1,1,64,113,0.0559606,"\int e^{i \tan ^{-1}(a x)} x^4 \, dx","Integrate[E^(I*ArcTan[a*x])*x^4,x]","\frac{45 \sinh ^{-1}(a x)+\sqrt{a^2 x^2+1} \left(24 i a^4 x^4+30 a^3 x^3-32 i a^2 x^2-45 a x+64 i\right)}{120 a^5}","\frac{3 \sinh ^{-1}(a x)}{8 a^5}+\frac{i x^4 \sqrt{a^2 x^2+1}}{5 a}+\frac{x^3 \sqrt{a^2 x^2+1}}{4 a^2}+\frac{(-45 a x+64 i) \sqrt{a^2 x^2+1}}{120 a^5}-\frac{4 i x^2 \sqrt{a^2 x^2+1}}{15 a^3}",1,"(Sqrt[1 + a^2*x^2]*(64*I - 45*a*x - (32*I)*a^2*x^2 + 30*a^3*x^3 + (24*I)*a^4*x^4) + 45*ArcSinh[a*x])/(120*a^5)","A",1
2,1,56,90,0.0435713,"\int e^{i \tan ^{-1}(a x)} x^3 \, dx","Integrate[E^(I*ArcTan[a*x])*x^3,x]","\frac{\sqrt{a^2 x^2+1} \left(6 i a^3 x^3+8 a^2 x^2-9 i a x-16\right)+9 i \sinh ^{-1}(a x)}{24 a^4}","\frac{3 i \sinh ^{-1}(a x)}{8 a^4}+\frac{x^2 \sqrt{a^2 x^2+1}}{3 a^2}+\frac{i x^3 \sqrt{a^2 x^2+1}}{4 a}-\frac{(16+9 i a x) \sqrt{a^2 x^2+1}}{24 a^4}",1,"(Sqrt[1 + a^2*x^2]*(-16 - (9*I)*a*x + 8*a^2*x^2 + (6*I)*a^3*x^3) + (9*I)*ArcSinh[a*x])/(24*a^4)","A",1
3,1,46,75,0.0342443,"\int e^{i \tan ^{-1}(a x)} x^2 \, dx","Integrate[E^(I*ArcTan[a*x])*x^2,x]","\frac{-3 \sinh ^{-1}(a x)+\left(2 i a^2 x^2+3 a x-4 i\right) \sqrt{a^2 x^2+1}}{6 a^3}","-\frac{\sinh ^{-1}(a x)}{2 a^3}+\frac{x \sqrt{a^2 x^2+1}}{2 a^2}+\frac{i \left(a^2 x^2+1\right)^{3/2}}{3 a^3}-\frac{i \sqrt{a^2 x^2+1}}{a^3}",1,"((-4*I + 3*a*x + (2*I)*a^2*x^2)*Sqrt[1 + a^2*x^2] - 3*ArcSinh[a*x])/(6*a^3)","A",1
4,1,38,42,0.0275293,"\int e^{i \tan ^{-1}(a x)} x \, dx","Integrate[E^(I*ArcTan[a*x])*x,x]","\frac{(2+i a x) \sqrt{a^2 x^2+1}-i \sinh ^{-1}(a x)}{2 a^2}","\frac{(2+i a x) \sqrt{a^2 x^2+1}}{2 a^2}-\frac{i \sinh ^{-1}(a x)}{2 a^2}",1,"((2 + I*a*x)*Sqrt[1 + a^2*x^2] - I*ArcSinh[a*x])/(2*a^2)","A",1
5,1,26,29,0.0147594,"\int e^{i \tan ^{-1}(a x)} \, dx","Integrate[E^(I*ArcTan[a*x]),x]","\frac{\sinh ^{-1}(a x)+i \sqrt{a^2 x^2+1}}{a}","\frac{\sinh ^{-1}(a x)}{a}+\frac{i \sqrt{a^2 x^2+1}}{a}",1,"(I*Sqrt[1 + a^2*x^2] + ArcSinh[a*x])/a","A",1
6,1,29,25,0.0151468,"\int \frac{e^{i \tan ^{-1}(a x)}}{x} \, dx","Integrate[E^(I*ArcTan[a*x])/x,x]","-\log \left(\sqrt{a^2 x^2+1}+1\right)+i \sinh ^{-1}(a x)+\log (x)","-\tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)+i \sinh ^{-1}(a x)",1,"I*ArcSinh[a*x] + Log[x] - Log[1 + Sqrt[1 + a^2*x^2]]","A",0
7,1,47,38,0.0300537,"\int \frac{e^{i \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[E^(I*ArcTan[a*x])/x^2,x]","-\frac{\sqrt{a^2 x^2+1}}{x}-i a \log \left(\sqrt{a^2 x^2+1}+1\right)+i a \log (x)","-\frac{\sqrt{a^2 x^2+1}}{x}-i a \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)",1,"-(Sqrt[1 + a^2*x^2]/x) + I*a*Log[x] - I*a*Log[1 + Sqrt[1 + a^2*x^2]]","A",0
8,1,57,63,0.0422234,"\int \frac{e^{i \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[E^(I*ArcTan[a*x])/x^3,x]","\frac{1}{2} \left(\frac{(-1-2 i a x) \sqrt{a^2 x^2+1}}{x^2}+a^2 \log \left(\sqrt{a^2 x^2+1}+1\right)+a^2 (-\log (x))\right)","-\frac{i a \sqrt{a^2 x^2+1}}{x}-\frac{\sqrt{a^2 x^2+1}}{2 x^2}+\frac{1}{2} a^2 \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)",1,"(((-1 - (2*I)*a*x)*Sqrt[1 + a^2*x^2])/x^2 - a^2*Log[x] + a^2*Log[1 + Sqrt[1 + a^2*x^2]])/2","A",0
9,1,70,90,0.0625825,"\int \frac{e^{i \tan ^{-1}(a x)}}{x^4} \, dx","Integrate[E^(I*ArcTan[a*x])/x^4,x]","\frac{1}{6} \left(-3 i a^3 \log (x)+\frac{\sqrt{a^2 x^2+1} \left(4 a^2 x^2-3 i a x-2\right)}{x^3}+3 i a^3 \log \left(\sqrt{a^2 x^2+1}+1\right)\right)","\frac{2 a^2 \sqrt{a^2 x^2+1}}{3 x}-\frac{i a \sqrt{a^2 x^2+1}}{2 x^2}-\frac{\sqrt{a^2 x^2+1}}{3 x^3}+\frac{1}{2} i a^3 \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)",1,"((Sqrt[1 + a^2*x^2]*(-2 - (3*I)*a*x + 4*a^2*x^2))/x^3 - (3*I)*a^3*Log[x] + (3*I)*a^3*Log[1 + Sqrt[1 + a^2*x^2]])/6","A",0
10,1,76,113,0.0630903,"\int \frac{e^{i \tan ^{-1}(a x)}}{x^5} \, dx","Integrate[E^(I*ArcTan[a*x])/x^5,x]","\frac{1}{24} \left(9 a^4 \log (x)-9 a^4 \log \left(\sqrt{a^2 x^2+1}+1\right)+\frac{\sqrt{a^2 x^2+1} \left(16 i a^3 x^3+9 a^2 x^2-8 i a x-6\right)}{x^4}\right)","\frac{3 a^2 \sqrt{a^2 x^2+1}}{8 x^2}-\frac{\sqrt{a^2 x^2+1}}{4 x^4}-\frac{i a \sqrt{a^2 x^2+1}}{3 x^3}-\frac{3}{8} a^4 \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)+\frac{2 i a^3 \sqrt{a^2 x^2+1}}{3 x}",1,"((Sqrt[1 + a^2*x^2]*(-6 - (8*I)*a*x + 9*a^2*x^2 + (16*I)*a^3*x^3))/x^4 + 9*a^4*Log[x] - 9*a^4*Log[1 + Sqrt[1 + a^2*x^2]])/24","A",0
11,1,48,48,0.0222597,"\int e^{2 i \tan ^{-1}(a x)} x^3 \, dx","Integrate[E^((2*I)*ArcTan[a*x])*x^3,x]","-\frac{2 \log (a x+i)}{a^4}-\frac{2 i x}{a^3}+\frac{x^2}{a^2}+\frac{2 i x^3}{3 a}-\frac{x^4}{4}","-\frac{2 \log (a x+i)}{a^4}-\frac{2 i x}{a^3}+\frac{x^2}{a^2}+\frac{2 i x^3}{3 a}-\frac{x^4}{4}",1,"((-2*I)*x)/a^3 + x^2/a^2 + (((2*I)/3)*x^3)/a - x^4/4 - (2*Log[I + a*x])/a^4","A",1
12,1,39,39,0.0147132,"\int e^{2 i \tan ^{-1}(a x)} x^2 \, dx","Integrate[E^((2*I)*ArcTan[a*x])*x^2,x]","-\frac{2 i \log (a x+i)}{a^3}+\frac{2 x}{a^2}+\frac{i x^2}{a}-\frac{x^3}{3}","-\frac{2 i \log (a x+i)}{a^3}+\frac{2 x}{a^2}+\frac{i x^2}{a}-\frac{x^3}{3}",1,"(2*x)/a^2 + (I*x^2)/a - x^3/3 - ((2*I)*Log[I + a*x])/a^3","A",1
13,1,29,29,0.011901,"\int e^{2 i \tan ^{-1}(a x)} x \, dx","Integrate[E^((2*I)*ArcTan[a*x])*x,x]","\frac{2 \log (a x+i)}{a^2}+\frac{2 i x}{a}-\frac{x^2}{2}","\frac{2 \log (a x+i)}{a^2}+\frac{2 i x}{a}-\frac{x^2}{2}",1,"((2*I)*x)/a - x^2/2 + (2*Log[I + a*x])/a^2","A",1
14,1,30,19,0.0097733,"\int e^{2 i \tan ^{-1}(a x)} \, dx","Integrate[E^((2*I)*ArcTan[a*x]),x]","\frac{i \log \left(a^2 x^2+1\right)}{a}+\frac{2 \tan ^{-1}(a x)}{a}-x","-x+\frac{2 i \log (a x+i)}{a}",1,"-x + (2*ArcTan[a*x])/a + (I*Log[1 + a^2*x^2])/a","A",0
15,1,13,13,0.0070004,"\int \frac{e^{2 i \tan ^{-1}(a x)}}{x} \, dx","Integrate[E^((2*I)*ArcTan[a*x])/x,x]","\log (x)-2 \log (a x+i)","\log (x)-2 \log (a x+i)",1,"Log[x] - 2*Log[I + a*x]","A",1
16,1,26,26,0.0095044,"\int \frac{e^{2 i \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[E^((2*I)*ArcTan[a*x])/x^2,x]","2 i a \log (x)-2 i a \log (a x+i)-\frac{1}{x}","2 i a \log (x)-2 i a \log (a x+i)-\frac{1}{x}",1,"-x^(-1) + (2*I)*a*Log[x] - (2*I)*a*Log[I + a*x]","A",1
17,1,36,36,0.0122697,"\int \frac{e^{2 i \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[E^((2*I)*ArcTan[a*x])/x^3,x]","-2 a^2 \log (x)+2 a^2 \log (a x+i)-\frac{2 i a}{x}-\frac{1}{2 x^2}","-2 a^2 \log (x)+2 a^2 \log (a x+i)-\frac{2 i a}{x}-\frac{1}{2 x^2}",1,"-1/2*1/x^2 - ((2*I)*a)/x - 2*a^2*Log[x] + 2*a^2*Log[I + a*x]","A",1
18,1,48,48,0.0142196,"\int \frac{e^{2 i \tan ^{-1}(a x)}}{x^4} \, dx","Integrate[E^((2*I)*ArcTan[a*x])/x^4,x]","-2 i a^3 \log (x)+2 i a^3 \log (a x+i)+\frac{2 a^2}{x}-\frac{i a}{x^2}-\frac{1}{3 x^3}","-2 i a^3 \log (x)+2 i a^3 \log (a x+i)+\frac{2 a^2}{x}-\frac{i a}{x^2}-\frac{1}{3 x^3}",1,"-1/3*1/x^3 - (I*a)/x^2 + (2*a^2)/x - (2*I)*a^3*Log[x] + (2*I)*a^3*Log[I + a*x]","A",1
19,1,80,137,0.0627348,"\int e^{3 i \tan ^{-1}(a x)} x^3 \, dx","Integrate[E^((3*I)*ArcTan[a*x])*x^3,x]","\sqrt{a^2 x^2+1} \left(\frac{4 i}{a^4 (a x+i)}+\frac{6}{a^4}+\frac{19 i x}{8 a^3}-\frac{x^2}{a^2}-\frac{i x^3}{4 a}\right)-\frac{51 i \sinh ^{-1}(a x)}{8 a^4}","-\frac{51 i \sinh ^{-1}(a x)}{8 a^4}-\frac{x^2 \sqrt{a^2 x^2+1}}{a^2}-\frac{i x^3 \sqrt{a^2 x^2+1}}{4 a}-\frac{9 i (-3 a x+2 i) \sqrt{a^2 x^2+1}}{8 a^4}+\frac{27 \sqrt{a^2 x^2+1}}{4 a^4}+\frac{(1+i a x)^3}{a^4 \sqrt{a^2 x^2+1}}",1,"Sqrt[1 + a^2*x^2]*(6/a^4 + (((19*I)/8)*x)/a^3 - x^2/a^2 - ((I/4)*x^3)/a + (4*I)/(a^4*(I + a*x))) - (((51*I)/8)*ArcSinh[a*x])/a^4","A",1
20,1,63,102,0.0558074,"\int e^{3 i \tan ^{-1}(a x)} x^2 \, dx","Integrate[E^((3*I)*ArcTan[a*x])*x^2,x]","\frac{33 \sinh ^{-1}(a x)+\frac{\sqrt{a^2 x^2+1} \left(-2 i a^3 x^3-7 a^2 x^2+19 i a x-52\right)}{a x+i}}{6 a^3}","\frac{11 \sinh ^{-1}(a x)}{2 a^3}+\frac{i (1+i a x)^3}{a^3 \sqrt{a^2 x^2+1}}+\frac{i (3+i a x)^2 \sqrt{a^2 x^2+1}}{3 a^3}+\frac{(-3 a x+28 i) \sqrt{a^2 x^2+1}}{6 a^3}",1,"((Sqrt[1 + a^2*x^2]*(-52 + (19*I)*a*x - 7*a^2*x^2 - (2*I)*a^3*x^3))/(I + a*x) + 33*ArcSinh[a*x])/(6*a^3)","A",1
