqrt{c-a c x}",1,"(2*c^2*Sqrt[1 + 1/(a*x)]*(1 + a*x)*Sqrt[c - a*c*x]*(71 - 54*a*x + 15*a^2*x^2))/(105*a*Sqrt[1 - 1/(a*x)])","A",0
229,1,57,77,0.0349382,"\int e^{\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - a*c*x)^(3/2),x]","-\frac{2 c \sqrt{\frac{1}{a x}+1} (a x+1) (3 a x-7) \sqrt{c-a c x}}{15 a \sqrt{1-\frac{1}{a x}}}","\frac{8 a^2 c^3 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{15 (c-a c x)^{3/2}}+\frac{2 a^2 c^2 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{5 \sqrt{c-a c x}}",1,"(-2*c*Sqrt[1 + 1/(a*x)]*(1 + a*x)*(-7 + 3*a*x)*Sqrt[c - a*c*x])/(15*a*Sqrt[1 - 1/(a*x)])","A",0
230,1,43,29,0.0217,"\int e^{\coth ^{-1}(a x)} \sqrt{c-a c x} \, dx","Integrate[E^ArcCoth[a*x]*Sqrt[c - a*c*x],x]","\frac{2 x \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}","\frac{2 (a x+1) \sqrt{c-a c x} e^{\coth ^{-1}(a x)}}{3 a}",1,"(2*(1 + 1/(a*x))^(3/2)*x*Sqrt[c - a*c*x])/(3*Sqrt[1 - 1/(a*x)])","A",0
231,1,99,118,0.069209,"\int \frac{e^{\coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx","Integrate[E^ArcCoth[a*x]/Sqrt[c - a*c*x],x]","\frac{2 x \sqrt{1-\frac{1}{a x}} \left(\sqrt{a} \sqrt{\frac{1}{a x}+1}-\sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{\sqrt{a} \sqrt{c-a c x}}","\frac{2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\sqrt{c-a c x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x}}",1,"(2*Sqrt[1 - 1/(a*x)]*x*(Sqrt[a]*Sqrt[1 + 1/(a*x)] - Sqrt[2]*Sqrt[x^(-1)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(Sqrt[a]*Sqrt[c - a*c*x])","A",1
232,1,116,128,0.1151494,"\int \frac{e^{\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx","Integrate[E^ArcCoth[a*x]/(c - a*c*x)^(3/2),x]","\frac{x \sqrt{1-\frac{1}{a x}} \left(2 \sqrt{a} \sqrt{\frac{1}{a x}+1}+\sqrt{2} \sqrt{\frac{1}{x}} (a x-1) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{2 \sqrt{a} c (a x-1) \sqrt{c-a c x}}","-\frac{a x \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{3/2}}{\left(a-\frac{1}{x}\right) (c-a c x)^{3/2}}-\frac{\sqrt{a} \left(1-\frac{1}{a x}\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{\sqrt{2} \left(\frac{1}{x}\right)^{3/2} (c-a c x)^{3/2}}",1,"(Sqrt[1 - 1/(a*x)]*x*(2*Sqrt[a]*Sqrt[1 + 1/(a*x)] + Sqrt[2]*Sqrt[x^(-1)]*(-1 + a*x)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(2*Sqrt[a]*c*(-1 + a*x)*Sqrt[c - a*c*x])","A",1
233,1,123,193,0.1641777,"\int \frac{e^{\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx","Integrate[E^ArcCoth[a*x]/(c - a*c*x)^(5/2),x]","\frac{x \sqrt{1-\frac{1}{a x}} \left(\sqrt{2} \sqrt{\frac{1}{x}} (a x-1)^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)-2 \sqrt{a} \sqrt{\frac{1}{a x}+1} (a x+3)\right)}{16 \sqrt{a} c^2 (a x-1)^2 \sqrt{c-a c x}}","\frac{a^{3/2} \left(1-\frac{1}{a x}\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{8 \sqrt{2} \left(\frac{1}{x}\right)^{5/2} (c-a c x)^{5/2}}-\frac{a^3 x^2 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{3/2}}{4 \left(a-\frac{1}{x}\right)^2 (c-a c x)^{5/2}}+\frac{a^2 x^2 \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}{8 \left(a-\frac{1}{x}\right) (c-a c x)^{5/2}}",1,"(Sqrt[1 - 1/(a*x)]*x*(-2*Sqrt[a]*Sqrt[1 + 1/(a*x)]*(3 + a*x) + Sqrt[2]*Sqrt[x^(-1)]*(-1 + a*x)^2*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(16*Sqrt[a]*c^2*(-1 + a*x)^2*Sqrt[c - a*c*x])","A",1
234,1,139,250,0.1735948,"\int \frac{e^{\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx","Integrate[E^ArcCoth[a*x]/(c - a*c*x)^(7/2),x]","\frac{\sqrt{1-\frac{1}{a x}} \left(\frac{2 \sqrt{a} \sqrt{\frac{1}{a x}+1} \left(-3 a^2 x^2+10 a x+25\right)}{\sqrt{\frac{1}{x}}}+3 \sqrt{2} (a x-1)^3 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{192 \sqrt{a} c^3 \sqrt{\frac{1}{x}} (a x-1)^3 \sqrt{c-a c x}}","-\frac{a^{5/2} \left(1-\frac{1}{a x}\right)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{32 \sqrt{2} \left(\frac{1}{x}\right)^{7/2} (c-a c x)^{7/2}}+\frac{a^4 x^3 \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{3/2}}{16 \left(a-\frac{1}{x}\right)^2 (c-a c x)^{7/2}}-\frac{a^4 x^2 \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{3/2}}{6 \left(a-\frac{1}{x}\right)^3 (c-a c x)^{7/2}}-\frac{a^3 x^3 \left(1-\frac{1}{a x}\right)^{7/2} \sqrt{\frac{1}{a x}+1}}{32 \left(a-\frac{1}{x}\right) (c-a c x)^{7/2}}",1,"(Sqrt[1 - 1/(a*x)]*((2*Sqrt[a]*Sqrt[1 + 1/(a*x)]*(25 + 10*a*x - 3*a^2*x^2))/Sqrt[x^(-1)] + 3*Sqrt[2]*(-1 + a*x)^3*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(192*Sqrt[a]*c^3*Sqrt[x^(-1)]*(-1 + a*x)^3*Sqrt[c - a*c*x])","A",1
235,1,34,40,0.044236,"\int e^{2 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a*c*x)^(7/2),x]","-\frac{2 c^3 (a x-1)^3 (7 a x+11) \sqrt{c-a c x}}{63 a}","\frac{4 (c-a c x)^{7/2}}{7 a}-\frac{2 (c-a c x)^{9/2}}{9 a c}",1,"(-2*c^3*(-1 + a*x)^3*(11 + 7*a*x)*Sqrt[c - a*c*x])/(63*a)","A",1
236,1,34,40,0.0440244,"\int e^{2 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a*c*x)^(5/2),x]","\frac{2 c^2 (a x-1)^2 (5 a x+9) \sqrt{c-a c x}}{35 a}","\frac{4 (c-a c x)^{5/2}}{5 a}-\frac{2 (c-a c x)^{7/2}}{7 a c}",1,"(2*c^2*(-1 + a*x)^2*(9 + 5*a*x)*Sqrt[c - a*c*x])/(35*a)","A",1
237,1,30,40,0.0351556,"\int e^{2 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a*c*x)^(3/2),x]","-\frac{2 c (a x-1) (3 a x+7) \sqrt{c-a c x}}{15 a}","\frac{4 (c-a c x)^{3/2}}{3 a}-\frac{2 (c-a c x)^{5/2}}{5 a c}",1,"(-2*c*(-1 + a*x)*(7 + 3*a*x)*Sqrt[c - a*c*x])/(15*a)","A",1
238,1,23,38,0.0294536,"\int e^{2 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx","Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - a*c*x],x]","\frac{2 (a x+5) \sqrt{c-a c x}}{3 a}","\frac{4 \sqrt{c-a c x}}{a}-\frac{2 (c-a c x)^{3/2}}{3 a c}",1,"(2*(5 + a*x)*Sqrt[c - a*c*x])/(3*a)","A",1
239,1,21,36,0.0253308,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx","Integrate[E^(2*ArcCoth[a*x])/Sqrt[c - a*c*x],x]","\frac{2 a x-6}{a \sqrt{c-a c x}}","-\frac{2 \sqrt{c-a c x}}{a c}-\frac{4}{a \sqrt{c-a c x}}",1,"(-6 + 2*a*x)/(a*Sqrt[c - a*c*x])","A",1
240,1,34,38,0.0503407,"\int \frac{e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a*c*x)^(3/2),x]","-\frac{2 (3 a x-1) \sqrt{c-a c x}}{3 a c^2 (a x-1)^2}","\frac{2}{a c \sqrt{c-a c x}}-\frac{4}{3 a (c-a c x)^{3/2}}",1,"(-2*(-1 + 3*a*x)*Sqrt[c - a*c*x])/(3*a*c^2*(-1 + a*x)^2)","A",1
241,1,34,40,0.0633733,"\int \frac{e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a*c*x)^(5/2),x]","\frac{2 (5 a x+1) \sqrt{c-a c x}}{15 a c^3 (a x-1)^3}","\frac{2}{3 a c (c-a c x)^{3/2}}-\frac{4}{5 a (c-a c x)^{5/2}}",1,"(2*(1 + 5*a*x)*Sqrt[c - a*c*x])/(15*a*c^3*(-1 + a*x)^3)","A",1
242,1,34,40,0.0609618,"\int \frac{e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a*c*x)^(7/2),x]","-\frac{2 (7 a x+3) \sqrt{c-a c x}}{35 a c^4 (a x-1)^4}","\frac{2}{5 a c (c-a c x)^{5/2}}-\frac{4}{7 a (c-a c x)^{7/2}}",1,"(-2*(3 + 7*a*x)*Sqrt[c - a*c*x])/(35*a*c^4*(-1 + a*x)^4)","A",1
243,1,77,197,0.0516732,"\int e^{3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a*c*x)^(9/2),x]","\frac{2 c^4 \sqrt{\frac{1}{a x}+1} (a x+1)^2 \left(105 a^3 x^3-455 a^2 x^2+755 a x-533\right) \sqrt{c-a c x}}{1155 a \sqrt{1-\frac{1}{a x}}}","-\frac{856 \left(\frac{1}{a x}+1\right)^{5/2} (c-a c x)^{9/2}}{1155 a^3 x^2 \left(1-\frac{1}{a x}\right)^{9/2}}+\frac{2 x \left(a-\frac{1}{x}\right)^3 \left(\frac{1}{a x}+1\right)^{5/2} (c-a c x)^{9/2}}{11 a^3 \left(1-\frac{1}{a x}\right)^{9/2}}+\frac{16 \left(\frac{1}{a x}+1\right)^{5/2} (c-a c x)^{9/2}}{21 a^2 x \left(1-\frac{1}{a x}\right)^{9/2}}-\frac{8 \left(\frac{1}{a x}+1\right)^{5/2} (c-a c x)^{9/2}}{33 a \left(1-\frac{1}{a x}\right)^{9/2}}",1,"(2*c^4*Sqrt[1 + 1/(a*x)]*(1 + a*x)^2*Sqrt[c - a*c*x]*(-533 + 755*a*x - 455*a^2*x^2 + 105*a^3*x^3))/(1155*a*Sqrt[1 - 1/(a*x)])","A",1
244,1,69,137,0.0425844,"\int e^{3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a*c*x)^(7/2),x]","-\frac{2 c^3 \sqrt{\frac{1}{a x}+1} (a x+1)^2 \left(35 a^2 x^2-110 a x+107\right) \sqrt{c-a c x}}{315 a \sqrt{1-\frac{1}{a x}}}","\frac{214 \left(\frac{1}{a x}+1\right)^{5/2} (c-a c x)^{7/2}}{315 a^2 x \left(1-\frac{1}{a x}\right)^{7/2}}-\frac{44 \left(\frac{1}{a x}+1\right)^{5/2} (c-a c x)^{7/2}}{63 a \left(1-\frac{1}{a x}\right)^{7/2}}+\frac{2 x \left(\frac{1}{a x}+1\right)^{5/2} (c-a c x)^{7/2}}{9 \left(1-\frac{1}{a x}\right)^{7/2}}",1,"(-2*c^3*Sqrt[1 + 1/(a*x)]*(1 + a*x)^2*Sqrt[c - a*c*x]*(107 - 110*a*x + 35*a^2*x^2))/(315*a*Sqrt[1 - 1/(a*x)])","A",1
245,1,59,89,0.0431983,"\int e^{3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a*c*x)^(5/2),x]","\frac{2 \sqrt{\frac{1}{a x}+1} (5 a x-9) \sqrt{c-a c x} (a c x+c)^2}{35 a \sqrt{1-\frac{1}{a x}}}","\frac{2 x \left(\frac{1}{a x}+1\right)^{5/2} (c-a c x)^{5/2}}{7 \left(1-\frac{1}{a x}\right)^{5/2}}-\frac{18 \left(\frac{1}{a x}+1\right)^{5/2} (c-a c x)^{5/2}}{35 a \left(1-\frac{1}{a x}\right)^{5/2}}",1,"(2*Sqrt[1 + 1/(a*x)]*(-9 + 5*a*x)*Sqrt[c - a*c*x]*(c + a*c*x)^2)/(35*a*Sqrt[1 - 1/(a*x)])","A",1
246,1,43,31,0.0339424,"\int e^{3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a*c*x)^(3/2),x]","\frac{2 x \left(\frac{1}{a x}+1\right)^{5/2} (c-a c x)^{3/2}}{5 \left(1-\frac{1}{a x}\right)^{3/2}}","\frac{2 (a x+1) (c-a c x)^{3/2} e^{3 \coth ^{-1}(a x)}}{5 a}",1,"(2*(1 + 1/(a*x))^(5/2)*x*(c - a*c*x)^(3/2))/(5*(1 - 1/(a*x))^(3/2))","A",0
247,1,105,163,0.0756278,"\int e^{3 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx","Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x],x]","\frac{2 \sqrt{c-a c x} \left(\sqrt{a} \sqrt{\frac{1}{a x}+1} (a x+7)-6 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{3 a^{3/2} \sqrt{1-\frac{1}{a x}}}","-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{a^{3/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{a \sqrt{1-\frac{1}{a x}}}",1,"(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[1 + 1/(a*x)]*(7 + a*x) - 6*Sqrt[2]*Sqrt[x^(-1)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(3*a^(3/2)*Sqrt[1 - 1/(a*x)])","A",1
248,1,116,177,0.1544497,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx","Integrate[E^(3*ArcCoth[a*x])/Sqrt[c - a*c*x],x]","\frac{x \sqrt{1-\frac{1}{a x}} \left(2 \sqrt{a} \sqrt{\frac{1}{a x}+1} (a x-2)-3 \sqrt{2} \sqrt{\frac{1}{x}} (a x-1) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{\sqrt{a} (a x-1) \sqrt{c-a c x}}","\frac{2 a x \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}}{\left(a-\frac{1}{x}\right) \sqrt{c-a c x}}-\frac{6 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\left(a-\frac{1}{x}\right) \sqrt{c-a c x}}-\frac{3 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x}}",1,"(Sqrt[1 - 1/(a*x)]*x*(2*Sqrt[a]*Sqrt[1 + 1/(a*x)]*(-2 + a*x) - 3*Sqrt[2]*Sqrt[x^(-1)]*(-1 + a*x)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(Sqrt[a]*(-1 + a*x)*Sqrt[c - a*c*x])","A",1
249,1,125,187,0.1565615,"\int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a*c*x)^(3/2),x]","\frac{x \sqrt{1-\frac{1}{a x}} \left(2 \sqrt{a} \sqrt{\frac{1}{a x}+1} (5 a x-1)+3 \sqrt{2} \sqrt{\frac{1}{x}} (a x-1)^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{8 \sqrt{a} c (a x-1)^2 \sqrt{c-a c x}}","-\frac{a^2 x \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{3/2}}{2 \left(a-\frac{1}{x}\right)^2 (c-a c x)^{3/2}}-\frac{3 a x \left(1-\frac{1}{a x}\right)^{3/2} \sqrt{\frac{1}{a x}+1}}{4 \left(a-\frac{1}{x}\right) (c-a c x)^{3/2}}-\frac{3 \sqrt{a} \left(1-\frac{1}{a x}\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{4 \sqrt{2} \left(\frac{1}{x}\right)^{3/2} (c-a c x)^{3/2}}",1,"(Sqrt[1 - 1/(a*x)]*x*(2*Sqrt[a]*Sqrt[1 + 1/(a*x)]*(-1 + 5*a*x) + 3*Sqrt[2]*Sqrt[x^(-1)]*(-1 + a*x)^2*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(8*Sqrt[a]*c*(-1 + a*x)^2*Sqrt[c - a*c*x])","A",1
250,1,142,250,0.1538246,"\int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a*c*x)^(5/2),x]","-\frac{\sqrt{1-\frac{1}{a x}} \left(2 \sqrt{a} \sqrt{\frac{1}{x}} \sqrt{\frac{1}{a x}+1} \left(3 a^2 x^2+22 a x+7\right)-\frac{3 \sqrt{2} (a x-1)^3 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{x}\right)}{96 \sqrt{a} c^2 \left(\frac{1}{x}\right)^{3/2} (a x-1)^3 \sqrt{c-a c x}}","\frac{a^{3/2} \left(1-\frac{1}{a x}\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{16 \sqrt{2} \left(\frac{1}{x}\right)^{5/2} (c-a c x)^{5/2}}-\frac{a^4 x^2 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{5/2}}{6 \left(a-\frac{1}{x}\right)^3 (c-a c x)^{5/2}}+\frac{a^3 x^2 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{3/2}}{24 \left(a-\frac{1}{x}\right)^2 (c-a c x)^{5/2}}+\frac{a^2 x^2 \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}{16 \left(a-\frac{1}{x}\right) (c-a c x)^{5/2}}",1,"-1/96*(Sqrt[1 - 1/(a*x)]*(2*Sqrt[a]*Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]*(7 + 22*a*x + 3*a^2*x^2) - (3*Sqrt[2]*(-1 + a*x)^3*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/x))/(Sqrt[a]*c^2*(x^(-1))^(3/2)*(-1 + a*x)^3*Sqrt[c - a*c*x])","A",1
251,1,147,307,0.2071352,"\int \frac{e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a*c*x)^(7/2),x]","\frac{\sqrt{1-\frac{1}{a x}} \left(\frac{2 \sqrt{a} \sqrt{\frac{1}{a x}+1} \left(-3 a^3 x^3+13 a^2 x^2+79 a x+39\right)}{\sqrt{\frac{1}{x}}}+3 \sqrt{2} (a x-1)^4 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{512 \sqrt{a} c^3 \sqrt{\frac{1}{x}} (a x-1)^4 \sqrt{c-a c x}}","-\frac{3 a^{5/2} \left(1-\frac{1}{a x}\right)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{256 \sqrt{2} \left(\frac{1}{x}\right)^{7/2} (c-a c x)^{7/2}}+\frac{a^5 x^3 \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{5/2}}{32 \left(a-\frac{1}{x}\right)^3 (c-a c x)^{7/2}}-\frac{a^5 x^2 \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{5/2}}{8 \left(a-\frac{1}{x}\right)^4 (c-a c x)^{7/2}}-\frac{a^4 x^3 \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{3/2}}{128 \left(a-\frac{1}{x}\right)^2 (c-a c x)^{7/2}}-\frac{3 a^3 x^3 \left(1-\frac{1}{a x}\right)^{7/2} \sqrt{\frac{1}{a x}+1}}{256 \left(a-\frac{1}{x}\right) (c-a c x)^{7/2}}",1,"(Sqrt[1 - 1/(a*x)]*((2*Sqrt[a]*Sqrt[1 + 1/(a*x)]*(39 + 79*a*x + 13*a^2*x^2 - 3*a^3*x^3))/Sqrt[x^(-1)] + 3*Sqrt[2]*(-1 + a*x)^4*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(512*Sqrt[a]*c^3*Sqrt[x^(-1)]*(-1 + a*x)^4*Sqrt[c - a*c*x])","A",1
252,1,86,194,0.054389,"\int e^{-\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx","Integrate[(c - a*c*x)^(9/2)/E^ArcCoth[a*x],x]","\frac{2 c^4 \sqrt{\frac{1}{a x}+1} \left(63 a^5 x^5-455 a^4 x^4+1510 a^3 x^3-3198 a^2 x^2+5419 a x-11531\right) \sqrt{c-a c x}}{693 a \sqrt{1-\frac{1}{a x}}}","\frac{16384 c^5 x \sqrt{1-\frac{1}{a^2 x^2}}}{693 \sqrt{c-a c x}}+\frac{4096}{693} c^4 x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}+\frac{512}{231} c^3 x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}+\frac{640}{693} c^2 x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{5/2}+\frac{40}{99} c x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{7/2}+\frac{2}{11} x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{9/2}",1,"(2*c^4*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(-11531 + 5419*a*x - 3198*a^2*x^2 + 1510*a^3*x^3 - 455*a^4*x^4 + 63*a^5*x^5))/(693*a*Sqrt[1 - 1/(a*x)])","A",0
253,1,78,161,0.0525383,"\int e^{-\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx","Integrate[(c - a*c*x)^(7/2)/E^ArcCoth[a*x],x]","-\frac{2 c^3 \sqrt{\frac{1}{a x}+1} \left(35 a^4 x^4-220 a^3 x^3+642 a^2 x^2-1276 a x+2867\right) \sqrt{c-a c x}}{315 a \sqrt{1-\frac{1}{a x}}}","\frac{4096 c^4 x \sqrt{1-\frac{1}{a^2 x^2}}}{315 \sqrt{c-a c x}}+\frac{1024}{315} c^3 x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}+\frac{128}{105} c^2 x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}+\frac{32}{63} c x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{5/2}+\frac{2}{9} x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{7/2}",1,"(-2*c^3*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(2867 - 1276*a*x + 642*a^2*x^2 - 220*a^3*x^3 + 35*a^4*x^4))/(315*a*Sqrt[1 - 1/(a*x)])","A",0
254,1,70,128,0.0446176,"\int e^{-\coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx","Integrate[(c - a*c*x)^(5/2)/E^ArcCoth[a*x],x]","\frac{2 c^2 \sqrt{\frac{1}{a x}+1} \left(5 a^3 x^3-27 a^2 x^2+71 a x-177\right) \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}}","\frac{256 c^3 x \sqrt{1-\frac{1}{a^2 x^2}}}{35 \sqrt{c-a c x}}+\frac{64}{35} c^2 x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}+\frac{24}{35} c x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}+\frac{2}{7} x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{5/2}",1,"(2*c^2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(-177 + 71*a*x - 27*a^2*x^2 + 5*a^3*x^3))/(35*a*Sqrt[1 - 1/(a*x)])","A",0
255,1,60,95,0.0382209,"\int e^{-\coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx","Integrate[(c - a*c*x)^(3/2)/E^ArcCoth[a*x],x]","-\frac{2 c \sqrt{\frac{1}{a x}+1} \left(3 a^2 x^2-14 a x+43\right) \sqrt{c-a c x}}{15 a \sqrt{1-\frac{1}{a x}}}","\frac{64 c^2 x \sqrt{1-\frac{1}{a^2 x^2}}}{15 \sqrt{c-a c x}}+\frac{16}{15} c x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}+\frac{2}{5} x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}",1,"(-2*c*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(43 - 14*a*x + 3*a^2*x^2))/(15*a*Sqrt[1 - 1/(a*x)])","A",0
256,1,50,62,0.0304101,"\int e^{-\coth ^{-1}(a x)} \sqrt{c-a c x} \, dx","Integrate[Sqrt[c - a*c*x]/E^ArcCoth[a*x],x]","\frac{2 \sqrt{\frac{1}{a x}+1} (a x-5) \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}","\frac{2}{3} x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}+\frac{8 c x \sqrt{1-\frac{1}{a^2 x^2}}}{3 \sqrt{c-a c x}}",1,"(2*Sqrt[1 + 1/(a*x)]*(-5 + a*x)*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - 1/(a*x)])","A",0
257,1,28,29,0.0287679,"\int \frac{e^{-\coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx","Integrate[1/(E^ArcCoth[a*x]*Sqrt[c - a*c*x]),x]","\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-a c x}}","\frac{2 (a x+1) e^{-\coth ^{-1}(a x)}}{a \sqrt{c-a c x}}",1,"(2*Sqrt[1 - 1/(a^2*x^2)]*x)/Sqrt[c - a*c*x]","A",0
258,1,76,76,0.064182,"\int \frac{e^{-\coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^(3/2)),x]","-\frac{\sqrt{2} \sqrt{a} \left(1-\frac{1}{a x}\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{\left(\frac{1}{x}\right)^{3/2} (c-a c x)^{3/2}}","-\frac{\sqrt{2} \sqrt{a} \left(1-\frac{1}{a x}\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{\left(\frac{1}{x}\right)^{3/2} (c-a c x)^{3/2}}",1,"-((Sqrt[2]*Sqrt[a]*(1 - 1/(a*x))^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/((x^(-1))^(3/2)*(c - a*c*x)^(3/2)))","A",1
259,1,116,136,0.1367423,"\int \frac{e^{-\coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^(5/2)),x]","\frac{x \sqrt{1-\frac{1}{a x}} \left(\sqrt{2} \sqrt{\frac{1}{x}} (a x-1) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)-2 \sqrt{a} \sqrt{\frac{1}{a x}+1}\right)}{4 \sqrt{a} c^2 (a x-1) \sqrt{c-a c x}}","\frac{a^{3/2} \left(1-\frac{1}{a x}\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{2 \sqrt{2} \left(\frac{1}{x}\right)^{5/2} (c-a c x)^{5/2}}-\frac{a^2 x^2 \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}{2 \left(a-\frac{1}{x}\right) (c-a c x)^{5/2}}",1,"(Sqrt[1 - 1/(a*x)]*x*(-2*Sqrt[a]*Sqrt[1 + 1/(a*x)] + Sqrt[2]*Sqrt[x^(-1)]*(-1 + a*x)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(4*Sqrt[a]*c^2*(-1 + a*x)*Sqrt[c - a*c*x])","A",1
260,1,125,193,0.1757735,"\int \frac{e^{-\coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a*c*x)^(7/2)),x]","\frac{x \sqrt{1-\frac{1}{a x}} \left(2 \sqrt{a} \sqrt{\frac{1}{a x}+1} (7-3 a x)+3 \sqrt{2} \sqrt{\frac{1}{x}} (a x-1)^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{32 \sqrt{a} c^3 (a x-1)^2 \sqrt{c-a c x}}","-\frac{3 a^{5/2} \left(1-\frac{1}{a x}\right)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{16 \sqrt{2} \left(\frac{1}{x}\right)^{7/2} (c-a c x)^{7/2}}+\frac{3 a^3 x^3 \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{7/2}}{16 \left(a-\frac{1}{x}\right) (c-a c x)^{7/2}}-\frac{a^3 x^2 \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{7/2}}{4 \left(a-\frac{1}{x}\right)^2 (c-a c x)^{7/2}}",1,"(Sqrt[1 - 1/(a*x)]*x*(2*Sqrt[a]*Sqrt[1 + 1/(a*x)]*(7 - 3*a*x) + 3*Sqrt[2]*Sqrt[x^(-1)]*(-1 + a*x)^2*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(32*Sqrt[a]*c^3*(-1 + a*x)^2*Sqrt[c - a*c*x])","A",1
261,1,88,137,0.0861795,"\int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx","Integrate[(c - a*c*x)^(7/2)/E^(2*ArcCoth[a*x]),x]","\frac{2 c^3 \left(\left(-35 a^4 x^4+230 a^3 x^3-732 a^2 x^2+1754 a x-6257\right) \sqrt{c-a c x}+5040 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)\right)}{315 a}","\frac{32 \sqrt{2} c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{a}-\frac{32 c^3 \sqrt{c-a c x}}{a}-\frac{16 c^2 (c-a c x)^{3/2}}{3 a}-\frac{2 (c-a c x)^{9/2}}{9 a c}-\frac{4 (c-a c x)^{7/2}}{7 a}-\frac{8 c (c-a c x)^{5/2}}{5 a}",1,"(2*c^3*(Sqrt[c - a*c*x]*(-6257 + 1754*a*x - 732*a^2*x^2 + 230*a^3*x^3 - 35*a^4*x^4) + 5040*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]))/(315*a)","A",1
262,1,80,116,0.0655278,"\int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx","Integrate[(c - a*c*x)^(5/2)/E^(2*ArcCoth[a*x]),x]","\frac{2 c^2 \left(\left(15 a^3 x^3-87 a^2 x^2+269 a x-1037\right) \sqrt{c-a c x}+840 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)\right)}{105 a}","\frac{16 \sqrt{2} c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{a}-\frac{16 c^2 \sqrt{c-a c x}}{a}-\frac{2 (c-a c x)^{7/2}}{7 a c}-\frac{4 (c-a c x)^{5/2}}{5 a}-\frac{8 c (c-a c x)^{3/2}}{3 a}",1,"(2*c^2*(Sqrt[c - a*c*x]*(-1037 + 269*a*x - 87*a^2*x^2 + 15*a^3*x^3) + 840*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]))/(105*a)","A",1
263,1,71,95,0.047066,"\int e^{-2 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx","Integrate[(c - a*c*x)^(3/2)/E^(2*ArcCoth[a*x]),x]","\frac{120 \sqrt{2} c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)-2 c \left(3 a^2 x^2-16 a x+73\right) \sqrt{c-a c x}}{15 a}","\frac{8 \sqrt{2} c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{a}-\frac{2 (c-a c x)^{5/2}}{5 a c}-\frac{4 (c-a c x)^{3/2}}{3 a}-\frac{8 c \sqrt{c-a c x}}{a}",1,"(-2*c*Sqrt[c - a*c*x]*(73 - 16*a*x + 3*a^2*x^2) + 120*Sqrt[2]*c^(3/2)*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/(15*a)","A",1
264,1,61,76,0.0363757,"\int e^{-2 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx","Integrate[Sqrt[c - a*c*x]/E^(2*ArcCoth[a*x]),x]","\frac{2 (a x-7) \sqrt{c-a c x}+12 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{3 a}","-\frac{2 (c-a c x)^{3/2}}{3 a c}-\frac{4 \sqrt{c-a c x}}{a}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{a}",1,"(2*(-7 + a*x)*Sqrt[c - a*c*x] + 12*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/(3*a)","A",1
265,1,58,58,0.0284146,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*Sqrt[c - a*c*x]),x]","\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{a \sqrt{c}}-\frac{2 \sqrt{c-a c x}}{a c}","\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{a \sqrt{c}}-\frac{2 \sqrt{c-a c x}}{a c}",1,"(-2*Sqrt[c - a*c*x])/(a*c) + (2*Sqrt[2]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/(a*Sqrt[c])","A",1
266,1,37,37,0.0187422,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^(3/2)),x]","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{a c^{3/2}}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{a c^{3/2}}",1,"(Sqrt[2]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/(a*c^(3/2))","A",1
267,1,37,57,0.0243621,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^(5/2)),x]","-\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{1}{2} (1-a x)\right)}{a c^2 \sqrt{c-a c x}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{\sqrt{2} a c^{5/2}}-\frac{1}{a c^2 \sqrt{c-a c x}}",1,"-(Hypergeometric2F1[-1/2, 1, 1/2, (1 - a*x)/2]/(a*c^2*Sqrt[c - a*c*x]))","C",0
268,1,39,83,0.0285611,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^(7/2)),x]","-\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{1}{2} (1-a x)\right)}{3 a c^2 (c-a c x)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{2 \sqrt{2} a c^{7/2}}-\frac{1}{2 a c^3 \sqrt{c-a c x}}-\frac{1}{3 a c^2 (c-a c x)^{3/2}}",1,"-1/3*Hypergeometric2F1[-3/2, 1, -1/2, (1 - a*x)/2]/(a*c^2*(c - a*c*x)^(3/2))","C",0
269,1,39,104,0.0373581,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{(c-a c x)^{9/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a*c*x)^(9/2)),x]","-\frac{\, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\frac{1}{2} (1-a x)\right)}{5 a c^2 (c-a c x)^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{4 \sqrt{2} a c^{9/2}}-\frac{1}{4 a c^4 \sqrt{c-a c x}}-\frac{1}{6 a c^3 (c-a c x)^{3/2}}-\frac{1}{5 a c^2 (c-a c x)^{5/2}}",1,"-1/5*Hypergeometric2F1[-5/2, 1, -3/2, (1 - a*x)/2]/(a*c^2*(c - a*c*x)^(5/2))","C",0
270,1,84,368,0.0615955,"\int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx","Integrate[(c - a*c*x)^(9/2)/E^(3*ArcCoth[a*x]),x]","\frac{2 c^4 \left(21 a^6 x^6-182 a^5 x^5+755 a^4 x^4-2132 a^3 x^3+5419 a^2 x^2-23062 a x-46355\right) \sqrt{c-a c x}}{231 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{94208 (c-a c x)^{9/2}}{231 a^6 x^5 \left(1-\frac{1}{a x}\right)^{9/2} \sqrt{\frac{1}{a x}+1}}-\frac{1024 \left(a-\frac{1}{x}\right)^3 (c-a c x)^{9/2}}{231 a^6 x^2 \left(1-\frac{1}{a x}\right)^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left(a-\frac{1}{x}\right)^6 (c-a c x)^{9/2}}{11 a^6 \left(1-\frac{1}{a x}\right)^{9/2} \sqrt{\frac{1}{a x}+1}}-\frac{16 \left(a-\frac{1}{x}\right)^5 (c-a c x)^{9/2}}{33 a^6 \left(1-\frac{1}{a x}\right)^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{320 \left(a-\frac{1}{x}\right)^4 (c-a c x)^{9/2}}{231 a^6 x \left(1-\frac{1}{a x}\right)^{9/2} \sqrt{\frac{1}{a x}+1}}-\frac{40960 (c-a c x)^{9/2}}{231 a^5 x^4 \left(1-\frac{1}{a x}\right)^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{4096 (c-a c x)^{9/2}}{231 a^4 x^3 \left(1-\frac{1}{a x}\right)^{9/2} \sqrt{\frac{1}{a x}+1}}",1,"(2*c^4*Sqrt[c - a*c*x]*(-46355 - 23062*a*x + 5419*a^2*x^2 - 2132*a^3*x^3 + 755*a^4*x^4 - 182*a^5*x^5 + 21*a^6*x^6))/(231*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
271,1,76,311,0.049817,"\int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx","Integrate[(c - a*c*x)^(7/2)/E^(3*ArcCoth[a*x]),x]","-\frac{2 c^3 \left(7 a^5 x^5-55 a^4 x^4+214 a^3 x^3-638 a^2 x^2+2867 a x+5797\right) \sqrt{c-a c x}}{63 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{11776 (c-a c x)^{7/2}}{63 a^5 x^4 \left(1-\frac{1}{a x}\right)^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left(a-\frac{1}{x}\right)^5 (c-a c x)^{7/2}}{9 a^5 \left(1-\frac{1}{a x}\right)^{7/2} \sqrt{\frac{1}{a x}+1}}-\frac{40 \left(a-\frac{1}{x}\right)^4 (c-a c x)^{7/2}}{63 a^5 \left(1-\frac{1}{a x}\right)^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{128 \left(a-\frac{1}{x}\right)^3 (c-a c x)^{7/2}}{63 a^5 x \left(1-\frac{1}{a x}\right)^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{5120 (c-a c x)^{7/2}}{63 a^4 x^3 \left(1-\frac{1}{a x}\right)^{7/2} \sqrt{\frac{1}{a x}+1}}-\frac{512 (c-a c x)^{7/2}}{63 a^3 x^2 \left(1-\frac{1}{a x}\right)^{7/2} \sqrt{\frac{1}{a x}+1}}",1,"(-2*c^3*Sqrt[c - a*c*x]*(5797 + 2867*a*x - 638*a^2*x^2 + 214*a^3*x^3 - 55*a^4*x^4 + 7*a^5*x^5))/(63*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
272,1,68,254,0.0413677,"\int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx","Integrate[(c - a*c*x)^(5/2)/E^(3*ArcCoth[a*x]),x]","\frac{2 c^2 \left(5 a^4 x^4-36 a^3 x^3+142 a^2 x^2-708 a x-1451\right) \sqrt{c-a c x}}{35 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{2944 (c-a c x)^{5/2}}{35 a^4 x^3 \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left(a-\frac{1}{x}\right)^4 (c-a c x)^{5/2}}{7 a^4 \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}-\frac{32 \left(a-\frac{1}{x}\right)^3 (c-a c x)^{5/2}}{35 a^4 \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}-\frac{256 (c-a c x)^{5/2}}{7 a^3 x^2 \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}+\frac{128 (c-a c x)^{5/2}}{35 a^2 x \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}",1,"(2*c^2*Sqrt[c - a*c*x]*(-1451 - 708*a*x + 142*a^2*x^2 - 36*a^3*x^3 + 5*a^4*x^4))/(35*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
273,1,57,195,0.0357495,"\int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx","Integrate[(c - a*c*x)^(3/2)/E^(3*ArcCoth[a*x]),x]","-\frac{2 c \left(a^3 x^3-7 a^2 x^2+43 a x+91\right) \sqrt{c-a c x}}{5 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{184 (c-a c x)^{3/2}}{5 a^3 x^2 \left(1-\frac{1}{a x}\right)^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \left(a-\frac{1}{x}\right)^3 (c-a c x)^{3/2}}{5 a^3 \left(1-\frac{1}{a x}\right)^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{16 (c-a c x)^{3/2}}{a^2 x \left(1-\frac{1}{a x}\right)^{3/2} \sqrt{\frac{1}{a x}+1}}-\frac{8 (c-a c x)^{3/2}}{5 a \left(1-\frac{1}{a x}\right)^{3/2} \sqrt{\frac{1}{a x}+1}}",1,"(-2*c*Sqrt[c - a*c*x]*(91 + 43*a*x - 7*a^2*x^2 + a^3*x^3))/(5*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
274,1,48,137,0.0303359,"\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx","Integrate[Sqrt[c - a*c*x]/E^(3*ArcCoth[a*x]),x]","\frac{2 \left(a^2 x^2-10 a x-23\right) \sqrt{c-a c x}}{3 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{46 \sqrt{c-a c x}}{3 a^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{20 \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}",1,"(2*Sqrt[c - a*c*x]*(-23 - 10*a*x + a^2*x^2))/(3*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
275,1,48,85,0.0330904,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x]),x]","\frac{2 \sqrt{1-\frac{1}{a x}} (a x+3)}{a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}","\frac{2 x \sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}+\frac{6 \sqrt{1-\frac{1}{a x}}}{a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}",1,"(2*Sqrt[1 - 1/(a*x)]*(3 + a*x))/(a*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])","A",1
276,1,41,29,0.0340225,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^(3/2)),x]","-\frac{2 x \left(1-\frac{1}{a x}\right)^{3/2}}{\sqrt{\frac{1}{a x}+1} (c-a c x)^{3/2}}","-\frac{2 (a x+1) e^{-3 \coth ^{-1}(a x)}}{a (c-a c x)^{3/2}}",1,"(-2*(1 - 1/(a*x))^(3/2)*x)/(Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(3/2))","A",0
277,1,122,120,0.0957067,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^(5/2)),x]","\frac{\sqrt{1-\frac{1}{a x}} \left(2 \sqrt{\frac{1}{x}}-\sqrt{2} \sqrt{a} \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{2 a c^2 \sqrt{\frac{1}{x}} \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}","\frac{a x^2 \left(1-\frac{1}{a x}\right)^{5/2}}{\sqrt{\frac{1}{a x}+1} (c-a c x)^{5/2}}-\frac{a^{3/2} \left(1-\frac{1}{a x}\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{\sqrt{2} \left(\frac{1}{x}\right)^{5/2} (c-a c x)^{5/2}}",1,"(Sqrt[1 - 1/(a*x)]*(2*Sqrt[x^(-1)] - Sqrt[2]*Sqrt[a]*Sqrt[1 + 1/(a*x)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(2*a*c^2*Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]*Sqrt[c - a*c*x])","A",1
278,1,140,184,0.1572101,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a*c*x)^(7/2)),x]","\frac{\sqrt{1-\frac{1}{a x}} \left(2 \sqrt{\frac{1}{x}} (3 a x-1)-3 \sqrt{2} \sqrt{a} \sqrt{\frac{1}{a x}+1} (a x-1) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{8 a c^3 \sqrt{\frac{1}{x}} \sqrt{\frac{1}{a x}+1} (a x-1) \sqrt{c-a c x}}","\frac{3 a^{5/2} \left(1-\frac{1}{a x}\right)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{4 \sqrt{2} \left(\frac{1}{x}\right)^{7/2} (c-a c x)^{7/2}}-\frac{3 a^2 x^3 \left(1-\frac{1}{a x}\right)^{7/2}}{4 \sqrt{\frac{1}{a x}+1} (c-a c x)^{7/2}}-\frac{a^2 x^2 \left(1-\frac{1}{a x}\right)^{7/2}}{2 \left(a-\frac{1}{x}\right) \sqrt{\frac{1}{a x}+1} (c-a c x)^{7/2}}",1,"(Sqrt[1 - 1/(a*x)]*(2*Sqrt[x^(-1)]*(-1 + 3*a*x) - 3*Sqrt[2]*Sqrt[a]*Sqrt[1 + 1/(a*x)]*(-1 + a*x)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(8*a*c^3*Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]*(-1 + a*x)*Sqrt[c - a*c*x])","A",1
279,1,41,99,0.0466358,"\int e^{\coth ^{-1}(x)} x (1+x) \, dx","Integrate[E^ArcCoth[x]*x*(1 + x),x]","\frac{1}{3} \sqrt{1-\frac{1}{x^2}} x \left(x^2+3 x+5\right)+\log \left(\left(\sqrt{1-\frac{1}{x^2}}+1\right) x\right)","\frac{1}{3} \left(\frac{1}{x}+1\right)^{5/2} \sqrt{\frac{x-1}{x}} x^3+\frac{1}{3} \left(\frac{1}{x}+1\right)^{3/2} \sqrt{\frac{x-1}{x}} x^2+\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}} x+\tanh ^{-1}\left(\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}}\right)",1,"(Sqrt[1 - x^(-2)]*x*(5 + 3*x + x^2))/3 + Log[(1 + Sqrt[1 - x^(-2)])*x]","A",0
280,1,40,79,0.0290883,"\int e^{\coth ^{-1}(x)} (1+x) \, dx","Integrate[E^ArcCoth[x]*(1 + x),x]","\frac{1}{2} \sqrt{1-\frac{1}{x^2}} x (x+4)+\frac{3}{2} \log \left(\left(\sqrt{1-\frac{1}{x^2}}+1\right) x\right)","\frac{1}{2} \left(\frac{1}{x}+1\right)^{3/2} \sqrt{\frac{x-1}{x}} x^2+\frac{3}{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}} x+\frac{3}{2} \tanh ^{-1}\left(\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}}\right)",1,"(Sqrt[1 - x^(-2)]*x*(4 + x))/2 + (3*Log[(1 + Sqrt[1 - x^(-2)])*x])/2","A",0
281,1,21,18,0.0221585,"\int e^{\coth ^{-1}(x)} (1-x) x \, dx","Integrate[E^ArcCoth[x]*(1 - x)*x,x]","-\frac{1}{3} \sqrt{1-\frac{1}{x^2}} x \left(x^2-1\right)","-\frac{1}{3} \left(1-\frac{1}{x^2}\right)^{3/2} x^3",1,"-1/3*(Sqrt[1 - x^(-2)]*x*(-1 + x^2))","A",1
282,1,39,35,0.0229581,"\int e^{\coth ^{-1}(x)} (1-x) \, dx","Integrate[E^ArcCoth[x]*(1 - x),x]","\frac{1}{2} \log \left(\left(\sqrt{1-\frac{1}{x^2}}+1\right) x\right)-\frac{1}{2} \sqrt{1-\frac{1}{x^2}} x^2","\frac{1}{2} \tanh ^{-1}\left(\sqrt{1-\frac{1}{x^2}}\right)-\frac{1}{2} \sqrt{1-\frac{1}{x^2}} x^2",1,"-1/2*(Sqrt[1 - x^(-2)]*x^2) + Log[(1 + Sqrt[1 - x^(-2)])*x]/2","A",0
283,1,52,133,0.0494086,"\int e^{\coth ^{-1}(x)} x (1+x)^2 \, dx","Integrate[E^ArcCoth[x]*x*(1 + x)^2,x]","\frac{15}{8} \log \left(\left(\sqrt{1-\frac{1}{x^2}}+1\right) x\right)+\frac{1}{8} \sqrt{1-\frac{1}{x^2}} x \left(2 x^3+8 x^2+15 x+24\right)","\frac{1}{4} \left(\frac{1}{x}+1\right)^{7/2} \sqrt{\frac{x-1}{x}} x^4+\frac{1}{4} \left(\frac{1}{x}+1\right)^{5/2} \sqrt{\frac{x-1}{x}} x^3+\frac{5}{8} \left(\frac{1}{x}+1\right)^{3/2} \sqrt{\frac{x-1}{x}} x^2+\frac{15}{8} \sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}} x+\frac{15}{8} \tanh ^{-1}\left(\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}}\right)",1,"(Sqrt[1 - x^(-2)]*x*(24 + 15*x + 8*x^2 + 2*x^3))/8 + (15*Log[(1 + Sqrt[1 - x^(-2)])*x])/8","A",0
284,1,47,106,0.0368132,"\int e^{\coth ^{-1}(x)} (1+x)^2 \, dx","Integrate[E^ArcCoth[x]*(1 + x)^2,x]","\frac{1}{6} \sqrt{1-\frac{1}{x^2}} x \left(2 x^2+9 x+22\right)+\frac{5}{2} \log \left(\left(\sqrt{1-\frac{1}{x^2}}+1\right) x\right)","\frac{1}{3} \left(\frac{1}{x}+1\right)^{5/2} \sqrt{\frac{x-1}{x}} x^3+\frac{5}{6} \left(\frac{1}{x}+1\right)^{3/2} \sqrt{\frac{x-1}{x}} x^2+\frac{5}{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}} x+\frac{5}{2} \tanh ^{-1}\left(\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}}\right)",1,"(Sqrt[1 - x^(-2)]*x*(22 + 9*x + 2*x^2))/6 + (5*Log[(1 + Sqrt[1 - x^(-2)])*x])/2","A",0
285,1,52,71,0.0471109,"\int e^{\coth ^{-1}(x)} (1-x)^2 x \, dx","Integrate[E^ArcCoth[x]*(1 - x)^2*x,x]","\frac{1}{24} \sqrt{1-\frac{1}{x^2}} x \left(6 x^3-8 x^2-3 x+8\right)-\frac{1}{8} \log \left(\left(\sqrt{1-\frac{1}{x^2}}+1\right) x\right)","\frac{1}{8} \sqrt{1-\frac{1}{x^2}} x^2-\frac{1}{8} \tanh ^{-1}\left(\sqrt{1-\frac{1}{x^2}}\right)+\frac{1}{4} \left(1-\frac{1}{x^2}\right)^{3/2} x^4-\frac{1}{3} \left(1-\frac{1}{x^2}\right)^{3/2} x^3",1,"(Sqrt[1 - x^(-2)]*x*(8 - 3*x - 8*x^2 + 6*x^3))/24 - Log[(1 + Sqrt[1 - x^(-2)])*x]/8","A",0
286,1,47,53,0.033961,"\int e^{\coth ^{-1}(x)} (1-x)^2 \, dx","Integrate[E^ArcCoth[x]*(1 - x)^2,x]","\frac{1}{6} \sqrt{1-\frac{1}{x^2}} x \left(2 x^2-3 x-2\right)+\frac{1}{2} \log \left(\left(\sqrt{1-\frac{1}{x^2}}+1\right) x\right)","-\frac{1}{2} \sqrt{1-\frac{1}{x^2}} x^2+\frac{1}{2} \tanh ^{-1}\left(\sqrt{1-\frac{1}{x^2}}\right)+\frac{1}{3} \left(1-\frac{1}{x^2}\right)^{3/2} x^3",1,"(Sqrt[1 - x^(-2)]*x*(-2 - 3*x + 2*x^2))/6 + Log[(1 + Sqrt[1 - x^(-2)])*x]/2","A",0
287,1,15,22,0.0262985,"\int \frac{e^{\coth ^{-1}(x)} x}{1+x} \, dx","Integrate[(E^ArcCoth[x]*x)/(1 + x),x]","x \sqrt{\frac{x^2-1}{x^2}}","\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}} x",1,"x*Sqrt[(-1 + x^2)/x^2]","A",1
288,1,18,22,0.0109385,"\int \frac{e^{\coth ^{-1}(x)}}{1+x} \, dx","Integrate[E^ArcCoth[x]/(1 + x),x]","\log \left(x \left(\sqrt{\frac{x^2-1}{x^2}}+1\right)\right)","\tanh ^{-1}\left(\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}}\right)",1,"Log[x*(1 + Sqrt[(-1 + x^2)/x^2])]","A",0
289,1,41,47,0.0592949,"\int \frac{e^{\coth ^{-1}(x)} x}{1-x} \, dx","Integrate[(E^ArcCoth[x]*x)/(1 - x),x]","-\frac{\sqrt{1-\frac{1}{x^2}} (x-3) x}{x-1}-2 \log \left(\left(\sqrt{1-\frac{1}{x^2}}+1\right) x\right)","\frac{2 \left(\frac{1}{x}+1\right)}{\sqrt{1-\frac{1}{x^2}}}-\sqrt{1-\frac{1}{x^2}} x-2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{x^2}}\right)",1,"-((Sqrt[1 - x^(-2)]*(-3 + x)*x)/(-1 + x)) - 2*Log[(1 + Sqrt[1 - x^(-2)])*x]","A",0
290,1,38,33,0.0261577,"\int \frac{e^{\coth ^{-1}(x)}}{1-x} \, dx","Integrate[E^ArcCoth[x]/(1 - x),x]","\frac{2 \sqrt{1-\frac{1}{x^2}} x}{x-1}-\log \left(\left(\sqrt{1-\frac{1}{x^2}}+1\right) x\right)","\frac{2 \left(\frac{1}{x}+1\right)}{\sqrt{1-\frac{1}{x^2}}}-\tanh ^{-1}\left(\sqrt{1-\frac{1}{x^2}}\right)",1,"(2*Sqrt[1 - x^(-2)]*x)/(-1 + x) - Log[(1 + Sqrt[1 - x^(-2)])*x]","A",0
291,1,36,45,0.0478269,"\int \frac{e^{\coth ^{-1}(x)} x}{(1+x)^2} \, dx","Integrate[(E^ArcCoth[x]*x)/(1 + x)^2,x]","\log \left(\left(\sqrt{1-\frac{1}{x^2}}+1\right) x\right)-\frac{\sqrt{1-\frac{1}{x^2}} x}{x+1}","\tanh ^{-1}\left(\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}}\right)-\frac{\sqrt{\frac{x-1}{x}}}{\sqrt{\frac{1}{x}+1}}",1,"-((Sqrt[1 - x^(-2)]*x)/(1 + x)) + Log[(1 + Sqrt[1 - x^(-2)])*x]","A",0
292,1,18,21,0.0122626,"\int \frac{e^{\coth ^{-1}(x)}}{(1+x)^2} \, dx","Integrate[E^ArcCoth[x]/(1 + x)^2,x]","\frac{\sqrt{1-\frac{1}{x^2}} x}{x+1}","\frac{\sqrt{\frac{x-1}{x}}}{\sqrt{\frac{1}{x}+1}}",1,"(Sqrt[1 - x^(-2)]*x)/(1 + x)","A",0
293,1,43,55,0.0651375,"\int \frac{e^{\coth ^{-1}(x)} x}{(1-x)^2} \, dx","Integrate[(E^ArcCoth[x]*x)/(1 - x)^2,x]","\frac{\sqrt{1-\frac{1}{x^2}} (5-7 x) x}{3 (x-1)^2}+\log \left(\left(\sqrt{1-\frac{1}{x^2}}+1\right) x\right)","-\frac{4 \left(\frac{1}{x}+1\right)}{3 \left(1-\frac{1}{x^2}\right)^{3/2}}-\frac{\frac{5}{x}+3}{3 \sqrt{1-\frac{1}{x^2}}}+\tanh ^{-1}\left(\sqrt{1-\frac{1}{x^2}}\right)",1,"(Sqrt[1 - x^(-2)]*(5 - 7*x)*x)/(3*(-1 + x)^2) + Log[(1 + Sqrt[1 - x^(-2)])*x]","A",0
294,1,24,24,0.0155066,"\int \frac{e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx","Integrate[E^ArcCoth[x]/(1 - x)^2,x]","-\frac{\sqrt{1-\frac{1}{x^2}} x (x+1)}{3 (x-1)^2}","-\frac{\left(1-\frac{1}{x^2}\right)^{3/2}}{3 \left(1-\frac{1}{x}\right)^3}",1,"-1/3*(Sqrt[1 - x^(-2)]*x*(1 + x))/(-1 + x)^2","A",1
295,1,67,65,0.0306866,"\int e^{\coth ^{-1}(a x)} x^m \sqrt{c-a c x} \, dx","Integrate[E^ArcCoth[a*x]*x^m*Sqrt[c - a*c*x],x]","-\frac{x^{m+1} \sqrt{c-a c x} \, _2F_1\left(-\frac{1}{2},-m-\frac{3}{2};-m-\frac{1}{2};-\frac{1}{a x}\right)}{\left(-m-\frac{3}{2}\right) \sqrt{1-\frac{1}{a x}}}","\frac{2 x^{m+1} \sqrt{c-a c x} \, _2F_1\left(-\frac{1}{2},-m-\frac{3}{2};-m-\frac{1}{2};-\frac{1}{a x}\right)}{(2 m+3) \sqrt{1-\frac{1}{a x}}}",1,"-((x^(1 + m)*Sqrt[c - a*c*x]*Hypergeometric2F1[-1/2, -3/2 - m, -1/2 - m, -(1/(a*x))])/((-3/2 - m)*Sqrt[1 - 1/(a*x)]))","A",1
296,1,64,140,0.0373228,"\int e^{\coth ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx","Integrate[E^ArcCoth[a*x]*x^2*Sqrt[c - a*c*x],x]","\frac{2 \sqrt{\frac{1}{a x}+1} (a x+1) \left(15 a^2 x^2-12 a x+8\right) \sqrt{c-a c x}}{105 a^3 \sqrt{1-\frac{1}{a x}}}","\frac{16 x \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{2 x^3 \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}-\frac{8 x^2 \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}}}",1,"(2*Sqrt[1 + 1/(a*x)]*(1 + a*x)*Sqrt[c - a*c*x]*(8 - 12*a*x + 15*a^2*x^2))/(105*a^3*Sqrt[1 - 1/(a*x)])","A",1
297,1,56,92,0.0284403,"\int e^{\coth ^{-1}(a x)} x \sqrt{c-a c x} \, dx","Integrate[E^ArcCoth[a*x]*x*Sqrt[c - a*c*x],x]","\frac{2 \sqrt{\frac{1}{a x}+1} (a x+1) (3 a x-2) \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}}}","\frac{2 x^2 \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}-\frac{4 x \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{15 a \sqrt{1-\frac{1}{a x}}}",1,"(2*Sqrt[1 + 1/(a*x)]*(1 + a*x)*(-2 + 3*a*x)*Sqrt[c - a*c*x])/(15*a^2*Sqrt[1 - 1/(a*x)])","A",1
298,1,43,29,0.019812,"\int e^{\coth ^{-1}(a x)} \sqrt{c-a c x} \, dx","Integrate[E^ArcCoth[a*x]*Sqrt[c - a*c*x],x]","\frac{2 x \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}","\frac{2 (a x+1) \sqrt{c-a c x} e^{\coth ^{-1}(a x)}}{3 a}",1,"(2*(1 + 1/(a*x))^(3/2)*x*Sqrt[c - a*c*x])/(3*Sqrt[1 - 1/(a*x)])","A",0
299,1,75,94,0.0556675,"\int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-a c x}}{x} \, dx","Integrate[(E^ArcCoth[a*x]*Sqrt[c - a*c*x])/x,x]","\frac{2 \sqrt{c-a c x} \left(\sqrt{a} \sqrt{\frac{a+\frac{1}{x}}{a}}-\sqrt{\frac{1}{x}} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)\right)}{\sqrt{a} \sqrt{1-\frac{1}{a x}}}","\frac{2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}}}-\frac{2 \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{1-\frac{1}{a x}}}",1,"(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[(a + x^(-1))/a] - Sqrt[x^(-1)]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]]))/(Sqrt[a]*Sqrt[1 - 1/(a*x)])","A",1
300,1,76,97,0.0433558,"\int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-a c x}}{x^2} \, dx","Integrate[(E^ArcCoth[a*x]*Sqrt[c - a*c*x])/x^2,x]","-\frac{\sqrt{\frac{1}{x}} \sqrt{c-a c x} \left(\sqrt{\frac{1}{x}} \sqrt{\frac{1}{a x}+1}+\sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)\right)}{\sqrt{1-\frac{1}{a x}}}","-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{x \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\sqrt{1-\frac{1}{a x}}}",1,"-((Sqrt[x^(-1)]*Sqrt[c - a*c*x]*(Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)] + Sqrt[a]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]]))/Sqrt[1 - 1/(a*x)])","A",1
301,1,48,101,0.075496,"\int e^{2 \coth ^{-1}(a x)} x^3 \sqrt{c-a c x} \, dx","Integrate[E^(2*ArcCoth[a*x])*x^3*Sqrt[c - a*c*x],x]","\frac{2 \left(35 a^4 x^4+85 a^3 x^3+102 a^2 x^2+136 a x+272\right) \sqrt{c-a c x}}{315 a^4}","\frac{2 (c-a c x)^{9/2}}{9 a^4 c^4}-\frac{10 (c-a c x)^{7/2}}{7 a^4 c^3}+\frac{18 (c-a c x)^{5/2}}{5 a^4 c^2}-\frac{14 (c-a c x)^{3/2}}{3 a^4 c}+\frac{4 \sqrt{c-a c x}}{a^4}",1,"(2*Sqrt[c - a*c*x]*(272 + 136*a*x + 102*a^2*x^2 + 85*a^3*x^3 + 35*a^4*x^4))/(315*a^4)","A",1
302,1,40,80,0.0569882,"\int e^{2 \coth ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx","Integrate[E^(2*ArcCoth[a*x])*x^2*Sqrt[c - a*c*x],x]","\frac{2 \left(15 a^3 x^3+39 a^2 x^2+52 a x+104\right) \sqrt{c-a c x}}{105 a^3}","-\frac{2 (c-a c x)^{7/2}}{7 a^3 c^3}+\frac{8 (c-a c x)^{5/2}}{5 a^3 c^2}-\frac{10 (c-a c x)^{3/2}}{3 a^3 c}+\frac{4 \sqrt{c-a c x}}{a^3}",1,"(2*Sqrt[c - a*c*x]*(104 + 52*a*x + 39*a^2*x^2 + 15*a^3*x^3))/(105*a^3)","A",1
303,1,31,57,0.0477423,"\int e^{2 \coth ^{-1}(a x)} x \sqrt{c-a c x} \, dx","Integrate[E^(2*ArcCoth[a*x])*x*Sqrt[c - a*c*x],x]","\frac{2 \left(a^2 x^2+3 a x+6\right) \sqrt{c-a c x}}{5 a^2}","\frac{2 (c-a c x)^{5/2}}{5 a^2 c^2}-\frac{2 (c-a c x)^{3/2}}{a^2 c}+\frac{4 \sqrt{c-a c x}}{a^2}",1,"(2*Sqrt[c - a*c*x]*(6 + 3*a*x + a^2*x^2))/(5*a^2)","A",1
304,1,23,38,0.0285695,"\int e^{2 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx","Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - a*c*x],x]","\frac{2 (a x+5) \sqrt{c-a c x}}{3 a}","\frac{4 \sqrt{c-a c x}}{a}-\frac{2 (c-a c x)^{3/2}}{3 a c}",1,"(2*(5 + a*x)*Sqrt[c - a*c*x])/(3*a)","A",1
305,1,39,39,0.0318908,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - a*c*x])/x,x]","2 \sqrt{c-a c x}+2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)","2 \sqrt{c-a c x}+2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)",1,"2*Sqrt[c - a*c*x] + 2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]]","A",1
306,1,42,42,0.0316439,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^2} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^2,x]","\frac{\sqrt{c-a c x}}{x}+3 a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)","\frac{\sqrt{c-a c x}}{x}+3 a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)",1,"Sqrt[c - a*c*x]/x + 3*a*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]]","A",1
307,1,55,68,0.0531654,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^3} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^3,x]","\frac{7}{4} a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)+\frac{(7 a x+2) \sqrt{c-a c x}}{4 x^2}","\frac{7}{4} a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)+\frac{\sqrt{c-a c x}}{2 x^2}+\frac{7 a \sqrt{c-a c x}}{4 x}",1,"((2 + 7*a*x)*Sqrt[c - a*c*x])/(4*x^2) + (7*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/4","A",1
308,1,63,89,0.0660237,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^4,x]","\frac{11}{8} a^3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)+\frac{\left(33 a^2 x^2+22 a x+8\right) \sqrt{c-a c x}}{24 x^3}","\frac{11}{8} a^3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)+\frac{11 a^2 \sqrt{c-a c x}}{8 x}+\frac{\sqrt{c-a c x}}{3 x^3}+\frac{11 a \sqrt{c-a c x}}{12 x^2}",1,"(Sqrt[c - a*c*x]*(8 + 22*a*x + 33*a^2*x^2))/(24*x^3) + (11*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/8","A",1
309,1,71,110,0.0742259,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^5} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^5,x]","\frac{75}{64} a^4 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)+\frac{\left(75 a^3 x^3+50 a^2 x^2+40 a x+16\right) \sqrt{c-a c x}}{64 x^4}","\frac{75}{64} a^4 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)+\frac{75 a^3 \sqrt{c-a c x}}{64 x}+\frac{25 a^2 \sqrt{c-a c x}}{32 x^2}+\frac{\sqrt{c-a c x}}{4 x^4}+\frac{5 a \sqrt{c-a c x}}{8 x^3}",1,"(Sqrt[c - a*c*x]*(16 + 40*a*x + 50*a^2*x^2 + 75*a^3*x^3))/(64*x^4) + (75*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/64","A",1
310,1,130,309,0.1313085,"\int e^{3 \coth ^{-1}(a x)} x^3 \sqrt{c-a c x} \, dx","Integrate[E^(3*ArcCoth[a*x])*x^3*Sqrt[c - a*c*x],x]","\frac{2 \sqrt{c-a c x} \left(\sqrt{a} \sqrt{\frac{1}{a x}+1} \left(35 a^4 x^4+95 a^3 x^3+138 a^2 x^2+236 a x+788\right)-630 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{315 a^{9/2} \sqrt{1-\frac{1}{a x}}}","-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{a^{9/2} \sqrt{1-\frac{1}{a x}}}+\frac{1576 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{315 a^4 \sqrt{1-\frac{1}{a x}}}+\frac{472 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{92 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{2 x^4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}+\frac{38 x^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}",1,"(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[1 + 1/(a*x)]*(788 + 236*a*x + 138*a^2*x^2 + 95*a^3*x^3 + 35*a^4*x^4) - 630*Sqrt[2]*Sqrt[x^(-1)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(315*a^(9/2)*Sqrt[1 - 1/(a*x)])","A",1
311,1,122,261,0.0943032,"\int e^{3 \coth ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx","Integrate[E^(3*ArcCoth[a*x])*x^2*Sqrt[c - a*c*x],x]","\frac{2 \sqrt{c-a c x} \left(\sqrt{a} \sqrt{\frac{1}{a x}+1} \left(3 a^3 x^3+9 a^2 x^2+16 a x+52\right)-42 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{21 a^{7/2} \sqrt{1-\frac{1}{a x}}}","-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{a^{7/2} \sqrt{1-\frac{1}{a x}}}+\frac{104 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{21 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{32 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{21 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{2 x^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{6 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{7 a \sqrt{1-\frac{1}{a x}}}",1,"(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[1 + 1/(a*x)]*(52 + 16*a*x + 9*a^2*x^2 + 3*a^3*x^3) - 42*Sqrt[2]*Sqrt[x^(-1)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(21*a^(7/2)*Sqrt[1 - 1/(a*x)])","A",1
312,1,114,211,0.0811661,"\int e^{3 \coth ^{-1}(a x)} x \sqrt{c-a c x} \, dx","Integrate[E^(3*ArcCoth[a*x])*x*Sqrt[c - a*c*x],x]","\frac{2 \sqrt{c-a c x} \left(\sqrt{a} \sqrt{\frac{1}{a x}+1} \left(3 a^2 x^2+11 a x+38\right)-30 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{15 a^{5/2} \sqrt{1-\frac{1}{a x}}}","-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{a^{5/2} \sqrt{1-\frac{1}{a x}}}+\frac{4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{a^2 \sqrt{1-\frac{1}{a x}}}+\frac{2 x^2 \left(\frac{1}{a x}+1\right)^{5/2} \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{2 x \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}",1,"(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[1 + 1/(a*x)]*(38 + 11*a*x + 3*a^2*x^2) - 30*Sqrt[2]*Sqrt[x^(-1)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(15*a^(5/2)*Sqrt[1 - 1/(a*x)])","A",1
313,1,105,163,0.0657272,"\int e^{3 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx","Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x],x]","\frac{2 \sqrt{c-a c x} \left(\sqrt{a} \sqrt{\frac{1}{a x}+1} (a x+7)-6 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{3 a^{3/2} \sqrt{1-\frac{1}{a x}}}","-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{a^{3/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{a \sqrt{1-\frac{1}{a x}}}",1,"(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[1 + 1/(a*x)]*(7 + a*x) - 6*Sqrt[2]*Sqrt[x^(-1)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(3*a^(3/2)*Sqrt[1 - 1/(a*x)])","A",1
314,1,120,170,0.0683675,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x,x]","\frac{2 \sqrt{c-a c x} \left(\sqrt{a} \sqrt{\frac{1}{a x}+1}+\sqrt{\frac{1}{x}} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)-2 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{\sqrt{a} \sqrt{1-\frac{1}{a x}}}","\frac{2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{\sqrt{a} \sqrt{1-\frac{1}{a x}}}",1,"(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[1 + 1/(a*x)] + Sqrt[x^(-1)]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]] - 2*Sqrt[2]*Sqrt[x^(-1)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(Sqrt[a]*Sqrt[1 - 1/(a*x)])","A",1
315,1,120,172,0.0842509,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^2} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^2,x]","\frac{\sqrt{\frac{1}{x}} \sqrt{c-a c x} \left(\sqrt{\frac{1}{x}} \sqrt{\frac{1}{a x}+1}+5 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)-4 \sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)\right)}{\sqrt{1-\frac{1}{a x}}}","\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{x \sqrt{1-\frac{1}{a x}}}+\frac{5 \sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{\sqrt{1-\frac{1}{a x}}}",1,"(Sqrt[x^(-1)]*Sqrt[c - a*c*x]*(Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)] + 5*Sqrt[a]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]] - 4*Sqrt[2]*Sqrt[a]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/Sqrt[1 - 1/(a*x)]","A",1
316,1,132,224,0.1649765,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^3} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^3,x]","\frac{\sqrt{c-a c x} \left(\frac{23 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\left(\frac{1}{x}\right)^{3/2}}-\frac{16 \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{\left(\frac{1}{x}\right)^{3/2}}+\sqrt{\frac{1}{a x}+1} (9 a x+2)\right)}{4 x^2 \sqrt{1-\frac{1}{a x}}}","\frac{23 a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{4 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{\sqrt{1-\frac{1}{a x}}}+\frac{a \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{2 x \sqrt{1-\frac{1}{a x}}}+\frac{7 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{4 x \sqrt{1-\frac{1}{a x}}}",1,"(Sqrt[c - a*c*x]*(Sqrt[1 + 1/(a*x)]*(2 + 9*a*x) + (23*a^(3/2)*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x^(-1))^(3/2) - (16*Sqrt[2]*a^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(x^(-1))^(3/2)))/(4*Sqrt[1 - 1/(a*x)]*x^2)","A",1
317,1,140,274,0.2277163,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^4,x]","\frac{\sqrt{c-a c x} \left(\frac{135 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\left(\frac{1}{x}\right)^{5/2}}-\frac{96 \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{\left(\frac{1}{x}\right)^{5/2}}+\sqrt{\frac{1}{a x}+1} \left(57 a^2 x^2+26 a x+8\right)\right)}{24 x^3 \sqrt{1-\frac{1}{a x}}}","\frac{45 a^{5/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{8 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{5/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{\sqrt{1-\frac{1}{a x}}}+\frac{3 a^2 \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{4 x \sqrt{1-\frac{1}{a x}}}+\frac{13 a^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{8 x \sqrt{1-\frac{1}{a x}}}+\frac{a \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{3 x^2 \sqrt{1-\frac{1}{a x}}}",1,"(Sqrt[c - a*c*x]*(Sqrt[1 + 1/(a*x)]*(8 + 26*a*x + 57*a^2*x^2) + (135*a^(5/2)*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x^(-1))^(5/2) - (96*Sqrt[2]*a^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(x^(-1))^(5/2)))/(24*Sqrt[1 - 1/(a*x)]*x^3)","A",1
318,1,148,322,0.272203,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^5} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - a*c*x])/x^5,x]","\frac{\sqrt{c-a c x} \left(\frac{1089 a^{7/2} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\left(\frac{1}{x}\right)^{7/2}}-\frac{768 \sqrt{2} a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{\left(\frac{1}{x}\right)^{7/2}}+\sqrt{\frac{1}{a x}+1} \left(447 a^3 x^3+214 a^2 x^2+136 a x+48\right)\right)}{192 x^4 \sqrt{1-\frac{1}{a x}}}","\frac{363 a^{7/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{64 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{7/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right)}{\sqrt{1-\frac{1}{a x}}}+\frac{21 a^3 \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{32 x \sqrt{1-\frac{1}{a x}}}+\frac{107 a^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{64 x \sqrt{1-\frac{1}{a x}}}+\frac{11 a^2 \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{24 x^2 \sqrt{1-\frac{1}{a x}}}+\frac{a \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-a c x}}{4 x^3 \sqrt{1-\frac{1}{a x}}}",1,"(Sqrt[c - a*c*x]*(Sqrt[1 + 1/(a*x)]*(48 + 136*a*x + 214*a^2*x^2 + 447*a^3*x^3) + (1089*a^(7/2)*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x^(-1))^(7/2) - (768*Sqrt[2]*a^(7/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(x^(-1))^(7/2)))/(192*Sqrt[1 - 1/(a*x)]*x^4)","A",1
319,1,46,144,0.0241014,"\int e^{\coth ^{-1}(x)} x (1+x)^{3/2} \, dx","Integrate[E^ArcCoth[x]*x*(1 + x)^(3/2),x]","\frac{2 \sqrt{\frac{x-1}{x}} \sqrt{x+1} \left(3 x^3+12 x^2+23 x+46\right)}{21 \sqrt{\frac{1}{x}+1}}","\frac{2 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2} x^2}{7 \left(\frac{1}{x}+1\right)^{3/2}}+\frac{8 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2} x}{7 \left(\frac{1}{x}+1\right)^{3/2}}+\frac{46 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2}}{21 \left(\frac{1}{x}+1\right)^{3/2}}+\frac{92 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2}}{21 \left(\frac{1}{x}+1\right)^{3/2} x}",1,"(2*Sqrt[(-1 + x)/x]*Sqrt[1 + x]*(46 + 23*x + 12*x^2 + 3*x^3))/(21*Sqrt[1 + x^(-1)])","A",1
320,1,41,107,0.0177219,"\int e^{\coth ^{-1}(x)} (1+x)^{3/2} \, dx","Integrate[E^ArcCoth[x]*(1 + x)^(3/2),x]","\frac{2 \sqrt{\frac{x-1}{x}} \sqrt{x+1} \left(3 x^2+14 x+43\right)}{15 \sqrt{\frac{1}{x}+1}}","\frac{2 \sqrt{-\frac{1-x}{x}} x (x+1)^{3/2}}{5 \left(\frac{1}{x}+1\right)^{3/2}}+\frac{28 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2}}{15 \left(\frac{1}{x}+1\right)^{3/2}}+\frac{86 \sqrt{-\frac{1-x}{x}} (x+1)^{3/2}}{15 \left(\frac{1}{x}+1\right)^{3/2} x}",1,"(2*Sqrt[(-1 + x)/x]*Sqrt[1 + x]*(43 + 14*x + 3*x^2))/(15*Sqrt[1 + x^(-1)])","A",1
321,1,46,104,0.0242002,"\int e^{\coth ^{-1}(x)} (1-x)^{3/2} x \, dx","Integrate[E^ArcCoth[x]*(1 - x)^(3/2)*x,x]","-\frac{2 \sqrt{\frac{1}{x}+1} \sqrt{1-x} (x+1) \left(15 x^2-33 x+22\right)}{105 \sqrt{\frac{x-1}{x}}}","\frac{2 \left(\frac{1}{x}+1\right)^{3/2} (1-x)^{3/2} x^2}{7 \left(1-\frac{1}{x}\right)^{3/2}}-\frac{22 \left(\frac{1}{x}+1\right)^{3/2} (1-x)^{3/2} x}{35 \left(1-\frac{1}{x}\right)^{3/2}}+\frac{44 \left(\frac{1}{x}+1\right)^{3/2} (1-x)^{3/2}}{105 \left(1-\frac{1}{x}\right)^{3/2}}",1,"(-2*Sqrt[1 + x^(-1)]*Sqrt[1 - x]*(1 + x)*(22 - 33*x + 15*x^2))/(105*Sqrt[(-1 + x)/x])","A",1
322,1,41,68,0.0181474,"\int e^{\coth ^{-1}(x)} (1-x)^{3/2} \, dx","Integrate[E^ArcCoth[x]*(1 - x)^(3/2),x]","-\frac{2 \sqrt{\frac{1}{x}+1} \sqrt{1-x} (x+1) (3 x-7)}{15 \sqrt{\frac{x-1}{x}}}","\frac{2 \left(\frac{1}{x}+1\right)^{3/2} (1-x)^{3/2} x}{5 \left(1-\frac{1}{x}\right)^{3/2}}-\frac{14 \left(\frac{1}{x}+1\right)^{3/2} (1-x)^{3/2}}{15 \left(1-\frac{1}{x}\right)^{3/2}}",1,"(-2*Sqrt[1 + x^(-1)]*Sqrt[1 - x]*(1 + x)*(-7 + 3*x))/(15*Sqrt[(-1 + x)/x])","A",1
323,1,39,107,0.0159156,"\int e^{\coth ^{-1}(x)} x \sqrt{1+x} \, dx","Integrate[E^ArcCoth[x]*x*Sqrt[1 + x],x]","\frac{2 \sqrt{\frac{x-1}{x}} \sqrt{x+1} \left(x^2+3 x+6\right)}{5 \sqrt{\frac{1}{x}+1}}","\frac{2 \sqrt{-\frac{1-x}{x}} \sqrt{x+1} x^2}{5 \sqrt{\frac{1}{x}+1}}+\frac{6 \sqrt{-\frac{1-x}{x}} \sqrt{x+1} x}{5 \sqrt{\frac{1}{x}+1}}+\frac{12 \sqrt{-\frac{1-x}{x}} \sqrt{x+1}}{5 \sqrt{\frac{1}{x}+1}}",1,"(2*Sqrt[(-1 + x)/x]*Sqrt[1 + x]*(6 + 3*x + x^2))/(5*Sqrt[1 + x^(-1)])","A",1
324,1,34,70,0.0168716,"\int e^{\coth ^{-1}(x)} \sqrt{1+x} \, dx","Integrate[E^ArcCoth[x]*Sqrt[1 + x],x]","\frac{2 \sqrt{\frac{x-1}{x}} \sqrt{x+1} (x+5)}{3 \sqrt{\frac{1}{x}+1}}","\frac{2 \sqrt{-\frac{1-x}{x}} \sqrt{x+1} x}{3 \sqrt{\frac{1}{x}+1}}+\frac{10 \sqrt{-\frac{1-x}{x}} \sqrt{x+1}}{3 \sqrt{\frac{1}{x}+1}}",1,"(2*Sqrt[(-1 + x)/x]*Sqrt[1 + x]*(5 + x))/(3*Sqrt[1 + x^(-1)])","A",1
325,1,41,71,0.0163623,"\int e^{\coth ^{-1}(x)} \sqrt{1-x} x \, dx","Integrate[E^ArcCoth[x]*Sqrt[1 - x]*x,x]","\frac{2 \sqrt{\frac{1}{x}+1} \sqrt{1-x} (x+1) (3 x-2)}{15 \sqrt{\frac{x-1}{x}}}","\frac{2 \left(\frac{1}{x}+1\right)^{3/2} \sqrt{1-x} x^2}{5 \sqrt{1-\frac{1}{x}}}-\frac{4 \left(\frac{1}{x}+1\right)^{3/2} \sqrt{1-x} x}{15 \sqrt{1-\frac{1}{x}}}",1,"(2*Sqrt[1 + x^(-1)]*Sqrt[1 - x]*(1 + x)*(-2 + 3*x))/(15*Sqrt[(-1 + x)/x])","A",1
326,1,34,20,0.0137444,"\int e^{\coth ^{-1}(x)} \sqrt{1-x} \, dx","Integrate[E^ArcCoth[x]*Sqrt[1 - x],x]","\frac{2 \left(\frac{1}{x}+1\right)^{3/2} \sqrt{1-x} x}{3 \sqrt{1-\frac{1}{x}}}","\frac{2}{3} \sqrt{1-x} (x+1) e^{\coth ^{-1}(x)}",1,"(2*(1 + x^(-1))^(3/2)*Sqrt[1 - x]*x)/(3*Sqrt[1 - x^(-1)])","A",0
327,1,26,73,0.0172203,"\int \frac{e^{\coth ^{-1}(x)} x}{\sqrt{1+x}} \, dx","Integrate[(E^ArcCoth[x]*x)/Sqrt[1 + x],x]","\frac{2 \sqrt{1-\frac{1}{x^2}} x (x+2)}{3 \sqrt{x+1}}","\frac{2 \sqrt{\frac{1}{x}+1} \sqrt{-\frac{1-x}{x}} x^2}{3 \sqrt{x+1}}+\frac{4 \sqrt{\frac{1}{x}+1} \sqrt{-\frac{1-x}{x}} x}{3 \sqrt{x+1}}",1,"(2*Sqrt[1 - x^(-2)]*x*(2 + x))/(3*Sqrt[1 + x])","A",1
328,1,21,33,0.009912,"\int \frac{e^{\coth ^{-1}(x)}}{\sqrt{1+x}} \, dx","Integrate[E^ArcCoth[x]/Sqrt[1 + x],x]","\frac{2 \sqrt{1-\frac{1}{x^2}} x}{\sqrt{x+1}}","\frac{2 \sqrt{\frac{1}{x}+1} \sqrt{-\frac{1-x}{x}} x}{\sqrt{x+1}}",1,"(2*Sqrt[1 - x^(-2)]*x)/Sqrt[1 + x]","A",1
329,1,69,126,0.0534524,"\int \frac{e^{\coth ^{-1}(x)} x}{\sqrt{1-x}} \, dx","Integrate[(E^ArcCoth[x]*x)/Sqrt[1 - x],x]","\frac{2 \sqrt{\frac{x-1}{x}} x \left(\sqrt{\frac{1}{x}+1} (x+4)-3 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left(\sqrt{2} \sqrt{\frac{1}{x+1}}\right)\right)}{3 \sqrt{1-x}}","\frac{2 \sqrt{1-\frac{1}{x}} \left(\frac{1}{x}+1\right)^{3/2} x^2}{3 \sqrt{1-x}}+\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{\frac{1}{x}+1} x}{\sqrt{1-x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{x}} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right)}{\sqrt{1-x} \sqrt{\frac{1}{x}}}",1,"(2*Sqrt[(-1 + x)/x]*x*(Sqrt[1 + x^(-1)]*(4 + x) - 3*Sqrt[2]*Sqrt[x^(-1)]*ArcTanh[Sqrt[2]*Sqrt[(1 + x)^(-1)]]))/(3*Sqrt[1 - x])","A",1
330,1,63,90,0.0304066,"\int \frac{e^{\coth ^{-1}(x)}}{\sqrt{1-x}} \, dx","Integrate[E^ArcCoth[x]/Sqrt[1 - x],x]","\frac{2 \sqrt{\frac{x-1}{x}} x \left(\sqrt{\frac{1}{x}+1}-\sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left(\sqrt{2} \sqrt{\frac{1}{x+1}}\right)\right)}{\sqrt{1-x}}","\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{\frac{1}{x}+1} x}{\sqrt{1-x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{x}} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right)}{\sqrt{1-x} \sqrt{\frac{1}{x}}}",1,"(2*Sqrt[(-1 + x)/x]*x*(Sqrt[1 + x^(-1)] - Sqrt[2]*Sqrt[x^(-1)]*ArcTanh[Sqrt[2]*Sqrt[(1 + x)^(-1)]]))/Sqrt[1 - x]","A",1
331,1,65,93,0.0433838,"\int \frac{e^{\coth ^{-1}(x)} x}{(1+x)^{3/2}} \, dx","Integrate[(E^ArcCoth[x]*x)/(1 + x)^(3/2),x]","\frac{\sqrt{\frac{1}{x}+1} x \left(2 \sqrt{\frac{x-1}{x}}-\sqrt{2} \sqrt{\frac{1}{x}} \tan ^{-1}\left(\frac{\sqrt{\frac{x-1}{x^2}} x}{\sqrt{2}}\right)\right)}{\sqrt{x+1}}","\frac{2 \left(\frac{1}{x}+1\right)^{3/2} \sqrt{-\frac{1-x}{x}} x^2}{(x+1)^{3/2}}+\frac{\sqrt{2} \left(\frac{1}{x}+1\right)^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{-\frac{1-x}{x}}}\right)}{\left(\frac{1}{x}\right)^{3/2} (x+1)^{3/2}}",1,"(Sqrt[1 + x^(-1)]*x*(2*Sqrt[(-1 + x)/x] - Sqrt[2]*Sqrt[x^(-1)]*ArcTan[(Sqrt[(-1 + x)/x^2]*x)/Sqrt[2]]))/Sqrt[1 + x]","A",0
332,1,41,58,0.0210546,"\int \frac{e^{\coth ^{-1}(x)}}{(1+x)^{3/2}} \, dx","Integrate[E^ArcCoth[x]/(1 + x)^(3/2),x]","\sqrt{2} \sqrt{\frac{1}{x+1}} \sqrt{x+1} \tan ^{-1}\left(\frac{\sqrt{\frac{x-1}{x^2}} x}{\sqrt{2}}\right)","-\frac{\sqrt{2} \left(\frac{1}{x}+1\right)^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{-\frac{1-x}{x}}}\right)}{\left(\frac{1}{x}\right)^{3/2} (x+1)^{3/2}}",1,"Sqrt[2]*Sqrt[(1 + x)^(-1)]*Sqrt[1 + x]*ArcTan[(Sqrt[(-1 + x)/x^2]*x)/Sqrt[2]]","A",0
333,1,75,130,0.0814136,"\int \frac{e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx","Integrate[(E^ArcCoth[x]*x)/(1 - x)^(3/2),x]","-\frac{\sqrt{\frac{x-1}{x}} x \left(2 \sqrt{\frac{1}{x}+1} (3-2 x)+5 \sqrt{2} (x-1) \sqrt{\frac{1}{x}} \tanh ^{-1}\left(\sqrt{2} \sqrt{\frac{1}{x+1}}\right)\right)}{2 (1-x)^{3/2}}","-\frac{\sqrt{1-\frac{1}{x}} \left(\frac{1}{x}+1\right)^{3/2} x^2}{2 (1-x)^{3/2}}+\frac{5 \left(1-\frac{1}{x}\right)^{3/2} \sqrt{\frac{1}{x}+1} x^2}{2 (1-x)^{3/2}}-\frac{5 \left(1-\frac{1}{x}\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right)}{\sqrt{2} (1-x)^{3/2} \left(\frac{1}{x}\right)^{3/2}}",1,"-1/2*(Sqrt[(-1 + x)/x]*x*(2*Sqrt[1 + x^(-1)]*(3 - 2*x) + 5*Sqrt[2]*(-1 + x)*Sqrt[x^(-1)]*ArcTanh[Sqrt[2]*Sqrt[(1 + x)^(-1)]]))/(1 - x)^(3/2)","A",1
334,1,58,90,0.0998188,"\int \frac{e^{\coth ^{-1}(x)}}{(1-x)^{3/2}} \, dx","Integrate[E^ArcCoth[x]/(1 - x)^(3/2),x]","\frac{\frac{2}{\sqrt{\frac{1}{x+1}}}+\sqrt{2} (x-1) \tanh ^{-1}\left(\sqrt{2} \sqrt{\frac{1}{x+1}}\right)}{2 \sqrt{-\frac{(x-1)^2}{x^2}} x}","-\frac{\sqrt{\frac{1}{x}+1} x \sqrt{1-\frac{1}{x}}}{(1-x)^{3/2}}-\frac{\left(1-\frac{1}{x}\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right)}{\sqrt{2} (1-x)^{3/2} \left(\frac{1}{x}\right)^{3/2}}",1,"(2/Sqrt[(1 + x)^(-1)] + Sqrt[2]*(-1 + x)*ArcTanh[Sqrt[2]*Sqrt[(1 + x)^(-1)]])/(2*Sqrt[-((-1 + x)^2/x^2)]*x)","A",0
335,1,102,131,0.0841864,"\int e^{-\coth ^{-1}(a x)} x^m \sqrt{c-a c x} \, dx","Integrate[(x^m*Sqrt[c - a*c*x])/E^ArcCoth[a*x],x]","\frac{2 x^m \sqrt{c-a c x} \left(a (2 m+1) x \sqrt{\frac{1}{a x}+1}-(4 m+5) \, _2F_1\left(\frac{1}{2},-m-\frac{1}{2};\frac{1}{2}-m;-\frac{1}{a x}\right)\right)}{a (2 m+1) (2 m+3) \sqrt{1-\frac{1}{a x}}}","\frac{2 \sqrt{\frac{1}{a x}+1} x^{m+1} \sqrt{c-a c x}}{(2 m+3) \sqrt{1-\frac{1}{a x}}}-\frac{2 (4 m+5) x^m \sqrt{c-a c x} \, _2F_1\left(\frac{1}{2},-m-\frac{1}{2};\frac{1}{2}-m;-\frac{1}{a x}\right)}{a (2 m+1) (2 m+3) \sqrt{1-\frac{1}{a x}}}",1,"(2*x^m*Sqrt[c - a*c*x]*(a*(1 + 2*m)*Sqrt[1 + 1/(a*x)]*x - (5 + 4*m)*Hypergeometric2F1[1/2, -1/2 - m, 1/2 - m, -(1/(a*x))]))/(a*(1 + 2*m)*(3 + 2*m)*Sqrt[1 - 1/(a*x)])","A",1
