1,1,71,0,0.014901,"\int e^{\cot ^{-1}(x)} \, dx","Int[E^ArcCot[x],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)","\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,"((4/5 + (8*I)/5)*((-I + x)/x)^(1 + I/2)*Hypergeometric2F1[1 + I/2, 2, 2 + I/2, (1 - I/x)/(1 + I/x)])/((I + x)/x)^(1 + I/2)","A",2,2,4,0.5000,1,"{5102, 131}"
2,1,9,0,0.0204457,"\int \frac{e^{\cot ^{-1}(x)}}{a+a x^2} \, dx","Int[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",1,1,14,0.07143,1,"{5113}"
3,1,35,0,0.0431212,"\int \frac{e^{\cot ^{-1}(x)}}{\left(a+a x^2\right)^2} \, dx","Int[E^ArcCot[x]/(a + a*x^2)^2,x]","-\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}","-\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,"(-2*E^ArcCot[x])/(5*a^2) - (E^ArcCot[x]*(1 - 2*x))/(5*a^2*(1 + x^2))","A",2,2,14,0.1429,1,"{5115, 5113}"
4,1,58,0,0.0668886,"\int \frac{e^{\cot ^{-1}(x)}}{\left(a+a x^2\right)^3} \, dx","Int[E^ArcCot[x]/(a + a*x^2)^3,x]","-\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}","-\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,"(-24*E^ArcCot[x])/(85*a^3) - (E^ArcCot[x]*(1 - 4*x))/(17*a^3*(1 + x^2)^2) - (12*E^ArcCot[x]*(1 - 2*x))/(85*a^3*(1 + x^2))","A",3,2,14,0.1429,1,"{5115, 5113}"
5,1,27,0,0.028173,"\int \frac{e^{\cot ^{-1}(x)}}{\left(a+a x^2\right)^{3/2}} \, dx","Int[E^ArcCot[x]/(a + a*x^2)^(3/2),x]","-\frac{(1-x) e^{\cot ^{-1}(x)}}{2 a \sqrt{a x^2+a}}","-\frac{(1-x) e^{\cot ^{-1}(x)}}{2 a \sqrt{a x^2+a}}",1,"-(E^ArcCot[x]*(1 - x))/(2*a*Sqrt[a + a*x^2])","A",1,1,16,0.06250,1,"{5114}"
6,1,55,0,0.0588136,"\int \frac{e^{\cot ^{-1}(x)}}{\left(a+a x^2\right)^{5/2}} \, dx","Int[E^ArcCot[x]/(a + a*x^2)^(5/2),x]","-\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}}","-\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]*(1 - 3*x))/(10*a*(a + a*x^2)^(3/2)) - (3*E^ArcCot[x]*(1 - x))/(10*a^2*Sqrt[a + a*x^2])","A",2,2,16,0.1250,1,"{5115, 5114}"
7,1,82,0,0.090235,"\int \frac{e^{\cot ^{-1}(x)}}{\left(a+a x^2\right)^{7/2}} \, dx","Int[E^ArcCot[x]/(a + a*x^2)^(7/2),x]","-\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}}","-\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]*(1 - 5*x))/(26*a*(a + a*x^2)^(5/2)) - (E^ArcCot[x]*(1 - 3*x))/(13*a^2*(a + a*x^2)^(3/2)) - (3*E^ArcCot[x]*(1 - x))/(13*a^3*Sqrt[a + a*x^2])","A",3,2,16,0.1250,1,"{5115, 5114}"
8,1,147,0,0.1821029,"\int \frac{e^{n \cot ^{-1}(a x)}}{\sqrt[3]{c+a^2 c x^2}} \, dx","Int[E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(1/3),x]","\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}}","\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*(1 + 1/(a^2*x^2))^(1/3)*((a - I/x)/(a + I/x))^((2 - (3*I)*n)/6)*(1 - I/(a*x))^((-2 + (3*I)*n)/6)*(1 + I/(a*x))^((4 - (3*I)*n)/6)*x*Hypergeometric2F1[-1/3, (2 - (3*I)*n)/6, 2/3, (2*I)/((a + I/x)*x)])/(c + a^2*c*x^2)^(1/3)","A",3,3,23,0.1304,1,"{5122, 5126, 132}"