21,1,54,92,0.0665256,"\int e^{3 i \tan ^{-1}(a x)} x \, dx","Integrate[E^((3*I)*ArcTan[a*x])*x,x]","-\frac{i \left(-9 \sinh ^{-1}(a x)+\frac{\sqrt{a^2 x^2+1} \left(a^2 x^2-5 i a x+14\right)}{a x+i}\right)}{2 a^2}","-\frac{\left(a^2 x^2+1\right)^{5/2}}{a^2 (1-i a x)^3}-\frac{3 \left(a^2 x^2+1\right)^{3/2}}{2 a^2 (1-i a x)}-\frac{9 \sqrt{a^2 x^2+1}}{2 a^2}+\frac{9 i \sinh ^{-1}(a x)}{2 a^2}",1,"((-1/2*I)*((Sqrt[1 + a^2*x^2]*(14 - (5*I)*a*x + a^2*x^2))/(I + a*x) - 9*ArcSinh[a*x]))/a^2","A",1
22,1,42,60,0.0348332,"\int e^{3 i \tan ^{-1}(a x)} \, dx","Integrate[E^((3*I)*ArcTan[a*x]),x]","-\frac{3 \sinh ^{-1}(a x)}{a}+\frac{\sqrt{a^2 x^2+1} \left(\frac{4}{a x+i}-i\right)}{a}","-\frac{2 i (1+i a x)^2}{a \sqrt{a^2 x^2+1}}-\frac{3 i \sqrt{a^2 x^2+1}}{a}-\frac{3 \sinh ^{-1}(a x)}{a}",1,"(Sqrt[1 + a^2*x^2]*(-I + 4/(I + a*x)))/a - (3*ArcSinh[a*x])/a","A",1
23,1,55,51,0.0439376,"\int \frac{e^{3 i \tan ^{-1}(a x)}}{x} \, dx","Integrate[E^((3*I)*ArcTan[a*x])/x,x]","\frac{4 i \sqrt{a^2 x^2+1}}{a x+i}-\log \left(\sqrt{a^2 x^2+1}+1\right)-i \sinh ^{-1}(a x)+\log (x)","\frac{4 i \sqrt{a^2 x^2+1}}{a x+i}-\tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)-i \sinh ^{-1}(a x)",1,"((4*I)*Sqrt[1 + a^2*x^2])/(I + a*x) - I*ArcSinh[a*x] + Log[x] - Log[1 + Sqrt[1 + a^2*x^2]]","A",0
24,1,61,63,0.0499641,"\int \frac{e^{3 i \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[E^((3*I)*ArcTan[a*x])/x^2,x]","\sqrt{a^2 x^2+1} \left(-\frac{1}{x}-\frac{4 a}{a x+i}\right)-3 i a \log \left(\sqrt{a^2 x^2+1}+1\right)+3 i a \log (x)","-\frac{4 a \sqrt{a^2 x^2+1}}{a x+i}-\frac{\sqrt{a^2 x^2+1}}{x}-3 i a \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)",1,"Sqrt[1 + a^2*x^2]*(-x^(-1) - (4*a)/(I + a*x)) + (3*I)*a*Log[x] - (3*I)*a*Log[1 + Sqrt[1 + a^2*x^2]]","A",0
25,1,79,92,0.0819831,"\int \frac{e^{3 i \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[E^((3*I)*ArcTan[a*x])/x^3,x]","\sqrt{a^2 x^2+1} \left(-\frac{4 i a^2}{a x+i}-\frac{3 i a}{x}-\frac{1}{2 x^2}\right)+\frac{9}{2} a^2 \log \left(\sqrt{a^2 x^2+1}+1\right)-\frac{9}{2} a^2 \log (x)","-\frac{4 i a^2 \sqrt{a^2 x^2+1}}{a x+i}-\frac{3 i a \sqrt{a^2 x^2+1}}{x}-\frac{\sqrt{a^2 x^2+1}}{2 x^2}+\frac{9}{2} a^2 \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)",1,"Sqrt[1 + a^2*x^2]*(-1/2*1/x^2 - ((3*I)*a)/x - ((4*I)*a^2)/(I + a*x)) - (9*a^2*Log[x])/2 + (9*a^2*Log[1 + Sqrt[1 + a^2*x^2]])/2","A",0
26,1,89,117,0.0806424,"\int \frac{e^{3 i \tan ^{-1}(a x)}}{x^4} \, dx","Integrate[E^((3*I)*ArcTan[a*x])/x^4,x]","\frac{1}{6} \left(-33 i a^3 \log (x)+33 i a^3 \log \left(\sqrt{a^2 x^2+1}+1\right)+\frac{\sqrt{a^2 x^2+1} \left(52 a^3 x^3+19 i a^2 x^2+7 a x-2 i\right)}{x^3 (a x+i)}\right)","\frac{14 a^2 \sqrt{a^2 x^2+1}}{3 x}-\frac{3 i a \sqrt{a^2 x^2+1}}{2 x^2}-\frac{\sqrt{a^2 x^2+1}}{3 x^3}+\frac{4 a^3 \sqrt{a^2 x^2+1}}{a x+i}+\frac{11}{2} i a^3 \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)",1,"((Sqrt[1 + a^2*x^2]*(-2*I + 7*a*x + (19*I)*a^2*x^2 + 52*a^3*x^3))/(x^3*(I + a*x)) - (33*I)*a^3*Log[x] + (33*I)*a^3*Log[1 + Sqrt[1 + a^2*x^2]])/6","A",0
27,1,65,65,0.0479815,"\int e^{4 i \tan ^{-1}(a x)} x^3 \, dx","Integrate[E^((4*I)*ArcTan[a*x])*x^3,x]","\frac{4 i}{a^4 (a x+i)}+\frac{16 \log (a x+i)}{a^4}+\frac{12 i x}{a^3}-\frac{4 x^2}{a^2}-\frac{4 i x^3}{3 a}+\frac{x^4}{4}","\frac{4 i}{a^4 (a x+i)}+\frac{16 \log (a x+i)}{a^4}+\frac{12 i x}{a^3}-\frac{4 x^2}{a^2}-\frac{4 i x^3}{3 a}+\frac{x^4}{4}",1,"((12*I)*x)/a^3 - (4*x^2)/a^2 - (((4*I)/3)*x^3)/a + x^4/4 + (4*I)/(a^4*(I + a*x)) + (16*Log[I + a*x])/a^4","A",1
28,1,53,53,0.0348492,"\int e^{4 i \tan ^{-1}(a x)} x^2 \, dx","Integrate[E^((4*I)*ArcTan[a*x])*x^2,x]","-\frac{4}{a^3 (a x+i)}+\frac{12 i \log (a x+i)}{a^3}-\frac{8 x}{a^2}-\frac{2 i x^2}{a}+\frac{x^3}{3}","-\frac{4}{a^3 (a x+i)}+\frac{12 i \log (a x+i)}{a^3}-\frac{8 x}{a^2}-\frac{2 i x^2}{a}+\frac{x^3}{3}",1,"(-8*x)/a^2 - ((2*I)*x^2)/a + x^3/3 - 4/(a^3*(I + a*x)) + ((12*I)*Log[I + a*x])/a^3","A",1
29,1,45,45,0.0275778,"\int e^{4 i \tan ^{-1}(a x)} x \, dx","Integrate[E^((4*I)*ArcTan[a*x])*x,x]","-\frac{4 i}{a^2 (a x+i)}-\frac{8 \log (a x+i)}{a^2}-\frac{4 i x}{a}+\frac{x^2}{2}","-\frac{4 i}{a^2 (a x+i)}-\frac{8 \log (a x+i)}{a^2}-\frac{4 i x}{a}+\frac{x^2}{2}",1,"((-4*I)*x)/a + x^2/2 - (4*I)/(a^2*(I + a*x)) - (8*Log[I + a*x])/a^2","A",1
30,1,42,31,0.0213682,"\int e^{4 i \tan ^{-1}(a x)} \, dx","Integrate[E^((4*I)*ArcTan[a*x]),x]","-\frac{2 i \log \left(a^2 x^2+1\right)}{a}+\frac{4}{a (a x+i)}-\frac{4 \tan ^{-1}(a x)}{a}+x","\frac{4}{a (a x+i)}-\frac{4 i \log (a x+i)}{a}+x",1,"x + 4/(a*(I + a*x)) - (4*ArcTan[a*x])/a - ((2*I)*Log[1 + a^2*x^2])/a","A",0
31,1,16,16,0.0095304,"\int \frac{e^{4 i \tan ^{-1}(a x)}}{x} \, dx","Integrate[E^((4*I)*ArcTan[a*x])/x,x]","\log (x)+\frac{4 i}{a x+i}","\log (x)+\frac{4 i}{a x+i}",1,"(4*I)/(I + a*x) + Log[x]","A",1
32,1,38,38,0.0260092,"\int \frac{e^{4 i \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[E^((4*I)*ArcTan[a*x])/x^2,x]","-\frac{4 a}{a x+i}+4 i a \log (x)-4 i a \log (a x+i)-\frac{1}{x}","-\frac{4 a}{a x+i}+4 i a \log (x)-4 i a \log (a x+i)-\frac{1}{x}",1,"-x^(-1) - (4*a)/(I + a*x) + (4*I)*a*Log[x] - (4*I)*a*Log[I + a*x]","A",1
33,1,52,52,0.0407375,"\int \frac{e^{4 i \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[E^((4*I)*ArcTan[a*x])/x^3,x]","-\frac{4 i a^2}{a x+i}-8 a^2 \log (x)+8 a^2 \log (a x+i)-\frac{4 i a}{x}-\frac{1}{2 x^2}","-\frac{4 i a^2}{a x+i}-8 a^2 \log (x)+8 a^2 \log (a x+i)-\frac{4 i a}{x}-\frac{1}{2 x^2}",1,"-1/2*1/x^2 - ((4*I)*a)/x - ((4*I)*a^2)/(I + a*x) - 8*a^2*Log[x] + 8*a^2*Log[I + a*x]","A",1
34,1,62,62,0.0397093,"\int \frac{e^{4 i \tan ^{-1}(a x)}}{x^4} \, dx","Integrate[E^((4*I)*ArcTan[a*x])/x^4,x]","\frac{4 a^3}{a x+i}-12 i a^3 \log (x)+12 i a^3 \log (a x+i)+\frac{8 a^2}{x}-\frac{2 i a}{x^2}-\frac{1}{3 x^3}","\frac{4 a^3}{a x+i}-12 i a^3 \log (x)+12 i a^3 \log (a x+i)+\frac{8 a^2}{x}-\frac{2 i a}{x^2}-\frac{1}{3 x^3}",1,"-1/3*1/x^3 - ((2*I)*a)/x^2 + (8*a^2)/x + (4*a^3)/(I + a*x) - (12*I)*a^3*Log[x] + (12*I)*a^3*Log[I + a*x]","A",1
35,1,56,90,0.0470086,"\int e^{-i \tan ^{-1}(a x)} x^3 \, dx","Integrate[x^3/E^(I*ArcTan[a*x]),x]","\frac{\sqrt{a^2 x^2+1} \left(-6 i a^3 x^3+8 a^2 x^2+9 i a x-16\right)-9 i \sinh ^{-1}(a x)}{24 a^4}","-\frac{3 i \sinh ^{-1}(a x)}{8 a^4}+\frac{x^2 \sqrt{a^2 x^2+1}}{3 a^2}-\frac{i x^3 \sqrt{a^2 x^2+1}}{4 a}-\frac{(16-9 i a x) \sqrt{a^2 x^2+1}}{24 a^4}",1,"(Sqrt[1 + a^2*x^2]*(-16 + (9*I)*a*x + 8*a^2*x^2 - (6*I)*a^3*x^3) - (9*I)*ArcSinh[a*x])/(24*a^4)","A",1
36,1,46,75,0.0351431,"\int e^{-i \tan ^{-1}(a x)} x^2 \, dx","Integrate[x^2/E^(I*ArcTan[a*x]),x]","\frac{-3 \sinh ^{-1}(a x)+\left(-2 i a^2 x^2+3 a x+4 i\right) \sqrt{a^2 x^2+1}}{6 a^3}","-\frac{\sinh ^{-1}(a x)}{2 a^3}+\frac{x \sqrt{a^2 x^2+1}}{2 a^2}-\frac{i \left(a^2 x^2+1\right)^{3/2}}{3 a^3}+\frac{i \sqrt{a^2 x^2+1}}{a^3}",1,"((4*I + 3*a*x - (2*I)*a^2*x^2)*Sqrt[1 + a^2*x^2] - 3*ArcSinh[a*x])/(6*a^3)","A",1
37,1,38,42,0.0282892,"\int e^{-i \tan ^{-1}(a x)} x \, dx","Integrate[x/E^(I*ArcTan[a*x]),x]","\frac{\sqrt{a^2 x^2+1} (2-i a x)+i \sinh ^{-1}(a x)}{2 a^2}","\frac{\sqrt{a^2 x^2+1} (2-i a x)}{2 a^2}+\frac{i \sinh ^{-1}(a x)}{2 a^2}",1,"((2 - I*a*x)*Sqrt[1 + a^2*x^2] + I*ArcSinh[a*x])/(2*a^2)","A",1
38,1,26,29,0.0146748,"\int e^{-i \tan ^{-1}(a x)} \, dx","Integrate[E^((-I)*ArcTan[a*x]),x]","\frac{\sinh ^{-1}(a x)-i \sqrt{a^2 x^2+1}}{a}","\frac{\sinh ^{-1}(a x)}{a}-\frac{i \sqrt{a^2 x^2+1}}{a}",1,"((-I)*Sqrt[1 + a^2*x^2] + ArcSinh[a*x])/a","A",1
39,1,29,25,0.0150671,"\int \frac{e^{-i \tan ^{-1}(a x)}}{x} \, dx","Integrate[1/(E^(I*ArcTan[a*x])*x),x]","-\log \left(\sqrt{a^2 x^2+1}+1\right)-i \sinh ^{-1}(a x)+\log (x)","-\tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)-i \sinh ^{-1}(a x)",1,"(-I)*ArcSinh[a*x] + Log[x] - Log[1 + Sqrt[1 + a^2*x^2]]","A",0
40,1,47,38,0.0269347,"\int \frac{e^{-i \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[1/(E^(I*ArcTan[a*x])*x^2),x]","-\frac{\sqrt{a^2 x^2+1}}{x}+i a \log \left(\sqrt{a^2 x^2+1}+1\right)-i a \log (x)","-\frac{\sqrt{a^2 x^2+1}}{x}+i a \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)",1,"-(Sqrt[1 + a^2*x^2]/x) - I*a*Log[x] + I*a*Log[1 + Sqrt[1 + a^2*x^2]]","A",0