336,1,67,142,0.0391385,"\int e^{-\coth ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx","Integrate[(x^2*Sqrt[c - a*c*x])/E^ArcCoth[a*x],x]","\frac{2 \sqrt{\frac{1}{a x}+1} \left(15 a^3 x^3-39 a^2 x^2+52 a x-104\right) \sqrt{c-a c x}}{105 a^3 \sqrt{1-\frac{1}{a x}}}","-\frac{2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}}{7 a c}+\frac{6 x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}}{35 a^2 c}+\frac{38 x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}}{105 a^2}+\frac{152 c x \sqrt{1-\frac{1}{a^2 x^2}}}{105 a^2 \sqrt{c-a c x}}",1,"(2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(-104 + 52*a*x - 39*a^2*x^2 + 15*a^3*x^3))/(105*a^3*Sqrt[1 - 1/(a*x)])","A",0
337,1,58,104,0.0355883,"\int e^{-\coth ^{-1}(a x)} x \sqrt{c-a c x} \, dx","Integrate[(x*Sqrt[c - a*c*x])/E^ArcCoth[a*x],x]","\frac{2 \sqrt{\frac{1}{a x}+1} \left(a^2 x^2-3 a x+6\right) \sqrt{c-a c x}}{5 a^2 \sqrt{1-\frac{1}{a x}}}","-\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}}{5 a c}-\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}}{5 a}-\frac{8 c x \sqrt{1-\frac{1}{a^2 x^2}}}{5 a \sqrt{c-a c x}}",1,"(2*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(6 - 3*a*x + a^2*x^2))/(5*a^2*Sqrt[1 - 1/(a*x)])","A",0
338,1,50,62,0.0269586,"\int e^{-\coth ^{-1}(a x)} \sqrt{c-a c x} \, dx","Integrate[Sqrt[c - a*c*x]/E^ArcCoth[a*x],x]","\frac{2 \sqrt{\frac{1}{a x}+1} (a x-5) \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}}}","\frac{2}{3} x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}+\frac{8 c x \sqrt{1-\frac{1}{a^2 x^2}}}{3 \sqrt{c-a c x}}",1,"(2*Sqrt[1 + 1/(a*x)]*(-5 + a*x)*Sqrt[c - a*c*x])/(3*a*Sqrt[1 - 1/(a*x)])","A",0
339,1,74,94,0.0530062,"\int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-a c x}}{x} \, dx","Integrate[Sqrt[c - a*c*x]/(E^ArcCoth[a*x]*x),x]","\frac{2 \sqrt{c-a c x} \left(\sqrt{a} \sqrt{\frac{1}{a x}+1}+\sqrt{\frac{1}{x}} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)\right)}{\sqrt{a} \sqrt{1-\frac{1}{a x}}}","\frac{2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{1-\frac{1}{a x}}}",1,"(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[1 + 1/(a*x)] + Sqrt[x^(-1)]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]]))/(Sqrt[a]*Sqrt[1 - 1/(a*x)])","A",1
340,1,78,96,0.0443394,"\int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-a c x}}{x^2} \, dx","Integrate[Sqrt[c - a*c*x]/(E^ArcCoth[a*x]*x^2),x]","-\frac{\sqrt{\frac{1}{x}} \sqrt{c-a c x} \left(3 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)-\sqrt{\frac{1}{x}} \sqrt{\frac{1}{a x}+1}\right)}{\sqrt{1-\frac{1}{a x}}}","\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{x \sqrt{1-\frac{1}{a x}}}-\frac{3 \sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\sqrt{1-\frac{1}{a x}}}",1,"-((Sqrt[x^(-1)]*Sqrt[c - a*c*x]*(-(Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]) + 3*Sqrt[a]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]]))/Sqrt[1 - 1/(a*x)])","A",1
341,1,85,139,0.1376632,"\int e^{-2 \coth ^{-1}(a x)} x^3 \sqrt{c-a c x} \, dx","Integrate[(x^3*Sqrt[c - a*c*x])/E^(2*ArcCoth[a*x]),x]","\frac{2 \left(\left(35 a^4 x^4-95 a^3 x^3+138 a^2 x^2-236 a x+788\right) \sqrt{c-a c x}-630 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)\right)}{315 a^4}","\frac{2 (c-a c x)^{9/2}}{9 a^4 c^4}-\frac{2 (c-a c x)^{7/2}}{7 a^4 c^3}+\frac{2 (c-a c x)^{5/2}}{5 a^4 c^2}+\frac{2 (c-a c x)^{3/2}}{3 a^4 c}+\frac{4 \sqrt{c-a c x}}{a^4}-\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{a^4}",1,"(2*(Sqrt[c - a*c*x]*(788 - 236*a*x + 138*a^2*x^2 - 95*a^3*x^3 + 35*a^4*x^4) - 630*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]))/(315*a^4)","A",1
342,1,78,97,0.1005934,"\int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx","Integrate[(x^2*Sqrt[c - a*c*x])/E^(2*ArcCoth[a*x]),x]","\frac{2 \left(3 a^3 x^3-9 a^2 x^2+16 a x-52\right) \sqrt{c-a c x}+84 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{21 a^3}","-\frac{2 (c-a c x)^{7/2}}{7 a^3 c^3}-\frac{2 (c-a c x)^{3/2}}{3 a^3 c}-\frac{4 \sqrt{c-a c x}}{a^3}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{a^3}",1,"(2*Sqrt[c - a*c*x]*(-52 + 16*a*x - 9*a^2*x^2 + 3*a^3*x^3) + 84*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/(21*a^3)","A",1
343,1,70,97,0.0944343,"\int e^{-2 \coth ^{-1}(a x)} x \sqrt{c-a c x} \, dx","Integrate[(x*Sqrt[c - a*c*x])/E^(2*ArcCoth[a*x]),x]","\frac{2 \left(3 a^2 x^2-11 a x+38\right) \sqrt{c-a c x}-60 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{15 a^2}","\frac{2 (c-a c x)^{5/2}}{5 a^2 c^2}+\frac{2 (c-a c x)^{3/2}}{3 a^2 c}+\frac{4 \sqrt{c-a c x}}{a^2}-\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{a^2}",1,"(2*Sqrt[c - a*c*x]*(38 - 11*a*x + 3*a^2*x^2) - 60*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/(15*a^2)","A",1
344,1,61,76,0.033257,"\int e^{-2 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx","Integrate[Sqrt[c - a*c*x]/E^(2*ArcCoth[a*x]),x]","\frac{2 (a x-7) \sqrt{c-a c x}+12 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{3 a}","-\frac{2 (c-a c x)^{3/2}}{3 a c}-\frac{4 \sqrt{c-a c x}}{a}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)}{a}",1,"(2*(-7 + a*x)*Sqrt[c - a*c*x] + 12*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/(3*a)","A",1
345,1,74,74,0.0342207,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x} \, dx","Integrate[Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x),x]","2 \sqrt{c-a c x}+2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)-4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)","2 \sqrt{c-a c x}+2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)-4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)",1,"2*Sqrt[c - a*c*x] + 2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]] - 4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]","A",1
346,1,78,78,0.0442925,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^2} \, dx","Integrate[Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x^2),x]","\frac{\sqrt{c-a c x}}{x}-5 a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)+4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)","\frac{\sqrt{c-a c x}}{x}-5 a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)+4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)",1,"Sqrt[c - a*c*x]/x - 5*a*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]] + 4*Sqrt[2]*a*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]","A",1
347,1,93,106,0.0844678,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^3} \, dx","Integrate[Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x^3),x]","\frac{23}{4} a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)-4 \sqrt{2} a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)+\frac{(2-9 a x) \sqrt{c-a c x}}{4 x^2}","\frac{23}{4} a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)-4 \sqrt{2} a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)+\frac{\sqrt{c-a c x}}{2 x^2}-\frac{9 a \sqrt{c-a c x}}{4 x}",1,"((2 - 9*a*x)*Sqrt[c - a*c*x])/(4*x^2) + (23*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/4 - 4*Sqrt[2]*a^2*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]","A",1
348,1,101,127,0.1028417,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx","Integrate[Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x^4),x]","-\frac{45}{8} a^3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)+\frac{\left(57 a^2 x^2-26 a x+8\right) \sqrt{c-a c x}}{24 x^3}","-\frac{45}{8} a^3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)+\frac{19 a^2 \sqrt{c-a c x}}{8 x}+\frac{\sqrt{c-a c x}}{3 x^3}-\frac{13 a \sqrt{c-a c x}}{12 x^2}",1,"(Sqrt[c - a*c*x]*(8 - 26*a*x + 57*a^2*x^2))/(24*x^3) - (45*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/8 + 4*Sqrt[2]*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]","A",1
349,1,109,148,0.1257546,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^5} \, dx","Integrate[Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x^5),x]","\frac{363}{64} a^4 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)-4 \sqrt{2} a^4 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)+\frac{\left(-447 a^3 x^3+214 a^2 x^2-136 a x+48\right) \sqrt{c-a c x}}{192 x^4}","\frac{363}{64} a^4 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{c}}\right)-4 \sqrt{2} a^4 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right)-\frac{149 a^3 \sqrt{c-a c x}}{64 x}+\frac{107 a^2 \sqrt{c-a c x}}{96 x^2}+\frac{\sqrt{c-a c x}}{4 x^4}-\frac{17 a \sqrt{c-a c x}}{24 x^3}",1,"(Sqrt[c - a*c*x]*(48 - 136*a*x + 214*a^2*x^2 - 447*a^3*x^3))/(192*x^4) + (363*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/64 - 4*Sqrt[2]*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]","A",1
350,1,73,281,0.0466616,"\int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt{c-a c x} \, dx","Integrate[(x^3*Sqrt[c - a*c*x])/E^(3*ArcCoth[a*x]),x]","\frac{2 \left(5 a^5 x^5-20 a^4 x^4+41 a^3 x^3-82 a^2 x^2+328 a x+656\right) \sqrt{c-a c x}}{45 a^5 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{1312 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{45 a^4 \sqrt{1-\frac{1}{a x}}}-\frac{656 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{45 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{164 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{82 x^2 \sqrt{c-a c x}}{9 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{2 x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{8 x^3 \sqrt{c-a c x}}{9 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}",1,"(2*Sqrt[c - a*c*x]*(656 + 328*a*x - 82*a^2*x^2 + 41*a^3*x^3 - 20*a^4*x^4 + 5*a^5*x^5))/(45*a^5*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
351,1,65,231,0.038723,"\int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx","Integrate[(x^2*Sqrt[c - a*c*x])/E^(3*ArcCoth[a*x]),x]","\frac{2 \left(15 a^4 x^4-66 a^3 x^3+167 a^2 x^2-668 a x-1336\right) \sqrt{c-a c x}}{105 a^4 x \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{2672 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{105 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{1336 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{334 x \sqrt{c-a c x}}{35 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{2 x^3 \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{44 x^2 \sqrt{c-a c x}}{35 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}",1,"(2*Sqrt[c - a*c*x]*(-1336 - 668*a*x + 167*a^2*x^2 - 66*a^3*x^3 + 15*a^4*x^4))/(105*a^4*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
352,1,57,182,0.0350985,"\int e^{-3 \coth ^{-1}(a x)} x \sqrt{c-a c x} \, dx","Integrate[(x*Sqrt[c - a*c*x])/E^(3*ArcCoth[a*x]),x]","\frac{2 \left(3 a^3 x^3-16 a^2 x^2+79 a x+158\right) \sqrt{c-a c x}}{15 a^3 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{316 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{158 \sqrt{c-a c x}}{15 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{2 x^2 \sqrt{c-a c x}}{5 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{32 x \sqrt{c-a c x}}{15 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}",1,"(2*Sqrt[c - a*c*x]*(158 + 79*a*x - 16*a^2*x^2 + 3*a^3*x^3))/(15*a^3*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
353,1,48,137,0.0294336,"\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx","Integrate[Sqrt[c - a*c*x]/E^(3*ArcCoth[a*x]),x]","\frac{2 \left(a^2 x^2-10 a x-23\right) \sqrt{c-a c x}}{3 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{46 \sqrt{c-a c x}}{3 a^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{2 x \sqrt{c-a c x}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{20 \sqrt{c-a c x}}{3 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}",1,"(2*Sqrt[c - a*c*x]*(-23 - 10*a*x + a^2*x^2))/(3*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
354,1,78,140,0.122023,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x} \, dx","Integrate[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x),x]","\frac{2 \sqrt{c-a c x} \left(-\sqrt{a} \sqrt{\frac{1}{x}} \sqrt{\frac{1}{a x}+1} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)+a+\frac{5}{x}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{2 \sqrt{c-a c x}}{\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{10 \sqrt{c-a c x}}{a x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{2 \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{1-\frac{1}{a x}}}",1,"(2*Sqrt[c - a*c*x]*(a + 5/x - Sqrt[a]*Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
355,1,79,140,0.0847479,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^2} \, dx","Integrate[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^2),x]","\frac{\sqrt{c-a c x} \left(\frac{7 a^{3/2} \sqrt{\frac{1}{a x}+1} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\left(\frac{1}{x}\right)^{3/2}}-9 a x-1\right)}{a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{x \sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-a c x}}{x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{7 \sqrt{a} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\sqrt{1-\frac{1}{a x}}}",1,"(Sqrt[c - a*c*x]*(-1 - 9*a*x + (7*a^(3/2)*Sqrt[1 + 1/(a*x)]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x^(-1))^(3/2)))/(a*Sqrt[1 - 1/(a^2*x^2)]*x^2)","A",0
356,1,90,190,0.0905341,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^3} \, dx","Integrate[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^3),x]","-\frac{\sqrt{c-a c x} \left(\frac{47 a^{5/2} \sqrt{\frac{1}{a x}+1} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\left(\frac{1}{x}\right)^{5/2}}-47 a^2 x^2-13 a x+2\right)}{4 a x^3 \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{47 a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{4 \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{2 x^2 \sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-a c x}}{x^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{47 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{4 x \sqrt{1-\frac{1}{a x}}}",1,"-1/4*(Sqrt[c - a*c*x]*(2 - 13*a*x - 47*a^2*x^2 + (47*a^(5/2)*Sqrt[1 + 1/(a*x)]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x^(-1))^(5/2)))/(a*Sqrt[1 - 1/(a^2*x^2)]*x^3)","A",0
357,1,98,238,0.0997308,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx","Integrate[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^4),x]","\frac{\sqrt{c-a c x} \left(\frac{357 a^{7/2} \sqrt{\frac{1}{a x}+1} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\left(\frac{1}{x}\right)^{7/2}}-357 a^3 x^3-119 a^2 x^2+38 a x-8\right)}{24 a x^4 \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{119 a^{5/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{8 \sqrt{1-\frac{1}{a x}}}-\frac{119 a^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{8 x \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{3 x^3 \sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-a c x}}{x^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{119 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{12 x^2 \sqrt{1-\frac{1}{a x}}}",1,"(Sqrt[c - a*c*x]*(-8 + 38*a*x - 119*a^2*x^2 - 357*a^3*x^3 + (357*a^(7/2)*Sqrt[1 + 1/(a*x)]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x^(-1))^(7/2)))/(24*a*Sqrt[1 - 1/(a^2*x^2)]*x^4)","A",0
358,1,106,286,0.1012963,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^5} \, dx","Integrate[Sqrt[c - a*c*x]/(E^(3*ArcCoth[a*x])*x^5),x]","-\frac{\sqrt{c-a c x} \left(\frac{3345 a^{9/2} \sqrt{\frac{1}{a x}+1} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{\left(\frac{1}{x}\right)^{9/2}}-3345 a^4 x^4-1115 a^3 x^3+446 a^2 x^2-200 a x+48\right)}{192 a x^5 \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{1115 a^{7/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left(\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right)}{64 \sqrt{1-\frac{1}{a x}}}+\frac{1115 a^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{64 x \sqrt{1-\frac{1}{a x}}}-\frac{1115 a^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{96 x^2 \sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{4 x^4 \sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-a c x}}{x^4 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{223 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{24 x^3 \sqrt{1-\frac{1}{a x}}}",1,"-1/192*(Sqrt[c - a*c*x]*(48 - 200*a*x + 446*a^2*x^2 - 1115*a^3*x^3 - 3345*a^4*x^4 + (3345*a^(9/2)*Sqrt[1 + 1/(a*x)]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x^(-1))^(9/2)))/(a*Sqrt[1 - 1/(a^2*x^2)]*x^5)","A",0
359,1,116,278,0.0805592,"\int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac{n}{2}} \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^(2 + n/2),x]","\frac{2 c^2 (a x+1) \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} \left(2 n \left(3 a^2 x^2-10 a x+7\right)+8 \left(a^2 x^2-4 a x+7\right)+n^2 (a x-1)^2\right) (c-a c x)^{n/2}}{a (n+2) (n+4) (n+6)}","\frac{2 \left(n^2+14 n+56\right) \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{a^2 (n+6) \left(n^2+6 n+8\right) x}-\frac{\left(n^2+14 n+56\right) \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{a (n+4) (n+6)}+\frac{(n+8) x \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{n+6}-\frac{x \left(a-\frac{1}{x}\right) \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{a}",1,"(2*c^2*(1 + 1/(a*x))^(n/2)*(1 + a*x)*(c - a*c*x)^(n/2)*(n^2*(-1 + a*x)^2 + 8*(7 - 4*a*x + a^2*x^2) + 2*n*(7 - 10*a*x + 3*a^2*x^2)))/(a*(2 + n)*(4 + n)*(6 + n)*(1 - 1/(a*x))^(n/2))","A",1
360,1,78,127,0.0510396,"\int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac{n}{2}} \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^(1 + n/2),x]","-\frac{2 c (a x+1) \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} (n (a x-1)+2 a x-6) (c-a c x)^{n/2}}{a (n+2) (n+4)}","\frac{2 x \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-1} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} (c-a c x)^{\frac{n+2}{2}}}{n+4}-\frac{2 (n+6) \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-1} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} (c-a c x)^{\frac{n+2}{2}}}{a (n+2) (n+4)}",1,"(-2*c*(1 + 1/(a*x))^(n/2)*(1 + a*x)*(c - a*c*x)^(n/2)*(-6 + 2*a*x + n*(-1 + a*x)))/(a*(2 + n)*(4 + n)*(1 - 1/(a*x))^(n/2))","A",1
361,1,58,36,0.0187502,"\int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^(n/2),x]","-\frac{x \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{\frac{n}{2}+1} (c-a c x)^{n/2}}{-\frac{n}{2}-1}","\frac{2 (a x+1) (c-a c x)^{n/2} e^{n \coth ^{-1}(a x)}}{a (n+2)}",1,"-(((1 + 1/(a*x))^(1 + n/2)*x*(c - a*c*x)^(n/2))/((-1 - n/2)*(1 - 1/(a*x))^(n/2)))","A",0
362,1,78,80,0.0291474,"\int e^{n \coth ^{-1}(a x)} (c-a c x)^{-1+\frac{n}{2}} \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^(-1 + n/2),x]","-\frac{2 \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} (c-a c x)^{n/2} \, _2F_1\left(1,-\frac{n}{2};1-\frac{n}{2};\frac{2}{a x+1}\right)}{a c n}","\frac{2 x \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}} \left(\frac{1}{a x}+1\right)^{n/2} (c-a c x)^{\frac{n-2}{2}} \, _2F_1\left(1,-\frac{n}{2};1-\frac{n}{2};\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{n}",1,"(-2*(1 + 1/(a*x))^(n/2)*(c - a*c*x)^(n/2)*Hypergeometric2F1[1, -1/2*n, 1 - n/2, 2/(1 + a*x)])/(a*c*n*(1 - 1/(a*x))^(n/2))","A",1
363,1,89,88,0.041952,"\int e^{n \coth ^{-1}(a x)} (c-a c x)^{-2+\frac{n}{2}} \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^(-2 + n/2),x]","\frac{2 \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} (c-a c x)^{n/2} \, _2F_1\left(2,1-\frac{n}{2};2-\frac{n}{2};\frac{2}{a x+1}\right)}{a c^2 (n-2) (a x+1)}","-\frac{2 x \left(1-\frac{1}{a x}\right)^{2-\frac{n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n-2}{2}} (c-a c x)^{\frac{n-4}{2}} \, _2F_1\left(2,1-\frac{n}{2};2-\frac{n}{2};\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{2-n}",1,"(2*(1 + 1/(a*x))^(n/2)*(c - a*c*x)^(n/2)*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, 2/(1 + a*x)])/(a*c^2*(-2 + n)*(1 - 1/(a*x))^(n/2)*(1 + a*x))","A",1
364,1,104,104,0.0546824,"\int e^{n \coth ^{-1}(a x)} (c-a c x)^p \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^p,x]","\frac{(a x+1) \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} (c-a c x)^p \left(\frac{a x-1}{a x+1}\right)^{\frac{1}{2} (n-2 p)} \, _2F_1\left(-p-1,\frac{n}{2}-p;-p;\frac{2}{a x+1}\right)}{a (p+1)}","\frac{x \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} (c-a c x)^p \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{\frac{1}{2} (n-2 p)} \, _2F_1\left(\frac{1}{2} (n-2 p),-p-1;-p;\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{p+1}",1,"((1 + 1/(a*x))^(n/2)*((-1 + a*x)/(1 + a*x))^((n - 2*p)/2)*(1 + a*x)*(c - a*c*x)^p*Hypergeometric2F1[-1 - p, n/2 - p, -p, 2/(1 + a*x)])/(a*(1 + p)*(1 - 1/(a*x))^(n/2))","A",1
365,1,190,81,1.9758424,"\int e^{n \coth ^{-1}(a x)} (c-a c x)^3 \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^3,x]","-\frac{c^3 e^{n \coth ^{-1}(a x)} \left((n+2) \left(n^2 \left(a^2 x^2-12 a x-1\right)+2 n \left(a^3 x^3-6 a^2 x^2+21 a x+6\right)+6 \left(a^4 x^4-4 a^3 x^3+6 a^2 x^2-4 a x-7\right)+\left(n^3-12 n^2+44 n-48\right) \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)+a n^3 x\right)+n \left(n^3-12 n^2+44 n-48\right) e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)\right)}{24 a (n+2)}","-\frac{32 c^3 \left(1-\frac{1}{a x}\right)^{4-\frac{n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n-8}{2}} \, _2F_1\left(5,4-\frac{n}{2};5-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a (8-n)}",1,"-1/24*(c^3*E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n*(-48 + 44*n - 12*n^2 + n^3)*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(a*n^3*x + n^2*(-1 - 12*a*x + a^2*x^2) + 2*n*(6 + 21*a*x - 6*a^2*x^2 + a^3*x^3) + 6*(-7 - 4*a*x + 6*a^2*x^2 - 4*a^3*x^3 + a^4*x^4) + (-48 + 44*n - 12*n^2 + n^3)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(a*(2 + n))","B",0
366,1,144,81,1.2275062,"\int e^{n \coth ^{-1}(a x)} (c-a c x)^2 \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^2,x]","\frac{c^2 e^{n \coth ^{-1}(a x)} \left((n+2) \left(2 a^3 x^3+n \left(a^2 x^2-6 a x-1\right)-6 a^2 x^2+\left(n^2-6 n+8\right) \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)+a n^2 x+6 a x+6\right)+n \left(n^2-6 n+8\right) e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)\right)}{6 a (n+2)}","\frac{16 c^2 \left(1-\frac{1}{a x}\right)^{3-\frac{n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n-6}{2}} \, _2F_1\left(4,3-\frac{n}{2};4-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a (6-n)}",1,"(c^2*E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n*(8 - 6*n + n^2)*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(6 + 6*a*x + a*n^2*x - 6*a^2*x^2 + 2*a^3*x^3 + n*(-1 - 6*a*x + a^2*x^2) + (8 - 6*n + n^2)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(6*a*(2 + n))","A",0
367,1,104,79,0.5494919,"\int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x),x]","-\frac{c e^{n \coth ^{-1}(a x)} \left((n+2) \left(a^2 x^2+(n-2) \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)+a (n-2) x-1\right)+(n-2) n e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)\right)}{2 a (n+2)}","-\frac{8 c \left(1-\frac{1}{a x}\right)^{2-\frac{n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n-4}{2}} \, _2F_1\left(3,2-\frac{n}{2};3-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a (4-n)}",1,"-1/2*(c*E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*(-2 + n)*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(-1 + a*(-2 + n)*x + a^2*x^2 + (-2 + n)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(a*(2 + n))","A",0
368,1,87,71,0.2076801,"\int \frac{e^{n \coth ^{-1}(a x)}}{c-a c x} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a*c*x),x]","-\frac{e^{n \coth ^{-1}(a x)} \left(n e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)+(n+2) \left(\, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)-1\right)\right)}{a c n (n+2)}","\frac{2 \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} \, _2F_1\left(1,-\frac{n}{2};1-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a c n}",1,"-((E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(-1 + Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(a*c*n*(2 + n)))","A",0
369,1,33,48,0.1797958,"\int \frac{e^{n \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a*c*x)^2,x]","-\frac{(a x+1) e^{n \coth ^{-1}(a x)}}{a c^2 (n+2) (a x-1)}","-\frac{\left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-1} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}}}{a c^2 (n+2)}",1,"-((E^(n*ArcCoth[a*x])*(1 + a*x))/(a*c^2*(2 + n)*(-1 + a*x)))","A",1
370,1,64,104,0.246939,"\int \frac{e^{n \coth ^{-1}(a x)}}{(c-a c x)^3} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a*c*x)^3,x]","\frac{(-a x+n+3) e^{n \coth ^{-1}(a x)} \left(\cosh \left(3 \coth ^{-1}(a x)\right)+\sinh \left(3 \coth ^{-1}(a x)\right)\right)}{a^2 c^3 (n+2) (n+4) x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{\left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-2} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}}}{a c^3 (n+4)}-\frac{(n+3) \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-1} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}}}{a c^3 (n+2) (n+4)}",1,"(E^(n*ArcCoth[a*x])*(3 + n - a*x)*(Cosh[3*ArcCoth[a*x]] + Sinh[3*ArcCoth[a*x]]))/(a^2*c^3*(2 + n)*(4 + n)*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
371,1,83,224,0.311528,"\int \frac{e^{n \coth ^{-1}(a x)}}{(c-a c x)^4} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a*c*x)^4,x]","-\frac{e^{n \coth ^{-1}(a x)} \left(\cosh \left(4 \coth ^{-1}(a x)\right)+\sinh \left(4 \coth ^{-1}(a x)\right)\right) \left((n+4)^2 \cosh \left(2 \coth ^{-1}(a x)\right)-2 (n+4) \sinh \left(2 \coth ^{-1}(a x)\right)-n^2-8 n-12\right)}{2 a c^4 (n+2) (n+4) (n+6)}","-\frac{\left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-3}}{a^2 c^4 x}-\frac{\left(n^2+8 n+14\right) \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-2}}{a c^4 (n+4) (n+6)}-\frac{\left(n^2+8 n+14\right) \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-1}}{a c^4 (n+6) \left(n^2+6 n+8\right)}+\frac{(n+5) \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-3}}{a c^4 (n+6)}",1,"-1/2*(E^(n*ArcCoth[a*x])*(-12 - 8*n - n^2 + (4 + n)^2*Cosh[2*ArcCoth[a*x]] - 2*(4 + n)*Sinh[2*ArcCoth[a*x]])*(Cosh[4*ArcCoth[a*x]] + Sinh[4*ArcCoth[a*x]]))/(a*c^4*(2 + n)*(4 + n)*(6 + n))","A",1
372,1,103,98,0.1026087,"\int e^{n \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^(5/2),x]","\frac{2 c^2 (a x+1)^3 \sqrt{c-a c x} \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} \left(\frac{a x-1}{a x+1}\right)^{\frac{n-1}{2}} \, _2F_1\left(-\frac{7}{2},\frac{n-5}{2};-\frac{5}{2};\frac{2}{a x+1}\right)}{7 a}","\frac{2}{7} x (c-a c x)^{5/2} \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{\frac{n-5}{2}} \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \, _2F_1\left(-\frac{7}{2},\frac{n-5}{2};-\frac{5}{2};\frac{2}{\left(a+\frac{1}{x}\right) x}\right)",1,"(2*c^2*(1 + 1/(a*x))^(n/2)*((-1 + a*x)/(1 + a*x))^((-1 + n)/2)*(1 + a*x)^3*Sqrt[c - a*c*x]*Hypergeometric2F1[-7/2, (-5 + n)/2, -5/2, 2/(1 + a*x)])/(7*a*(1 - 1/(a*x))^(n/2))","A",1
373,1,101,98,0.0757376,"\int e^{n \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^(3/2),x]","-\frac{2 c (a x+1)^2 \sqrt{c-a c x} \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} \left(\frac{a x-1}{a x+1}\right)^{\frac{n-1}{2}} \, _2F_1\left(-\frac{5}{2},\frac{n-3}{2};-\frac{3}{2};\frac{2}{a x+1}\right)}{5 a}","\frac{2}{5} x (c-a c x)^{3/2} \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{\frac{n-3}{2}} \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \, _2F_1\left(-\frac{5}{2},\frac{n-3}{2};-\frac{3}{2};\frac{2}{\left(a+\frac{1}{x}\right) x}\right)",1,"(-2*c*(1 + 1/(a*x))^(n/2)*((-1 + a*x)/(1 + a*x))^((-1 + n)/2)*(1 + a*x)^2*Sqrt[c - a*c*x]*Hypergeometric2F1[-5/2, (-3 + n)/2, -3/2, 2/(1 + a*x)])/(5*a*(1 - 1/(a*x))^(n/2))","A",1
374,1,98,98,0.0646483,"\int e^{n \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx","Integrate[E^(n*ArcCoth[a*x])*Sqrt[c - a*c*x],x]","\frac{2 (a x+1) \sqrt{c-a c x} \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} \left(\frac{a x-1}{a x+1}\right)^{\frac{n-1}{2}} \, _2F_1\left(-\frac{3}{2},\frac{n-1}{2};-\frac{1}{2};\frac{2}{a x+1}\right)}{3 a}","\frac{2}{3} x \sqrt{c-a c x} \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{\frac{n-1}{2}} \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \, _2F_1\left(-\frac{3}{2},\frac{n-1}{2};-\frac{1}{2};\frac{2}{\left(a+\frac{1}{x}\right) x}\right)",1,"(2*(1 + 1/(a*x))^(n/2)*((-1 + a*x)/(1 + a*x))^((-1 + n)/2)*(1 + a*x)*Sqrt[c - a*c*x]*Hypergeometric2F1[-3/2, (-1 + n)/2, -1/2, 2/(1 + a*x)])/(3*a*(1 - 1/(a*x))^(n/2))","A",1
375,1,96,96,0.0587655,"\int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx","Integrate[E^(n*ArcCoth[a*x])/Sqrt[c - a*c*x],x]","\frac{2 (a x+1) \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} \left(\frac{a x-1}{a x+1}\right)^{\frac{n+1}{2}} \, _2F_1\left(-\frac{1}{2},\frac{n+1}{2};\frac{1}{2};\frac{2}{a x+1}\right)}{a \sqrt{c-a c x}}","\frac{2 x \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{\frac{n+1}{2}} \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \, _2F_1\left(-\frac{1}{2},\frac{n+1}{2};\frac{1}{2};\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{\sqrt{c-a c x}}",1,"(2*(1 + 1/(a*x))^(n/2)*((-1 + a*x)/(1 + a*x))^((1 + n)/2)*(1 + a*x)*Hypergeometric2F1[-1/2, (1 + n)/2, 1/2, 2/(1 + a*x)])/(a*(1 - 1/(a*x))^(n/2)*Sqrt[c - a*c*x])","A",1
376,1,94,96,0.0415034,"\int \frac{e^{n \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a*c*x)^(3/2),x]","\frac{2 \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} \left(\frac{a x-1}{a x+1}\right)^{\frac{n+1}{2}} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{3}{2};\frac{2}{a x+1}\right)}{a c \sqrt{c-a c x}}","-\frac{2 x \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{\frac{n+3}{2}} \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{3}{2};\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{(c-a c x)^{3/2}}",1,"(2*(1 + 1/(a*x))^(n/2)*((-1 + a*x)/(1 + a*x))^((1 + n)/2)*Hypergeometric2F1[1/2, (3 + n)/2, 3/2, 2/(1 + a*x)])/(a*c*(1 - 1/(a*x))^(n/2)*Sqrt[c - a*c*x])","A",1
377,1,117,167,0.116105,"\int \frac{e^{n \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a*c*x)^(5/2),x]","\frac{\left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} \left((a x-1) \left(\frac{a x-1}{a x+1}\right)^{\frac{n+1}{2}} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{3}{2};\frac{2}{a x+1}\right)-a x-1\right)}{a c^2 (n+3) (a x-1) \sqrt{c-a c x}}","\frac{a x^2 \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{\frac{n+3}{2}} \left(1-\frac{1}{a x}\right)^{\frac{2-n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{3}{2};\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{(n+3) (c-a c x)^{5/2}}-\frac{a x^2 \left(1-\frac{1}{a x}\right)^{\frac{2-n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}}}{(n+3) (c-a c x)^{5/2}}",1,"((1 + 1/(a*x))^(n/2)*(-1 - a*x + (-1 + a*x)*((-1 + a*x)/(1 + a*x))^((1 + n)/2)*Hypergeometric2F1[1/2, (3 + n)/2, 3/2, 2/(1 + a*x)]))/(a*c^2*(3 + n)*(1 - 1/(a*x))^(n/2)*(-1 + a*x)*Sqrt[c - a*c*x])","A",1
378,1,138,245,0.1719062,"\int \frac{e^{n \coth ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a*c*x)^(7/2),x]","\frac{\left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} \left(3 (a x-1)^2 \left(\frac{a x-1}{a x+1}\right)^{\frac{n+1}{2}} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{3}{2};\frac{2}{a x+1}\right)+(a x+1) (-3 a x+2 n+9)\right)}{2 a c^3 (n+3) (n+5) (a x-1)^2 \sqrt{c-a c x}}","-\frac{3 a^2 x^3 \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{\frac{n+3}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{\frac{4-n}{2}} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{3}{2};\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{2 \left(n^2+8 n+15\right) (c-a c x)^{7/2}}+\frac{3 a^2 x^3 \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{\frac{4-n}{2}}}{2 \left(n^2+8 n+15\right) (c-a c x)^{7/2}}-\frac{a x^2 \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{\frac{2-n}{2}}}{(n+5) (c-a c x)^{7/2}}",1,"((1 + 1/(a*x))^(n/2)*((9 + 2*n - 3*a*x)*(1 + a*x) + 3*(-1 + a*x)^2*((-1 + a*x)/(1 + a*x))^((1 + n)/2)*Hypergeometric2F1[1/2, (3 + n)/2, 3/2, 2/(1 + a*x)]))/(2*a*c^3*(3 + n)*(5 + n)*(1 - 1/(a*x))^(n/2)*(-1 + a*x)^2*Sqrt[c - a*c*x])","A",1
379,1,175,114,0.1997343,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^4 \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a*x))^4,x]","\frac{c^4 \left(6 a^5 x^5+16 a^4 x^4-15 a^3 x^3-14 a^2 x^2+24 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)+9 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left(\frac{1}{a x}\right)-18 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+9 a x-2\right)}{6 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{c^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{3 a}+\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left(6 a-\frac{1}{x}\right)}{2 a^2}+c^4 x \left(1-\frac{1}{a^2 x^2}\right)^{3/2}-\frac{3 c^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}-\frac{c^4 \csc ^{-1}(a x)}{2 a}",1,"(c^4*(-2 + 9*a*x - 14*a^2*x^2 - 15*a^3*x^3 + 16*a^4*x^4 + 6*a^5*x^5 + 24*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] + 9*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4*ArcSin[1/(a*x)] - 18*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/(6*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4)","A",0
380,1,167,88,0.1613297,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^3 \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a*x))^3,x]","\frac{c^3 \left(2 a^4 x^4+4 a^3 x^3-3 a^2 x^2+2 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)+2 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left(\frac{1}{a x}\right)-4 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)-4 a x+1\right)}{2 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}} \left(4 a+\frac{1}{x}\right)}{2 a^2}+c^3 x \left(1-\frac{1}{a^2 x^2}\right)^{3/2}-\frac{2 c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}+\frac{c^3 \csc ^{-1}(a x)}{2 a}",1,"(c^3*(1 - 4*a*x - 3*a^2*x^2 + 4*a^3*x^3 + 2*a^4*x^4 + 2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] + 2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3*ArcSin[1/(a*x)] - 4*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/(2*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3)","A",0
381,1,158,62,0.1744619,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^2 \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a*x))^2,x]","\frac{c^2 \left(2 a^3 x^3+2 a^2 x^2-2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)+a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left(\frac{1}{a x}\right)-2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)-2 a x-2\right)}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{c^2 x \sqrt{1-\frac{1}{a^2 x^2}} \left(a+\frac{1}{x}\right)}{a}-\frac{c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}+\frac{c^2 \csc ^{-1}(a x)}{a}",1,"(c^2*(-2 - 2*a*x + 2*a^2*x^2 + 2*a^3*x^3 - 2*a^2*Sqrt[1 - 1/(a^2*x^2)]*x^2*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] + a^2*Sqrt[1 - 1/(a^2*x^2)]*x^2*ArcSin[1/(a*x)] - 2*a^2*Sqrt[1 - 1/(a^2*x^2)]*x^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/(2*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2)","B",0
382,1,31,27,0.0431285,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right) \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a*x)),x]","\frac{c \left(a x \sqrt{1-\frac{1}{a^2 x^2}}+\sin ^{-1}\left(\frac{1}{a x}\right)\right)}{a}","c x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{c \csc ^{-1}(a x)}{a}",1,"(c*(a*Sqrt[1 - 1/(a^2*x^2)]*x + ArcSin[1/(a*x)]))/a","A",0
383,1,63,70,0.103142,"\int \frac{e^{\coth ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx","Integrate[E^ArcCoth[a*x]/(c - c/(a*x)),x]","\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-3)+2 (a x-1) \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a c (a x-1)}","-\frac{2 \left(a+\frac{1}{x}\right)}{a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c}+\frac{2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c}",1,"(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-3 + a*x) + 2*(-1 + a*x)*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c*(-1 + a*x))","A",0
384,1,94,105,0.0698326,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^2} \, dx","Integrate[E^ArcCoth[a*x]/(c - c/(a*x))^2,x]","\frac{3 a^3 x^3-16 a^2 x^2+9 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1) \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)-5 a x+14}{3 a^2 c^2 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)}","-\frac{4 \left(a+\frac{1}{x}\right)}{3 a^2 c^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}-\frac{9 a+\frac{11}{x}}{3 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2}+\frac{3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c^2}",1,"(14 - 5*a*x - 16*a^2*x^2 + 3*a^3*x^3 + 9*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(3*a^2*c^2*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x))","A",1
385,1,104,138,0.0821296,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^3} \, dx","Integrate[E^ArcCoth[a*x]/(c - c/(a*x))^3,x]","\frac{15 a^4 x^4-134 a^3 x^3+73 a^2 x^2+60 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+128 a x-94}{15 a^2 c^3 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2}","-\frac{8 \left(a+\frac{1}{x}\right)}{5 a^2 c^3 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}-\frac{4 \left(5 a+\frac{8}{x}\right)}{15 a^2 c^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}-\frac{60 a+\frac{79}{x}}{15 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3}+\frac{4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c^3}",1,"(-94 + 128*a*x + 73*a^2*x^2 - 134*a^3*x^3 + 15*a^4*x^4 + 60*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(15*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2)","A",1
386,1,112,171,0.0945654,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^4} \, dx","Integrate[E^ArcCoth[a*x]/(c - c/(a*x))^4,x]","\frac{105 a^5 x^5-1339 a^4 x^4+1812 a^3 x^3+485 a^2 x^2+525 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)-1947 a x+824}{105 a^2 c^4 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^3}","-\frac{16 \left(a+\frac{1}{x}\right)}{7 a^2 c^4 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}-\frac{4 \left(7 a+\frac{17}{x}\right)}{35 a^2 c^4 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}-\frac{175 a+\frac{307}{x}}{105 a^2 c^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}-\frac{525 a+\frac{719}{x}}{105 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^4}+\frac{5 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c^4}",1,"(824 - 1947*a*x + 485*a^2*x^2 + 1812*a^3*x^3 - 1339*a^4*x^4 + 105*a^5*x^5 + 525*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(105*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^3)","A",1
387,1,63,61,0.2414432,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^5 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^5,x]","-\frac{c^5}{4 a^5 x^4}+\frac{c^5}{a^4 x^3}-\frac{c^5}{a^3 x^2}-\frac{2 c^5}{a^2 x}-\frac{3 c^5 \log (a x)}{a}+c^5 x","-\frac{c^5}{4 a^5 x^4}+\frac{c^5}{a^4 x^3}-\frac{c^5}{a^3 x^2}-\frac{2 c^5}{a^2 x}-\frac{3 c^5 \log (x)}{a}+c^5 x",1,"-1/4*c^5/(a^5*x^4) + c^5/(a^4*x^3) - c^5/(a^3*x^2) - (2*c^5)/(a^2*x) + c^5*x - (3*c^5*Log[a*x])/a","A",0
388,1,42,40,0.1734829,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^4 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^4,x]","\frac{c^4}{3 a^4 x^3}-\frac{c^4}{a^3 x^2}-\frac{2 c^4 \log (a x)}{a}+c^4 x","\frac{c^4}{3 a^4 x^3}-\frac{c^4}{a^3 x^2}-\frac{2 c^4 \log (x)}{a}+c^4 x",1,"c^4/(3*a^4*x^3) - c^4/(a^3*x^2) + c^4*x - (2*c^4*Log[a*x])/a","A",0
389,1,41,39,0.1331865,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^3 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^3,x]","-\frac{c^3}{2 a^3 x^2}+\frac{c^3}{a^2 x}-\frac{c^3 \log (a x)}{a}+c^3 x","-\frac{c^3}{2 a^3 x^2}+\frac{c^3}{a^2 x}-\frac{c^3 \log (x)}{a}+c^3 x",1,"-1/2*c^3/(a^3*x^2) + c^3/(a^2*x) + c^3*x - (c^3*Log[a*x])/a","A",0
390,1,16,16,0.1026283,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^2 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^2,x]","\frac{c^2}{a^2 x}+c^2 x","\frac{c^2}{a^2 x}+c^2 x",1,"c^2/(a^2*x) + c^2*x","A",1
391,1,11,11,0.0426161,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right) \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x)),x]","\frac{c \log (x)}{a}+c x","\frac{c \log (x)}{a}+c x",1,"c*x + (c*Log[x])/a","A",1
392,1,30,37,0.0324645,"\int \frac{e^{2 \coth ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - c/(a*x)),x]","\frac{a x+\frac{2}{1-a x}+3 \log (1-a x)}{a c}","\frac{2}{a c (1-a x)}+\frac{3 \log (1-a x)}{a c}+\frac{x}{c}",1,"(a*x + 2/(1 - a*x) + 3*Log[1 - a*x])/(a*c)","A",1
393,1,51,53,0.1119428,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^2} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - c/(a*x))^2,x]","-\frac{5}{a c^2 (a x-1)}-\frac{1}{a c^2 (a x-1)^2}+\frac{4 \log (1-a x)}{a c^2}+\frac{x}{c^2}","\frac{5}{a c^2 (1-a x)}-\frac{1}{a c^2 (1-a x)^2}+\frac{4 \log (1-a x)}{a c^2}+\frac{x}{c^2}",1,"x/c^2 - 1/(a*c^2*(-1 + a*x)^2) - 5/(a*c^2*(-1 + a*x)) + (4*Log[1 - a*x])/(a*c^2)","A",1
394,1,63,73,0.1320654,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^3} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - c/(a*x))^3,x]","\frac{6 a^4 x^4-18 a^3 x^3-36 a^2 x^2+81 a x+30 (a x-1)^3 \log (1-a x)-37}{6 a c^3 (a x-1)^3}","\frac{9}{a c^3 (1-a x)}-\frac{7}{2 a c^3 (1-a x)^2}+\frac{2}{3 a c^3 (1-a x)^3}+\frac{5 \log (1-a x)}{a c^3}+\frac{x}{c^3}",1,"(-37 + 81*a*x - 36*a^2*x^2 - 18*a^3*x^3 + 6*a^4*x^4 + 30*(-1 + a*x)^3*Log[1 - a*x])/(6*a*c^3*(-1 + a*x)^3)","A",1
395,1,71,87,0.1707804,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^4} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - c/(a*x))^4,x]","\frac{2 a^5 x^5-8 a^4 x^4-16 a^3 x^3+60 a^2 x^2-56 a x+12 (a x-1)^4 \log (1-a x)+17}{2 a c^4 (a x-1)^4}","\frac{14}{a c^4 (1-a x)}-\frac{8}{a c^4 (1-a x)^2}+\frac{3}{a c^4 (1-a x)^3}-\frac{1}{2 a c^4 (1-a x)^4}+\frac{6 \log (1-a x)}{a c^4}+\frac{x}{c^4}",1,"(17 - 56*a*x + 60*a^2*x^2 - 16*a^3*x^3 - 8*a^4*x^4 + 2*a^5*x^5 + 12*(-1 + a*x)^4*Log[1 - a*x])/(2*a*c^4*(-1 + a*x)^4)","A",1
396,1,175,103,0.2326634,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^4 \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^4,x]","-\frac{c^4 \left(-24 a^5 x^5-32 a^4 x^4+12 a^3 x^3+40 a^2 x^2+42 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)-15 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left(\frac{1}{a x}\right)+24 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+12 a x-8\right)}{24 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a+\frac{3}{x}\right)}{2 a^2}+\frac{c^4 x \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(3 a+\frac{1}{x}\right)}{3 a}-\frac{c^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}+\frac{3 c^4 \csc ^{-1}(a x)}{2 a}",1,"-1/24*(c^4*(-8 + 12*a*x + 40*a^2*x^2 + 12*a^3*x^3 - 32*a^4*x^4 - 24*a^5*x^5 + 42*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 15*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4*ArcSin[1/(a*x)] + 24*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/(a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4)","A",0
397,1,51,61,0.0822094,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^3 \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^3,x]","\frac{c^3 \left(\sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2+1\right)+3 a x \sin ^{-1}\left(\frac{1}{a x}\right)\right)}{2 a^2 x}","c^3 x \left(1-\frac{1}{a^2 x^2}\right)^{3/2}+\frac{3 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+\frac{3 c^3 \csc ^{-1}(a x)}{2 a}",1,"(c^3*(Sqrt[1 - 1/(a^2*x^2)]*(1 + 2*a^2*x^2) + 3*a*x*ArcSin[1/(a*x)]))/(2*a^2*x)","A",0
398,1,154,63,0.163217,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^2 \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^2,x]","-\frac{c^2 \left(-a^3 x^3+a^2 x^2+4 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)+a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left(\frac{1}{a x}\right)-a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+a x-1\right)}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{c^2 x \sqrt{1-\frac{1}{a^2 x^2}} \left(a-\frac{1}{x}\right)}{a}+\frac{c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}+\frac{c^2 \csc ^{-1}(a x)}{a}",1,"-((c^2*(-1 + a*x + a^2*x^2 - a^3*x^3 + 4*a^2*Sqrt[1 - 1/(a^2*x^2)]*x^2*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] + a^2*Sqrt[1 - 1/(a^2*x^2)]*x^2*ArcSin[1/(a*x)] - a^2*Sqrt[1 - 1/(a^2*x^2)]*x^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/(a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2))","B",0
399,1,73,49,0.118161,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right) \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x)),x]","\frac{c \left(a x \sqrt{1-\frac{1}{a^2 x^2}}+2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)-2 \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)-2 \sin ^{-1}\left(\frac{1}{a x}\right)\right)}{a}","c x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{2 c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}-\frac{c \csc ^{-1}(a x)}{a}",1,"(c*(a*Sqrt[1 - 1/(a^2*x^2)]*x - 2*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 2*ArcSin[1/(a*x)] + 2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/a","A",0
400,1,70,105,0.1371484,"\int \frac{e^{3 \coth ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - c/(a*x)),x]","\frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(3 a^2 x^2-26 a x+19\right)}{(a x-1)^2}+12 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{3 a c}","-\frac{8 \left(a+\frac{1}{x}\right)}{3 a^2 c \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}-\frac{4 \left(3 a+\frac{4}{x}\right)}{3 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c}+\frac{4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(19 - 26*a*x + 3*a^2*x^2))/(-1 + a*x)^2 + 12*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(3*a*c)","A",0
401,1,104,138,0.080654,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^2} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - c/(a*x))^2,x]","\frac{15 a^4 x^4-173 a^3 x^3+91 a^2 x^2+75 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+161 a x-118}{15 a^2 c^2 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2}","-\frac{16 \left(a+\frac{1}{x}\right)}{5 a^2 c^2 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}-\frac{4 \left(5 a+\frac{11}{x}\right)}{15 a^2 c^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}-\frac{75 a+\frac{103}{x}}{15 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2}+\frac{5 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c^2}",1,"(-118 + 161*a*x + 91*a^2*x^2 - 173*a^3*x^3 + 15*a^4*x^4 + 75*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(15*a^2*c^2*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2)","A",1
402,1,112,165,0.0914779,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^3} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - c/(a*x))^3,x]","\frac{7 a^5 x^5-109 a^4 x^4+145 a^3 x^3+39 a^2 x^2+42 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)-156 a x+66}{7 a^2 c^3 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^3}","-\frac{32 \left(a+\frac{1}{x}\right)}{7 a^2 c^3 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}-\frac{2 \left(7 a+\frac{13}{x}\right)}{7 a^2 c^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}-\frac{42 a+\frac{59}{x}}{7 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3}-\frac{16}{7 a^2 c^3 x \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}+\frac{6 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c^3}",1,"(66 - 156*a*x + 39*a^2*x^2 + 145*a^3*x^3 - 109*a^4*x^4 + 7*a^5*x^5 + 42*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(7*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^3)","A",1
403,1,120,204,0.1035494,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^4} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - c/(a*x))^4,x]","\frac{315 a^6 x^6-6224 a^5 x^5+13241 a^4 x^4-5567 a^3 x^3-10232 a^2 x^2+2205 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+11651 a x-3464}{315 a^2 c^4 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^4}","\frac{16 \left(9 a-\frac{5}{x}\right)}{63 a^2 c^4 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}-\frac{64 \left(a+\frac{1}{x}\right)}{9 a^2 c^4 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}-\frac{8 \left(21 a+\frac{41}{x}\right)}{105 a^2 c^4 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}-\frac{735 a+\frac{1417}{x}}{315 a^2 c^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}-\frac{2205 a+\frac{3149}{x}}{315 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^4}+\frac{7 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c^4}",1,"(-3464 + 11651*a*x - 10232*a^2*x^2 - 5567*a^3*x^3 + 13241*a^4*x^4 - 6224*a^5*x^5 + 315*a^6*x^6 + 2205*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(315*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^4)","A",1
404,1,66,64,0.2991181,"\int e^{4 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^5 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - c/(a*x))^5,x]","\frac{c^5}{4 a^5 x^4}-\frac{c^5}{3 a^4 x^3}-\frac{c^5}{a^3 x^2}+\frac{2 c^5}{a^2 x}-\frac{c^5 \log (a x)}{a}+c^5 x","\frac{c^5}{4 a^5 x^4}-\frac{c^5}{3 a^4 x^3}-\frac{c^5}{a^3 x^2}+\frac{2 c^5}{a^2 x}-\frac{c^5 \log (x)}{a}+c^5 x",1,"c^5/(4*a^5*x^4) - c^5/(3*a^4*x^3) - c^5/(a^3*x^2) + (2*c^5)/(a^2*x) + c^5*x - (c^5*Log[a*x])/a","A",0
405,1,30,30,0.2010218,"\int e^{4 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^4 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - c/(a*x))^4,x]","-\frac{c^4}{3 a^4 x^3}+\frac{2 c^4}{a^2 x}+c^4 x","-\frac{c^4}{3 a^4 x^3}+\frac{2 c^4}{a^2 x}+c^4 x",1,"-1/3*c^4/(a^4*x^3) + (2*c^4)/(a^2*x) + c^4*x","A",1
406,1,40,38,0.1510414,"\int e^{4 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^3 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - c/(a*x))^3,x]","\frac{c^3}{2 a^3 x^2}+\frac{c^3}{a^2 x}+\frac{c^3 \log (a x)}{a}+c^3 x","\frac{c^3}{2 a^3 x^2}+\frac{c^3}{a^2 x}+\frac{c^3 \log (x)}{a}+c^3 x",1,"c^3/(2*a^3*x^2) + c^3/(a^2*x) + c^3*x + (c^3*Log[a*x])/a","A",0
407,1,29,27,0.1138158,"\int e^{4 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^2 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - c/(a*x))^2,x]","-\frac{c^2}{a^2 x}+\frac{2 c^2 \log (a x)}{a}+c^2 x","-\frac{c^2}{a^2 x}+\frac{2 c^2 \log (x)}{a}+c^2 x",1,"-(c^2/(a^2*x)) + c^2*x + (2*c^2*Log[a*x])/a","A",0
408,1,25,25,0.0422241,"\int e^{4 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right) \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - c/(a*x)),x]","-\frac{c \log (x)}{a}+\frac{4 c \log (1-a x)}{a}+c x","-\frac{c \log (x)}{a}+\frac{4 c \log (1-a x)}{a}+c x",1,"c*x - (c*Log[x])/a + (4*c*Log[1 - a*x])/a","A",1
409,1,51,53,0.0348714,"\int \frac{e^{4 \coth ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - c/(a*x)),x]","-\frac{a \left(-\frac{8}{a^2 (1-a x)}+\frac{2}{a^2 (1-a x)^2}-\frac{5 \log (1-a x)}{a^2}-\frac{x}{a}\right)}{c}","\frac{8}{a c (1-a x)}-\frac{2}{a c (1-a x)^2}+\frac{5 \log (1-a x)}{a c}+\frac{x}{c}",1,"-((a*(-(x/a) + 2/(a^2*(1 - a*x)^2) - 8/(a^2*(1 - a*x)) - (5*Log[1 - a*x])/a^2))/c)","A",1
410,1,63,71,0.119669,"\int \frac{e^{4 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^2} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - c/(a*x))^2,x]","\frac{3 a^4 x^4-9 a^3 x^3-30 a^2 x^2+57 a x+18 (a x-1)^3 \log (1-a x)-25}{3 a c^2 (a x-1)^3}","\frac{13}{a c^2 (1-a x)}-\frac{6}{a c^2 (1-a x)^2}+\frac{4}{3 a c^2 (1-a x)^3}+\frac{6 \log (1-a x)}{a c^2}+\frac{x}{c^2}",1,"(-25 + 57*a*x - 30*a^2*x^2 - 9*a^3*x^3 + 3*a^4*x^4 + 18*(-1 + a*x)^3*Log[1 - a*x])/(3*a*c^2*(-1 + a*x)^3)","A",1
411,1,71,89,0.1566618,"\int \frac{e^{4 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^3} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - c/(a*x))^3,x]","\frac{6 a^5 x^5-24 a^4 x^4-78 a^3 x^3+243 a^2 x^2-218 a x+42 (a x-1)^4 \log (1-a x)+65}{6 a c^3 (a x-1)^4}","\frac{19}{a c^3 (1-a x)}-\frac{25}{2 a c^3 (1-a x)^2}+\frac{16}{3 a c^3 (1-a x)^3}-\frac{1}{a c^3 (1-a x)^4}+\frac{7 \log (1-a x)}{a c^3}+\frac{x}{c^3}",1,"(65 - 218*a*x + 243*a^2*x^2 - 78*a^3*x^3 - 24*a^4*x^4 + 6*a^5*x^5 + 42*(-1 + a*x)^4*Log[1 - a*x])/(6*a*c^3*(-1 + a*x)^4)","A",1
412,1,79,105,0.1878698,"\int \frac{e^{4 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^4} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - c/(a*x))^4,x]","\frac{15 a^6 x^6-75 a^5 x^5-240 a^4 x^4+1080 a^3 x^3-1480 a^2 x^2+890 a x+120 (a x-1)^5 \log (1-a x)-202}{15 a c^4 (a x-1)^5}","\frac{26}{a c^4 (1-a x)}-\frac{22}{a c^4 (1-a x)^2}+\frac{41}{3 a c^4 (1-a x)^3}-\frac{5}{a c^4 (1-a x)^4}+\frac{4}{5 a c^4 (1-a x)^5}+\frac{8 \log (1-a x)}{a c^4}+\frac{x}{c^4}",1,"(-202 + 890*a*x - 1480*a^2*x^2 + 1080*a^3*x^3 - 240*a^4*x^4 - 75*a^5*x^5 + 15*a^6*x^6 + 120*(-1 + a*x)^5*Log[1 - a*x])/(15*a*c^4*(-1 + a*x)^5)","A",1
413,1,175,135,0.1550586,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^4 \, dx","Integrate[(c - c/(a*x))^4/E^ArcCoth[a*x],x]","\frac{c^4 \left(6 a^5 x^5-64 a^4 x^4+9 a^3 x^3+62 a^2 x^2+90 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)-30 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left(\frac{1}{a x}\right)-30 a^4 x^4 \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)-15 a x+2\right)}{6 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}","c^4 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{32 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}+\frac{5 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}-\frac{5 c^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}-\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}-\frac{25 c^4 \csc ^{-1}(a x)}{2 a}",1,"(c^4*(2 - 15*a*x + 62*a^2*x^2 + 9*a^3*x^3 - 64*a^4*x^4 + 6*a^5*x^5 + 90*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 30*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4*ArcSin[1/(a*x)] - 30*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/(6*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4)","A",0
414,1,167,106,0.1322512,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^3 \, dx","Integrate[(c - c/(a*x))^3/E^ArcCoth[a*x],x]","\frac{c^3 \left(2 a^4 x^4-8 a^3 x^3-a^2 x^2+10 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)-8 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left(\frac{1}{a x}\right)-8 a^3 x^3 \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+8 a x-1\right)}{2 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}","c^3 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{4 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}-\frac{4 c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}-\frac{13 c^3 \csc ^{-1}(a x)}{2 a}",1,"(c^3*(-1 + 8*a*x - a^2*x^2 - 8*a^3*x^3 + 2*a^4*x^4 + 10*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 8*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3*ArcSin[1/(a*x)] - 8*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/(2*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3)","A",0
415,1,55,77,0.2179194,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^2 \, dx","Integrate[(c - c/(a*x))^2/E^ArcCoth[a*x],x]","\frac{c^2 \left(\sqrt{1-\frac{1}{a^2 x^2}} (a x-1)-3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)-3 \sin ^{-1}\left(\frac{1}{a x}\right)\right)}{a}","c^2 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{3 c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}-\frac{3 c^2 \csc ^{-1}(a x)}{a}",1,"(c^2*(Sqrt[1 - 1/(a^2*x^2)]*(-1 + a*x) - 3*ArcSin[1/(a*x)] - 3*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/a","A",0
416,1,73,49,0.1100024,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right) \, dx","Integrate[(c - c/(a*x))/E^ArcCoth[a*x],x]","\frac{c \left(a x \sqrt{1-\frac{1}{a^2 x^2}}-2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)-2 \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)-2 \sin ^{-1}\left(\frac{1}{a x}\right)\right)}{a}","c x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{2 c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}-\frac{c \csc ^{-1}(a x)}{a}",1,"(c*(a*Sqrt[1 - 1/(a^2*x^2)]*x - 2*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 2*ArcSin[1/(a*x)] - 2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]))/a","A",0
417,1,19,19,0.072573,"\int \frac{e^{-\coth ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - c/(a*x))),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x)/c","A",1
418,1,69,73,0.0408882,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^2} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - c/(a*x))^2),x]","\frac{a^2 x^2+a x \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)-a x-2}{a^2 c^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2}-\frac{a x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \left(a-\frac{1}{x}\right)}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c^2}",1,"(-2 - a*x + a^2*x^2 + a*Sqrt[1 - 1/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a^2*c^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
419,1,94,105,0.0709793,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^3} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - c/(a*x))^3),x]","\frac{3 a^3 x^3-11 a^2 x^2+6 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1) \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)-4 a x+10}{3 a^2 c^3 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)}","-\frac{2 \left(a+\frac{1}{x}\right)}{3 a^2 c^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}-\frac{6 a+\frac{7}{x}}{3 a^2 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3}+\frac{2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c^3}",1,"(10 - 4*a*x - 11*a^2*x^2 + 3*a^3*x^3 + 6*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(3*a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x))","A",1
420,1,104,138,0.0878039,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^4} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - c/(a*x))^4),x]","\frac{5 a^4 x^4-34 a^3 x^3+18 a^2 x^2+15 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+33 a x-24}{5 a^2 c^4 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2}","-\frac{4 \left(a+\frac{1}{x}\right)}{5 a^2 c^4 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}-\frac{5 a+\frac{7}{x}}{5 a^2 c^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}-\frac{15 a+\frac{19}{x}}{5 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^4}+\frac{3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c^4}",1,"(-24 + 33*a*x + 18*a^2*x^2 - 34*a^3*x^3 + 5*a^4*x^4 + 15*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(5*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2)","A",1
421,1,67,65,0.1935893,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^4 \, dx","Integrate[(c - c/(a*x))^4/E^(2*ArcCoth[a*x]),x]","\frac{c^4}{3 a^4 x^3}-\frac{3 c^4}{a^3 x^2}+\frac{16 c^4}{a^2 x}+\frac{26 c^4 \log (a x)}{a}-\frac{32 c^4 \log (a x+1)}{a}+c^4 x","\frac{c^4}{3 a^4 x^3}-\frac{3 c^4}{a^3 x^2}+\frac{16 c^4}{a^2 x}+\frac{26 c^4 \log (x)}{a}-\frac{32 c^4 \log (a x+1)}{a}+c^4 x",1,"c^4/(3*a^4*x^3) - (3*c^4)/(a^3*x^2) + (16*c^4)/(a^2*x) + c^4*x + (26*c^4*Log[a*x])/a - (32*c^4*Log[1 + a*x])/a","A",0
422,1,56,54,0.1480315,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^3 \, dx","Integrate[(c - c/(a*x))^3/E^(2*ArcCoth[a*x]),x]","-\frac{c^3}{2 a^3 x^2}+\frac{5 c^3}{a^2 x}+\frac{11 c^3 \log (a x)}{a}-\frac{16 c^3 \log (a x+1)}{a}+c^3 x","-\frac{c^3}{2 a^3 x^2}+\frac{5 c^3}{a^2 x}+\frac{11 c^3 \log (x)}{a}-\frac{16 c^3 \log (a x+1)}{a}+c^3 x",1,"-1/2*c^3/(a^3*x^2) + (5*c^3)/(a^2*x) + c^3*x + (11*c^3*Log[a*x])/a - (16*c^3*Log[1 + a*x])/a","A",0
423,1,42,40,0.1130436,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^2 \, dx","Integrate[(c - c/(a*x))^2/E^(2*ArcCoth[a*x]),x]","\frac{c^2}{a^2 x}+\frac{4 c^2 \log (a x)}{a}-\frac{8 c^2 \log (a x+1)}{a}+c^2 x","\frac{c^2}{a^2 x}+\frac{4 c^2 \log (x)}{a}-\frac{8 c^2 \log (a x+1)}{a}+c^2 x",1,"c^2/(a^2*x) + c^2*x + (4*c^2*Log[a*x])/a - (8*c^2*Log[1 + a*x])/a","A",0
424,1,23,23,0.0437004,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right) \, dx","Integrate[(c - c/(a*x))/E^(2*ArcCoth[a*x]),x]","\frac{c \log (x)}{a}-\frac{4 c \log (a x+1)}{a}+c x","\frac{c \log (x)}{a}-\frac{4 c \log (a x+1)}{a}+c x",1,"c*x + (c*Log[x])/a - (4*c*Log[1 + a*x])/a","A",1
425,1,22,20,0.019085,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))),x]","\frac{a \left(\frac{x}{a}-\frac{\log (a x+1)}{a^2}\right)}{c}","\frac{x}{c}-\frac{\log (a x+1)}{a c}",1,"(a*(x/a - Log[1 + a*x]/a^2))/c","A",1
426,1,39,18,0.0914148,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^2} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^2),x]","\frac{\log (1-a x)}{2 a c^2}-\frac{\log (a x+1)}{2 a c^2}+\frac{x}{c^2}","\frac{x}{c^2}-\frac{\tanh ^{-1}(a x)}{a c^2}",1,"x/c^2 + Log[1 - a*x]/(2*a*c^2) - Log[1 + a*x]/(2*a*c^2)","B",0
427,1,56,57,0.1251231,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^3} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^3),x]","-\frac{1}{2 a c^3 (a x-1)}+\frac{5 \log (1-a x)}{4 a c^3}-\frac{\log (a x+1)}{4 a c^3}+\frac{x}{c^3}","\frac{1}{2 a c^3 (1-a x)}+\frac{5 \log (1-a x)}{4 a c^3}-\frac{\log (a x+1)}{4 a c^3}+\frac{x}{c^3}",1,"x/c^3 - 1/(2*a*c^3*(-1 + a*x)) + (5*Log[1 - a*x])/(4*a*c^3) - Log[1 + a*x]/(4*a*c^3)","A",1
428,1,73,75,0.1706255,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^4} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^4),x]","-\frac{7}{4 a c^4 (a x-1)}-\frac{1}{4 a c^4 (a x-1)^2}+\frac{17 \log (1-a x)}{8 a c^4}-\frac{\log (a x+1)}{8 a c^4}+\frac{x}{c^4}","\frac{7}{4 a c^4 (1-a x)}-\frac{1}{4 a c^4 (1-a x)^2}+\frac{17 \log (1-a x)}{8 a c^4}-\frac{\log (a x+1)}{8 a c^4}+\frac{x}{c^4}",1,"x/c^4 - 1/(4*a*c^4*(-1 + a*x)^2) - 7/(4*a*c^4*(-1 + a*x)) + (17*Log[1 - a*x])/(8*a*c^4) - Log[1 + a*x]/(8*a*c^4)","A",1
429,1,567,164,1.1915483,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^4 \, dx","Integrate[(c - c/(a*x))^4/E^(3*ArcCoth[a*x]),x]","\frac{c^4 \left(2772 \sqrt{2} a^3 x^3 (a x+1) (a x-1)^3 \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)+1980 \sqrt{2} a^2 x^2 (a x+1) (a x-1)^4 \, _2F_1\left(\frac{3}{2},\frac{7}{2};\frac{9}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)+35 \left(396 a^8 x^8 \sqrt{\frac{1}{a x}+1}-50160 a^7 x^7 \sqrt{\frac{1}{a x}+1}+66726 a^7 x^7 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)-1980 a^7 x^7 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left(\frac{1}{a x}\right)+29403 a^6 x^6 \sqrt{\frac{1}{a x}+1}+66726 a^6 x^6 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)-1980 a^6 x^6 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left(\frac{1}{a x}\right)+26268 a^5 x^5 \sqrt{\frac{1}{a x}+1}-7425 a^4 x^4 \sqrt{\frac{1}{a x}+1}+1716 a^3 x^3 \sqrt{\frac{1}{a x}+1}-198 a^2 x^2 \sqrt{\frac{1}{a x}+1}-2772 a^7 x^7 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+44 \sqrt{2} a x (a x-1)^5 (a x+1) \, _2F_1\left(\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)+36 \sqrt{2} (a x-1)^6 (a x+1) \, _2F_1\left(\frac{3}{2},\frac{11}{2};\frac{13}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)\right)\right)}{13860 a^7 x^6 \sqrt{1-\frac{1}{a x}} (a x+1)}","\frac{64 c^4 \left(a-\frac{1}{x}\right)}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^4 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{68 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}-\frac{7 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}-\frac{7 c^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}+\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}+\frac{91 c^4 \csc ^{-1}(a x)}{2 a}",1,"(c^4*(2772*Sqrt[2]*a^3*x^3*(-1 + a*x)^3*(1 + a*x)*Hypergeometric2F1[3/2, 5/2, 7/2, (1 - 1/(a*x))/2] + 1980*Sqrt[2]*a^2*x^2*(-1 + a*x)^4*(1 + a*x)*Hypergeometric2F1[3/2, 7/2, 9/2, (1 - 1/(a*x))/2] + 35*(-198*a^2*Sqrt[1 + 1/(a*x)]*x^2 + 1716*a^3*Sqrt[1 + 1/(a*x)]*x^3 - 7425*a^4*Sqrt[1 + 1/(a*x)]*x^4 + 26268*a^5*Sqrt[1 + 1/(a*x)]*x^5 + 29403*a^6*Sqrt[1 + 1/(a*x)]*x^6 - 50160*a^7*Sqrt[1 + 1/(a*x)]*x^7 + 396*a^8*Sqrt[1 + 1/(a*x)]*x^8 + 66726*a^6*Sqrt[1 - 1/(a*x)]*x^6*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] + 66726*a^7*Sqrt[1 - 1/(a*x)]*x^7*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 1980*a^6*Sqrt[1 - 1/(a*x)]*x^6*ArcSin[1/(a*x)] - 1980*a^7*Sqrt[1 - 1/(a*x)]*x^7*ArcSin[1/(a*x)] - 2772*a^7*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[1 + 1/(a*x)]*x^7*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]] + 44*Sqrt[2]*a*x*(-1 + a*x)^5*(1 + a*x)*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] + 36*Sqrt[2]*(-1 + a*x)^6*(1 + a*x)*Hypergeometric2F1[3/2, 11/2, 13/2, (1 - 1/(a*x))/2])))/(13860*a^7*Sqrt[1 - 1/(a*x)]*x^6*(1 + a*x))","C",0
