1,1,71,71,0.0716132,"\int e^{\cot ^{-1}(x)} \, dx","Integrate[E^ArcCot[x],x]","i e^{\cot ^{-1}(x)} \, _2F_1\left(-\frac{i}{2},1;1-\frac{i}{2};e^{2 i \cot ^{-1}(x)}\right)+\left(\frac{2}{5}+\frac{i}{5}\right) e^{(1+2 i) \cot ^{-1}(x)} \, _2F_1\left(1,1-\frac{i}{2};2-\frac{i}{2};e^{2 i \cot ^{-1}(x)}\right)+x e^{\cot ^{-1}(x)}","\left(\frac{4}{5}+\frac{8 i}{5}\right) \left(\frac{x-i}{x}\right)^{1+\frac{i}{2}} \left(\frac{x+i}{x}\right)^{-1-\frac{i}{2}} \, _2F_1\left(1+\frac{i}{2},2;2+\frac{i}{2};\frac{1-\frac{i}{x}}{1+\frac{i}{x}}\right)",1,"E^ArcCot[x]*x + I*E^ArcCot[x]*Hypergeometric2F1[-1/2*I, 1, 1 - I/2, E^((2*I)*ArcCot[x])] + (2/5 + I/5)*E^((1 + 2*I)*ArcCot[x])*Hypergeometric2F1[1, 1 - I/2, 2 - I/2, E^((2*I)*ArcCot[x])]","A",0
2,1,9,9,0.0359279,"\int \frac{e^{\cot ^{-1}(x)}}{a+a x^2} \, dx","Integrate[E^ArcCot[x]/(a + a*x^2),x]","-\frac{e^{\cot ^{-1}(x)}}{a}","-\frac{e^{\cot ^{-1}(x)}}{a}",1,"-(E^ArcCot[x]/a)","A",1
3,1,28,35,0.1165191,"\int \frac{e^{\cot ^{-1}(x)}}{\left(a+a x^2\right)^2} \, dx","Integrate[E^ArcCot[x]/(a + a*x^2)^2,x]","-\frac{\left(2 x^2-2 x+3\right) e^{\cot ^{-1}(x)}}{5 a^2 \left(x^2+1\right)}","-\frac{(1-2 x) e^{\cot ^{-1}(x)}}{5 a^2 \left(x^2+1\right)}-\frac{2 e^{\cot ^{-1}(x)}}{5 a^2}",1,"-1/5*(E^ArcCot[x]*(3 - 2*x + 2*x^2))/(a^2*(1 + x^2))","A",1
4,1,38,58,0.1589437,"\int \frac{e^{\cot ^{-1}(x)}}{\left(a+a x^2\right)^3} \, dx","Integrate[E^ArcCot[x]/(a + a*x^2)^3,x]","-\frac{\left(24 x^4-24 x^3+60 x^2-44 x+41\right) e^{\cot ^{-1}(x)}}{85 a^3 \left(x^2+1\right)^2}","-\frac{(1-4 x) e^{\cot ^{-1}(x)}}{17 a^3 \left(x^2+1\right)^2}-\frac{12 (1-2 x) e^{\cot ^{-1}(x)}}{85 a^3 \left(x^2+1\right)}-\frac{24 e^{\cot ^{-1}(x)}}{85 a^3}",1,"-1/85*(E^ArcCot[x]*(41 - 44*x + 60*x^2 - 24*x^3 + 24*x^4))/(a^3*(1 + x^2)^2)","A",1
5,1,25,27,0.0733985,"\int \frac{e^{\cot ^{-1}(x)}}{\left(a+a x^2\right)^{3/2}} \, dx","Integrate[E^ArcCot[x]/(a + a*x^2)^(3/2),x]","\frac{(x-1) e^{\cot ^{-1}(x)}}{2 a \sqrt{a \left(x^2+1\right)}}","-\frac{(1-x) e^{\cot ^{-1}(x)}}{2 a \sqrt{a x^2+a}}",1,"(E^ArcCot[x]*(-1 + x))/(2*a*Sqrt[a*(1 + x^2)])","A",1