41,1,57,63,0.0437193,"\int \frac{e^{-i \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[1/(E^(I*ArcTan[a*x])*x^3),x]","\frac{1}{2} \left(\frac{(-1+2 i a x) \sqrt{a^2 x^2+1}}{x^2}+a^2 \log \left(\sqrt{a^2 x^2+1}+1\right)+a^2 (-\log (x))\right)","\frac{i a \sqrt{a^2 x^2+1}}{x}-\frac{\sqrt{a^2 x^2+1}}{2 x^2}+\frac{1}{2} a^2 \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)",1,"(((-1 + (2*I)*a*x)*Sqrt[1 + a^2*x^2])/x^2 - a^2*Log[x] + a^2*Log[1 + Sqrt[1 + a^2*x^2]])/2","A",0
42,1,70,90,0.053569,"\int \frac{e^{-i \tan ^{-1}(a x)}}{x^4} \, dx","Integrate[1/(E^(I*ArcTan[a*x])*x^4),x]","\frac{1}{6} \left(3 i a^3 \log (x)+\frac{\sqrt{a^2 x^2+1} \left(4 a^2 x^2+3 i a x-2\right)}{x^3}-3 i a^3 \log \left(\sqrt{a^2 x^2+1}+1\right)\right)","\frac{2 a^2 \sqrt{a^2 x^2+1}}{3 x}+\frac{i a \sqrt{a^2 x^2+1}}{2 x^2}-\frac{\sqrt{a^2 x^2+1}}{3 x^3}-\frac{1}{2} i a^3 \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)",1,"((Sqrt[1 + a^2*x^2]*(-2 + (3*I)*a*x + 4*a^2*x^2))/x^3 + (3*I)*a^3*Log[x] - (3*I)*a^3*Log[1 + Sqrt[1 + a^2*x^2]])/6","A",0
43,1,76,113,0.0644894,"\int \frac{e^{-i \tan ^{-1}(a x)}}{x^5} \, dx","Integrate[1/(E^(I*ArcTan[a*x])*x^5),x]","\frac{1}{24} \left(9 a^4 \log (x)-9 a^4 \log \left(\sqrt{a^2 x^2+1}+1\right)+\frac{\sqrt{a^2 x^2+1} \left(-16 i a^3 x^3+9 a^2 x^2+8 i a x-6\right)}{x^4}\right)","\frac{3 a^2 \sqrt{a^2 x^2+1}}{8 x^2}-\frac{\sqrt{a^2 x^2+1}}{4 x^4}+\frac{i a \sqrt{a^2 x^2+1}}{3 x^3}-\frac{3}{8} a^4 \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)-\frac{2 i a^3 \sqrt{a^2 x^2+1}}{3 x}",1,"((Sqrt[1 + a^2*x^2]*(-6 + (8*I)*a*x + 9*a^2*x^2 - (16*I)*a^3*x^3))/x^4 + 9*a^4*Log[x] - 9*a^4*Log[1 + Sqrt[1 + a^2*x^2]])/24","A",0
44,1,49,49,0.0211588,"\int e^{-2 i \tan ^{-1}(a x)} x^3 \, dx","Integrate[x^3/E^((2*I)*ArcTan[a*x]),x]","-\frac{2 \log (-a x+i)}{a^4}+\frac{2 i x}{a^3}+\frac{x^2}{a^2}-\frac{2 i x^3}{3 a}-\frac{x^4}{4}","-\frac{2 \log (-a x+i)}{a^4}+\frac{2 i x}{a^3}+\frac{x^2}{a^2}-\frac{2 i x^3}{3 a}-\frac{x^4}{4}",1,"((2*I)*x)/a^3 + x^2/a^2 - (((2*I)/3)*x^3)/a - x^4/4 - (2*Log[I - a*x])/a^4","A",1
45,1,40,40,0.0144172,"\int e^{-2 i \tan ^{-1}(a x)} x^2 \, dx","Integrate[x^2/E^((2*I)*ArcTan[a*x]),x]","\frac{2 i \log (-a x+i)}{a^3}+\frac{2 x}{a^2}-\frac{i x^2}{a}-\frac{x^3}{3}","\frac{2 i \log (-a x+i)}{a^3}+\frac{2 x}{a^2}-\frac{i x^2}{a}-\frac{x^3}{3}",1,"(2*x)/a^2 - (I*x^2)/a - x^3/3 + ((2*I)*Log[I - a*x])/a^3","A",1
46,1,30,30,0.011895,"\int e^{-2 i \tan ^{-1}(a x)} x \, dx","Integrate[x/E^((2*I)*ArcTan[a*x]),x]","\frac{2 \log (-a x+i)}{a^2}-\frac{2 i x}{a}-\frac{x^2}{2}","\frac{2 \log (-a x+i)}{a^2}-\frac{2 i x}{a}-\frac{x^2}{2}",1,"((-2*I)*x)/a - x^2/2 + (2*Log[I - a*x])/a^2","A",1
47,1,30,20,0.0110692,"\int e^{-2 i \tan ^{-1}(a x)} \, dx","Integrate[E^((-2*I)*ArcTan[a*x]),x]","-\frac{i \log \left(a^2 x^2+1\right)}{a}+\frac{2 \tan ^{-1}(a x)}{a}-x","-x-\frac{2 i \log (-a x+i)}{a}",1,"-x + (2*ArcTan[a*x])/a - (I*Log[1 + a^2*x^2])/a","A",0
48,1,14,14,0.0071781,"\int \frac{e^{-2 i \tan ^{-1}(a x)}}{x} \, dx","Integrate[1/(E^((2*I)*ArcTan[a*x])*x),x]","\log (x)-2 \log (-a x+i)","\log (x)-2 \log (-a x+i)",1,"Log[x] - 2*Log[I - a*x]","A",1
49,1,27,27,0.0107259,"\int \frac{e^{-2 i \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[1/(E^((2*I)*ArcTan[a*x])*x^2),x]","-2 i a \log (x)+2 i a \log (-a x+i)-\frac{1}{x}","-2 i a \log (x)+2 i a \log (-a x+i)-\frac{1}{x}",1,"-x^(-1) - (2*I)*a*Log[x] + (2*I)*a*Log[I - a*x]","A",1
50,1,37,37,0.0123703,"\int \frac{e^{-2 i \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[1/(E^((2*I)*ArcTan[a*x])*x^3),x]","-2 a^2 \log (x)+2 a^2 \log (-a x+i)+\frac{2 i a}{x}-\frac{1}{2 x^2}","-2 a^2 \log (x)+2 a^2 \log (-a x+i)+\frac{2 i a}{x}-\frac{1}{2 x^2}",1,"-1/2*1/x^2 + ((2*I)*a)/x - 2*a^2*Log[x] + 2*a^2*Log[I - a*x]","A",1
51,1,49,49,0.015223,"\int \frac{e^{-2 i \tan ^{-1}(a x)}}{x^4} \, dx","Integrate[1/(E^((2*I)*ArcTan[a*x])*x^4),x]","2 i a^3 \log (x)-2 i a^3 \log (-a x+i)+\frac{2 a^2}{x}+\frac{i a}{x^2}-\frac{1}{3 x^3}","2 i a^3 \log (x)-2 i a^3 \log (-a x+i)+\frac{2 a^2}{x}+\frac{i a}{x^2}-\frac{1}{3 x^3}",1,"-1/3*1/x^3 + (I*a)/x^2 + (2*a^2)/x + (2*I)*a^3*Log[x] - (2*I)*a^3*Log[I - a*x]","A",1
52,1,80,137,0.0616659,"\int e^{-3 i \tan ^{-1}(a x)} x^3 \, dx","Integrate[x^3/E^((3*I)*ArcTan[a*x]),x]","\frac{51 i \sinh ^{-1}(a x)}{8 a^4}+\sqrt{a^2 x^2+1} \left(-\frac{4 i}{a^4 (a x-i)}+\frac{6}{a^4}-\frac{19 i x}{8 a^3}-\frac{x^2}{a^2}+\frac{i x^3}{4 a}\right)","\frac{51 i \sinh ^{-1}(a x)}{8 a^4}-\frac{x^2 \sqrt{a^2 x^2+1}}{a^2}+\frac{i x^3 \sqrt{a^2 x^2+1}}{4 a}-\frac{9 i (3 a x+2 i) \sqrt{a^2 x^2+1}}{8 a^4}+\frac{27 \sqrt{a^2 x^2+1}}{4 a^4}+\frac{(1-i a x)^3}{a^4 \sqrt{a^2 x^2+1}}",1,"Sqrt[1 + a^2*x^2]*(6/a^4 - (((19*I)/8)*x)/a^3 - x^2/a^2 + ((I/4)*x^3)/a - (4*I)/(a^4*(-I + a*x))) + (((51*I)/8)*ArcSinh[a*x])/a^4","A",1
53,1,63,102,0.0559268,"\int e^{-3 i \tan ^{-1}(a x)} x^2 \, dx","Integrate[x^2/E^((3*I)*ArcTan[a*x]),x]","\frac{33 \sinh ^{-1}(a x)+\frac{\sqrt{a^2 x^2+1} \left(2 i a^3 x^3-7 a^2 x^2-19 i a x-52\right)}{a x-i}}{6 a^3}","\frac{11 \sinh ^{-1}(a x)}{2 a^3}-\frac{i (1-i a x)^3}{a^3 \sqrt{a^2 x^2+1}}-\frac{i (3-i a x)^2 \sqrt{a^2 x^2+1}}{3 a^3}-\frac{(3 a x+28 i) \sqrt{a^2 x^2+1}}{6 a^3}",1,"((Sqrt[1 + a^2*x^2]*(-52 - (19*I)*a*x - 7*a^2*x^2 + (2*I)*a^3*x^3))/(-I + a*x) + 33*ArcSinh[a*x])/(6*a^3)","A",1
54,1,60,92,0.0456071,"\int e^{-3 i \tan ^{-1}(a x)} x \, dx","Integrate[x/E^((3*I)*ArcTan[a*x]),x]","\sqrt{a^2 x^2+1} \left(\frac{4 i}{a^2 (a x-i)}-\frac{3}{a^2}+\frac{i x}{2 a}\right)-\frac{9 i \sinh ^{-1}(a x)}{2 a^2}","-\frac{\left(a^2 x^2+1\right)^{5/2}}{a^2 (1+i a x)^3}-\frac{3 \left(a^2 x^2+1\right)^{3/2}}{2 a^2 (1+i a x)}-\frac{9 \sqrt{a^2 x^2+1}}{2 a^2}-\frac{9 i \sinh ^{-1}(a x)}{2 a^2}",1,"Sqrt[1 + a^2*x^2]*(-3/a^2 + ((I/2)*x)/a + (4*I)/(a^2*(-I + a*x))) - (((9*I)/2)*ArcSinh[a*x])/a^2","A",1
55,1,42,60,0.0356007,"\int e^{-3 i \tan ^{-1}(a x)} \, dx","Integrate[E^((-3*I)*ArcTan[a*x]),x]","-\frac{3 \sinh ^{-1}(a x)}{a}+\frac{\sqrt{a^2 x^2+1} \left(\frac{4}{a x-i}+i\right)}{a}","\frac{2 i (1-i a x)^2}{a \sqrt{a^2 x^2+1}}+\frac{3 i \sqrt{a^2 x^2+1}}{a}-\frac{3 \sinh ^{-1}(a x)}{a}",1,"(Sqrt[1 + a^2*x^2]*(I + 4/(-I + a*x)))/a - (3*ArcSinh[a*x])/a","A",1
56,1,55,52,0.0448121,"\int \frac{e^{-3 i \tan ^{-1}(a x)}}{x} \, dx","Integrate[1/(E^((3*I)*ArcTan[a*x])*x),x]","-\frac{4 i \sqrt{a^2 x^2+1}}{a x-i}-\log \left(\sqrt{a^2 x^2+1}+1\right)+i \sinh ^{-1}(a x)+\log (x)","\frac{4 i \sqrt{a^2 x^2+1}}{-a x+i}-\tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)+i \sinh ^{-1}(a x)",1,"((-4*I)*Sqrt[1 + a^2*x^2])/(-I + a*x) + I*ArcSinh[a*x] + Log[x] - Log[1 + Sqrt[1 + a^2*x^2]]","A",0
57,1,61,64,0.0526461,"\int \frac{e^{-3 i \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[1/(E^((3*I)*ArcTan[a*x])*x^2),x]","\sqrt{a^2 x^2+1} \left(-\frac{1}{x}-\frac{4 a}{a x-i}\right)+3 i a \log \left(\sqrt{a^2 x^2+1}+1\right)-3 i a \log (x)","\frac{4 a \sqrt{a^2 x^2+1}}{-a x+i}-\frac{\sqrt{a^2 x^2+1}}{x}+3 i a \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)",1,"Sqrt[1 + a^2*x^2]*(-x^(-1) - (4*a)/(-I + a*x)) - (3*I)*a*Log[x] + (3*I)*a*Log[1 + Sqrt[1 + a^2*x^2]]","A",0
58,1,79,93,0.0871734,"\int \frac{e^{-3 i \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[1/(E^((3*I)*ArcTan[a*x])*x^3),x]","\sqrt{a^2 x^2+1} \left(\frac{4 i a^2}{a x-i}+\frac{3 i a}{x}-\frac{1}{2 x^2}\right)+\frac{9}{2} a^2 \log \left(\sqrt{a^2 x^2+1}+1\right)-\frac{9}{2} a^2 \log (x)","-\frac{4 i a^2 \sqrt{a^2 x^2+1}}{-a x+i}+\frac{3 i a \sqrt{a^2 x^2+1}}{x}-\frac{\sqrt{a^2 x^2+1}}{2 x^2}+\frac{9}{2} a^2 \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)",1,"Sqrt[1 + a^2*x^2]*(-1/2*1/x^2 + ((3*I)*a)/x + ((4*I)*a^2)/(-I + a*x)) - (9*a^2*Log[x])/2 + (9*a^2*Log[1 + Sqrt[1 + a^2*x^2]])/2","A",0