430,1,663,135,0.459366,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^3 \, dx","Integrate[(c - c/(a*x))^3/E^(3*ArcCoth[a*x]),x]","\frac{c^3 \left(630 a^7 x^7 \sqrt{\frac{1}{a x}+1}+70 \sqrt{2} a^6 x^6 \, _2F_1\left(\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)-32340 a^6 x^6 \sqrt{\frac{1}{a x}+1}+44730 a^6 x^6 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)-2520 a^6 x^6 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left(\frac{1}{a x}\right)-280 \sqrt{2} a^5 x^5 \, _2F_1\left(\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)+17955 a^5 x^5 \sqrt{\frac{1}{a x}+1}+44730 a^5 x^5 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)-2520 a^5 x^5 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left(\frac{1}{a x}\right)+350 \sqrt{2} a^4 x^4 \, _2F_1\left(\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)+16800 a^4 x^4 \sqrt{\frac{1}{a x}+1}-3465 a^3 x^3 \sqrt{\frac{1}{a x}+1}+126 \sqrt{2} a^2 x^2 (a x-1)^3 (a x+1) \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)-350 \sqrt{2} a^2 x^2 \, _2F_1\left(\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)+420 a^2 x^2 \sqrt{\frac{1}{a x}+1}-3780 a^6 x^6 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+90 \sqrt{2} a x (a x-1)^4 (a x+1) \, _2F_1\left(\frac{3}{2},\frac{7}{2};\frac{9}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)+280 \sqrt{2} a x \, _2F_1\left(\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)-70 \sqrt{2} \, _2F_1\left(\frac{3}{2},\frac{9}{2};\frac{11}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)\right)}{630 a^6 x^5 \sqrt{1-\frac{1}{a x}} (a x+1)}","\frac{32 c^3 \left(a-\frac{1}{x}\right)}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^3 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{6 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}-\frac{6 c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}+\frac{33 c^3 \csc ^{-1}(a x)}{2 a}",1,"(c^3*(420*a^2*Sqrt[1 + 1/(a*x)]*x^2 - 3465*a^3*Sqrt[1 + 1/(a*x)]*x^3 + 16800*a^4*Sqrt[1 + 1/(a*x)]*x^4 + 17955*a^5*Sqrt[1 + 1/(a*x)]*x^5 - 32340*a^6*Sqrt[1 + 1/(a*x)]*x^6 + 630*a^7*Sqrt[1 + 1/(a*x)]*x^7 + 44730*a^5*Sqrt[1 - 1/(a*x)]*x^5*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] + 44730*a^6*Sqrt[1 - 1/(a*x)]*x^6*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 2520*a^5*Sqrt[1 - 1/(a*x)]*x^5*ArcSin[1/(a*x)] - 2520*a^6*Sqrt[1 - 1/(a*x)]*x^6*ArcSin[1/(a*x)] - 3780*a^6*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[1 + 1/(a*x)]*x^6*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]] + 126*Sqrt[2]*a^2*x^2*(-1 + a*x)^3*(1 + a*x)*Hypergeometric2F1[3/2, 5/2, 7/2, (1 - 1/(a*x))/2] + 90*Sqrt[2]*a*x*(-1 + a*x)^4*(1 + a*x)*Hypergeometric2F1[3/2, 7/2, 9/2, (1 - 1/(a*x))/2] - 70*Sqrt[2]*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] + 280*Sqrt[2]*a*x*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] - 350*Sqrt[2]*a^2*x^2*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] + 350*Sqrt[2]*a^4*x^4*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] - 280*Sqrt[2]*a^5*x^5*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2] + 70*Sqrt[2]*a^6*x^6*Hypergeometric2F1[3/2, 9/2, 11/2, (1 - 1/(a*x))/2]))/(630*a^6*Sqrt[1 - 1/(a*x)]*x^5*(1 + a*x))","C",0
431,1,424,105,0.4235475,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^2 \, dx","Integrate[(c - c/(a*x))^2/E^(3*ArcCoth[a*x]),x]","\frac{c^2 \left(35 a^6 x^6 \sqrt{\frac{1}{a x}+1}-595 a^5 x^5 \sqrt{\frac{1}{a x}+1}+910 a^5 x^5 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)-105 a^5 x^5 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left(\frac{1}{a x}\right)+280 a^4 x^4 \sqrt{\frac{1}{a x}+1}+910 a^4 x^4 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)-105 a^4 x^4 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left(\frac{1}{a x}\right)+315 a^3 x^3 \sqrt{\frac{1}{a x}+1}-35 a^2 x^2 \sqrt{\frac{1}{a x}+1}-175 a^5 x^5 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+7 \sqrt{2} a x (a x-1)^3 (a x+1) \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)+5 \sqrt{2} (a x-1)^4 (a x+1) \, _2F_1\left(\frac{3}{2},\frac{7}{2};\frac{9}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)\right)}{35 a^5 x^4 \sqrt{1-\frac{1}{a x}} (a x+1)}","\frac{16 c^2 \left(a-\frac{1}{x}\right)}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^2 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{5 c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}+\frac{5 c^2 \csc ^{-1}(a x)}{a}",1,"(c^2*(-35*a^2*Sqrt[1 + 1/(a*x)]*x^2 + 315*a^3*Sqrt[1 + 1/(a*x)]*x^3 + 280*a^4*Sqrt[1 + 1/(a*x)]*x^4 - 595*a^5*Sqrt[1 + 1/(a*x)]*x^5 + 35*a^6*Sqrt[1 + 1/(a*x)]*x^6 + 910*a^4*Sqrt[1 - 1/(a*x)]*x^4*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] + 910*a^5*Sqrt[1 - 1/(a*x)]*x^5*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 105*a^4*Sqrt[1 - 1/(a*x)]*x^4*ArcSin[1/(a*x)] - 105*a^5*Sqrt[1 - 1/(a*x)]*x^5*ArcSin[1/(a*x)] - 175*a^5*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[1 + 1/(a*x)]*x^5*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]] + 7*Sqrt[2]*a*x*(-1 + a*x)^3*(1 + a*x)*Hypergeometric2F1[3/2, 5/2, 7/2, (1 - 1/(a*x))/2] + 5*Sqrt[2]*(-1 + a*x)^4*(1 + a*x)*Hypergeometric2F1[3/2, 7/2, 9/2, (1 - 1/(a*x))/2]))/(35*a^5*Sqrt[1 - 1/(a*x)]*x^4*(1 + a*x))","C",0
432,1,234,75,0.5444939,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right) \, dx","Integrate[(c - c/(a*x))/E^(3*ArcCoth[a*x]),x]","\frac{5 a^2 c x^2 \left((a x+1) \left(\sqrt{\frac{1}{a x}+1} \left(a^2 x^2-3 a x+2\right)+6 a x \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left(\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right)-2 a x \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left(\frac{1}{a x}\right)\right)-4 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)\right)+\sqrt{2} c (a x+1) (a x-1)^3 \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{1}{2} \left(1-\frac{1}{a x}\right)\right)}{5 a^4 x^3 \sqrt{1-\frac{1}{a x}} (a x+1)}","\frac{8 c \left(a-\frac{1}{x}\right)}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{4 c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a}+\frac{c \csc ^{-1}(a x)}{a}",1,"(5*a^2*c*x^2*((1 + a*x)*(Sqrt[1 + 1/(a*x)]*(2 - 3*a*x + a^2*x^2) + 6*a*Sqrt[1 - 1/(a*x)]*x*ArcSin[Sqrt[1 - 1/(a*x)]/Sqrt[2]] - 2*a*Sqrt[1 - 1/(a*x)]*x*ArcSin[1/(a*x)]) - 4*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[1 + 1/(a*x)]*x^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]]) + Sqrt[2]*c*(-1 + a*x)^3*(1 + a*x)*Hypergeometric2F1[3/2, 5/2, 7/2, (1 - 1/(a*x))/2])/(5*a^4*Sqrt[1 - 1/(a*x)]*x^3*(1 + a*x))","C",0
433,1,61,72,0.1057898,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))),x]","\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} (a x+3)-2 (a x+1) \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a (a c x+c)}","\frac{2 \left(a-\frac{1}{x}\right)}{a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c}-\frac{2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c}",1,"(a*Sqrt[1 - 1/(a^2*x^2)]*x*(3 + a*x) - 2*(1 + a*x)*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*(c + a*c*x))","A",0
434,1,69,74,0.0449898,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^2} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^2),x]","\frac{a^2 x^2-a x \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+a x-2}{a^2 c^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{x \left(a-\frac{1}{x}\right)}{a c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2}-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c^2}",1,"(-2 + a*x + a^2*x^2 - a*Sqrt[1 - 1/(a^2*x^2)]*x*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(a^2*c^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
435,1,33,45,0.0246021,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^3} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^3),x]","\frac{a^2 x^2-2}{a^2 c^3 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2}{a^2 c^3 x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(-2 + a^2*x^2)/(a^2*c^3*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
436,1,94,111,0.0689185,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^4} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^4),x]","\frac{3 a^3 x^3-7 a^2 x^2+3 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1) \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)-5 a x+8}{3 a^2 c^4 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)}","-\frac{x \left(4 a+\frac{3}{x}\right)}{3 a c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{8 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^4}-\frac{a x}{3 c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left(a-\frac{1}{x}\right)}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c^4}",1,"(8 - 5*a*x - 7*a^2*x^2 + 3*a^3*x^3 + 3*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(3*a^2*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x))","A",1
437,1,104,138,0.0925649,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^5} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^5),x]","\frac{15 a^4 x^4-76 a^3 x^3+32 a^2 x^2+30 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)+82 a x-56}{15 a^2 c^5 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2}","-\frac{2 \left(a+\frac{1}{x}\right)}{5 a^2 c^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}-\frac{10 a+\frac{13}{x}}{15 a^2 c^5 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}-\frac{30 a+\frac{41}{x}}{15 a^2 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^5}+\frac{2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a^2 x^2}}\right)}{a c^5}",1,"(-56 + 82*a*x + 32*a^2*x^2 - 76*a^3*x^3 + 15*a^4*x^4 + 30*a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/(15*a^2*c^5*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*x)^2)","A",1
438,1,109,235,0.1246999,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{9/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a*x))^(9/2),x]","\frac{c^4 \sqrt{c-\frac{c}{a x}} \left(\sqrt{\frac{1}{a x}+1} \left(105 a^4 x^4+292 a^3 x^3-356 a^2 x^2+162 a x-30\right)-735 a^3 x^3 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)\right)}{105 a^4 x^3 \sqrt{1-\frac{1}{a x}}}","-\frac{7 c^{9/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{a}+\frac{173 c^5 \sqrt{1-\frac{1}{a^2 x^2}}}{105 a \sqrt{c-\frac{c}{a x}}}+\frac{227 c^4 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}}{105 a}+\frac{59 c^3 \sqrt{1-\frac{1}{a^2 x^2}} \left(c-\frac{c}{a x}\right)^{3/2}}{35 a}+\frac{9 c^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(c-\frac{c}{a x}\right)^{5/2}}{7 a}+c x \sqrt{1-\frac{1}{a^2 x^2}} \left(c-\frac{c}{a x}\right)^{7/2}",1,"(c^4*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(-30 + 162*a*x - 356*a^2*x^2 + 292*a^3*x^3 + 105*a^4*x^4) - 735*a^3*x^3*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(105*a^4*Sqrt[1 - 1/(a*x)]*x^3)","A",0
439,1,101,196,0.0922651,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{7/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a*x))^(7/2),x]","\frac{c^3 \sqrt{c-\frac{c}{a x}} \left(\sqrt{\frac{1}{a x}+1} \left(15 a^3 x^3+56 a^2 x^2-28 a x+6\right)-75 a^2 x^2 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)\right)}{15 a^3 x^2 \sqrt{1-\frac{1}{a x}}}","-\frac{5 c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{a}+\frac{49 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{15 a \sqrt{c-\frac{c}{a x}}}+\frac{31 c^3 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}}{15 a}+\frac{7 c^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(c-\frac{c}{a x}\right)^{3/2}}{5 a}+c x \sqrt{1-\frac{1}{a^2 x^2}} \left(c-\frac{c}{a x}\right)^{5/2}",1,"(c^3*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(6 - 28*a*x + 56*a^2*x^2 + 15*a^3*x^3) - 75*a^2*x^2*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(15*a^3*Sqrt[1 - 1/(a*x)]*x^2)","A",0
440,1,89,157,0.0870391,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{5/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a*x))^(5/2),x]","\frac{c^2 \sqrt{c-\frac{c}{a x}} \left(\sqrt{\frac{1}{a x}+1} \left(3 a^2 x^2+10 a x-2\right)-9 a x \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)\right)}{3 a^2 x \sqrt{1-\frac{1}{a x}}}","-\frac{3 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{a}+\frac{c^4 x \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{\left(c-\frac{c}{a x}\right)^{3/2}}-\frac{2 c^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{3 a \left(c-\frac{c}{a x}\right)^{3/2}}+\frac{3 c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}",1,"(c^2*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(-2 + 10*a*x + 3*a^2*x^2) - 9*a*x*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(3*a^2*Sqrt[1 - 1/(a*x)]*x)","A",0
441,1,70,117,0.0582815,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{3/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a*x))^(3/2),x]","\frac{c \sqrt{c-\frac{c}{a x}} \left(\sqrt{\frac{1}{a x}+1} (a x+2)-\tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)\right)}{a \sqrt{1-\frac{1}{a x}}}","-\frac{c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{a}+\frac{c^3 x \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{\left(c-\frac{c}{a x}\right)^{3/2}}+\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}",1,"(c*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(2 + a*x) - ArcTanh[Sqrt[1 + 1/(a*x)]]))/(a*Sqrt[1 - 1/(a*x)])","A",0
442,1,66,78,0.0489048,"\int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx","Integrate[E^ArcCoth[a*x]*Sqrt[c - c/(a*x)],x]","\frac{\sqrt{c-\frac{c}{a x}} \left(a x+\sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)+1\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{c x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{a}",1,"(Sqrt[c - c/(a*x)]*(1 + a*x + Sqrt[1 + 1/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",0
443,1,93,152,0.0719992,"\int \frac{e^{\coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx","Integrate[E^ArcCoth[a*x]/Sqrt[c - c/(a*x)],x]","\frac{\sqrt{1-\frac{1}{a x}} \left(a x \sqrt{\frac{1}{a x}+1}+3 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)\right)}{a \sqrt{c-\frac{c}{a x}}}","\frac{x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\sqrt{c-\frac{c}{a x}}}+\frac{3 \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \sqrt{c-\frac{c}{a x}}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{a \sqrt{c-\frac{c}{a x}}}",1,"(Sqrt[1 - 1/(a*x)]*(a*Sqrt[1 + 1/(a*x)]*x + 3*ArcTanh[Sqrt[1 + 1/(a*x)]] - 2*Sqrt[2]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(a*Sqrt[c - c/(a*x)])","A",1
444,1,122,215,0.1025278,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{3/2}} \, dx","Integrate[E^ArcCoth[a*x]/(c - c/(a*x))^(3/2),x]","\frac{\sqrt{1-\frac{1}{a x}} \left(2 a x \sqrt{\frac{1}{a x}+1} (a x-2)+10 (a x-1) \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)-7 \sqrt{2} (a x-1) \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)\right)}{2 a c (a x-1) \sqrt{c-\frac{c}{a x}}}","\frac{a x \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{3/2}}{\left(a-\frac{1}{x}\right) \left(c-\frac{c}{a x}\right)^{3/2}}-\frac{2 \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{3/2}}{\left(a-\frac{1}{x}\right) \left(c-\frac{c}{a x}\right)^{3/2}}+\frac{5 \left(1-\frac{1}{a x}\right)^{3/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(c-\frac{c}{a x}\right)^{3/2}}-\frac{7 \left(1-\frac{1}{a x}\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{\sqrt{2} a \left(c-\frac{c}{a x}\right)^{3/2}}",1,"(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(-2 + a*x) + 10*(-1 + a*x)*ArcTanh[Sqrt[1 + 1/(a*x)]] - 7*Sqrt[2]*(-1 + a*x)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(2*a*c*Sqrt[c - c/(a*x)]*(-1 + a*x))","A",1
445,1,135,277,0.128427,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{5/2}} \, dx","Integrate[E^ArcCoth[a*x]/(c - c/(a*x))^(5/2),x]","\frac{\sqrt{1-\frac{1}{a x}} \left(2 a x \sqrt{\frac{1}{a x}+1} \left(8 a^2 x^2-35 a x+23\right)+112 (a x-1)^2 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)-79 \sqrt{2} (a x-1)^2 \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)\right)}{16 a c^2 (a x-1)^2 \sqrt{c-\frac{c}{a x}}}","\frac{a^2 x \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{5/2}}{\left(a-\frac{1}{x}\right)^2 \left(c-\frac{c}{a x}\right)^{5/2}}-\frac{23 \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{5/2}}{8 \left(a-\frac{1}{x}\right) \left(c-\frac{c}{a x}\right)^{5/2}}-\frac{3 a \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{5/2}}{2 \left(a-\frac{1}{x}\right)^2 \left(c-\frac{c}{a x}\right)^{5/2}}+\frac{7 \left(1-\frac{1}{a x}\right)^{5/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(c-\frac{c}{a x}\right)^{5/2}}-\frac{79 \left(1-\frac{1}{a x}\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{8 \sqrt{2} a \left(c-\frac{c}{a x}\right)^{5/2}}",1,"(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(23 - 35*a*x + 8*a^2*x^2) + 112*(-1 + a*x)^2*ArcTanh[Sqrt[1 + 1/(a*x)]] - 79*Sqrt[2]*(-1 + a*x)^2*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(16*a*c^2*Sqrt[c - c/(a*x)]*(-1 + a*x)^2)","A",1
446,1,91,143,0.1667979,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{9/2} \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^(9/2),x]","\frac{c^4 \left(21 a^4 x^4+92 a^3 x^3+4 a^2 x^2-18 a x+6\right) \sqrt{c-\frac{c}{a x}}}{21 a^4 x^3}-\frac{5 c^{9/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}","-\frac{5 c^{9/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}+\frac{5 c^4 \sqrt{c-\frac{c}{a x}}}{a}+\frac{5 c^3 \left(c-\frac{c}{a x}\right)^{3/2}}{3 a}+\frac{c^2 \left(c-\frac{c}{a x}\right)^{5/2}}{a}+\frac{5 c \left(c-\frac{c}{a x}\right)^{7/2}}{7 a}+x \left(c-\frac{c}{a x}\right)^{9/2}",1,"(c^4*Sqrt[c - c/(a*x)]*(6 - 18*a*x + 4*a^2*x^2 + 92*a^3*x^3 + 21*a^4*x^4))/(21*a^4*x^3) - (5*c^(9/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a","A",1
447,1,83,118,0.1294925,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{7/2} \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^(7/2),x]","\frac{c^3 \left(5 a^3 x^3+8 a^2 x^2+4 a x-2\right) \sqrt{c-\frac{c}{a x}}}{5 a^3 x^2}-\frac{3 c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}","-\frac{3 c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}+\frac{3 c^3 \sqrt{c-\frac{c}{a x}}}{a}+\frac{c^2 \left(c-\frac{c}{a x}\right)^{3/2}}{a}+\frac{3 c \left(c-\frac{c}{a x}\right)^{5/2}}{5 a}+x \left(c-\frac{c}{a x}\right)^{7/2}",1,"(c^3*Sqrt[c - c/(a*x)]*(-2 + 4*a*x + 8*a^2*x^2 + 5*a^3*x^3))/(5*a^3*x^2) - (3*c^(7/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a","A",1
448,1,75,95,0.0723101,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{5/2} \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^(5/2),x]","\frac{c^2 \left(3 a^2 x^2-2 a x+2\right) \sqrt{c-\frac{c}{a x}}-3 a c^{5/2} x \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{3 a^2 x}","-\frac{c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}+\frac{c^2 \sqrt{c-\frac{c}{a x}}}{a}+\frac{c \left(c-\frac{c}{a x}\right)^{3/2}}{3 a}+x \left(c-\frac{c}{a x}\right)^{5/2}",1,"(c^2*Sqrt[c - c/(a*x)]*(2 - 2*a*x + 3*a^2*x^2) - 3*a*c^(5/2)*x*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(3*a^2*x)","A",1
449,1,55,70,0.0507581,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{3/2} \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^(3/2),x]","\frac{c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)+c (a x-2) \sqrt{c-\frac{c}{a x}}}{a}","\frac{c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}-\frac{c \sqrt{c-\frac{c}{a x}}}{a}+x \left(c-\frac{c}{a x}\right)^{3/2}",1,"(c*Sqrt[c - c/(a*x)]*(-2 + a*x) + c^(3/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a","A",1
450,1,50,50,0.0351799,"\int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx","Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)],x]","x \sqrt{c-\frac{c}{a x}}+\frac{3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}","x \sqrt{c-\frac{c}{a x}}+\frac{3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}",1,"Sqrt[c - c/(a*x)]*x + (3*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a","A",1
451,1,43,70,0.0323968,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx","Integrate[E^(2*ArcCoth[a*x])/Sqrt[c - c/(a*x)],x]","\frac{a x-5 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1-\frac{1}{a x}\right)}{a \sqrt{c-\frac{c}{a x}}}","\frac{x}{\sqrt{c-\frac{c}{a x}}}-\frac{5}{a \sqrt{c-\frac{c}{a x}}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a \sqrt{c}}",1,"(a*x - 5*Hypergeometric2F1[-1/2, 1, 1/2, 1 - 1/(a*x)])/(a*Sqrt[c - c/(a*x)])","C",0
452,1,55,95,0.0356181,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{3/2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - c/(a*x))^(3/2),x]","\frac{x \left(3 a x-7 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};1-\frac{1}{a x}\right)\right)}{3 c (a x-1) \sqrt{c-\frac{c}{a x}}}","\frac{7 \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a c^{3/2}}+\frac{x}{\left(c-\frac{c}{a x}\right)^{3/2}}-\frac{7}{a c \sqrt{c-\frac{c}{a x}}}-\frac{7}{3 a \left(c-\frac{c}{a x}\right)^{3/2}}",1,"(x*(3*a*x - 7*Hypergeometric2F1[-3/2, 1, -1/2, 1 - 1/(a*x)]))/(3*c*Sqrt[c - c/(a*x)]*(-1 + a*x))","C",0
453,1,58,118,0.0390072,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{5/2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - c/(a*x))^(5/2),x]","\frac{x}{\left(c-\frac{c}{a x}\right)^{5/2}}-\frac{9 \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};1-\frac{1}{a x}\right)}{5 a \left(c-\frac{c}{a x}\right)^{5/2}}","\frac{9 \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a c^{5/2}}-\frac{9}{a c^2 \sqrt{c-\frac{c}{a x}}}+\frac{x}{\left(c-\frac{c}{a x}\right)^{5/2}}-\frac{3}{a c \left(c-\frac{c}{a x}\right)^{3/2}}-\frac{9}{5 a \left(c-\frac{c}{a x}\right)^{5/2}}",1,"x/(c - c/(a*x))^(5/2) - (9*Hypergeometric2F1[-5/2, 1, -3/2, 1 - 1/(a*x)])/(5*a*(c - c/(a*x))^(5/2))","C",0
454,1,46,145,0.0462636,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{7/2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - c/(a*x))^(7/2),x]","\frac{7 x-\frac{11 \, _2F_1\left(-\frac{7}{2},1;-\frac{5}{2};1-\frac{1}{a x}\right)}{a}}{7 \left(c-\frac{c}{a x}\right)^{7/2}}","\frac{11 \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a c^{7/2}}-\frac{11}{a c^3 \sqrt{c-\frac{c}{a x}}}-\frac{11}{3 a c^2 \left(c-\frac{c}{a x}\right)^{3/2}}+\frac{x}{\left(c-\frac{c}{a x}\right)^{7/2}}-\frac{11}{5 a c \left(c-\frac{c}{a x}\right)^{5/2}}-\frac{11}{7 a \left(c-\frac{c}{a x}\right)^{7/2}}",1,"(7*x - (11*Hypergeometric2F1[-7/2, 1, -5/2, 1 - 1/(a*x)])/a)/(7*(c - c/(a*x))^(7/2))","C",0
455,1,109,268,0.1074512,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{9/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^(9/2),x]","\frac{c^4 \sqrt{c-\frac{c}{a x}} \left(\sqrt{\frac{1}{a x}+1} \left(35 a^4 x^4+164 a^3 x^3-12 a^2 x^2-26 a x+10\right)-105 a^3 x^3 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)\right)}{35 a^4 x^3 \sqrt{1-\frac{1}{a x}}}","\frac{9 \left(a-\frac{1}{x}\right)^2 \left(\frac{1}{a x}+1\right)^{3/2} \left(c-\frac{c}{a x}\right)^{9/2}}{7 a^3 \left(1-\frac{1}{a x}\right)^{9/2}}+\frac{x \left(a-\frac{1}{x}\right)^3 \left(\frac{1}{a x}+1\right)^{3/2} \left(c-\frac{c}{a x}\right)^{9/2}}{a^3 \left(1-\frac{1}{a x}\right)^{9/2}}+\frac{3 \left(28 a-\frac{17}{x}\right) \left(\frac{1}{a x}+1\right)^{3/2} \left(c-\frac{c}{a x}\right)^{9/2}}{35 a^2 \left(1-\frac{1}{a x}\right)^{9/2}}+\frac{3 \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{9/2}}{a \left(1-\frac{1}{a x}\right)^{9/2}}-\frac{3 \left(c-\frac{c}{a x}\right)^{9/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(1-\frac{1}{a x}\right)^{9/2}}",1,"(c^4*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(10 - 26*a*x - 12*a^2*x^2 + 164*a^3*x^3 + 35*a^4*x^4) - 105*a^3*x^3*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(35*a^4*Sqrt[1 - 1/(a*x)]*x^3)","A",1
456,1,101,237,0.0866579,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{7/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^(7/2),x]","\frac{c^3 \sqrt{c-\frac{c}{a x}} \left(\sqrt{\frac{1}{a x}+1} \left(15 a^3 x^3+44 a^2 x^2+8 a x-6\right)-15 a^2 x^2 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)\right)}{15 a^3 x^2 \sqrt{1-\frac{1}{a x}}}","-\frac{2 \left(\frac{1}{a x}+1\right)^{5/2} \left(c-\frac{c}{a x}\right)^{7/2}}{5 a \left(1-\frac{1}{a x}\right)^{7/2}}+\frac{\left(\frac{1}{a x}+1\right)^{3/2} \left(c-\frac{c}{a x}\right)^{7/2}}{3 a \left(1-\frac{1}{a x}\right)^{7/2}}+\frac{x \left(\frac{1}{a x}+1\right)^{5/2} \left(c-\frac{c}{a x}\right)^{7/2}}{\left(1-\frac{1}{a x}\right)^{7/2}}+\frac{\sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{7/2}}{a \left(1-\frac{1}{a x}\right)^{7/2}}-\frac{\left(c-\frac{c}{a x}\right)^{7/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(1-\frac{1}{a x}\right)^{7/2}}",1,"(c^3*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(-6 + 8*a*x + 44*a^2*x^2 + 15*a^3*x^3) - 15*a^2*x^2*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(15*a^3*Sqrt[1 - 1/(a*x)]*x^2)","A",1
457,1,89,156,0.0700701,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{5/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^(5/2),x]","\frac{c^2 \sqrt{c-\frac{c}{a x}} \left(\sqrt{\frac{1}{a x}+1} \left(3 a^2 x^2+2 a x+2\right)+3 a x \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)\right)}{3 a^2 x \sqrt{1-\frac{1}{a x}}}","\frac{c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{a}+\frac{c^5 x \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{\left(c-\frac{c}{a x}\right)^{5/2}}-\frac{c^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{3 a \left(c-\frac{c}{a x}\right)^{3/2}}-\frac{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}",1,"(c^2*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(2 + 2*a*x + 3*a^2*x^2) + 3*a*x*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(3*a^2*Sqrt[1 - 1/(a*x)]*x)","A",0
458,1,66,118,0.0497383,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{3/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a*x))^(3/2),x]","-\frac{2 \left(\frac{1}{a x}+1\right)^{5/2} \left(c-\frac{c}{a x}\right)^{3/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};1+\frac{1}{a x}\right)}{5 a \left(1-\frac{1}{a x}\right)^{3/2}}","\frac{3 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{a}+\frac{c^3 x \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{\left(c-\frac{c}{a x}\right)^{3/2}}-\frac{3 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a \sqrt{c-\frac{c}{a x}}}",1,"(-2*(1 + 1/(a*x))^(5/2)*(c - c/(a*x))^(3/2)*Hypergeometric2F1[2, 5/2, 7/2, 1 + 1/(a*x)])/(5*a*(1 - 1/(a*x))^(3/2))","C",0
459,1,93,152,0.0823105,"\int e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx","Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)],x]","\frac{\sqrt{c-\frac{c}{a x}} \left(a x \sqrt{\frac{1}{a x}+1}+5 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)-4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)\right)}{a \sqrt{1-\frac{1}{a x}}}","\frac{x \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{5 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{a \sqrt{1-\frac{1}{a x}}}",1,"(Sqrt[c - c/(a*x)]*(a*Sqrt[1 + 1/(a*x)]*x + 5*ArcTanh[Sqrt[1 + 1/(a*x)]] - 4*Sqrt[2]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(a*Sqrt[1 - 1/(a*x)])","A",1
460,1,115,215,0.1030721,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx","Integrate[E^(3*ArcCoth[a*x])/Sqrt[c - c/(a*x)],x]","\frac{\sqrt{1-\frac{1}{a x}} \left(a x \sqrt{\frac{1}{a x}+1} (a x-3)+7 (a x-1) \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)-5 \sqrt{2} (a x-1) \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)\right)}{a (a x-1) \sqrt{c-\frac{c}{a x}}}","\frac{a x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\left(a-\frac{1}{x}\right) \sqrt{c-\frac{c}{a x}}}-\frac{3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\left(a-\frac{1}{x}\right) \sqrt{c-\frac{c}{a x}}}+\frac{7 \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \sqrt{c-\frac{c}{a x}}}-\frac{5 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{a \sqrt{c-\frac{c}{a x}}}",1,"(Sqrt[1 - 1/(a*x)]*(a*Sqrt[1 + 1/(a*x)]*x*(-3 + a*x) + 7*(-1 + a*x)*ArcTanh[Sqrt[1 + 1/(a*x)]] - 5*Sqrt[2]*(-1 + a*x)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(a*Sqrt[c - c/(a*x)]*(-1 + a*x))","A",1
461,1,135,275,0.1325476,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{3/2}} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - c/(a*x))^(3/2),x]","\frac{\sqrt{1-\frac{1}{a x}} \left(2 a x \sqrt{\frac{1}{a x}+1} \left(4 a^2 x^2-23 a x+15\right)+72 (a x-1)^2 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)-51 \sqrt{2} (a x-1)^2 \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)\right)}{8 a c (a x-1)^2 \sqrt{c-\frac{c}{a x}}}","\frac{a^2 x \left(1-\frac{1}{a x}\right)^{3/2} \sqrt{\frac{1}{a x}+1}}{\left(a-\frac{1}{x}\right)^2 \left(c-\frac{c}{a x}\right)^{3/2}}-\frac{2 a \left(1-\frac{1}{a x}\right)^{3/2} \sqrt{\frac{1}{a x}+1}}{\left(a-\frac{1}{x}\right)^2 \left(c-\frac{c}{a x}\right)^{3/2}}-\frac{15 \left(1-\frac{1}{a x}\right)^{3/2} \sqrt{\frac{1}{a x}+1}}{4 \left(a-\frac{1}{x}\right) \left(c-\frac{c}{a x}\right)^{3/2}}+\frac{9 \left(1-\frac{1}{a x}\right)^{3/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(c-\frac{c}{a x}\right)^{3/2}}-\frac{51 \left(1-\frac{1}{a x}\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{4 \sqrt{2} a \left(c-\frac{c}{a x}\right)^{3/2}}",1,"(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(15 - 23*a*x + 4*a^2*x^2) + 72*(-1 + a*x)^2*ArcTanh[Sqrt[1 + 1/(a*x)]] - 51*Sqrt[2]*(-1 + a*x)^2*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(8*a*c*Sqrt[c - c/(a*x)]*(-1 + a*x)^2)","A",1
462,1,143,335,0.1625467,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{5/2}} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - c/(a*x))^(5/2),x]","\frac{\sqrt{1-\frac{1}{a x}} \left(2 a x \sqrt{\frac{1}{a x}+1} \left(48 a^3 x^3-415 a^2 x^2+554 a x-219\right)+1056 (a x-1)^3 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)-747 \sqrt{2} (a x-1)^3 \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)\right)}{96 a c^2 (a x-1)^3 \sqrt{c-\frac{c}{a x}}}","\frac{a^3 x \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}{\left(a-\frac{1}{x}\right)^3 \left(c-\frac{c}{a x}\right)^{5/2}}-\frac{5 a^2 \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}{3 \left(a-\frac{1}{x}\right)^3 \left(c-\frac{c}{a x}\right)^{5/2}}-\frac{29 a \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}{12 \left(a-\frac{1}{x}\right)^2 \left(c-\frac{c}{a x}\right)^{5/2}}-\frac{73 \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}{16 \left(a-\frac{1}{x}\right) \left(c-\frac{c}{a x}\right)^{5/2}}+\frac{11 \left(1-\frac{1}{a x}\right)^{5/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(c-\frac{c}{a x}\right)^{5/2}}-\frac{249 \left(1-\frac{1}{a x}\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{16 \sqrt{2} a \left(c-\frac{c}{a x}\right)^{5/2}}",1,"(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(-219 + 554*a*x - 415*a^2*x^2 + 48*a^3*x^3) + 1056*(-1 + a*x)^3*ArcTanh[Sqrt[1 + 1/(a*x)]] - 747*Sqrt[2]*(-1 + a*x)^3*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(96*a*c^2*Sqrt[c - c/(a*x)]*(-1 + a*x)^3)","A",1
463,1,101,221,0.1107989,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{7/2} \, dx","Integrate[(c - c/(a*x))^(7/2)/E^ArcCoth[a*x],x]","\frac{c^3 \sqrt{c-\frac{c}{a x}} \left(\sqrt{\frac{1}{a x}+1} \left(5 a^3 x^3-92 a^2 x^2+16 a x-2\right)-45 a^2 x^2 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)\right)}{5 a^3 x^2 \sqrt{1-\frac{1}{a x}}}","\frac{x \left(a-\frac{1}{x}\right)^3 \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{7/2}}{a^3 \left(1-\frac{1}{a x}\right)^{7/2}}+\frac{3 \left(a-\frac{1}{x}\right)^2 \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{7/2}}{5 a^3 \left(1-\frac{1}{a x}\right)^{7/2}}-\frac{\left(80 a-\frac{7}{x}\right) \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{7/2}}{5 a^2 \left(1-\frac{1}{a x}\right)^{7/2}}-\frac{9 \left(c-\frac{c}{a x}\right)^{7/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(1-\frac{1}{a x}\right)^{7/2}}",1,"(c^3*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(-2 + 16*a*x - 92*a^2*x^2 + 5*a^3*x^3) - 45*a^2*x^2*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(5*a^3*Sqrt[1 - 1/(a*x)]*x^2)","A",1
464,1,89,161,0.0703186,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{5/2} \, dx","Integrate[(c - c/(a*x))^(5/2)/E^ArcCoth[a*x],x]","\frac{c^2 \sqrt{c-\frac{c}{a x}} \left(\sqrt{\frac{1}{a x}+1} \left(3 a^2 x^2-22 a x+2\right)-21 a x \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)\right)}{3 a^2 x \sqrt{1-\frac{1}{a x}}}","\frac{x \left(a-\frac{1}{x}\right)^2 \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{5/2}}{a^2 \left(1-\frac{1}{a x}\right)^{5/2}}-\frac{\left(16 a+\frac{1}{x}\right) \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{5/2}}{3 a^2 \left(1-\frac{1}{a x}\right)^{5/2}}-\frac{7 \left(c-\frac{c}{a x}\right)^{5/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(1-\frac{1}{a x}\right)^{5/2}}",1,"(c^2*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(2 - 22*a*x + 3*a^2*x^2) - 21*a*x*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(3*a^2*Sqrt[1 - 1/(a*x)]*x)","A",1
465,1,70,140,0.0548479,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{3/2} \, dx","Integrate[(c - c/(a*x))^(3/2)/E^ArcCoth[a*x],x]","\frac{c \sqrt{c-\frac{c}{a x}} \left(\sqrt{\frac{1}{a x}+1} (a x-2)-5 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)\right)}{a \sqrt{1-\frac{1}{a x}}}","\frac{x \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{3/2}}{\left(1-\frac{1}{a x}\right)^{3/2}}-\frac{2 \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{3/2}}{a \left(1-\frac{1}{a x}\right)^{3/2}}-\frac{5 \left(c-\frac{c}{a x}\right)^{3/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(1-\frac{1}{a x}\right)^{3/2}}",1,"(c*Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*(-2 + a*x) - 5*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(a*Sqrt[1 - 1/(a*x)])","A",1
466,1,65,79,0.0427401,"\int e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx","Integrate[Sqrt[c - c/(a*x)]/E^ArcCoth[a*x],x]","\frac{\sqrt{c-\frac{c}{a x}} \left(x \sqrt{\frac{1}{a x}+1}-\frac{3 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a}\right)}{\sqrt{1-\frac{1}{a x}}}","\frac{c x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}-\frac{3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{a}",1,"(Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*x - (3*ArcTanh[Sqrt[1 + 1/(a*x)]])/a))/Sqrt[1 - 1/(a*x)]","A",0
467,1,66,78,0.0509844,"\int \frac{e^{-\coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx","Integrate[1/(E^ArcCoth[a*x]*Sqrt[c - c/(a*x)]),x]","\frac{\sqrt{1-\frac{1}{a x}} \left(a x \sqrt{\frac{1}{a x}+1}-\tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)\right)}{a \sqrt{c-\frac{c}{a x}}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{a \sqrt{c}}",1,"(Sqrt[1 - 1/(a*x)]*(a*Sqrt[1 + 1/(a*x)]*x - ArcTanh[Sqrt[1 + 1/(a*x)]]))/(a*Sqrt[c - c/(a*x)])","A",0
468,1,91,151,0.0773887,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{3/2}} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - c/(a*x))^(3/2)),x]","\frac{\left(1-\frac{1}{a x}\right)^{3/2} \left(a x \sqrt{\frac{1}{a x}+1}+\tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)\right)}{a \left(c-\frac{c}{a x}\right)^{3/2}}","\frac{x \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{3/2}}{\left(c-\frac{c}{a x}\right)^{3/2}}+\frac{\left(1-\frac{1}{a x}\right)^{3/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(c-\frac{c}{a x}\right)^{3/2}}-\frac{\sqrt{2} \left(1-\frac{1}{a x}\right)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{a \left(c-\frac{c}{a x}\right)^{3/2}}",1,"((1 - 1/(a*x))^(3/2)*(a*Sqrt[1 + 1/(a*x)]*x + ArcTanh[Sqrt[1 + 1/(a*x)]] - Sqrt[2]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(a*(c - c/(a*x))^(3/2))","A",1
469,1,123,219,0.1117772,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{5/2}} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - c/(a*x))^(5/2)),x]","\frac{\sqrt{1-\frac{1}{a x}} \left(2 a x \sqrt{\frac{1}{a x}+1} (2 a x-3)+12 (a x-1) \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)-9 \sqrt{2} (a x-1) \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)\right)}{4 a c^2 (a x-1) \sqrt{c-\frac{c}{a x}}}","\frac{a x \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{5/2}}{\left(a-\frac{1}{x}\right) \left(c-\frac{c}{a x}\right)^{5/2}}-\frac{3 \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{5/2}}{2 \left(a-\frac{1}{x}\right) \left(c-\frac{c}{a x}\right)^{5/2}}+\frac{3 \left(1-\frac{1}{a x}\right)^{5/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(c-\frac{c}{a x}\right)^{5/2}}-\frac{9 \left(1-\frac{1}{a x}\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{2 \sqrt{2} a \left(c-\frac{c}{a x}\right)^{5/2}}",1,"(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(-3 + 2*a*x) + 12*(-1 + a*x)*ArcTanh[Sqrt[1 + 1/(a*x)]] - 9*Sqrt[2]*(-1 + a*x)*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(4*a*c^2*Sqrt[c - c/(a*x)]*(-1 + a*x))","A",1
470,1,135,277,0.1404628,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{7/2}} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - c/(a*x))^(7/2)),x]","\frac{\sqrt{1-\frac{1}{a x}} \left(2 a x \sqrt{\frac{1}{a x}+1} \left(16 a^2 x^2-55 a x+35\right)+160 (a x-1)^2 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)-115 \sqrt{2} (a x-1)^2 \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)\right)}{32 a c^3 (a x-1)^2 \sqrt{c-\frac{c}{a x}}}","\frac{a^2 x \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{7/2}}{\left(a-\frac{1}{x}\right)^2 \left(c-\frac{c}{a x}\right)^{7/2}}-\frac{35 \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{7/2}}{16 \left(a-\frac{1}{x}\right) \left(c-\frac{c}{a x}\right)^{7/2}}-\frac{5 a \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{7/2}}{4 \left(a-\frac{1}{x}\right)^2 \left(c-\frac{c}{a x}\right)^{7/2}}+\frac{5 \left(1-\frac{1}{a x}\right)^{7/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(c-\frac{c}{a x}\right)^{7/2}}-\frac{115 \left(1-\frac{1}{a x}\right)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{16 \sqrt{2} a \left(c-\frac{c}{a x}\right)^{7/2}}",1,"(Sqrt[1 - 1/(a*x)]*(2*a*Sqrt[1 + 1/(a*x)]*x*(35 - 55*a*x + 16*a^2*x^2) + 160*(-1 + a*x)^2*ArcTanh[Sqrt[1 + 1/(a*x)]] - 115*Sqrt[2]*(-1 + a*x)^2*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(32*a*c^3*Sqrt[c - c/(a*x)]*(-1 + a*x)^2)","A",1
471,1,125,163,0.2731299,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{7/2} \, dx","Integrate[(c - c/(a*x))^(7/2)/E^(2*ArcCoth[a*x]),x]","\frac{c^3 \left(15 a^3 x^3-376 a^2 x^2+52 a x-6\right) \sqrt{c-\frac{c}{a x}}}{15 a^3 x^2}-\frac{11 c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}+\frac{32 \sqrt{2} c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a}","-\frac{11 c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}+\frac{32 \sqrt{2} c^{7/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a}-\frac{21 c^3 \sqrt{c-\frac{c}{a x}}}{a}-\frac{5 c^2 \left(c-\frac{c}{a x}\right)^{3/2}}{3 a}+\frac{3 c \left(c-\frac{c}{a x}\right)^{5/2}}{5 a}+x \left(c-\frac{c}{a x}\right)^{7/2}",1,"(c^3*Sqrt[c - c/(a*x)]*(-6 + 52*a*x - 376*a^2*x^2 + 15*a^3*x^3))/(15*a^3*x^2) - (11*c^(7/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a + (32*Sqrt[2]*c^(7/2)*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a","A",1
472,1,116,138,0.1110646,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{5/2} \, dx","Integrate[(c - c/(a*x))^(5/2)/E^(2*ArcCoth[a*x]),x]","\frac{c^2 \left(3 a^2 x^2-26 a x+2\right) \sqrt{c-\frac{c}{a x}}-27 a c^{5/2} x \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)+48 \sqrt{2} a c^{5/2} x \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{3 a^2 x}","-\frac{9 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}+\frac{16 \sqrt{2} c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a}-\frac{7 c^2 \sqrt{c-\frac{c}{a x}}}{a}+\frac{c \left(c-\frac{c}{a x}\right)^{3/2}}{3 a}+x \left(c-\frac{c}{a x}\right)^{5/2}",1,"(c^2*Sqrt[c - c/(a*x)]*(2 - 26*a*x + 3*a^2*x^2) - 27*a*c^(5/2)*x*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]] + 48*Sqrt[2]*a*c^(5/2)*x*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/(3*a^2*x)","A",1
473,1,95,113,0.0885198,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{3/2} \, dx","Integrate[(c - c/(a*x))^(3/2)/E^(2*ArcCoth[a*x]),x]","\frac{-7 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)+8 \sqrt{2} c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)+c (a x-2) \sqrt{c-\frac{c}{a x}}}{a}","-\frac{7 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}+\frac{8 \sqrt{2} c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a}-\frac{c \sqrt{c-\frac{c}{a x}}}{a}+x \left(c-\frac{c}{a x}\right)^{3/2}",1,"(c*Sqrt[c - c/(a*x)]*(-2 + a*x) - 7*c^(3/2)*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]] + 8*Sqrt[2]*c^(3/2)*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a","A",1
474,1,92,92,0.0539028,"\int e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx","Integrate[Sqrt[c - c/(a*x)]/E^(2*ArcCoth[a*x]),x]","x \sqrt{c-\frac{c}{a x}}-\frac{5 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a}","x \sqrt{c-\frac{c}{a x}}-\frac{5 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a}",1,"Sqrt[c - c/(a*x)]*x - (5*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a + (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a","A",1
475,1,95,95,0.0513573,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)]),x]","\frac{x \sqrt{c-\frac{c}{a x}}}{c}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a \sqrt{c}}+\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a \sqrt{c}}","\frac{x \sqrt{c-\frac{c}{a x}}}{c}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a \sqrt{c}}+\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a \sqrt{c}}",1,"(Sqrt[c - c/(a*x)]*x)/c - (3*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*Sqrt[c]) + (2*Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/(a*Sqrt[c])","A",1
476,1,94,94,0.0574689,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{3/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^(3/2)),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a c^{3/2}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a c^{3/2}}+\frac{x \sqrt{c-\frac{c}{a x}}}{c^2}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a c^{3/2}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a c^{3/2}}+\frac{x \sqrt{c-\frac{c}{a x}}}{c^2}",1,"(Sqrt[c - c/(a*x)]*x)/c^2 - ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]]/(a*c^(3/2)) + (Sqrt[2]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/(a*c^(3/2))","A",1
477,1,70,116,0.0580591,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{5/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^(5/2)),x]","\frac{-\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a-\frac{1}{x}}{2 a}\right)-\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1-\frac{1}{a x}\right)+a x}{a c^2 \sqrt{c-\frac{c}{a x}}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a c^{5/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{\sqrt{2} a c^{5/2}}+\frac{x}{c^2 \sqrt{c-\frac{c}{a x}}}-\frac{2}{a c^2 \sqrt{c-\frac{c}{a x}}}",1,"(a*x - Hypergeometric2F1[-1/2, 1, 1/2, (a - x^(-1))/(2*a)] - Hypergeometric2F1[-1/2, 1, 1/2, 1 - 1/(a*x)])/(a*c^2*Sqrt[c - c/(a*x)])","C",0
478,1,79,147,0.0634519,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{7/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^(7/2)),x]","\frac{x \left(-\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a-\frac{1}{x}}{2 a}\right)-3 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};1-\frac{1}{a x}\right)+3 a x\right)}{3 c^3 (a x-1) \sqrt{c-\frac{c}{a x}}}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a c^{7/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{2 \sqrt{2} a c^{7/2}}-\frac{7}{2 a c^3 \sqrt{c-\frac{c}{a x}}}+\frac{x}{c^2 \left(c-\frac{c}{a x}\right)^{3/2}}-\frac{4}{3 a c^2 \left(c-\frac{c}{a x}\right)^{3/2}}",1,"(x*(3*a*x - Hypergeometric2F1[-3/2, 1, -1/2, (a - x^(-1))/(2*a)] - 3*Hypergeometric2F1[-3/2, 1, -1/2, 1 - 1/(a*x)]))/(3*c^3*Sqrt[c - c/(a*x)]*(-1 + a*x))","C",0
479,1,82,172,0.0728111,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{9/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a*x))^(9/2)),x]","\frac{a x^2 \left(-\, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\frac{a-\frac{1}{x}}{2 a}\right)-5 \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};1-\frac{1}{a x}\right)+5 a x\right)}{5 c^4 (a x-1)^2 \sqrt{c-\frac{c}{a x}}}","\frac{5 \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a c^{9/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{4 \sqrt{2} a c^{9/2}}-\frac{21}{4 a c^4 \sqrt{c-\frac{c}{a x}}}-\frac{11}{6 a c^3 \left(c-\frac{c}{a x}\right)^{3/2}}+\frac{x}{c^2 \left(c-\frac{c}{a x}\right)^{5/2}}-\frac{6}{5 a c^2 \left(c-\frac{c}{a x}\right)^{5/2}}",1,"(a*x^2*(5*a*x - Hypergeometric2F1[-5/2, 1, -3/2, (a - x^(-1))/(2*a)] - 5*Hypergeometric2F1[-5/2, 1, -3/2, 1 - 1/(a*x)]))/(5*c^4*Sqrt[c - c/(a*x)]*(-1 + a*x)^2)","C",0
480,1,140,335,0.1322046,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{9/2} \, dx","Integrate[(c - c/(a*x))^(9/2)/E^(3*ArcCoth[a*x]),x]","\frac{c^4 \sqrt{c-\frac{c}{a x}} \left(7 a^5 x^5+70 a^4 x^4 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1+\frac{1}{a x}\right)+1685 a^4 x^4-35 a^4 x^4 \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)+720 a^3 x^3-110 a^2 x^2+20 a x-2\right)}{7 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \left(a-\frac{1}{x}\right)^5 \left(c-\frac{c}{a x}\right)^{9/2}}{a^5 \left(1-\frac{1}{a x}\right)^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{10 \left(a-\frac{1}{x}\right)^4 \left(c-\frac{c}{a x}\right)^{9/2}}{a^5 \left(1-\frac{1}{a x}\right)^{9/2} \sqrt{\frac{1}{a x}+1}}+\frac{65 \sqrt{\frac{1}{a x}+1} \left(a-\frac{1}{x}\right)^3 \left(c-\frac{c}{a x}\right)^{9/2}}{7 a^4 \left(1-\frac{1}{a x}\right)^{9/2}}+\frac{135 \sqrt{\frac{1}{a x}+1} \left(a-\frac{1}{x}\right)^2 \left(c-\frac{c}{a x}\right)^{9/2}}{7 a^3 \left(1-\frac{1}{a x}\right)^{9/2}}+\frac{5 \left(304 a-\frac{65}{x}\right) \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{9/2}}{7 a^2 \left(1-\frac{1}{a x}\right)^{9/2}}-\frac{15 \left(c-\frac{c}{a x}\right)^{9/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(1-\frac{1}{a x}\right)^{9/2}}",1,"(c^4*Sqrt[c - c/(a*x)]*(-2 + 20*a*x - 110*a^2*x^2 + 720*a^3*x^3 + 1685*a^4*x^4 + 7*a^5*x^5 - 35*a^4*Sqrt[1 + 1/(a*x)]*x^4*ArcTanh[Sqrt[1 + 1/(a*x)]] + 70*a^4*x^4*Hypergeometric2F1[-1/2, 1, 1/2, 1 + 1/(a*x)]))/(7*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4)","C",0
481,1,132,277,0.1138708,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{7/2} \, dx","Integrate[(c - c/(a*x))^(7/2)/E^(3*ArcCoth[a*x]),x]","\frac{c^3 \sqrt{c-\frac{c}{a x}} \left(15 a^4 x^4+150 a^3 x^3 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1+\frac{1}{a x}\right)+1441 a^3 x^3-45 a^3 x^3 \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)+548 a^2 x^2-62 a x+6\right)}{15 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \left(a-\frac{1}{x}\right)^4 \left(c-\frac{c}{a x}\right)^{7/2}}{a^4 \left(1-\frac{1}{a x}\right)^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{10 \left(a-\frac{1}{x}\right)^3 \left(c-\frac{c}{a x}\right)^{7/2}}{a^4 \left(1-\frac{1}{a x}\right)^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{47 \sqrt{\frac{1}{a x}+1} \left(a-\frac{1}{x}\right)^2 \left(c-\frac{c}{a x}\right)^{7/2}}{5 a^3 \left(1-\frac{1}{a x}\right)^{7/2}}+\frac{\left(1360 a-\frac{311}{x}\right) \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{7/2}}{15 a^2 \left(1-\frac{1}{a x}\right)^{7/2}}-\frac{13 \left(c-\frac{c}{a x}\right)^{7/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(1-\frac{1}{a x}\right)^{7/2}}",1,"(c^3*Sqrt[c - c/(a*x)]*(6 - 62*a*x + 548*a^2*x^2 + 1441*a^3*x^3 + 15*a^4*x^4 - 45*a^3*Sqrt[1 + 1/(a*x)]*x^3*ArcTanh[Sqrt[1 + 1/(a*x)]] + 150*a^3*x^3*Hypergeometric2F1[-1/2, 1, 1/2, 1 + 1/(a*x)]))/(15*a^4*Sqrt[1 - 1/(a^2*x^2)]*x^3)","C",0
482,1,124,219,0.1016728,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{5/2} \, dx","Integrate[(c - c/(a*x))^(5/2)/E^(3*ArcCoth[a*x]),x]","\frac{c^2 \sqrt{c-\frac{c}{a x}} \left(3 a^3 x^3+30 a^2 x^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1+\frac{1}{a x}\right)+103 a^2 x^2-3 a^2 x^2 \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)+32 a x-2\right)}{3 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \left(a-\frac{1}{x}\right)^3 \left(c-\frac{c}{a x}\right)^{5/2}}{a^3 \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}+\frac{10 \left(a-\frac{1}{x}\right)^2 \left(c-\frac{c}{a x}\right)^{5/2}}{a^3 \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}+\frac{\left(112 a-\frac{29}{x}\right) \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{5/2}}{3 a^2 \left(1-\frac{1}{a x}\right)^{5/2}}-\frac{11 \left(c-\frac{c}{a x}\right)^{5/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(1-\frac{1}{a x}\right)^{5/2}}",1,"(c^2*Sqrt[c - c/(a*x)]*(-2 + 32*a*x + 103*a^2*x^2 + 3*a^3*x^3 - 3*a^2*Sqrt[1 + 1/(a*x)]*x^2*ArcTanh[Sqrt[1 + 1/(a*x)]] + 30*a^2*x^2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + 1/(a*x)]))/(3*a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2)","C",0
483,1,71,158,0.0456312,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{3/2} \, dx","Integrate[(c - c/(a*x))^(3/2)/E^(3*ArcCoth[a*x]),x]","\frac{c \sqrt{c-\frac{c}{a x}} \left(a^2 x^2+9 a x \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1+\frac{1}{a x}\right)+10 a x+2\right)}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \left(a-\frac{1}{x}\right)^2 \left(c-\frac{c}{a x}\right)^{3/2}}{a^2 \left(1-\frac{1}{a x}\right)^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{\left(21 a+\frac{1}{x}\right) \left(c-\frac{c}{a x}\right)^{3/2}}{a^2 \left(1-\frac{1}{a x}\right)^{3/2} \sqrt{\frac{1}{a x}+1}}-\frac{9 \left(c-\frac{c}{a x}\right)^{3/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(1-\frac{1}{a x}\right)^{3/2}}",1,"(c*Sqrt[c - c/(a*x)]*(2 + 10*a*x + a^2*x^2 + 9*a*x*Hypergeometric2F1[-1/2, 1, 1/2, 1 + 1/(a*x)]))/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","C",0
484,1,67,140,0.0489219,"\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx","Integrate[Sqrt[c - c/(a*x)]/E^(3*ArcCoth[a*x]),x]","\frac{\sqrt{c-\frac{c}{a x}} \left(a x-7 \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)+9\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{9 \sqrt{c-\frac{c}{a x}}}{a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{7 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \sqrt{1-\frac{1}{a x}}}",1,"(Sqrt[c - c/(a*x)]*(9 + a*x - 7*Sqrt[1 + 1/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
485,1,69,118,0.0452232,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)]),x]","\frac{\sqrt{1-\frac{1}{a x}} \left(5 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1+\frac{1}{a x}\right)+a x\right)}{a \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}","\frac{x \sqrt{c-\frac{c}{a x}}}{c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{5 \sqrt{c-\frac{c}{a x}}}{a c \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{a \sqrt{c}}",1,"(Sqrt[1 - 1/(a*x)]*(a*x + 5*Hypergeometric2F1[-1/2, 1, 1/2, 1 + 1/(a*x)]))/(a*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)])","C",0
486,1,64,117,0.0499364,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{3/2}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^(3/2)),x]","\frac{2 \left(1-\frac{1}{a x}\right)^{3/2} \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};1+\frac{1}{a x}\right)}{a \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{3/2}}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{a c^{3/2}}-\frac{2 x \sqrt{c-\frac{c}{a x}}}{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 x \sqrt{1-\frac{1}{a^2 x^2}}}{c \sqrt{c-\frac{c}{a x}}}",1,"(2*(1 - 1/(a*x))^(3/2)*Hypergeometric2F1[-1/2, 2, 1/2, 1 + 1/(a*x)])/(a*Sqrt[1 + 1/(a*x)]*(c - c/(a*x))^(3/2))","C",0
487,1,90,199,0.0680837,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{5/2}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^(5/2)),x]","\frac{\sqrt{1-\frac{1}{a x}} \left(\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+\frac{1}{x}}{2 a}\right)+\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1+\frac{1}{a x}\right)+a x\right)}{a c^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}","\frac{x \left(1-\frac{1}{a x}\right)^{5/2}}{\sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{5/2}}+\frac{2 \left(1-\frac{1}{a x}\right)^{5/2}}{a \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{5/2}}-\frac{\left(1-\frac{1}{a x}\right)^{5/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(c-\frac{c}{a x}\right)^{5/2}}-\frac{\left(1-\frac{1}{a x}\right)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{\sqrt{2} a \left(c-\frac{c}{a x}\right)^{5/2}}",1,"(Sqrt[1 - 1/(a*x)]*(a*x + Hypergeometric2F1[-1/2, 1, 1/2, (a + x^(-1))/(2*a)] + Hypergeometric2F1[-1/2, 1, 1/2, 1 + 1/(a*x)]))/(a*c^2*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)])","C",0
488,1,121,267,0.0882217,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{7/2}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a*x))^(7/2)),x]","\frac{\sqrt{1-\frac{1}{a x}} \left(11 (a x-1) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+\frac{1}{x}}{2 a}\right)+(4-4 a x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};1+\frac{1}{a x}\right)+2 a x (2 a x-3)\right)}{4 a c^3 \sqrt{\frac{1}{a x}+1} (a x-1) \sqrt{c-\frac{c}{a x}}}","\frac{a x \left(1-\frac{1}{a x}\right)^{7/2}}{\left(a-\frac{1}{x}\right) \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{7/2}}+\frac{7 \left(1-\frac{1}{a x}\right)^{7/2}}{4 a \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{7/2}}-\frac{3 \left(1-\frac{1}{a x}\right)^{7/2}}{2 \left(a-\frac{1}{x}\right) \sqrt{\frac{1}{a x}+1} \left(c-\frac{c}{a x}\right)^{7/2}}+\frac{\left(1-\frac{1}{a x}\right)^{7/2} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \left(c-\frac{c}{a x}\right)^{7/2}}-\frac{11 \left(1-\frac{1}{a x}\right)^{7/2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{4 \sqrt{2} a \left(c-\frac{c}{a x}\right)^{7/2}}",1,"(Sqrt[1 - 1/(a*x)]*(2*a*x*(-3 + 2*a*x) + 11*(-1 + a*x)*Hypergeometric2F1[-1/2, 1, 1/2, (a + x^(-1))/(2*a)] + (4 - 4*a*x)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + 1/(a*x)]))/(4*a*c^3*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*(-1 + a*x))","C",0
489,1,60,60,0.0335086,"\int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^m \, dx","Integrate[E^ArcCoth[a*x]*Sqrt[c - c/(a*x)]*x^m,x]","\frac{x^{m+1} \sqrt{c-\frac{c}{a x}} \, _2F_1\left(-\frac{1}{2},-m-1;-m;-\frac{1}{a x}\right)}{(m+1) \sqrt{1-\frac{1}{a x}}}","\frac{x^{m+1} \sqrt{c-\frac{c}{a x}} \, _2F_1\left(-\frac{1}{2},-m-1;-m;-\frac{1}{a x}\right)}{(m+1) \sqrt{1-\frac{1}{a x}}}",1,"(Sqrt[c - c/(a*x)]*x^(1 + m)*Hypergeometric2F1[-1/2, -1 - m, -m, -(1/(a*x))])/((1 + m)*Sqrt[1 - 1/(a*x)])","A",1
490,1,147,164,0.5075319,"\int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx","Integrate[E^ArcCoth[a*x]*Sqrt[c - c/(a*x)]*x^2,x]","\frac{\frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(8 a^2 x^2+2 a x-3\right) \sqrt{c-\frac{c}{a x}}}{a x-1}+3 \sqrt{c} \log \left(2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(2 a^2 x^2-a x-1\right)\right)-3 \sqrt{c} \log (1-a x)}{48 a^3}","\frac{c x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{12 a \sqrt{c-\frac{c}{a x}}}-\frac{c x \sqrt{1-\frac{1}{a^2 x^2}}}{8 a^2 \sqrt{c-\frac{c}{a x}}}+\frac{c x^3 \sqrt{1-\frac{1}{a^2 x^2}}}{3 \sqrt{c-\frac{c}{a x}}}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{8 a^3}",1,"((2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(-3 + 2*a*x + 8*a^2*x^2))/(-1 + a*x) - 3*Sqrt[c]*Log[1 - a*x] + 3*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)])/(48*a^3)","A",0
491,1,148,124,0.3449208,"\int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x \, dx","Integrate[E^ArcCoth[a*x]*Sqrt[c - c/(a*x)]*x,x]","\frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} (2 a x+1) \sqrt{c-\frac{c}{a x}}+\sqrt{c} (1-a x) \log \left(2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(2 a^2 x^2-a x-1\right)\right)+\sqrt{c} (a x-1) \log (1-a x)}{8 a^2 (a x-1)}","\frac{c x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 \sqrt{c-\frac{c}{a x}}}+\frac{c x \sqrt{1-\frac{1}{a^2 x^2}}}{4 a \sqrt{c-\frac{c}{a x}}}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{4 a^2}",1,"(2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(1 + 2*a*x) + Sqrt[c]*(-1 + a*x)*Log[1 - a*x] + Sqrt[c]*(1 - a*x)*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)])/(8*a^2*(-1 + a*x))","A",0
492,1,66,78,0.0488465,"\int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx","Integrate[E^ArcCoth[a*x]*Sqrt[c - c/(a*x)],x]","\frac{\sqrt{c-\frac{c}{a x}} \left(a x+\sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)+1\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{c x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{a}",1,"(Sqrt[c - c/(a*x)]*(1 + a*x + Sqrt[1 + 1/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",0
493,1,132,76,0.2414008,"\int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x} \, dx","Integrate[(E^ArcCoth[a*x]*Sqrt[c - c/(a*x)])/x,x]","\frac{-2 a x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+\sqrt{c} (a x-1) \log \left(2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(2 a^2 x^2-a x-1\right)\right)+\sqrt{c} (1-a x) \log (1-a x)}{a x-1}","2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)-\frac{2 c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}",1,"(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x + Sqrt[c]*(1 - a*x)*Log[1 - a*x] + Sqrt[c]*(-1 + a*x)*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)])/(-1 + a*x)","A",0
494,1,45,37,0.092884,"\int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx","Integrate[(E^ArcCoth[a*x]*Sqrt[c - c/(a*x)])/x^2,x]","-\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} (a x+1) \sqrt{c-\frac{c}{a x}}}{3 a x-3}","-\frac{2 a c^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{3 \left(c-\frac{c}{a x}\right)^{3/2}}",1,"(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(1 + a*x))/(-3 + 3*a*x)","A",1
495,1,58,77,0.1087797,"\int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^3} \, dx","Integrate[(E^ArcCoth[a*x]*Sqrt[c - c/(a*x)])/x^3,x]","\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2-a x-3\right) \sqrt{c-\frac{c}{a x}}}{15 x (a x-1)}","\frac{2 a^2 c \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{5 \sqrt{c-\frac{c}{a x}}}-\frac{2 a^2 c^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{15 \left(c-\frac{c}{a x}\right)^{3/2}}",1,"(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(-3 - a*x + 2*a^2*x^2))/(15*x*(-1 + a*x))","A",1
496,1,66,117,0.1212141,"\int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx","Integrate[(E^ArcCoth[a*x]*Sqrt[c - c/(a*x)])/x^4,x]","-\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left(8 a^3 x^3-4 a^2 x^2+3 a x+15\right) \sqrt{c-\frac{c}{a x}}}{105 x^2 (a x-1)}","-\frac{2 a c^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{7 x^2 \left(c-\frac{c}{a x}\right)^{3/2}}+\frac{8 a^3 c^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{105 \left(c-\frac{c}{a x}\right)^{3/2}}-\frac{8 a^3 c \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{35 \sqrt{c-\frac{c}{a x}}}",1,"(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(15 + 3*a*x - 4*a^2*x^2 + 8*a^3*x^3))/(105*x^2*(-1 + a*x))","A",1
497,1,74,159,0.1286991,"\int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^5} \, dx","Integrate[(E^ArcCoth[a*x]*Sqrt[c - c/(a*x)])/x^5,x]","\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left(16 a^4 x^4-8 a^3 x^3+6 a^2 x^2-5 a x-35\right) \sqrt{c-\frac{c}{a x}}}{315 x^3 (a x-1)}","\frac{4 a^2 c^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{21 x^2 \left(c-\frac{c}{a x}\right)^{3/2}}-\frac{2 a c^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{9 x^3 \left(c-\frac{c}{a x}\right)^{3/2}}-\frac{16 a^4 c^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{315 \left(c-\frac{c}{a x}\right)^{3/2}}+\frac{16 a^4 c \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{105 \sqrt{c-\frac{c}{a x}}}",1,"(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(-35 - 5*a*x + 6*a^2*x^2 - 8*a^3*x^3 + 16*a^4*x^4))/(315*x^3*(-1 + a*x))","A",1
498,1,50,130,0.0404331,"\int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^3 \, dx","Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)]*x^3,x]","\frac{\sqrt{c-\frac{c}{a x}} \left(a^4 x^4+15 \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};1-\frac{1}{a x}\right)\right)}{4 a^4}","\frac{75 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{64 a^4}+\frac{75 x \sqrt{c-\frac{c}{a x}}}{64 a^3}+\frac{25 x^2 \sqrt{c-\frac{c}{a x}}}{32 a^2}+\frac{1}{4} x^4 \sqrt{c-\frac{c}{a x}}+\frac{5 x^3 \sqrt{c-\frac{c}{a x}}}{8 a}",1,"(Sqrt[c - c/(a*x)]*(a^4*x^4 + 15*Hypergeometric2F1[1/2, 4, 3/2, 1 - 1/(a*x)]))/(4*a^4)","C",0
499,1,50,105,0.0383102,"\int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx","Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)]*x^2,x]","\frac{\sqrt{c-\frac{c}{a x}} \left(a^3 x^3+11 \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\frac{1}{a x}\right)\right)}{3 a^3}","\frac{11 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{8 a^3}+\frac{11 x \sqrt{c-\frac{c}{a x}}}{8 a^2}+\frac{1}{3} x^3 \sqrt{c-\frac{c}{a x}}+\frac{11 x^2 \sqrt{c-\frac{c}{a x}}}{12 a}",1,"(Sqrt[c - c/(a*x)]*(a^3*x^3 + 11*Hypergeometric2F1[1/2, 3, 3/2, 1 - 1/(a*x)]))/(3*a^3)","C",0
500,1,77,80,0.0813733,"\int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x \, dx","Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)]*x,x]","\frac{\sqrt{c-\frac{c}{a x}} \left(a x \sqrt{1-\frac{1}{a x}} (2 a x+7)+7 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}}\right)\right)}{4 a^2 \sqrt{1-\frac{1}{a x}}}","\frac{7 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{4 a^2}+\frac{1}{2} x^2 \sqrt{c-\frac{c}{a x}}+\frac{7 x \sqrt{c-\frac{c}{a x}}}{4 a}",1,"(Sqrt[c - c/(a*x)]*(a*Sqrt[1 - 1/(a*x)]*x*(7 + 2*a*x) + 7*ArcTanh[Sqrt[1 - 1/(a*x)]]))/(4*a^2*Sqrt[1 - 1/(a*x)])","A",0
501,1,50,50,0.0440508,"\int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx","Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)],x]","x \sqrt{c-\frac{c}{a x}}+\frac{3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}","x \sqrt{c-\frac{c}{a x}}+\frac{3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}",1,"Sqrt[c - c/(a*x)]*x + (3*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a","A",1
502,1,47,47,0.0365235,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x,x]","2 \sqrt{c-\frac{c}{a x}}+2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)","2 \sqrt{c-\frac{c}{a x}}+2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)",1,"2*Sqrt[c - c/(a*x)] + 2*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]]","A",1
503,1,28,42,0.0387423,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^2,x]","\frac{2 (5 a x+1) \sqrt{c-\frac{c}{a x}}}{3 x}","4 a \sqrt{c-\frac{c}{a x}}-\frac{2 a \left(c-\frac{c}{a x}\right)^{3/2}}{3 c}",1,"(2*Sqrt[c - c/(a*x)]*(1 + 5*a*x))/(3*x)","A",1
504,1,36,69,0.0470248,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^3} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^3,x]","\frac{2 \left(6 a^2 x^2+3 a x+1\right) \sqrt{c-\frac{c}{a x}}}{5 x^2}","\frac{2 a^2 \left(c-\frac{c}{a x}\right)^{5/2}}{5 c^2}-\frac{2 a^2 \left(c-\frac{c}{a x}\right)^{3/2}}{c}+4 a^2 \sqrt{c-\frac{c}{a x}}",1,"(2*Sqrt[c - c/(a*x)]*(1 + 3*a*x + 6*a^2*x^2))/(5*x^2)","A",1
505,1,44,96,0.054506,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^4,x]","\frac{2 \left(104 a^3 x^3+52 a^2 x^2+39 a x+15\right) \sqrt{c-\frac{c}{a x}}}{105 x^3}","-\frac{2 a^3 \left(c-\frac{c}{a x}\right)^{7/2}}{7 c^3}+\frac{8 a^3 \left(c-\frac{c}{a x}\right)^{5/2}}{5 c^2}-\frac{10 a^3 \left(c-\frac{c}{a x}\right)^{3/2}}{3 c}+4 a^3 \sqrt{c-\frac{c}{a x}}",1,"(2*Sqrt[c - c/(a*x)]*(15 + 39*a*x + 52*a^2*x^2 + 104*a^3*x^3))/(105*x^3)","A",1