6,1,51,55,0.1685411,"\int \frac{e^{\cot ^{-1}(x)}}{\left(a+a x^2\right)^{5/2}} \, dx","Integrate[E^ArcCot[x]/(a + a*x^2)^(5/2),x]","\frac{e^{\cot ^{-1}(x)} \left(-3 \sqrt{\frac{1}{x^2}+1} x \cos \left(3 \cot ^{-1}(x)\right)+15 x+2 \cos \left(2 \cot ^{-1}(x)\right)-14\right)}{40 a^2 \sqrt{a \left(x^2+1\right)}}","-\frac{3 (1-x) e^{\cot ^{-1}(x)}}{10 a^2 \sqrt{a x^2+a}}-\frac{(1-3 x) e^{\cot ^{-1}(x)}}{10 a \left(a x^2+a\right)^{3/2}}",1,"(E^ArcCot[x]*(-14 + 15*x + 2*Cos[2*ArcCot[x]] - 3*Sqrt[1 + x^(-2)]*x*Cos[3*ArcCot[x]]))/(40*a^2*Sqrt[a*(1 + x^2)])","A",1
7,1,95,82,0.2119352,"\int \frac{e^{\cot ^{-1}(x)}}{\left(a+a x^2\right)^{7/2}} \, dx","Integrate[E^ArcCot[x]/(a + a*x^2)^(7/2),x]","\frac{e^{\cot ^{-1}(x)} \left(-39 \sqrt{\frac{1}{x^2}+1} x \cos \left(3 \cot ^{-1}(x)\right)+5 \sqrt{\frac{1}{x^2}+1} x \cos \left(5 \cot ^{-1}(x)\right)+13 \sqrt{\frac{1}{x^2}+1} x \sin \left(3 \cot ^{-1}(x)\right)-\sqrt{\frac{1}{x^2}+1} x \sin \left(5 \cot ^{-1}(x)\right)+130 x-130\right)}{416 a^3 \sqrt{a \left(x^2+1\right)}}","-\frac{3 (1-x) e^{\cot ^{-1}(x)}}{13 a^3 \sqrt{a x^2+a}}-\frac{(1-3 x) e^{\cot ^{-1}(x)}}{13 a^2 \left(a x^2+a\right)^{3/2}}-\frac{(1-5 x) e^{\cot ^{-1}(x)}}{26 a \left(a x^2+a\right)^{5/2}}",1,"(E^ArcCot[x]*(-130 + 130*x - 39*Sqrt[1 + x^(-2)]*x*Cos[3*ArcCot[x]] + 5*Sqrt[1 + x^(-2)]*x*Cos[5*ArcCot[x]] + 13*Sqrt[1 + x^(-2)]*x*Sin[3*ArcCot[x]] - Sqrt[1 + x^(-2)]*x*Sin[5*ArcCot[x]]))/(416*a^3*Sqrt[a*(1 + x^2)])","A",1
8,1,89,147,0.1755078,"\int \frac{e^{n \cot ^{-1}(a x)}}{\sqrt[3]{c+a^2 c x^2}} \, dx","Integrate[E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(1/3),x]","-\frac{3 \left(a^2 c x^2+c\right)^{2/3} \left(-1+e^{2 i \cot ^{-1}(a x)}\right) e^{(n-2 i) \cot ^{-1}(a x)} \, _2F_1\left(1,\frac{i n}{2}+\frac{1}{3};\frac{i n}{2}+\frac{5}{3};e^{-2 i \cot ^{-1}(a x)}\right)}{a c (3 n-4 i)}","\frac{3 x \sqrt[3]{\frac{1}{a^2 x^2}+1} \left(\frac{a-\frac{i}{x}}{a+\frac{i}{x}}\right)^{\frac{1}{6} (2-3 i n)} \left(1-\frac{i}{a x}\right)^{\frac{1}{6} (-2+3 i n)} \left(1+\frac{i}{a x}\right)^{\frac{1}{6} (4-3 i n)} \, _2F_1\left(-\frac{1}{3},\frac{1}{6} (2-3 i n);\frac{2}{3};\frac{2 i}{\left(a+\frac{i}{x}\right) x}\right)}{\sqrt[3]{a^2 c x^2+c}}",1,"(-3*E^((-2*I + n)*ArcCot[a*x])*(-1 + E^((2*I)*ArcCot[a*x]))*(c + a^2*c*x^2)^(2/3)*Hypergeometric2F1[1, 1/3 + (I/2)*n, 5/3 + (I/2)*n, E^((-2*I)*ArcCot[a*x])])/(a*c*(-4*I + 3*n))","A",0