59,1,89,118,0.0816633,"\int \frac{e^{-3 i \tan ^{-1}(a x)}}{x^4} \, dx","Integrate[1/(E^((3*I)*ArcTan[a*x])*x^4),x]","\frac{1}{6} \left(33 i a^3 \log (x)-33 i a^3 \log \left(\sqrt{a^2 x^2+1}+1\right)+\frac{\sqrt{a^2 x^2+1} \left(52 a^3 x^3-19 i a^2 x^2+7 a x+2 i\right)}{x^3 (a x-i)}\right)","\frac{14 a^2 \sqrt{a^2 x^2+1}}{3 x}+\frac{3 i a \sqrt{a^2 x^2+1}}{2 x^2}-\frac{\sqrt{a^2 x^2+1}}{3 x^3}-\frac{4 a^3 \sqrt{a^2 x^2+1}}{-a x+i}-\frac{11}{2} i a^3 \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)",1,"((Sqrt[1 + a^2*x^2]*(2*I + 7*a*x - (19*I)*a^2*x^2 + 52*a^3*x^3))/(x^3*(-I + a*x)) + (33*I)*a^3*Log[x] - (33*I)*a^3*Log[1 + Sqrt[1 + a^2*x^2]])/6","A",0
60,1,95,139,0.0883298,"\int \frac{e^{-3 i \tan ^{-1}(a x)}}{x^5} \, dx","Integrate[1/(E^((3*I)*ArcTan[a*x])*x^5),x]","\frac{1}{8} \left(51 a^4 \log (x)-51 a^4 \log \left(\sqrt{a^2 x^2+1}+1\right)+\frac{\sqrt{a^2 x^2+1} \left(-80 i a^4 x^4-29 a^3 x^3-11 i a^2 x^2+6 a x+2 i\right)}{x^4 (a x-i)}\right)","\frac{19 a^2 \sqrt{a^2 x^2+1}}{8 x^2}-\frac{\sqrt{a^2 x^2+1}}{4 x^4}+\frac{i a \sqrt{a^2 x^2+1}}{x^3}+\frac{4 i a^4 \sqrt{a^2 x^2+1}}{-a x+i}-\frac{51}{8} a^4 \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)-\frac{6 i a^3 \sqrt{a^2 x^2+1}}{x}",1,"((Sqrt[1 + a^2*x^2]*(2*I + 6*a*x - (11*I)*a^2*x^2 - 29*a^3*x^3 - (80*I)*a^4*x^4))/(x^4*(-I + a*x)) + 51*a^4*Log[x] - 51*a^4*Log[1 + Sqrt[1 + a^2*x^2]])/8","A",0
61,1,82,339,0.0406201,"\int e^{\frac{1}{2} i \tan ^{-1}(a x)} x^2 \, dx","Integrate[E^((I/2)*ArcTan[a*x])*x^2,x]","\frac{(1-i a x)^{3/4} \left(\sqrt[4]{1+i a x} \left(4 i a^2 x^2+5 a x-i\right)-6 i \sqrt[4]{2} \, _2F_1\left(-\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-i a x)\right)\right)}{12 a^3}","-\frac{i (1-i a x)^{3/4} (1+i a x)^{5/4}}{12 a^3}-\frac{3 i (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{8 a^3}-\frac{3 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{16 \sqrt{2} a^3}+\frac{3 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{16 \sqrt{2} a^3}+\frac{3 i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{8 \sqrt{2} a^3}-\frac{3 i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{8 \sqrt{2} a^3}+\frac{x (1-i a x)^{3/4} (1+i a x)^{5/4}}{3 a^2}",1,"((1 - I*a*x)^(3/4)*((1 + I*a*x)^(1/4)*(-I + 5*a*x + (4*I)*a^2*x^2) - (6*I)*2^(1/4)*Hypergeometric2F1[-1/4, 3/4, 7/4, (1 - I*a*x)/2]))/(12*a^3)","C",0
62,1,63,295,0.0232239,"\int e^{\frac{1}{2} i \tan ^{-1}(a x)} x \, dx","Integrate[E^((I/2)*ArcTan[a*x])*x,x]","\frac{(1-i a x)^{3/4} \left(2 \sqrt[4]{2} \, _2F_1\left(-\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-i a x)\right)+3 (1+i a x)^{5/4}\right)}{6 a^2}","\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 a^2}+\frac{(1-i a x)^{3/4} \sqrt[4]{1+i a x}}{4 a^2}+\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{8 \sqrt{2} a^2}-\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{8 \sqrt{2} a^2}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{4 \sqrt{2} a^2}+\frac{\tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{4 \sqrt{2} a^2}",1,"((1 - I*a*x)^(3/4)*(3*(1 + I*a*x)^(5/4) + 2*2^(1/4)*Hypergeometric2F1[-1/4, 3/4, 7/4, (1 - I*a*x)/2]))/(6*a^2)","C",0
63,1,41,268,0.0296402,"\int e^{\frac{1}{2} i \tan ^{-1}(a x)} \, dx","Integrate[E^((I/2)*ArcTan[a*x]),x]","-\frac{8 i e^{\frac{5}{2} i \tan ^{-1}(a x)} \, _2F_1\left(\frac{5}{4},2;\frac{9}{4};-e^{2 i \tan ^{-1}(a x)}\right)}{5 a}","\frac{i (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{a}+\frac{i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{2 \sqrt{2} a}-\frac{i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{2 \sqrt{2} a}-\frac{i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{\sqrt{2} a}+\frac{i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{\sqrt{2} a}",1,"(((-8*I)/5)*E^(((5*I)/2)*ArcTan[a*x])*Hypergeometric2F1[5/4, 2, 9/4, -E^((2*I)*ArcTan[a*x])])/a","C",0
64,1,97,267,0.0400371,"\int \frac{e^{\frac{1}{2} i \tan ^{-1}(a x)}}{x} \, dx","Integrate[E^((I/2)*ArcTan[a*x])/x,x]","-\frac{2 (1-i a x)^{3/4} \left(\sqrt[4]{2} (1+i a x)^{3/4} \, _2F_1\left(\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-i a x)\right)+2 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)\right)}{3 (1+i a x)^{3/4}}","-\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{\sqrt{2}}+\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{\sqrt{2}}-2 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)-\sqrt{2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)-2 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)",1,"(-2*(1 - I*a*x)^(3/4)*(2^(1/4)*(1 + I*a*x)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, (1 - I*a*x)/2] + 2*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)]))/(3*(1 + I*a*x)^(3/4))","C",0
65,1,71,92,0.0196761,"\int \frac{e^{\frac{1}{2} i \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[E^((I/2)*ArcTan[a*x])/x^2,x]","-\frac{i (1-i a x)^{3/4} \left(2 a x \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)+3 a x-3 i\right)}{3 x (1+i a x)^{3/4}}","-\frac{(1-i a x)^{3/4} \sqrt[4]{1+i a x}}{x}-i a \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-i a \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)",1,"((-1/3*I)*(1 - I*a*x)^(3/4)*(-3*I + 3*a*x + 2*a*x*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)]))/(x*(1 + I*a*x)^(3/4))","C",0
66,1,81,132,0.024341,"\int \frac{e^{\frac{1}{2} i \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[E^((I/2)*ArcTan[a*x])/x^3,x]","\frac{(1-i a x)^{3/4} \left(2 a^2 x^2 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)+9 a^2 x^2-15 i a x-6\right)}{12 x^2 (1+i a x)^{3/4}}","\frac{1}{4} a^2 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{1}{4} a^2 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 x^2}-\frac{i a (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{4 x}",1,"((1 - I*a*x)^(3/4)*(-6 - (15*I)*a*x + 9*a^2*x^2 + 2*a^2*x^2*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)]))/(12*x^2*(1 + I*a*x)^(3/4))","C",0
67,1,93,170,0.028471,"\int \frac{e^{\frac{1}{2} i \tan ^{-1}(a x)}}{x^4} \, dx","Integrate[E^((I/2)*ArcTan[a*x])/x^4,x]","\frac{(1-i a x)^{3/4} \left(6 i a^3 x^3 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)+11 i a^3 x^3+21 a^2 x^2-18 i a x-8\right)}{24 x^3 (1+i a x)^{3/4}}","\frac{3}{8} i a^3 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{3}{8} i a^3 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{11 a^2 (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{24 x}-\frac{(1-i a x)^{3/4} \sqrt[4]{1+i a x}}{3 x^3}-\frac{5 i a (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{12 x^2}",1,"((1 - I*a*x)^(3/4)*(-8 - (18*I)*a*x + 21*a^2*x^2 + (11*I)*a^3*x^3 + (6*I)*a^3*x^3*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)]))/(24*x^3*(1 + I*a*x)^(3/4))","C",0
68,1,99,202,0.0325583,"\int \frac{e^{\frac{1}{2} i \tan ^{-1}(a x)}}{x^5} \, dx","Integrate[E^((I/2)*ArcTan[a*x])/x^5,x]","-\frac{(1-i a x)^{3/4} \left(22 a^4 x^4 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)+83 a^4 x^4-141 i a^3 x^3-114 a^2 x^2+104 i a x+48\right)}{192 x^4 (1+i a x)^{3/4}}","-\frac{11}{64} a^4 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{11}{64} a^4 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{83 i a^3 (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{192 x}+\frac{29 a^2 (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{96 x^2}-\frac{(1-i a x)^{3/4} \sqrt[4]{1+i a x}}{4 x^4}-\frac{7 i a (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{24 x^3}",1,"-1/192*((1 - I*a*x)^(3/4)*(48 + (104*I)*a*x - 114*a^2*x^2 - (141*I)*a^3*x^3 + 83*a^4*x^4 + 22*a^4*x^4*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)]))/(x^4*(1 + I*a*x)^(3/4))","C",0
69,1,111,240,0.0367613,"\int \frac{e^{\frac{1}{2} i \tan ^{-1}(a x)}}{x^6} \, dx","Integrate[E^((I/2)*ArcTan[a*x])/x^6,x]","\frac{(1-i a x)^{3/4} \left(-310 i a^5 x^5 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)-611 i a^5 x^5-1149 a^4 x^4+978 i a^3 x^3+872 a^2 x^2-816 i a x-384\right)}{1920 x^5 (1+i a x)^{3/4}}","-\frac{31}{128} i a^5 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{31}{128} i a^5 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{611 a^4 (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{1920 x}+\frac{269 i a^3 (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{960 x^2}+\frac{11 a^2 (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{48 x^3}-\frac{(1-i a x)^{3/4} \sqrt[4]{1+i a x}}{5 x^5}-\frac{9 i a (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{40 x^4}",1,"((1 - I*a*x)^(3/4)*(-384 - (816*I)*a*x + 872*a^2*x^2 + (978*I)*a^3*x^3 - 1149*a^4*x^4 - (611*I)*a^5*x^5 - (310*I)*a^5*x^5*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)]))/(1920*x^5*(1 + I*a*x)^(3/4))","C",0