506,1,52,121,0.0682832,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^5} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^5,x]","\frac{2 \left(272 a^4 x^4+136 a^3 x^3+102 a^2 x^2+85 a x+35\right) \sqrt{c-\frac{c}{a x}}}{315 x^4}","\frac{2 a^4 \left(c-\frac{c}{a x}\right)^{9/2}}{9 c^4}-\frac{10 a^4 \left(c-\frac{c}{a x}\right)^{7/2}}{7 c^3}+\frac{18 a^4 \left(c-\frac{c}{a x}\right)^{5/2}}{5 c^2}-\frac{14 a^4 \left(c-\frac{c}{a x}\right)^{3/2}}{3 c}+4 a^4 \sqrt{c-\frac{c}{a x}}",1,"(2*Sqrt[c - c/(a*x)]*(35 + 85*a*x + 102*a^2*x^2 + 136*a^3*x^3 + 272*a^4*x^4))/(315*x^4)","A",1
507,1,252,313,0.9875872,"\int e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^3 \, dx","Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)]*x^3,x]","\frac{1089 \sqrt{c} \log \left(2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(2 a^2 x^2-a x-1\right)\right)-768 \sqrt{2} \sqrt{c} \log \left(2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(3 a^2 x^2-2 a x-1\right)\right)+\frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(48 a^3 x^3+136 a^2 x^2+214 a x+447\right) \sqrt{c-\frac{c}{a x}}}{a x-1}-1089 \sqrt{c} \log (1-a x)+768 \sqrt{2} \sqrt{c} \log \left((a x-1)^2\right)}{384 a^4}","\frac{363 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{64 a^4 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{a^4 \sqrt{1-\frac{1}{a x}}}+\frac{149 x \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{64 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{107 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{96 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{x^4 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{4 \sqrt{1-\frac{1}{a x}}}+\frac{17 x^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{24 a \sqrt{1-\frac{1}{a x}}}",1,"((2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(447 + 214*a*x + 136*a^2*x^2 + 48*a^3*x^3))/(-1 + a*x) - 1089*Sqrt[c]*Log[1 - a*x] + 768*Sqrt[2]*Sqrt[c]*Log[(-1 + a*x)^2] + 1089*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)] - 768*Sqrt[2]*Sqrt[c]*Log[2*Sqrt[2]*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - 2*a*x + 3*a^2*x^2)])/(384*a^4)","A",0
508,1,244,261,0.6601646,"\int e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx","Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)]*x^2,x]","\frac{\frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(8 a^2 x^2+26 a x+57\right) \sqrt{c-\frac{c}{a x}}}{a x-1}+135 \sqrt{c} \log \left(2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(2 a^2 x^2-a x-1\right)\right)-96 \sqrt{2} \sqrt{c} \log \left(2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(3 a^2 x^2-2 a x-1\right)\right)-135 \sqrt{c} \log (1-a x)+96 \sqrt{2} \sqrt{c} \log \left((a x-1)^2\right)}{48 a^3}","\frac{45 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{8 a^3 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{a^3 \sqrt{1-\frac{1}{a x}}}+\frac{19 x \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{8 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{x^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{13 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{12 a \sqrt{1-\frac{1}{a x}}}",1,"((2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(57 + 26*a*x + 8*a^2*x^2))/(-1 + a*x) - 135*Sqrt[c]*Log[1 - a*x] + 96*Sqrt[2]*Sqrt[c]*Log[(-1 + a*x)^2] + 135*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)] - 96*Sqrt[2]*Sqrt[c]*Log[2*Sqrt[2]*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - 2*a*x + 3*a^2*x^2)])/(48*a^3)","A",0
509,1,236,209,0.5763399,"\int e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x \, dx","Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)]*x,x]","\frac{\frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} (2 a x+9) \sqrt{c-\frac{c}{a x}}}{a x-1}+23 \sqrt{c} \log \left(2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(2 a^2 x^2-a x-1\right)\right)-16 \sqrt{2} \sqrt{c} \log \left(2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(3 a^2 x^2-2 a x-1\right)\right)-23 \sqrt{c} \log (1-a x)+16 \sqrt{2} \sqrt{c} \log \left((a x-1)^2\right)}{8 a^2}","\frac{23 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{4 a^2 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{a^2 \sqrt{1-\frac{1}{a x}}}+\frac{x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{2 \sqrt{1-\frac{1}{a x}}}+\frac{9 x \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{4 a \sqrt{1-\frac{1}{a x}}}",1,"((2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(9 + 2*a*x))/(-1 + a*x) - 23*Sqrt[c]*Log[1 - a*x] + 16*Sqrt[2]*Sqrt[c]*Log[(-1 + a*x)^2] + 23*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)] - 16*Sqrt[2]*Sqrt[c]*Log[2*Sqrt[2]*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - 2*a*x + 3*a^2*x^2)])/(8*a^2)","A",0
510,1,93,152,0.0863271,"\int e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx","Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)],x]","\frac{\sqrt{c-\frac{c}{a x}} \left(a x \sqrt{\frac{1}{a x}+1}+5 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)-4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)\right)}{a \sqrt{1-\frac{1}{a x}}}","\frac{x \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{5 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{a \sqrt{1-\frac{1}{a x}}}",1,"(Sqrt[c - c/(a*x)]*(a*Sqrt[1 + 1/(a*x)]*x + 5*ArcTanh[Sqrt[1 + 1/(a*x)]] - 4*Sqrt[2]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]]))/(a*Sqrt[1 - 1/(a*x)])","A",1
511,1,218,146,0.3953502,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x,x]","\frac{2 a x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}}{a x-1}+\sqrt{c} \log \left(2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(2 a^2 x^2-a x-1\right)\right)-2 \sqrt{2} \sqrt{c} \log \left(2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(3 a^2 x^2-2 a x-1\right)\right)-\sqrt{c} \log (1-a x)+2 \sqrt{2} \sqrt{c} \log \left((a x-1)^2\right)","\frac{2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{\sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{\sqrt{1-\frac{1}{a x}}}",1,"(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x)/(-1 + a*x) - Sqrt[c]*Log[1 - a*x] + 2*Sqrt[2]*Sqrt[c]*Log[(-1 + a*x)^2] + Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)] - 2*Sqrt[2]*Sqrt[c]*Log[2*Sqrt[2]*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - 2*a*x + 3*a^2*x^2)]","A",0
512,1,155,125,0.2479033,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^2,x]","\frac{2 a \left(\sqrt{1-\frac{1}{a^2 x^2}} (7 a x+1) \sqrt{c-\frac{c}{a x}}-3 \sqrt{2} \sqrt{c} (a x-1) \log \left(2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(3 a^2 x^2-2 a x-1\right)\right)+3 \sqrt{2} \sqrt{c} (a x-1) \log \left((a x-1)^2\right)\right)}{3 a x-3}","\frac{2 a c^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{3 \left(c-\frac{c}{a x}\right)^{3/2}}+\frac{4 a c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}-4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{2} \sqrt{c-\frac{c}{a x}}}\right)",1,"(2*a*(Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(1 + 7*a*x) + 3*Sqrt[2]*Sqrt[c]*(-1 + a*x)*Log[(-1 + a*x)^2] - 3*Sqrt[2]*Sqrt[c]*(-1 + a*x)*Log[2*Sqrt[2]*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - 2*a*x + 3*a^2*x^2)]))/(-3 + 3*a*x)","A",0
513,1,162,170,0.280616,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^3} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^3,x]","\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left(38 a^2 x^2+11 a x+3\right) \sqrt{c-\frac{c}{a x}}}{15 x (a x-1)}-2 \sqrt{2} a^2 \sqrt{c} \log \left(2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(3 a^2 x^2-2 a x-1\right)\right)+2 \sqrt{2} a^2 \sqrt{c} \log \left((a x-1)^2\right)","\frac{2 a^2 c^3 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{5 \left(c-\frac{c}{a x}\right)^{5/2}}+\frac{2 a^2 c^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{3 \left(c-\frac{c}{a x}\right)^{3/2}}+\frac{4 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}-4 \sqrt{2} a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{2} \sqrt{c-\frac{c}{a x}}}\right)",1,"(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(3 + 11*a*x + 38*a^2*x^2))/(15*x*(-1 + a*x)) + 2*Sqrt[2]*a^2*Sqrt[c]*Log[(-1 + a*x)^2] - 2*Sqrt[2]*a^2*Sqrt[c]*Log[2*Sqrt[2]*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - 2*a*x + 3*a^2*x^2)]","A",0
514,1,170,209,0.3089697,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^4,x]","2 \sqrt{2} a^3 \sqrt{c} \log \left((a x-1)^2\right)-2 \sqrt{2} a^3 \sqrt{c} \log \left(2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(3 a^2 x^2-2 a x-1\right)\right)+\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left(52 a^3 x^3+16 a^2 x^2+9 a x+3\right) \sqrt{c-\frac{c}{a x}}}{21 x^2 (a x-1)}","\frac{4 a^3 c^3 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{7 \left(c-\frac{c}{a x}\right)^{5/2}}-\frac{2 a^3 c^2 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{7 \left(c-\frac{c}{a x}\right)^{3/2}}+\frac{2 a^3 c^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{3 \left(c-\frac{c}{a x}\right)^{3/2}}+\frac{4 a^3 c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}-4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{2} \sqrt{c-\frac{c}{a x}}}\right)",1,"(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(3 + 9*a*x + 16*a^2*x^2 + 52*a^3*x^3))/(21*x^2*(-1 + a*x)) + 2*Sqrt[2]*a^3*Sqrt[c]*Log[(-1 + a*x)^2] - 2*Sqrt[2]*a^3*Sqrt[c]*Log[2*Sqrt[2]*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - 2*a*x + 3*a^2*x^2)]","A",0
515,1,178,303,0.3145759,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^5} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)])/x^5,x]","2 \sqrt{2} a^4 \sqrt{c} \log \left((a x-1)^2\right)-2 \sqrt{2} a^4 \sqrt{c} \log \left(2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(3 a^2 x^2-2 a x-1\right)\right)+\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left(788 a^4 x^4+236 a^3 x^3+138 a^2 x^2+95 a x+35\right) \sqrt{c-\frac{c}{a x}}}{315 x^3 (a x-1)}","\frac{2 a^4 \left(\frac{1}{a x}+1\right)^{9/2} \sqrt{c-\frac{c}{a x}}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{2 a^4 \left(\frac{1}{a x}+1\right)^{7/2} \sqrt{c-\frac{c}{a x}}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left(\frac{1}{a x}+1\right)^{5/2} \sqrt{c-\frac{c}{a x}}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{4 a^4 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^4 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right)}{\sqrt{1-\frac{1}{a x}}}",1,"(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(35 + 95*a*x + 138*a^2*x^2 + 236*a^3*x^3 + 788*a^4*x^4))/(315*x^3*(-1 + a*x)) + 2*Sqrt[2]*a^4*Sqrt[c]*Log[(-1 + a*x)^2] - 2*Sqrt[2]*a^4*Sqrt[c]*Log[2*Sqrt[2]*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - 2*a*x + 3*a^2*x^2)]","A",0
516,1,93,126,0.0670145,"\int e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^m \, dx","Integrate[(Sqrt[c - c/(a*x)]*x^m)/E^ArcCoth[a*x],x]","\frac{x^m \sqrt{c-\frac{c}{a x}} \left(2 a m x \sqrt{\frac{1}{a x}+1}-(4 m+3) \, _2F_1\left(\frac{1}{2},-m;1-m;-\frac{1}{a x}\right)\right)}{2 a m (m+1) \sqrt{1-\frac{1}{a x}}}","\frac{\sqrt{\frac{1}{a x}+1} x^{m+1} \sqrt{c-\frac{c}{a x}}}{(m+1) \sqrt{1-\frac{1}{a x}}}-\frac{(4 m+3) x^m \sqrt{c-\frac{c}{a x}} \, _2F_1\left(\frac{1}{2},-m;1-m;-\frac{1}{a x}\right)}{2 a m (m+1) \sqrt{1-\frac{1}{a x}}}",1,"(Sqrt[c - c/(a*x)]*x^m*(2*a*m*Sqrt[1 + 1/(a*x)]*x - (3 + 4*m)*Hypergeometric2F1[1/2, -m, 1 - m, -(1/(a*x))]))/(2*a*m*(1 + m)*Sqrt[1 - 1/(a*x)])","A",1
517,1,147,164,0.5107029,"\int e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx","Integrate[(Sqrt[c - c/(a*x)]*x^2)/E^ArcCoth[a*x],x]","\frac{\frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(8 a^2 x^2-22 a x+33\right) \sqrt{c-\frac{c}{a x}}}{a x-1}-33 \sqrt{c} \log \left(2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(2 a^2 x^2-a x-1\right)\right)+33 \sqrt{c} \log (1-a x)}{48 a^3}","-\frac{11 c x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{12 a \sqrt{c-\frac{c}{a x}}}+\frac{11 c x \sqrt{1-\frac{1}{a^2 x^2}}}{8 a^2 \sqrt{c-\frac{c}{a x}}}+\frac{c x^3 \sqrt{1-\frac{1}{a^2 x^2}}}{3 \sqrt{c-\frac{c}{a x}}}-\frac{11 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{8 a^3}",1,"((2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(33 - 22*a*x + 8*a^2*x^2))/(-1 + a*x) + 33*Sqrt[c]*Log[1 - a*x] - 33*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)])/(48*a^3)","A",0
518,1,139,124,0.4865754,"\int e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x \, dx","Integrate[(Sqrt[c - c/(a*x)]*x)/E^ArcCoth[a*x],x]","\frac{x^2 \sqrt{1-\frac{1}{a^2 x^2}} (2 a x-7) \sqrt{c-\frac{c}{a x}}}{4 a x-4}+\frac{7 \sqrt{c} \log \left(2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(2 a^2 x^2-a x-1\right)\right)}{8 a^2}-\frac{7 \sqrt{c} \log (1-a x)}{8 a^2}","\frac{c x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 \sqrt{c-\frac{c}{a x}}}-\frac{7 c x \sqrt{1-\frac{1}{a^2 x^2}}}{4 a \sqrt{c-\frac{c}{a x}}}+\frac{7 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{4 a^2}",1,"(Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(-7 + 2*a*x))/(-4 + 4*a*x) - (7*Sqrt[c]*Log[1 - a*x])/(8*a^2) + (7*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)])/(8*a^2)","A",0
519,1,65,79,0.045514,"\int e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx","Integrate[Sqrt[c - c/(a*x)]/E^ArcCoth[a*x],x]","\frac{\sqrt{c-\frac{c}{a x}} \left(x \sqrt{\frac{1}{a x}+1}-\frac{3 \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a}\right)}{\sqrt{1-\frac{1}{a x}}}","\frac{c x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}-\frac{3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)}{a}",1,"(Sqrt[c - c/(a*x)]*(Sqrt[1 + 1/(a*x)]*x - (3*ArcTanh[Sqrt[1 + 1/(a*x)]])/a))/Sqrt[1 - 1/(a*x)]","A",0
520,1,132,76,0.2522871,"\int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x} \, dx","Integrate[Sqrt[c - c/(a*x)]/(E^ArcCoth[a*x]*x),x]","\frac{2 a x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+\sqrt{c} (a x-1) \log \left(2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(2 a^2 x^2-a x-1\right)\right)+\sqrt{c} (1-a x) \log (1-a x)}{a x-1}","\frac{2 c \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}+2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right)",1,"(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x + Sqrt[c]*(1 - a*x)*Log[1 - a*x] + Sqrt[c]*(-1 + a*x)*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)])/(-1 + a*x)","A",0
521,1,46,70,0.0886015,"\int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx","Integrate[Sqrt[c - c/(a*x)]/(E^ArcCoth[a*x]*x^2),x]","-\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} (5 a x-1) \sqrt{c-\frac{c}{a x}}}{3 a x-3}","-\frac{2}{3} a \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{8 a c \sqrt{1-\frac{1}{a^2 x^2}}}{3 \sqrt{c-\frac{c}{a x}}}",1,"(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(-1 + 5*a*x))/(-3 + 3*a*x)","A",1
522,1,58,113,0.1190164,"\int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^3} \, dx","Integrate[Sqrt[c - c/(a*x)]/(E^ArcCoth[a*x]*x^3),x]","\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left(6 a^2 x^2-3 a x+1\right) \sqrt{c-\frac{c}{a x}}}{5 x (a x-1)}","\frac{2 a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(c-\frac{c}{a x}\right)^{3/2}}{5 c}+\frac{2}{5} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+\frac{8 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}{5 \sqrt{c-\frac{c}{a x}}}",1,"(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(1 - 3*a*x + 6*a^2*x^2))/(5*x*(-1 + a*x))","A",1
523,1,66,149,0.1284805,"\int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx","Integrate[Sqrt[c - c/(a*x)]/(E^ArcCoth[a*x]*x^4),x]","-\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left(104 a^3 x^3-52 a^2 x^2+39 a x-15\right) \sqrt{c-\frac{c}{a x}}}{105 x^2 (a x-1)}","-\frac{26 a c \sqrt{1-\frac{1}{a^2 x^2}}}{35 x^2 \sqrt{c-\frac{c}{a x}}}+\frac{2 c \sqrt{1-\frac{1}{a^2 x^2}}}{7 x^3 \sqrt{c-\frac{c}{a x}}}-\frac{104}{105} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{104 a^3 c \sqrt{1-\frac{1}{a^2 x^2}}}{105 \sqrt{c-\frac{c}{a x}}}",1,"(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(-15 + 39*a*x - 52*a^2*x^2 + 104*a^3*x^3))/(105*x^2*(-1 + a*x))","A",1
524,1,116,172,0.1628292,"\int e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^3 \, dx","Integrate[(Sqrt[c - c/(a*x)]*x^3)/E^(2*ArcCoth[a*x]),x]","\frac{a x \left(48 a^3 x^3-136 a^2 x^2+214 a x-447\right) \sqrt{c-\frac{c}{a x}}+1089 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)-768 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{192 a^4}","\frac{363 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{64 a^4}-\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a^4}-\frac{149 x \sqrt{c-\frac{c}{a x}}}{64 a^3}+\frac{107 x^2 \sqrt{c-\frac{c}{a x}}}{96 a^2}+\frac{1}{4} x^4 \sqrt{c-\frac{c}{a x}}-\frac{17 x^3 \sqrt{c-\frac{c}{a x}}}{24 a}",1,"(a*Sqrt[c - c/(a*x)]*x*(-447 + 214*a*x - 136*a^2*x^2 + 48*a^3*x^3) + 1089*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]] - 768*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/(192*a^4)","A",1
525,1,108,147,0.1173956,"\int e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx","Integrate[(Sqrt[c - c/(a*x)]*x^2)/E^(2*ArcCoth[a*x]),x]","\frac{a x \left(8 a^2 x^2-26 a x+57\right) \sqrt{c-\frac{c}{a x}}-135 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)+96 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{24 a^3}","-\frac{45 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{8 a^3}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a^3}+\frac{19 x \sqrt{c-\frac{c}{a x}}}{8 a^2}+\frac{1}{3} x^3 \sqrt{c-\frac{c}{a x}}-\frac{13 x^2 \sqrt{c-\frac{c}{a x}}}{12 a}",1,"(a*Sqrt[c - c/(a*x)]*x*(57 - 26*a*x + 8*a^2*x^2) - 135*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]] + 96*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/(24*a^3)","A",1
526,1,100,122,0.0952473,"\int e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x \, dx","Integrate[(Sqrt[c - c/(a*x)]*x)/E^(2*ArcCoth[a*x]),x]","\frac{a x (2 a x-9) \sqrt{c-\frac{c}{a x}}+23 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)-16 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{4 a^2}","\frac{23 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{4 a^2}-\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a^2}+\frac{1}{2} x^2 \sqrt{c-\frac{c}{a x}}-\frac{9 x \sqrt{c-\frac{c}{a x}}}{4 a}",1,"(a*Sqrt[c - c/(a*x)]*x*(-9 + 2*a*x) + 23*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]] - 16*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/(4*a^2)","A",1
527,1,92,92,0.0524858,"\int e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx","Integrate[Sqrt[c - c/(a*x)]/E^(2*ArcCoth[a*x]),x]","x \sqrt{c-\frac{c}{a x}}-\frac{5 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a}","x \sqrt{c-\frac{c}{a x}}-\frac{5 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)}{a}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)}{a}",1,"Sqrt[c - c/(a*x)]*x - (5*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/a + (4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])])/a","A",1
528,1,86,86,0.0435656,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x} \, dx","Integrate[Sqrt[c - c/(a*x)]/(E^(2*ArcCoth[a*x])*x),x]","2 \sqrt{c-\frac{c}{a x}}+2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)-4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)","2 \sqrt{c-\frac{c}{a x}}+2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right)-4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)",1,"2*Sqrt[c - c/(a*x)] + 2*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]] - 4*Sqrt[2]*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]","A",1
529,1,69,82,0.0727379,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx","Integrate[Sqrt[c - c/(a*x)]/(E^(2*ArcCoth[a*x])*x^2),x]","\frac{2 (1-7 a x) \sqrt{c-\frac{c}{a x}}}{3 x}+4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)","-\frac{2 a \left(c-\frac{c}{a x}\right)^{3/2}}{3 c}-4 a \sqrt{c-\frac{c}{a x}}+4 \sqrt{2} a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)",1,"(2*Sqrt[c - c/(a*x)]*(1 - 7*a*x))/(3*x) + 4*Sqrt[2]*a*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]","A",1
530,1,79,113,0.0858618,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^3} \, dx","Integrate[Sqrt[c - c/(a*x)]/(E^(2*ArcCoth[a*x])*x^3),x]","\frac{2 \left(38 a^2 x^2-11 a x+3\right) \sqrt{c-\frac{c}{a x}}}{15 x^2}-4 \sqrt{2} a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)","\frac{2 a^2 \left(c-\frac{c}{a x}\right)^{5/2}}{5 c^2}+\frac{2 a^2 \left(c-\frac{c}{a x}\right)^{3/2}}{3 c}+4 a^2 \sqrt{c-\frac{c}{a x}}-4 \sqrt{2} a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)",1,"(2*Sqrt[c - c/(a*x)]*(3 - 11*a*x + 38*a^2*x^2))/(15*x^2) - 4*Sqrt[2]*a^2*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]","A",1
531,1,87,113,0.1421409,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx","Integrate[Sqrt[c - c/(a*x)]/(E^(2*ArcCoth[a*x])*x^4),x]","4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)+\frac{2 \left(-52 a^3 x^3+16 a^2 x^2-9 a x+3\right) \sqrt{c-\frac{c}{a x}}}{21 x^3}","-\frac{2 a^3 \left(c-\frac{c}{a x}\right)^{7/2}}{7 c^3}-\frac{2 a^3 \left(c-\frac{c}{a x}\right)^{3/2}}{3 c}-4 a^3 \sqrt{c-\frac{c}{a x}}+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)",1,"(2*Sqrt[c - c/(a*x)]*(3 - 9*a*x + 16*a^2*x^2 - 52*a^3*x^3))/(21*x^3) + 4*Sqrt[2]*a^3*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]","A",1
532,1,95,163,0.1483215,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^5} \, dx","Integrate[Sqrt[c - c/(a*x)]/(E^(2*ArcCoth[a*x])*x^5),x]","\frac{2 \left(788 a^4 x^4-236 a^3 x^3+138 a^2 x^2-95 a x+35\right) \sqrt{c-\frac{c}{a x}}}{315 x^4}-4 \sqrt{2} a^4 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)","\frac{2 a^4 \left(c-\frac{c}{a x}\right)^{9/2}}{9 c^4}-\frac{2 a^4 \left(c-\frac{c}{a x}\right)^{7/2}}{7 c^3}+\frac{2 a^4 \left(c-\frac{c}{a x}\right)^{5/2}}{5 c^2}+\frac{2 a^4 \left(c-\frac{c}{a x}\right)^{3/2}}{3 c}+4 a^4 \sqrt{c-\frac{c}{a x}}-4 \sqrt{2} a^4 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right)",1,"(2*Sqrt[c - c/(a*x)]*(35 - 95*a*x + 138*a^2*x^2 - 236*a^3*x^3 + 788*a^4*x^4))/(315*x^4) - 4*Sqrt[2]*a^4*Sqrt[c]*ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*Sqrt[c])]","A",1
533,1,167,303,0.6331082,"\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^3 \, dx","Integrate[(Sqrt[c - c/(a*x)]*x^3)/E^(3*ArcCoth[a*x]),x]","\frac{3345 \sqrt{c} \log \left(2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(2 a^2 x^2-a x-1\right)\right)+\frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(48 a^4 x^4-200 a^3 x^3+446 a^2 x^2-1115 a x-3345\right) \sqrt{c-\frac{c}{a x}}}{a^2 x^2-1}-3345 \sqrt{c} \log (1-a x)}{384 a^4}","-\frac{1115 \sqrt{c-\frac{c}{a x}}}{64 a^4 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{1115 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{64 a^4 \sqrt{1-\frac{1}{a x}}}-\frac{1115 x \sqrt{c-\frac{c}{a x}}}{192 a^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{223 x^2 \sqrt{c-\frac{c}{a x}}}{96 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{x^4 \sqrt{c-\frac{c}{a x}}}{4 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{25 x^3 \sqrt{c-\frac{c}{a x}}}{24 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}",1,"((2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(-3345 - 1115*a*x + 446*a^2*x^2 - 200*a^3*x^3 + 48*a^4*x^4))/(-1 + a^2*x^2) - 3345*Sqrt[c]*Log[1 - a*x] + 3345*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)])/(384*a^4)","A",0
534,1,159,251,0.5407619,"\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx","Integrate[(Sqrt[c - c/(a*x)]*x^2)/E^(3*ArcCoth[a*x]),x]","\frac{-357 \sqrt{c} \log \left(2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(2 a^2 x^2-a x-1\right)\right)+\frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(8 a^3 x^3-38 a^2 x^2+119 a x+357\right) \sqrt{c-\frac{c}{a x}}}{a^2 x^2-1}+357 \sqrt{c} \log (1-a x)}{48 a^3}","\frac{119 \sqrt{c-\frac{c}{a x}}}{8 a^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{119 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{8 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{119 x \sqrt{c-\frac{c}{a x}}}{24 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{x^3 \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{19 x^2 \sqrt{c-\frac{c}{a x}}}{12 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}",1,"((2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(357 + 119*a*x - 38*a^2*x^2 + 8*a^3*x^3))/(-1 + a^2*x^2) + 357*Sqrt[c]*Log[1 - a*x] - 357*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)])/(48*a^3)","A",0
535,1,151,199,0.5411218,"\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x \, dx","Integrate[(Sqrt[c - c/(a*x)]*x)/E^(3*ArcCoth[a*x]),x]","\frac{x^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2-13 a x-47\right) \sqrt{c-\frac{c}{a x}}}{4 a^2 x^2-4}+\frac{47 \sqrt{c} \log \left(2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(2 a^2 x^2-a x-1\right)\right)}{8 a^2}-\frac{47 \sqrt{c} \log (1-a x)}{8 a^2}","-\frac{47 \sqrt{c-\frac{c}{a x}}}{4 a^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{47 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{4 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{x^2 \sqrt{c-\frac{c}{a x}}}{2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{13 x \sqrt{c-\frac{c}{a x}}}{4 a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(-47 - 13*a*x + 2*a^2*x^2))/(-4 + 4*a^2*x^2) - (47*Sqrt[c]*Log[1 - a*x])/(8*a^2) + (47*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)])/(8*a^2)","A",0
536,1,67,140,0.0529616,"\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx","Integrate[Sqrt[c - c/(a*x)]/E^(3*ArcCoth[a*x]),x]","\frac{\sqrt{c-\frac{c}{a x}} \left(a x-7 \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)+9\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{9 \sqrt{c-\frac{c}{a x}}}{a \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{7 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{a \sqrt{1-\frac{1}{a x}}}",1,"(Sqrt[c - c/(a*x)]*(9 + a*x - 7*Sqrt[1 + 1/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
537,1,131,134,0.4083917,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x} \, dx","Integrate[Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x),x]","-\frac{2 a x \sqrt{1-\frac{1}{a^2 x^2}} (5 a x+1) \sqrt{c-\frac{c}{a x}}}{a^2 x^2-1}+\sqrt{c} \log \left(2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left(2 a^2 x^2-a x-1\right)\right)-\sqrt{c} \log (1-a x)","-\frac{2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{8 \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}+\frac{2 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left(\sqrt{\frac{1}{a x}+1}\right)}{\sqrt{1-\frac{1}{a x}}}",1,"(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x*(1 + 5*a*x))/(-1 + a^2*x^2) - Sqrt[c]*Log[1 - a*x] + Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)]","A",0
538,1,58,109,0.1045671,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^2} \, dx","Integrate[Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^2),x]","\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left(23 a^2 x^2+10 a x-1\right) \sqrt{c-\frac{c}{a x}}}{3 a^2 x^2-3}","-\frac{2 a \left(c-\frac{c}{a x}\right)^{5/2}}{3 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{16 a \left(c-\frac{c}{a x}\right)^{3/2}}{3 c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{64 a \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(-1 + 10*a*x + 23*a^2*x^2))/(-3 + 3*a^2*x^2)","A",1
539,1,70,150,0.1191806,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^3} \, dx","Integrate[Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^3),x]","-\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left(158 a^3 x^3+79 a^2 x^2-16 a x+3\right) \sqrt{c-\frac{c}{a x}}}{15 x \left(a^2 x^2-1\right)}","-\frac{a^2 \left(c-\frac{c}{a x}\right)^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{7 a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left(c-\frac{c}{a x}\right)^{3/2}}{5 c}-\frac{56}{15} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}-\frac{224 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}{15 \sqrt{c-\frac{c}{a x}}}",1,"(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(3 - 16*a*x + 79*a^2*x^2 + 158*a^3*x^3))/(15*x*(-1 + a^2*x^2))","A",1
540,1,78,188,0.1335722,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx","Integrate[Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^4),x]","\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left(1336 a^4 x^4+668 a^3 x^3-167 a^2 x^2+66 a x-15\right) \sqrt{c-\frac{c}{a x}}}{105 x^2 \left(a^2 x^2-1\right)}","\frac{a^3 \left(c-\frac{c}{a x}\right)^{7/2}}{c^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left(c-\frac{c}{a x}\right)^{5/2}}{7 c^2}+\frac{59 a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left(c-\frac{c}{a x}\right)^{3/2}}{35 c}+\frac{472}{105} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+\frac{1888 a^3 c \sqrt{1-\frac{1}{a^2 x^2}}}{105 \sqrt{c-\frac{c}{a x}}}",1,"(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(-15 + 66*a*x - 167*a^2*x^2 + 668*a^3*x^3 + 1336*a^4*x^4))/(105*x^2*(-1 + a^2*x^2))","A",1
541,1,86,289,0.1506093,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^5} \, dx","Integrate[Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^5),x]","-\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left(656 a^5 x^5+328 a^4 x^4-82 a^3 x^3+41 a^2 x^2-20 a x+5\right) \sqrt{c-\frac{c}{a x}}}{45 x^3 \left(a^2 x^2-1\right)}","-\frac{2 a^4 \left(\frac{1}{a x}+1\right)^{9/2} \sqrt{c-\frac{c}{a x}}}{9 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left(\frac{1}{a x}+1\right)^{7/2} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{38 a^4 \left(\frac{1}{a x}+1\right)^{5/2} \sqrt{c-\frac{c}{a x}}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{50 a^4 \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}-\frac{32 a^4 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{8 a^4 \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}",1,"(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(5 - 20*a*x + 41*a^2*x^2 - 82*a^3*x^3 + 328*a^4*x^4 + 656*a^5*x^5))/(45*x^3*(-1 + a^2*x^2))","A",1
542,1,155,185,0.37986,"\int e^{n \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right) \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - c/(a*x)),x]","\frac{c e^{n \coth ^{-1}(a x)} \left(n \left(-e^{2 \coth ^{-1}(a x)}\right) \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;-e^{2 \coth ^{-1}(a x)}\right)+(n-1) n e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)+(n+2) \left(\, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;-e^{2 \coth ^{-1}(a x)}\right)+(n-1) \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)+a n x\right)\right)}{a n (n+2)}","-\frac{c 2^{n/2} \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}} \, _2F_1\left(1-\frac{n}{2},1-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{2 a}\right)}{a (2-n)}-\frac{2 c (1-n) \left(\frac{1}{a x}+1\right)^{n/2} \left(1-\frac{1}{a x}\right)^{-n/2} \, _2F_1\left(1,\frac{n}{2};\frac{n+2}{2};\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\right)}{a n}+c x \left(\frac{1}{a x}+1\right)^{n/2} \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}}",1,"(c*E^(n*ArcCoth[a*x])*(-(E^(2*ArcCoth[a*x])*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, -E^(2*ArcCoth[a*x])]) + E^(2*ArcCoth[a*x])*(-1 + n)*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(a*n*x + Hypergeometric2F1[1, n/2, 1 + n/2, -E^(2*ArcCoth[a*x])] + (-1 + n)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(a*n*(2 + n))","A",0
543,1,97,113,0.4078408,"\int \frac{e^{n \coth ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - c/(a*x)),x]","\frac{e^{n \coth ^{-1}(a x)} \left(n (n+1) e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)+(n+2) \left((n+1) \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)+a n x-1\right)\right)}{a c n (n+2)}","\frac{x \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}}}{c}-\frac{2 (n+1) \left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} \, _2F_1\left(1,-\frac{n}{2};1-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a c n}",1,"(E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n*(1 + n)*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(-1 + a*n*x + (1 + n)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(a*c*n*(2 + n))","A",0
544,1,113,166,0.0721259,"\int \frac{e^{n \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^2} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - c/(a*x))^2,x]","\frac{\left(1-\frac{1}{a x}\right)^{-n/2} \left(\frac{1}{a x}+1\right)^{n/2} \left(n (a x+1) (n (a x-1)+2 a x-3)-2 (n+2)^2 (a x-1) \, _2F_1\left(1,-\frac{n}{2};1-\frac{n}{2};\frac{a x-1}{a x+1}\right)\right)}{a c^2 n (n+2) (a x-1)}","-\frac{2 (n+2) \left(\frac{1}{a x}+1\right)^{n/2} \left(1-\frac{1}{a x}\right)^{-n/2} \, _2F_1\left(1,-\frac{n}{2};1-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a c^2 n}-\frac{(n+3) \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-1}}{a c^2 (n+2)}+\frac{x \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-1}}{c^2}",1,"((1 + 1/(a*x))^(n/2)*(n*(1 + a*x)*(-3 + 2*a*x + n*(-1 + a*x)) - 2*(2 + n)^2*(-1 + a*x)*Hypergeometric2F1[1, -1/2*n, 1 - n/2, (-1 + a*x)/(1 + a*x)]))/(a*c^2*n*(2 + n)*(1 - 1/(a*x))^(n/2)*(-1 + a*x))","A",1
545,-1,0,111,180.0052046,"\int e^{n \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^{3/2} \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - c/(a*x))^(3/2),x]","\text{\$Aborted}","-\frac{2^{\frac{5}{2}-\frac{n}{2}} \left(c-\frac{c}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} F_1\left(\frac{n+2}{2};\frac{n-3}{2},2;\frac{n+4}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right)}{a (n+2) \left(1-\frac{1}{a x}\right)^{3/2}}",1,"$Aborted","F",-1
546,-1,0,111,180.0051463,"\int e^{n \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} \, dx","Integrate[E^(n*ArcCoth[a*x])*Sqrt[c - c/(a*x)],x]","\text{\$Aborted}","-\frac{2^{\frac{3}{2}-\frac{n}{2}} \sqrt{c-\frac{c}{a x}} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} F_1\left(\frac{n+2}{2};\frac{n-1}{2},2;\frac{n+4}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right)}{a (n+2) \sqrt{1-\frac{1}{a x}}}",1,"$Aborted","F",-1
547,-1,0,111,180.0080951,"\int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a x}}} \, dx","Integrate[E^(n*ArcCoth[a*x])/Sqrt[c - c/(a*x)],x]","\text{\$Aborted}","-\frac{2^{\frac{1}{2}-\frac{n}{2}} \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} F_1\left(\frac{n+2}{2};\frac{n+1}{2},2;\frac{n+4}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right)}{a (n+2) \sqrt{c-\frac{c}{a x}}}",1,"$Aborted","F",-1
548,-1,0,111,180.0053637,"\int \frac{e^{n \coth ^{-1}(a x)}}{\left(c-\frac{c}{a x}\right)^{3/2}} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - c/(a*x))^(3/2),x]","\text{\$Aborted}","-\frac{2^{-\frac{n}{2}-\frac{1}{2}} \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} F_1\left(\frac{n+2}{2};\frac{n+3}{2},2;\frac{n+4}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right)}{a (n+2) \left(c-\frac{c}{a x}\right)^{3/2}}",1,"$Aborted","F",-1
549,0,0,110,0.9657617,"\int e^{n \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^p \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - c/(a*x))^p,x]","\int e^{n \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^p \, dx","-\frac{2^{-\frac{n}{2}+p+1} \left(\frac{1}{a x}+1\right)^{\frac{n+2}{2}} \left(1-\frac{1}{a x}\right)^{-p} \left(c-\frac{c}{a x}\right)^p F_1\left(\frac{n+2}{2};\frac{1}{2} (n-2 p),2;\frac{n+4}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right)}{a (n+2)}",1,"Integrate[E^(n*ArcCoth[a*x])*(c - c/(a*x))^p, x]","F",-1
550,1,67,67,0.0280755,"\int e^{2 p \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^p \, dx","Integrate[E^(2*p*ArcCoth[a*x])*(c - c/(a*x))^p,x]","-\frac{\left(1-\frac{1}{a x}\right)^{-p} \left(\frac{1}{a x}+1\right)^{p+1} \left(c-\frac{c}{a x}\right)^p \, _2F_1\left(2,p+1;p+2;1+\frac{1}{a x}\right)}{a (p+1)}","-\frac{\left(1-\frac{1}{a x}\right)^{-p} \left(\frac{1}{a x}+1\right)^{p+1} \left(c-\frac{c}{a x}\right)^p \, _2F_1\left(2,p+1;p+2;1+\frac{1}{a x}\right)}{a (p+1)}",1,"-(((1 + 1/(a*x))^(1 + p)*(c - c/(a*x))^p*Hypergeometric2F1[2, 1 + p, 2 + p, 1 + 1/(a*x)])/(a*(1 + p)*(1 - 1/(a*x))^p))","A",1
551,0,0,93,0.6593774,"\int e^{-2 p \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^p \, dx","Integrate[(c - c/(a*x))^p/E^(2*p*ArcCoth[a*x]),x]","\int e^{-2 p \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^p \, dx","-\frac{4^p \left(1-\frac{1}{a x}\right)^{-p} \left(\frac{1}{a x}+1\right)^{1-p} F_1\left(1-p;-2 p,2;2-p;\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right) \left(c-\frac{c}{a x}\right)^p}{a (1-p)}",1,"Integrate[(c - c/(a*x))^p/E^(2*p*ArcCoth[a*x]), x]","F",-1
552,1,46,57,0.0242004,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^p \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a*x))^p,x]","\frac{\left(c-\frac{c}{a x}\right)^p \left(a p x-(p-2) \, _2F_1\left(1,p;p+1;1-\frac{1}{a x}\right)\right)}{a p}","\frac{(2-p) \left(c-\frac{c}{a x}\right)^p \, _2F_1\left(1,p;p+1;1-\frac{1}{a x}\right)}{a p}+x \left(c-\frac{c}{a x}\right)^p",1,"((c - c/(a*x))^p*(a*p*x - (-2 + p)*Hypergeometric2F1[1, p, 1 + p, 1 - 1/(a*x)]))/(a*p)","A",1
553,0,0,90,1.076197,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^p \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a*x))^p,x]","\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^p \, dx","-\frac{2^{p+\frac{1}{2}} \left(\frac{1}{a x}+1\right)^{3/2} \left(1-\frac{1}{a x}\right)^{-p} F_1\left(\frac{3}{2};\frac{1}{2}-p,2;\frac{5}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right) \left(c-\frac{c}{a x}\right)^p}{3 a}",1,"Integrate[E^ArcCoth[a*x]*(c - c/(a*x))^p, x]","F",-1
554,0,0,88,1.2300417,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^p \, dx","Integrate[(c - c/(a*x))^p/E^ArcCoth[a*x],x]","\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^p \, dx","-\frac{2^{p+\frac{3}{2}} \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{-p} F_1\left(\frac{1}{2};-p-\frac{1}{2},2;\frac{3}{2};\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right) \left(c-\frac{c}{a x}\right)^p}{a}",1,"Integrate[(c - c/(a*x))^p/E^ArcCoth[a*x], x]","F",-1
555,1,87,114,0.0508047,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a x}\right)^p \, dx","Integrate[(c - c/(a*x))^p/E^(2*ArcCoth[a*x]),x]","\frac{(a x-1)^2 \left(c-\frac{c}{a x}\right)^p \left(\, _2F_1\left(1,p+2;p+3;\frac{a-\frac{1}{x}}{2 a}\right)+2 (p+2) \left(a x-\, _2F_1\left(1,p+2;p+3;1-\frac{1}{a x}\right)\right)\right)}{2 a^3 (p+2) x^2}","\frac{\left(c-\frac{c}{a x}\right)^{p+2} \, _2F_1\left(1,p+2;p+3;\frac{a-\frac{1}{x}}{2 a}\right)}{2 a c^2 (p+2)}-\frac{\left(c-\frac{c}{a x}\right)^{p+2} \, _2F_1\left(1,p+2;p+3;1-\frac{1}{a x}\right)}{a c^2}+\frac{x \left(c-\frac{c}{a x}\right)^{p+2}}{c^2}",1,"((c - c/(a*x))^p*(-1 + a*x)^2*(Hypergeometric2F1[1, 2 + p, 3 + p, (a - x^(-1))/(2*a)] + 2*(2 + p)*(a*x - Hypergeometric2F1[1, 2 + p, 3 + p, 1 - 1/(a*x)])))/(2*a^3*(2 + p)*x^2)","A",1
556,1,111,393,0.1999873,"\int e^{\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^4 \, dx","Integrate[E^ArcCoth[a*x]*(c - a^2*c*x^2)^4,x]","\frac{c^4 \left(315 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+a x \sqrt{1-\frac{1}{a^2 x^2}} \left(128 a^8 x^8+144 a^7 x^7-512 a^6 x^6-600 a^5 x^5+768 a^4 x^4+978 a^3 x^3-512 a^2 x^2-837 a x+128\right)\right)}{1152 a}","\frac{1}{9} a^8 c^4 x^9 \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{11/2}-\frac{7}{72} a^7 c^4 x^8 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{11/2}+\frac{5}{72} a^6 c^4 x^7 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{11/2}-\frac{5}{144} a^5 c^4 x^6 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{11/2}+\frac{1}{144} a^4 c^4 x^5 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{9/2}+\frac{1}{64} a^3 c^4 x^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}+\frac{7}{192} a^2 c^4 x^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}+\frac{35}{384} a c^4 x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}+\frac{35}{128} c^4 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}+\frac{35 c^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{128 a}",1,"(c^4*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(128 - 837*a*x - 512*a^2*x^2 + 978*a^3*x^3 + 768*a^4*x^4 - 600*a^5*x^5 - 512*a^6*x^6 + 144*a^7*x^7 + 128*a^8*x^8) + 315*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(1152*a)","A",0
557,1,95,313,0.1829346,"\int e^{\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^3 \, dx","Integrate[E^ArcCoth[a*x]*(c - a^2*c*x^2)^3,x]","\frac{c^3 \left(105 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+a x \sqrt{1-\frac{1}{a^2 x^2}} \left(-48 a^6 x^6-56 a^5 x^5+144 a^4 x^4+182 a^3 x^3-144 a^2 x^2-231 a x+48\right)\right)}{336 a}","-\frac{1}{7} a^6 c^3 x^7 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{9/2}+\frac{5}{42} a^5 c^3 x^6 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{9/2}-\frac{1}{14} a^4 c^3 x^5 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{9/2}+\frac{1}{56} a^3 c^3 x^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}+\frac{1}{24} a^2 c^3 x^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}+\frac{5}{48} a c^3 x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}+\frac{5}{16} c^3 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}+\frac{5 c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{16 a}",1,"(c^3*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(48 - 231*a*x - 144*a^2*x^2 + 182*a^3*x^3 + 144*a^4*x^4 - 56*a^5*x^5 - 48*a^6*x^6) + 105*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(336*a)","A",0
558,1,79,233,0.1051196,"\int e^{\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^2 \, dx","Integrate[E^ArcCoth[a*x]*(c - a^2*c*x^2)^2,x]","\frac{c^2 \left(15 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+a x \sqrt{1-\frac{1}{a^2 x^2}} \left(8 a^4 x^4+10 a^3 x^3-16 a^2 x^2-25 a x+8\right)\right)}{40 a}","\frac{1}{5} a^4 c^2 x^5 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{7/2}-\frac{3}{20} a^3 c^2 x^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}+\frac{1}{20} a^2 c^2 x^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}+\frac{1}{8} a c^2 x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}+\frac{3}{8} c^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}+\frac{3 c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{8 a}",1,"(c^2*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(8 - 25*a*x - 16*a^2*x^2 + 10*a^3*x^3 + 8*a^4*x^4) + 15*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(40*a)","A",0
559,1,61,145,0.101857,"\int e^{\coth ^{-1}(a x)} \left(c-a^2 c x^2\right) \, dx","Integrate[E^ArcCoth[a*x]*(c - a^2*c*x^2),x]","\frac{c \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(-2 a^2 x^2-3 a x+2\right)+3 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{6 a}","-\frac{1}{3} a^2 c x^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}+\frac{1}{6} a c x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}+\frac{1}{2} c x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}+\frac{c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{2 a}",1,"(c*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(2 - 3*a*x - 2*a^2*x^2) + 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(6*a)","A",0
560,1,13,13,0.0458177,"\int \frac{e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx","Integrate[E^ArcCoth[a*x]/(c - a^2*c*x^2),x]","\frac{e^{\coth ^{-1}(a x)}}{a c}","\frac{e^{\coth ^{-1}(a x)}}{a c}",1,"E^ArcCoth[a*x]/(a*c)","A",1
561,1,50,51,0.1590131,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^2} \, dx","Integrate[E^ArcCoth[a*x]/(c - a^2*c*x^2)^2,x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2-2 a x-1\right)}{3 c^2 (a x-1)^2 (a x+1)}","\frac{2 e^{\coth ^{-1}(a x)}}{3 a c^2}-\frac{(1-2 a x) e^{\coth ^{-1}(a x)}}{3 a c^2 \left(1-a^2 x^2\right)}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(-1 - 2*a*x + 2*a^2*x^2))/(3*c^2*(-1 + a*x)^2*(1 + a*x))","A",0
562,1,66,85,0.1972346,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^3} \, dx","Integrate[E^ArcCoth[a*x]/(c - a^2*c*x^2)^3,x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(8 a^4 x^4-8 a^3 x^3-12 a^2 x^2+12 a x+3\right)}{15 c^3 (a x-1)^3 (a x+1)^2}","-\frac{(1-4 a x) e^{\coth ^{-1}(a x)}}{15 a c^3 \left(1-a^2 x^2\right)^2}-\frac{4 (1-2 a x) e^{\coth ^{-1}(a x)}}{15 a c^3 \left(1-a^2 x^2\right)}+\frac{8 e^{\coth ^{-1}(a x)}}{15 a c^3}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(3 + 12*a*x - 12*a^2*x^2 - 8*a^3*x^3 + 8*a^4*x^4))/(15*c^3*(-1 + a*x)^3*(1 + a*x)^2)","A",0
563,1,82,119,0.2792392,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^4} \, dx","Integrate[E^ArcCoth[a*x]/(c - a^2*c*x^2)^4,x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(16 a^6 x^6-16 a^5 x^5-40 a^4 x^4+40 a^3 x^3+30 a^2 x^2-30 a x-5\right)}{35 c^4 (a x-1)^4 (a x+1)^3}","-\frac{(1-6 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left(1-a^2 x^2\right)^3}-\frac{8 (1-2 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left(1-a^2 x^2\right)}-\frac{2 (1-4 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left(1-a^2 x^2\right)^2}+\frac{16 e^{\coth ^{-1}(a x)}}{35 a c^4}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(-5 - 30*a*x + 30*a^2*x^2 + 40*a^3*x^3 - 40*a^4*x^4 - 16*a^5*x^5 + 16*a^6*x^6))/(35*c^4*(-1 + a*x)^4*(1 + a*x)^3)","A",0
564,1,47,84,0.0407057,"\int e^{2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^5 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^5,x]","-\frac{c^5 (a x+1)^7 \left(105 a^4 x^4-504 a^3 x^3+938 a^2 x^2-812 a x+281\right)}{1155 a}","-\frac{c^5 (a x+1)^{11}}{11 a}+\frac{4 c^5 (a x+1)^{10}}{5 a}-\frac{8 c^5 (a x+1)^9}{3 a}+\frac{4 c^5 (a x+1)^8}{a}-\frac{16 c^5 (a x+1)^7}{7 a}",1,"-1/1155*(c^5*(1 + a*x)^7*(281 - 812*a*x + 938*a^2*x^2 - 504*a^3*x^3 + 105*a^4*x^4))/a","A",1
565,1,39,69,0.0314194,"\int e^{2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^4 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^4,x]","\frac{c^4 (a x+1)^6 \left(28 a^3 x^3-105 a^2 x^2+138 a x-65\right)}{252 a}","\frac{c^4 (a x+1)^9}{9 a}-\frac{3 c^4 (a x+1)^8}{4 a}+\frac{12 c^4 (a x+1)^7}{7 a}-\frac{4 c^4 (a x+1)^6}{3 a}",1,"(c^4*(1 + a*x)^6*(-65 + 138*a*x - 105*a^2*x^2 + 28*a^3*x^3))/(252*a)","A",1
566,1,31,52,0.0280836,"\int e^{2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^3 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^3,x]","-\frac{c^3 (a x+1)^5 \left(15 a^2 x^2-40 a x+29\right)}{105 a}","-\frac{c^3 (a x+1)^7}{7 a}+\frac{2 c^3 (a x+1)^6}{3 a}-\frac{4 c^3 (a x+1)^5}{5 a}",1,"-1/105*(c^3*(1 + a*x)^5*(29 - 40*a*x + 15*a^2*x^2))/a","A",1
567,1,23,35,0.0184487,"\int e^{2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^2 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^2,x]","\frac{c^2 (a x+1)^4 (2 a x-3)}{10 a}","\frac{c^2 (a x+1)^5}{5 a}-\frac{c^2 (a x+1)^4}{2 a}",1,"(c^2*(1 + a*x)^4*(-3 + 2*a*x))/(10*a)","A",1
568,1,20,15,0.0138448,"\int e^{2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right) \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2),x]","-c \left(\frac{a^2 x^3}{3}+a x^2+x\right)","-\frac{c (a x+1)^3}{3 a}",1,"-(c*(x + a*x^2 + (a^2*x^3)/3))","A",1
569,1,18,16,0.0156794,"\int \frac{e^{2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a^2*c*x^2),x]","\frac{e^{2 \coth ^{-1}(a x)}}{2 a c}","-\frac{1}{a c (1-a x)}",1,"E^(2*ArcCoth[a*x])/(2*a*c)","C",1
570,1,35,51,0.0281944,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^2} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^2,x]","\frac{a x+(a x-1)^2 \left(-\tanh ^{-1}(a x)\right)-2}{4 a c^2 (a x-1)^2}","-\frac{1}{4 a c^2 (1-a x)}-\frac{1}{4 a c^2 (1-a x)^2}-\frac{\tanh ^{-1}(a x)}{4 a c^2}",1,"(-2 + a*x - (-1 + a*x)^2*ArcTanh[a*x])/(4*a*c^2*(-1 + a*x)^2)","A",1
571,1,63,86,0.0411076,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^3} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^3,x]","\frac{3 a^3 x^3-6 a^2 x^2+a x-3 (a x-1)^3 (a x+1) \tanh ^{-1}(a x)+4}{12 a c^3 (a x-1)^3 (a x+1)}","-\frac{3}{16 a c^3 (1-a x)}+\frac{1}{16 a c^3 (a x+1)}-\frac{1}{8 a c^3 (1-a x)^2}-\frac{1}{12 a c^3 (1-a x)^3}-\frac{\tanh ^{-1}(a x)}{4 a c^3}",1,"(4 + a*x - 6*a^2*x^2 + 3*a^3*x^3 - 3*(-1 + a*x)^3*(1 + a*x)*ArcTanh[a*x])/(12*a*c^3*(-1 + a*x)^3*(1 + a*x))","A",1
572,1,82,121,0.071491,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^4} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^4,x]","-\frac{-15 a^5 x^5+30 a^4 x^4+10 a^3 x^3-50 a^2 x^2+17 a x+15 (a x-1)^4 (a x+1)^2 \tanh ^{-1}(a x)+16}{64 a c^4 (a x-1)^4 (a x+1)^2}","-\frac{5}{32 a c^4 (1-a x)}+\frac{5}{64 a c^4 (a x+1)}-\frac{3}{32 a c^4 (1-a x)^2}+\frac{1}{64 a c^4 (a x+1)^2}-\frac{1}{16 a c^4 (1-a x)^3}-\frac{1}{32 a c^4 (1-a x)^4}-\frac{15 \tanh ^{-1}(a x)}{64 a c^4}",1,"-1/64*(16 + 17*a*x - 50*a^2*x^2 + 10*a^3*x^3 + 30*a^4*x^4 - 15*a^5*x^5 + 15*(-1 + a*x)^4*(1 + a*x)^2*ArcTanh[a*x])/(a*c^4*(-1 + a*x)^4*(1 + a*x)^2)","A",1
573,1,111,393,0.2028341,"\int e^{3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^4 \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^4,x]","\frac{c^4 \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(896 a^8 x^8+3024 a^7 x^7+1024 a^6 x^6-7224 a^5 x^5-8448 a^4 x^4+3066 a^3 x^3+10240 a^2 x^2+4599 a x-3712\right)-3465 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{8064 a}","\frac{1}{9} a^8 c^4 x^9 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{13/2}-\frac{5}{72} a^7 c^4 x^8 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{13/2}+\frac{5}{168} a^6 c^4 x^7 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{13/2}-\frac{5 a^5 c^4 x^6 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{11/2}}{1008}-\frac{11 a^4 c^4 x^5 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{9/2}}{1008}-\frac{11}{448} a^3 c^4 x^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}-\frac{11}{192} a^2 c^4 x^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}-\frac{55}{384} a c^4 x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}-\frac{55}{128} c^4 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{55 c^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{128 a}",1,"(c^4*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-3712 + 4599*a*x + 10240*a^2*x^2 + 3066*a^3*x^3 - 8448*a^4*x^4 - 7224*a^5*x^5 + 1024*a^6*x^6 + 3024*a^7*x^7 + 896*a^8*x^8) - 3465*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(8064*a)","A",0
574,1,95,313,0.1476534,"\int e^{3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^3 \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^3,x]","-\frac{c^3 \left(315 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+a x \sqrt{1-\frac{1}{a^2 x^2}} \left(80 a^6 x^6+280 a^5 x^5+208 a^4 x^4-350 a^3 x^3-656 a^2 x^2-245 a x+368\right)\right)}{560 a}","-\frac{1}{7} a^6 c^3 x^7 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{11/2}+\frac{1}{14} a^5 c^3 x^6 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{11/2}-\frac{1}{70} a^4 c^3 x^5 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{9/2}-\frac{9}{280} a^3 c^3 x^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}-\frac{3}{40} a^2 c^3 x^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}-\frac{3}{16} a c^3 x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}-\frac{9}{16} c^3 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{9 c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{16 a}",1,"-1/560*(c^3*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(368 - 245*a*x - 656*a^2*x^2 - 350*a^3*x^3 + 208*a^4*x^4 + 280*a^5*x^5 + 80*a^6*x^6) + 315*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/a","A",0
575,1,79,233,0.1100117,"\int e^{3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^2 \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^2,x]","\frac{c^2 \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(24 a^4 x^4+90 a^3 x^3+112 a^2 x^2+15 a x-136\right)-105 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{120 a}","\frac{1}{5} a^4 c^2 x^5 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{9/2}-\frac{1}{20} a^3 c^2 x^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}-\frac{7}{60} a^2 c^2 x^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}-\frac{7}{24} a c^2 x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}-\frac{7}{8} c^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{7 c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{8 a}",1,"(c^2*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-136 + 15*a*x + 112*a^2*x^2 + 90*a^3*x^3 + 24*a^4*x^4) - 105*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(120*a)","A",0
576,1,61,145,0.0825927,"\int e^{3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right) \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a^2*c*x^2),x]","-\frac{c \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2+9 a x+22\right)+15 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{6 a}","-\frac{1}{3} a^2 c x^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}-\frac{5}{6} a c x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}-\frac{5}{2} c x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{5 c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{2 a}",1,"-1/6*(c*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(22 + 9*a*x + 2*a^2*x^2) + 15*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/a","A",0
577,1,18,18,0.048918,"\int \frac{e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a^2*c*x^2),x]","\frac{e^{3 \coth ^{-1}(a x)}}{3 a c}","\frac{e^{3 \coth ^{-1}(a x)}}{3 a c}",1,"E^(3*ArcCoth[a*x])/(3*a*c)","A",1
578,1,43,55,0.1620946,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^2} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a^2*c*x^2)^2,x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2-6 a x+7\right)}{15 c^2 (a x-1)^3}","\frac{(3-2 a x) e^{3 \coth ^{-1}(a x)}}{5 a c^2 \left(1-a^2 x^2\right)}-\frac{2 e^{3 \coth ^{-1}(a x)}}{15 a c^2}",1,"-1/15*(Sqrt[1 - 1/(a^2*x^2)]*x*(7 - 6*a*x + 2*a^2*x^2))/(c^2*(-1 + a*x)^3)","A",0
579,1,66,91,0.2242249,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^3} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a^2*c*x^2)^3,x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(8 a^4 x^4-24 a^3 x^3+20 a^2 x^2+4 a x-13\right)}{35 c^3 (a x-1)^4 (a x+1)}","-\frac{(3-4 a x) e^{3 \coth ^{-1}(a x)}}{7 a c^3 \left(1-a^2 x^2\right)^2}+\frac{12 (3-2 a x) e^{3 \coth ^{-1}(a x)}}{35 a c^3 \left(1-a^2 x^2\right)}-\frac{8 e^{3 \coth ^{-1}(a x)}}{35 a c^3}",1,"-1/35*(Sqrt[1 - 1/(a^2*x^2)]*x*(-13 + 4*a*x + 20*a^2*x^2 - 24*a^3*x^3 + 8*a^4*x^4))/(c^3*(-1 + a*x)^4*(1 + a*x))","A",0
580,1,82,127,0.2842213,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^4} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a^2*c*x^2)^4,x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(16 a^6 x^6-48 a^5 x^5+24 a^4 x^4+56 a^3 x^3-66 a^2 x^2+6 a x+19\right)}{63 c^4 (a x-1)^5 (a x+1)^2}","-\frac{10 (3-4 a x) e^{3 \coth ^{-1}(a x)}}{63 a c^4 \left(1-a^2 x^2\right)^2}+\frac{8 (3-2 a x) e^{3 \coth ^{-1}(a x)}}{21 a c^4 \left(1-a^2 x^2\right)}-\frac{(1-2 a x) e^{3 \coth ^{-1}(a x)}}{9 a c^4 \left(1-a^2 x^2\right)^3}-\frac{16 e^{3 \coth ^{-1}(a x)}}{63 a c^4}",1,"-1/63*(Sqrt[1 - 1/(a^2*x^2)]*x*(19 + 6*a*x - 66*a^2*x^2 + 56*a^3*x^3 + 24*a^4*x^4 - 48*a^5*x^5 + 16*a^6*x^6))/(c^4*(-1 + a*x)^5*(1 + a*x)^2)","A",0
581,1,39,66,0.039486,"\int e^{4 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^5 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^5,x]","-\frac{c^5 (a x+1)^8 \left(15 a^3 x^3-54 a^2 x^2+67 a x-29\right)}{165 a}","-\frac{c^5 (a x+1)^{11}}{11 a}+\frac{3 c^5 (a x+1)^{10}}{5 a}-\frac{4 c^5 (a x+1)^9}{3 a}+\frac{c^5 (a x+1)^8}{a}",1,"-1/165*(c^5*(1 + a*x)^8*(-29 + 67*a*x - 54*a^2*x^2 + 15*a^3*x^3))/a","A",1
582,1,31,52,0.0293091,"\int e^{4 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^4 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^4,x]","\frac{c^4 (a x+1)^7 \left(14 a^2 x^2-35 a x+23\right)}{126 a}","\frac{c^4 (a x+1)^9}{9 a}-\frac{c^4 (a x+1)^8}{2 a}+\frac{4 c^4 (a x+1)^7}{7 a}",1,"(c^4*(1 + a*x)^7*(23 - 35*a*x + 14*a^2*x^2))/(126*a)","A",1
583,1,23,35,0.0217972,"\int e^{4 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^3 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^3,x]","-\frac{c^3 (a x+1)^6 (3 a x-4)}{21 a}","\frac{c^3 (a x+1)^6}{3 a}-\frac{c^3 (a x+1)^7}{7 a}",1,"-1/21*(c^3*(1 + a*x)^6*(-4 + 3*a*x))/a","A",1
584,1,37,17,0.0231213,"\int e^{4 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^2 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^2,x]","c^2 \left(\frac{a^4 x^5}{5}+a^3 x^4+2 a^2 x^3+2 a x^2+x\right)","\frac{c^2 (a x+1)^5}{5 a}",1,"c^2*(x + 2*a*x^2 + 2*a^2*x^3 + a^3*x^4 + (a^4*x^5)/5)","B",1
585,1,36,46,0.0144455,"\int e^{4 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right) \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2),x]","-\frac{1}{3} a^2 c x^3-2 a c x^2-\frac{8 c \log (1-a x)}{a}-7 c x","-\frac{c (a x+1)^3}{3 a}-\frac{c (a x+1)^2}{a}-\frac{8 c \log (1-a x)}{a}-4 c x",1,"-7*c*x - 2*a*c*x^2 - (a^2*c*x^3)/3 - (8*c*Log[1 - a*x])/a","A",1
586,1,25,13,0.0115952,"\int \frac{e^{4 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - a^2*c*x^2),x]","\frac{(a x+1)^2}{4 a c (1-a x)^2}","\frac{x}{c (1-a x)^2}",1,"(1 + a*x)^2/(4*a*c*(1 - a*x)^2)","A",1
587,1,17,18,0.0199768,"\int \frac{e^{4 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^2} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - a^2*c*x^2)^2,x]","-\frac{1}{3 a c^2 (a x-1)^3}","\frac{1}{3 a c^2 (1-a x)^3}",1,"-1/3*1/(a*c^2*(-1 + a*x)^3)","A",1
588,1,52,87,0.0386941,"\int \frac{e^{4 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^3} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - a^2*c*x^2)^3,x]","\frac{-3 a^3 x^3+12 a^2 x^2-19 a x+3 (a x-1)^4 \tanh ^{-1}(a x)+16}{48 a c^3 (a x-1)^4}","\frac{1}{16 a c^3 (1-a x)}+\frac{1}{16 a c^3 (1-a x)^2}+\frac{1}{12 a c^3 (1-a x)^3}+\frac{1}{8 a c^3 (1-a x)^4}+\frac{\tanh ^{-1}(a x)}{16 a c^3}",1,"(16 - 19*a*x + 12*a^2*x^2 - 3*a^3*x^3 + 3*(-1 + a*x)^4*ArcTanh[a*x])/(48*a*c^3*(-1 + a*x)^4)","A",1
589,1,80,122,0.0601383,"\int \frac{e^{4 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^4} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - a^2*c*x^2)^4,x]","\frac{-15 a^5 x^5+60 a^4 x^4-80 a^3 x^3+20 a^2 x^2+47 a x+15 (a x-1)^5 (a x+1) \tanh ^{-1}(a x)-48}{160 a c^4 (a x-1)^5 (a x+1)}","\frac{5}{64 a c^4 (1-a x)}-\frac{1}{64 a c^4 (a x+1)}+\frac{1}{16 a c^4 (1-a x)^2}+\frac{1}{16 a c^4 (1-a x)^3}+\frac{1}{16 a c^4 (1-a x)^4}+\frac{1}{20 a c^4 (1-a x)^5}+\frac{3 \tanh ^{-1}(a x)}{32 a c^4}",1,"(-48 + 47*a*x + 20*a^2*x^2 - 80*a^3*x^3 + 60*a^4*x^4 - 15*a^5*x^5 + 15*(-1 + a*x)^5*(1 + a*x)*ArcTanh[a*x])/(160*a*c^4*(-1 + a*x)^5*(1 + a*x))","A",1
590,1,111,393,0.2044039,"\int e^{-\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^4 \, dx","Integrate[(c - a^2*c*x^2)^4/E^ArcCoth[a*x],x]","\frac{c^4 \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(128 a^8 x^8-144 a^7 x^7-512 a^6 x^6+600 a^5 x^5+768 a^4 x^4-978 a^3 x^3-512 a^2 x^2+837 a x+128\right)-315 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{1152 a}","\frac{1}{9} a^8 c^4 x^9 \left(1-\frac{1}{a x}\right)^{9/2} \left(\frac{1}{a x}+1\right)^{9/2}-\frac{1}{8} a^7 c^4 x^8 \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{9/2}+\frac{1}{8} a^6 c^4 x^7 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{9/2}-\frac{5}{48} a^5 c^4 x^6 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{9/2}+\frac{1}{16} a^4 c^4 x^5 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{9/2}-\frac{1}{64} a^3 c^4 x^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}-\frac{7}{192} a^2 c^4 x^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}-\frac{35}{384} a c^4 x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}-\frac{35}{128} c^4 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{35 c^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{128 a}",1,"(c^4*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(128 + 837*a*x - 512*a^2*x^2 - 978*a^3*x^3 + 768*a^4*x^4 + 600*a^5*x^5 - 512*a^6*x^6 - 144*a^7*x^7 + 128*a^8*x^8) - 315*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(1152*a)","A",0
591,1,95,313,0.1993456,"\int e^{-\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^3 \, dx","Integrate[(c - a^2*c*x^2)^3/E^ArcCoth[a*x],x]","\frac{c^3 \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(-48 a^6 x^6+56 a^5 x^5+144 a^4 x^4-182 a^3 x^3-144 a^2 x^2+231 a x+48\right)-105 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{336 a}","-\frac{1}{7} a^6 c^3 x^7 \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{7/2}+\frac{1}{6} a^5 c^3 x^6 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{7/2}-\frac{1}{6} a^4 c^3 x^5 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{7/2}+\frac{1}{8} a^3 c^3 x^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}-\frac{1}{24} a^2 c^3 x^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}-\frac{5}{48} a c^3 x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}-\frac{5}{16} c^3 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{5 c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{16 a}",1,"(c^3*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(48 + 231*a*x - 144*a^2*x^2 - 182*a^3*x^3 + 144*a^4*x^4 + 56*a^5*x^5 - 48*a^6*x^6) - 105*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(336*a)","A",0
592,1,79,233,0.1091256,"\int e^{-\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^2 \, dx","Integrate[(c - a^2*c*x^2)^2/E^ArcCoth[a*x],x]","\frac{c^2 \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(8 a^4 x^4-10 a^3 x^3-16 a^2 x^2+25 a x+8\right)-15 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{40 a}","\frac{1}{5} a^4 c^2 x^5 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{5/2}-\frac{1}{4} a^3 c^2 x^4 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{5/2}+\frac{1}{4} a^2 c^2 x^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}-\frac{1}{8} a c^2 x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}-\frac{3}{8} c^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{3 c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{8 a}",1,"(c^2*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(8 + 25*a*x - 16*a^2*x^2 - 10*a^3*x^3 + 8*a^4*x^4) - 15*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(40*a)","A",0
593,1,61,145,0.0959954,"\int e^{-\coth ^{-1}(a x)} \left(c-a^2 c x^2\right) \, dx","Integrate[(c - a^2*c*x^2)/E^ArcCoth[a*x],x]","\frac{c \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(-2 a^2 x^2+3 a x+2\right)-3 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{6 a}","-\frac{1}{3} a^2 c x^3 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{3/2}+\frac{1}{2} a c x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}-\frac{1}{2} c x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{2 a}",1,"(c*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(2 + 3*a*x - 2*a^2*x^2) - 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(6*a)","A",0
594,1,16,16,0.0470179,"\int \frac{e^{-\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)),x]","-\frac{e^{-\coth ^{-1}(a x)}}{a c}","-\frac{e^{-\coth ^{-1}(a x)}}{a c}",1,"-(1/(a*c*E^ArcCoth[a*x]))","A",1
595,1,48,55,0.1558175,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^2} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^2),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2+2 a x-1\right)}{3 (a x-1) (a c x+c)^2}","\frac{(2 a x+1) e^{-\coth ^{-1}(a x)}}{3 a c^2 \left(1-a^2 x^2\right)}-\frac{2 e^{-\coth ^{-1}(a x)}}{3 a c^2}",1,"-1/3*(Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + 2*a*x + 2*a^2*x^2))/((-1 + a*x)*(c + a*c*x)^2)","A",0