9,1,89,147,0.1622937,"\int \frac{e^{n \cot ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{2/3}} \, dx","Integrate[E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(2/3),x]","-\frac{3 \sqrt[3]{a^2 c x^2+c} \left(-1+e^{2 i \cot ^{-1}(a x)}\right) e^{(n-2 i) \cot ^{-1}(a x)} \, _2F_1\left(1,\frac{i n}{2}+\frac{2}{3};\frac{i n}{2}+\frac{4}{3};e^{-2 i \cot ^{-1}(a x)}\right)}{a c (3 n-2 i)}","-\frac{3 x \left(\frac{1}{a^2 x^2}+1\right)^{2/3} \left(\frac{a-\frac{i}{x}}{a+\frac{i}{x}}\right)^{\frac{1}{6} (4-3 i n)} \left(1-\frac{i}{a x}\right)^{\frac{1}{6} (-4+3 i n)} \left(1+\frac{i}{a x}\right)^{\frac{1}{6} (2-3 i n)} \, _2F_1\left(\frac{1}{3},\frac{1}{6} (4-3 i n);\frac{4}{3};\frac{2 i}{\left(a+\frac{i}{x}\right) x}\right)}{\left(a^2 c x^2+c\right)^{2/3}}",1,"(-3*E^((-2*I + n)*ArcCot[a*x])*(-1 + E^((2*I)*ArcCot[a*x]))*(c + a^2*c*x^2)^(1/3)*Hypergeometric2F1[1, 2/3 + (I/2)*n, 4/3 + (I/2)*n, E^((-2*I)*ArcCot[a*x])])/(a*c*(-2*I + 3*n))","A",0
10,1,89,207,0.1771353,"\int \frac{e^{n \cot ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{4/3}} \, dx","Integrate[E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(4/3),x]","-\frac{3 \left(-1+e^{2 i \cot ^{-1}(a x)}\right) e^{(n-2 i) \cot ^{-1}(a x)} \, _2F_1\left(1,\frac{i n}{2}+\frac{4}{3};\frac{i n}{2}+\frac{2}{3};e^{-2 i \cot ^{-1}(a x)}\right)}{a c (3 n+2 i) \sqrt[3]{a^2 c x^2+c}}","-\frac{3 (3 n-2 a x) e^{n \cot ^{-1}(a x)}}{a c \left(9 n^2+4\right) \sqrt[3]{a^2 c x^2+c}}-\frac{6 x \sqrt[3]{\frac{1}{a^2 x^2}+1} \left(1-\frac{i}{a x}\right)^{\frac{1}{6} (-2+3 i n)} \left(1+\frac{i}{a x}\right)^{\frac{1}{6} (4-3 i n)} \left(\frac{a-\frac{i}{x}}{a+\frac{i}{x}}\right)^{\frac{1}{6} (2-3 i n)} \, _2F_1\left(-\frac{1}{3},\frac{1}{6} (2-3 i n);\frac{2}{3};\frac{2 i}{\left(a+\frac{i}{x}\right) x}\right)}{c \left(9 n^2+4\right) \sqrt[3]{a^2 c x^2+c}}",1,"(-3*E^((-2*I + n)*ArcCot[a*x])*(-1 + E^((2*I)*ArcCot[a*x]))*Hypergeometric2F1[1, 4/3 + (I/2)*n, 2/3 + (I/2)*n, E^((-2*I)*ArcCot[a*x])])/(a*c*(2*I + 3*n)*(c + a^2*c*x^2)^(1/3))","A",0