70,1,148,337,0.1460638,"\int e^{\frac{3}{2} i \tan ^{-1}(a x)} x^3 \, dx","Integrate[E^(((3*I)/2)*ArcTan[a*x])*x^3,x]","\frac{\sqrt[4]{1-i a x} \left(i a^3 x^3 (1+i a x)^{3/4}+a^2 x^2 (1+i a x)^{3/4}-24\ 2^{3/4} \, _2F_1\left(-\frac{11}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-i a x)\right)+8\ 2^{3/4} \, _2F_1\left(-\frac{7}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-i a x)\right)+2\ 2^{3/4} \, _2F_1\left(-\frac{3}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-i a x)\right)\right)}{4 a^4}","-\frac{\sqrt[4]{1-i a x} (1+i a x)^{7/4} (11+4 i a x)}{32 a^4}-\frac{41 \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac{123 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{128 \sqrt{2} a^4}-\frac{123 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{128 \sqrt{2} a^4}+\frac{123 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{64 \sqrt{2} a^4}-\frac{123 \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{64 \sqrt{2} a^4}+\frac{x^2 \sqrt[4]{1-i a x} (1+i a x)^{7/4}}{4 a^2}",1,"((1 - I*a*x)^(1/4)*(a^2*x^2*(1 + I*a*x)^(3/4) + I*a^3*x^3*(1 + I*a*x)^(3/4) - 24*2^(3/4)*Hypergeometric2F1[-11/4, 1/4, 5/4, (1 - I*a*x)/2] + 8*2^(3/4)*Hypergeometric2F1[-7/4, 1/4, 5/4, (1 - I*a*x)/2] + 2*2^(3/4)*Hypergeometric2F1[-3/4, 1/4, 5/4, (1 - I*a*x)/2]))/(4*a^4)","C",0
71,1,82,339,0.0434176,"\int e^{\frac{3}{2} i \tan ^{-1}(a x)} x^2 \, dx","Integrate[E^(((3*I)/2)*ArcTan[a*x])*x^2,x]","\frac{\sqrt[4]{1-i a x} \left((1+i a x)^{3/4} \left(4 i a^2 x^2+7 a x-3 i\right)-34 i 2^{3/4} \, _2F_1\left(-\frac{3}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-i a x)\right)\right)}{12 a^3}","-\frac{i \sqrt[4]{1-i a x} (1+i a x)^{7/4}}{4 a^3}-\frac{17 i \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{24 a^3}+\frac{17 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{16 \sqrt{2} a^3}-\frac{17 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{16 \sqrt{2} a^3}+\frac{17 i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{8 \sqrt{2} a^3}-\frac{17 i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{8 \sqrt{2} a^3}+\frac{x \sqrt[4]{1-i a x} (1+i a x)^{7/4}}{3 a^2}",1,"((1 - I*a*x)^(1/4)*((1 + I*a*x)^(3/4)*(-3*I + 7*a*x + (4*I)*a^2*x^2) - (34*I)*2^(3/4)*Hypergeometric2F1[-3/4, 1/4, 5/4, (1 - I*a*x)/2]))/(12*a^3)","C",0
72,1,61,295,0.0198738,"\int e^{\frac{3}{2} i \tan ^{-1}(a x)} x \, dx","Integrate[E^(((3*I)/2)*ArcTan[a*x])*x,x]","\frac{\sqrt[4]{1-i a x} \left(6\ 2^{3/4} \, _2F_1\left(-\frac{3}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-i a x)\right)+(1+i a x)^{7/4}\right)}{2 a^2}","\frac{\sqrt[4]{1-i a x} (1+i a x)^{7/4}}{2 a^2}+\frac{3 \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac{9 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{8 \sqrt{2} a^2}+\frac{9 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{8 \sqrt{2} a^2}-\frac{9 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{4 \sqrt{2} a^2}+\frac{9 \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{4 \sqrt{2} a^2}",1,"((1 - I*a*x)^(1/4)*((1 + I*a*x)^(7/4) + 6*2^(3/4)*Hypergeometric2F1[-3/4, 1/4, 5/4, (1 - I*a*x)/2]))/(2*a^2)","C",0
73,1,41,268,0.029951,"\int e^{\frac{3}{2} i \tan ^{-1}(a x)} \, dx","Integrate[E^(((3*I)/2)*ArcTan[a*x]),x]","-\frac{8 i e^{\frac{7}{2} i \tan ^{-1}(a x)} \, _2F_1\left(\frac{7}{4},2;\frac{11}{4};-e^{2 i \tan ^{-1}(a x)}\right)}{7 a}","\frac{i \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{a}-\frac{3 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{2 \sqrt{2} a}+\frac{3 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{2 \sqrt{2} a}-\frac{3 i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{\sqrt{2} a}+\frac{3 i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{\sqrt{2} a}",1,"(((-8*I)/7)*E^(((7*I)/2)*ArcTan[a*x])*Hypergeometric2F1[7/4, 2, 11/4, -E^((2*I)*ArcTan[a*x])])/a","C",0
74,1,96,267,0.0348016,"\int \frac{e^{\frac{3}{2} i \tan ^{-1}(a x)}}{x} \, dx","Integrate[E^(((3*I)/2)*ArcTan[a*x])/x,x]","-2\ 2^{3/4} \sqrt[4]{1-i a x} \, _2F_1\left(\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-i a x)\right)-\frac{4 \sqrt[4]{1-i a x} \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};-\frac{1-i a x}{-i a x-1}\right)}{\sqrt[4]{1+i a x}}","\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{\sqrt{2}}-\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{\sqrt{2}}+2 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)-\sqrt{2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)-2 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)",1,"-2*2^(3/4)*(1 - I*a*x)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (1 - I*a*x)/2] - (4*(1 - I*a*x)^(1/4)*Hypergeometric2F1[1/4, 1, 5/4, -((1 - I*a*x)/(-1 - I*a*x))])/(1 + I*a*x)^(1/4)","C",0
75,1,68,92,0.0161134,"\int \frac{e^{\frac{3}{2} i \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[E^(((3*I)/2)*ArcTan[a*x])/x^2,x]","-\frac{i \sqrt[4]{1-i a x} \left(6 a x \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a x+i}{i-a x}\right)+a x-i\right)}{x \sqrt[4]{1+i a x}}","-\frac{\sqrt[4]{1-i a x} (1+i a x)^{3/4}}{x}+3 i a \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-3 i a \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)",1,"((-I)*(1 - I*a*x)^(1/4)*(-I + a*x + 6*a*x*Hypergeometric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)]))/(x*(1 + I*a*x)^(1/4))","C",0
76,1,81,132,0.0199974,"\int \frac{e^{\frac{3}{2} i \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[E^(((3*I)/2)*ArcTan[a*x])/x^3,x]","\frac{\sqrt[4]{1-i a x} \left(18 a^2 x^2 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a x+i}{i-a x}\right)+5 a^2 x^2-7 i a x-2\right)}{4 x^2 \sqrt[4]{1+i a x}}","-\frac{9}{4} a^2 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{9}{4} a^2 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{\sqrt[4]{1-i a x} (1+i a x)^{7/4}}{2 x^2}-\frac{3 i a \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{4 x}",1,"((1 - I*a*x)^(1/4)*(-2 - (7*I)*a*x + 5*a^2*x^2 + 18*a^2*x^2*Hypergeometric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)]))/(4*x^2*(1 + I*a*x)^(1/4))","C",0
77,1,93,170,0.0248411,"\int \frac{e^{\frac{3}{2} i \tan ^{-1}(a x)}}{x^4} \, dx","Integrate[E^(((3*I)/2)*ArcTan[a*x])/x^4,x]","\frac{\sqrt[4]{1-i a x} \left(102 i a^3 x^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a x+i}{i-a x}\right)+23 i a^3 x^3+37 a^2 x^2-22 i a x-8\right)}{24 x^3 \sqrt[4]{1+i a x}}","-\frac{17}{8} i a^3 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{17}{8} i a^3 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{23 a^2 \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{24 x}-\frac{\sqrt[4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}-\frac{7 i a \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}",1,"((1 - I*a*x)^(1/4)*(-8 - (22*I)*a*x + 37*a^2*x^2 + (23*I)*a^3*x^3 + (102*I)*a^3*x^3*Hypergeometric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)]))/(24*x^3*(1 + I*a*x)^(1/4))","C",0
78,1,99,202,0.0334617,"\int \frac{e^{\frac{3}{2} i \tan ^{-1}(a x)}}{x^5} \, dx","Integrate[E^(((3*I)/2)*ArcTan[a*x])/x^5,x]","-\frac{\sqrt[4]{1-i a x} \left(246 a^4 x^4 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a x+i}{i-a x}\right)+63 a^4 x^4-93 i a^3 x^3-54 a^2 x^2+40 i a x+16\right)}{64 x^4 \sqrt[4]{1+i a x}}","\frac{123}{64} a^4 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{123}{64} a^4 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{63 i a^3 \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{64 x}+\frac{15 a^2 \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{32 x^2}-\frac{\sqrt[4]{1-i a x} (1+i a x)^{3/4}}{4 x^4}-\frac{3 i a \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{8 x^3}",1,"-1/64*((1 - I*a*x)^(1/4)*(16 + (40*I)*a*x - 54*a^2*x^2 - (93*I)*a^3*x^3 + 63*a^4*x^4 + 246*a^4*x^4*Hypergeometric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)]))/(x^4*(1 + I*a*x)^(1/4))","C",0