596,1,64,91,0.2136119,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^3} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^3),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(8 a^4 x^4+8 a^3 x^3-12 a^2 x^2-12 a x+3\right)}{15 (a x-1)^2 (a c x+c)^3}","\frac{4 (2 a x+1) e^{-\coth ^{-1}(a x)}}{15 a c^3 \left(1-a^2 x^2\right)}+\frac{(4 a x+1) e^{-\coth ^{-1}(a x)}}{15 a c^3 \left(1-a^2 x^2\right)^2}-\frac{8 e^{-\coth ^{-1}(a x)}}{15 a c^3}",1,"-1/15*(Sqrt[1 - 1/(a^2*x^2)]*x*(3 - 12*a*x - 12*a^2*x^2 + 8*a^3*x^3 + 8*a^4*x^4))/((-1 + a*x)^2*(c + a*c*x)^3)","A",0
597,1,80,127,0.288952,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^4} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^4),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(16 a^6 x^6+16 a^5 x^5-40 a^4 x^4-40 a^3 x^3+30 a^2 x^2+30 a x-5\right)}{35 (a x-1)^3 (a c x+c)^4}","\frac{8 (2 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left(1-a^2 x^2\right)}+\frac{2 (4 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left(1-a^2 x^2\right)^2}+\frac{(6 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left(1-a^2 x^2\right)^3}-\frac{16 e^{-\coth ^{-1}(a x)}}{35 a c^4}",1,"-1/35*(Sqrt[1 - 1/(a^2*x^2)]*x*(-5 + 30*a*x + 30*a^2*x^2 - 40*a^3*x^3 - 40*a^4*x^4 + 16*a^5*x^5 + 16*a^6*x^6))/((-1 + a*x)^3*(c + a*c*x)^4)","A",0
598,1,39,73,0.0335131,"\int e^{-2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^4 \, dx","Integrate[(c - a^2*c*x^2)^4/E^(2*ArcCoth[a*x]),x]","\frac{c^4 (a x-1)^6 \left(28 a^3 x^3+105 a^2 x^2+138 a x+65\right)}{252 a}","-\frac{c^4 (1-a x)^9}{9 a}+\frac{3 c^4 (1-a x)^8}{4 a}-\frac{12 c^4 (1-a x)^7}{7 a}+\frac{4 c^4 (1-a x)^6}{3 a}",1,"(c^4*(-1 + a*x)^6*(65 + 138*a*x + 105*a^2*x^2 + 28*a^3*x^3))/(252*a)","A",1
599,1,31,55,0.027631,"\int e^{-2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^3 \, dx","Integrate[(c - a^2*c*x^2)^3/E^(2*ArcCoth[a*x]),x]","-\frac{c^3 (a x-1)^5 \left(15 a^2 x^2+40 a x+29\right)}{105 a}","\frac{c^3 (1-a x)^7}{7 a}-\frac{2 c^3 (1-a x)^6}{3 a}+\frac{4 c^3 (1-a x)^5}{5 a}",1,"-1/105*(c^3*(-1 + a*x)^5*(29 + 40*a*x + 15*a^2*x^2))/a","A",1
600,1,30,37,0.0175832,"\int e^{-2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^2 \, dx","Integrate[(c - a^2*c*x^2)^2/E^(2*ArcCoth[a*x]),x]","\frac{1}{10} c^2 x \left(2 a^4 x^4-5 a^3 x^3+10 a x-10\right)","\frac{c^2 (1-a x)^4}{2 a}-\frac{c^2 (1-a x)^5}{5 a}",1,"(c^2*x*(-10 + 10*a*x - 5*a^3*x^3 + 2*a^4*x^4))/10","A",1
601,1,21,16,0.0142092,"\int e^{-2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right) \, dx","Integrate[(c - a^2*c*x^2)/E^(2*ArcCoth[a*x]),x]","-c \left(\frac{a^2 x^3}{3}-a x^2+x\right)","\frac{c (1-a x)^3}{3 a}",1,"-(c*(x - a*x^2 + (a^2*x^3)/3))","A",1
602,1,18,14,0.0160932,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)),x]","-\frac{e^{-2 \coth ^{-1}(a x)}}{2 a c}","\frac{1}{a c (a x+1)}",1,"-1/2*1/(a*c*E^(2*ArcCoth[a*x]))","C",1
603,1,33,49,0.0310916,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^2} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^2),x]","\frac{a x+(a x+1)^2 \left(-\tanh ^{-1}(a x)\right)+2}{4 a (a c x+c)^2}","\frac{1}{4 a c^2 (a x+1)}+\frac{1}{4 a c^2 (a x+1)^2}-\frac{\tanh ^{-1}(a x)}{4 a c^2}",1,"(2 + a*x - (1 + a*x)^2*ArcTanh[a*x])/(4*a*(c + a*c*x)^2)","A",1
604,1,61,84,0.0446113,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^3} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^3),x]","\frac{3 a^3 x^3+6 a^2 x^2+a x-3 (a x-1) (a x+1)^3 \tanh ^{-1}(a x)-4}{12 a (a x-1) (a c x+c)^3}","-\frac{1}{16 a c^3 (1-a x)}+\frac{3}{16 a c^3 (a x+1)}+\frac{1}{8 a c^3 (a x+1)^2}+\frac{1}{12 a c^3 (a x+1)^3}-\frac{\tanh ^{-1}(a x)}{4 a c^3}",1,"(-4 + a*x + 6*a^2*x^2 + 3*a^3*x^3 - 3*(-1 + a*x)*(1 + a*x)^3*ArcTanh[a*x])/(12*a*(-1 + a*x)*(c + a*c*x)^3)","A",1
605,1,80,119,0.0681258,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^4} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^4),x]","\frac{15 a^5 x^5+30 a^4 x^4-10 a^3 x^3-50 a^2 x^2-17 a x-15 (a x-1)^2 (a x+1)^4 \tanh ^{-1}(a x)+16}{64 a (a x-1)^2 (a c x+c)^4}","-\frac{5}{64 a c^4 (1-a x)}+\frac{5}{32 a c^4 (a x+1)}-\frac{1}{64 a c^4 (1-a x)^2}+\frac{3}{32 a c^4 (a x+1)^2}+\frac{1}{16 a c^4 (a x+1)^3}+\frac{1}{32 a c^4 (a x+1)^4}-\frac{15 \tanh ^{-1}(a x)}{64 a c^4}",1,"(16 - 17*a*x - 50*a^2*x^2 - 10*a^3*x^3 + 30*a^4*x^4 + 15*a^5*x^5 - 15*(-1 + a*x)^2*(1 + a*x)^4*ArcTanh[a*x])/(64*a*(-1 + a*x)^2*(c + a*c*x)^4)","A",1
606,1,111,393,0.2088505,"\int e^{-3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^4 \, dx","Integrate[(c - a^2*c*x^2)^4/E^(3*ArcCoth[a*x]),x]","\frac{c^4 \left(3465 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+a x \sqrt{1-\frac{1}{a^2 x^2}} \left(896 a^8 x^8-3024 a^7 x^7+1024 a^6 x^6+7224 a^5 x^5-8448 a^4 x^4-3066 a^3 x^3+10240 a^2 x^2-4599 a x-3712\right)\right)}{8064 a}","\frac{1}{9} a^8 c^4 x^9 \left(1-\frac{1}{a x}\right)^{11/2} \left(\frac{1}{a x}+1\right)^{7/2}-\frac{11}{72} a^7 c^4 x^8 \left(1-\frac{1}{a x}\right)^{9/2} \left(\frac{1}{a x}+1\right)^{7/2}+\frac{11}{56} a^6 c^4 x^7 \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{7/2}-\frac{11}{48} a^5 c^4 x^6 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{7/2}+\frac{11}{48} a^4 c^4 x^5 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{7/2}-\frac{11}{64} a^3 c^4 x^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}+\frac{11}{192} a^2 c^4 x^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}+\frac{55}{384} a c^4 x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}+\frac{55}{128} c^4 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}+\frac{55 c^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{128 a}",1,"(c^4*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-3712 - 4599*a*x + 10240*a^2*x^2 - 3066*a^3*x^3 - 8448*a^4*x^4 + 7224*a^5*x^5 + 1024*a^6*x^6 - 3024*a^7*x^7 + 896*a^8*x^8) + 3465*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(8064*a)","A",0
607,1,95,313,0.1559985,"\int e^{-3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^3 \, dx","Integrate[(c - a^2*c*x^2)^3/E^(3*ArcCoth[a*x]),x]","-\frac{c^3 \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(80 a^6 x^6-280 a^5 x^5+208 a^4 x^4+350 a^3 x^3-656 a^2 x^2+245 a x+368\right)-315 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{560 a}","-\frac{1}{7} a^6 c^3 x^7 \left(1-\frac{1}{a x}\right)^{9/2} \left(\frac{1}{a x}+1\right)^{5/2}+\frac{3}{14} a^5 c^3 x^6 \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{5/2}-\frac{3}{10} a^4 c^3 x^5 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{5/2}+\frac{3}{8} a^3 c^3 x^4 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{5/2}-\frac{3}{8} a^2 c^3 x^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}+\frac{3}{16} a c^3 x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}+\frac{9}{16} c^3 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}+\frac{9 c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{16 a}",1,"-1/560*(c^3*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(368 + 245*a*x - 656*a^2*x^2 + 350*a^3*x^3 + 208*a^4*x^4 - 280*a^5*x^5 + 80*a^6*x^6) - 315*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/a","A",0
608,1,79,233,0.1168419,"\int e^{-3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^2 \, dx","Integrate[(c - a^2*c*x^2)^2/E^(3*ArcCoth[a*x]),x]","\frac{c^2 \left(105 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+a x \sqrt{1-\frac{1}{a^2 x^2}} \left(24 a^4 x^4-90 a^3 x^3+112 a^2 x^2-15 a x-136\right)\right)}{120 a}","\frac{1}{5} a^4 c^2 x^5 \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{3/2}-\frac{7}{20} a^3 c^2 x^4 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{3/2}+\frac{7}{12} a^2 c^2 x^3 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{3/2}-\frac{7}{8} a c^2 x^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}+\frac{7}{8} c^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}+\frac{7 c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{8 a}",1,"(c^2*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-136 - 15*a*x + 112*a^2*x^2 - 90*a^3*x^3 + 24*a^4*x^4) + 105*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(120*a)","A",0
609,1,61,145,0.1100512,"\int e^{-3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right) \, dx","Integrate[(c - a^2*c*x^2)/E^(3*ArcCoth[a*x]),x]","\frac{c \left(a x \sqrt{1-\frac{1}{a^2 x^2}} \left(-2 a^2 x^2+9 a x-22\right)+15 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)\right)}{6 a}","-\frac{1}{3} a^2 c x^3 \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}+\frac{5}{6} a c x^2 \left(1-\frac{1}{a x}\right)^{3/2} \sqrt{\frac{1}{a x}+1}-\frac{5}{2} c x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}+\frac{5 c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{2 a}",1,"(c*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-22 + 9*a*x - 2*a^2*x^2) + 15*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(6*a)","A",0
610,1,18,18,0.0503974,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)),x]","-\frac{e^{-3 \coth ^{-1}(a x)}}{3 a c}","-\frac{e^{-3 \coth ^{-1}(a x)}}{3 a c}",1,"-1/3*1/(a*c*E^(3*ArcCoth[a*x]))","A",1
611,1,43,55,0.1562793,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^2} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^2),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(2 a^2 x^2+6 a x+7\right)}{15 c^2 (a x+1)^3}","\frac{2 e^{-3 \coth ^{-1}(a x)}}{15 a c^2}-\frac{(2 a x+3) e^{-3 \coth ^{-1}(a x)}}{5 a c^2 \left(1-a^2 x^2\right)}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(7 + 6*a*x + 2*a^2*x^2))/(15*c^2*(1 + a*x)^3)","A",0
612,1,66,91,0.2179137,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^3} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^3),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(8 a^4 x^4+24 a^3 x^3+20 a^2 x^2-4 a x-13\right)}{35 c^3 (a x-1) (a x+1)^4}","-\frac{12 (2 a x+3) e^{-3 \coth ^{-1}(a x)}}{35 a c^3 \left(1-a^2 x^2\right)}+\frac{(4 a x+3) e^{-3 \coth ^{-1}(a x)}}{7 a c^3 \left(1-a^2 x^2\right)^2}+\frac{8 e^{-3 \coth ^{-1}(a x)}}{35 a c^3}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(-13 - 4*a*x + 20*a^2*x^2 + 24*a^3*x^3 + 8*a^4*x^4))/(35*c^3*(-1 + a*x)*(1 + a*x)^4)","A",0
613,1,82,127,0.2801582,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^4} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^4),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(16 a^6 x^6+48 a^5 x^5+24 a^4 x^4-56 a^3 x^3-66 a^2 x^2-6 a x+19\right)}{63 c^4 (a x-1)^2 (a x+1)^5}","\frac{(2 a x+1) e^{-3 \coth ^{-1}(a x)}}{9 a c^4 \left(1-a^2 x^2\right)^3}-\frac{8 (2 a x+3) e^{-3 \coth ^{-1}(a x)}}{21 a c^4 \left(1-a^2 x^2\right)}+\frac{10 (4 a x+3) e^{-3 \coth ^{-1}(a x)}}{63 a c^4 \left(1-a^2 x^2\right)^2}+\frac{16 e^{-3 \coth ^{-1}(a x)}}{63 a c^4}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(19 - 6*a*x - 66*a^2*x^2 - 56*a^3*x^3 + 24*a^4*x^4 + 48*a^5*x^5 + 16*a^6*x^6))/(63*c^4*(-1 + a*x)^2*(1 + a*x)^5)","A",0
614,1,79,229,0.0720474,"\int e^{\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{9/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - a^2*c*x^2)^(9/2),x]","\frac{c^4 (a x+1)^6 \left(63 a^4 x^4-308 a^3 x^3+588 a^2 x^2-528 a x+193\right) \sqrt{c-a^2 c x^2}}{630 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(a x+1)^{10} \left(c-a^2 c x^2\right)^{9/2}}{10 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}-\frac{8 (a x+1)^9 \left(c-a^2 c x^2\right)^{9/2}}{9 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}+\frac{3 (a x+1)^8 \left(c-a^2 c x^2\right)^{9/2}}{a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}-\frac{32 (a x+1)^7 \left(c-a^2 c x^2\right)^{9/2}}{7 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}+\frac{8 (a x+1)^6 \left(c-a^2 c x^2\right)^{9/2}}{3 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}",1,"(c^4*(1 + a*x)^6*Sqrt[c - a^2*c*x^2]*(193 - 528*a*x + 588*a^2*x^2 - 308*a^3*x^3 + 63*a^4*x^4))/(630*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
615,1,71,183,0.0608262,"\int e^{\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{7/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - a^2*c*x^2)^(7/2),x]","-\frac{c^3 (a x+1)^5 \left(35 a^3 x^3-135 a^2 x^2+185 a x-93\right) \sqrt{c-a^2 c x^2}}{280 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(a x+1)^8 \left(c-a^2 c x^2\right)^{7/2}}{8 a^8 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}-\frac{6 (a x+1)^7 \left(c-a^2 c x^2\right)^{7/2}}{7 a^8 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}+\frac{2 (a x+1)^6 \left(c-a^2 c x^2\right)^{7/2}}{a^8 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}-\frac{8 (a x+1)^5 \left(c-a^2 c x^2\right)^{7/2}}{5 a^8 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}",1,"-1/280*(c^3*(1 + a*x)^5*Sqrt[c - a^2*c*x^2]*(-93 + 185*a*x - 135*a^2*x^2 + 35*a^3*x^3))/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
616,1,63,136,0.0469281,"\int e^{\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{5/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - a^2*c*x^2)^(5/2),x]","\frac{c^2 (a x+1)^4 \left(5 a^2 x^2-14 a x+11\right) \sqrt{c-a^2 c x^2}}{30 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(a x+1)^6 \left(c-a^2 c x^2\right)^{5/2}}{6 a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}-\frac{4 (a x+1)^5 \left(c-a^2 c x^2\right)^{5/2}}{5 a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}+\frac{(a x+1)^4 \left(c-a^2 c x^2\right)^{5/2}}{a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}",1,"(c^2*(1 + a*x)^4*(11 - 14*a*x + 5*a^2*x^2)*Sqrt[c - a^2*c*x^2])/(30*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
617,1,53,93,0.034453,"\int e^{\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{3/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - a^2*c*x^2)^(3/2),x]","-\frac{c (a x+1)^3 (3 a x-5) \sqrt{c-a^2 c x^2}}{12 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(a x+1)^4 \left(c-a^2 c x^2\right)^{3/2}}{4 a^4 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}-\frac{2 (a x+1)^3 \left(c-a^2 c x^2\right)^{3/2}}{3 a^4 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}",1,"-1/12*(c*(1 + a*x)^3*(-5 + 3*a*x)*Sqrt[c - a^2*c*x^2])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
618,1,41,68,0.0197493,"\int e^{\coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx","Integrate[E^ArcCoth[a*x]*Sqrt[c - a^2*c*x^2],x]","\frac{(a x+2) \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((2 + a*x)*Sqrt[c - a^2*c*x^2])/(2*a*Sqrt[1 - 1/(a^2*x^2)])","A",1
619,1,38,38,0.020261,"\int \frac{e^{\coth ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[E^ArcCoth[a*x]/Sqrt[c - a^2*c*x^2],x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{\sqrt{c-a^2 c x^2}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{\sqrt{c-a^2 c x^2}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*Log[1 - a*x])/Sqrt[c - a^2*c*x^2]","A",1
620,1,56,91,0.0420267,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^ArcCoth[a*x]/(c - a^2*c*x^2)^(3/2),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left((a x-1) \tanh ^{-1}(a x)-1\right)}{2 c (a x-1) \sqrt{c-a^2 c x^2}}","\frac{a^2 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 (1-a x) \left(c-a^2 c x^2\right)^{3/2}}+\frac{a^2 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \tanh ^{-1}(a x)}{2 \left(c-a^2 c x^2\right)^{3/2}}",1,"-1/2*(Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + (-1 + a*x)*ArcTanh[a*x]))/(c*(-1 + a*x)*Sqrt[c - a^2*c*x^2])","A",1
621,1,83,184,0.0636719,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[E^ArcCoth[a*x]/(c - a^2*c*x^2)^(5/2),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(-3 a^2 x^2+3 a x+3 (a x-1)^2 (a x+1) \tanh ^{-1}(a x)+2\right)}{8 c^2 (a x-1)^2 (a x+1) \sqrt{c-a^2 c x^2}}","-\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{4 (1-a x) \left(c-a^2 c x^2\right)^{5/2}}+\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (a x+1) \left(c-a^2 c x^2\right)^{5/2}}-\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (1-a x)^2 \left(c-a^2 c x^2\right)^{5/2}}-\frac{3 a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \tanh ^{-1}(a x)}{8 \left(c-a^2 c x^2\right)^{5/2}}",1,"-1/8*(Sqrt[1 - 1/(a^2*x^2)]*x*(2 + 3*a*x - 3*a^2*x^2 + 3*(-1 + a*x)^2*(1 + a*x)*ArcTanh[a*x]))/(c^2*(-1 + a*x)^2*(1 + a*x)*Sqrt[c - a^2*c*x^2])","A",1
622,1,101,277,0.0895127,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{7/2}} \, dx","Integrate[E^ArcCoth[a*x]/(c - a^2*c*x^2)^(7/2),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(-15 a^4 x^4+15 a^3 x^3+25 a^2 x^2-25 a x+15 (a x-1)^3 (a x+1)^2 \tanh ^{-1}(a x)-8\right)}{48 c^3 (a x-1)^3 (a x+1)^2 \sqrt{c-a^2 c x^2}}","\frac{3 a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{16 (1-a x) \left(c-a^2 c x^2\right)^{7/2}}-\frac{a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{8 (a x+1) \left(c-a^2 c x^2\right)^{7/2}}+\frac{3 a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{32 (1-a x)^2 \left(c-a^2 c x^2\right)^{7/2}}-\frac{a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{32 (a x+1)^2 \left(c-a^2 c x^2\right)^{7/2}}+\frac{a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{24 (1-a x)^3 \left(c-a^2 c x^2\right)^{7/2}}+\frac{5 a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2} \tanh ^{-1}(a x)}{16 \left(c-a^2 c x^2\right)^{7/2}}",1,"-1/48*(Sqrt[1 - 1/(a^2*x^2)]*x*(-8 - 25*a*x + 25*a^2*x^2 + 15*a^3*x^3 - 15*a^4*x^4 + 15*(-1 + a*x)^3*(1 + a*x)^2*ArcTanh[a*x]))/(c^3*(-1 + a*x)^3*(1 + a*x)^2*Sqrt[c - a^2*c*x^2])","A",1
623,1,167,176,0.1644751,"\int e^{2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{9/2} \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(9/2),x]","\frac{c^4 \sqrt{c-a^2 c x^2} \left(\sqrt{a x+1} \left(-1152 a^{10} x^{10}-1408 a^9 x^9+5584 a^8 x^8+7216 a^7 x^7-10552 a^6 x^6-15048 a^5 x^5+9210 a^4 x^4+16390 a^3 x^3-2185 a^2 x^2-10615 a x+2560\right)+6930 \sqrt{1-a x} \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)\right)}{11520 a \sqrt{1-a x} \sqrt{1-a^2 x^2}}","-\frac{77 c^{9/2} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{256 a}-\frac{77}{256} c^4 x \sqrt{c-a^2 c x^2}-\frac{77}{384} c^3 x \left(c-a^2 c x^2\right)^{3/2}-\frac{77}{480} c^2 x \left(c-a^2 c x^2\right)^{5/2}-\frac{11}{80} c x \left(c-a^2 c x^2\right)^{7/2}+\frac{(a x+1) \left(c-a^2 c x^2\right)^{9/2}}{10 a}+\frac{11 \left(c-a^2 c x^2\right)^{9/2}}{90 a}",1,"(c^4*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(2560 - 10615*a*x - 2185*a^2*x^2 + 16390*a^3*x^3 + 9210*a^4*x^4 - 15048*a^5*x^5 - 10552*a^6*x^6 + 7216*a^7*x^7 + 5584*a^8*x^8 - 1408*a^9*x^9 - 1152*a^10*x^10) + 6930*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(11520*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])","A",0
624,1,151,153,0.1408498,"\int e^{2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{7/2} \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(7/2),x]","\frac{c^3 \sqrt{c-a^2 c x^2} \left(\sqrt{a x+1} \left(112 a^8 x^8+144 a^7 x^7-424 a^6 x^6-600 a^5 x^5+558 a^4 x^4+978 a^3 x^3-187 a^2 x^2-837 a x+256\right)+630 \sqrt{1-a x} \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)\right)}{896 a \sqrt{1-a x} \sqrt{1-a^2 x^2}}","-\frac{45 c^{7/2} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{128 a}-\frac{45}{128} c^3 x \sqrt{c-a^2 c x^2}-\frac{15}{64} c^2 x \left(c-a^2 c x^2\right)^{3/2}-\frac{3}{16} c x \left(c-a^2 c x^2\right)^{5/2}+\frac{(a x+1) \left(c-a^2 c x^2\right)^{7/2}}{8 a}+\frac{9 \left(c-a^2 c x^2\right)^{7/2}}{56 a}",1,"(c^3*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(256 - 837*a*x - 187*a^2*x^2 + 978*a^3*x^3 + 558*a^4*x^4 - 600*a^5*x^5 - 424*a^6*x^6 + 144*a^7*x^7 + 112*a^8*x^8) + 630*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(896*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])","A",0
625,1,135,130,0.1285244,"\int e^{2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{5/2} \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(5/2),x]","\frac{c^2 \sqrt{c-a^2 c x^2} \left(\sqrt{a x+1} \left(-40 a^6 x^6-56 a^5 x^5+106 a^4 x^4+182 a^3 x^3-57 a^2 x^2-231 a x+96\right)+210 \sqrt{1-a x} \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)\right)}{240 a \sqrt{1-a x} \sqrt{1-a^2 x^2}}","-\frac{7 c^{5/2} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{16 a}-\frac{7}{16} c^2 x \sqrt{c-a^2 c x^2}-\frac{7}{24} c x \left(c-a^2 c x^2\right)^{3/2}+\frac{(a x+1) \left(c-a^2 c x^2\right)^{5/2}}{6 a}+\frac{7 \left(c-a^2 c x^2\right)^{5/2}}{30 a}",1,"(c^2*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(96 - 231*a*x - 57*a^2*x^2 + 182*a^3*x^3 + 106*a^4*x^4 - 56*a^5*x^5 - 40*a^6*x^6) + 210*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(240*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])","A",0
626,1,117,107,0.100499,"\int e^{2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{3/2} \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(3/2),x]","\frac{c \sqrt{c-a^2 c x^2} \left(\sqrt{a x+1} \left(6 a^4 x^4+10 a^3 x^3-7 a^2 x^2-25 a x+16\right)+30 \sqrt{1-a x} \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)\right)}{24 a \sqrt{1-a x} \sqrt{1-a^2 x^2}}","-\frac{5 c^{3/2} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{8 a}-\frac{5}{8} c x \sqrt{c-a^2 c x^2}+\frac{(a x+1) \left(c-a^2 c x^2\right)^{3/2}}{4 a}+\frac{5 \left(c-a^2 c x^2\right)^{3/2}}{12 a}",1,"(c*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(16 - 25*a*x - 7*a^2*x^2 + 10*a^3*x^3 + 6*a^4*x^4) + 30*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(24*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])","A",0
627,1,76,86,0.0598102,"\int e^{2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx","Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2],x]","\frac{\sqrt{c-a^2 c x^2} \left(\sqrt{1-a^2 x^2} (a x+4)+6 \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)\right)}{2 a \sqrt{1-a^2 x^2}}","\frac{(a x+1) \sqrt{c-a^2 c x^2}}{2 a}+\frac{3 \sqrt{c-a^2 c x^2}}{2 a}-\frac{3 \sqrt{c} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{2 a}",1,"(Sqrt[c - a^2*c*x^2]*((4 + a*x)*Sqrt[1 - a^2*x^2] + 6*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(2*a*Sqrt[1 - a^2*x^2])","A",0
628,1,82,59,0.0414644,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/Sqrt[c - a^2*c*x^2],x]","-\frac{2 \sqrt{1-a^2 x^2} \left(\sqrt{a x+1}+\sqrt{1-a x} \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)\right)}{a \sqrt{1-a x} \sqrt{c-a^2 c x^2}}","\frac{\tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{a \sqrt{c}}-\frac{2 (a x+1)}{a \sqrt{c-a^2 c x^2}}",1,"(-2*Sqrt[1 - a^2*x^2]*(Sqrt[1 + a*x] + Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(a*Sqrt[1 - a*x]*Sqrt[c - a^2*c*x^2])","A",0
629,1,64,51,0.0397705,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^(3/2),x]","-\frac{(2-a x) \sqrt{a x+1} \sqrt{1-a^2 x^2}}{3 a c (1-a x)^{3/2} \sqrt{c-a^2 c x^2}}","-\frac{x}{3 c \sqrt{c-a^2 c x^2}}-\frac{2 (a x+1)}{3 a \left(c-a^2 c x^2\right)^{3/2}}",1,"-1/3*((2 - a*x)*Sqrt[1 + a*x]*Sqrt[1 - a^2*x^2])/(a*c*(1 - a*x)^(3/2)*Sqrt[c - a^2*c*x^2])","A",0
630,1,53,74,0.0432224,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^(5/2),x]","-\frac{2 a^3 x^3-4 a^2 x^2+a x+2}{5 a c^2 (a x-1)^2 \sqrt{c-a^2 c x^2}}","-\frac{2 x}{5 c^2 \sqrt{c-a^2 c x^2}}-\frac{x}{5 c \left(c-a^2 c x^2\right)^{3/2}}-\frac{2 (a x+1)}{5 a \left(c-a^2 c x^2\right)^{5/2}}",1,"-1/5*(2 + a*x - 4*a^2*x^2 + 2*a^3*x^3)/(a*c^2*(-1 + a*x)^2*Sqrt[c - a^2*c*x^2])","A",1
631,1,96,97,0.0588955,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{7/2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^(7/2),x]","-\frac{\sqrt{1-a^2 x^2} \left(-8 a^5 x^5+16 a^4 x^4+4 a^3 x^3-24 a^2 x^2+9 a x+6\right)}{21 a c^3 (1-a x)^{7/2} (a x+1)^{3/2} \sqrt{c-a^2 c x^2}}","-\frac{8 x}{21 c^3 \sqrt{c-a^2 c x^2}}-\frac{4 x}{21 c^2 \left(c-a^2 c x^2\right)^{3/2}}-\frac{x}{7 c \left(c-a^2 c x^2\right)^{5/2}}-\frac{2 (a x+1)}{7 a \left(c-a^2 c x^2\right)^{7/2}}",1,"-1/21*(Sqrt[1 - a^2*x^2]*(6 + 9*a*x - 24*a^2*x^2 + 4*a^3*x^3 + 16*a^4*x^4 - 8*a^5*x^5))/(a*c^3*(1 - a*x)^(7/2)*(1 + a*x)^(3/2)*Sqrt[c - a^2*c*x^2])","A",1
632,1,112,120,0.0666016,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{9/2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^(9/2),x]","\frac{\sqrt{1-a^2 x^2} \left(-16 a^7 x^7+32 a^6 x^6+24 a^5 x^5-80 a^4 x^4+10 a^3 x^3+60 a^2 x^2-25 a x-10\right)}{45 a c^4 (1-a x)^{9/2} (a x+1)^{5/2} \sqrt{c-a^2 c x^2}}","-\frac{16 x}{45 c^4 \sqrt{c-a^2 c x^2}}-\frac{8 x}{45 c^3 \left(c-a^2 c x^2\right)^{3/2}}-\frac{2 x}{15 c^2 \left(c-a^2 c x^2\right)^{5/2}}-\frac{x}{9 c \left(c-a^2 c x^2\right)^{7/2}}-\frac{2 (a x+1)}{9 a \left(c-a^2 c x^2\right)^{9/2}}",1,"(Sqrt[1 - a^2*x^2]*(-10 - 25*a*x + 60*a^2*x^2 + 10*a^3*x^3 - 80*a^4*x^4 + 24*a^5*x^5 + 32*a^6*x^6 - 16*a^7*x^7))/(45*a*c^4*(1 - a*x)^(9/2)*(1 + a*x)^(5/2)*Sqrt[c - a^2*c*x^2])","A",1
633,1,71,185,0.0650692,"\int e^{3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{9/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^(9/2),x]","\frac{c^4 (a x+1)^7 \left(21 a^3 x^3-77 a^2 x^2+98 a x-44\right) \sqrt{c-a^2 c x^2}}{210 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(a x+1)^{10} \left(c-a^2 c x^2\right)^{9/2}}{10 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}-\frac{2 (a x+1)^9 \left(c-a^2 c x^2\right)^{9/2}}{3 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}+\frac{3 (a x+1)^8 \left(c-a^2 c x^2\right)^{9/2}}{2 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}-\frac{8 (a x+1)^7 \left(c-a^2 c x^2\right)^{9/2}}{7 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}",1,"(c^4*(1 + a*x)^7*Sqrt[c - a^2*c*x^2]*(-44 + 98*a*x - 77*a^2*x^2 + 21*a^3*x^3))/(210*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
634,1,63,139,0.0541782,"\int e^{3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{7/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^(7/2),x]","-\frac{c^3 (a x+1)^6 \left(21 a^2 x^2-54 a x+37\right) \sqrt{c-a^2 c x^2}}{168 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(a x+1)^8 \left(c-a^2 c x^2\right)^{7/2}}{8 a^8 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}-\frac{4 (a x+1)^7 \left(c-a^2 c x^2\right)^{7/2}}{7 a^8 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}+\frac{2 (a x+1)^6 \left(c-a^2 c x^2\right)^{7/2}}{3 a^8 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}",1,"-1/168*(c^3*(1 + a*x)^6*(37 - 54*a*x + 21*a^2*x^2)*Sqrt[c - a^2*c*x^2])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
635,1,55,93,0.0449886,"\int e^{3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{5/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^(5/2),x]","\frac{c^2 (a x+1)^5 (5 a x-7) \sqrt{c-a^2 c x^2}}{30 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(a x+1)^6 \left(c-a^2 c x^2\right)^{5/2}}{6 a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}-\frac{2 (a x+1)^5 \left(c-a^2 c x^2\right)^{5/2}}{5 a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}",1,"(c^2*(1 + a*x)^5*(-7 + 5*a*x)*Sqrt[c - a^2*c*x^2])/(30*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
636,1,58,46,0.0348517,"\int e^{3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{3/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^(3/2),x]","-\frac{c \left(a^3 x^3+4 a^2 x^2+6 a x+4\right) \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(a x+1)^4 \left(c-a^2 c x^2\right)^{3/2}}{4 a^4 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}",1,"-1/4*(c*Sqrt[c - a^2*c*x^2]*(4 + 6*a*x + 4*a^2*x^2 + a^3*x^3))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",0
637,1,57,113,0.0296299,"\int e^{3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx","Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2],x]","\frac{\sqrt{c-a^2 c x^2} (a x (a x+6)+8 \log (1-a x))}{2 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 \sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(a*x*(6 + a*x) + 8*Log[1 - a*x]))/(2*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
638,1,53,79,0.0326861,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[E^(3*ArcCoth[a*x])/Sqrt[c - a^2*c*x^2],x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} ((a x-1) \log (1-a x)-2)}{(a x-1) \sqrt{c-a^2 c x^2}}","\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{(1-a x) \sqrt{c-a^2 c x^2}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{\sqrt{c-a^2 c x^2}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(-2 + (-1 + a*x)*Log[1 - a*x]))/((-1 + a*x)*Sqrt[c - a^2*c*x^2])","A",1
639,1,51,47,0.0489239,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a^2*c*x^2)^(3/2),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a^2 c x^2}}{2 c^2 (a x-1)^3 (a x+1)}","-\frac{a^2 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 (1-a x)^2 \left(c-a^2 c x^2\right)^{3/2}}",1,"-1/2*(Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a^2*c*x^2])/(c^2*(-1 + a*x)^3*(1 + a*x))","A",1
640,1,71,185,0.0662942,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a^2*c*x^2)^(5/2),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(-3 a^2 x^2+9 a x+3 (a x-1)^3 \tanh ^{-1}(a x)-10\right)}{24 c^2 (a x-1)^3 \sqrt{c-a^2 c x^2}}","\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (1-a x) \left(c-a^2 c x^2\right)^{5/2}}+\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (1-a x)^2 \left(c-a^2 c x^2\right)^{5/2}}+\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{6 (1-a x)^3 \left(c-a^2 c x^2\right)^{5/2}}+\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \tanh ^{-1}(a x)}{8 \left(c-a^2 c x^2\right)^{5/2}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(-10 + 9*a*x - 3*a^2*x^2 + 3*(-1 + a*x)^3*ArcTanh[a*x]))/(24*c^2*(-1 + a*x)^3*Sqrt[c - a^2*c*x^2])","A",1
641,1,99,278,0.0917169,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{7/2}} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - a^2*c*x^2)^(7/2),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(-15 a^4 x^4+45 a^3 x^3-35 a^2 x^2-15 a x+15 (a x-1)^4 (a x+1) \tanh ^{-1}(a x)+32\right)}{96 c^3 (a x-1)^4 (a x+1) \sqrt{c-a^2 c x^2}}","-\frac{a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{8 (1-a x) \left(c-a^2 c x^2\right)^{7/2}}+\frac{a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{32 (a x+1) \left(c-a^2 c x^2\right)^{7/2}}-\frac{3 a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{32 (1-a x)^2 \left(c-a^2 c x^2\right)^{7/2}}-\frac{a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{12 (1-a x)^3 \left(c-a^2 c x^2\right)^{7/2}}-\frac{a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{16 (1-a x)^4 \left(c-a^2 c x^2\right)^{7/2}}-\frac{5 a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2} \tanh ^{-1}(a x)}{32 \left(c-a^2 c x^2\right)^{7/2}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(32 - 15*a*x - 35*a^2*x^2 + 45*a^3*x^3 - 15*a^4*x^4 + 15*(-1 + a*x)^4*(1 + a*x)*ArcTanh[a*x]))/(96*c^3*(-1 + a*x)^4*(1 + a*x)*Sqrt[c - a^2*c*x^2])","A",1
642,1,79,234,0.0661503,"\int e^{-\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{9/2} \, dx","Integrate[(c - a^2*c*x^2)^(9/2)/E^ArcCoth[a*x],x]","\frac{c^4 (a x-1)^6 \left(63 a^4 x^4+308 a^3 x^3+588 a^2 x^2+528 a x+193\right) \sqrt{c-a^2 c x^2}}{630 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(1-a x)^{10} \left(c-a^2 c x^2\right)^{9/2}}{10 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}-\frac{8 (1-a x)^9 \left(c-a^2 c x^2\right)^{9/2}}{9 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}+\frac{3 (1-a x)^8 \left(c-a^2 c x^2\right)^{9/2}}{a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}-\frac{32 (1-a x)^7 \left(c-a^2 c x^2\right)^{9/2}}{7 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}+\frac{8 (1-a x)^6 \left(c-a^2 c x^2\right)^{9/2}}{3 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}",1,"(c^4*(-1 + a*x)^6*Sqrt[c - a^2*c*x^2]*(193 + 528*a*x + 588*a^2*x^2 + 308*a^3*x^3 + 63*a^4*x^4))/(630*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
643,1,71,187,0.0549805,"\int e^{-\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{7/2} \, dx","Integrate[(c - a^2*c*x^2)^(7/2)/E^ArcCoth[a*x],x]","-\frac{c^3 (a x-1)^5 \left(35 a^3 x^3+135 a^2 x^2+185 a x+93\right) \sqrt{c-a^2 c x^2}}{280 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(1-a x)^8 \left(c-a^2 c x^2\right)^{7/2}}{8 a^8 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}-\frac{6 (1-a x)^7 \left(c-a^2 c x^2\right)^{7/2}}{7 a^8 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}+\frac{2 (1-a x)^6 \left(c-a^2 c x^2\right)^{7/2}}{a^8 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}-\frac{8 (1-a x)^5 \left(c-a^2 c x^2\right)^{7/2}}{5 a^8 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}",1,"-1/280*(c^3*(-1 + a*x)^5*Sqrt[c - a^2*c*x^2]*(93 + 185*a*x + 135*a^2*x^2 + 35*a^3*x^3))/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
644,1,63,139,0.0455178,"\int e^{-\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{5/2} \, dx","Integrate[(c - a^2*c*x^2)^(5/2)/E^ArcCoth[a*x],x]","\frac{c^2 (a x-1)^4 \left(5 a^2 x^2+14 a x+11\right) \sqrt{c-a^2 c x^2}}{30 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(1-a x)^6 \left(c-a^2 c x^2\right)^{5/2}}{6 a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}-\frac{4 (1-a x)^5 \left(c-a^2 c x^2\right)^{5/2}}{5 a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}+\frac{(1-a x)^4 \left(c-a^2 c x^2\right)^{5/2}}{a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}",1,"(c^2*(-1 + a*x)^4*(11 + 14*a*x + 5*a^2*x^2)*Sqrt[c - a^2*c*x^2])/(30*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
645,1,53,95,0.0357368,"\int e^{-\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{3/2} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)/E^ArcCoth[a*x],x]","-\frac{c (a x-1)^3 (3 a x+5) \sqrt{c-a^2 c x^2}}{12 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(1-a x)^4 \left(c-a^2 c x^2\right)^{3/2}}{4 a^4 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}-\frac{2 (1-a x)^3 \left(c-a^2 c x^2\right)^{3/2}}{3 a^4 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}",1,"-1/12*(c*(-1 + a*x)^3*(5 + 3*a*x)*Sqrt[c - a^2*c*x^2])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
646,1,41,69,0.0226109,"\int e^{-\coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/E^ArcCoth[a*x],x]","\frac{(a x-2) \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((-2 + a*x)*Sqrt[c - a^2*c*x^2])/(2*a*Sqrt[1 - 1/(a^2*x^2)])","A",1
647,1,37,37,0.0212585,"\int \frac{e^{-\coth ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[1/(E^ArcCoth[a*x]*Sqrt[c - a^2*c*x^2]),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{\sqrt{c-a^2 c x^2}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{\sqrt{c-a^2 c x^2}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*Log[1 + a*x])/Sqrt[c - a^2*c*x^2]","A",1
648,1,54,90,0.0538451,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^(3/2)),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left((a x+1) \tanh ^{-1}(a x)-1\right)}{2 (a c x+c) \sqrt{c-a^2 c x^2}}","\frac{a^2 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 (a x+1) \left(c-a^2 c x^2\right)^{3/2}}-\frac{a^2 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \tanh ^{-1}(a x)}{2 \left(c-a^2 c x^2\right)^{3/2}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + (1 + a*x)*ArcTanh[a*x]))/(2*(c + a*c*x)*Sqrt[c - a^2*c*x^2])","A",1
649,1,81,183,0.0732655,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^(5/2)),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(-3 a^2 x^2-3 a x+3 (a x-1) (a x+1)^2 \tanh ^{-1}(a x)+2\right)}{8 (a x-1) (a c x+c)^2 \sqrt{c-a^2 c x^2}}","\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (1-a x) \left(c-a^2 c x^2\right)^{5/2}}-\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{4 (a x+1) \left(c-a^2 c x^2\right)^{5/2}}-\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (a x+1)^2 \left(c-a^2 c x^2\right)^{5/2}}+\frac{3 a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \tanh ^{-1}(a x)}{8 \left(c-a^2 c x^2\right)^{5/2}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(2 - 3*a*x - 3*a^2*x^2 + 3*(-1 + a*x)*(1 + a*x)^2*ArcTanh[a*x]))/(8*(-1 + a*x)*(c + a*c*x)^2*Sqrt[c - a^2*c*x^2])","A",1
650,1,99,276,0.098528,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{7/2}} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - a^2*c*x^2)^(7/2)),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(-15 a^4 x^4-15 a^3 x^3+25 a^2 x^2+25 a x+15 (a x-1)^2 (a x+1)^3 \tanh ^{-1}(a x)-8\right)}{48 (a x-1)^2 (a c x+c)^3 \sqrt{c-a^2 c x^2}}","-\frac{a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{8 (1-a x) \left(c-a^2 c x^2\right)^{7/2}}+\frac{3 a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{16 (a x+1) \left(c-a^2 c x^2\right)^{7/2}}-\frac{a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{32 (1-a x)^2 \left(c-a^2 c x^2\right)^{7/2}}+\frac{3 a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{32 (a x+1)^2 \left(c-a^2 c x^2\right)^{7/2}}+\frac{a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{24 (a x+1)^3 \left(c-a^2 c x^2\right)^{7/2}}-\frac{5 a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2} \tanh ^{-1}(a x)}{16 \left(c-a^2 c x^2\right)^{7/2}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(-8 + 25*a*x + 25*a^2*x^2 - 15*a^3*x^3 - 15*a^4*x^4 + 15*(-1 + a*x)^2*(1 + a*x)^3*ArcTanh[a*x]))/(48*(-1 + a*x)^2*(c + a*c*x)^3*Sqrt[c - a^2*c*x^2])","A",1
651,1,136,131,0.1515315,"\int e^{-2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{5/2} \, dx","Integrate[(c - a^2*c*x^2)^(5/2)/E^(2*ArcCoth[a*x]),x]","\frac{c^2 \sqrt{c-a^2 c x^2} \left(210 \sqrt{1-a x} \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)-\sqrt{a x+1} \left(40 a^6 x^6-136 a^5 x^5+86 a^4 x^4+202 a^3 x^3-327 a^2 x^2+39 a x+96\right)\right)}{240 a \sqrt{1-a x} \sqrt{1-a^2 x^2}}","-\frac{7 c^{5/2} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{16 a}-\frac{7}{16} c^2 x \sqrt{c-a^2 c x^2}-\frac{7}{24} c x \left(c-a^2 c x^2\right)^{3/2}-\frac{(1-a x) \left(c-a^2 c x^2\right)^{5/2}}{6 a}-\frac{7 \left(c-a^2 c x^2\right)^{5/2}}{30 a}",1,"(c^2*Sqrt[c - a^2*c*x^2]*(-(Sqrt[1 + a*x]*(96 + 39*a*x - 327*a^2*x^2 + 202*a^3*x^3 + 86*a^4*x^4 - 136*a^5*x^5 + 40*a^6*x^6)) + 210*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(240*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])","A",0
652,1,117,108,0.0990969,"\int e^{-2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{3/2} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)/E^(2*ArcCoth[a*x]),x]","\frac{c \sqrt{c-a^2 c x^2} \left(\sqrt{a x+1} \left(6 a^4 x^4-22 a^3 x^3+25 a^2 x^2+7 a x-16\right)+30 \sqrt{1-a x} \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)\right)}{24 a \sqrt{1-a x} \sqrt{1-a^2 x^2}}","-\frac{5 c^{3/2} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{8 a}-\frac{5}{8} c x \sqrt{c-a^2 c x^2}-\frac{(1-a x) \left(c-a^2 c x^2\right)^{3/2}}{4 a}-\frac{5 \left(c-a^2 c x^2\right)^{3/2}}{12 a}",1,"(c*Sqrt[c - a^2*c*x^2]*(Sqrt[1 + a*x]*(-16 + 7*a*x + 25*a^2*x^2 - 22*a^3*x^3 + 6*a^4*x^4) + 30*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(24*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])","A",0
653,1,100,87,0.0650872,"\int e^{-2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/E^(2*ArcCoth[a*x]),x]","\frac{\sqrt{c-a^2 c x^2} \left(6 \sqrt{1-a x} \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)-\sqrt{a x+1} \left(a^2 x^2-5 a x+4\right)\right)}{2 a \sqrt{1-a x} \sqrt{1-a^2 x^2}}","-\frac{(1-a x) \sqrt{c-a^2 c x^2}}{2 a}-\frac{3 \sqrt{c-a^2 c x^2}}{2 a}-\frac{3 \sqrt{c} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{2 a}",1,"(Sqrt[c - a^2*c*x^2]*(-(Sqrt[1 + a*x]*(4 - 5*a*x + a^2*x^2)) + 6*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(2*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])","A",0
654,1,100,60,0.0676743,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2]),x]","-\frac{2 \sqrt{1-a^2 x^2} \left(\sqrt{a x+1} (a x-1)+\sqrt{1-a x} (a x+1) \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)\right)}{a \sqrt{1-a x} (a x+1) \sqrt{c-a^2 c x^2}}","\frac{2 (1-a x)}{a \sqrt{c-a^2 c x^2}}+\frac{\tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{a \sqrt{c}}",1,"(-2*Sqrt[1 - a^2*x^2]*((-1 + a*x)*Sqrt[1 + a*x] + Sqrt[1 - a*x]*(1 + a*x)*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(a*Sqrt[1 - a*x]*(1 + a*x)*Sqrt[c - a^2*c*x^2])","A",0
655,1,63,52,0.0409603,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(3/2)),x]","\frac{\sqrt{1-a x} (a x+2) \sqrt{1-a^2 x^2}}{3 a c (a x+1)^{3/2} \sqrt{c-a^2 c x^2}}","\frac{2 (1-a x)}{3 a \left(c-a^2 c x^2\right)^{3/2}}-\frac{x}{3 c \sqrt{c-a^2 c x^2}}",1,"(Sqrt[1 - a*x]*(2 + a*x)*Sqrt[1 - a^2*x^2])/(3*a*c*(1 + a*x)^(3/2)*Sqrt[c - a^2*c*x^2])","A",0
656,1,79,75,0.054475,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(5/2)),x]","-\frac{\sqrt{1-a^2 x^2} \left(2 a^3 x^3+4 a^2 x^2+a x-2\right)}{5 a c^2 \sqrt{1-a x} (a x+1)^{5/2} \sqrt{c-a^2 c x^2}}","-\frac{2 x}{5 c^2 \sqrt{c-a^2 c x^2}}-\frac{x}{5 c \left(c-a^2 c x^2\right)^{3/2}}+\frac{2 (1-a x)}{5 a \left(c-a^2 c x^2\right)^{5/2}}",1,"-1/5*(Sqrt[1 - a^2*x^2]*(-2 + a*x + 4*a^2*x^2 + 2*a^3*x^3))/(a*c^2*Sqrt[1 - a*x]*(1 + a*x)^(5/2)*Sqrt[c - a^2*c*x^2])","A",1
657,1,96,98,0.0663271,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{7/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(7/2)),x]","-\frac{\sqrt{1-a^2 x^2} \left(-8 a^5 x^5-16 a^4 x^4+4 a^3 x^3+24 a^2 x^2+9 a x-6\right)}{21 a c^3 (1-a x)^{3/2} (a x+1)^{7/2} \sqrt{c-a^2 c x^2}}","-\frac{8 x}{21 c^3 \sqrt{c-a^2 c x^2}}-\frac{4 x}{21 c^2 \left(c-a^2 c x^2\right)^{3/2}}-\frac{x}{7 c \left(c-a^2 c x^2\right)^{5/2}}+\frac{2 (1-a x)}{7 a \left(c-a^2 c x^2\right)^{7/2}}",1,"-1/21*(Sqrt[1 - a^2*x^2]*(-6 + 9*a*x + 24*a^2*x^2 + 4*a^3*x^3 - 16*a^4*x^4 - 8*a^5*x^5))/(a*c^3*(1 - a*x)^(3/2)*(1 + a*x)^(7/2)*Sqrt[c - a^2*c*x^2])","A",1
658,1,112,121,0.0762126,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{9/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^(9/2)),x]","\frac{\sqrt{1-a^2 x^2} \left(-16 a^7 x^7-32 a^6 x^6+24 a^5 x^5+80 a^4 x^4+10 a^3 x^3-60 a^2 x^2-25 a x+10\right)}{45 a c^4 (1-a x)^{5/2} (a x+1)^{9/2} \sqrt{c-a^2 c x^2}}","-\frac{16 x}{45 c^4 \sqrt{c-a^2 c x^2}}-\frac{8 x}{45 c^3 \left(c-a^2 c x^2\right)^{3/2}}-\frac{2 x}{15 c^2 \left(c-a^2 c x^2\right)^{5/2}}-\frac{x}{9 c \left(c-a^2 c x^2\right)^{7/2}}+\frac{2 (1-a x)}{9 a \left(c-a^2 c x^2\right)^{9/2}}",1,"(Sqrt[1 - a^2*x^2]*(10 - 25*a*x - 60*a^2*x^2 + 10*a^3*x^3 + 80*a^4*x^4 + 24*a^5*x^5 - 32*a^6*x^6 - 16*a^7*x^7))/(45*a*c^4*(1 - a*x)^(5/2)*(1 + a*x)^(9/2)*Sqrt[c - a^2*c*x^2])","A",1
659,1,71,189,0.0624551,"\int e^{-3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{9/2} \, dx","Integrate[(c - a^2*c*x^2)^(9/2)/E^(3*ArcCoth[a*x]),x]","\frac{c^4 (a x-1)^7 \left(21 a^3 x^3+77 a^2 x^2+98 a x+44\right) \sqrt{c-a^2 c x^2}}{210 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(1-a x)^{10} \left(c-a^2 c x^2\right)^{9/2}}{10 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}-\frac{2 (1-a x)^9 \left(c-a^2 c x^2\right)^{9/2}}{3 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}+\frac{3 (1-a x)^8 \left(c-a^2 c x^2\right)^{9/2}}{2 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}-\frac{8 (1-a x)^7 \left(c-a^2 c x^2\right)^{9/2}}{7 a^{10} x^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}",1,"(c^4*(-1 + a*x)^7*Sqrt[c - a^2*c*x^2]*(44 + 98*a*x + 77*a^2*x^2 + 21*a^3*x^3))/(210*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
660,1,63,142,0.0540032,"\int e^{-3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{7/2} \, dx","Integrate[(c - a^2*c*x^2)^(7/2)/E^(3*ArcCoth[a*x]),x]","-\frac{c^3 (a x-1)^6 \left(21 a^2 x^2+54 a x+37\right) \sqrt{c-a^2 c x^2}}{168 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(1-a x)^8 \left(c-a^2 c x^2\right)^{7/2}}{8 a^8 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}-\frac{4 (1-a x)^7 \left(c-a^2 c x^2\right)^{7/2}}{7 a^8 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}+\frac{2 (1-a x)^6 \left(c-a^2 c x^2\right)^{7/2}}{3 a^8 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}",1,"-1/168*(c^3*(-1 + a*x)^6*(37 + 54*a*x + 21*a^2*x^2)*Sqrt[c - a^2*c*x^2])/(a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
661,1,55,95,0.043185,"\int e^{-3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{5/2} \, dx","Integrate[(c - a^2*c*x^2)^(5/2)/E^(3*ArcCoth[a*x]),x]","\frac{c^2 (a x-1)^5 (5 a x+7) \sqrt{c-a^2 c x^2}}{30 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(1-a x)^6 \left(c-a^2 c x^2\right)^{5/2}}{6 a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}-\frac{2 (1-a x)^5 \left(c-a^2 c x^2\right)^{5/2}}{5 a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}",1,"(c^2*(-1 + a*x)^5*(7 + 5*a*x)*Sqrt[c - a^2*c*x^2])/(30*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
662,1,58,47,0.0354574,"\int e^{-3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{3/2} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)/E^(3*ArcCoth[a*x]),x]","-\frac{c \left(a^3 x^3-4 a^2 x^2+6 a x-4\right) \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(1-a x)^4 \left(c-a^2 c x^2\right)^{3/2}}{4 a^4 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}",1,"-1/4*(c*Sqrt[c - a^2*c*x^2]*(-4 + 6*a*x - 4*a^2*x^2 + a^3*x^3))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",0
663,1,56,112,0.0295667,"\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/E^(3*ArcCoth[a*x]),x]","\frac{\sqrt{c-a^2 c x^2} (a x (a x-6)+8 \log (a x+1))}{2 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 \sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(a*x*(-6 + a*x) + 8*Log[1 + a*x]))/(2*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
664,1,52,77,0.0364324,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2]),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} ((a x+1) \log (a x+1)+2)}{(a x+1) \sqrt{c-a^2 c x^2}}","\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{(a x+1) \sqrt{c-a^2 c x^2}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{\sqrt{c-a^2 c x^2}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(2 + (1 + a*x)*Log[1 + a*x]))/((1 + a*x)*Sqrt[c - a^2*c*x^2])","A",1
665,1,51,46,0.055127,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^(3/2)),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a^2 c x^2}}{2 c^2 (a x-1) (a x+1)^3}","-\frac{a^2 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 (a x+1)^2 \left(c-a^2 c x^2\right)^{3/2}}",1,"-1/2*(Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a^2*c*x^2])/(c^2*(-1 + a*x)*(1 + a*x)^3)","A",1
666,1,71,182,0.0682822,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^(5/2)),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(-3 a^2 x^2-9 a x+3 (a x+1)^3 \tanh ^{-1}(a x)-10\right)}{24 c^2 (a x+1)^3 \sqrt{c-a^2 c x^2}}","\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (a x+1) \left(c-a^2 c x^2\right)^{5/2}}+\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (a x+1)^2 \left(c-a^2 c x^2\right)^{5/2}}+\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{6 (a x+1)^3 \left(c-a^2 c x^2\right)^{5/2}}-\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \tanh ^{-1}(a x)}{8 \left(c-a^2 c x^2\right)^{5/2}}",1,"-1/24*(Sqrt[1 - 1/(a^2*x^2)]*x*(-10 - 9*a*x - 3*a^2*x^2 + 3*(1 + a*x)^3*ArcTanh[a*x]))/(c^2*(1 + a*x)^3*Sqrt[c - a^2*c*x^2])","A",1
667,1,99,275,0.1017565,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{7/2}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^(7/2)),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(-15 a^4 x^4-45 a^3 x^3-35 a^2 x^2+15 a x+15 (a x-1) (a x+1)^4 \tanh ^{-1}(a x)+32\right)}{96 c^3 (a x-1) (a x+1)^4 \sqrt{c-a^2 c x^2}}","\frac{a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{32 (1-a x) \left(c-a^2 c x^2\right)^{7/2}}-\frac{a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{8 (a x+1) \left(c-a^2 c x^2\right)^{7/2}}-\frac{3 a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{32 (a x+1)^2 \left(c-a^2 c x^2\right)^{7/2}}-\frac{a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{12 (a x+1)^3 \left(c-a^2 c x^2\right)^{7/2}}-\frac{a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}{16 (a x+1)^4 \left(c-a^2 c x^2\right)^{7/2}}+\frac{5 a^6 x^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2} \tanh ^{-1}(a x)}{32 \left(c-a^2 c x^2\right)^{7/2}}",1,"-1/96*(Sqrt[1 - 1/(a^2*x^2)]*x*(32 + 15*a*x - 35*a^2*x^2 - 45*a^3*x^3 - 15*a^4*x^4 + 15*(-1 + a*x)*(1 + a*x)^4*ArcTanh[a*x]))/(c^3*(-1 + a*x)*(1 + a*x)^4*Sqrt[c - a^2*c*x^2])","A",1
668,1,45,76,0.0253821,"\int e^{\coth ^{-1}(a x)} x^2 \sqrt{c-a^2 c x^2} \, dx","Integrate[E^ArcCoth[a*x]*x^2*Sqrt[c - a^2*c*x^2],x]","\frac{x^2 (3 a x+4) \sqrt{c-a^2 c x^2}}{12 a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^2 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^3 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(x^2*(4 + 3*a*x)*Sqrt[c - a^2*c*x^2])/(12*a*Sqrt[1 - 1/(a^2*x^2)])","A",1
669,1,43,74,0.0229851,"\int e^{\coth ^{-1}(a x)} x \sqrt{c-a^2 c x^2} \, dx","Integrate[E^ArcCoth[a*x]*x*Sqrt[c - a^2*c*x^2],x]","\frac{x (2 a x+3) \sqrt{c-a^2 c x^2}}{6 a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^2 \sqrt{c-a^2 c x^2}}{3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(x*(3 + 2*a*x)*Sqrt[c - a^2*c*x^2])/(6*a*Sqrt[1 - 1/(a^2*x^2)])","A",1
670,1,41,68,0.0195289,"\int e^{\coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx","Integrate[E^ArcCoth[a*x]*Sqrt[c - a^2*c*x^2],x]","\frac{(a x+2) \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((2 + a*x)*Sqrt[c - a^2*c*x^2])/(2*a*Sqrt[1 - 1/(a^2*x^2)])","A",1
671,1,42,69,0.0233818,"\int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x} \, dx","Integrate[(E^ArcCoth[a*x]*Sqrt[c - a^2*c*x^2])/x,x]","\frac{\sqrt{c-a^2 c x^2} (a x+\log (x))}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{\sqrt{c-a^2 c x^2}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\log (x) \sqrt{c-a^2 c x^2}}{a x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(a*x + Log[x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
672,1,43,73,0.0251657,"\int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^2} \, dx","Integrate[(E^ArcCoth[a*x]*Sqrt[c - a^2*c*x^2])/x^2,x]","\frac{\sqrt{c-a^2 c x^2} (a x \log (x)-1)}{a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{\log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-a^2 c x^2}}{a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(-1 + a*x*Log[x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x^2)","A",1
673,1,96,137,0.1314872,"\int e^{2 \coth ^{-1}(a x)} x^3 \sqrt{c-a^2 c x^2} \, dx","Integrate[E^(2*ArcCoth[a*x])*x^3*Sqrt[c - a^2*c*x^2],x]","\frac{15 \sqrt{c} \tan ^{-1}\left(\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left(a^2 x^2-1\right)}\right)+\left(4 a^4 x^4+10 a^3 x^3+12 a^2 x^2+15 a x+24\right) \sqrt{c-a^2 c x^2}}{20 a^4}","\frac{3 x^2 \sqrt{c-a^2 c x^2}}{5 a^2}+\frac{1}{5} x^4 \sqrt{c-a^2 c x^2}+\frac{x^3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{3 (5 a x+8) \sqrt{c-a^2 c x^2}}{20 a^4}-\frac{3 \sqrt{c} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{4 a^4}",1,"(Sqrt[c - a^2*c*x^2]*(24 + 15*a*x + 12*a^2*x^2 + 10*a^3*x^3 + 4*a^4*x^4) + 15*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(20*a^4)","A",1
674,1,88,112,0.1061729,"\int e^{2 \coth ^{-1}(a x)} x^2 \sqrt{c-a^2 c x^2} \, dx","Integrate[E^(2*ArcCoth[a*x])*x^2*Sqrt[c - a^2*c*x^2],x]","\frac{21 \sqrt{c} \tan ^{-1}\left(\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left(a^2 x^2-1\right)}\right)+\left(6 a^3 x^3+16 a^2 x^2+21 a x+32\right) \sqrt{c-a^2 c x^2}}{24 a^3}","\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}+\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}+\frac{(21 a x+32) \sqrt{c-a^2 c x^2}}{24 a^3}-\frac{7 \sqrt{c} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{8 a^3}",1,"(Sqrt[c - a^2*c*x^2]*(32 + 21*a*x + 16*a^2*x^2 + 6*a^3*x^3) + 21*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(24*a^3)","A",1
675,1,79,85,0.0889796,"\int e^{2 \coth ^{-1}(a x)} x \sqrt{c-a^2 c x^2} \, dx","Integrate[E^(2*ArcCoth[a*x])*x*Sqrt[c - a^2*c*x^2],x]","\frac{\left(a^2 x^2+3 a x+5\right) \sqrt{c-a^2 c x^2}+3 \sqrt{c} \tan ^{-1}\left(\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left(a^2 x^2-1\right)}\right)}{3 a^2}","\frac{1}{3} x^2 \sqrt{c-a^2 c x^2}+\frac{(3 a x+5) \sqrt{c-a^2 c x^2}}{3 a^2}-\frac{\sqrt{c} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{a^2}",1,"((5 + 3*a*x + a^2*x^2)*Sqrt[c - a^2*c*x^2] + 3*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(3*a^2)","A",1
676,1,76,86,0.0588851,"\int e^{2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx","Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2],x]","\frac{\sqrt{c-a^2 c x^2} \left(\sqrt{1-a^2 x^2} (a x+4)+6 \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)\right)}{2 a \sqrt{1-a^2 x^2}}","\frac{(a x+1) \sqrt{c-a^2 c x^2}}{2 a}+\frac{3 \sqrt{c-a^2 c x^2}}{2 a}-\frac{3 \sqrt{c} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{2 a}",1,"(Sqrt[c - a^2*c*x^2]*((4 + a*x)*Sqrt[1 - a^2*x^2] + 6*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(2*a*Sqrt[1 - a^2*x^2])","A",0
677,1,97,75,0.0817414,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x,x]","\sqrt{c-a^2 c x^2}+\sqrt{c} \log \left(\sqrt{c} \sqrt{c-a^2 c x^2}+c\right)+2 \sqrt{c} \tan ^{-1}\left(\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left(a^2 x^2-1\right)}\right)-\sqrt{c} \log (x)","\sqrt{c-a^2 c x^2}-2 \sqrt{c} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)+\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right)",1,"Sqrt[c - a^2*c*x^2] + 2*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))] - Sqrt[c]*Log[x] + Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]]","A",0
678,1,104,82,0.1002774,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^2} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^2,x]","\frac{\sqrt{c-a^2 c x^2}}{x}+2 a \sqrt{c} \log \left(\sqrt{c} \sqrt{c-a^2 c x^2}+c\right)+a \sqrt{c} \tan ^{-1}\left(\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left(a^2 x^2-1\right)}\right)-2 a \sqrt{c} \log (x)","\frac{\sqrt{c-a^2 c x^2}}{x}-a \sqrt{c} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)+2 a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right)",1,"Sqrt[c - a^2*c*x^2]/x + a*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))] - 2*a*Sqrt[c]*Log[x] + 2*a*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]]","A",0
679,1,76,78,0.1253574,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^3} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^3,x]","\frac{1}{2} \left(\frac{(4 a x+1) \sqrt{c-a^2 c x^2}}{x^2}+3 a^2 \sqrt{c} \log \left(\sqrt{c} \sqrt{c-a^2 c x^2}+c\right)-3 a^2 \sqrt{c} \log (x)\right)","\frac{2 a \sqrt{c-a^2 c x^2}}{x}+\frac{\sqrt{c-a^2 c x^2}}{2 x^2}+\frac{3}{2} a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right)",1,"(((1 + 4*a*x)*Sqrt[c - a^2*c*x^2])/x^2 - 3*a^2*Sqrt[c]*Log[x] + 3*a^2*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]])/2","A",0
680,1,82,99,0.1265005,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^4} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^4,x]","a^3 \left(-\sqrt{c}\right) \log (x)+\frac{\left(5 a^2 x^2+3 a x+1\right) \sqrt{c-a^2 c x^2}}{3 x^3}+a^3 \sqrt{c} \log \left(\sqrt{c} \sqrt{c-a^2 c x^2}+c\right)","\frac{5 a^2 \sqrt{c-a^2 c x^2}}{3 x}+\frac{a \sqrt{c-a^2 c x^2}}{x^2}+\frac{\sqrt{c-a^2 c x^2}}{3 x^3}+a^3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right)",1,"((1 + 3*a*x + 5*a^2*x^2)*Sqrt[c - a^2*c*x^2])/(3*x^3) - a^3*Sqrt[c]*Log[x] + a^3*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]]","A",0
681,1,95,130,0.1516213,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^5} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^5,x]","-\frac{7}{8} a^4 \sqrt{c} \log (x)+\frac{7}{8} a^4 \sqrt{c} \log \left(\sqrt{c} \sqrt{c-a^2 c x^2}+c\right)+\frac{\left(32 a^3 x^3+21 a^2 x^2+16 a x+6\right) \sqrt{c-a^2 c x^2}}{24 x^4}","\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{7}{8} a^4 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right)+\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}",1,"(Sqrt[c - a^2*c*x^2]*(6 + 16*a*x + 21*a^2*x^2 + 32*a^3*x^3))/(24*x^4) - (7*a^4*Sqrt[c]*Log[x])/8 + (7*a^4*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]])/8","A",0
682,1,88,228,0.0642943,"\int e^{3 \coth ^{-1}(a x)} x^3 \sqrt{c-a^2 c x^2} \, dx","Integrate[E^(3*ArcCoth[a*x])*x^3*Sqrt[c - a^2*c*x^2],x]","\frac{\sqrt{c-a^2 c x^2} \left(\frac{4 \log (1-a x)}{a^4}+\frac{4 x}{a^3}+\frac{2 x^2}{a^2}+\frac{a x^5}{5}+\frac{4 x^3}{3 a}+\frac{3 x^4}{4}\right)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{4 x^2 \sqrt{c-a^2 c x^2}}{3 a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^4 \sqrt{c-a^2 c x^2}}{5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 x^3 \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{a^5 x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2}}{a^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 x \sqrt{c-a^2 c x^2}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*((4*x)/a^3 + (2*x^2)/a^2 + (4*x^3)/(3*a) + (3*x^4)/4 + (a*x^5)/5 + (4*Log[1 - a*x])/a^4))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
683,1,74,186,0.0481947,"\int e^{3 \coth ^{-1}(a x)} x^2 \sqrt{c-a^2 c x^2} \, dx","Integrate[E^(3*ArcCoth[a*x])*x^2*Sqrt[c - a^2*c*x^2],x]","\frac{\sqrt{c-a^2 c x^2} \left(\frac{4 \log (1-a x)}{a^3}+\frac{4 x}{a^2}+\frac{a x^4}{4}+\frac{2 x^2}{a}+x^3\right)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^2 \sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 x \sqrt{c-a^2 c x^2}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^3 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{a^4 x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*((4*x)/a^2 + (2*x^2)/a + x^3 + (a*x^4)/4 + (4*Log[1 - a*x])/a^3))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
684,1,66,152,0.0405204,"\int e^{3 \coth ^{-1}(a x)} x \sqrt{c-a^2 c x^2} \, dx","Integrate[E^(3*ArcCoth[a*x])*x*Sqrt[c - a^2*c*x^2],x]","\frac{\sqrt{c-a^2 c x^2} \left(a x \left(2 a^2 x^2+9 a x+24\right)+24 \log (1-a x)\right)}{6 a^3 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^2 \sqrt{c-a^2 c x^2}}{3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 x \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{a^3 x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(a*x*(24 + 9*a*x + 2*a^2*x^2) + 24*Log[1 - a*x]))/(6*a^3*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
685,1,57,113,0.0270054,"\int e^{3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx","Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2],x]","\frac{\sqrt{c-a^2 c x^2} (a x (a x+6)+8 \log (1-a x))}{2 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 \sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(a*x*(6 + a*x) + 8*Log[1 - a*x]))/(2*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
686,1,53,114,0.0290757,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x,x]","\frac{\sqrt{c-a^2 c x^2} (a x+4 \log (1-a x)-\log (x))}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{\sqrt{c-a^2 c x^2}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\log (x) \sqrt{c-a^2 c x^2}}{a x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(a*x - Log[x] + 4*Log[1 - a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
687,1,55,114,0.0355819,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^2} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^2,x]","\frac{\sqrt{c-a^2 c x^2} \left(-3 a \log (x)+4 a \log (1-a x)+\frac{1}{x}\right)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{\sqrt{c-a^2 c x^2}}{a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 \log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (1-a x)}{x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(x^(-1) - 3*a*Log[x] + 4*a*Log[1 - a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
688,1,69,153,0.0415075,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^3} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^3,x]","\frac{\sqrt{c-a^2 c x^2} \left(-4 a^2 \log (x)+4 a^2 \log (1-a x)+\frac{3 a}{x}+\frac{1}{2 x^2}\right)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{3 \sqrt{c-a^2 c x^2}}{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a \log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a \sqrt{c-a^2 c x^2} \log (1-a x)}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-a^2 c x^2}}{2 a x^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(1/(2*x^2) + (3*a)/x - 4*a^2*Log[x] + 4*a^2*Log[1 - a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
689,1,79,194,0.0449622,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^4} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^4,x]","\frac{\sqrt{c-a^2 c x^2} \left(-4 a^3 \log (x)+4 a^3 \log (1-a x)+\frac{4 a^2}{x}+\frac{3 a}{2 x^2}+\frac{1}{3 x^3}\right)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{4 a \sqrt{c-a^2 c x^2}}{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a^2 \log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a^2 \sqrt{c-a^2 c x^2} \log (1-a x)}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-a^2 c x^2}}{3 a x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 \sqrt{c-a^2 c x^2}}{2 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(1/(3*x^3) + (3*a)/(2*x^2) + (4*a^2)/x - 4*a^3*Log[x] + 4*a^3*Log[1 - a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
690,1,84,228,0.0631235,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^5} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2])/x^5,x]","\frac{\sqrt{c-a^2 c x^2} \left(-4 a^4 \log (x)+4 a^4 \log (1-a x)+\frac{4 a^3}{x}+\frac{2 a^2}{x^2}+\frac{a}{x^3}+\frac{1}{4 x^4}\right)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{4 a^2 \sqrt{c-a^2 c x^2}}{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-a^2 c x^2}}{4 a x^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-a^2 c x^2}}{x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 a \sqrt{c-a^2 c x^2}}{x^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a^3 \log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a^3 \sqrt{c-a^2 c x^2} \log (1-a x)}{x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(1/(4*x^4) + a/x^3 + (2*a^2)/x^2 + (4*a^3)/x - 4*a^4*Log[x] + 4*a^4*Log[1 - a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
691,1,77,211,0.0909608,"\int \frac{e^{\coth ^{-1}(a x)} x^4}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[(E^ArcCoth[a*x]*x^4)/(c - a^2*c*x^2)^(3/2),x]","\frac{x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(2 \left(a^2 x^2+2 a x+\frac{1}{1-a x}\right)+7 \log (1-a x)+\log (a x+1)\right)}{4 a^2 \left(c-a^2 c x^2\right)^{3/2}}","\frac{x^5 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 \left(c-a^2 c x^2\right)^{3/2}}+\frac{x^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{a \left(c-a^2 c x^2\right)^{3/2}}+\frac{x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 a^2 (1-a x) \left(c-a^2 c x^2\right)^{3/2}}+\frac{7 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (1-a x)}{4 a^2 \left(c-a^2 c x^2\right)^{3/2}}+\frac{x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (a x+1)}{4 a^2 \left(c-a^2 c x^2\right)^{3/2}}",1,"((1 - 1/(a^2*x^2))^(3/2)*x^3*(2*(2*a*x + a^2*x^2 + (1 - a*x)^(-1)) + 7*Log[1 - a*x] + Log[1 + a*x]))/(4*a^2*(c - a^2*c*x^2)^(3/2))","A",1
692,1,71,172,0.0613353,"\int \frac{e^{\coth ^{-1}(a x)} x^3}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[(E^ArcCoth[a*x]*x^3)/(c - a^2*c*x^2)^(3/2),x]","\frac{x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(4 a x+\frac{2}{1-a x}+5 \log (1-a x)-\log (a x+1)\right)}{4 a \left(c-a^2 c x^2\right)^{3/2}}","\frac{x^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{\left(c-a^2 c x^2\right)^{3/2}}+\frac{x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 a (1-a x) \left(c-a^2 c x^2\right)^{3/2}}+\frac{5 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (1-a x)}{4 a \left(c-a^2 c x^2\right)^{3/2}}-\frac{x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (a x+1)}{4 a \left(c-a^2 c x^2\right)^{3/2}}",1,"((1 - 1/(a^2*x^2))^(3/2)*x^3*(4*a*x + 2/(1 - a*x) + 5*Log[1 - a*x] - Log[1 + a*x]))/(4*a*(c - a^2*c*x^2)^(3/2))","A",1
693,1,75,130,0.0489692,"\int \frac{e^{\coth ^{-1}(a x)} x^2}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[(E^ArcCoth[a*x]*x^2)/(c - a^2*c*x^2)^(3/2),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (3 (a x-1) \log (1-a x)+(a x-1) \log (a x+1)-2)}{4 a^2 c (a x-1) \sqrt{c-a^2 c x^2}}","\frac{x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 (1-a x) \left(c-a^2 c x^2\right)^{3/2}}+\frac{3 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (1-a x)}{4 \left(c-a^2 c x^2\right)^{3/2}}+\frac{x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (a x+1)}{4 \left(c-a^2 c x^2\right)^{3/2}}",1,"-1/4*(Sqrt[1 - 1/(a^2*x^2)]*x*(-2 + 3*(-1 + a*x)*Log[1 - a*x] + (-1 + a*x)*Log[1 + a*x]))/(a^2*c*(-1 + a*x)*Sqrt[c - a^2*c*x^2])","A",1