11,1,89,207,0.1823899,"\int \frac{e^{n \cot ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/3}} \, dx","Integrate[E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(5/3),x]","-\frac{3 \left(-1+e^{2 i \cot ^{-1}(a x)}\right) e^{(n-2 i) \cot ^{-1}(a x)} \, _2F_1\left(1,\frac{i n}{2}+\frac{5}{3};\frac{i n}{2}+\frac{1}{3};e^{-2 i \cot ^{-1}(a x)}\right)}{a c (3 n+4 i) \left(a^2 c x^2+c\right)^{2/3}}","-\frac{3 (3 n-4 a x) e^{n \cot ^{-1}(a x)}}{a c \left(9 n^2+16\right) \left(a^2 c x^2+c\right)^{2/3}}-\frac{12 x \left(\frac{1}{a^2 x^2}+1\right)^{2/3} \left(\frac{a-\frac{i}{x}}{a+\frac{i}{x}}\right)^{\frac{1}{6} (4-3 i n)} \left(1-\frac{i}{a x}\right)^{\frac{1}{6} (-4+3 i n)} \left(1+\frac{i}{a x}\right)^{\frac{1}{6} (2-3 i n)} \, _2F_1\left(\frac{1}{3},\frac{1}{6} (4-3 i n);\frac{4}{3};\frac{2 i}{\left(a+\frac{i}{x}\right) x}\right)}{c \left(9 n^2+16\right) \left(a^2 c x^2+c\right)^{2/3}}",1,"(-3*E^((-2*I + n)*ArcCot[a*x])*(-1 + E^((2*I)*ArcCot[a*x]))*Hypergeometric2F1[1, 5/3 + (I/2)*n, 1/3 + (I/2)*n, E^((-2*I)*ArcCot[a*x])])/(a*c*(4*I + 3*n)*(c + a^2*c*x^2)^(2/3))","A",0
12,1,100,272,0.1938832,"\int \frac{e^{n \cot ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{7/3}} \, dx","Integrate[E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(7/3),x]","-\frac{3 \left(a^2 c x^2+c\right)^{2/3} \left(-1+e^{2 i \cot ^{-1}(a x)}\right) e^{(n-2 i) \cot ^{-1}(a x)} \, _2F_1\left(1,\frac{i n}{2}+\frac{7}{3};\frac{i n}{2}-\frac{1}{3};e^{-2 i \cot ^{-1}(a x)}\right)}{a c^3 (3 n+8 i) \left(a^2 x^2+1\right)^2}","-\frac{240 x \sqrt[3]{\frac{1}{a^2 x^2}+1} \left(1-\frac{i}{a x}\right)^{\frac{1}{6} (-2+3 i n)} \left(1+\frac{i}{a x}\right)^{\frac{1}{6} (4-3 i n)} \left(\frac{a-\frac{i}{x}}{a+\frac{i}{x}}\right)^{\frac{1}{6} (2-3 i n)} \, _2F_1\left(-\frac{1}{3},\frac{1}{6} (2-3 i n);\frac{2}{3};\frac{2 i}{\left(a+\frac{i}{x}\right) x}\right)}{c^2 \left(9 n^2+4\right) \left(9 n^2+64\right) \sqrt[3]{a^2 c x^2+c}}-\frac{120 (3 n-2 a x) e^{n \cot ^{-1}(a x)}}{a c^2 \left(9 n^2+4\right) \left(9 n^2+64\right) \sqrt[3]{a^2 c x^2+c}}-\frac{3 (3 n-8 a x) e^{n \cot ^{-1}(a x)}}{a c \left(9 n^2+64\right) \left(a^2 c x^2+c\right)^{4/3}}",1,"(-3*E^((-2*I + n)*ArcCot[a*x])*(-1 + E^((2*I)*ArcCot[a*x]))*(c + a^2*c*x^2)^(2/3)*Hypergeometric2F1[1, 7/3 + (I/2)*n, -1/3 + (I/2)*n, E^((-2*I)*ArcCot[a*x])])/(a*c^3*(8*I + 3*n)*(1 + a^2*x^2)^2)","A",0