79,1,96,373,0.0461629,"\int e^{\frac{5}{2} i \tan ^{-1}(a x)} x^3 \, dx","Integrate[E^(((5*I)/2)*ArcTan[a*x])*x^3,x]","\frac{380 \sqrt[4]{2} (1-i a x) \, _2F_1\left(-\frac{5}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-i a x)\right)-\sqrt[4]{1+i a x} (a x-i)^2 \left(6 a^2 x^2-5 i a x+59\right)}{24 a^4 \sqrt[4]{1-i a x}}","-\frac{i (-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac{475 (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{64 a^4}+\frac{475 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{128 \sqrt{2} a^4}-\frac{475 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{128 \sqrt{2} a^4}-\frac{475 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{64 \sqrt{2} a^4}+\frac{475 \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{64 \sqrt{2} a^4}-\frac{17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac{4 i x^3 (1+i a x)^{5/4}}{a \sqrt[4]{1-i a x}}",1,"(-((1 + I*a*x)^(1/4)*(-I + a*x)^2*(59 - (5*I)*a*x + 6*a^2*x^2)) + 380*2^(1/4)*(1 - I*a*x)*Hypergeometric2F1[-5/4, 3/4, 7/4, (1 - I*a*x)/2])/(24*a^4*(1 - I*a*x)^(1/4))","C",0
80,1,86,371,0.0372084,"\int e^{\frac{5}{2} i \tan ^{-1}(a x)} x^2 \, dx","Integrate[E^(((5*I)/2)*ArcTan[a*x])*x^2,x]","\frac{44 \sqrt[4]{2} (a x+i) \, _2F_1\left(-\frac{5}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-i a x)\right)-\sqrt[4]{1+i a x} (a x-i)^2 (a x+7 i)}{3 a^3 \sqrt[4]{1-i a x}}","\frac{i (1-i a x)^{3/4} (1+i a x)^{9/4}}{3 a^3}+\frac{2 i (1+i a x)^{9/4}}{a^3 \sqrt[4]{1-i a x}}+\frac{11 i (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^3}+\frac{55 i (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{8 a^3}+\frac{55 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{16 \sqrt{2} a^3}-\frac{55 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{16 \sqrt{2} a^3}-\frac{55 i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{8 \sqrt{2} a^3}+\frac{55 i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{8 \sqrt{2} a^3}",1,"(-((1 + I*a*x)^(1/4)*(-I + a*x)^2*(7*I + a*x)) + 44*2^(1/4)*(I + a*x)*Hypergeometric2F1[-5/4, 3/4, 7/4, (1 - I*a*x)/2])/(3*a^3*(1 - I*a*x)^(1/4))","C",0
81,1,72,324,0.0388381,"\int e^{\frac{5}{2} i \tan ^{-1}(a x)} x \, dx","Integrate[E^(((5*I)/2)*ArcTan[a*x])*x,x]","\frac{2 \left(20 i \sqrt[4]{2} (a x+i) \, _2F_1\left(-\frac{5}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-i a x)\right)-3 (1+i a x)^{9/4}\right)}{3 a^2 \sqrt[4]{1-i a x}}","-\frac{2 (1+i a x)^{9/4}}{a^2 \sqrt[4]{1-i a x}}-\frac{5 (1-i a x)^{3/4} (1+i a x)^{5/4}}{2 a^2}-\frac{25 (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{4 a^2}-\frac{25 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{8 \sqrt{2} a^2}+\frac{25 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{8 \sqrt{2} a^2}+\frac{25 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{4 \sqrt{2} a^2}-\frac{25 \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{4 \sqrt{2} a^2}",1,"(2*(-3*(1 + I*a*x)^(9/4) + (20*I)*2^(1/4)*(I + a*x)*Hypergeometric2F1[-5/4, 3/4, 7/4, (1 - I*a*x)/2]))/(3*a^2*(1 - I*a*x)^(1/4))","C",0
82,1,41,299,0.0411125,"\int e^{\frac{5}{2} i \tan ^{-1}(a x)} \, dx","Integrate[E^(((5*I)/2)*ArcTan[a*x]),x]","-\frac{8 i e^{\frac{9}{2} i \tan ^{-1}(a x)} \, _2F_1\left(2,\frac{9}{4};\frac{13}{4};-e^{2 i \tan ^{-1}(a x)}\right)}{9 a}","-\frac{4 i (1+i a x)^{5/4}}{a \sqrt[4]{1-i a x}}-\frac{5 i (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{a}-\frac{5 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{2 \sqrt{2} a}+\frac{5 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{2 \sqrt{2} a}+\frac{5 i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{\sqrt{2} a}-\frac{5 i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{\sqrt{2} a}",1,"(((-8*I)/9)*E^(((9*I)/2)*ArcTan[a*x])*Hypergeometric2F1[2, 9/4, 13/4, -E^((2*I)*ArcTan[a*x])])/a","C",0
83,1,112,293,0.0443969,"\int \frac{e^{\frac{5}{2} i \tan ^{-1}(a x)}}{x} \, dx","Integrate[E^(((5*I)/2)*ArcTan[a*x])/x,x]","\frac{4 \left(3 \sqrt[4]{2} (1+i a x)^{3/4} \, _2F_1\left(-\frac{1}{4},-\frac{1}{4};\frac{3}{4};\frac{1}{2} (1-i a x)\right)+(-1+i a x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)+3 i a x+3\right)}{3 \sqrt[4]{1-i a x} (1+i a x)^{3/4}}","\frac{8 \sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}+\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{\sqrt{2}}-\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{\sqrt{2}}-2 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)+\sqrt{2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)-2 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)",1,"(4*(3 + (3*I)*a*x + 3*2^(1/4)*(1 + I*a*x)^(3/4)*Hypergeometric2F1[-1/4, -1/4, 3/4, (1 - I*a*x)/2] + (-1 + I*a*x)*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)]))/(3*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))","C",0
84,1,87,121,0.0214189,"\int \frac{e^{\frac{5}{2} i \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[E^(((5*I)/2)*ArcTan[a*x])/x^2,x]","\frac{-3 \left(9 a^2 x^2-8 i a x+1\right)-10 a x (a x+i) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)}{3 x \sqrt[4]{1-i a x} (1+i a x)^{3/4}}","-\frac{(1+i a x)^{5/4}}{x \sqrt[4]{1-i a x}}+\frac{10 i a \sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}-5 i a \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-5 i a \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)",1,"(-3*(1 - (8*I)*a*x + 9*a^2*x^2) - 10*a*x*(I + a*x)*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)])/(3*x*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))","C",0
85,1,99,163,0.0252971,"\int \frac{e^{\frac{5}{2} i \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[E^(((5*I)/2)*ArcTan[a*x])/x^3,x]","\frac{-129 i a^3 x^3+50 a^2 x^2 (1-i a x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)-102 a^2 x^2-33 i a x-6}{12 x^2 \sqrt[4]{1-i a x} (1+i a x)^{3/4}}","-\frac{25 a^2 \sqrt[4]{1+i a x}}{2 \sqrt[4]{1-i a x}}+\frac{25}{4} a^2 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{25}{4} a^2 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{(1+i a x)^{9/4}}{2 x^2 \sqrt[4]{1-i a x}}-\frac{5 i a (1+i a x)^{5/4}}{4 x \sqrt[4]{1-i a x}}",1,"(-6 - (33*I)*a*x - 102*a^2*x^2 - (129*I)*a^3*x^3 + 50*a^2*x^2*(1 - I*a*x)*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)])/(12*x^2*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))","C",0
86,1,106,203,0.0311315,"\int \frac{e^{\frac{5}{2} i \tan ^{-1}(a x)}}{x^4} \, dx","Integrate[E^(((5*I)/2)*ArcTan[a*x])/x^4,x]","\frac{287 a^4 x^4+110 a^3 x^3 (a x+i) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)-226 i a^3 x^3+87 a^2 x^2-34 i a x-8}{24 x^3 \sqrt[4]{1-i a x} (1+i a x)^{3/4}}","-\frac{287 i a^3 \sqrt[4]{1+i a x}}{24 \sqrt[4]{1-i a x}}+\frac{55}{8} i a^3 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{55}{8} i a^3 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{61 a^2 \sqrt[4]{1+i a x}}{24 x \sqrt[4]{1-i a x}}-\frac{\sqrt[4]{1+i a x}}{3 x^3 \sqrt[4]{1-i a x}}-\frac{13 i a \sqrt[4]{1+i a x}}{12 x^2 \sqrt[4]{1-i a x}}",1,"(-8 - (34*I)*a*x + 87*a^2*x^2 - (226*I)*a^3*x^3 + 287*a^4*x^4 + 110*a^3*x^3*(I + a*x)*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)])/(24*x^3*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))","C",0
87,1,118,233,0.037464,"\int \frac{e^{\frac{5}{2} i \tan ^{-1}(a x)}}{x^5} \, dx","Integrate[E^(((5*I)/2)*ArcTan[a*x])/x^5,x]","\frac{2467 i a^5 x^5+950 i a^4 x^4 (a x+i) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)+1946 a^4 x^4+747 i a^3 x^3+362 a^2 x^2-184 i a x-48}{192 x^4 \sqrt[4]{1-i a x} (1+i a x)^{3/4}}","\frac{2467 a^4 \sqrt[4]{1+i a x}}{192 \sqrt[4]{1-i a x}}-\frac{475}{64} a^4 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{475}{64} a^4 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{521 i a^3 \sqrt[4]{1+i a x}}{192 x \sqrt[4]{1-i a x}}+\frac{113 a^2 \sqrt[4]{1+i a x}}{96 x^2 \sqrt[4]{1-i a x}}-\frac{\sqrt[4]{1+i a x}}{4 x^4 \sqrt[4]{1-i a x}}-\frac{17 i a \sqrt[4]{1+i a x}}{24 x^3 \sqrt[4]{1-i a x}}",1,"(-48 - (184*I)*a*x + 362*a^2*x^2 + (747*I)*a^3*x^3 + 1946*a^4*x^4 + (2467*I)*a^5*x^5 + (950*I)*a^4*x^4*(I + a*x)*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)])/(192*x^4*(1 - I*a*x)^(1/4)*(1 + I*a*x)^(3/4))","C",0