694,1,59,87,0.0467731,"\int \frac{e^{\coth ^{-1}(a x)} x}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[(E^ArcCoth[a*x]*x)/(c - a^2*c*x^2)^(3/2),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left((a x-1) \tanh ^{-1}(a x)+1\right)}{2 a c (a x-1) \sqrt{c-a^2 c x^2}}","\frac{a x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 (1-a x) \left(c-a^2 c x^2\right)^{3/2}}-\frac{a x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \tanh ^{-1}(a x)}{2 \left(c-a^2 c x^2\right)^{3/2}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(1 + (-1 + a*x)*ArcTanh[a*x]))/(2*a*c*(-1 + a*x)*Sqrt[c - a^2*c*x^2])","A",1
695,1,56,91,0.0335855,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^ArcCoth[a*x]/(c - a^2*c*x^2)^(3/2),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left((a x-1) \tanh ^{-1}(a x)-1\right)}{2 c (a x-1) \sqrt{c-a^2 c x^2}}","\frac{a^2 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 (1-a x) \left(c-a^2 c x^2\right)^{3/2}}+\frac{a^2 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \tanh ^{-1}(a x)}{2 \left(c-a^2 c x^2\right)^{3/2}}",1,"-1/2*(Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + (-1 + a*x)*ArcTanh[a*x]))/(c*(-1 + a*x)*Sqrt[c - a^2*c*x^2])","A",1
696,1,68,177,0.0688759,"\int \frac{e^{\coth ^{-1}(a x)}}{x \left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^ArcCoth[a*x]/(x*(c - a^2*c*x^2)^(3/2)),x]","\frac{a^3 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(\frac{1}{2-2 a x}-\frac{3}{4} \log (1-a x)-\frac{1}{4} \log (a x+1)+\log (x)\right)}{\left(c-a^2 c x^2\right)^{3/2}}","\frac{a^3 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 (1-a x) \left(c-a^2 c x^2\right)^{3/2}}+\frac{a^3 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (x)}{\left(c-a^2 c x^2\right)^{3/2}}-\frac{3 a^3 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (1-a x)}{4 \left(c-a^2 c x^2\right)^{3/2}}-\frac{a^3 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (a x+1)}{4 \left(c-a^2 c x^2\right)^{3/2}}",1,"(a^3*(1 - 1/(a^2*x^2))^(3/2)*x^3*((2 - 2*a*x)^(-1) + Log[x] - (3*Log[1 - a*x])/4 - Log[1 + a*x]/4))/(c - a^2*c*x^2)^(3/2)","A",1
697,1,79,214,0.0700636,"\int \frac{e^{\coth ^{-1}(a x)}}{x^2 \left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^ArcCoth[a*x]/(x^2*(c - a^2*c*x^2)^(3/2)),x]","\frac{a^3 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(\frac{2 a}{1-a x}+4 a \log (x)-5 a \log (1-a x)+a \log (a x+1)-\frac{4}{x}\right)}{4 \left(c-a^2 c x^2\right)^{3/2}}","\frac{a^4 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 (1-a x) \left(c-a^2 c x^2\right)^{3/2}}+\frac{a^4 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (x)}{\left(c-a^2 c x^2\right)^{3/2}}-\frac{5 a^4 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (1-a x)}{4 \left(c-a^2 c x^2\right)^{3/2}}+\frac{a^4 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (a x+1)}{4 \left(c-a^2 c x^2\right)^{3/2}}-\frac{a^3 x^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{\left(c-a^2 c x^2\right)^{3/2}}",1,"(a^3*(1 - 1/(a^2*x^2))^(3/2)*x^3*(-4/x + (2*a)/(1 - a*x) + 4*a*Log[x] - 5*a*Log[1 - a*x] + a*Log[1 + a*x]))/(4*(c - a^2*c*x^2)^(3/2))","A",1
698,1,94,252,0.0923531,"\int \frac{e^{\coth ^{-1}(a x)}}{x^3 \left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^ArcCoth[a*x]/(x^3*(c - a^2*c*x^2)^(3/2)),x]","\frac{a^3 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(\frac{2 a^2}{1-a x}+8 a^2 \log (x)-7 a^2 \log (1-a x)-a^2 \log (a x+1)-\frac{4 a}{x}-\frac{2}{x^2}\right)}{4 \left(c-a^2 c x^2\right)^{3/2}}","\frac{a^5 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 (1-a x) \left(c-a^2 c x^2\right)^{3/2}}+\frac{2 a^5 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (x)}{\left(c-a^2 c x^2\right)^{3/2}}-\frac{7 a^5 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (1-a x)}{4 \left(c-a^2 c x^2\right)^{3/2}}-\frac{a^5 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \log (a x+1)}{4 \left(c-a^2 c x^2\right)^{3/2}}-\frac{a^4 x^2 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{\left(c-a^2 c x^2\right)^{3/2}}-\frac{a^3 x \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}{2 \left(c-a^2 c x^2\right)^{3/2}}",1,"(a^3*(1 - 1/(a^2*x^2))^(3/2)*x^3*(-2/x^2 - (4*a)/x + (2*a^2)/(1 - a*x) + 8*a^2*Log[x] - 7*a^2*Log[1 - a*x] - a^2*Log[1 + a*x]))/(4*(c - a^2*c*x^2)^(3/2))","A",1
699,1,89,262,0.1133817,"\int \frac{e^{\coth ^{-1}(a x)} x^5}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[(E^ArcCoth[a*x]*x^5)/(c - a^2*c*x^2)^(5/2),x]","\frac{x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(16 a x+\frac{16}{1-a x}-\frac{2}{a x+1}-\frac{2}{(a x-1)^2}+23 \log (1-a x)-7 \log (a x+1)\right)}{16 a \left(c-a^2 c x^2\right)^{5/2}}","\frac{x^6 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{\left(c-a^2 c x^2\right)^{5/2}}+\frac{x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{a (1-a x) \left(c-a^2 c x^2\right)^{5/2}}-\frac{x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 a (a x+1) \left(c-a^2 c x^2\right)^{5/2}}-\frac{x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 a (1-a x)^2 \left(c-a^2 c x^2\right)^{5/2}}+\frac{23 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \log (1-a x)}{16 a \left(c-a^2 c x^2\right)^{5/2}}-\frac{7 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \log (a x+1)}{16 a \left(c-a^2 c x^2\right)^{5/2}}",1,"((1 - 1/(a^2*x^2))^(5/2)*x^5*(16*a*x + 16/(1 - a*x) - 2/(-1 + a*x)^2 - 2/(1 + a*x) + 23*Log[1 - a*x] - 7*Log[1 + a*x]))/(16*a*(c - a^2*c*x^2)^(5/2))","A",1
700,1,84,217,0.1123515,"\int \frac{e^{\coth ^{-1}(a x)} x^4}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[(E^ArcCoth[a*x]*x^4)/(c - a^2*c*x^2)^(5/2),x]","\frac{x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(-\frac{2 \left(5 a^2 x^2+3 a x-6\right)}{(a x-1)^2 (a x+1)}+11 \log (1-a x)+5 \log (a x+1)\right)}{16 \left(c-a^2 c x^2\right)^{5/2}}","\frac{3 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{4 (1-a x) \left(c-a^2 c x^2\right)^{5/2}}+\frac{x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (a x+1) \left(c-a^2 c x^2\right)^{5/2}}-\frac{x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (1-a x)^2 \left(c-a^2 c x^2\right)^{5/2}}+\frac{11 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \log (1-a x)}{16 \left(c-a^2 c x^2\right)^{5/2}}+\frac{5 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \log (a x+1)}{16 \left(c-a^2 c x^2\right)^{5/2}}",1,"((1 - 1/(a^2*x^2))^(5/2)*x^5*((-2*(-6 + 3*a*x + 5*a^2*x^2))/((-1 + a*x)^2*(1 + a*x)) + 11*Log[1 - a*x] + 5*Log[1 + a*x]))/(16*(c - a^2*c*x^2)^(5/2))","A",1
701,1,86,176,0.0701403,"\int \frac{e^{\coth ^{-1}(a x)} x^3}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[(E^ArcCoth[a*x]*x^3)/(c - a^2*c*x^2)^(5/2),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(5 a^2 x^2-a x+3 (a x-1)^2 (a x+1) \tanh ^{-1}(a x)-2\right)}{8 a^3 c^2 (a x-1)^2 (a x+1) \sqrt{c-a^2 c x^2}}","\frac{a x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{2 (1-a x) \left(c-a^2 c x^2\right)^{5/2}}-\frac{a x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (a x+1) \left(c-a^2 c x^2\right)^{5/2}}-\frac{a x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (1-a x)^2 \left(c-a^2 c x^2\right)^{5/2}}-\frac{3 a x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \tanh ^{-1}(a x)}{8 \left(c-a^2 c x^2\right)^{5/2}}",1,"-1/8*(Sqrt[1 - 1/(a^2*x^2)]*x*(-2 - a*x + 5*a^2*x^2 + 3*(-1 + a*x)^2*(1 + a*x)*ArcTanh[a*x]))/(a^3*c^2*(-1 + a*x)^2*(1 + a*x)*Sqrt[c - a^2*c*x^2])","A",1
702,1,85,184,0.062186,"\int \frac{e^{\coth ^{-1}(a x)} x^2}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[(E^ArcCoth[a*x]*x^2)/(c - a^2*c*x^2)^(5/2),x]","\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(-a^2 x^2-3 a x+(a x-1)^2 (a x+1) \tanh ^{-1}(a x)+2\right)}{8 a^2 c^2 (a x-1)^2 (a x+1) \sqrt{c-a^2 c x^2}}","\frac{a^2 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{4 (1-a x) \left(c-a^2 c x^2\right)^{5/2}}+\frac{a^2 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (a x+1) \left(c-a^2 c x^2\right)^{5/2}}-\frac{a^2 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (1-a x)^2 \left(c-a^2 c x^2\right)^{5/2}}+\frac{a^2 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \tanh ^{-1}(a x)}{8 \left(c-a^2 c x^2\right)^{5/2}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*x*(2 - 3*a*x - a^2*x^2 + (-1 + a*x)^2*(1 + a*x)*ArcTanh[a*x]))/(8*a^2*c^2*(-1 + a*x)^2*(1 + a*x)*Sqrt[c - a^2*c*x^2])","A",1
703,1,60,137,0.0725409,"\int \frac{e^{\coth ^{-1}(a x)} x}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[(E^ArcCoth[a*x]*x)/(c - a^2*c*x^2)^(5/2),x]","-\frac{a^3 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(\frac{1}{a x+1}+\frac{1}{(a x-1)^2}-\tanh ^{-1}(a x)\right)}{8 \left(c-a^2 c x^2\right)^{5/2}}","-\frac{a^3 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (a x+1) \left(c-a^2 c x^2\right)^{5/2}}-\frac{a^3 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (1-a x)^2 \left(c-a^2 c x^2\right)^{5/2}}+\frac{a^3 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \tanh ^{-1}(a x)}{8 \left(c-a^2 c x^2\right)^{5/2}}",1,"-1/8*(a^3*(1 - 1/(a^2*x^2))^(5/2)*x^5*((-1 + a*x)^(-2) + (1 + a*x)^(-1) - ArcTanh[a*x]))/(c - a^2*c*x^2)^(5/2)","A",1
704,1,83,184,0.0565908,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[E^ArcCoth[a*x]/(c - a^2*c*x^2)^(5/2),x]","-\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(-3 a^2 x^2+3 a x+3 (a x-1)^2 (a x+1) \tanh ^{-1}(a x)+2\right)}{8 c^2 (a x-1)^2 (a x+1) \sqrt{c-a^2 c x^2}}","-\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{4 (1-a x) \left(c-a^2 c x^2\right)^{5/2}}+\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (a x+1) \left(c-a^2 c x^2\right)^{5/2}}-\frac{a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (1-a x)^2 \left(c-a^2 c x^2\right)^{5/2}}-\frac{3 a^4 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \tanh ^{-1}(a x)}{8 \left(c-a^2 c x^2\right)^{5/2}}",1,"-1/8*(Sqrt[1 - 1/(a^2*x^2)]*x*(2 + 3*a*x - 3*a^2*x^2 + 3*(-1 + a*x)^2*(1 + a*x)*ArcTanh[a*x]))/(c^2*(-1 + a*x)^2*(1 + a*x)*Sqrt[c - a^2*c*x^2])","A",1
705,1,88,271,0.1097162,"\int \frac{e^{\coth ^{-1}(a x)}}{x \left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[E^ArcCoth[a*x]/(x*(c - a^2*c*x^2)^(5/2)),x]","\frac{a^5 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(\frac{8}{a x-1}-\frac{2}{a x+1}-\frac{2}{(a x-1)^2}+11 \log (1-a x)+5 \log (a x+1)-16 \log (x)\right)}{16 \left(c-a^2 c x^2\right)^{5/2}}","-\frac{a^5 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{2 (1-a x) \left(c-a^2 c x^2\right)^{5/2}}-\frac{a^5 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (a x+1) \left(c-a^2 c x^2\right)^{5/2}}-\frac{a^5 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (1-a x)^2 \left(c-a^2 c x^2\right)^{5/2}}-\frac{a^5 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \log (x)}{\left(c-a^2 c x^2\right)^{5/2}}+\frac{11 a^5 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \log (1-a x)}{16 \left(c-a^2 c x^2\right)^{5/2}}+\frac{5 a^5 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \log (a x+1)}{16 \left(c-a^2 c x^2\right)^{5/2}}",1,"(a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5*(-2/(-1 + a*x)^2 + 8/(-1 + a*x) - 2/(1 + a*x) - 16*Log[x] + 11*Log[1 - a*x] + 5*Log[1 + a*x]))/(16*(c - a^2*c*x^2)^(5/2))","A",1
706,1,99,307,0.1039283,"\int \frac{e^{\coth ^{-1}(a x)}}{x^2 \left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[E^ArcCoth[a*x]/(x^2*(c - a^2*c*x^2)^(5/2)),x]","\frac{a^5 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(\frac{12 a}{a x-1}+\frac{2 a}{a x+1}-\frac{2 a}{(a x-1)^2}-16 a \log (x)+23 a \log (1-a x)-7 a \log (a x+1)+\frac{16}{x}\right)}{16 \left(c-a^2 c x^2\right)^{5/2}}","-\frac{3 a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{4 (1-a x) \left(c-a^2 c x^2\right)^{5/2}}+\frac{a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (a x+1) \left(c-a^2 c x^2\right)^{5/2}}-\frac{a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{8 (1-a x)^2 \left(c-a^2 c x^2\right)^{5/2}}-\frac{a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \log (x)}{\left(c-a^2 c x^2\right)^{5/2}}+\frac{23 a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \log (1-a x)}{16 \left(c-a^2 c x^2\right)^{5/2}}-\frac{7 a^6 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \log (a x+1)}{16 \left(c-a^2 c x^2\right)^{5/2}}+\frac{a^5 x^4 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}{\left(c-a^2 c x^2\right)^{5/2}}",1,"(a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5*(16/x - (2*a)/(-1 + a*x)^2 + (12*a)/(-1 + a*x) + (2*a)/(1 + a*x) - 16*a*Log[x] + 23*a*Log[1 - a*x] - 7*a*Log[1 + a*x]))/(16*(c - a^2*c*x^2)^(5/2))","A",1
707,1,45,76,0.0313509,"\int e^{-\coth ^{-1}(a x)} x^2 \sqrt{c-a^2 c x^2} \, dx","Integrate[(x^2*Sqrt[c - a^2*c*x^2])/E^ArcCoth[a*x],x]","\frac{x^2 (3 a x-4) \sqrt{c-a^2 c x^2}}{12 a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^3 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x^2 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(x^2*(-4 + 3*a*x)*Sqrt[c - a^2*c*x^2])/(12*a*Sqrt[1 - 1/(a^2*x^2)])","A",1
708,1,43,74,0.0255466,"\int e^{-\coth ^{-1}(a x)} x \sqrt{c-a^2 c x^2} \, dx","Integrate[(x*Sqrt[c - a^2*c*x^2])/E^ArcCoth[a*x],x]","\frac{x (2 a x-3) \sqrt{c-a^2 c x^2}}{6 a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^2 \sqrt{c-a^2 c x^2}}{3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(x*(-3 + 2*a*x)*Sqrt[c - a^2*c*x^2])/(6*a*Sqrt[1 - 1/(a^2*x^2)])","A",1
709,1,41,69,0.0221604,"\int e^{-\coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/E^ArcCoth[a*x],x]","\frac{(a x-2) \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((-2 + a*x)*Sqrt[c - a^2*c*x^2])/(2*a*Sqrt[1 - 1/(a^2*x^2)])","A",1
710,1,44,70,0.0218876,"\int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/(E^ArcCoth[a*x]*x),x]","\frac{\sqrt{c-a^2 c x^2} (a x-\log (x))}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{\sqrt{c-a^2 c x^2}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\log (x) \sqrt{c-a^2 c x^2}}{a x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(a*x - Log[x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
711,1,44,72,0.0252019,"\int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^2} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/(E^ArcCoth[a*x]*x^2),x]","\frac{\sqrt{c-a^2 c x^2} \left(a \log (x)+\frac{1}{x}\right)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{\sqrt{c-a^2 c x^2}}{a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(x^(-1) + a*Log[x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
712,1,96,137,0.1439908,"\int e^{-2 \coth ^{-1}(a x)} x^3 \sqrt{c-a^2 c x^2} \, dx","Integrate[(x^3*Sqrt[c - a^2*c*x^2])/E^(2*ArcCoth[a*x]),x]","\frac{\left(4 a^4 x^4-10 a^3 x^3+12 a^2 x^2-15 a x+24\right) \sqrt{c-a^2 c x^2}-15 \sqrt{c} \tan ^{-1}\left(\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left(a^2 x^2-1\right)}\right)}{20 a^4}","\frac{3 x^2 \sqrt{c-a^2 c x^2}}{5 a^2}+\frac{1}{5} x^4 \sqrt{c-a^2 c x^2}-\frac{x^3 \sqrt{c-a^2 c x^2}}{2 a}+\frac{3 (8-5 a x) \sqrt{c-a^2 c x^2}}{20 a^4}+\frac{3 \sqrt{c} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{4 a^4}",1,"(Sqrt[c - a^2*c*x^2]*(24 - 15*a*x + 12*a^2*x^2 - 10*a^3*x^3 + 4*a^4*x^4) - 15*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(20*a^4)","A",1
713,1,88,112,0.1062227,"\int e^{-2 \coth ^{-1}(a x)} x^2 \sqrt{c-a^2 c x^2} \, dx","Integrate[(x^2*Sqrt[c - a^2*c*x^2])/E^(2*ArcCoth[a*x]),x]","\frac{21 \sqrt{c} \tan ^{-1}\left(\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left(a^2 x^2-1\right)}\right)+\left(6 a^3 x^3-16 a^2 x^2+21 a x-32\right) \sqrt{c-a^2 c x^2}}{24 a^3}","-\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}+\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}-\frac{(32-21 a x) \sqrt{c-a^2 c x^2}}{24 a^3}-\frac{7 \sqrt{c} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{8 a^3}",1,"(Sqrt[c - a^2*c*x^2]*(-32 + 21*a*x - 16*a^2*x^2 + 6*a^3*x^3) + 21*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(24*a^3)","A",1
714,1,79,84,0.0908187,"\int e^{-2 \coth ^{-1}(a x)} x \sqrt{c-a^2 c x^2} \, dx","Integrate[(x*Sqrt[c - a^2*c*x^2])/E^(2*ArcCoth[a*x]),x]","\frac{\left(a^2 x^2-3 a x+5\right) \sqrt{c-a^2 c x^2}-3 \sqrt{c} \tan ^{-1}\left(\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left(a^2 x^2-1\right)}\right)}{3 a^2}","\frac{1}{3} x^2 \sqrt{c-a^2 c x^2}+\frac{(5-3 a x) \sqrt{c-a^2 c x^2}}{3 a^2}+\frac{\sqrt{c} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{a^2}",1,"((5 - 3*a*x + a^2*x^2)*Sqrt[c - a^2*c*x^2] - 3*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))])/(3*a^2)","A",1
715,1,100,87,0.0595198,"\int e^{-2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/E^(2*ArcCoth[a*x]),x]","\frac{\sqrt{c-a^2 c x^2} \left(6 \sqrt{1-a x} \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)-\sqrt{a x+1} \left(a^2 x^2-5 a x+4\right)\right)}{2 a \sqrt{1-a x} \sqrt{1-a^2 x^2}}","-\frac{(1-a x) \sqrt{c-a^2 c x^2}}{2 a}-\frac{3 \sqrt{c-a^2 c x^2}}{2 a}-\frac{3 \sqrt{c} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)}{2 a}",1,"(Sqrt[c - a^2*c*x^2]*(-(Sqrt[1 + a*x]*(4 - 5*a*x + a^2*x^2)) + 6*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(2*a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])","A",0
716,1,97,75,0.0863676,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/(E^(2*ArcCoth[a*x])*x),x]","\sqrt{c-a^2 c x^2}+\sqrt{c} \log \left(\sqrt{c} \sqrt{c-a^2 c x^2}+c\right)-2 \sqrt{c} \tan ^{-1}\left(\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left(a^2 x^2-1\right)}\right)-\sqrt{c} \log (x)","\sqrt{c-a^2 c x^2}+2 \sqrt{c} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)+\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right)",1,"Sqrt[c - a^2*c*x^2] - 2*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))] - Sqrt[c]*Log[x] + Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]]","A",0
717,1,104,82,0.1030066,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^2} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/(E^(2*ArcCoth[a*x])*x^2),x]","\frac{\sqrt{c-a^2 c x^2}}{x}-2 a \sqrt{c} \log \left(\sqrt{c} \sqrt{c-a^2 c x^2}+c\right)+a \sqrt{c} \tan ^{-1}\left(\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left(a^2 x^2-1\right)}\right)+2 a \sqrt{c} \log (x)","\frac{\sqrt{c-a^2 c x^2}}{x}-a \sqrt{c} \tan ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right)-2 a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right)",1,"Sqrt[c - a^2*c*x^2]/x + a*Sqrt[c]*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + a^2*x^2))] + 2*a*Sqrt[c]*Log[x] - 2*a*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]]","A",0
718,1,76,78,0.1296225,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^3} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/(E^(2*ArcCoth[a*x])*x^3),x]","\frac{1}{2} \left(\frac{(1-4 a x) \sqrt{c-a^2 c x^2}}{x^2}+3 a^2 \sqrt{c} \log \left(\sqrt{c} \sqrt{c-a^2 c x^2}+c\right)-3 a^2 \sqrt{c} \log (x)\right)","-\frac{2 a \sqrt{c-a^2 c x^2}}{x}+\frac{\sqrt{c-a^2 c x^2}}{2 x^2}+\frac{3}{2} a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right)",1,"(((1 - 4*a*x)*Sqrt[c - a^2*c*x^2])/x^2 - 3*a^2*Sqrt[c]*Log[x] + 3*a^2*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]])/2","A",0
719,1,82,101,0.1359235,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^4} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/(E^(2*ArcCoth[a*x])*x^4),x]","a^3 \sqrt{c} \log (x)+\frac{\left(5 a^2 x^2-3 a x+1\right) \sqrt{c-a^2 c x^2}}{3 x^3}-a^3 \sqrt{c} \log \left(\sqrt{c} \sqrt{c-a^2 c x^2}+c\right)","\frac{5 a^2 \sqrt{c-a^2 c x^2}}{3 x}-\frac{a \sqrt{c-a^2 c x^2}}{x^2}+\frac{\sqrt{c-a^2 c x^2}}{3 x^3}+a^3 \left(-\sqrt{c}\right) \tanh ^{-1}\left(\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right)",1,"((1 - 3*a*x + 5*a^2*x^2)*Sqrt[c - a^2*c*x^2])/(3*x^3) + a^3*Sqrt[c]*Log[x] - a^3*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]]","A",0
720,1,95,130,0.1557409,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^5} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/(E^(2*ArcCoth[a*x])*x^5),x]","-\frac{7}{8} a^4 \sqrt{c} \log (x)+\frac{7}{8} a^4 \sqrt{c} \log \left(\sqrt{c} \sqrt{c-a^2 c x^2}+c\right)+\frac{\left(-32 a^3 x^3+21 a^2 x^2-16 a x+6\right) \sqrt{c-a^2 c x^2}}{24 x^4}","\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{\sqrt{c-a^2 c x^2}}{4 x^4}-\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{7}{8} a^4 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right)-\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}",1,"(Sqrt[c - a^2*c*x^2]*(6 - 16*a*x + 21*a^2*x^2 - 32*a^3*x^3))/(24*x^4) - (7*a^4*Sqrt[c]*Log[x])/8 + (7*a^4*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]])/8","A",0
721,1,87,227,0.063191,"\int e^{-3 \coth ^{-1}(a x)} x^3 \sqrt{c-a^2 c x^2} \, dx","Integrate[(x^3*Sqrt[c - a^2*c*x^2])/E^(3*ArcCoth[a*x]),x]","\frac{\sqrt{c-a^2 c x^2} \left(-\frac{4 \log (a x+1)}{a^4}+\frac{4 x}{a^3}-\frac{2 x^2}{a^2}+\frac{a x^5}{5}+\frac{4 x^3}{3 a}-\frac{3 x^4}{4}\right)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{4 x^2 \sqrt{c-a^2 c x^2}}{3 a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^4 \sqrt{c-a^2 c x^2}}{5 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 x^3 \sqrt{c-a^2 c x^2}}{4 a \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{a^5 x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2}}{a^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2 x \sqrt{c-a^2 c x^2}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*((4*x)/a^3 - (2*x^2)/a^2 + (4*x^3)/(3*a) - (3*x^4)/4 + (a*x^5)/5 - (4*Log[1 + a*x])/a^4))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
722,1,72,186,0.0492045,"\int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt{c-a^2 c x^2} \, dx","Integrate[(x^2*Sqrt[c - a^2*c*x^2])/E^(3*ArcCoth[a*x]),x]","\frac{\sqrt{c-a^2 c x^2} \left(a x \left(a^3 x^3-4 a^2 x^2+8 a x-16\right)+16 \log (a x+1)\right)}{4 a^4 x \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{x^2 \sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 x \sqrt{c-a^2 c x^2}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^3 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{a^4 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-a^2 c x^2}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(a*x*(-16 + 8*a*x - 4*a^2*x^2 + a^3*x^3) + 16*Log[1 + a*x]))/(4*a^4*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
723,1,65,151,0.0428662,"\int e^{-3 \coth ^{-1}(a x)} x \sqrt{c-a^2 c x^2} \, dx","Integrate[(x*Sqrt[c - a^2*c*x^2])/E^(3*ArcCoth[a*x]),x]","\frac{\sqrt{c-a^2 c x^2} \left(a x \left(2 a^2 x^2-9 a x+24\right)-24 \log (a x+1)\right)}{6 a^3 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^2 \sqrt{c-a^2 c x^2}}{3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 x \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{a^3 x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(a*x*(24 - 9*a*x + 2*a^2*x^2) - 24*Log[1 + a*x]))/(6*a^3*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
724,1,56,112,0.0259171,"\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/E^(3*ArcCoth[a*x]),x]","\frac{\sqrt{c-a^2 c x^2} (a x (a x-6)+8 \log (a x+1))}{2 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 \sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(a*x*(-6 + a*x) + 8*Log[1 + a*x]))/(2*a^2*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
725,1,50,112,0.0284134,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/(E^(3*ArcCoth[a*x])*x),x]","\frac{\sqrt{c-a^2 c x^2} (a x-4 \log (a x+1)+\log (x))}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{\sqrt{c-a^2 c x^2}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\log (x) \sqrt{c-a^2 c x^2}}{a x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(a*x + Log[x] - 4*Log[1 + a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
726,1,56,114,0.0323518,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^2} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/(E^(3*ArcCoth[a*x])*x^2),x]","\frac{\sqrt{c-a^2 c x^2} \left(-3 a \log (x)+4 a \log (a x+1)-\frac{1}{x}\right)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{\sqrt{c-a^2 c x^2}}{a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 \log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(-x^(-1) - 3*a*Log[x] + 4*a*Log[1 + a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
727,1,68,152,0.0384044,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^3} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/(E^(3*ArcCoth[a*x])*x^3),x]","\frac{\sqrt{c-a^2 c x^2} \left(4 a^2 \log (x)-4 a^2 \log (a x+1)+\frac{3 a}{x}-\frac{1}{2 x^2}\right)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{3 \sqrt{c-a^2 c x^2}}{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a \log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a \sqrt{c-a^2 c x^2} \log (a x+1)}{x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-a^2 c x^2}}{2 a x^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(-1/2*1/x^2 + (3*a)/x + 4*a^2*Log[x] - 4*a^2*Log[1 + a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
728,1,78,193,0.045225,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^4} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/(E^(3*ArcCoth[a*x])*x^4),x]","\frac{\sqrt{c-a^2 c x^2} \left(-4 a^3 \log (x)+4 a^3 \log (a x+1)-\frac{4 a^2}{x}+\frac{3 a}{2 x^2}-\frac{1}{3 x^3}\right)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{4 a \sqrt{c-a^2 c x^2}}{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a^2 \log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a^2 \sqrt{c-a^2 c x^2} \log (a x+1)}{x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-a^2 c x^2}}{3 a x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 \sqrt{c-a^2 c x^2}}{2 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(-1/3*1/x^3 + (3*a)/(2*x^2) - (4*a^2)/x - 4*a^3*Log[x] + 4*a^3*Log[1 + a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
729,1,83,227,0.0574877,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^5} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/(E^(3*ArcCoth[a*x])*x^5),x]","\frac{\sqrt{c-a^2 c x^2} \left(4 a^4 \log (x)-4 a^4 \log (a x+1)+\frac{4 a^3}{x}-\frac{2 a^2}{x^2}+\frac{a}{x^3}-\frac{1}{4 x^4}\right)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{4 a^2 \sqrt{c-a^2 c x^2}}{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-a^2 c x^2}}{4 a x^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-a^2 c x^2}}{x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2 a \sqrt{c-a^2 c x^2}}{x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a^3 \log (x) \sqrt{c-a^2 c x^2}}{x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a^3 \sqrt{c-a^2 c x^2} \log (a x+1)}{x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - a^2*c*x^2]*(-1/4*1/x^4 + a/x^3 - (2*a^2)/x^2 + (4*a^3)/x + 4*a^4*Log[x] - 4*a^4*Log[1 + a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)]*x)","A",1
730,1,74,136,0.057484,"\int e^{3 \coth ^{-1}(a x)} x^m \sqrt{c-a^2 c x^2} \, dx","Integrate[E^(3*ArcCoth[a*x])*x^m*Sqrt[c - a^2*c*x^2],x]","\frac{x^m \sqrt{c-a^2 c x^2} (-4 (m+2) \, _2F_1(1,m+1;m+2;a x)+m (a x+3)+a x+6)}{a (m+1) (m+2) \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{4 x^m \sqrt{c-a^2 c x^2} \, _2F_1(1,m+1;m+2;a x)}{a (m+1) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^{m+1} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 x^m \sqrt{c-a^2 c x^2}}{a (m+1) \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(x^m*Sqrt[c - a^2*c*x^2]*(6 + a*x + m*(3 + a*x) - 4*(2 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, a*x]))/(a*(1 + m)*(2 + m)*Sqrt[1 - 1/(a^2*x^2)])","A",1
731,1,129,172,0.2271167,"\int e^{2 \coth ^{-1}(a x)} x^m \sqrt{c-a^2 c x^2} \, dx","Integrate[E^(2*ArcCoth[a*x])*x^m*Sqrt[c - a^2*c*x^2],x]","\frac{x^{m+1} \left(\frac{\sqrt{c-a^2 c x^2} \, _2F_1\left(-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right)}{\sqrt{1-a^2 x^2}}+\frac{2 \sqrt{1-a x} \sqrt{-c (a x+1)} F_1\left(m+1;\frac{1}{2},-\frac{1}{2};m+2;a x,-a x\right)}{\sqrt{a x-1} \sqrt{a x+1}}\right)}{m+1}","-\frac{c (2 m+3) \sqrt{1-a^2 x^2} x^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right)}{(m+1) (m+2) \sqrt{c-a^2 c x^2}}-\frac{2 a c \sqrt{1-a^2 x^2} x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right)}{(m+2) \sqrt{c-a^2 c x^2}}+\frac{x^{m+1} \sqrt{c-a^2 c x^2}}{m+2}",1,"(x^(1 + m)*((2*Sqrt[1 - a*x]*Sqrt[-(c*(1 + a*x))]*AppellF1[1 + m, 1/2, -1/2, 2 + m, a*x, -(a*x)])/(Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[-1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/Sqrt[1 - a^2*x^2]))/(1 + m)","C",0
732,1,56,82,0.0341548,"\int e^{\coth ^{-1}(a x)} x^m \sqrt{c-a^2 c x^2} \, dx","Integrate[E^ArcCoth[a*x]*x^m*Sqrt[c - a^2*c*x^2],x]","\frac{x^m \sqrt{c-a^2 c x^2} (a m x+a x+m+2)}{a (m+1) (m+2) \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^{m+1} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^m \sqrt{c-a^2 c x^2}}{a (m+1) \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(x^m*(2 + m + a*x + a*m*x)*Sqrt[c - a^2*c*x^2])/(a*(1 + m)*(2 + m)*Sqrt[1 - 1/(a^2*x^2)])","A",1
733,1,58,83,0.0367683,"\int e^{-\coth ^{-1}(a x)} x^m \sqrt{c-a^2 c x^2} \, dx","Integrate[(x^m*Sqrt[c - a^2*c*x^2])/E^ArcCoth[a*x],x]","\frac{x^m \sqrt{c-a^2 c x^2} (m (a x-1)+a x-2)}{a (m+1) (m+2) \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^{m+1} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x^m \sqrt{c-a^2 c x^2}}{a (m+1) \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(x^m*Sqrt[c - a^2*c*x^2]*(-2 + a*x + m*(-1 + a*x)))/(a*(1 + m)*(2 + m)*Sqrt[1 - 1/(a^2*x^2)])","A",1
734,1,110,172,0.162644,"\int e^{-2 \coth ^{-1}(a x)} x^m \sqrt{c-a^2 c x^2} \, dx","Integrate[(x^m*Sqrt[c - a^2*c*x^2])/E^(2*ArcCoth[a*x]),x]","\frac{x^{m+1} \left(\frac{\sqrt{c-a^2 c x^2} \, _2F_1\left(-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right)}{\sqrt{1-a^2 x^2}}-\frac{2 \sqrt{c-a c x} F_1\left(m+1;\frac{1}{2},-\frac{1}{2};m+2;-a x,a x\right)}{\sqrt{1-a x}}\right)}{m+1}","-\frac{c (2 m+3) \sqrt{1-a^2 x^2} x^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right)}{(m+1) (m+2) \sqrt{c-a^2 c x^2}}+\frac{2 a c \sqrt{1-a^2 x^2} x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};a^2 x^2\right)}{(m+2) \sqrt{c-a^2 c x^2}}+\frac{x^{m+1} \sqrt{c-a^2 c x^2}}{m+2}",1,"(x^(1 + m)*((-2*Sqrt[c - a*c*x]*AppellF1[1 + m, 1/2, -1/2, 2 + m, -(a*x), a*x])/Sqrt[1 - a*x] + (Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[-1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/Sqrt[1 - a^2*x^2]))/(1 + m)","C",0
735,1,75,137,0.0559842,"\int e^{-3 \coth ^{-1}(a x)} x^m \sqrt{c-a^2 c x^2} \, dx","Integrate[(x^m*Sqrt[c - a^2*c*x^2])/E^(3*ArcCoth[a*x]),x]","\frac{x^m \sqrt{c-a^2 c x^2} (4 (m+2) \, _2F_1(1,m+1;m+2;-a x)+m (a x-3)+a x-6)}{a (m+1) (m+2) \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{4 x^m \sqrt{c-a^2 c x^2} \, _2F_1(1,m+1;m+2;-a x)}{a (m+1) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^{m+1} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 x^m \sqrt{c-a^2 c x^2}}{a (m+1) \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(x^m*Sqrt[c - a^2*c*x^2]*(-6 + a*x + m*(-3 + a*x) + 4*(2 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -(a*x)]))/(a*(1 + m)*(2 + m)*Sqrt[1 - 1/(a^2*x^2)])","A",1
736,1,267,81,2.458529,"\int e^{n \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^3 \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a^2*c*x^2)^3,x]","-\frac{c^3 e^{n \coth ^{-1}(a x)} \left(720 a^7 x^7+120 a^6 n x^6+24 a^5 n^2 x^5-3024 a^5 x^5+6 a^4 n^3 x^4-576 a^4 n x^4+2 a^3 n^4 x^3-152 a^3 n^2 x^3+5040 a^3 x^3+a^2 n^5 x^2-64 a^2 n^3 x^2+1368 a^2 n x^2+\left(n^6-56 n^4+784 n^2-2304\right) \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)+n \left(n^5-2 n^4-52 n^3+104 n^2+576 n-1152\right) e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)+a n^6 x-58 a n^4 x+912 a n^2 x-5040 a x-n^5+58 n^3-912 n\right)}{5040 a}","-\frac{256 c^3 \left(1-\frac{1}{a x}\right)^{4-\frac{n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n-8}{2}} \, _2F_1\left(8,4-\frac{n}{2};5-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a (8-n)}",1,"-1/5040*(c^3*E^(n*ArcCoth[a*x])*(-912*n + 58*n^3 - n^5 - 5040*a*x + 912*a*n^2*x - 58*a*n^4*x + a*n^6*x + 1368*a^2*n*x^2 - 64*a^2*n^3*x^2 + a^2*n^5*x^2 + 5040*a^3*x^3 - 152*a^3*n^2*x^3 + 2*a^3*n^4*x^3 - 576*a^4*n*x^4 + 6*a^4*n^3*x^4 - 3024*a^5*x^5 + 24*a^5*n^2*x^5 + 120*a^6*n*x^6 + 720*a^7*x^7 + E^(2*ArcCoth[a*x])*n*(-1152 + 576*n + 104*n^2 - 52*n^3 - 2*n^4 + n^5)*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (-2304 + 784*n^2 - 56*n^4 + n^6)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])]))/a","B",0
737,1,179,81,1.4239213,"\int e^{n \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^2 \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a^2*c*x^2)^2,x]","\frac{c^2 e^{n \coth ^{-1}(a x)} \left(24 a^5 x^5+6 a^4 n x^4+2 a^3 n^2 x^3-80 a^3 x^3+a^2 n^3 x^2-28 a^2 n x^2+\left(n^4-20 n^2+64\right) \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)+n \left(n^3-2 n^2-16 n+32\right) e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)+a n^4 x-22 a n^2 x+120 a x-n^3+22 n\right)}{120 a}","\frac{64 c^2 \left(1-\frac{1}{a x}\right)^{3-\frac{n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n-6}{2}} \, _2F_1\left(6,3-\frac{n}{2};4-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a (6-n)}",1,"(c^2*E^(n*ArcCoth[a*x])*(22*n - n^3 + 120*a*x - 22*a*n^2*x + a*n^4*x - 28*a^2*n*x^2 + a^2*n^3*x^2 - 80*a^3*x^3 + 2*a^3*n^2*x^3 + 6*a^4*n*x^4 + 24*a^5*x^5 + E^(2*ArcCoth[a*x])*n*(32 - 16*n - 2*n^2 + n^3)*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (64 - 20*n^2 + n^4)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])]))/(120*a)","B",0
738,1,111,79,0.7706459,"\int e^{n \coth ^{-1}(a x)} \left(c-a^2 c x^2\right) \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a^2*c*x^2),x]","-\frac{c e^{n \coth ^{-1}(a x)} \left(2 a^3 x^3+a^2 n x^2+\left(n^2-4\right) \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)+(n-2) n e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)+a n^2 x-6 a x-n\right)}{6 a}","-\frac{16 c \left(1-\frac{1}{a x}\right)^{2-\frac{n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n-4}{2}} \, _2F_1\left(4,2-\frac{n}{2};3-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a (4-n)}",1,"-1/6*(c*E^(n*ArcCoth[a*x])*(-n - 6*a*x + a*n^2*x + a^2*n*x^2 + 2*a^3*x^3 + E^(2*ArcCoth[a*x])*(-2 + n)*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (-4 + n^2)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])]))/a","A",0
739,1,82,78,0.0380665,"\int e^{n \coth ^{-1}(a x)} \, dx","Integrate[E^(n*ArcCoth[a*x]),x]","\frac{e^{n \coth ^{-1}(a x)} \left(n e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)+(n+2) \left(\, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)+a x\right)\right)}{a (n+2)}","\frac{4 \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n-2}{2}} \, _2F_1\left(2,1-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a (2-n)}",1,"(E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(a*x + Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(a*(2 + n))","A",0
740,1,18,18,0.054045,"\int \frac{e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2),x]","\frac{e^{n \coth ^{-1}(a x)}}{a c n}","\frac{e^{n \coth ^{-1}(a x)}}{a c n}",1,"E^(n*ArcCoth[a*x])/(a*c*n)","A",1
741,1,55,72,0.1765385,"\int \frac{e^{n \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^2} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^2,x]","-\frac{\left(2 a^2 x^2-2 a n x+n^2-2\right) e^{n \coth ^{-1}(a x)}}{a c^2 n \left(n^2-4\right) \left(a^2 x^2-1\right)}","\frac{2 e^{n \coth ^{-1}(a x)}}{a c^2 n \left(4-n^2\right)}-\frac{(n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left(4-n^2\right) \left(1-a^2 x^2\right)}",1,"-((E^(n*ArcCoth[a*x])*(-2 + n^2 - 2*a*n*x + 2*a^2*x^2))/(a*c^2*n*(-4 + n^2)*(-1 + a^2*x^2)))","A",1
742,1,97,127,0.2272114,"\int \frac{e^{n \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^3} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^3,x]","\frac{\left(4 n^2 \left(3 a^2 x^2-4\right)-8 a n x \left(3 a^2 x^2-5\right)+24 \left(a^2 x^2-1\right)^2-4 a n^3 x+n^4\right) e^{n \coth ^{-1}(a x)}}{a c^3 n \left(n^2-16\right) \left(n^2-4\right) \left(a^2 x^2-1\right)^2}","-\frac{(n-4 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left(16-n^2\right) \left(1-a^2 x^2\right)^2}-\frac{12 (n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left(4-n^2\right) \left(16-n^2\right) \left(1-a^2 x^2\right)}+\frac{24 e^{n \coth ^{-1}(a x)}}{a c^3 n \left(n^4-20 n^2+64\right)}",1,"(E^(n*ArcCoth[a*x])*(n^4 - 4*a*n^3*x + 24*(-1 + a^2*x^2)^2 - 8*a*n*x*(-5 + 3*a^2*x^2) + 4*n^2*(-4 + 3*a^2*x^2)))/(a*c^3*n*(-16 + n^2)*(-4 + n^2)*(-1 + a^2*x^2)^2)","A",1
743,1,152,197,0.3158615,"\int \frac{e^{n \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^4} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^4,x]","-\frac{\left(10 n^4 \left(3 a^2 x^2-5\right)-120 a n^3 x \left(a^2 x^2-2\right)+720 \left(a^2 x^2-1\right)^3+8 n^2 \left(45 a^4 x^4-105 a^2 x^2+68\right)-48 a n x \left(15 a^4 x^4-40 a^2 x^2+33\right)-6 a n^5 x+n^6\right) e^{n \coth ^{-1}(a x)}}{a c^4 n \left(n^2-36\right) \left(n^2-16\right) \left(n^2-4\right) \left(a^2 x^2-1\right)^3}","-\frac{(n-6 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left(36-n^2\right) \left(1-a^2 x^2\right)^3}-\frac{360 (n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left(4-n^2\right) \left(16-n^2\right) \left(36-n^2\right) \left(1-a^2 x^2\right)}-\frac{30 (n-4 a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left(16-n^2\right) \left(36-n^2\right) \left(1-a^2 x^2\right)^2}+\frac{720 e^{n \coth ^{-1}(a x)}}{a c^4 n \left(36-n^2\right) \left(n^4-20 n^2+64\right)}",1,"-((E^(n*ArcCoth[a*x])*(n^6 - 6*a*n^5*x - 120*a*n^3*x*(-2 + a^2*x^2) + 720*(-1 + a^2*x^2)^3 + 10*n^4*(-5 + 3*a^2*x^2) - 48*a*n*x*(33 - 40*a^2*x^2 + 15*a^4*x^4) + 8*n^2*(68 - 105*a^2*x^2 + 45*a^4*x^4)))/(a*c^4*n*(-36 + n^2)*(-16 + n^2)*(-4 + n^2)*(-1 + a^2*x^2)^3))","A",1
744,1,280,116,2.410355,"\int e^{n \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^{3/2} \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a^2*c*x^2)^(3/2),x]","\frac{c^2 \left(96 a^3 c x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(a x \sqrt{1-\frac{1}{a^2 x^2}} (a x+n) e^{n \coth ^{-1}(a x)}+2 (n-1) e^{(n+1) \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};e^{2 \coth ^{-1}(a x)}\right)\right)-c \left(a^2 x^2-1\right) \left(16 a \left(n^3-n^2+3 n-3\right) x \sqrt{1-\frac{1}{a^2 x^2}} e^{(n+1) \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};e^{2 \coth ^{-1}(a x)}\right)+2 \left(a^2 x^2-1\right)^2 e^{n \coth ^{-1}(a x)} \left(a \left(n^2+3\right) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left(3 \coth ^{-1}(a x)\right)-a \left(n^2-21\right) x+2 n \left(\left(n^2+3\right) \cosh \left(2 \coth ^{-1}(a x)\right)-n^2+1\right)\right)\right)\right)}{192 a \left(c-a^2 c x^2\right)^{3/2}}","\frac{32 \left(c-a^2 c x^2\right)^{3/2} \left(1-\frac{1}{a x}\right)^{\frac{5-n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n-5}{2}} \, _2F_1\left(5,\frac{5-n}{2};\frac{7-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a^4 (5-n) x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}",1,"(c^2*(96*a^3*c*(1 - 1/(a^2*x^2))^(3/2)*x^3*(a*E^(n*ArcCoth[a*x])*Sqrt[1 - 1/(a^2*x^2)]*x*(n + a*x) + 2*E^((1 + n)*ArcCoth[a*x])*(-1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])]) - c*(-1 + a^2*x^2)*(2*E^(n*ArcCoth[a*x])*(-1 + a^2*x^2)^2*(-(a*(-21 + n^2)*x) + 2*n*(1 - n^2 + (3 + n^2)*Cosh[2*ArcCoth[a*x]]) + a*(3 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[3*ArcCoth[a*x]]) + 16*a*E^((1 + n)*ArcCoth[a*x])*(-3 + 3*n - n^2 + n^3)*Sqrt[1 - 1/(a^2*x^2)]*x*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])])))/(192*a*(c - a^2*c*x^2)^(3/2))","B",0
745,1,101,116,0.5977042,"\int e^{n \coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx","Integrate[E^(n*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2],x]","-\frac{c x \sqrt{1-\frac{1}{a^2 x^2}} e^{n \coth ^{-1}(a x)} \left(a x \sqrt{1-\frac{1}{a^2 x^2}} (a x+n)+2 (n-1) e^{\coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};e^{2 \coth ^{-1}(a x)}\right)\right)}{2 \sqrt{c-a^2 c x^2}}","\frac{8 \sqrt{c-a^2 c x^2} \left(1-\frac{1}{a x}\right)^{\frac{3-n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n-3}{2}} \, _2F_1\left(3,\frac{3-n}{2};\frac{5-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a^2 (3-n) x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"-1/2*(c*E^(n*ArcCoth[a*x])*Sqrt[1 - 1/(a^2*x^2)]*x*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(n + a*x) + 2*E^ArcCoth[a*x]*(-1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])]))/Sqrt[c - a^2*c*x^2]","A",0
746,1,81,111,0.195631,"\int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[E^(n*ArcCoth[a*x])/Sqrt[c - a^2*c*x^2],x]","-\frac{2 \sqrt{c-a^2 c x^2} e^{(n+1) \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};e^{2 \coth ^{-1}(a x)}\right)}{\sqrt{1-\frac{1}{a^2 x^2}} \left(a^2 c n x+a^2 c x\right)}","\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}} \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n-1}{2}} \, _2F_1\left(1,\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{(1-n) \sqrt{c-a^2 c x^2}}",1,"(-2*E^((1 + n)*ArcCoth[a*x])*Sqrt[c - a^2*c*x^2]*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])])/(Sqrt[1 - 1/(a^2*x^2)]*(a^2*c*x + a^2*c*n*x))","A",0
747,1,43,46,0.200821,"\int \frac{e^{n \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(3/2),x]","\frac{(n-a x) e^{n \coth ^{-1}(a x)}}{a c \left(n^2-1\right) \sqrt{c-a^2 c x^2}}","-\frac{(n-a x) e^{n \coth ^{-1}(a x)}}{a c \left(1-n^2\right) \sqrt{c-a^2 c x^2}}",1,"(E^(n*ArcCoth[a*x])*(n - a*x))/(a*c*(-1 + n^2)*Sqrt[c - a^2*c*x^2])","A",1
748,1,110,102,0.6291682,"\int \frac{e^{n \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(5/2),x]","\frac{e^{n \coth ^{-1}(a x)} \left(3 a \left(n^2-1\right) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left(3 \coth ^{-1}(a x)\right)-3 a n^2 x-2 \left(n^2-1\right) n \cosh \left(2 \coth ^{-1}(a x)\right)+27 a x+2 n^3-26 n\right)}{4 a c^2 \left(n^4-10 n^2+9\right) \sqrt{c-a^2 c x^2}}","-\frac{6 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left(1-n^2\right) \left(9-n^2\right) \sqrt{c-a^2 c x^2}}-\frac{(n-3 a x) e^{n \coth ^{-1}(a x)}}{a c \left(9-n^2\right) \left(c-a^2 c x^2\right)^{3/2}}",1,"(E^(n*ArcCoth[a*x])*(-26*n + 2*n^3 + 27*a*x - 3*a*n^2*x - 2*n*(-1 + n^2)*Cosh[2*ArcCoth[a*x]] + 3*a*(-1 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[3*ArcCoth[a*x]]))/(4*a*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])","A",0
749,1,299,166,1.5676603,"\int \frac{e^{n \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{7/2}} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(7/2),x]","-\frac{a^2 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} e^{n \coth ^{-1}(a x)} \left(\frac{10 n^5}{a x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{340 n^3}{a x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{10 \left(n^4-34 n^2+225\right)}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2250 n}{a x \sqrt{1-\frac{1}{a^2 x^2}}}-5 n^5 \sinh \left(3 \coth ^{-1}(a x)\right)+n^5 \sinh \left(5 \coth ^{-1}(a x)\right)-5 n^4 \cosh \left(5 \coth ^{-1}(a x)\right)+130 n^3 \sinh \left(3 \coth ^{-1}(a x)\right)-10 n^3 \sinh \left(5 \coth ^{-1}(a x)\right)+50 n^2 \cosh \left(5 \coth ^{-1}(a x)\right)+15 \left(n^4-26 n^2+25\right) \cosh \left(3 \coth ^{-1}(a x)\right)-125 n \sinh \left(3 \coth ^{-1}(a x)\right)+9 n \sinh \left(5 \coth ^{-1}(a x)\right)-45 \cosh \left(5 \coth ^{-1}(a x)\right)\right)}{16 c^2 (n-5) (n-3) (n-1) (n+1) (n+3) (n+5) \left(c-a^2 c x^2\right)^{3/2}}","-\frac{120 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left(1-n^2\right) \left(9-n^2\right) \left(25-n^2\right) \sqrt{c-a^2 c x^2}}-\frac{20 (n-3 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left(9-n^2\right) \left(25-n^2\right) \left(c-a^2 c x^2\right)^{3/2}}-\frac{(n-5 a x) e^{n \coth ^{-1}(a x)}}{a c \left(25-n^2\right) \left(c-a^2 c x^2\right)^{5/2}}",1,"-1/16*(a^2*E^(n*ArcCoth[a*x])*(1 - 1/(a^2*x^2))^(3/2)*x^3*((-10*(225 - 34*n^2 + n^4))/Sqrt[1 - 1/(a^2*x^2)] + (2250*n)/(a*Sqrt[1 - 1/(a^2*x^2)]*x) - (340*n^3)/(a*Sqrt[1 - 1/(a^2*x^2)]*x) + (10*n^5)/(a*Sqrt[1 - 1/(a^2*x^2)]*x) + 15*(25 - 26*n^2 + n^4)*Cosh[3*ArcCoth[a*x]] - 45*Cosh[5*ArcCoth[a*x]] + 50*n^2*Cosh[5*ArcCoth[a*x]] - 5*n^4*Cosh[5*ArcCoth[a*x]] - 125*n*Sinh[3*ArcCoth[a*x]] + 130*n^3*Sinh[3*ArcCoth[a*x]] - 5*n^5*Sinh[3*ArcCoth[a*x]] + 9*n*Sinh[5*ArcCoth[a*x]] - 10*n^3*Sinh[5*ArcCoth[a*x]] + n^5*Sinh[5*ArcCoth[a*x]]))/(c^2*(-5 + n)*(-3 + n)*(-1 + n)*(1 + n)*(3 + n)*(5 + n)*(c - a^2*c*x^2)^(3/2))","A",0
750,1,260,239,1.6785827,"\int \frac{e^{n \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{9/2}} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(9/2),x]","\frac{a x^2 \left(1-\frac{1}{a^2 x^2}\right) e^{n \coth ^{-1}(a x)} \left(-\frac{63 a x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left(3 \coth ^{-1}(a x)\right)}{n^2-9}+\frac{35 a x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left(5 \coth ^{-1}(a x)\right)}{n^2-25}-\frac{7 a x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left(7 \coth ^{-1}(a x)\right)}{n^2-49}+\frac{21 a n x \sqrt{1-\frac{1}{a^2 x^2}} \sinh \left(3 \coth ^{-1}(a x)\right)}{n^2-9}-\frac{7 a n x \sqrt{1-\frac{1}{a^2 x^2}} \sinh \left(5 \coth ^{-1}(a x)\right)}{n^2-25}+\frac{a n x \sqrt{1-\frac{1}{a^2 x^2}} \sinh \left(7 \coth ^{-1}(a x)\right)}{n^2-49}+\frac{35 a x}{n^2-1}-\frac{35 n}{n^2-1}\right)}{64 c^3 \left(c-a^2 c x^2\right)^{3/2}}","-\frac{5040 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left(1-n^2\right) \left(9-n^2\right) \left(25-n^2\right) \left(49-n^2\right) \sqrt{c-a^2 c x^2}}-\frac{840 (n-3 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left(9-n^2\right) \left(25-n^2\right) \left(49-n^2\right) \left(c-a^2 c x^2\right)^{3/2}}-\frac{42 (n-5 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left(25-n^2\right) \left(49-n^2\right) \left(c-a^2 c x^2\right)^{5/2}}-\frac{(n-7 a x) e^{n \coth ^{-1}(a x)}}{a c \left(49-n^2\right) \left(c-a^2 c x^2\right)^{7/2}}",1,"(a*E^(n*ArcCoth[a*x])*(1 - 1/(a^2*x^2))*x^2*((-35*n)/(-1 + n^2) + (35*a*x)/(-1 + n^2) - (63*a*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[3*ArcCoth[a*x]])/(-9 + n^2) + (35*a*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[5*ArcCoth[a*x]])/(-25 + n^2) - (7*a*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[7*ArcCoth[a*x]])/(-49 + n^2) + (21*a*n*Sqrt[1 - 1/(a^2*x^2)]*x*Sinh[3*ArcCoth[a*x]])/(-9 + n^2) - (7*a*n*Sqrt[1 - 1/(a^2*x^2)]*x*Sinh[5*ArcCoth[a*x]])/(-25 + n^2) + (a*n*Sqrt[1 - 1/(a^2*x^2)]*x*Sinh[7*ArcCoth[a*x]])/(-49 + n^2)))/(64*c^3*(c - a^2*c*x^2)^(3/2))","A",0
751,1,133,359,0.6078531,"\int \frac{e^{n \coth ^{-1}(a x)} x^3}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[(E^(n*ArcCoth[a*x])*x^3)/(c - a^2*c*x^2)^(3/2),x]","\frac{\frac{c (a n x-1) e^{n \coth ^{-1}(a x)}}{n^2-1}-\frac{c \left(a^2 x^2-1\right) \left(\frac{2 n e^{(n+1) \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};e^{2 \coth ^{-1}(a x)}\right)}{a x \sqrt{1-\frac{1}{a^2 x^2}}}+(n+1) e^{n \coth ^{-1}(a x)}\right)}{n+1}}{a^4 c^2 \sqrt{c-a^2 c x^2}}","-\frac{2 n x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{\frac{n-1}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}} \, _2F_1\left(1,\frac{n-1}{2};\frac{n+1}{2};\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\right)}{a (1-n) \left(c-a^2 c x^2\right)^{3/2}}+\frac{\left(n^2+2 n+2\right) x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{\frac{n-1}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}}}{a (1-n) (n+1) \left(c-a^2 c x^2\right)^{3/2}}+\frac{x^4 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{\frac{n-1}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-1)}}{\left(c-a^2 c x^2\right)^{3/2}}-\frac{(n+2) x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{\frac{n-1}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-1)}}{a (n+1) \left(c-a^2 c x^2\right)^{3/2}}",1,"((c*E^(n*ArcCoth[a*x])*(-1 + a*n*x))/(-1 + n^2) - (c*(-1 + a^2*x^2)*(E^(n*ArcCoth[a*x])*(1 + n) + (2*E^((1 + n)*ArcCoth[a*x])*n*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])])/(a*Sqrt[1 - 1/(a^2*x^2)]*x)))/(1 + n))/(a^4*c^2*Sqrt[c - a^2*c*x^2])","A",0
752,1,127,164,0.4360392,"\int \frac{e^{n \coth ^{-1}(a x)} x^2}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[(E^(n*ArcCoth[a*x])*x^2)/(c - a^2*c*x^2)^(3/2),x]","-\frac{e^{n \coth ^{-1}(a x)} \left(2 (n-1) \left(a^2 x^2-1\right) e^{\coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};e^{2 \coth ^{-1}(a x)}\right)+a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-n)\right)}{a^4 c (n-1) (n+1) x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a^2 c x^2}}","-\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}} \left(\frac{1}{a x}+1\right)^{\frac{n-1}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}} \, _2F_1\left(1,\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a^2 c (1-n) \sqrt{c-a^2 c x^2}}-\frac{(n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left(1-n^2\right) \sqrt{c-a^2 c x^2}}",1,"-((E^(n*ArcCoth[a*x])*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-n + a*x) + 2*E^ArcCoth[a*x]*(-1 + n)*(-1 + a^2*x^2)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])]))/(a^4*c*(-1 + n)*(1 + n)*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a^2*c*x^2]))","A",0
753,1,43,46,0.1972684,"\int \frac{e^{n \coth ^{-1}(a x)} x}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[(E^(n*ArcCoth[a*x])*x)/(c - a^2*c*x^2)^(3/2),x]","\frac{(a n x-1) e^{n \coth ^{-1}(a x)}}{a^2 c \left(n^2-1\right) \sqrt{c-a^2 c x^2}}","\frac{(1-a n x) e^{n \coth ^{-1}(a x)}}{a^2 c \left(1-n^2\right) \sqrt{c-a^2 c x^2}}",1,"(E^(n*ArcCoth[a*x])*(-1 + a*n*x))/(a^2*c*(-1 + n^2)*Sqrt[c - a^2*c*x^2])","A",1
754,1,43,46,0.1949531,"\int \frac{e^{n \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(3/2),x]","\frac{(n-a x) e^{n \coth ^{-1}(a x)}}{a c \left(n^2-1\right) \sqrt{c-a^2 c x^2}}","-\frac{(n-a x) e^{n \coth ^{-1}(a x)}}{a c \left(1-n^2\right) \sqrt{c-a^2 c x^2}}",1,"(E^(n*ArcCoth[a*x])*(n - a*x))/(a*c*(-1 + n^2)*Sqrt[c - a^2*c*x^2])","A",1
755,1,127,277,0.8893672,"\int \frac{e^{n \coth ^{-1}(a x)}}{x \left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^(n*ArcCoth[a*x])/(x*(c - a^2*c*x^2)^(3/2)),x]","\frac{e^{n \coth ^{-1}(a x)} \left(a x \sqrt{1-\frac{1}{a^2 x^2}} (a n x-1)-2 (n-1) \left(a^2 x^2-1\right) e^{\coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-e^{2 \coth ^{-1}(a x)}\right)\right)}{a c (n-1) (n+1) x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a^2 c x^2}}","-\frac{a^3 2^{\frac{n+1}{2}} x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}} \, _2F_1\left(\frac{1-n}{2},\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{2 a}\right)}{(1-n) \left(c-a^2 c x^2\right)^{3/2}}+\frac{a^3 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{\frac{n-1}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}}}{\left(1-n^2\right) \left(c-a^2 c x^2\right)^{3/2}}-\frac{a^3 x^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{\frac{n-1}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-1)}}{(n+1) \left(c-a^2 c x^2\right)^{3/2}}",1,"(E^(n*ArcCoth[a*x])*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + a*n*x) - 2*E^ArcCoth[a*x]*(-1 + n)*(-1 + a^2*x^2)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -E^(2*ArcCoth[a*x])]))/(a*c*(-1 + n)*(1 + n)*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a^2*c*x^2])","A",0
756,1,201,463,2.1690523,"\int \frac{e^{n \coth ^{-1}(a x)} x^4}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[(E^(n*ArcCoth[a*x])*x^4)/(c - a^2*c*x^2)^(5/2),x]","\frac{\left(a^2 x^2-1\right) \left(-\frac{8 a x \sqrt{1-\frac{1}{a^2 x^2}} e^{(n+1) \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};e^{2 \coth ^{-1}(a x)}\right)}{n+1}+\frac{e^{n \coth ^{-1}(a x)} \left(-3 a \left(n^2-1\right) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left(3 \coth ^{-1}(a x)\right)+3 a n^2 x+2 \left(n^2-1\right) n \cosh \left(2 \coth ^{-1}(a x)\right)-27 a x-2 n^3+26 n\right)}{n^4-10 n^2+9}+\frac{8 (n-a x) e^{n \coth ^{-1}(a x)}}{n^2-1}\right)}{4 a^5 c \left(c-a^2 c x^2\right)^{3/2}}","-\frac{2 x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{\frac{n-1}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}} \, _2F_1\left(1,\frac{n-1}{2};\frac{n+1}{2};\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\right)}{(1-n) \left(c-a^2 c x^2\right)^{5/2}}+\frac{\left(n^2+6 n+15\right) x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{\frac{n-3}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}}}{(1-n) (n+1) (n+3) \left(c-a^2 c x^2\right)^{5/2}}-\frac{\left(-n^3-2 n^2+7 n+18\right) x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{\frac{n-3}{2}} \left(1-\frac{1}{a x}\right)^{\frac{3-n}{2}}}{\left(n^4-10 n^2+9\right) \left(c-a^2 c x^2\right)^{5/2}}-\frac{x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{\frac{n-3}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-3)}}{(n+3) \left(c-a^2 c x^2\right)^{5/2}}-\frac{(n+6) x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{\frac{n-3}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-1)}}{(n+1) (n+3) \left(c-a^2 c x^2\right)^{5/2}}",1,"((-1 + a^2*x^2)*((8*E^(n*ArcCoth[a*x])*(n - a*x))/(-1 + n^2) + (E^(n*ArcCoth[a*x])*(26*n - 2*n^3 - 27*a*x + 3*a*n^2*x + 2*n*(-1 + n^2)*Cosh[2*ArcCoth[a*x]] - 3*a*(-1 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[3*ArcCoth[a*x]]))/(9 - 10*n^2 + n^4) - (8*a*E^((1 + n)*ArcCoth[a*x])*Sqrt[1 - 1/(a^2*x^2)]*x*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])])/(1 + n)))/(4*a^5*c*(c - a^2*c*x^2)^(3/2))","A",0
757,1,110,330,0.7300301,"\int \frac{e^{n \coth ^{-1}(a x)} x^3}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[(E^(n*ArcCoth[a*x])*x^3)/(c - a^2*c*x^2)^(5/2),x]","-\frac{e^{n \coth ^{-1}(a x)} \left(a n \left(n^2-1\right) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left(3 \coth ^{-1}(a x)\right)-6 \left(n^2-1\right) \cosh \left(2 \coth ^{-1}(a x)\right)+3 \left(a n^3 x-9 a n x-2 n^2+10\right)\right)}{4 a^4 c^2 \left(n^4-10 n^2+9\right) \sqrt{c-a^2 c x^2}}","-\frac{3 a x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{\frac{n-3}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-1)}}{\left(n^2+4 n+3\right) \left(c-a^2 c x^2\right)^{5/2}}+\frac{6 a x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{\frac{n-3}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}}}{(n+3) \left(1-n^2\right) \left(c-a^2 c x^2\right)^{5/2}}-\frac{6 a x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{\frac{n-3}{2}} \left(1-\frac{1}{a x}\right)^{\frac{3-n}{2}}}{\left(n^4-10 n^2+9\right) \left(c-a^2 c x^2\right)^{5/2}}-\frac{a x^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{\frac{n-3}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-3)}}{(n+3) \left(c-a^2 c x^2\right)^{5/2}}",1,"-1/4*(E^(n*ArcCoth[a*x])*(3*(10 - 2*n^2 - 9*a*n*x + a*n^3*x) - 6*(-1 + n^2)*Cosh[2*ArcCoth[a*x]] + a*n*(-1 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[3*ArcCoth[a*x]]))/(a^4*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])","A",0
758,1,109,102,0.7571421,"\int \frac{e^{n \coth ^{-1}(a x)} x^2}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[(E^(n*ArcCoth[a*x])*x^2)/(c - a^2*c*x^2)^(5/2),x]","\frac{e^{n \coth ^{-1}(a x)} \left(3 a \left(n^2-1\right) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left(3 \coth ^{-1}(a x)\right)+a n^2 x-2 \left(n^2-1\right) n \cosh \left(2 \coth ^{-1}(a x)\right)-9 a x-2 n^3+10 n\right)}{4 a^3 c^2 \left(n^4-10 n^2+9\right) \sqrt{c-a^2 c x^2}}","\frac{\left(3-n^2\right) (n-a x) e^{n \coth ^{-1}(a x)}}{a^3 c^2 \left(n^4-10 n^2+9\right) \sqrt{c-a^2 c x^2}}-\frac{(n-3 a x) e^{n \coth ^{-1}(a x)}}{a^3 c \left(9-n^2\right) \left(c-a^2 c x^2\right)^{3/2}}",1,"(E^(n*ArcCoth[a*x])*(10*n - 2*n^3 - 9*a*x + a*n^2*x - 2*n*(-1 + n^2)*Cosh[2*ArcCoth[a*x]] + 3*a*(-1 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[3*ArcCoth[a*x]]))/(4*a^3*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])","A",0
759,1,108,97,0.6592221,"\int \frac{e^{n \coth ^{-1}(a x)} x}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[(E^(n*ArcCoth[a*x])*x)/(c - a^2*c*x^2)^(5/2),x]","\frac{e^{n \coth ^{-1}(a x)} \left(-a \left(n^2-1\right) n x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left(3 \coth ^{-1}(a x)\right)+a n^3 x+6 \left(n^2-1\right) \cosh \left(2 \coth ^{-1}(a x)\right)-9 a n x+2 n^2+6\right)}{4 a^2 c^2 \left(n^4-10 n^2+9\right) \sqrt{c-a^2 c x^2}}","\frac{2 n (n-a x) e^{n \coth ^{-1}(a x)}}{a^2 c^2 \left(n^4-10 n^2+9\right) \sqrt{c-a^2 c x^2}}+\frac{(3-a n x) e^{n \coth ^{-1}(a x)}}{a^2 c \left(9-n^2\right) \left(c-a^2 c x^2\right)^{3/2}}",1,"(E^(n*ArcCoth[a*x])*(6 + 2*n^2 - 9*a*n*x + a*n^3*x + 6*(-1 + n^2)*Cosh[2*ArcCoth[a*x]] - a*n*(-1 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[3*ArcCoth[a*x]]))/(4*a^2*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])","A",0
760,1,110,102,0.2397118,"\int \frac{e^{n \coth ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - a^2*c*x^2)^(5/2),x]","\frac{e^{n \coth ^{-1}(a x)} \left(3 a \left(n^2-1\right) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left(3 \coth ^{-1}(a x)\right)-3 a n^2 x-2 \left(n^2-1\right) n \cosh \left(2 \coth ^{-1}(a x)\right)+27 a x+2 n^3-26 n\right)}{4 a c^2 \left(n^4-10 n^2+9\right) \sqrt{c-a^2 c x^2}}","-\frac{6 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left(1-n^2\right) \left(9-n^2\right) \sqrt{c-a^2 c x^2}}-\frac{(n-3 a x) e^{n \coth ^{-1}(a x)}}{a c \left(9-n^2\right) \left(c-a^2 c x^2\right)^{3/2}}",1,"(E^(n*ArcCoth[a*x])*(-26*n + 2*n^3 + 27*a*x - 3*a*n^2*x - 2*n*(-1 + n^2)*Cosh[2*ArcCoth[a*x]] + 3*a*(-1 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[3*ArcCoth[a*x]]))/(4*a*c^2*(9 - 10*n^2 + n^4)*Sqrt[c - a^2*c*x^2])","A",0
761,1,220,944,1.6877347,"\int \frac{e^{n \coth ^{-1}(a x)}}{x \left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[E^(n*ArcCoth[a*x])/(x*(c - a^2*c*x^2)^(5/2)),x]","\frac{e^{n \coth ^{-1}(a x)} \left(6 a \left(n^2-1\right) x \sqrt{1-\frac{1}{a^2 x^2}} \cosh \left(2 \coth ^{-1}(a x)\right)-n \left(n^2-1\right) \left(a^2 x^2-1\right) \cosh \left(3 \coth ^{-1}(a x)\right)+a x \sqrt{1-\frac{1}{a^2 x^2}} \left(5 a n^3 x-45 a n x-2 n^2+42\right)\right)-8 \left(n^3-n^2-9 n+9\right) \left(a^2 x^2-1\right) e^{(n+1) \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-e^{2 \coth ^{-1}(a x)}\right)}{4 a c^2 (n-1) (n+1) \left(n^2-9\right) x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a^2 c x^2}}","-\frac{2^{\frac{n+5}{2}} a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} x^5 \, _2F_1\left(\frac{1}{2} (-n-3),\frac{1}{2} (-n-3);\frac{1}{2} (-n-1);\frac{a-\frac{1}{x}}{2 a}\right) \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-3)}}{(n+3) \left(c-a^2 c x^2\right)^{5/2}}-\frac{a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(1+\frac{1}{a x}\right)^{\frac{n-3}{2}} x^5 \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-3)}}{(n+3) \left(c-a^2 c x^2\right)^{5/2}}+\frac{4 a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(1+\frac{1}{a x}\right)^{\frac{n-1}{2}} x^5 \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-3)}}{(n+3) \left(c-a^2 c x^2\right)^{5/2}}-\frac{6 a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(1+\frac{1}{a x}\right)^{\frac{n+1}{2}} x^5 \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-3)}}{(n+3) \left(c-a^2 c x^2\right)^{5/2}}+\frac{4 a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(1+\frac{1}{a x}\right)^{\frac{n+3}{2}} x^5 \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-3)}}{(n+3) \left(c-a^2 c x^2\right)^{5/2}}-\frac{3 a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(1+\frac{1}{a x}\right)^{\frac{n-3}{2}} x^5 \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-1)}}{\left(n^2+4 n+3\right) \left(c-a^2 c x^2\right)^{5/2}}+\frac{8 a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(1+\frac{1}{a x}\right)^{\frac{n-1}{2}} x^5 \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-1)}}{\left(n^2+4 n+3\right) \left(c-a^2 c x^2\right)^{5/2}}-\frac{6 a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(1+\frac{1}{a x}\right)^{\frac{n+1}{2}} x^5 \left(1-\frac{1}{a x}\right)^{\frac{1}{2} (-n-1)}}{\left(n^2+4 n+3\right) \left(c-a^2 c x^2\right)^{5/2}}+\frac{6 a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(1+\frac{1}{a x}\right)^{\frac{n-3}{2}} x^5 \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}}}{(n+3) \left(1-n^2\right) \left(c-a^2 c x^2\right)^{5/2}}-\frac{8 a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(1+\frac{1}{a x}\right)^{\frac{n-1}{2}} x^5 \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}}}{(n+3) \left(1-n^2\right) \left(c-a^2 c x^2\right)^{5/2}}-\frac{6 a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(1+\frac{1}{a x}\right)^{\frac{n-3}{2}} x^5 \left(1-\frac{1}{a x}\right)^{\frac{3-n}{2}}}{\left(n^4-10 n^2+9\right) \left(c-a^2 c x^2\right)^{5/2}}",1,"(E^(n*ArcCoth[a*x])*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(42 - 2*n^2 - 45*a*n*x + 5*a*n^3*x) + 6*a*(-1 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Cosh[2*ArcCoth[a*x]] - n*(-1 + n^2)*(-1 + a^2*x^2)*Cosh[3*ArcCoth[a*x]]) - 8*E^((1 + n)*ArcCoth[a*x])*(9 - 9*n - n^2 + n^3)*(-1 + a^2*x^2)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -E^(2*ArcCoth[a*x])])/(4*a*c^2*(-1 + n)*(1 + n)*(-9 + n^2)*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a^2*c*x^2])","A",0
762,1,83,127,0.2002694,"\int e^{n \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^p \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - a^2*c*x^2)^p,x]","-\frac{\left(a^2 x^2-1\right) \left(e^{2 \coth ^{-1}(a x)}-1\right) \left(c-a^2 c x^2\right)^p e^{(n-2) \coth ^{-1}(a x)} \, _2F_1\left(1,-\frac{n}{2}-p;-\frac{n}{2}+p+2;e^{-2 \coth ^{-1}(a x)}\right)}{a (n-2 (p+1))}","\frac{x \left(1-\frac{1}{a^2 x^2}\right)^{-p} \left(c-a^2 c x^2\right)^p \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{\frac{1}{2} (n-2 p)} \left(1-\frac{1}{a x}\right)^{p-\frac{n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n}{2}+p+1} \, _2F_1\left(-2 p-1,\frac{1}{2} (n-2 p);-2 p;\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{2 p+1}",1,"-((E^((-2 + n)*ArcCoth[a*x])*(-1 + E^(2*ArcCoth[a*x]))*(-1 + a^2*x^2)*(c - a^2*c*x^2)^p*Hypergeometric2F1[1, -1/2*n - p, 2 - n/2 + p, E^(-2*ArcCoth[a*x])])/(a*(n - 2*(1 + p))))","A",0
763,1,36,51,0.0723912,"\int e^{2 p \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^p \, dx","Integrate[E^(2*p*ArcCoth[a*x])*(c - a^2*c*x^2)^p,x]","\frac{(a x+1) \left(c-a^2 c x^2\right)^p e^{2 p \coth ^{-1}(a x)}}{2 a p+a}","\frac{x \left(1-\frac{1}{a^2 x^2}\right)^{-p} \left(\frac{1}{a x}+1\right)^{2 p+1} \left(c-a^2 c x^2\right)^p}{2 p+1}",1,"(E^(2*p*ArcCoth[a*x])*(1 + a*x)*(c - a^2*c*x^2)^p)/(a + 2*a*p)","A",1
764,1,36,52,0.0664671,"\int e^{-2 p \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^p \, dx","Integrate[(c - a^2*c*x^2)^p/E^(2*p*ArcCoth[a*x]),x]","\frac{(a x-1) \left(c-a^2 c x^2\right)^p e^{-2 p \coth ^{-1}(a x)}}{2 a p+a}","\frac{x \left(1-\frac{1}{a^2 x^2}\right)^{-p} \left(1-\frac{1}{a x}\right)^{2 p+1} \left(c-a^2 c x^2\right)^p}{2 p+1}",1,"((-1 + a*x)*(c - a^2*c*x^2)^p)/(E^(2*p*ArcCoth[a*x])*(a + 2*a*p))","A",1
765,1,72,63,0.0259856,"\int e^{4 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^p \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^p,x]","-\frac{2^{p+2} (1-a x)^{p-1} \left(1-a^2 x^2\right)^{-p} \left(c-a^2 c x^2\right)^p \, _2F_1\left(-p-2,p-1;p;\frac{1}{2} (1-a x)\right)}{a (p-1)}","\frac{c 2^{p+2} (a x+1)^{1-p} \left(c-a^2 c x^2\right)^{p-1} \, _2F_1\left(-p-2,p-1;p;\frac{1}{2} (1-a x)\right)}{a (1-p)}",1,"-((2^(2 + p)*(1 - a*x)^(-1 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-2 - p, -1 + p, p, (1 - a*x)/2])/(a*(-1 + p)*(1 - a^2*x^2)^p))","A",1