9,1,147,0,0.1968723,"\int \frac{e^{n \cot ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{2/3}} \, dx","Int[E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(2/3),x]","-\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}}","-\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*(1 + 1/(a^2*x^2))^(2/3)*((a - I/x)/(a + I/x))^((4 - (3*I)*n)/6)*(1 - I/(a*x))^((-4 + (3*I)*n)/6)*(1 + I/(a*x))^((2 - (3*I)*n)/6)*x*Hypergeometric2F1[1/3, (4 - (3*I)*n)/6, 4/3, (2*I)/((a + I/x)*x)])/(c + a^2*c*x^2)^(2/3)","A",3,3,23,0.1304,1,"{5122, 5126, 132}"
10,1,207,0,0.2351556,"\int \frac{e^{n \cot ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{4/3}} \, dx","Int[E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(4/3),x]","-\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}}","-\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^(n*ArcCot[a*x])*(3*n - 2*a*x))/(a*c*(4 + 9*n^2)*(c + a^2*c*x^2)^(1/3)) - (6*(1 + 1/(a^2*x^2))^(1/3)*((a - I/x)/(a + I/x))^((2 - (3*I)*n)/6)*(1 - I/(a*x))^((-2 + (3*I)*n)/6)*(1 + I/(a*x))^((4 - (3*I)*n)/6)*x*Hypergeometric2F1[-1/3, (2 - (3*I)*n)/6, 2/3, (2*I)/((a + I/x)*x)])/(c*(4 + 9*n^2)*(c + a^2*c*x^2)^(1/3))","A",4,4,23,0.1739,1,"{5115, 5122, 5126, 132}"
11,1,207,0,0.2499643,"\int \frac{e^{n \cot ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/3}} \, dx","Int[E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(5/3),x]","-\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}}","-\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^(n*ArcCot[a*x])*(3*n - 4*a*x))/(a*c*(16 + 9*n^2)*(c + a^2*c*x^2)^(2/3)) - (12*(1 + 1/(a^2*x^2))^(2/3)*((a - I/x)/(a + I/x))^((4 - (3*I)*n)/6)*(1 - I/(a*x))^((-4 + (3*I)*n)/6)*(1 + I/(a*x))^((2 - (3*I)*n)/6)*x*Hypergeometric2F1[1/3, (4 - (3*I)*n)/6, 4/3, (2*I)/((a + I/x)*x)])/(c*(16 + 9*n^2)*(c + a^2*c*x^2)^(2/3))","A",4,4,23,0.1739,1,"{5115, 5122, 5126, 132}"
12,1,272,0,0.3002287,"\int \frac{e^{n \cot ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{7/3}} \, dx","Int[E^(n*ArcCot[a*x])/(c + a^2*c*x^2)^(7/3),x]","-\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}}","-\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^(n*ArcCot[a*x])*(3*n - 8*a*x))/(a*c*(64 + 9*n^2)*(c + a^2*c*x^2)^(4/3)) - (120*E^(n*ArcCot[a*x])*(3*n - 2*a*x))/(a*c^2*(4 + 9*n^2)*(64 + 9*n^2)*(c + a^2*c*x^2)^(1/3)) - (240*(1 + 1/(a^2*x^2))^(1/3)*((a - I/x)/(a + I/x))^((2 - (3*I)*n)/6)*(1 - I/(a*x))^((-2 + (3*I)*n)/6)*(1 + I/(a*x))^((4 - (3*I)*n)/6)*x*Hypergeometric2F1[-1/3, (2 - (3*I)*n)/6, 2/3, (2*I)/((a + I/x)*x)])/(c^2*(4 + 9*n^2)*(64 + 9*n^2)*(c + a^2*c*x^2)^(1/3))","A",5,4,23,0.1739,1,"{5115, 5122, 5126, 132}"