88,1,127,337,0.1079976,"\int e^{-\frac{1}{2} i \tan ^{-1}(a x)} x^3 \, dx","Integrate[x^3/E^((I/2)*ArcTan[a*x]),x]","\frac{(1-i a x)^{5/4} \left(5 a^2 x^2 (1+i a x)^{3/4}+4\ 2^{3/4} \, _2F_1\left(-\frac{7}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-i a x)\right)-12\ 2^{3/4} \, _2F_1\left(-\frac{3}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-i a x)\right)+5\ 2^{3/4} \, _2F_1\left(\frac{1}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-i a x)\right)\right)}{20 a^4}","-\frac{(1-i a x)^{5/4} (1+i a x)^{3/4} (25-4 i a x)}{96 a^4}-\frac{11 \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}-\frac{11 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{128 \sqrt{2} a^4}+\frac{11 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{128 \sqrt{2} a^4}-\frac{11 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{64 \sqrt{2} a^4}+\frac{11 \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{64 \sqrt{2} a^4}+\frac{x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}",1,"((1 - I*a*x)^(5/4)*(5*a^2*x^2*(1 + I*a*x)^(3/4) + 4*2^(3/4)*Hypergeometric2F1[-7/4, 5/4, 9/4, (1 - I*a*x)/2] - 12*2^(3/4)*Hypergeometric2F1[-3/4, 5/4, 9/4, (1 - I*a*x)/2] + 5*2^(3/4)*Hypergeometric2F1[1/4, 5/4, 9/4, (1 - I*a*x)/2]))/(20*a^4)","C",0
89,1,73,339,0.0316952,"\int e^{-\frac{1}{2} i \tan ^{-1}(a x)} x^2 \, dx","Integrate[x^2/E^((I/2)*ArcTan[a*x]),x]","\frac{(1-i a x)^{5/4} \left(5 (1+i a x)^{3/4} (4 a x+i)-9 i 2^{3/4} \, _2F_1\left(\frac{1}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-i a x)\right)\right)}{60 a^3}","\frac{i (1+i a x)^{3/4} (1-i a x)^{5/4}}{12 a^3}+\frac{3 i (1+i a x)^{3/4} \sqrt[4]{1-i a x}}{8 a^3}+\frac{3 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{16 \sqrt{2} a^3}-\frac{3 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{16 \sqrt{2} a^3}+\frac{3 i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{8 \sqrt{2} a^3}-\frac{3 i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{8 \sqrt{2} a^3}+\frac{x (1+i a x)^{3/4} (1-i a x)^{5/4}}{3 a^2}",1,"((1 - I*a*x)^(5/4)*(5*(1 + I*a*x)^(3/4)*(I + 4*a*x) - (9*I)*2^(3/4)*Hypergeometric2F1[1/4, 5/4, 9/4, (1 - I*a*x)/2]))/(60*a^3)","C",0
90,1,63,295,0.0180358,"\int e^{-\frac{1}{2} i \tan ^{-1}(a x)} x \, dx","Integrate[x/E^((I/2)*ArcTan[a*x]),x]","\frac{(1-i a x)^{5/4} \left(5 (1+i a x)^{3/4}-2^{3/4} \, _2F_1\left(\frac{1}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-i a x)\right)\right)}{10 a^2}","\frac{(1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}+\frac{(1+i a x)^{3/4} \sqrt[4]{1-i a x}}{4 a^2}+\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{8 \sqrt{2} a^2}-\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{8 \sqrt{2} a^2}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{4 \sqrt{2} a^2}-\frac{\tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{4 \sqrt{2} a^2}",1,"((1 - I*a*x)^(5/4)*(5*(1 + I*a*x)^(3/4) - 2^(3/4)*Hypergeometric2F1[1/4, 5/4, 9/4, (1 - I*a*x)/2]))/(10*a^2)","C",0
91,1,41,268,0.0297868,"\int e^{-\frac{1}{2} i \tan ^{-1}(a x)} \, dx","Integrate[E^((-1/2*I)*ArcTan[a*x]),x]","-\frac{8 i e^{\frac{3}{2} i \tan ^{-1}(a x)} \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-e^{2 i \tan ^{-1}(a x)}\right)}{3 a}","-\frac{i \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{a}-\frac{i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{2 \sqrt{2} a}+\frac{i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{2 \sqrt{2} a}-\frac{i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{\sqrt{2} a}+\frac{i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{\sqrt{2} a}",1,"(((-8*I)/3)*E^(((3*I)/2)*ArcTan[a*x])*Hypergeometric2F1[3/4, 2, 7/4, -E^((2*I)*ArcTan[a*x])])/a","C",0
92,1,96,267,0.0333288,"\int \frac{e^{-\frac{1}{2} i \tan ^{-1}(a x)}}{x} \, dx","Integrate[1/(E^((I/2)*ArcTan[a*x])*x),x]","2\ 2^{3/4} \sqrt[4]{1-i a x} \, _2F_1\left(\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-i a x)\right)-\frac{4 \sqrt[4]{1-i a x} \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};-\frac{1-i a x}{-i a x-1}\right)}{\sqrt[4]{1+i a x}}","-\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{\sqrt{2}}+\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{\sqrt{2}}+2 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)+\sqrt{2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)-2 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)",1,"2*2^(3/4)*(1 - I*a*x)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (1 - I*a*x)/2] - (4*(1 - I*a*x)^(1/4)*Hypergeometric2F1[1/4, 1, 5/4, -((1 - I*a*x)/(-1 - I*a*x))])/(1 + I*a*x)^(1/4)","C",0
93,1,69,92,0.0158972,"\int \frac{e^{-\frac{1}{2} i \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[1/(E^((I/2)*ArcTan[a*x])*x^2),x]","\frac{i \sqrt[4]{1-i a x} \left(2 a x \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a x+i}{i-a x}\right)-a x+i\right)}{x \sqrt[4]{1+i a x}}","-\frac{\sqrt[4]{1-i a x} (1+i a x)^{3/4}}{x}-i a \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+i a \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)",1,"(I*(1 - I*a*x)^(1/4)*(I - a*x + 2*a*x*Hypergeometric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)]))/(x*(1 + I*a*x)^(1/4))","C",0
94,1,81,132,0.0195054,"\int \frac{e^{-\frac{1}{2} i \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[1/(E^((I/2)*ArcTan[a*x])*x^3),x]","\frac{\sqrt[4]{1-i a x} \left(2 a^2 x^2 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a x+i}{i-a x}\right)-3 a^2 x^2+i a x-2\right)}{4 x^2 \sqrt[4]{1+i a x}}","-\frac{1}{4} a^2 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{1}{4} a^2 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{(1-i a x)^{5/4} (1+i a x)^{3/4}}{2 x^2}+\frac{i a \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{4 x}",1,"((1 - I*a*x)^(1/4)*(-2 + I*a*x - 3*a^2*x^2 + 2*a^2*x^2*Hypergeometric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)]))/(4*x^2*(1 + I*a*x)^(1/4))","C",0
95,1,92,170,0.0273703,"\int \frac{e^{-\frac{1}{2} i \tan ^{-1}(a x)}}{x^4} \, dx","Integrate[1/(E^((I/2)*ArcTan[a*x])*x^4),x]","\frac{\sqrt[4]{1-i a x} \left(-18 i a^3 x^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a x+i}{i-a x}\right)+11 i a^3 x^3+a^2 x^2+2 i a x-8\right)}{24 x^3 \sqrt[4]{1+i a x}}","\frac{3}{8} i a^3 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{3}{8} i a^3 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{11 a^2 \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{24 x}-\frac{\sqrt[4]{1-i a x} (1+i a x)^{3/4}}{3 x^3}+\frac{5 i a \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{12 x^2}",1,"((1 - I*a*x)^(1/4)*(-8 + (2*I)*a*x + a^2*x^2 + (11*I)*a^3*x^3 - (18*I)*a^3*x^3*Hypergeometric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)]))/(24*x^3*(1 + I*a*x)^(1/4))","C",0
96,1,99,202,0.0340908,"\int \frac{e^{-\frac{1}{2} i \tan ^{-1}(a x)}}{x^5} \, dx","Integrate[1/(E^((I/2)*ArcTan[a*x])*x^5),x]","\frac{\sqrt[4]{1-i a x} \left(-66 a^4 x^4 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a x+i}{i-a x}\right)+83 a^4 x^4-25 i a^3 x^3+2 a^2 x^2+8 i a x-48\right)}{192 x^4 \sqrt[4]{1+i a x}}","\frac{11}{64} a^4 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{11}{64} a^4 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{83 i a^3 \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{192 x}+\frac{29 a^2 \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{96 x^2}-\frac{\sqrt[4]{1-i a x} (1+i a x)^{3/4}}{4 x^4}+\frac{7 i a \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{24 x^3}",1,"((1 - I*a*x)^(1/4)*(-48 + (8*I)*a*x + 2*a^2*x^2 - (25*I)*a^3*x^3 + 83*a^4*x^4 - 66*a^4*x^4*Hypergeometric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)]))/(192*x^4*(1 + I*a*x)^(1/4))","C",0