766,1,122,118,0.2794161,"\int e^{3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^p \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - a^2*c*x^2)^p,x]","-\frac{4^{p+1} \left(a x \sqrt{1-\frac{1}{a^2 x^2}}\right)^{-2 p} e^{5 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^p \left(1-e^{2 \coth ^{-1}(a x)}\right)^{2 p} \left(\frac{e^{\coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}\right)^{2 p} \, _2F_1\left(p+\frac{5}{2},2 p+2;p+\frac{7}{2};e^{2 \coth ^{-1}(a x)}\right)}{2 a p+5 a}","\frac{x \left(1-\frac{1}{a^2 x^2}\right)^{-p} \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{\frac{3}{2}-p} \left(1-\frac{1}{a x}\right)^{p-\frac{3}{2}} \left(\frac{1}{a x}+1\right)^{p+\frac{5}{2}} \left(c-a^2 c x^2\right)^p \, _2F_1\left(-2 p-1,\frac{3}{2}-p;-2 p;\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{2 p+1}",1,"-((4^(1 + p)*E^(5*ArcCoth[a*x])*(1 - E^(2*ArcCoth[a*x]))^(2*p)*(E^ArcCoth[a*x]/(-1 + E^(2*ArcCoth[a*x])))^(2*p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[5/2 + p, 2 + 2*p, 7/2 + p, E^(2*ArcCoth[a*x])])/((5*a + 2*a*p)*(a*Sqrt[1 - 1/(a^2*x^2)]*x)^(2*p)))","A",0
767,1,67,54,0.0190858,"\int e^{2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^p \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - a^2*c*x^2)^p,x]","\frac{2^{p+1} (1-a x)^p \left(1-a^2 x^2\right)^{-p} \left(c-a^2 c x^2\right)^p \, _2F_1\left(-p-1,p;p+1;\frac{1}{2} (1-a x)\right)}{a p}","\frac{2^{p+1} (a x+1)^{-p} \left(c-a^2 c x^2\right)^p \, _2F_1\left(-p-1,p;p+1;\frac{1}{2} (1-a x)\right)}{a p}",1,"(2^(1 + p)*(1 - a*x)^p*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - p, p, 1 + p, (1 - a*x)/2])/(a*p*(1 - a^2*x^2)^p)","A",1
768,1,122,118,0.2220368,"\int e^{\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^p \, dx","Integrate[E^ArcCoth[a*x]*(c - a^2*c*x^2)^p,x]","-\frac{4^{p+1} \left(a x \sqrt{1-\frac{1}{a^2 x^2}}\right)^{-2 p} e^{3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^p \left(1-e^{2 \coth ^{-1}(a x)}\right)^{2 p} \left(\frac{e^{\coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}\right)^{2 p} \, _2F_1\left(p+\frac{3}{2},2 p+2;p+\frac{5}{2};e^{2 \coth ^{-1}(a x)}\right)}{2 a p+3 a}","\frac{x \left(1-\frac{1}{a^2 x^2}\right)^{-p} \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{\frac{1}{2}-p} \left(1-\frac{1}{a x}\right)^{p-\frac{1}{2}} \left(\frac{1}{a x}+1\right)^{p+\frac{3}{2}} \left(c-a^2 c x^2\right)^p \, _2F_1\left(-2 p-1,\frac{1}{2}-p;-2 p;\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{2 p+1}",1,"-((4^(1 + p)*E^(3*ArcCoth[a*x])*(1 - E^(2*ArcCoth[a*x]))^(2*p)*(E^ArcCoth[a*x]/(-1 + E^(2*ArcCoth[a*x])))^(2*p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[3/2 + p, 2 + 2*p, 5/2 + p, E^(2*ArcCoth[a*x])])/((3*a + 2*a*p)*(a*Sqrt[1 - 1/(a^2*x^2)]*x)^(2*p)))","A",0
769,1,118,118,0.2375364,"\int e^{-\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^p \, dx","Integrate[(c - a^2*c*x^2)^p/E^ArcCoth[a*x],x]","-\frac{4^{p+1} \left(a x \sqrt{1-\frac{1}{a^2 x^2}}\right)^{-2 p} e^{\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^p \left(1-e^{2 \coth ^{-1}(a x)}\right)^{2 p} \left(\frac{e^{\coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}\right)^{2 p} \, _2F_1\left(p+\frac{1}{2},2 p+2;p+\frac{3}{2};e^{2 \coth ^{-1}(a x)}\right)}{2 a p+a}","\frac{x \left(1-\frac{1}{a^2 x^2}\right)^{-p} \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{-p-\frac{1}{2}} \left(1-\frac{1}{a x}\right)^{p+\frac{1}{2}} \left(\frac{1}{a x}+1\right)^{p+\frac{1}{2}} \left(c-a^2 c x^2\right)^p \, _2F_1\left(-2 p-1,-p-\frac{1}{2};-2 p;\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{2 p+1}",1,"-((4^(1 + p)*E^ArcCoth[a*x]*(1 - E^(2*ArcCoth[a*x]))^(2*p)*(E^ArcCoth[a*x]/(-1 + E^(2*ArcCoth[a*x])))^(2*p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[1/2 + p, 2 + 2*p, 3/2 + p, E^(2*ArcCoth[a*x])])/((a + 2*a*p)*(a*Sqrt[1 - 1/(a^2*x^2)]*x)^(2*p)))","A",0
770,1,73,55,0.0267854,"\int e^{-2 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^p \, dx","Integrate[(c - a^2*c*x^2)^p/E^(2*ArcCoth[a*x]),x]","\frac{2^{p-1} (1-a x)^{p+2} \left(1-a^2 x^2\right)^{-p} \left(c-a^2 c x^2\right)^p \, _2F_1\left(1-p,p+2;p+3;\frac{1}{2} (1-a x)\right)}{a (p+2)}","-\frac{2^{p+1} (1-a x)^{-p} \left(c-a^2 c x^2\right)^p \, _2F_1\left(-p-1,p;p+1;\frac{1}{2} (a x+1)\right)}{a p}",1,"(2^(-1 + p)*(1 - a*x)^(2 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[1 - p, 2 + p, 3 + p, (1 - a*x)/2])/(a*(2 + p)*(1 - a^2*x^2)^p)","A",0
771,1,119,118,0.2827815,"\int e^{-3 \coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^p \, dx","Integrate[(c - a^2*c*x^2)^p/E^(3*ArcCoth[a*x]),x]","\frac{4^{p+1} \left(a x \sqrt{1-\frac{1}{a^2 x^2}}\right)^{-2 p} e^{-\coth ^{-1}(a x)} \left(c-a^2 c x^2\right)^p \left(1-e^{2 \coth ^{-1}(a x)}\right)^{2 p} \left(\frac{e^{\coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}\right)^{2 p} \, _2F_1\left(p-\frac{1}{2},2 p+2;p+\frac{1}{2};e^{2 \coth ^{-1}(a x)}\right)}{a-2 a p}","\frac{x \left(1-\frac{1}{a^2 x^2}\right)^{-p} \left(\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)^{-p-\frac{3}{2}} \left(1-\frac{1}{a x}\right)^{p+\frac{3}{2}} \left(\frac{1}{a x}+1\right)^{p-\frac{1}{2}} \left(c-a^2 c x^2\right)^p \, _2F_1\left(-2 p-1,-p-\frac{3}{2};-2 p;\frac{2}{\left(a+\frac{1}{x}\right) x}\right)}{2 p+1}",1,"(4^(1 + p)*(1 - E^(2*ArcCoth[a*x]))^(2*p)*(E^ArcCoth[a*x]/(-1 + E^(2*ArcCoth[a*x])))^(2*p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1/2 + p, 2 + 2*p, 1/2 + p, E^(2*ArcCoth[a*x])])/(E^ArcCoth[a*x]*(a - 2*a*p)*(a*Sqrt[1 - 1/(a^2*x^2)]*x)^(2*p))","A",0
772,1,120,342,0.3180509,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^4 \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a^2*x^2))^4,x]","\frac{c^4 \left(3675 a^6 \sin ^{-1}\left(\frac{1}{a x}\right)+1680 a^6 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left(1680 a^7 x^7-2816 a^6 x^6+3045 a^5 x^5+1952 a^4 x^4-1330 a^3 x^3-1056 a^2 x^2+280 a x+240\right)}{x^6}\right)}{1680 a^7}","\frac{8 c^4 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{9/2}}{7 a}+\frac{47 c^4 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{9/2}}{42 a}+c^4 x \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{9/2}+\frac{61 c^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{9/2}}{70 a}-\frac{131 c^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}}{280 a}-\frac{91 c^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}}{120 a}-\frac{67 c^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}}{48 a}-\frac{51 c^4 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{16 a}+\frac{35 c^4 \csc ^{-1}(a x)}{16 a}+\frac{c^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c^4*((Sqrt[1 - 1/(a^2*x^2)]*(240 + 280*a*x - 1056*a^2*x^2 - 1330*a^3*x^3 + 1952*a^4*x^4 + 3045*a^5*x^5 - 2816*a^6*x^6 + 1680*a^7*x^7))/x^6 + 3675*a^6*ArcSin[1/(a*x)] + 1680*a^6*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(1680*a^7)","A",0
773,1,104,268,0.2473291,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^3 \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a^2*x^2))^3,x]","\frac{c^3 \left(225 a^4 \sin ^{-1}\left(\frac{1}{a x}\right)+120 a^4 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left(120 a^5 x^5-184 a^4 x^4+135 a^3 x^3+88 a^2 x^2-30 a x-24\right)}{x^4}\right)}{120 a^5}","\frac{6 c^3 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{7/2}}{5 a}+c^3 x \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{7/2}+\frac{23 c^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}}{20 a}-\frac{43 c^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}}{60 a}-\frac{31 c^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}}{24 a}-\frac{23 c^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{8 a}+\frac{15 c^3 \csc ^{-1}(a x)}{8 a}+\frac{c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c^3*((Sqrt[1 - 1/(a^2*x^2)]*(-24 - 30*a*x + 88*a^2*x^2 + 135*a^3*x^3 - 184*a^4*x^4 + 120*a^5*x^5))/x^4 + 225*a^4*ArcSin[1/(a*x)] + 120*a^4*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(120*a^5)","A",0
774,1,94,194,0.1642344,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^2 \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a^2*x^2))^2,x]","\frac{c^2 \left(6 a^2 x^2 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+9 a^2 x^2 \sin ^{-1}\left(\frac{1}{a x}\right)+\sqrt{1-\frac{1}{a^2 x^2}} \left(6 a^3 x^3-8 a^2 x^2+3 a x+2\right)\right)}{6 a^3 x^2}","c^2 x \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{5/2}+\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}}{3 a}-\frac{7 c^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}}{6 a}-\frac{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{2 a}+\frac{3 c^2 \csc ^{-1}(a x)}{2 a}+\frac{c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c^2*(Sqrt[1 - 1/(a^2*x^2)]*(2 + 3*a*x - 8*a^2*x^2 + 6*a^3*x^3) + 9*a^2*x^2*ArcSin[1/(a*x)] + 6*a^2*x^2*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(6*a^3*x^2)","A",0
775,1,53,107,0.123878,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right) \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a^2*x^2)),x]","\frac{c \left(\sqrt{1-\frac{1}{a^2 x^2}} (a x-1)+\log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\sin ^{-1}\left(\frac{1}{a x}\right)\right)}{a}","c x \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}-\frac{2 c \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{a}+\frac{c \csc ^{-1}(a x)}{a}+\frac{c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c*(Sqrt[1 - 1/(a^2*x^2)]*(-1 + a*x) + ArcSin[1/(a*x)] + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/a","A",0
776,1,56,104,0.1379468,"\int \frac{e^{\coth ^{-1}(a x)}}{c-\frac{c}{a^2 x^2}} \, dx","Integrate[E^ArcCoth[a*x]/(c - c/(a^2*x^2)),x]","\frac{\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x-2)}{a x-1}+\frac{\log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a}}{c}","\frac{x \sqrt{\frac{1}{a x}+1}}{c \sqrt{1-\frac{1}{a x}}}-\frac{2 \sqrt{\frac{1}{a x}+1}}{a c \sqrt{1-\frac{1}{a x}}}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c}",1,"((Sqrt[1 - 1/(a^2*x^2)]*x*(-2 + a*x))/(-1 + a*x) + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]/a)/c","A",0
777,1,83,180,0.2151809,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^2} \, dx","Integrate[E^ArcCoth[a*x]/(c - c/(a^2*x^2))^2,x]","\frac{\log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(3 a^3 x^3-7 a^2 x^2-5 a x+8\right)}{3 (a x-1)^2 (a x+1)}}{a c^2}","\frac{x}{c^2 \left(1-\frac{1}{a x}\right)^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{\frac{1}{a x}+1}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{4}{3 a c^2 \left(1-\frac{1}{a x}\right)^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c^2}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(8 - 5*a*x - 7*a^2*x^2 + 3*a^3*x^3))/(3*(-1 + a*x)^2*(1 + a*x)) + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^2)","A",0
778,1,99,254,0.274753,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^3} \, dx","Integrate[E^ArcCoth[a*x]/(c - c/(a^2*x^2))^3,x]","\frac{\log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(15 a^5 x^5-38 a^4 x^4-52 a^3 x^3+87 a^2 x^2+33 a x-48\right)}{15 (a x-1)^3 (a x+1)^2}}{a c^3}","\frac{x}{c^3 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{3/2}}+\frac{16 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \sqrt{\frac{1}{a x}+1}}+\frac{21 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \left(\frac{1}{a x}+1\right)^{3/2}}-\frac{34}{5 a c^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}}-\frac{29}{15 a c^3 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{3/2}}-\frac{6}{5 a c^3 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{3/2}}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c^3}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-48 + 33*a*x + 87*a^2*x^2 - 52*a^3*x^3 - 38*a^4*x^4 + 15*a^5*x^5))/(15*(-1 + a*x)^3*(1 + a*x)^2) + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^3)","A",0
779,1,115,328,0.3513657,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^4} \, dx","Integrate[E^ArcCoth[a*x]/(c - c/(a^2*x^2))^4,x]","\frac{\log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(105 a^7 x^7-281 a^6 x^6-559 a^5 x^5+965 a^4 x^4+715 a^3 x^3-1065 a^2 x^2-279 a x+384\right)}{105 (a x-1)^4 (a x+1)^3}}{a c^4}","\frac{x}{c^4 \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{5/2}}+\frac{128 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \sqrt{\frac{1}{a x}+1}}+\frac{163 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left(\frac{1}{a x}+1\right)^{3/2}}+\frac{262 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left(\frac{1}{a x}+1\right)^{5/2}}-\frac{269}{21 a c^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}}-\frac{62}{21 a c^4 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{5/2}}-\frac{11}{7 a c^4 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{5/2}}-\frac{8}{7 a c^4 \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{5/2}}+\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c^4}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(384 - 279*a*x - 1065*a^2*x^2 + 715*a^3*x^3 + 965*a^4*x^4 - 559*a^5*x^5 - 281*a^6*x^6 + 105*a^7*x^7))/(105*(-1 + a*x)^4*(1 + a*x)^3) + Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^4)","A",0
780,1,127,127,0.0519379,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^5 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^5,x]","-\frac{c^5}{9 a^{10} x^9}-\frac{c^5}{4 a^9 x^8}+\frac{3 c^5}{7 a^8 x^7}+\frac{4 c^5}{3 a^7 x^6}-\frac{2 c^5}{5 a^6 x^5}-\frac{3 c^5}{a^5 x^4}-\frac{2 c^5}{3 a^4 x^3}+\frac{4 c^5}{a^3 x^2}+\frac{3 c^5}{a^2 x}+\frac{2 c^5 \log (x)}{a}+c^5 x","-\frac{c^5}{9 a^{10} x^9}-\frac{c^5}{4 a^9 x^8}+\frac{3 c^5}{7 a^8 x^7}+\frac{4 c^5}{3 a^7 x^6}-\frac{2 c^5}{5 a^6 x^5}-\frac{3 c^5}{a^5 x^4}-\frac{2 c^5}{3 a^4 x^3}+\frac{4 c^5}{a^3 x^2}+\frac{3 c^5}{a^2 x}+\frac{2 c^5 \log (x)}{a}+c^5 x",1,"-1/9*c^5/(a^10*x^9) - c^5/(4*a^9*x^8) + (3*c^5)/(7*a^8*x^7) + (4*c^5)/(3*a^7*x^6) - (2*c^5)/(5*a^6*x^5) - (3*c^5)/(a^5*x^4) - (2*c^5)/(3*a^4*x^3) + (4*c^5)/(a^3*x^2) + (3*c^5)/(a^2*x) + c^5*x + (2*c^5*Log[x])/a","A",1
781,1,90,90,0.0358885,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^4 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^4,x]","\frac{c^4}{7 a^8 x^7}+\frac{c^4}{3 a^7 x^6}-\frac{2 c^4}{5 a^6 x^5}-\frac{3 c^4}{2 a^5 x^4}+\frac{3 c^4}{a^3 x^2}+\frac{2 c^4}{a^2 x}+\frac{2 c^4 \log (x)}{a}+c^4 x","\frac{c^4}{7 a^8 x^7}+\frac{c^4}{3 a^7 x^6}-\frac{2 c^4}{5 a^6 x^5}-\frac{3 c^4}{2 a^5 x^4}+\frac{3 c^4}{a^3 x^2}+\frac{2 c^4}{a^2 x}+\frac{2 c^4 \log (x)}{a}+c^4 x",1,"c^4/(7*a^8*x^7) + c^4/(3*a^7*x^6) - (2*c^4)/(5*a^6*x^5) - (3*c^4)/(2*a^5*x^4) + (3*c^4)/(a^3*x^2) + (2*c^4)/(a^2*x) + c^4*x + (2*c^4*Log[x])/a","A",1
782,1,76,76,0.0283428,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^3 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^3,x]","-\frac{c^3}{5 a^6 x^5}-\frac{c^3}{2 a^5 x^4}+\frac{c^3}{3 a^4 x^3}+\frac{2 c^3}{a^3 x^2}+\frac{c^3}{a^2 x}+\frac{2 c^3 \log (x)}{a}+c^3 x","-\frac{c^3}{5 a^6 x^5}-\frac{c^3}{2 a^5 x^4}+\frac{c^3}{3 a^4 x^3}+\frac{2 c^3}{a^3 x^2}+\frac{c^3}{a^2 x}+\frac{2 c^3 \log (x)}{a}+c^3 x",1,"-1/5*c^3/(a^6*x^5) - c^3/(2*a^5*x^4) + c^3/(3*a^4*x^3) + (2*c^3)/(a^3*x^2) + c^3/(a^2*x) + c^3*x + (2*c^3*Log[x])/a","A",1
783,1,39,39,0.0204061,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^2 \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^2,x]","\frac{c^2}{3 a^4 x^3}+\frac{c^2}{a^3 x^2}+\frac{2 c^2 \log (x)}{a}+c^2 x","\frac{c^2}{3 a^4 x^3}+\frac{c^2}{a^3 x^2}+\frac{2 c^2 \log (x)}{a}+c^2 x",1,"c^2/(3*a^4*x^3) + c^2/(a^3*x^2) + c^2*x + (2*c^2*Log[x])/a","A",1
784,1,21,21,0.0158622,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right) \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2)),x]","-\frac{c}{a^2 x}+\frac{2 c \log (x)}{a}+c x","-\frac{c}{a^2 x}+\frac{2 c \log (x)}{a}+c x",1,"-(c/(a^2*x)) + c*x + (2*c*Log[x])/a","A",1
785,1,28,36,0.0341053,"\int \frac{e^{2 \coth ^{-1}(a x)}}{c-\frac{c}{a^2 x^2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - c/(a^2*x^2)),x]","\frac{\frac{1}{a-a^2 x}+\frac{2 \log (1-a x)}{a}+x}{c}","\frac{1}{a c (1-a x)}+\frac{2 \log (1-a x)}{a c}+\frac{x}{c}",1,"(x + (a - a^2*x)^(-1) + (2*Log[1 - a*x])/a)/c","A",1
786,1,75,75,0.0527315,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^2} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - c/(a^2*x^2))^2,x]","\frac{7}{4 a c^2 (1-a x)}-\frac{1}{4 a c^2 (1-a x)^2}+\frac{17 \log (1-a x)}{8 a c^2}-\frac{\log (a x+1)}{8 a c^2}+\frac{x}{c^2}","\frac{7}{4 a c^2 (1-a x)}-\frac{1}{4 a c^2 (1-a x)^2}+\frac{17 \log (1-a x)}{8 a c^2}-\frac{\log (a x+1)}{8 a c^2}+\frac{x}{c^2}",1,"x/c^2 - 1/(4*a*c^2*(1 - a*x)^2) + 7/(4*a*c^2*(1 - a*x)) + (17*Log[1 - a*x])/(8*a*c^2) - Log[1 + a*x]/(8*a*c^2)","A",1
787,1,82,110,0.0804353,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^3} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - c/(a^2*x^2))^3,x]","\frac{\frac{2 \left(6 a^5 x^5-12 a^4 x^4-15 a^3 x^3+24 a^2 x^2+7 a x-11\right)}{(a x-1)^3 (a x+1)}+27 \log (1-a x)-3 \log (a x+1)}{12 a c^3}","\frac{39}{16 a c^3 (1-a x)}-\frac{1}{16 a c^3 (a x+1)}-\frac{5}{8 a c^3 (1-a x)^2}+\frac{1}{12 a c^3 (1-a x)^3}+\frac{9 \log (1-a x)}{4 a c^3}-\frac{\log (a x+1)}{4 a c^3}+\frac{x}{c^3}",1,"((2*(-11 + 7*a*x + 24*a^2*x^2 - 15*a^3*x^3 - 12*a^4*x^4 + 6*a^5*x^5))/((-1 + a*x)^3*(1 + a*x)) + 27*Log[1 - a*x] - 3*Log[1 + a*x])/(12*a*c^3)","A",1
788,1,98,145,0.1197134,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^4} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - c/(a^2*x^2))^4,x]","\frac{\frac{2 \left(192 a^7 x^7-384 a^6 x^6-819 a^5 x^5+1254 a^4 x^4+866 a^3 x^3-1258 a^2 x^2-275 a x+400\right)}{(a x-1)^4 (a x+1)^2}+909 \log (1-a x)-141 \log (a x+1)}{384 a c^4}","\frac{99}{32 a c^4 (1-a x)}-\frac{11}{64 a c^4 (a x+1)}-\frac{35}{32 a c^4 (1-a x)^2}+\frac{1}{64 a c^4 (a x+1)^2}+\frac{13}{48 a c^4 (1-a x)^3}-\frac{1}{32 a c^4 (1-a x)^4}+\frac{303 \log (1-a x)}{128 a c^4}-\frac{47 \log (a x+1)}{128 a c^4}+\frac{x}{c^4}",1,"((2*(400 - 275*a*x - 1258*a^2*x^2 + 866*a^3*x^3 + 1254*a^4*x^4 - 819*a^5*x^5 - 384*a^6*x^6 + 192*a^7*x^7))/((-1 + a*x)^4*(1 + a*x)^2) + 909*Log[1 - a*x] - 141*Log[1 + a*x])/(384*a*c^4)","A",1
789,1,126,343,0.2729168,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^4 \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^4,x]","\frac{c^4 \left(525 a^6 x^6 \sin ^{-1}\left(\frac{1}{a x}\right)+1680 a^6 x^6 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\sqrt{1-\frac{1}{a^2 x^2}} \left(560 a^7 x^7-2496 a^6 x^6-525 a^5 x^5+992 a^4 x^4+770 a^3 x^3-96 a^2 x^2-280 a x-80\right)\right)}{560 a^7 x^6}","\frac{8 c^4 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{11/2}}{7 a}+c^4 x \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{11/2}+\frac{15 c^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{11/2}}{14 a}-\frac{57 c^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{9/2}}{70 a}-\frac{303 c^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}}{280 a}-\frac{61 c^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}}{40 a}-\frac{37 c^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}}{16 a}-\frac{63 c^4 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{16 a}+\frac{15 c^4 \csc ^{-1}(a x)}{16 a}+\frac{3 c^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c^4*(Sqrt[1 - 1/(a^2*x^2)]*(-80 - 280*a*x - 96*a^2*x^2 + 770*a^3*x^3 + 992*a^4*x^4 - 525*a^5*x^5 - 2496*a^6*x^6 + 560*a^7*x^7) + 525*a^6*x^6*ArcSin[1/(a*x)] + 1680*a^6*x^6*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(560*a^7*x^6)","A",0
790,1,110,269,0.203005,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^3 \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^3,x]","\frac{c^3 \left(15 a^4 x^4 \sin ^{-1}\left(\frac{1}{a x}\right)+120 a^4 x^4 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\sqrt{1-\frac{1}{a^2 x^2}} \left(40 a^5 x^5-152 a^4 x^4-55 a^3 x^3+24 a^2 x^2+30 a x+8\right)\right)}{40 a^5 x^4}","c^3 x \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{9/2}+\frac{6 c^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{9/2}}{5 a}-\frac{21 c^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}}{20 a}-\frac{29 c^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}}{20 a}-\frac{17 c^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}}{8 a}-\frac{27 c^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{8 a}+\frac{3 c^3 \csc ^{-1}(a x)}{8 a}+\frac{3 c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c^3*(Sqrt[1 - 1/(a^2*x^2)]*(8 + 30*a*x + 24*a^2*x^2 - 55*a^3*x^3 - 152*a^4*x^4 + 40*a^5*x^5) + 15*a^4*x^4*ArcSin[1/(a*x)] + 120*a^4*x^4*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(40*a^5*x^4)","A",0
791,1,94,195,0.1459474,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^2 \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^2,x]","\frac{c^2 \left(18 a^2 x^2 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)-3 a^2 x^2 \sin ^{-1}\left(\frac{1}{a x}\right)+\sqrt{1-\frac{1}{a^2 x^2}} \left(6 a^3 x^3-16 a^2 x^2-9 a x-2\right)\right)}{6 a^3 x^2}","c^2 x \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}-\frac{4 c^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}}{3 a}-\frac{11 c^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}}{6 a}-\frac{5 c^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{2 a}-\frac{c^2 \csc ^{-1}(a x)}{2 a}+\frac{3 c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c^2*(Sqrt[1 - 1/(a^2*x^2)]*(-2 - 9*a*x - 16*a^2*x^2 + 6*a^3*x^3) - 3*a^2*x^2*ArcSin[1/(a*x)] + 18*a^2*x^2*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(6*a^3*x^2)","A",0
792,1,57,76,0.0985768,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right) \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2)),x]","\frac{c \left(\sqrt{1-\frac{1}{a^2 x^2}} (a x+1)+3 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)-3 \sin ^{-1}\left(\frac{1}{a x}\right)\right)}{a}","c x \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}-\frac{3 c \csc ^{-1}(a x)}{a}+\frac{3 c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c*(Sqrt[1 - 1/(a^2*x^2)]*(1 + a*x) - 3*ArcSin[1/(a*x)] + 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/a","A",0
793,1,69,144,0.1561295,"\int \frac{e^{3 \coth ^{-1}(a x)}}{c-\frac{c}{a^2 x^2}} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2)),x]","\frac{\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(3 a^2 x^2-19 a x+14\right)}{(a x-1)^2}+\frac{9 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a}}{3 c}","\frac{x \sqrt{\frac{1}{a x}+1}}{c \left(1-\frac{1}{a x}\right)^{3/2}}-\frac{14 \sqrt{\frac{1}{a x}+1}}{3 a c \sqrt{1-\frac{1}{a x}}}-\frac{5 \sqrt{\frac{1}{a x}+1}}{3 a c \left(1-\frac{1}{a x}\right)^{3/2}}+\frac{3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c}",1,"((Sqrt[1 - 1/(a^2*x^2)]*x*(14 - 19*a*x + 3*a^2*x^2))/(-1 + a*x)^2 + (9*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a)/(3*c)","A",0
794,1,78,181,0.1979504,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^2} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^2,x]","\frac{3 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(5 a^3 x^3-39 a^2 x^2+57 a x-24\right)}{5 (a x-1)^3}}{a c^2}","\frac{x \sqrt{\frac{1}{a x}+1}}{c^2 \left(1-\frac{1}{a x}\right)^{5/2}}-\frac{24 \sqrt{\frac{1}{a x}+1}}{5 a c^2 \sqrt{1-\frac{1}{a x}}}-\frac{9 \sqrt{\frac{1}{a x}+1}}{5 a c^2 \left(1-\frac{1}{a x}\right)^{3/2}}-\frac{6 \sqrt{\frac{1}{a x}+1}}{5 a c^2 \left(1-\frac{1}{a x}\right)^{5/2}}+\frac{3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c^2}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-24 + 57*a*x - 39*a^2*x^2 + 5*a^3*x^3))/(5*(-1 + a*x)^3) + 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^2)","A",0
795,1,101,255,0.2360799,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^3} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^3,x]","\frac{3 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(35 a^5 x^5-286 a^4 x^4+368 a^3 x^3+125 a^2 x^2-423 a x+176\right)}{35 (a x-1)^4 (a x+1)}}{a c^3}","\frac{x}{c^3 \left(1-\frac{1}{a x}\right)^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{176 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \sqrt{\frac{1}{a x}+1}}-\frac{281}{35 a c^3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{88}{35 a c^3 \left(1-\frac{1}{a x}\right)^{3/2} \sqrt{\frac{1}{a x}+1}}-\frac{53}{35 a c^3 \left(1-\frac{1}{a x}\right)^{5/2} \sqrt{\frac{1}{a x}+1}}-\frac{8}{7 a c^3 \left(1-\frac{1}{a x}\right)^{7/2} \sqrt{\frac{1}{a x}+1}}+\frac{3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c^3}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(176 - 423*a*x + 125*a^2*x^2 + 368*a^3*x^3 - 286*a^4*x^4 + 35*a^5*x^5))/(35*(-1 + a*x)^4*(1 + a*x)) + 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^3)","A",0
796,1,117,329,0.2996976,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^4} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^4,x]","\frac{3 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(315 a^7 x^7-2669 a^6 x^6+2967 a^5 x^5+4029 a^4 x^4-7399 a^3 x^3+339 a^2 x^2+4047 a x-1664\right)}{315 (a x-1)^5 (a x+1)^2}}{a c^4}","\frac{x}{c^4 \left(1-\frac{1}{a x}\right)^{9/2} \left(\frac{1}{a x}+1\right)^{3/2}}+\frac{1664 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \sqrt{\frac{1}{a x}+1}}+\frac{2609 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \left(\frac{1}{a x}+1\right)^{3/2}}-\frac{1462}{105 a c^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}}-\frac{1147}{315 a c^4 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{3/2}}-\frac{208}{105 a c^4 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{3/2}}-\frac{29}{21 a c^4 \left(1-\frac{1}{a x}\right)^{7/2} \left(\frac{1}{a x}+1\right)^{3/2}}-\frac{10}{9 a c^4 \left(1-\frac{1}{a x}\right)^{9/2} \left(\frac{1}{a x}+1\right)^{3/2}}+\frac{3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c^4}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-1664 + 4047*a*x + 339*a^2*x^2 - 7399*a^3*x^3 + 4029*a^4*x^4 + 2967*a^5*x^5 - 2669*a^6*x^6 + 315*a^7*x^7))/(315*(-1 + a*x)^5*(1 + a*x)^2) + 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^4)","A",0
797,1,116,116,0.0466899,"\int e^{4 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^5 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - c/(a^2*x^2))^5,x]","\frac{c^5}{9 a^{10} x^9}+\frac{c^5}{2 a^9 x^8}+\frac{3 c^5}{7 a^8 x^7}-\frac{4 c^5}{3 a^7 x^6}-\frac{14 c^5}{5 a^6 x^5}+\frac{14 c^5}{3 a^4 x^3}+\frac{4 c^5}{a^3 x^2}-\frac{3 c^5}{a^2 x}+\frac{4 c^5 \log (x)}{a}+c^5 x","\frac{c^5}{9 a^{10} x^9}+\frac{c^5}{2 a^9 x^8}+\frac{3 c^5}{7 a^8 x^7}-\frac{4 c^5}{3 a^7 x^6}-\frac{14 c^5}{5 a^6 x^5}+\frac{14 c^5}{3 a^4 x^3}+\frac{4 c^5}{a^3 x^2}-\frac{3 c^5}{a^2 x}+\frac{4 c^5 \log (x)}{a}+c^5 x",1,"c^5/(9*a^10*x^9) + c^5/(2*a^9*x^8) + (3*c^5)/(7*a^8*x^7) - (4*c^5)/(3*a^7*x^6) - (14*c^5)/(5*a^6*x^5) + (14*c^5)/(3*a^4*x^3) + (4*c^5)/(a^3*x^2) - (3*c^5)/(a^2*x) + c^5*x + (4*c^5*Log[x])/a","A",1
798,1,100,100,0.0359584,"\int e^{4 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^4 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - c/(a^2*x^2))^4,x]","-\frac{c^4}{7 a^8 x^7}-\frac{2 c^4}{3 a^7 x^6}-\frac{4 c^4}{5 a^6 x^5}+\frac{c^4}{a^5 x^4}+\frac{10 c^4}{3 a^4 x^3}+\frac{2 c^4}{a^3 x^2}-\frac{4 c^4}{a^2 x}+\frac{4 c^4 \log (x)}{a}+c^4 x","-\frac{c^4}{7 a^8 x^7}-\frac{2 c^4}{3 a^7 x^6}-\frac{4 c^4}{5 a^6 x^5}+\frac{c^4}{a^5 x^4}+\frac{10 c^4}{3 a^4 x^3}+\frac{2 c^4}{a^3 x^2}-\frac{4 c^4}{a^2 x}+\frac{4 c^4 \log (x)}{a}+c^4 x",1,"-1/7*c^4/(a^8*x^7) - (2*c^4)/(3*a^7*x^6) - (4*c^4)/(5*a^6*x^5) + c^4/(a^5*x^4) + (10*c^4)/(3*a^4*x^3) + (2*c^4)/(a^3*x^2) - (4*c^4)/(a^2*x) + c^4*x + (4*c^4*Log[x])/a","A",1
799,1,63,63,0.0267572,"\int e^{4 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^3 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - c/(a^2*x^2))^3,x]","\frac{c^3}{5 a^6 x^5}+\frac{c^3}{a^5 x^4}+\frac{5 c^3}{3 a^4 x^3}-\frac{5 c^3}{a^2 x}+\frac{4 c^3 \log (x)}{a}+c^3 x","\frac{c^3}{5 a^6 x^5}+\frac{c^3}{a^5 x^4}+\frac{5 c^3}{3 a^4 x^3}-\frac{5 c^3}{a^2 x}+\frac{4 c^3 \log (x)}{a}+c^3 x",1,"c^3/(5*a^6*x^5) + c^3/(a^5*x^4) + (5*c^3)/(3*a^4*x^3) - (5*c^3)/(a^2*x) + c^3*x + (4*c^3*Log[x])/a","A",1
800,1,51,51,0.0235414,"\int e^{4 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^2 \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - c/(a^2*x^2))^2,x]","-\frac{c^2}{3 a^4 x^3}-\frac{2 c^2}{a^3 x^2}-\frac{6 c^2}{a^2 x}+\frac{4 c^2 \log (x)}{a}+c^2 x","-\frac{c^2}{3 a^4 x^3}-\frac{2 c^2}{a^3 x^2}-\frac{6 c^2}{a^2 x}+\frac{4 c^2 \log (x)}{a}+c^2 x",1,"-1/3*c^2/(a^4*x^3) - (2*c^2)/(a^3*x^2) - (6*c^2)/(a^2*x) + c^2*x + (4*c^2*Log[x])/a","A",1
801,1,33,33,0.0232699,"\int e^{4 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right) \, dx","Integrate[E^(4*ArcCoth[a*x])*(c - c/(a^2*x^2)),x]","\frac{c}{a^2 x}-\frac{4 c \log (x)}{a}+\frac{8 c \log (1-a x)}{a}+c x","\frac{c}{a^2 x}-\frac{4 c \log (x)}{a}+\frac{8 c \log (1-a x)}{a}+c x",1,"c/(a^2*x) + c*x - (4*c*Log[x])/a + (8*c*Log[1 - a*x])/a","A",1
802,1,53,53,0.0370074,"\int \frac{e^{4 \coth ^{-1}(a x)}}{c-\frac{c}{a^2 x^2}} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - c/(a^2*x^2)),x]","\frac{5}{a c (1-a x)}-\frac{1}{a c (1-a x)^2}+\frac{4 \log (1-a x)}{a c}+\frac{x}{c}","\frac{5}{a c (1-a x)}-\frac{1}{a c (1-a x)^2}+\frac{4 \log (1-a x)}{a c}+\frac{x}{c}",1,"x/c - 1/(a*c*(1 - a*x)^2) + 5/(a*c*(1 - a*x)) + (4*Log[1 - a*x])/(a*c)","A",1
803,1,63,71,0.0418901,"\int \frac{e^{4 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^2} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - c/(a^2*x^2))^2,x]","\frac{3 a^4 x^4-9 a^3 x^3-9 a^2 x^2+27 a x+12 (a x-1)^3 \log (1-a x)-13}{3 a c^2 (a x-1)^3}","\frac{6}{a c^2 (1-a x)}-\frac{2}{a c^2 (1-a x)^2}+\frac{1}{3 a c^2 (1-a x)^3}+\frac{4 \log (1-a x)}{a c^2}+\frac{x}{c^2}",1,"(-13 + 27*a*x - 9*a^2*x^2 - 9*a^3*x^3 + 3*a^4*x^4 + 12*(-1 + a*x)^3*Log[1 - a*x])/(3*a*c^2*(-1 + a*x)^3)","A",1
804,1,89,111,0.0713089,"\int \frac{e^{4 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^3} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - c/(a^2*x^2))^3,x]","\frac{2 \left(48 a^5 x^5-192 a^4 x^4-45 a^3 x^3+660 a^2 x^2-701 a x+224\right)+387 (a x-1)^4 \log (1-a x)-3 (a x-1)^4 \log (a x+1)}{96 a c^3 (a x-1)^4}","\frac{111}{16 a c^3 (1-a x)}-\frac{49}{16 a c^3 (1-a x)^2}+\frac{11}{12 a c^3 (1-a x)^3}-\frac{1}{8 a c^3 (1-a x)^4}+\frac{129 \log (1-a x)}{32 a c^3}-\frac{\log (a x+1)}{32 a c^3}+\frac{x}{c^3}",1,"(2*(224 - 701*a*x + 660*a^2*x^2 - 45*a^3*x^3 - 192*a^4*x^4 + 48*a^5*x^5) + 387*(-1 + a*x)^4*Log[1 - a*x] - 3*(-1 + a*x)^4*Log[1 + a*x])/(96*a*c^3*(-1 + a*x)^4)","A",1
805,1,98,146,0.1071227,"\int \frac{e^{4 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^4} \, dx","Integrate[E^(4*ArcCoth[a*x])/(c - c/(a^2*x^2))^4,x]","\frac{\frac{2 \left(480 a^7 x^7-1920 a^6 x^6-1365 a^5 x^5+9300 a^4 x^4-6800 a^3 x^3-4900 a^2 x^2+7541 a x-2384\right)}{(a x-1)^5 (a x+1)}+3915 \log (1-a x)-75 \log (a x+1)}{960 a c^4}","\frac{501}{64 a c^4 (1-a x)}-\frac{1}{64 a c^4 (a x+1)}-\frac{67}{16 a c^4 (1-a x)^2}+\frac{83}{48 a c^4 (1-a x)^3}-\frac{7}{16 a c^4 (1-a x)^4}+\frac{1}{20 a c^4 (1-a x)^5}+\frac{261 \log (1-a x)}{64 a c^4}-\frac{5 \log (a x+1)}{64 a c^4}+\frac{x}{c^4}",1,"((2*(-2384 + 7541*a*x - 4900*a^2*x^2 - 6800*a^3*x^3 + 9300*a^4*x^4 - 1365*a^5*x^5 - 1920*a^6*x^6 + 480*a^7*x^7))/((-1 + a*x)^5*(1 + a*x)) + 3915*Log[1 - a*x] - 75*Log[1 + a*x])/(960*a*c^4)","A",1
806,1,120,343,0.3081933,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^4 \, dx","Integrate[(c - c/(a^2*x^2))^4/E^ArcCoth[a*x],x]","\frac{c^4 \left(3675 a^6 \sin ^{-1}\left(\frac{1}{a x}\right)-1680 a^6 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left(1680 a^7 x^7+2816 a^6 x^6+3045 a^5 x^5-1952 a^4 x^4-1330 a^3 x^3+1056 a^2 x^2+280 a x-240\right)}{x^6}\right)}{1680 a^7}","c^4 x \left(\frac{1}{a x}+1\right)^{7/2} \left(1-\frac{1}{a x}\right)^{9/2}+\frac{8 c^4 \left(\frac{1}{a x}+1\right)^{7/2} \left(1-\frac{1}{a x}\right)^{7/2}}{7 a}+\frac{7 c^4 \left(\frac{1}{a x}+1\right)^{7/2} \left(1-\frac{1}{a x}\right)^{5/2}}{6 a}+\frac{29 c^4 \left(\frac{1}{a x}+1\right)^{7/2} \left(1-\frac{1}{a x}\right)^{3/2}}{30 a}+\frac{19 c^4 \left(\frac{1}{a x}+1\right)^{7/2} \sqrt{1-\frac{1}{a x}}}{40 a}+\frac{7 c^4 \left(\frac{1}{a x}+1\right)^{5/2} \sqrt{1-\frac{1}{a x}}}{40 a}-\frac{c^4 \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{1-\frac{1}{a x}}}{16 a}-\frac{19 c^4 \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{16 a}+\frac{35 c^4 \csc ^{-1}(a x)}{16 a}-\frac{c^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c^4*((Sqrt[1 - 1/(a^2*x^2)]*(-240 + 280*a*x + 1056*a^2*x^2 - 1330*a^3*x^3 - 1952*a^4*x^4 + 3045*a^5*x^5 + 2816*a^6*x^6 + 1680*a^7*x^7))/x^6 + 3675*a^6*ArcSin[1/(a*x)] - 1680*a^6*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(1680*a^7)","A",0
807,1,110,269,0.2087792,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^3 \, dx","Integrate[(c - c/(a^2*x^2))^3/E^ArcCoth[a*x],x]","\frac{c^3 \left(225 a^4 x^4 \sin ^{-1}\left(\frac{1}{a x}\right)-120 a^4 x^4 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\sqrt{1-\frac{1}{a^2 x^2}} \left(120 a^5 x^5+184 a^4 x^4+135 a^3 x^3-88 a^2 x^2-30 a x+24\right)\right)}{120 a^5 x^4}","c^3 x \left(\frac{1}{a x}+1\right)^{5/2} \left(1-\frac{1}{a x}\right)^{7/2}+\frac{6 c^3 \left(\frac{1}{a x}+1\right)^{5/2} \left(1-\frac{1}{a x}\right)^{5/2}}{5 a}+\frac{5 c^3 \left(\frac{1}{a x}+1\right)^{5/2} \left(1-\frac{1}{a x}\right)^{3/2}}{4 a}+\frac{11 c^3 \left(\frac{1}{a x}+1\right)^{5/2} \sqrt{1-\frac{1}{a x}}}{12 a}+\frac{c^3 \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{1-\frac{1}{a x}}}{24 a}-\frac{7 c^3 \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{8 a}+\frac{15 c^3 \csc ^{-1}(a x)}{8 a}-\frac{c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c^3*(Sqrt[1 - 1/(a^2*x^2)]*(24 - 30*a*x - 88*a^2*x^2 + 135*a^3*x^3 + 184*a^4*x^4 + 120*a^5*x^5) + 225*a^4*x^4*ArcSin[1/(a*x)] - 120*a^4*x^4*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(120*a^5*x^4)","A",0
808,1,94,195,0.1533329,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^2 \, dx","Integrate[(c - c/(a^2*x^2))^2/E^ArcCoth[a*x],x]","\frac{c^2 \left(-6 a^2 x^2 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+9 a^2 x^2 \sin ^{-1}\left(\frac{1}{a x}\right)+\sqrt{1-\frac{1}{a^2 x^2}} \left(6 a^3 x^3+8 a^2 x^2+3 a x-2\right)\right)}{6 a^3 x^2}","c^2 x \left(\frac{1}{a x}+1\right)^{3/2} \left(1-\frac{1}{a x}\right)^{5/2}+\frac{4 c^2 \left(\frac{1}{a x}+1\right)^{3/2} \left(1-\frac{1}{a x}\right)^{3/2}}{3 a}+\frac{3 c^2 \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{1-\frac{1}{a x}}}{2 a}-\frac{c^2 \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{2 a}+\frac{3 c^2 \csc ^{-1}(a x)}{2 a}-\frac{c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c^2*(Sqrt[1 - 1/(a^2*x^2)]*(-2 + 3*a*x + 8*a^2*x^2 + 6*a^3*x^3) + 9*a^2*x^2*ArcSin[1/(a*x)] - 6*a^2*x^2*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(6*a^3*x^2)","A",0
809,1,55,108,0.0906676,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right) \, dx","Integrate[(c - c/(a^2*x^2))/E^ArcCoth[a*x],x]","\frac{c \left(\sqrt{1-\frac{1}{a^2 x^2}} (a x+1)-\log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\sin ^{-1}\left(\frac{1}{a x}\right)\right)}{a}","c x \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{3/2}+\frac{2 c \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{a}+\frac{c \csc ^{-1}(a x)}{a}-\frac{c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c*(Sqrt[1 - 1/(a^2*x^2)]*(1 + a*x) + ArcSin[1/(a*x)] - Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/a","A",0
810,1,57,105,0.1208924,"\int \frac{e^{-\coth ^{-1}(a x)}}{c-\frac{c}{a^2 x^2}} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))),x]","\frac{\frac{x \sqrt{1-\frac{1}{a^2 x^2}} (a x+2)}{a x+1}-\frac{\log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a}}{c}","\frac{x \sqrt{1-\frac{1}{a x}}}{c \sqrt{\frac{1}{a x}+1}}+\frac{2 \sqrt{1-\frac{1}{a x}}}{a c \sqrt{\frac{1}{a x}+1}}-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c}",1,"((Sqrt[1 - 1/(a^2*x^2)]*x*(2 + a*x))/(1 + a*x) - Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]/a)/c","A",0
811,1,85,179,0.1833332,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^2} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))^2),x]","\frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(3 a^3 x^3+7 a^2 x^2-5 a x-8\right)}{3 (a x-1) (a x+1)^2}-\log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a c^2}","\frac{x}{c^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{\frac{1}{a x}+1}}+\frac{5 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \left(\frac{1}{a x}+1\right)^{3/2}}-\frac{2}{a c^2 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{3/2}}-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c^2}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-8 - 5*a*x + 7*a^2*x^2 + 3*a^3*x^3))/(3*(-1 + a*x)*(1 + a*x)^2) - Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^2)","A",0
812,1,101,255,0.2317673,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^3} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))^3),x]","\frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(15 a^5 x^5+38 a^4 x^4-52 a^3 x^3-87 a^2 x^2+33 a x+48\right)}{15 (a x-1)^2 (a x+1)^3}-\log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a c^3}","\frac{x}{c^3 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{5/2}}+\frac{16 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \sqrt{\frac{1}{a x}+1}}+\frac{11 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \left(\frac{1}{a x}+1\right)^{3/2}}+\frac{14 \sqrt{1-\frac{1}{a x}}}{5 a c^3 \left(\frac{1}{a x}+1\right)^{5/2}}-\frac{13}{3 a c^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{5/2}}-\frac{4}{3 a c^3 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{5/2}}-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c^3}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(48 + 33*a*x - 87*a^2*x^2 - 52*a^3*x^3 + 38*a^4*x^4 + 15*a^5*x^5))/(15*(-1 + a*x)^2*(1 + a*x)^3) - Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^3)","A",0
813,1,117,329,0.3190048,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^4} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))^4),x]","\frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(105 a^7 x^7+281 a^6 x^6-559 a^5 x^5-965 a^4 x^4+715 a^3 x^3+1065 a^2 x^2-279 a x-384\right)}{105 (a x-1)^3 (a x+1)^4}-\log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a c^4}","\frac{x}{c^4 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{7/2}}+\frac{128 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \sqrt{\frac{1}{a x}+1}}+\frac{93 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left(\frac{1}{a x}+1\right)^{3/2}}+\frac{122 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left(\frac{1}{a x}+1\right)^{5/2}}+\frac{115 \sqrt{1-\frac{1}{a x}}}{21 a c^4 \left(\frac{1}{a x}+1\right)^{7/2}}-\frac{28}{3 a c^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}}-\frac{31}{15 a c^4 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{7/2}}-\frac{6}{5 a c^4 \left(1-\frac{1}{a x}\right)^{5/2} \left(\frac{1}{a x}+1\right)^{7/2}}-\frac{\tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c^4}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-384 - 279*a*x + 1065*a^2*x^2 + 715*a^3*x^3 - 965*a^4*x^4 - 559*a^5*x^5 + 281*a^6*x^6 + 105*a^7*x^7))/(105*(-1 + a*x)^3*(1 + a*x)^4) - Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^4)","A",0
814,1,90,90,0.0421792,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^4 \, dx","Integrate[(c - c/(a^2*x^2))^4/E^(2*ArcCoth[a*x]),x]","\frac{c^4}{7 a^8 x^7}-\frac{c^4}{3 a^7 x^6}-\frac{2 c^4}{5 a^6 x^5}+\frac{3 c^4}{2 a^5 x^4}-\frac{3 c^4}{a^3 x^2}+\frac{2 c^4}{a^2 x}-\frac{2 c^4 \log (x)}{a}+c^4 x","\frac{c^4}{7 a^8 x^7}-\frac{c^4}{3 a^7 x^6}-\frac{2 c^4}{5 a^6 x^5}+\frac{3 c^4}{2 a^5 x^4}-\frac{3 c^4}{a^3 x^2}+\frac{2 c^4}{a^2 x}-\frac{2 c^4 \log (x)}{a}+c^4 x",1,"c^4/(7*a^8*x^7) - c^4/(3*a^7*x^6) - (2*c^4)/(5*a^6*x^5) + (3*c^4)/(2*a^5*x^4) - (3*c^4)/(a^3*x^2) + (2*c^4)/(a^2*x) + c^4*x - (2*c^4*Log[x])/a","A",1
815,1,76,76,0.0325624,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^3 \, dx","Integrate[(c - c/(a^2*x^2))^3/E^(2*ArcCoth[a*x]),x]","-\frac{c^3}{5 a^6 x^5}+\frac{c^3}{2 a^5 x^4}+\frac{c^3}{3 a^4 x^3}-\frac{2 c^3}{a^3 x^2}+\frac{c^3}{a^2 x}-\frac{2 c^3 \log (x)}{a}+c^3 x","-\frac{c^3}{5 a^6 x^5}+\frac{c^3}{2 a^5 x^4}+\frac{c^3}{3 a^4 x^3}-\frac{2 c^3}{a^3 x^2}+\frac{c^3}{a^2 x}-\frac{2 c^3 \log (x)}{a}+c^3 x",1,"-1/5*c^3/(a^6*x^5) + c^3/(2*a^5*x^4) + c^3/(3*a^4*x^3) - (2*c^3)/(a^3*x^2) + c^3/(a^2*x) + c^3*x - (2*c^3*Log[x])/a","A",1
816,1,40,40,0.0269507,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^2 \, dx","Integrate[(c - c/(a^2*x^2))^2/E^(2*ArcCoth[a*x]),x]","\frac{c^2}{3 a^4 x^3}-\frac{c^2}{a^3 x^2}-\frac{2 c^2 \log (x)}{a}+c^2 x","\frac{c^2}{3 a^4 x^3}-\frac{c^2}{a^3 x^2}-\frac{2 c^2 \log (x)}{a}+c^2 x",1,"c^2/(3*a^4*x^3) - c^2/(a^3*x^2) + c^2*x - (2*c^2*Log[x])/a","A",1
817,1,21,21,0.0179969,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right) \, dx","Integrate[(c - c/(a^2*x^2))/E^(2*ArcCoth[a*x]),x]","-\frac{c}{a^2 x}-\frac{2 c \log (x)}{a}+c x","-\frac{c}{a^2 x}-\frac{2 c \log (x)}{a}+c x",1,"-(c/(a^2*x)) + c*x - (2*c*Log[x])/a","A",1
818,1,28,35,0.0398729,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{c-\frac{c}{a^2 x^2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))),x]","\frac{-\frac{1}{a^2 x+a}-\frac{2 \log (a x+1)}{a}+x}{c}","-\frac{1}{a c (a x+1)}-\frac{2 \log (a x+1)}{a c}+\frac{x}{c}",1,"(x - (a + a^2*x)^(-1) - (2*Log[1 + a*x])/a)/c","A",1
819,1,70,73,0.0600394,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^2} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^2),x]","\frac{2 \left(4 a^3 x^3+8 a^2 x^2-3 a x-6\right)+(a x+1)^2 \log (1-a x)-17 (a x+1)^2 \log (a x+1)}{8 a (a c x+c)^2}","-\frac{7}{4 a c^2 (a x+1)}+\frac{1}{4 a c^2 (a x+1)^2}+\frac{\log (1-a x)}{8 a c^2}-\frac{17 \log (a x+1)}{8 a c^2}+\frac{x}{c^2}",1,"(2*(-6 - 3*a*x + 8*a^2*x^2 + 4*a^3*x^3) + (1 + a*x)^2*Log[1 - a*x] - 17*(1 + a*x)^2*Log[1 + a*x])/(8*a*(c + a*c*x)^2)","A",1
820,1,104,108,0.1008895,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^3} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^3),x]","\frac{2 \left(6 a^5 x^5+12 a^4 x^4-15 a^3 x^3-24 a^2 x^2+7 a x+11\right)+3 (a x-1) (a x+1)^3 \log (1-a x)-27 (a x-1) (a x+1)^3 \log (a x+1)}{12 a (a x-1) (a c x+c)^3}","\frac{1}{16 a c^3 (1-a x)}-\frac{39}{16 a c^3 (a x+1)}+\frac{5}{8 a c^3 (a x+1)^2}-\frac{1}{12 a c^3 (a x+1)^3}+\frac{\log (1-a x)}{4 a c^3}-\frac{9 \log (a x+1)}{4 a c^3}+\frac{x}{c^3}",1,"(2*(11 + 7*a*x - 24*a^2*x^2 - 15*a^3*x^3 + 12*a^4*x^4 + 6*a^5*x^5) + 3*(-1 + a*x)*(1 + a*x)^3*Log[1 - a*x] - 27*(-1 + a*x)*(1 + a*x)^3*Log[1 + a*x])/(12*a*(-1 + a*x)*(c + a*c*x)^3)","A",1
821,1,124,143,0.1306188,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^4} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^4),x]","\frac{2 \left(192 a^7 x^7+384 a^6 x^6-819 a^5 x^5-1254 a^4 x^4+866 a^3 x^3+1258 a^2 x^2-275 a x-400\right)+141 (a x-1)^2 (a x+1)^4 \log (1-a x)-909 (a x-1)^2 (a x+1)^4 \log (a x+1)}{384 a (a x-1)^2 (a c x+c)^4}","\frac{11}{64 a c^4 (1-a x)}-\frac{99}{32 a c^4 (a x+1)}-\frac{1}{64 a c^4 (1-a x)^2}+\frac{35}{32 a c^4 (a x+1)^2}-\frac{13}{48 a c^4 (a x+1)^3}+\frac{1}{32 a c^4 (a x+1)^4}+\frac{47 \log (1-a x)}{128 a c^4}-\frac{303 \log (a x+1)}{128 a c^4}+\frac{x}{c^4}",1,"(2*(-400 - 275*a*x + 1258*a^2*x^2 + 866*a^3*x^3 - 1254*a^4*x^4 - 819*a^5*x^5 + 384*a^6*x^6 + 192*a^7*x^7) + 141*(-1 + a*x)^2*(1 + a*x)^4*Log[1 - a*x] - 909*(-1 + a*x)^2*(1 + a*x)^4*Log[1 + a*x])/(384*a*(-1 + a*x)^2*(c + a*c*x)^4)","A",1
822,1,126,343,0.3053923,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^4 \, dx","Integrate[(c - c/(a^2*x^2))^4/E^(3*ArcCoth[a*x]),x]","\frac{c^4 \left(525 a^6 x^6 \sin ^{-1}\left(\frac{1}{a x}\right)-1680 a^6 x^6 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\sqrt{1-\frac{1}{a^2 x^2}} \left(560 a^7 x^7+2496 a^6 x^6-525 a^5 x^5-992 a^4 x^4+770 a^3 x^3+96 a^2 x^2-280 a x+80\right)\right)}{560 a^7 x^6}","c^4 x \left(\frac{1}{a x}+1\right)^{5/2} \left(1-\frac{1}{a x}\right)^{11/2}+\frac{8 c^4 \left(\frac{1}{a x}+1\right)^{5/2} \left(1-\frac{1}{a x}\right)^{9/2}}{7 a}+\frac{17 c^4 \left(\frac{1}{a x}+1\right)^{5/2} \left(1-\frac{1}{a x}\right)^{7/2}}{14 a}+\frac{11 c^4 \left(\frac{1}{a x}+1\right)^{5/2} \left(1-\frac{1}{a x}\right)^{5/2}}{10 a}+\frac{5 c^4 \left(\frac{1}{a x}+1\right)^{5/2} \left(1-\frac{1}{a x}\right)^{3/2}}{8 a}-\frac{3 c^4 \left(\frac{1}{a x}+1\right)^{5/2} \sqrt{1-\frac{1}{a x}}}{8 a}+\frac{27 c^4 \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{1-\frac{1}{a x}}}{16 a}+\frac{33 c^4 \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{16 a}+\frac{15 c^4 \csc ^{-1}(a x)}{16 a}-\frac{3 c^4 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c^4*(Sqrt[1 - 1/(a^2*x^2)]*(80 - 280*a*x + 96*a^2*x^2 + 770*a^3*x^3 - 992*a^4*x^4 - 525*a^5*x^5 + 2496*a^6*x^6 + 560*a^7*x^7) + 525*a^6*x^6*ArcSin[1/(a*x)] - 1680*a^6*x^6*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(560*a^7*x^6)","A",0
823,1,110,269,0.2524854,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^3 \, dx","Integrate[(c - c/(a^2*x^2))^3/E^(3*ArcCoth[a*x]),x]","\frac{c^3 \left(15 a^4 x^4 \sin ^{-1}\left(\frac{1}{a x}\right)-120 a^4 x^4 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)+\sqrt{1-\frac{1}{a^2 x^2}} \left(40 a^5 x^5+152 a^4 x^4-55 a^3 x^3-24 a^2 x^2+30 a x-8\right)\right)}{40 a^5 x^4}","c^3 x \left(\frac{1}{a x}+1\right)^{3/2} \left(1-\frac{1}{a x}\right)^{9/2}+\frac{6 c^3 \left(\frac{1}{a x}+1\right)^{3/2} \left(1-\frac{1}{a x}\right)^{7/2}}{5 a}+\frac{27 c^3 \left(\frac{1}{a x}+1\right)^{3/2} \left(1-\frac{1}{a x}\right)^{5/2}}{20 a}+\frac{5 c^3 \left(\frac{1}{a x}+1\right)^{3/2} \left(1-\frac{1}{a x}\right)^{3/2}}{4 a}+\frac{3 c^3 \left(\frac{1}{a x}+1\right)^{3/2} \sqrt{1-\frac{1}{a x}}}{8 a}+\frac{21 c^3 \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{8 a}+\frac{3 c^3 \csc ^{-1}(a x)}{8 a}-\frac{3 c^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c^3*(Sqrt[1 - 1/(a^2*x^2)]*(-8 + 30*a*x - 24*a^2*x^2 - 55*a^3*x^3 + 152*a^4*x^4 + 40*a^5*x^5) + 15*a^4*x^4*ArcSin[1/(a*x)] - 120*a^4*x^4*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(40*a^5*x^4)","A",0
824,1,94,195,0.1717878,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^2 \, dx","Integrate[(c - c/(a^2*x^2))^2/E^(3*ArcCoth[a*x]),x]","\frac{c^2 \left(-18 a^2 x^2 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)-3 a^2 x^2 \sin ^{-1}\left(\frac{1}{a x}\right)+\sqrt{1-\frac{1}{a^2 x^2}} \left(6 a^3 x^3+16 a^2 x^2-9 a x+2\right)\right)}{6 a^3 x^2}","c^2 x \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{7/2}+\frac{4 c^2 \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{5/2}}{3 a}+\frac{11 c^2 \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{3/2}}{6 a}+\frac{5 c^2 \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{2 a}-\frac{c^2 \csc ^{-1}(a x)}{2 a}-\frac{3 c^2 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c^2*(Sqrt[1 - 1/(a^2*x^2)]*(2 - 9*a*x + 16*a^2*x^2 + 6*a^3*x^3) - 3*a^2*x^2*ArcSin[1/(a*x)] - 18*a^2*x^2*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(6*a^3*x^2)","A",0
825,1,57,76,0.1025493,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right) \, dx","Integrate[(c - c/(a^2*x^2))/E^(3*ArcCoth[a*x]),x]","\frac{c \left(\sqrt{1-\frac{1}{a^2 x^2}} (a x-1)-3 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)-3 \sin ^{-1}\left(\frac{1}{a x}\right)\right)}{a}","c x \sqrt{\frac{1}{a x}+1} \left(1-\frac{1}{a x}\right)^{3/2}-\frac{3 c \csc ^{-1}(a x)}{a}-\frac{3 c \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a}",1,"(c*(Sqrt[1 - 1/(a^2*x^2)]*(-1 + a*x) - 3*ArcSin[1/(a*x)] - 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/a","A",0
826,1,69,144,0.1493329,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{c-\frac{c}{a^2 x^2}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))),x]","\frac{\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(3 a^2 x^2+19 a x+14\right)}{(a x+1)^2}-\frac{9 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a}}{3 c}","\frac{x \sqrt{1-\frac{1}{a x}}}{c \left(\frac{1}{a x}+1\right)^{3/2}}+\frac{14 \sqrt{1-\frac{1}{a x}}}{3 a c \sqrt{\frac{1}{a x}+1}}+\frac{5 \sqrt{1-\frac{1}{a x}}}{3 a c \left(\frac{1}{a x}+1\right)^{3/2}}-\frac{3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c}",1,"((Sqrt[1 - 1/(a^2*x^2)]*x*(14 + 19*a*x + 3*a^2*x^2))/(1 + a*x)^2 - (9*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a)/(3*c)","A",0
827,1,78,181,0.1813997,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^2} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^2),x]","\frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(5 a^3 x^3+39 a^2 x^2+57 a x+24\right)}{5 (a x+1)^3}-3 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a c^2}","\frac{x \sqrt{1-\frac{1}{a x}}}{c^2 \left(\frac{1}{a x}+1\right)^{5/2}}+\frac{24 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \sqrt{\frac{1}{a x}+1}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left(\frac{1}{a x}+1\right)^{3/2}}+\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left(\frac{1}{a x}+1\right)^{5/2}}-\frac{3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c^2}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(24 + 57*a*x + 39*a^2*x^2 + 5*a^3*x^3))/(5*(1 + a*x)^3) - 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^2)","A",0
828,1,101,253,0.2428249,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^3} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^3),x]","\frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(35 a^5 x^5+286 a^4 x^4+368 a^3 x^3-125 a^2 x^2-423 a x-176\right)}{35 (a x-1) (a x+1)^4}-3 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a c^3}","\frac{x}{c^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}}+\frac{176 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \sqrt{\frac{1}{a x}+1}}+\frac{71 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left(\frac{1}{a x}+1\right)^{3/2}}+\frac{54 \sqrt{1-\frac{1}{a x}}}{35 a c^3 \left(\frac{1}{a x}+1\right)^{5/2}}+\frac{11 \sqrt{1-\frac{1}{a x}}}{7 a c^3 \left(\frac{1}{a x}+1\right)^{7/2}}-\frac{2}{a c^3 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{7/2}}-\frac{3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c^3}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(-176 - 423*a*x - 125*a^2*x^2 + 368*a^3*x^3 + 286*a^4*x^4 + 35*a^5*x^5))/(35*(-1 + a*x)*(1 + a*x)^4) - 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^3)","A",0
829,1,117,327,0.2943569,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^4} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^4),x]","\frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left(315 a^7 x^7+2669 a^6 x^6+2967 a^5 x^5-4029 a^4 x^4-7399 a^3 x^3-339 a^2 x^2+4047 a x+1664\right)}{315 (a x-1)^2 (a x+1)^5}-3 \log \left(x \left(\sqrt{1-\frac{1}{a^2 x^2}}+1\right)\right)}{a c^4}","\frac{x}{c^4 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{9/2}}+\frac{1664 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \sqrt{\frac{1}{a x}+1}}+\frac{719 \sqrt{1-\frac{1}{a x}}}{315 a c^4 \left(\frac{1}{a x}+1\right)^{3/2}}+\frac{202 \sqrt{1-\frac{1}{a x}}}{105 a c^4 \left(\frac{1}{a x}+1\right)^{5/2}}+\frac{139 \sqrt{1-\frac{1}{a x}}}{63 a c^4 \left(\frac{1}{a x}+1\right)^{7/2}}+\frac{28 \sqrt{1-\frac{1}{a x}}}{9 a c^4 \left(\frac{1}{a x}+1\right)^{9/2}}-\frac{5}{a c^4 \sqrt{1-\frac{1}{a x}} \left(\frac{1}{a x}+1\right)^{9/2}}-\frac{4}{3 a c^4 \left(1-\frac{1}{a x}\right)^{3/2} \left(\frac{1}{a x}+1\right)^{9/2}}-\frac{3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right)}{a c^4}",1,"((a*Sqrt[1 - 1/(a^2*x^2)]*x*(1664 + 4047*a*x - 339*a^2*x^2 - 7399*a^3*x^3 - 4029*a^4*x^4 + 2967*a^5*x^5 + 2669*a^6*x^6 + 315*a^7*x^7))/(315*(-1 + a*x)^2*(1 + a*x)^5) - 3*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/(a*c^4)","A",0
830,1,96,321,0.0827017,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{7/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(7/2),x]","\frac{\left(c-\frac{c}{a^2 x^2}\right)^{7/2} \left(a^7 x+a^6 \log (x)+\frac{3 a^5}{x}+\frac{3 a^4}{2 x^2}-\frac{a^3}{x^3}-\frac{3 a^2}{4 x^4}+\frac{a}{5 x^5}+\frac{1}{6 x^6}\right)}{a^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}","\frac{c^3 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^3 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^7 x^6 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((c - c/(a^2*x^2))^(7/2)*(1/(6*x^6) + a/(5*x^5) - (3*a^2)/(4*x^4) - a^3/x^3 + (3*a^4)/(2*x^2) + (3*a^5)/x + a^7*x + a^6*Log[x]))/(a^7*(1 - 1/(a^2*x^2))^(7/2))","A",1
831,1,75,236,0.0637433,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{5/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(5/2),x]","\frac{\left(c-\frac{c}{a^2 x^2}\right)^{5/2} \left(a^5 x+a^4 \log (x)+\frac{2 a^3}{x}+\frac{a^2}{x^2}-\frac{a}{3 x^3}-\frac{1}{4 x^4}\right)}{a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}","\frac{c^2 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^2 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((c - c/(a^2*x^2))^(5/2)*(-1/4*1/x^4 - a/(3*x^3) + a^2/x^2 + (2*a^3)/x + a^5*x + a^4*Log[x]))/(a^5*(1 - 1/(a^2*x^2))^(5/2))","A",1
832,1,64,146,0.0496441,"\int e^{\coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{3/2} \, dx","Integrate[E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(3/2),x]","\frac{\left(c-\frac{c}{a^2 x^2}\right)^{3/2} \left(a^3 x+a^2 \log (x)+\frac{3 a^2}{2}+\frac{a}{x}+\frac{1}{2 x^2}\right)}{a^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}","\frac{c x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((c - c/(a^2*x^2))^(3/2)*((3*a^2)/2 + 1/(2*x^2) + a/x + a^3*x + a^2*Log[x]))/(a^3*(1 - 1/(a^2*x^2))^(3/2))","A",0
833,1,39,67,0.02509,"\int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx","Integrate[E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)],x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} (a x+\log (x))}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(a*x + Log[x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
834,1,44,72,0.0323426,"\int \frac{e^{\coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a^2 x^2}}} \, dx","Integrate[E^ArcCoth[a*x]/Sqrt[c - c/(a^2*x^2)],x]","\frac{\sqrt{1-\frac{1}{a^2 x^2}} (a x+\log (1-a x))}{a \sqrt{c-\frac{c}{a^2 x^2}}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{a \sqrt{c-\frac{c}{a^2 x^2}}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*(a*x + Log[1 - a*x]))/(a*Sqrt[c - c/(a^2*x^2)])","A",1
835,1,68,173,0.0772836,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{3/2}} \, dx","Integrate[E^ArcCoth[a*x]/(c - c/(a^2*x^2))^(3/2),x]","\frac{\left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(4 a x+\frac{2}{1-a x}+5 \log (1-a x)-\log (a x+1)\right)}{4 a \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{5 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{4 a c \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{4 a c \sqrt{c-\frac{c}{a^2 x^2}}}",1,"((1 - 1/(a^2*x^2))^(3/2)*(4*a*x + 2/(1 - a*x) + 5*Log[1 - a*x] - Log[1 + a*x]))/(4*a*(c - c/(a^2*x^2))^(3/2))","A",1
836,1,86,263,0.1165541,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{5/2}} \, dx","Integrate[E^ArcCoth[a*x]/(c - c/(a^2*x^2))^(5/2),x]","\frac{\left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(16 a x+\frac{16}{1-a x}-\frac{2}{a x+1}-\frac{2}{(a x-1)^2}+23 \log (1-a x)-7 \log (a x+1)\right)}{16 a \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{a c^2 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{23 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{7 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}",1,"((1 - 1/(a^2*x^2))^(5/2)*(16*a*x + 16/(1 - a*x) - 2/(-1 + a*x)^2 - 2/(1 + a*x) + 23*Log[1 - a*x] - 7*Log[1 + a*x]))/(16*a*(c - c/(a^2*x^2))^(5/2))","A",1
837,1,104,359,0.1649227,"\int \frac{e^{\coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{7/2}} \, dx","Integrate[E^ArcCoth[a*x]/(c - c/(a^2*x^2))^(7/2),x]","\frac{\left(1-\frac{1}{a^2 x^2}\right)^{7/2} \left(96 a x+\frac{144}{1-a x}-\frac{30}{a x+1}-\frac{33}{(a x-1)^2}+\frac{3}{(a x+1)^2}-\frac{4}{(a x-1)^3}+153 \log (1-a x)-57 \log (a x+1)\right)}{96 a \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{5 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{11 \sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{24 a c^3 (1-a x)^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{51 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{19 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}",1,"((1 - 1/(a^2*x^2))^(7/2)*(96*a*x + 144/(1 - a*x) - 4/(-1 + a*x)^3 - 33/(-1 + a*x)^2 + 3/(1 + a*x)^2 - 30/(1 + a*x) + 153*Log[1 - a*x] - 57*Log[1 + a*x]))/(96*a*(c - c/(a^2*x^2))^(7/2))","A",1
838,1,150,372,0.1760468,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{7/2} \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2),x]","\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}} \left(480 a^6 x^6 \log \left(\sqrt{a^2 x^2-1}+a x\right)+375 a^6 x^6 \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)+\sqrt{a^2 x^2-1} \left(240 a^6 x^6-736 a^5 x^5+105 a^4 x^4+352 a^3 x^3+70 a^2 x^2-96 a x-40\right)\right)}{240 a^6 x^5 \sqrt{a^2 x^2-1}}","\frac{a x^2 (a x+1) \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}{15 (1-a x)^2}+\frac{x (a x+1) \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}{6 (1-a x)}-\frac{13 a^2 x^3 (a x+1) \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}{40 (1-a x)^3}-\frac{57 a^6 x^7 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}{16 (1-a x)^3 (a x+1)^3}+\frac{2 a^6 x^7 \left(c-\frac{c}{a^2 x^2}\right)^{7/2} \sin ^{-1}(a x)}{(1-a x)^{7/2} (a x+1)^{7/2}}+\frac{25 a^6 x^7 \left(c-\frac{c}{a^2 x^2}\right)^{7/2} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{16 (1-a x)^{7/2} (a x+1)^{7/2}}+\frac{41 a^5 x^6 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}{24 (1-a x)^3 (a x+1)^2}+\frac{57 a^4 x^5 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}{80 (1-a x)^3 (a x+1)}+\frac{11 a^3 x^4 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}{30 (1-a x)^3}",1,"(c^3*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(-40 - 96*a*x + 70*a^2*x^2 + 352*a^3*x^3 + 105*a^4*x^4 - 736*a^5*x^5 + 240*a^6*x^6) + 375*a^6*x^6*ArcTan[1/Sqrt[-1 + a^2*x^2]] + 480*a^6*x^6*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(240*a^6*x^5*Sqrt[-1 + a^2*x^2])","A",0
839,1,134,294,0.1463061,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{5/2} \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(5/2),x]","\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}} \left(48 a^4 x^4 \log \left(\sqrt{a^2 x^2-1}+a x\right)+27 a^4 x^4 \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)+\sqrt{a^2 x^2-1} \left(24 a^4 x^4-64 a^3 x^3-3 a^2 x^2+16 a x+6\right)\right)}{24 a^4 x^3 \sqrt{a^2 x^2-1}}","\frac{a x^2 (a x+1) \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}{6 (1-a x)^2}+\frac{x (a x+1) \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}{4 (1-a x)}-\frac{5 a^2 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}{8 (1-a x)^2}+\frac{25 a^4 x^5 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}{8 (1-a x)^2 (a x+1)^2}-\frac{2 a^4 x^5 \left(c-\frac{c}{a^2 x^2}\right)^{5/2} \sin ^{-1}(a x)}{(1-a x)^{5/2} (a x+1)^{5/2}}-\frac{9 a^4 x^5 \left(c-\frac{c}{a^2 x^2}\right)^{5/2} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{8 (1-a x)^{5/2} (a x+1)^{5/2}}-\frac{17 a^3 x^4 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}{12 (1-a x)^2 (a x+1)}",1,"(c^2*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(6 + 16*a*x - 3*a^2*x^2 - 64*a^3*x^3 + 24*a^4*x^4) + 27*a^4*x^4*ArcTan[1/Sqrt[-1 + a^2*x^2]] + 48*a^4*x^4*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(24*a^4*x^3*Sqrt[-1 + a^2*x^2])","A",0
840,1,115,213,0.1132873,"\int e^{2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{3/2} \, dx","Integrate[E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(3/2),x]","\frac{c \sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a^2 x^2-1} \left(2 a^2 x^2-4 a x-1\right)+4 a^2 x^2 \log \left(\sqrt{a^2 x^2-1}+a x\right)+a^2 x^2 \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)\right)}{2 a^2 x \sqrt{a^2 x^2-1}}","\frac{a x^2 \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}{1-a x}+\frac{x (a x+1) \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}{2 (1-a x)}-\frac{5 a^2 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}{2 (1-a x) (a x+1)}+\frac{2 a^2 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{3/2} \sin ^{-1}(a x)}{(1-a x)^{3/2} (a x+1)^{3/2}}+\frac{a^2 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{3/2} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{2 (1-a x)^{3/2} (a x+1)^{3/2}}",1,"(c*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(-1 - 4*a*x + 2*a^2*x^2) + a^2*x^2*ArcTan[1/Sqrt[-1 + a^2*x^2]] + 4*a^2*x^2*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(2*a^2*x*Sqrt[-1 + a^2*x^2])","A",0
841,1,80,116,0.0868662,"\int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx","Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)],x]","\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a^2 x^2-1}+2 \log \left(\sqrt{a^2 x^2-1}+a x\right)-\tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)\right)}{\sqrt{a^2 x^2-1}}","x \sqrt{c-\frac{c}{a^2 x^2}}-\frac{2 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{a x+1}}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{\sqrt{1-a x} \sqrt{a x+1}}",1,"(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[-1 + a^2*x^2] - ArcTan[1/Sqrt[-1 + a^2*x^2]] + 2*Log[a*x + Sqrt[-1 + a^2*x^2]]))/Sqrt[-1 + a^2*x^2]","A",0
842,1,68,111,0.0769419,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a^2 x^2}}} \, dx","Integrate[E^(2*ArcCoth[a*x])/Sqrt[c - c/(a^2*x^2)],x]","\frac{a^2 x^2+2 \sqrt{a^2 x^2-1} \log \left(\sqrt{a^2 x^2-1}+a x\right)-2 a x-3}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}","-\frac{(a x+1)^2}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{2 (1-a x) (a x+1)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{2 \sqrt{1-a x} \sqrt{a x+1} \sin ^{-1}(a x)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}",1,"(-3 - 2*a*x + a^2*x^2 + 2*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(a^2*Sqrt[c - c/(a^2*x^2)]*x)","A",0
843,1,95,123,0.0895611,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{3/2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - c/(a^2*x^2))^(3/2),x]","\frac{3 a^3 x^3-11 a^2 x^2+6 (a x-1) \sqrt{a^2 x^2-1} \log \left(\sqrt{a^2 x^2-1}+a x\right)-4 a x+10}{3 a^2 c x (a x-1) \sqrt{c-\frac{c}{a^2 x^2}}}","-\frac{(a x+1)^2}{3 a^2 x \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}+\frac{2 (5-2 a x) (1-a x) (a x+1)^2}{3 a^4 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}-\frac{2 (1-a x)^{3/2} (a x+1)^{3/2} \sin ^{-1}(a x)}{a^4 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}",1,"(10 - 4*a*x - 11*a^2*x^2 + 3*a^3*x^3 + 6*(-1 + a*x)*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(3*a^2*c*Sqrt[c - c/(a^2*x^2)]*x*(-1 + a*x))","A",0
844,1,105,203,0.1096186,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{5/2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - c/(a^2*x^2))^(5/2),x]","\frac{15 a^4 x^4-76 a^3 x^3+32 a^2 x^2+30 (a x-1)^2 \sqrt{a^2 x^2-1} \log \left(\sqrt{a^2 x^2-1}+a x\right)+82 a x-56}{15 a^2 c^2 x (a x-1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}","-\frac{(a x+1)^2}{5 a^2 x \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}-\frac{2 (a x+1)^2 (43 a x+28) (1-a x)^3}{15 a^6 x^5 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}+\frac{2 (a x+1)^{5/2} (1-a x)^{5/2} \sin ^{-1}(a x)}{a^6 x^5 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}-\frac{58 (a x+1)^2 (1-a x)^2}{15 a^4 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}+\frac{2 (a x+1)^2 (1-a x)}{3 a^3 x^2 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}",1,"(-56 + 82*a*x + 32*a^2*x^2 - 76*a^3*x^3 + 15*a^4*x^4 + 30*(-1 + a*x)^2*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(15*a^2*c^2*Sqrt[c - c/(a^2*x^2)]*x*(-1 + a*x)^2)","A",0
845,1,133,283,0.132984,"\int \frac{e^{2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{7/2}} \, dx","Integrate[E^(2*ArcCoth[a*x])/(c - c/(a^2*x^2))^(7/2),x]","\frac{105 a^6 x^6-562 a^5 x^5+74 a^4 x^4+1226 a^3 x^3-636 a^2 x^2+210 (a x-1)^3 (a x+1) \sqrt{a^2 x^2-1} \log \left(\sqrt{a^2 x^2-1}+a x\right)-654 a x+432}{105 a^2 c^3 x (a x-1)^3 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}","-\frac{(a x+1)^2}{7 a^2 x \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}+\frac{2 (a x+1)^3 (107 a x+72) (1-a x)^4}{35 a^8 x^7 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}-\frac{2 (a x+1)^{7/2} (1-a x)^{7/2} \sin ^{-1}(a x)}{a^8 x^7 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}+\frac{142 (a x+1)^2 (1-a x)^4}{35 a^6 x^5 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}+\frac{782 (a x+1)^2 (1-a x)^3}{105 a^5 x^4 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}-\frac{124 (a x+1)^2 (1-a x)^2}{105 a^4 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}+\frac{2 (a x+1)^2 (1-a x)}{5 a^3 x^2 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}",1,"(432 - 654*a*x - 636*a^2*x^2 + 1226*a^3*x^3 + 74*a^4*x^4 - 562*a^5*x^5 + 105*a^6*x^6 + 210*(-1 + a*x)^3*(1 + a*x)*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(105*a^2*c^3*Sqrt[c - c/(a^2*x^2)]*x*(-1 + a*x)^3*(1 + a*x))","A",0
846,1,97,322,0.0926781,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{9/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(9/2),x]","\frac{\left(c-\frac{c}{a^2 x^2}\right)^{9/2} \left(a^9 x+3 a^8 \log (x)+\frac{4 a^6}{x^2}+\frac{2 a^5}{x^3}-\frac{3 a^4}{2 x^4}-\frac{8 a^3}{5 x^5}+\frac{3 a}{7 x^7}+\frac{1}{8 x^8}\right)}{a^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}","\frac{c^4 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^4 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{8 a^9 x^8 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{7 a^8 x^7 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{8 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((c - c/(a^2*x^2))^(9/2)*(1/(8*x^8) + (3*a)/(7*x^7) - (8*a^3)/(5*x^5) - (3*a^4)/(2*x^4) + (2*a^5)/x^3 + (4*a^6)/x^2 + a^9*x + 3*a^8*Log[x]))/(a^9*(1 - 1/(a^2*x^2))^(9/2))","A",1
847,1,94,324,0.091704,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{7/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2),x]","\frac{\left(c-\frac{c}{a^2 x^2}\right)^{7/2} \left(3 a^6 \log (x)-\frac{-60 a^7 x^7+60 a^5 x^5-150 a^4 x^4-100 a^3 x^3+15 a^2 x^2+36 a x+10}{60 x^6}\right)}{a^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}","\frac{c^3 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^3 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^7 x^6 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((c - c/(a^2*x^2))^(7/2)*(-1/60*(10 + 36*a*x + 15*a^2*x^2 - 100*a^3*x^3 - 150*a^4*x^4 + 60*a^5*x^5 - 60*a^7*x^7)/x^6 + 3*a^6*Log[x]))/(a^7*(1 - 1/(a^2*x^2))^(7/2))","A",1
848,1,87,234,0.0581521,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{5/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(5/2),x]","\frac{\left(c-\frac{c}{a^2 x^2}\right)^{5/2} \left(\frac{3}{4} a \left(a^4 x+4 a^3 \log (x)-\frac{6 a^2}{x}-\frac{2 a}{x^2}-\frac{1}{3 x^3}\right)+\frac{(a x+1)^5}{4 x^4}\right)}{a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}","\frac{c^2 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2 c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^2 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((c - c/(a^2*x^2))^(5/2)*((1 + a*x)^5/(4*x^4) + (3*a*(-1/3*1/x^3 - (2*a)/x^2 - (6*a^2)/x + a^4*x + 4*a^3*Log[x]))/4))/(a^5*(1 - 1/(a^2*x^2))^(5/2))","A",0
849,1,59,148,0.047584,"\int e^{3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{3/2} \, dx","Integrate[E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(3/2),x]","\frac{\left(c-\frac{c}{a^2 x^2}\right)^{3/2} \left(a^3 x+3 a^2 \log (x)-\frac{3 a}{x}-\frac{1}{2 x^2}\right)}{a^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}","\frac{c x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((c - c/(a^2*x^2))^(3/2)*(-1/2*1/x^2 - (3*a)/x + a^3*x + 3*a^2*Log[x]))/(a^3*(1 - 1/(a^2*x^2))^(3/2))","A",1
850,1,50,109,0.0338527,"\int e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx","Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)],x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} (a x+4 \log (1-a x)-\log (x))}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(a*x - Log[x] + 4*Log[1 - a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
851,1,56,115,0.0522793,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a^2 x^2}}} \, dx","Integrate[E^(3*ArcCoth[a*x])/Sqrt[c - c/(a^2*x^2)],x]","\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left(a x+\frac{2}{1-a x}+3 \log (1-a x)\right)}{a \sqrt{c-\frac{c}{a^2 x^2}}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a^2 x^2}}}+\frac{2 \sqrt{1-\frac{1}{a^2 x^2}}}{a (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{a \sqrt{c-\frac{c}{a^2 x^2}}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*(a*x + 2/(1 - a*x) + 3*Log[1 - a*x]))/(a*Sqrt[c - c/(a^2*x^2)])","A",1
852,1,64,171,0.0815295,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{3/2}} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^(3/2),x]","\frac{\left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(2 a x+\frac{5-6 a x}{(a x-1)^2}+6 \log (1-a x)\right)}{2 a \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{3 \sqrt{1-\frac{1}{a^2 x^2}}}{a c (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{a c \sqrt{c-\frac{c}{a^2 x^2}}}",1,"((1 - 1/(a^2*x^2))^(3/2)*(2*a*x + (5 - 6*a*x)/(-1 + a*x)^2 + 6*Log[1 - a*x]))/(2*a*(c - c/(a^2*x^2))^(3/2))","A",1
853,1,86,267,0.1168031,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{5/2}} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^(5/2),x]","\frac{\left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(48 a x+\frac{186}{1-a x}-\frac{54}{(a x-1)^2}-\frac{8}{(a x-1)^3}+147 \log (1-a x)-3 \log (a x+1)\right)}{48 a \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{31 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{9 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{6 a c^2 (1-a x)^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{49 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}",1,"((1 - 1/(a^2*x^2))^(5/2)*(48*a*x + 186/(1 - a*x) - 8/(-1 + a*x)^3 - 54/(-1 + a*x)^2 + 147*Log[1 - a*x] - 3*Log[1 + a*x]))/(48*a*(c - c/(a^2*x^2))^(5/2))","A",1
854,1,140,360,0.1370832,"\int \frac{e^{3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{7/2}} \, dx","Integrate[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2))^(7/2),x]","\frac{a^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2} \left(\frac{75}{16 a^8 (1-a x)}-\frac{1}{32 a^8 (a x+1)}-\frac{59}{32 a^8 (1-a x)^2}+\frac{1}{2 a^8 (1-a x)^3}-\frac{1}{16 a^8 (1-a x)^4}+\frac{201 \log (1-a x)}{64 a^8}-\frac{9 \log (a x+1)}{64 a^8}+\frac{x}{a^7}\right)}{\left(c-\frac{c}{a^2 x^2}\right)^{7/2}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{75 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{59 \sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 (1-a x)^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (1-a x)^4 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{201 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{9 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}",1,"(a^7*(1 - 1/(a^2*x^2))^(7/2)*(x/a^7 - 1/(16*a^8*(1 - a*x)^4) + 1/(2*a^8*(1 - a*x)^3) - 59/(32*a^8*(1 - a*x)^2) + 75/(16*a^8*(1 - a*x)) - 1/(32*a^8*(1 + a*x)) + (201*Log[1 - a*x])/(64*a^8) - (9*Log[1 + a*x])/(64*a^8)))/(c - c/(a^2*x^2))^(7/2)","A",1
855,1,97,322,0.0752464,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{7/2} \, dx","Integrate[(c - c/(a^2*x^2))^(7/2)/E^ArcCoth[a*x],x]","\frac{\left(c-\frac{c}{a^2 x^2}\right)^{7/2} \left(a^7 x-a^6 \log (x)+\frac{3 a^5}{x}-\frac{3 a^4}{2 x^2}-\frac{a^3}{x^3}+\frac{3 a^2}{4 x^4}+\frac{a}{5 x^5}-\frac{1}{6 x^6}\right)}{a^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}","\frac{c^3 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^7 x^6 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((c - c/(a^2*x^2))^(7/2)*(-1/6*1/x^6 + a/(5*x^5) + (3*a^2)/(4*x^4) - a^3/x^3 - (3*a^4)/(2*x^2) + (3*a^5)/x + a^7*x - a^6*Log[x]))/(a^7*(1 - 1/(a^2*x^2))^(7/2))","A",1
856,1,77,238,0.0653293,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{5/2} \, dx","Integrate[(c - c/(a^2*x^2))^(5/2)/E^ArcCoth[a*x],x]","\frac{\left(c-\frac{c}{a^2 x^2}\right)^{5/2} \left(a^5 x-a^4 \log (x)+\frac{2 a^3}{x}-\frac{a^2}{x^2}-\frac{a}{3 x^3}+\frac{1}{4 x^4}\right)}{a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}","\frac{c^2 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((c - c/(a^2*x^2))^(5/2)*(1/(4*x^4) - a/(3*x^3) - a^2/x^2 + (2*a^3)/x + a^5*x - a^4*Log[x]))/(a^5*(1 - 1/(a^2*x^2))^(5/2))","A",1
857,1,65,147,0.0492865,"\int e^{-\coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{3/2} \, dx","Integrate[(c - c/(a^2*x^2))^(3/2)/E^ArcCoth[a*x],x]","\frac{\left(c-\frac{c}{a^2 x^2}\right)^{3/2} \left(a^3 x-a^2 \log (x)-\frac{3 a^2}{2}+\frac{a}{x}-\frac{1}{2 x^2}\right)}{a^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}","\frac{c x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((c - c/(a^2*x^2))^(3/2)*((-3*a^2)/2 - 1/(2*x^2) + a/x + a^3*x - a^2*Log[x]))/(a^3*(1 - 1/(a^2*x^2))^(3/2))","A",0
858,1,41,68,0.0267595,"\int e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/E^ArcCoth[a*x],x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} (a x-\log (x))}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(a*x - Log[x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
859,1,45,72,0.0346063,"\int \frac{e^{-\coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a^2 x^2}}} \, dx","Integrate[1/(E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)]),x]","\frac{\sqrt{1-\frac{1}{a^2 x^2}} (a x-\log (a x+1))}{a \sqrt{c-\frac{c}{a^2 x^2}}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{a \sqrt{c-\frac{c}{a^2 x^2}}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*(a*x - Log[1 + a*x]))/(a*Sqrt[c - c/(a^2*x^2)])","A",1
860,1,65,172,0.0693901,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{3/2}} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(3/2)),x]","\frac{\left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(4 a x-\frac{2}{a x+1}+\log (1-a x)-5 \log (a x+1)\right)}{4 a \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{4 a c \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{5 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{4 a c \sqrt{c-\frac{c}{a^2 x^2}}}",1,"((1 - 1/(a^2*x^2))^(3/2)*(4*a*x - 2/(1 + a*x) + Log[1 - a*x] - 5*Log[1 + a*x]))/(4*a*(c - c/(a^2*x^2))^(3/2))","A",1
861,1,85,263,0.1271071,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{5/2}} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(5/2)),x]","\frac{\left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(2 \left(8 a x+\frac{1}{1-a x}-\frac{8}{a x+1}+\frac{1}{(a x+1)^2}\right)+7 \log (1-a x)-23 \log (a x+1)\right)}{16 a \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{a c^2 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{7 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{23 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}",1,"((1 - 1/(a^2*x^2))^(5/2)*(2*(8*a*x + (1 - a*x)^(-1) + (1 + a*x)^(-2) - 8/(1 + a*x)) + 7*Log[1 - a*x] - 23*Log[1 + a*x]))/(16*a*(c - c/(a^2*x^2))^(5/2))","A",1
862,1,104,358,0.1652762,"\int \frac{e^{-\coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{7/2}} \, dx","Integrate[1/(E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(7/2)),x]","\frac{\left(1-\frac{1}{a^2 x^2}\right)^{7/2} \left(96 a x+\frac{30}{1-a x}-\frac{144}{a x+1}-\frac{3}{(a x-1)^2}+\frac{33}{(a x+1)^2}-\frac{4}{(a x+1)^3}+57 \log (1-a x)-153 \log (a x+1)\right)}{96 a \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{5 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{11 \sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{24 a c^3 (a x+1)^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{19 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{51 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{32 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}",1,"((1 - 1/(a^2*x^2))^(7/2)*(96*a*x + 30/(1 - a*x) - 3/(-1 + a*x)^2 - 4/(1 + a*x)^3 + 33/(1 + a*x)^2 - 144/(1 + a*x) + 57*Log[1 - a*x] - 153*Log[1 + a*x]))/(96*a*(c - c/(a^2*x^2))^(7/2))","A",1
863,1,150,375,0.1867581,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{7/2} \, dx","Integrate[(c - c/(a^2*x^2))^(7/2)/E^(2*ArcCoth[a*x]),x]","\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}} \left(-480 a^6 x^6 \log \left(\sqrt{a^2 x^2-1}+a x\right)+375 a^6 x^6 \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)+\sqrt{a^2 x^2-1} \left(240 a^6 x^6+736 a^5 x^5+105 a^4 x^4-352 a^3 x^3+70 a^2 x^2+96 a x-40\right)\right)}{240 a^6 x^5 \sqrt{a^2 x^2-1}}","-\frac{a x^2 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}{15 (a x+1)}+\frac{x (1-a x) \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}{6 (a x+1)}-\frac{23 a^2 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}{120 (1-a x) (a x+1)}+\frac{7 a^6 x^7 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}{16 (1-a x)^3 (a x+1)^3}-\frac{2 a^6 x^7 \left(c-\frac{c}{a^2 x^2}\right)^{7/2} \sin ^{-1}(a x)}{(1-a x)^{7/2} (a x+1)^{7/2}}+\frac{25 a^6 x^7 \left(c-\frac{c}{a^2 x^2}\right)^{7/2} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{16 (1-a x)^{7/2} (a x+1)^{7/2}}+\frac{3 a^5 x^6 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}{8 (1-a x)^3 (a x+1)^2}-\frac{19 a^4 x^5 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}{16 (1-a x)^3 (a x+1)}+\frac{2 a^3 x^4 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}{3 (1-a x)^2 (a x+1)}",1,"(c^3*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(-40 + 96*a*x + 70*a^2*x^2 - 352*a^3*x^3 + 105*a^4*x^4 + 736*a^5*x^5 + 240*a^6*x^6) + 375*a^6*x^6*ArcTan[1/Sqrt[-1 + a^2*x^2]] - 480*a^6*x^6*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(240*a^6*x^5*Sqrt[-1 + a^2*x^2])","A",0
864,1,134,293,0.1500026,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{5/2} \, dx","Integrate[(c - c/(a^2*x^2))^(5/2)/E^(2*ArcCoth[a*x]),x]","\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}} \left(-48 a^4 x^4 \log \left(\sqrt{a^2 x^2-1}+a x\right)+27 a^4 x^4 \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)+\sqrt{a^2 x^2-1} \left(24 a^4 x^4+64 a^3 x^3-3 a^2 x^2-16 a x+6\right)\right)}{24 a^4 x^3 \sqrt{a^2 x^2-1}}","-\frac{a x^2 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}{6 (a x+1)}+\frac{x (1-a x) \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}{4 (a x+1)}-\frac{7 a^2 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}{24 (1-a x) (a x+1)}-\frac{7 a^4 x^5 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}{8 (1-a x)^2 (a x+1)^2}+\frac{2 a^4 x^5 \left(c-\frac{c}{a^2 x^2}\right)^{5/2} \sin ^{-1}(a x)}{(1-a x)^{5/2} (a x+1)^{5/2}}-\frac{9 a^4 x^5 \left(c-\frac{c}{a^2 x^2}\right)^{5/2} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{8 (1-a x)^{5/2} (a x+1)^{5/2}}+\frac{2 a^3 x^4 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}{(1-a x)^2 (a x+1)}",1,"(c^2*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(6 - 16*a*x - 3*a^2*x^2 + 64*a^3*x^3 + 24*a^4*x^4) + 27*a^4*x^4*ArcTan[1/Sqrt[-1 + a^2*x^2]] - 48*a^4*x^4*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(24*a^4*x^3*Sqrt[-1 + a^2*x^2])","A",0
865,1,115,213,0.1159545,"\int e^{-2 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{3/2} \, dx","Integrate[(c - c/(a^2*x^2))^(3/2)/E^(2*ArcCoth[a*x]),x]","\frac{c \sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a^2 x^2-1} \left(2 a^2 x^2+4 a x-1\right)-4 a^2 x^2 \log \left(\sqrt{a^2 x^2-1}+a x\right)+a^2 x^2 \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)\right)}{2 a^2 x \sqrt{a^2 x^2-1}}","-\frac{a x^2 \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}{a x+1}+\frac{x (1-a x) \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}{2 (a x+1)}-\frac{5 a^2 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}{2 (1-a x) (a x+1)}-\frac{2 a^2 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{3/2} \sin ^{-1}(a x)}{(1-a x)^{3/2} (a x+1)^{3/2}}+\frac{a^2 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{3/2} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{2 (1-a x)^{3/2} (a x+1)^{3/2}}",1,"(c*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(-1 + 4*a*x + 2*a^2*x^2) + a^2*x^2*ArcTan[1/Sqrt[-1 + a^2*x^2]] - 4*a^2*x^2*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(2*a^2*x*Sqrt[-1 + a^2*x^2])","A",0
866,1,80,116,0.0905339,"\int e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/E^(2*ArcCoth[a*x]),x]","\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a^2 x^2-1}-2 \log \left(\sqrt{a^2 x^2-1}+a x\right)-\tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)\right)}{\sqrt{a^2 x^2-1}}","x \sqrt{c-\frac{c}{a^2 x^2}}+\frac{2 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{a x+1}}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{\sqrt{1-a x} \sqrt{a x+1}}",1,"(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[-1 + a^2*x^2] - ArcTan[1/Sqrt[-1 + a^2*x^2]] - 2*Log[a*x + Sqrt[-1 + a^2*x^2]]))/Sqrt[-1 + a^2*x^2]","A",0
867,1,68,112,0.0773872,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a^2 x^2}}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]),x]","\frac{a^2 x^2-2 \sqrt{a^2 x^2-1} \log \left(\sqrt{a^2 x^2-1}+a x\right)+2 a x-3}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}","-\frac{(1-a x)^2}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{2 (a x+1) (1-a x)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{2 \sqrt{a x+1} \sqrt{1-a x} \sin ^{-1}(a x)}{a^2 x \sqrt{c-\frac{c}{a^2 x^2}}}",1,"(-3 + 2*a*x + a^2*x^2 - 2*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(a^2*Sqrt[c - c/(a^2*x^2)]*x)","A",0
868,1,95,124,0.0922921,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{3/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(3/2)),x]","\frac{3 a^3 x^3+11 a^2 x^2-6 (a x+1) \sqrt{a^2 x^2-1} \log \left(\sqrt{a^2 x^2-1}+a x\right)-4 a x-10}{3 a^2 c x (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}","-\frac{(1-a x)^2}{3 a^2 x \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}+\frac{2 (a x+1) (2 a x+5) (1-a x)^2}{3 a^4 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}+\frac{2 (a x+1)^{3/2} (1-a x)^{3/2} \sin ^{-1}(a x)}{a^4 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}",1,"(-10 - 4*a*x + 11*a^2*x^2 + 3*a^3*x^3 - 6*(1 + a*x)*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(3*a^2*c*Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x))","A",0
869,1,105,195,0.1061777,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{5/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(5/2)),x]","\frac{15 a^4 x^4+76 a^3 x^3+32 a^2 x^2-30 (a x+1)^2 \sqrt{a^2 x^2-1} \log \left(\sqrt{a^2 x^2-1}+a x\right)-82 a x-56}{15 a^2 c^2 x (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}","-\frac{(1-a x)^2}{a^2 x \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}-\frac{2 (a x+1)^2 (13 a x+28) (1-a x)^3}{15 a^6 x^5 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}-\frac{2 (a x+1)^{5/2} (1-a x)^{5/2} \sin ^{-1}(a x)}{a^6 x^5 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}+\frac{2 (a x+1) (1-a x)^3}{15 a^4 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}-\frac{2 (1-a x)^3}{5 a^3 x^2 \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}",1,"(-56 - 82*a*x + 32*a^2*x^2 + 76*a^3*x^3 + 15*a^4*x^4 - 30*(1 + a*x)^2*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(15*a^2*c^2*Sqrt[c - c/(a^2*x^2)]*x*(1 + a*x)^2)","A",0
870,1,131,270,0.1284719,"\int \frac{e^{-2 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{7/2}} \, dx","Integrate[1/(E^(2*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2)),x]","\frac{105 a^6 x^6+562 a^5 x^5+74 a^4 x^4-1226 a^3 x^3-636 a^2 x^2-210 (a x-1) (a x+1)^3 \sqrt{a^2 x^2-1} \log \left(\sqrt{a^2 x^2-1}+a x\right)+654 a x+432}{105 a^2 x (a x-1) \sqrt{c-\frac{c}{a^2 x^2}} (a c x+c)^3}","-\frac{(1-a x)^2}{3 a^2 x \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}+\frac{2 (a x+1)^3 (37 a x+72) (1-a x)^4}{35 a^8 x^7 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}+\frac{2 (a x+1)^{7/2} (1-a x)^{7/2} \sin ^{-1}(a x)}{a^8 x^7 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}+\frac{2 (a x+1)^2 (1-a x)^4}{35 a^6 x^5 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}+\frac{82 (a x+1) (1-a x)^4}{105 a^5 x^4 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}+\frac{12 (1-a x)^4}{7 a^4 x^3 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}+\frac{10 (1-a x)^3}{3 a^3 x^2 \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}",1,"(432 + 654*a*x - 636*a^2*x^2 - 1226*a^3*x^3 + 74*a^4*x^4 + 562*a^5*x^5 + 105*a^6*x^6 - 210*(-1 + a*x)*(1 + a*x)^3*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(105*a^2*Sqrt[c - c/(a^2*x^2)]*x*(-1 + a*x)*(c + a*c*x)^3)","A",0
871,1,97,322,0.0914111,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{9/2} \, dx","Integrate[(c - c/(a^2*x^2))^(9/2)/E^(3*ArcCoth[a*x]),x]","\frac{\left(c-\frac{c}{a^2 x^2}\right)^{9/2} \left(a^9 x-3 a^8 \log (x)-\frac{4 a^6}{x^2}+\frac{2 a^5}{x^3}+\frac{3 a^4}{2 x^4}-\frac{8 a^3}{5 x^5}+\frac{3 a}{7 x^7}-\frac{1}{8 x^8}\right)}{a^9 \left(1-\frac{1}{a^2 x^2}\right)^{9/2}}","\frac{c^4 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^4 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{8 a^9 x^8 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{7 a^8 x^7 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{8 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 c^4 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((c - c/(a^2*x^2))^(9/2)*(-1/8*1/x^8 + (3*a)/(7*x^7) - (8*a^3)/(5*x^5) + (3*a^4)/(2*x^4) + (2*a^5)/x^3 - (4*a^6)/x^2 + a^9*x - 3*a^8*Log[x]))/(a^9*(1 - 1/(a^2*x^2))^(9/2))","A",1
872,1,94,324,0.090871,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{7/2} \, dx","Integrate[(c - c/(a^2*x^2))^(7/2)/E^(3*ArcCoth[a*x]),x]","\frac{\left(c-\frac{c}{a^2 x^2}\right)^{7/2} \left(\frac{60 a^7 x^7-60 a^5 x^5-150 a^4 x^4+100 a^3 x^3+15 a^2 x^2-36 a x+10}{60 x^6}-3 a^6 \log (x)\right)}{a^7 \left(1-\frac{1}{a^2 x^2}\right)^{7/2}}","\frac{c^3 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^3 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^7 x^6 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{5 a^6 x^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{5 c^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((c - c/(a^2*x^2))^(7/2)*((10 - 36*a*x + 15*a^2*x^2 + 100*a^3*x^3 - 150*a^4*x^4 - 60*a^5*x^5 + 60*a^7*x^7)/(60*x^6) - 3*a^6*Log[x]))/(a^7*(1 - 1/(a^2*x^2))^(7/2))","A",1
873,1,81,235,0.0706069,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{5/2} \, dx","Integrate[(c - c/(a^2*x^2))^(5/2)/E^(3*ArcCoth[a*x]),x]","\frac{\left(c-\frac{c}{a^2 x^2}\right)^{5/2} \left(a^5 x-3 a^4 \log (x)-\frac{5 a^4}{4}-\frac{2 a^3}{x}-\frac{a^2}{x^2}+\frac{a}{x^3}-\frac{1}{4 x^4}\right)}{a^5 \left(1-\frac{1}{a^2 x^2}\right)^{5/2}}","\frac{c^2 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2 c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c^2 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((c - c/(a^2*x^2))^(5/2)*((-5*a^4)/4 - 1/(4*x^4) + a/x^3 - a^2/x^2 - (2*a^3)/x + a^5*x - 3*a^4*Log[x]))/(a^5*(1 - 1/(a^2*x^2))^(5/2))","A",0
874,1,59,148,0.0692905,"\int e^{-3 \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^{3/2} \, dx","Integrate[(c - c/(a^2*x^2))^(3/2)/E^(3*ArcCoth[a*x]),x]","\frac{\left(c-\frac{c}{a^2 x^2}\right)^{3/2} \left(a^3 x-3 a^2 \log (x)-\frac{3 a}{x}+\frac{1}{2 x^2}\right)}{a^3 \left(1-\frac{1}{a^2 x^2}\right)^{3/2}}","\frac{c x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 c \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c \sqrt{c-\frac{c}{a^2 x^2}}}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"((c - c/(a^2*x^2))^(3/2)*(1/(2*x^2) - (3*a)/x + a^3*x - 3*a^2*Log[x]))/(a^3*(1 - 1/(a^2*x^2))^(3/2))","A",1
875,1,47,107,0.0631073,"\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/E^(3*ArcCoth[a*x]),x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} (a x-4 \log (a x+1)+\log (x))}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(a*x + Log[x] - 4*Log[1 + a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
876,1,54,113,0.054203,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a^2 x^2}}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]),x]","\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left(a x-\frac{2}{a x+1}-3 \log (a x+1)\right)}{a \sqrt{c-\frac{c}{a^2 x^2}}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a^2 x^2}}}-\frac{2 \sqrt{1-\frac{1}{a^2 x^2}}}{a (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{a \sqrt{c-\frac{c}{a^2 x^2}}}",1,"(Sqrt[1 - 1/(a^2*x^2)]*(a*x - 2/(1 + a*x) - 3*Log[1 + a*x]))/(a*Sqrt[c - c/(a^2*x^2)])","A",1
877,1,63,168,0.0925422,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{3/2}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(3/2)),x]","\frac{\left(1-\frac{1}{a^2 x^2}\right)^{3/2} \left(2 a x+\frac{-6 a x-5}{(a x+1)^2}-6 \log (a x+1)\right)}{2 a \left(c-\frac{c}{a^2 x^2}\right)^{3/2}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}}}{a c (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{a c \sqrt{c-\frac{c}{a^2 x^2}}}",1,"((1 - 1/(a^2*x^2))^(3/2)*(2*a*x + (-5 - 6*a*x)/(1 + a*x)^2 - 6*Log[1 + a*x]))/(2*a*(c - c/(a^2*x^2))^(3/2))","A",1
878,1,85,264,0.1364771,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{5/2}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(5/2)),x]","\frac{\left(1-\frac{1}{a^2 x^2}\right)^{5/2} \left(48 a x-\frac{186}{a x+1}+\frac{54}{(a x+1)^2}-\frac{8}{(a x+1)^3}+3 \log (1-a x)-147 \log (a x+1)\right)}{48 a \left(c-\frac{c}{a^2 x^2}\right)^{5/2}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{31 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{9 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a c^2 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{6 a c^2 (a x+1)^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{49 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{16 a c^2 \sqrt{c-\frac{c}{a^2 x^2}}}",1,"((1 - 1/(a^2*x^2))^(5/2)*(48*a*x - 8/(1 + a*x)^3 + 54/(1 + a*x)^2 - 186/(1 + a*x) + 3*Log[1 - a*x] - 147*Log[1 + a*x]))/(48*a*(c - c/(a^2*x^2))^(5/2))","A",1
879,1,105,357,0.2387342,"\int \frac{e^{-3 \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^{7/2}} \, dx","Integrate[1/(E^(3*ArcCoth[a*x])*(c - c/(a^2*x^2))^(7/2)),x]","\frac{\left(1-\frac{1}{a^2 x^2}\right)^{7/2} \left(2 \left(32 a x+\frac{1}{1-a x}-\frac{150}{a x+1}+\frac{59}{(a x+1)^2}-\frac{16}{(a x+1)^3}+\frac{2}{(a x+1)^4}\right)+9 \log (1-a x)-201 \log (a x+1)\right)}{64 a \left(c-\frac{c}{a^2 x^2}\right)^{7/2}}","\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{75 \sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{59 \sqrt{1-\frac{1}{a^2 x^2}}}{32 a c^3 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{2 a c^3 (a x+1)^3 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{16 a c^3 (a x+1)^4 \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{9 \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}-\frac{201 \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt{c-\frac{c}{a^2 x^2}}}",1,"((1 - 1/(a^2*x^2))^(7/2)*(2*(32*a*x + (1 - a*x)^(-1) + 2/(1 + a*x)^4 - 16/(1 + a*x)^3 + 59/(1 + a*x)^2 - 150/(1 + a*x)) + 9*Log[1 - a*x] - 201*Log[1 + a*x]))/(64*a*(c - c/(a^2*x^2))^(7/2))","A",1
880,1,53,80,0.0470473,"\int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^m \, dx","Integrate[E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)]*x^m,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(\frac{a x^{m+1}}{m+1}+\frac{x^m}{m}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^{m+1} \sqrt{c-\frac{c}{a^2 x^2}}}{(m+1) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^m \sqrt{c-\frac{c}{a^2 x^2}}}{a m \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(x^m/m + (a*x^(1 + m))/(1 + m)))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
881,1,45,76,0.0352584,"\int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^2 \, dx","Integrate[E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)]*x^2,x]","\frac{x^2 (2 a x+3) \sqrt{c-\frac{c}{a^2 x^2}}}{6 a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*x^2*(3 + 2*a*x))/(6*a*Sqrt[1 - 1/(a^2*x^2)])","A",1
882,1,43,71,0.0353981,"\int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x \, dx","Integrate[E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)]*x,x]","\frac{\left(\frac{a x^2}{2}+x\right) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(x + (a*x^2)/2))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
883,1,39,67,0.030389,"\int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx","Integrate[E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)],x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} (a x+\log (x))}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(a*x + Log[x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
884,1,43,70,0.0378526,"\int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x} \, dx","Integrate[(E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)])/x,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(a \log (x)-\frac{1}{x}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{\log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{a x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(-x^(-1) + a*Log[x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
885,1,47,46,0.0483537,"\int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^2} \, dx","Integrate[(E^ArcCoth[a*x]*Sqrt[c - c/(a^2*x^2)])/x^2,x]","\frac{\left(-\frac{a}{x}-\frac{1}{2 x^2}\right) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{(a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(-1/2*1/x^2 - a/x))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",0
886,1,93,160,0.1226717,"\int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^3 \, dx","Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x^3,x]","\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left(21 \log \left(\sqrt{a^2 x^2-1}+a x\right)+\sqrt{a^2 x^2-1} \left(6 a^3 x^3+16 a^2 x^2+21 a x+32\right)\right)}{24 a^3 \sqrt{a^2 x^2-1}}","\frac{x^2 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^2}+\frac{x (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^3}+\frac{7 x (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}{24 a^3}+\frac{7 x \sqrt{c-\frac{c}{a^2 x^2}}}{8 a^3}-\frac{7 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{8 a^3 \sqrt{1-a x} \sqrt{a x+1}}",1,"(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[-1 + a^2*x^2]*(32 + 21*a*x + 16*a^2*x^2 + 6*a^3*x^3) + 21*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(24*a^3*Sqrt[-1 + a^2*x^2])","A",0
887,1,84,123,0.0916838,"\int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^2 \, dx","Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x^2,x]","\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a^2 x^2-1} \left(a^2 x^2+3 a x+5\right)+3 \log \left(\sqrt{a^2 x^2-1}+a x\right)\right)}{3 a^2 \sqrt{a^2 x^2-1}}","\frac{x (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^2}+\frac{x (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^2}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{a^2}-\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{a^2 \sqrt{1-a x} \sqrt{a x+1}}",1,"(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[-1 + a^2*x^2]*(5 + 3*a*x + a^2*x^2) + 3*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(3*a^2*Sqrt[-1 + a^2*x^2])","A",0
888,1,77,98,0.083449,"\int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x \, dx","Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x,x]","\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{1-a^2 x^2} (a x+4)+6 \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)\right)}{2 a \sqrt{1-a^2 x^2}}","\frac{x (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}{2 a}+\frac{3 x \sqrt{c-\frac{c}{a^2 x^2}}}{2 a}-\frac{3 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{2 a \sqrt{1-a x} \sqrt{a x+1}}",1,"(Sqrt[c - c/(a^2*x^2)]*x*((4 + a*x)*Sqrt[1 - a^2*x^2] + 6*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(2*a*Sqrt[1 - a^2*x^2])","A",0
889,1,80,116,0.1001528,"\int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx","Integrate[E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)],x]","\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a^2 x^2-1}+2 \log \left(\sqrt{a^2 x^2-1}+a x\right)-\tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)\right)}{\sqrt{a^2 x^2-1}}","x \sqrt{c-\frac{c}{a^2 x^2}}-\frac{2 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{a x+1}}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{\sqrt{1-a x} \sqrt{a x+1}}",1,"(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[-1 + a^2*x^2] - ArcTan[1/Sqrt[-1 + a^2*x^2]] + 2*Log[a*x + Sqrt[-1 + a^2*x^2]]))/Sqrt[-1 + a^2*x^2]","A",0
890,1,82,117,0.1078072,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a^2 x^2-1}+a x \log \left(\sqrt{a^2 x^2-1}+a x\right)-2 a x \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)\right)}{\sqrt{a^2 x^2-1}}","\sqrt{c-\frac{c}{a^2 x^2}}-\frac{a x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{a x+1}}+\frac{2 a x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{\sqrt{1-a x} \sqrt{a x+1}}",1,"(Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2] - 2*a*x*ArcTan[1/Sqrt[-1 + a^2*x^2]] + a*x*Log[a*x + Sqrt[-1 + a^2*x^2]]))/Sqrt[-1 + a^2*x^2]","A",0
891,1,78,111,0.0950604,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^2} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^2,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left((4 a x+1) \sqrt{a^2 x^2-1}-3 a^2 x^2 \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)\right)}{2 x \sqrt{a^2 x^2-1}}","\frac{3}{2} a \sqrt{c-\frac{c}{a^2 x^2}}+\frac{(a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}{2 x}+\frac{3 a^2 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{2 \sqrt{1-a x} \sqrt{a x+1}}",1,"(Sqrt[c - c/(a^2*x^2)]*((1 + 4*a*x)*Sqrt[-1 + a^2*x^2] - 3*a^2*x^2*ArcTan[1/Sqrt[-1 + a^2*x^2]]))/(2*x*Sqrt[-1 + a^2*x^2])","A",0
892,1,86,137,0.1096702,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^3} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^3,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a^2 x^2-1} \left(5 a^2 x^2+3 a x+1\right)-3 a^3 x^3 \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)\right)}{3 x^2 \sqrt{a^2 x^2-1}}","a^2 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{a (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}{3 x}+\frac{(a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^2}+\frac{a^3 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{\sqrt{1-a x} \sqrt{a x+1}}",1,"(Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(1 + 3*a*x + 5*a^2*x^2) - 3*a^3*x^3*ArcTan[1/Sqrt[-1 + a^2*x^2]]))/(3*x^2*Sqrt[-1 + a^2*x^2])","A",0
893,1,94,156,0.142873,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^4} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^4,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a^2 x^2-1} \left(32 a^3 x^3+21 a^2 x^2+16 a x+6\right)-21 a^4 x^4 \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)\right)}{24 x^3 \sqrt{a^2 x^2-1}}","\frac{7 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{8 x}+\frac{2 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^2}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{4 x^3}+\frac{7 a^4 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{8 \sqrt{1-a x} \sqrt{a x+1}}+\frac{4}{3} a^3 \sqrt{c-\frac{c}{a^2 x^2}}",1,"(Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(6 + 16*a*x + 21*a^2*x^2 + 32*a^3*x^3) - 21*a^4*x^4*ArcTan[1/Sqrt[-1 + a^2*x^2]]))/(24*x^3*Sqrt[-1 + a^2*x^2])","A",0
894,1,102,181,0.1154089,"\int \frac{e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^5} \, dx","Integrate[(E^(2*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^5,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a^2 x^2-1} \left(24 a^4 x^4+15 a^3 x^3+12 a^2 x^2+10 a x+4\right)-15 a^5 x^5 \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)\right)}{20 x^4 \sqrt{a^2 x^2-1}}","\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}+\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}+\frac{3 a^5 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{4 \sqrt{1-a x} \sqrt{a x+1}}+\frac{6}{5} a^4 \sqrt{c-\frac{c}{a^2 x^2}}+\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}",1,"(Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(4 + 10*a*x + 12*a^2*x^2 + 15*a^3*x^3 + 24*a^4*x^4) - 15*a^5*x^5*ArcTan[1/Sqrt[-1 + a^2*x^2]]))/(20*x^4*Sqrt[-1 + a^2*x^2])","A",0
895,1,71,186,0.060297,"\int e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^3 \, dx","Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x^3,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(\frac{4 \log (1-a x)}{a^3}+\frac{4 x}{a^2}+\frac{a x^4}{4}+\frac{2 x^2}{a}+x^3\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{2 x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^4 \sqrt{c-\frac{c}{a^2 x^2}}}{4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{a^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 x \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*((4*x)/a^2 + (2*x^2)/a + x^3 + (a*x^4)/4 + (4*Log[1 - a*x])/a^3))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
896,1,63,152,0.0485309,"\int e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^2 \, dx","Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x^2,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(a x \left(2 a^2 x^2+9 a x+24\right)+24 \log (1-a x)\right)}{6 a^3 \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{3 x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 x \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(a*x*(24 + 9*a*x + 2*a^2*x^2) + 24*Log[1 - a*x]))/(6*a^3*Sqrt[1 - 1/(a^2*x^2)])","A",1
897,1,54,113,0.0317295,"\int e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x \, dx","Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)]*x,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} (a x (a x+6)+8 \log (1-a x))}{2 a^2 \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 x \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(a*x*(6 + a*x) + 8*Log[1 - a*x]))/(2*a^2*Sqrt[1 - 1/(a^2*x^2)])","A",1
898,1,50,109,0.0318327,"\int e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx","Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)],x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} (a x+4 \log (1-a x)-\log (x))}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(a*x - Log[x] + 4*Log[1 - a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
899,1,52,108,0.0410034,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(-3 a \log (x)+4 a \log (1-a x)+\frac{1}{x}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{a x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{\sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(x^(-1) - 3*a*Log[x] + 4*a*Log[1 - a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
900,1,66,147,0.0522965,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^2} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^2,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(-4 a^2 \log (x)+4 a^2 \log (1-a x)+\frac{3 a}{x}+\frac{1}{2 x^2}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{3 \sqrt{c-\frac{c}{a^2 x^2}}}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{2 a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{\sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(1/(2*x^2) + (3*a)/x - 4*a^2*Log[x] + 4*a^2*Log[1 - a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
901,1,76,188,0.0549151,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^3} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^3,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(-4 a^3 \log (x)+4 a^3 \log (1-a x)+\frac{4 a^2}{x}+\frac{3 a}{2 x^2}+\frac{1}{3 x^3}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{4 a \sqrt{c-\frac{c}{a^2 x^2}}}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a^2 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a^2 \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{3 a x^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(1/(3*x^3) + (3*a)/(2*x^2) + (4*a^2)/x - 4*a^3*Log[x] + 4*a^3*Log[1 - a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
902,1,81,222,0.0702678,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^4} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^4,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(-4 a^4 \log (x)+4 a^4 \log (1-a x)+\frac{4 a^3}{x}+\frac{2 a^2}{x^2}+\frac{a}{x^3}+\frac{1}{4 x^4}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{4 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 a \sqrt{c-\frac{c}{a^2 x^2}}}{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{4 a x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{x^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a^3 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a^3 \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{\sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(1/(4*x^4) + a/x^3 + (2*a^2)/x^2 + (4*a^3)/x - 4*a^4*Log[x] + 4*a^4*Log[1 - a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
903,1,90,264,0.0799947,"\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^5} \, dx","Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^5,x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(-4 a^5 \log (x)+4 a^5 \log (1-a x)+\frac{240 a^4 x^4+120 a^3 x^3+80 a^2 x^2+45 a x+12}{60 x^5}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{2 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 a x^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a^4 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a^4 \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*((12 + 45*a*x + 80*a^2*x^2 + 120*a^3*x^3 + 240*a^4*x^4)/(60*x^5) - 4*a^5*Log[x] + 4*a^5*Log[1 - a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
904,1,50,81,0.0465984,"\int e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^m \, dx","Integrate[(Sqrt[c - c/(a^2*x^2)]*x^m)/E^ArcCoth[a*x],x]","\frac{x^m \sqrt{c-\frac{c}{a^2 x^2}} \left(\frac{a x}{m+1}-\frac{1}{m}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^{m+1} \sqrt{c-\frac{c}{a^2 x^2}}}{(m+1) \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x^m \sqrt{c-\frac{c}{a^2 x^2}}}{a m \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*x^m*(-m^(-1) + (a*x)/(1 + m)))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
905,1,45,76,0.0301365,"\int e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^2 \, dx","Integrate[(Sqrt[c - c/(a^2*x^2)]*x^2)/E^ArcCoth[a*x],x]","\frac{x^2 (2 a x-3) \sqrt{c-\frac{c}{a^2 x^2}}}{6 a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*x^2*(-3 + 2*a*x))/(6*a*Sqrt[1 - 1/(a^2*x^2)])","A",1
906,1,42,72,0.0266614,"\int e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x \, dx","Integrate[(Sqrt[c - c/(a^2*x^2)]*x)/E^ArcCoth[a*x],x]","\frac{x (a x-2) \sqrt{c-\frac{c}{a^2 x^2}}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*x*(-2 + a*x))/(2*a*Sqrt[1 - 1/(a^2*x^2)])","A",1
907,1,41,68,0.0209921,"\int e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/E^ArcCoth[a*x],x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} (a x-\log (x))}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(a*x - Log[x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
908,1,41,69,0.0266429,"\int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/(E^ArcCoth[a*x]*x),x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(a \log (x)+\frac{1}{x}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{a x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(x^(-1) + a*Log[x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
909,1,47,47,0.0265485,"\int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^2} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/(E^ArcCoth[a*x]*x^2),x]","\frac{\left(\frac{1}{2 x^2}-\frac{a}{x}\right) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{(1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(1/(2*x^2) - a/x))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",0
910,1,93,163,0.0970816,"\int e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^3 \, dx","Integrate[(Sqrt[c - c/(a^2*x^2)]*x^3)/E^(2*ArcCoth[a*x]),x]","\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left(21 \log \left(\sqrt{a^2 x^2-1}+a x\right)+\sqrt{a^2 x^2-1} \left(6 a^3 x^3-16 a^2 x^2+21 a x-32\right)\right)}{24 a^3 \sqrt{a^2 x^2-1}}","\frac{x^2 (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^2}-\frac{x (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^3}-\frac{7 x (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}{24 a^3}-\frac{7 x \sqrt{c-\frac{c}{a^2 x^2}}}{8 a^3}-\frac{7 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{8 a^3 \sqrt{a x+1} \sqrt{1-a x}}",1,"(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[-1 + a^2*x^2]*(-32 + 21*a*x - 16*a^2*x^2 + 6*a^3*x^3) + 21*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(24*a^3*Sqrt[-1 + a^2*x^2])","A",0
911,1,84,124,0.0849091,"\int e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^2 \, dx","Integrate[(Sqrt[c - c/(a^2*x^2)]*x^2)/E^(2*ArcCoth[a*x]),x]","\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a^2 x^2-1} \left(a^2 x^2-3 a x+5\right)-3 \log \left(\sqrt{a^2 x^2-1}+a x\right)\right)}{3 a^2 \sqrt{a^2 x^2-1}}","\frac{x (1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^2}+\frac{x (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^2}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{a^2}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{a^2 \sqrt{a x+1} \sqrt{1-a x}}",1,"(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[-1 + a^2*x^2]*(5 - 3*a*x + a^2*x^2) - 3*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(3*a^2*Sqrt[-1 + a^2*x^2])","A",0
912,1,100,99,0.0675137,"\int e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x \, dx","Integrate[(Sqrt[c - c/(a^2*x^2)]*x)/E^(2*ArcCoth[a*x]),x]","-\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a x+1} \left(a^2 x^2-5 a x+4\right)-6 \sqrt{1-a x} \sin ^{-1}\left(\frac{\sqrt{1-a x}}{\sqrt{2}}\right)\right)}{2 a \sqrt{1-a x} \sqrt{1-a^2 x^2}}","-\frac{x (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}{2 a}-\frac{3 x \sqrt{c-\frac{c}{a^2 x^2}}}{2 a}-\frac{3 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{2 a \sqrt{1-a x} \sqrt{a x+1}}",1,"-1/2*(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[1 + a*x]*(4 - 5*a*x + a^2*x^2) - 6*Sqrt[1 - a*x]*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(a*Sqrt[1 - a*x]*Sqrt[1 - a^2*x^2])","A",0
913,1,80,116,0.0850012,"\int e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/E^(2*ArcCoth[a*x]),x]","\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a^2 x^2-1}-2 \log \left(\sqrt{a^2 x^2-1}+a x\right)-\tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)\right)}{\sqrt{a^2 x^2-1}}","x \sqrt{c-\frac{c}{a^2 x^2}}+\frac{2 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{a x+1}}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{\sqrt{1-a x} \sqrt{a x+1}}",1,"(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[-1 + a^2*x^2] - ArcTan[1/Sqrt[-1 + a^2*x^2]] - 2*Log[a*x + Sqrt[-1 + a^2*x^2]]))/Sqrt[-1 + a^2*x^2]","A",0
914,1,82,117,0.0933984,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcCoth[a*x])*x),x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a^2 x^2-1}+a x \log \left(\sqrt{a^2 x^2-1}+a x\right)+2 a x \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)\right)}{\sqrt{a^2 x^2-1}}","\sqrt{c-\frac{c}{a^2 x^2}}-\frac{a x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{\sqrt{1-a x} \sqrt{a x+1}}-\frac{2 a x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{\sqrt{1-a x} \sqrt{a x+1}}",1,"(Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2] + 2*a*x*ArcTan[1/Sqrt[-1 + a^2*x^2]] + a*x*Log[a*x + Sqrt[-1 + a^2*x^2]]))/Sqrt[-1 + a^2*x^2]","A",0
915,1,78,112,0.0752985,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^2} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcCoth[a*x])*x^2),x]","-\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left((4 a x-1) \sqrt{a^2 x^2-1}+3 a^2 x^2 \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)\right)}{2 x \sqrt{a^2 x^2-1}}","-\frac{3}{2} a \sqrt{c-\frac{c}{a^2 x^2}}+\frac{(1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}{2 x}+\frac{3 a^2 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{2 \sqrt{1-a x} \sqrt{a x+1}}",1,"-1/2*(Sqrt[c - c/(a^2*x^2)]*((-1 + 4*a*x)*Sqrt[-1 + a^2*x^2] + 3*a^2*x^2*ArcTan[1/Sqrt[-1 + a^2*x^2]]))/(x*Sqrt[-1 + a^2*x^2])","A",0
916,1,86,140,0.083648,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^3} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcCoth[a*x])*x^3),x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(\sqrt{a^2 x^2-1} \left(5 a^2 x^2-3 a x+1\right)+3 a^3 x^3 \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)\right)}{3 x^2 \sqrt{a^2 x^2-1}}","a^2 \sqrt{c-\frac{c}{a^2 x^2}}-\frac{a (1-a x) \sqrt{c-\frac{c}{a^2 x^2}}}{3 x}+\frac{(1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^2}-\frac{a^3 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{\sqrt{1-a x} \sqrt{a x+1}}",1,"(Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(1 - 3*a*x + 5*a^2*x^2) + 3*a^3*x^3*ArcTan[1/Sqrt[-1 + a^2*x^2]]))/(3*x^2*Sqrt[-1 + a^2*x^2])","A",0
917,1,94,156,0.0894051,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^4} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcCoth[a*x])*x^4),x]","-\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(21 a^4 x^4 \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)+\sqrt{a^2 x^2-1} \left(32 a^3 x^3-21 a^2 x^2+16 a x-6\right)\right)}{24 x^3 \sqrt{a^2 x^2-1}}","\frac{7 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{8 x}-\frac{2 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^2}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{4 x^3}+\frac{7 a^4 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{8 \sqrt{1-a x} \sqrt{a x+1}}-\frac{4}{3} a^3 \sqrt{c-\frac{c}{a^2 x^2}}",1,"-1/24*(Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(-6 + 16*a*x - 21*a^2*x^2 + 32*a^3*x^3) + 21*a^4*x^4*ArcTan[1/Sqrt[-1 + a^2*x^2]]))/(x^3*Sqrt[-1 + a^2*x^2])","A",0
918,1,102,181,0.1013731,"\int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^5} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(2*ArcCoth[a*x])*x^5),x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(15 a^5 x^5 \tan ^{-1}\left(\frac{1}{\sqrt{a^2 x^2-1}}\right)+\sqrt{a^2 x^2-1} \left(24 a^4 x^4-15 a^3 x^3+12 a^2 x^2-10 a x+4\right)\right)}{20 x^4 \sqrt{a^2 x^2-1}}","\frac{3 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{5 x^2}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 x^4}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^3}-\frac{3 a^5 x \sqrt{c-\frac{c}{a^2 x^2}} \tanh ^{-1}\left(\sqrt{1-a x} \sqrt{a x+1}\right)}{4 \sqrt{1-a x} \sqrt{a x+1}}+\frac{6}{5} a^4 \sqrt{c-\frac{c}{a^2 x^2}}-\frac{3 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x}",1,"(Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(4 - 10*a*x + 12*a^2*x^2 - 15*a^3*x^3 + 24*a^4*x^4) + 15*a^5*x^5*ArcTan[1/Sqrt[-1 + a^2*x^2]]))/(20*x^4*Sqrt[-1 + a^2*x^2])","A",0
919,1,69,186,0.0567105,"\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^3 \, dx","Integrate[(Sqrt[c - c/(a^2*x^2)]*x^3)/E^(3*ArcCoth[a*x]),x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(a x \left(a^3 x^3-4 a^2 x^2+8 a x-16\right)+16 \log (a x+1)\right)}{4 a^4 \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{2 x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^4 \sqrt{c-\frac{c}{a^2 x^2}}}{4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{a^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 x \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(a*x*(-16 + 8*a*x - 4*a^2*x^2 + a^3*x^3) + 16*Log[1 + a*x]))/(4*a^4*Sqrt[1 - 1/(a^2*x^2)])","A",1
920,1,62,151,0.0455263,"\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^2 \, dx","Integrate[(Sqrt[c - c/(a^2*x^2)]*x^2)/E^(3*ArcCoth[a*x]),x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(a x \left(2 a^2 x^2-9 a x+24\right)-24 \log (a x+1)\right)}{6 a^3 \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{3 x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 x \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{a^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(a*x*(24 - 9*a*x + 2*a^2*x^2) - 24*Log[1 + a*x]))/(6*a^3*Sqrt[1 - 1/(a^2*x^2)])","A",1
921,1,53,112,0.0328041,"\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x \, dx","Integrate[(Sqrt[c - c/(a^2*x^2)]*x)/E^(3*ArcCoth[a*x]),x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} (a x (a x-6)+8 \log (a x+1))}{2 a^2 \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 x \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(a*x*(-6 + a*x) + 8*Log[1 + a*x]))/(2*a^2*Sqrt[1 - 1/(a^2*x^2)])","A",1
922,1,47,107,0.030334,"\int e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/E^(3*ArcCoth[a*x]),x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} (a x-4 \log (a x+1)+\log (x))}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{a \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(a*x + Log[x] - 4*Log[1 + a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
923,1,53,108,0.0386872,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcCoth[a*x])*x),x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(-3 a \log (x)+4 a \log (a x+1)-\frac{1}{x}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{a x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{3 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{\sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(-x^(-1) - 3*a*Log[x] + 4*a*Log[1 + a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
924,1,65,146,0.0487935,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^2} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcCoth[a*x])*x^2),x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(4 a^2 \log (x)-4 a^2 \log (a x+1)+\frac{3 a}{x}-\frac{1}{2 x^2}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{3 \sqrt{c-\frac{c}{a^2 x^2}}}{x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{2 a x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{\sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(-1/2*1/x^2 + (3*a)/x + 4*a^2*Log[x] - 4*a^2*Log[1 + a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
925,1,75,187,0.0526259,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^3} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcCoth[a*x])*x^3),x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(-4 a^3 \log (x)+4 a^3 \log (a x+1)-\frac{4 a^2}{x}+\frac{3 a}{2 x^2}-\frac{1}{3 x^3}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","-\frac{4 a \sqrt{c-\frac{c}{a^2 x^2}}}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a^2 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a^2 \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{3 a x^3 \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(-1/3*1/x^3 + (3*a)/(2*x^2) - (4*a^2)/x - 4*a^3*Log[x] + 4*a^3*Log[1 + a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
926,1,80,221,0.0605931,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^4} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcCoth[a*x])*x^4),x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(4 a^4 \log (x)-4 a^4 \log (a x+1)+\frac{4 a^3}{x}-\frac{2 a^2}{x^2}+\frac{a}{x^3}-\frac{1}{4 x^4}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{4 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2 a \sqrt{c-\frac{c}{a^2 x^2}}}{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{4 a x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a^3 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a^3 \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{\sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(-1/4*1/x^4 + a/x^3 - (2*a^2)/x^2 + (4*a^3)/x + 4*a^4*Log[x] - 4*a^4*Log[1 + a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
927,1,89,263,0.077317,"\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^5} \, dx","Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcCoth[a*x])*x^5),x]","\frac{\sqrt{c-\frac{c}{a^2 x^2}} \left(-4 a^5 \log (x)+4 a^5 \log (a x+1)-\frac{240 a^4 x^4-120 a^3 x^3+80 a^2 x^2-45 a x+12}{60 x^5}\right)}{a \sqrt{1-\frac{1}{a^2 x^2}}}","\frac{2 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 a x^5 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a^4 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a^4 \sqrt{c-\frac{c}{a^2 x^2}} \log (a x+1)}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{x \sqrt{1-\frac{1}{a^2 x^2}}}",1,"(Sqrt[c - c/(a^2*x^2)]*(-1/60*(12 - 45*a*x + 80*a^2*x^2 - 120*a^3*x^3 + 240*a^4*x^4)/x^5 - 4*a^5*Log[x] + 4*a^5*Log[1 + a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])","A",1
928,1,123,154,0.2626362,"\int e^{n \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right) \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - c/(a^2*x^2)),x]","\frac{c e^{n \coth ^{-1}(a x)} \left(n e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)+(n+2) \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)+4 e^{2 \coth ^{-1}(a x)} \, _2F_1\left(2,\frac{n}{2}+1;\frac{n}{2}+2;-e^{2 \coth ^{-1}(a x)}\right)+a n x+2 a x\right)}{a (n+2)}","\frac{4 c \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}} \left(\frac{1}{a x}+1\right)^{\frac{n-2}{2}} \, _2F_1\left(2,1-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a (2-n)}-\frac{c 2^{\frac{n}{2}+1} \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}} \, _2F_1\left(1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{a-\frac{1}{x}}{2 a}\right)}{a (2-n)}",1,"(c*E^(n*ArcCoth[a*x])*(2*a*x + a*n*x + E^(2*ArcCoth[a*x])*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])] + 4*E^(2*ArcCoth[a*x])*Hypergeometric2F1[2, 1 + n/2, 2 + n/2, -E^(2*ArcCoth[a*x])]))/(a*(2 + n))","A",0
929,1,94,150,0.2747992,"\int \frac{e^{n \coth ^{-1}(a x)}}{c-\frac{c}{a^2 x^2}} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - c/(a^2*x^2)),x]","\frac{e^{n \coth ^{-1}(a x)} \left(n^2 e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)+(n+2) \left(n \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)+a n x-1\right)\right)}{a c n (n+2)}","\frac{2 \left(\frac{1}{a x}+1\right)^{n/2} \left(1-\frac{1}{a x}\right)^{-n/2} \, _2F_1\left(1,\frac{n}{2};\frac{n+2}{2};\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\right)}{a c}-\frac{(n+1) \left(\frac{1}{a x}+1\right)^{n/2} \left(1-\frac{1}{a x}\right)^{-n/2}}{a c n}+\frac{x \left(\frac{1}{a x}+1\right)^{n/2} \left(1-\frac{1}{a x}\right)^{-n/2}}{c}",1,"(E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n^2*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(-1 + a*n*x + n*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(a*c*n*(2 + n))","A",0
930,1,180,289,0.9297775,"\int \frac{e^{n \coth ^{-1}(a x)}}{\left(c-\frac{c}{a^2 x^2}\right)^2} \, dx","Integrate[E^(n*ArcCoth[a*x])/(c - c/(a^2*x^2))^2,x]","\frac{e^{n \coth ^{-1}(a x)} \left(a^3 n^3 x^3-4 a^3 n x^3+(n-2) n^2 \left(a^2 x^2-1\right) e^{2 \coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;e^{2 \coth ^{-1}(a x)}\right)+\left(n^2-4\right) n \left(a^2 x^2-1\right) \, _2F_1\left(1,\frac{n}{2};\frac{n}{2}+1;e^{2 \coth ^{-1}(a x)}\right)-2 a^2 n^2 x^2+6 a^2 x^2-a n^3 x+6 a n x+n^2-6\right)}{a c^2 (n-2) n (n+2) \left(a^2 x^2-1\right)}","\frac{2 \left(\frac{1}{a x}+1\right)^{n/2} \left(1-\frac{1}{a x}\right)^{-n/2} \, _2F_1\left(1,\frac{n}{2};\frac{n+2}{2};\frac{a+\frac{1}{x}}{a-\frac{1}{x}}\right)}{a c^2}-\frac{\left(n^2+4 n+6\right) \left(\frac{1}{a x}+1\right)^{\frac{n-2}{2}} \left(1-\frac{1}{a x}\right)^{-n/2}}{a c^2 n (n+2)}+\frac{\left(-n^3-n^2+4 n+6\right) \left(\frac{1}{a x}+1\right)^{\frac{n-2}{2}} \left(1-\frac{1}{a x}\right)^{1-\frac{n}{2}}}{a c^2 (2-n) n (n+2)}-\frac{(n+3) \left(\frac{1}{a x}+1\right)^{\frac{n-2}{2}} \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-1}}{a c^2 (n+2)}+\frac{x \left(\frac{1}{a x}+1\right)^{\frac{n-2}{2}} \left(1-\frac{1}{a x}\right)^{-\frac{n}{2}-1}}{c^2}",1,"(E^(n*ArcCoth[a*x])*(-6 + n^2 + 6*a*n*x - a*n^3*x + 6*a^2*x^2 - 2*a^2*n^2*x^2 - 4*a^3*n*x^3 + a^3*n^3*x^3 + E^(2*ArcCoth[a*x])*(-2 + n)*n^2*(-1 + a^2*x^2)*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[a*x])] + n*(-4 + n^2)*(-1 + a^2*x^2)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])]))/(a*c^2*(-2 + n)*n*(2 + n)*(-1 + a^2*x^2))","A",0
931,1,146,295,0.5166984,"\int e^{n \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} \, dx","Integrate[E^(n*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)],x]","\frac{a x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a^2 x^2}} e^{n \coth ^{-1}(a x)} \left(a (n+1) x \sqrt{1-\frac{1}{a^2 x^2}}+2 e^{\coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-e^{2 \coth ^{-1}(a x)}\right)+2 n e^{\coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};e^{2 \coth ^{-1}(a x)}\right)\right)}{(n+1) \left(a^2 x^2-1\right)}","\frac{2 n \sqrt{c-\frac{c}{a^2 x^2}} \left(\frac{1}{a x}+1\right)^{\frac{n-1}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}} \, _2F_1\left(1,\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a (1-n) \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{2^{\frac{n+1}{2}} \sqrt{c-\frac{c}{a^2 x^2}} \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}} \, _2F_1\left(\frac{1-n}{2},\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{2 a}\right)}{a (1-n) \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left(\frac{1}{a x}+1\right)^{\frac{n+1}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}",1,"(a*E^(n*ArcCoth[a*x])*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a^2*x^2)]*x^2*(a*(1 + n)*Sqrt[1 - 1/(a^2*x^2)]*x + 2*E^ArcCoth[a*x]*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -E^(2*ArcCoth[a*x])] + 2*E^ArcCoth[a*x]*n*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])]))/((1 + n)*(-1 + a^2*x^2))","A",0
932,1,112,183,0.4118856,"\int \frac{e^{n \coth ^{-1}(a x)}}{\sqrt{c-\frac{c}{a^2 x^2}}} \, dx","Integrate[E^(n*ArcCoth[a*x])/Sqrt[c - c/(a^2*x^2)],x]","\frac{\left(a^2 x^2-1\right) e^{n \coth ^{-1}(a x)} \left(a (n+1) x \sqrt{1-\frac{1}{a^2 x^2}}+2 n e^{\coth ^{-1}(a x)} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};e^{2 \coth ^{-1}(a x)}\right)\right)}{a^3 (n+1) x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a^2 x^2}}}","\frac{2 n \sqrt{1-\frac{1}{a^2 x^2}} \left(\frac{1}{a x}+1\right)^{\frac{n-1}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}} \, _2F_1\left(1,\frac{1-n}{2};\frac{3-n}{2};\frac{a-\frac{1}{x}}{a+\frac{1}{x}}\right)}{a (1-n) \sqrt{c-\frac{c}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left(\frac{1}{a x}+1\right)^{\frac{n+1}{2}} \left(1-\frac{1}{a x}\right)^{\frac{1-n}{2}}}{\sqrt{c-\frac{c}{a^2 x^2}}}",1,"(E^(n*ArcCoth[a*x])*(-1 + a^2*x^2)*(a*(1 + n)*Sqrt[1 - 1/(a^2*x^2)]*x + 2*E^ArcCoth[a*x]*n*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, E^(2*ArcCoth[a*x])]))/(a^3*(1 + n)*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a^2*x^2)]*x^2)","A",0
933,0,0,116,0.5931026,"\int e^{n \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^p \, dx","Integrate[E^(n*ArcCoth[a*x])*(c - c/(a^2*x^2))^p,x]","\int e^{n \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^p \, dx","-\frac{2^{-\frac{n}{2}+p+1} \left(1-\frac{1}{a^2 x^2}\right)^{-p} \left(c-\frac{c}{a^2 x^2}\right)^p \left(\frac{1}{a x}+1\right)^{\frac{n}{2}+p+1} F_1\left(\frac{n}{2}+p+1;\frac{1}{2} (n-2 p),2;\frac{n}{2}+p+2;\frac{a+\frac{1}{x}}{2 a},1+\frac{1}{a x}\right)}{a (n+2 p+2)}",1,"Integrate[E^(n*ArcCoth[a*x])*(c - c/(a^2*x^2))^p, x]","F",-1
934,0,0,76,0.6251342,"\int e^{-2 p \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^p \, dx","Integrate[(c - c/(a^2*x^2))^p/E^(2*p*ArcCoth[a*x]),x]","\int e^{-2 p \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^p \, dx","\frac{\left(1-\frac{1}{a^2 x^2}\right)^{-p} \left(1-\frac{1}{a x}\right)^{2 p+1} \left(c-\frac{c}{a^2 x^2}\right)^p \, _2F_1\left(2,2 p+1;2 (p+1);1-\frac{1}{a x}\right)}{a (2 p+1)}",1,"Integrate[(c - c/(a^2*x^2))^p/E^(2*p*ArcCoth[a*x]), x]","F",-1
935,0,0,75,0.6039286,"\int e^{2 p \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^p \, dx","Integrate[E^(2*p*ArcCoth[a*x])*(c - c/(a^2*x^2))^p,x]","\int e^{2 p \coth ^{-1}(a x)} \left(c-\frac{c}{a^2 x^2}\right)^p \, dx","-\frac{\left(1-\frac{1}{a^2 x^2}\right)^{-p} \left(\frac{1}{a x}+1\right)^{2 p+1} \left(c-\frac{c}{a^2 x^2}\right)^p \, _2F_1\left(2,2 p+1;2 (p+1);1+\frac{1}{a x}\right)}{a (2 p+1)}",1,"Integrate[E^(2*p*ArcCoth[a*x])*(c - c/(a^2*x^2))^p, x]","F",-1