97,1,127,337,0.1151066,"\int e^{-\frac{3}{2} i \tan ^{-1}(a x)} x^3 \, dx","Integrate[x^3/E^(((3*I)/2)*ArcTan[a*x]),x]","\frac{(1-i a x)^{7/4} \left(7 a^2 x^2 \sqrt[4]{1+i a x}+12 \sqrt[4]{2} \, _2F_1\left(-\frac{5}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-i a x)\right)-20 \sqrt[4]{2} \, _2F_1\left(-\frac{1}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-i a x)\right)+7 \sqrt[4]{2} \, _2F_1\left(\frac{3}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-i a x)\right)\right)}{28 a^4}","-\frac{(1-i a x)^{7/4} \sqrt[4]{1+i a x} (11-4 i a x)}{32 a^4}-\frac{41 (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{64 a^4}+\frac{123 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{128 \sqrt{2} a^4}-\frac{123 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{128 \sqrt{2} a^4}-\frac{123 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{64 \sqrt{2} a^4}+\frac{123 \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{64 \sqrt{2} a^4}+\frac{x^2 (1-i a x)^{7/4} \sqrt[4]{1+i a x}}{4 a^2}",1,"((1 - I*a*x)^(7/4)*(7*a^2*x^2*(1 + I*a*x)^(1/4) + 12*2^(1/4)*Hypergeometric2F1[-5/4, 7/4, 11/4, (1 - I*a*x)/2] - 20*2^(1/4)*Hypergeometric2F1[-1/4, 7/4, 11/4, (1 - I*a*x)/2] + 7*2^(1/4)*Hypergeometric2F1[3/4, 7/4, 11/4, (1 - I*a*x)/2]))/(28*a^4)","C",0
98,1,73,339,0.031409,"\int e^{-\frac{3}{2} i \tan ^{-1}(a x)} x^2 \, dx","Integrate[x^2/E^(((3*I)/2)*ArcTan[a*x]),x]","\frac{(1-i a x)^{7/4} \left(7 \sqrt[4]{1+i a x} (4 a x+3 i)-17 i \sqrt[4]{2} \, _2F_1\left(\frac{3}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-i a x)\right)\right)}{84 a^3}","\frac{i \sqrt[4]{1+i a x} (1-i a x)^{7/4}}{4 a^3}+\frac{17 i \sqrt[4]{1+i a x} (1-i a x)^{3/4}}{24 a^3}-\frac{17 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{16 \sqrt{2} a^3}+\frac{17 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{16 \sqrt{2} a^3}+\frac{17 i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{8 \sqrt{2} a^3}-\frac{17 i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{8 \sqrt{2} a^3}+\frac{x \sqrt[4]{1+i a x} (1-i a x)^{7/4}}{3 a^2}",1,"((1 - I*a*x)^(7/4)*(7*(1 + I*a*x)^(1/4)*(3*I + 4*a*x) - (17*I)*2^(1/4)*Hypergeometric2F1[3/4, 7/4, 11/4, (1 - I*a*x)/2]))/(84*a^3)","C",0
99,1,63,295,0.0162578,"\int e^{-\frac{3}{2} i \tan ^{-1}(a x)} x \, dx","Integrate[x/E^(((3*I)/2)*ArcTan[a*x]),x]","\frac{(1-i a x)^{7/4} \left(7 \sqrt[4]{1+i a x}-3 \sqrt[4]{2} \, _2F_1\left(\frac{3}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-i a x)\right)\right)}{14 a^2}","\frac{\sqrt[4]{1+i a x} (1-i a x)^{7/4}}{2 a^2}+\frac{3 \sqrt[4]{1+i a x} (1-i a x)^{3/4}}{4 a^2}-\frac{9 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{8 \sqrt{2} a^2}+\frac{9 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{8 \sqrt{2} a^2}+\frac{9 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{4 \sqrt{2} a^2}-\frac{9 \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{4 \sqrt{2} a^2}",1,"((1 - I*a*x)^(7/4)*(7*(1 + I*a*x)^(1/4) - 3*2^(1/4)*Hypergeometric2F1[3/4, 7/4, 11/4, (1 - I*a*x)/2]))/(14*a^2)","C",0
100,1,39,268,0.0410151,"\int e^{-\frac{3}{2} i \tan ^{-1}(a x)} \, dx","Integrate[E^(((-3*I)/2)*ArcTan[a*x]),x]","-\frac{8 i e^{\frac{1}{2} i \tan ^{-1}(a x)} \, _2F_1\left(\frac{1}{4},2;\frac{5}{4};-e^{2 i \tan ^{-1}(a x)}\right)}{a}","-\frac{i (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{a}+\frac{3 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{2 \sqrt{2} a}-\frac{3 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{2 \sqrt{2} a}-\frac{3 i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{\sqrt{2} a}+\frac{3 i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{\sqrt{2} a}",1,"((-8*I)*E^((I/2)*ArcTan[a*x])*Hypergeometric2F1[1/4, 2, 5/4, -E^((2*I)*ArcTan[a*x])])/a","C",0
101,1,97,267,0.0305719,"\int \frac{e^{-\frac{3}{2} i \tan ^{-1}(a x)}}{x} \, dx","Integrate[1/(E^(((3*I)/2)*ArcTan[a*x])*x),x]","\frac{2 (1-i a x)^{3/4} \left(\sqrt[4]{2} (1+i a x)^{3/4} \, _2F_1\left(\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-i a x)\right)-2 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)\right)}{3 (1+i a x)^{3/4}}","\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{\sqrt{2}}-\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{\sqrt{2}}-2 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)+\sqrt{2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)-2 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)",1,"(2*(1 - I*a*x)^(3/4)*(2^(1/4)*(1 + I*a*x)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, (1 - I*a*x)/2] - 2*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)]))/(3*(1 + I*a*x)^(3/4))","C",0
102,1,69,92,0.0176221,"\int \frac{e^{-\frac{3}{2} i \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[1/(E^(((3*I)/2)*ArcTan[a*x])*x^2),x]","\frac{i (1-i a x)^{3/4} \left(2 a x \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)-a x+i\right)}{x (1+i a x)^{3/4}}","-\frac{(1-i a x)^{3/4} \sqrt[4]{1+i a x}}{x}+3 i a \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+3 i a \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)",1,"(I*(1 - I*a*x)^(3/4)*(I - a*x + 2*a*x*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)]))/(x*(1 + I*a*x)^(3/4))","C",0
103,1,81,132,0.0354165,"\int \frac{e^{-\frac{3}{2} i \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[1/(E^(((3*I)/2)*ArcTan[a*x])*x^3),x]","\frac{(1-i a x)^{3/4} \left(6 a^2 x^2 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)-5 a^2 x^2+3 i a x-2\right)}{4 x^2 (1+i a x)^{3/4}}","\frac{9}{4} a^2 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{9}{4} a^2 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{(1-i a x)^{7/4} \sqrt[4]{1+i a x}}{2 x^2}+\frac{3 i a (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{4 x}",1,"((1 - I*a*x)^(3/4)*(-2 + (3*I)*a*x - 5*a^2*x^2 + 6*a^2*x^2*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)]))/(4*x^2*(1 + I*a*x)^(3/4))","C",0
104,1,93,170,0.0272969,"\int \frac{e^{-\frac{3}{2} i \tan ^{-1}(a x)}}{x^4} \, dx","Integrate[1/(E^(((3*I)/2)*ArcTan[a*x])*x^4),x]","\frac{(1-i a x)^{3/4} \left(-34 i a^3 x^3 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)+23 i a^3 x^3+9 a^2 x^2+6 i a x-8\right)}{24 x^3 (1+i a x)^{3/4}}","-\frac{17}{8} i a^3 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{17}{8} i a^3 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{23 a^2 (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{24 x}-\frac{(1-i a x)^{3/4} \sqrt[4]{1+i a x}}{3 x^3}+\frac{7 i a (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{12 x^2}",1,"((1 - I*a*x)^(3/4)*(-8 + (6*I)*a*x + 9*a^2*x^2 + (23*I)*a^3*x^3 - (34*I)*a^3*x^3*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)]))/(24*x^3*(1 + I*a*x)^(3/4))","C",0
105,1,99,202,0.0368022,"\int \frac{e^{-\frac{3}{2} i \tan ^{-1}(a x)}}{x^5} \, dx","Integrate[1/(E^(((3*I)/2)*ArcTan[a*x])*x^5),x]","\frac{(1-i a x)^{3/4} \left(-82 a^4 x^4 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a x+i}{i-a x}\right)+63 a^4 x^4-33 i a^3 x^3+6 a^2 x^2+8 i a x-16\right)}{64 x^4 (1+i a x)^{3/4}}","-\frac{123}{64} a^4 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{123}{64} a^4 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{63 i a^3 (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{64 x}+\frac{15 a^2 (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{32 x^2}-\frac{(1-i a x)^{3/4} \sqrt[4]{1+i a x}}{4 x^4}+\frac{3 i a (1-i a x)^{3/4} \sqrt[4]{1+i a x}}{8 x^3}",1,"((1 - I*a*x)^(3/4)*(-16 + (8*I)*a*x + 6*a^2*x^2 - (33*I)*a^3*x^3 + 63*a^4*x^4 - 82*a^4*x^4*Hypergeometric2F1[3/4, 1, 7/4, (I + a*x)/(I - a*x)]))/(64*x^4*(1 + I*a*x)^(3/4))","C",0
106,1,100,373,0.0454505,"\int e^{-\frac{5}{2} i \tan ^{-1}(a x)} x^3 \, dx","Integrate[x^3/E^(((5*I)/2)*ArcTan[a*x]),x]","-\frac{\sqrt[4]{1-i a x} (a x+i)^2 \left(3 \left(6 a^2 x^2+5 i a x+59\right)-95\ 2^{3/4} \sqrt[4]{1+i a x} \, _2F_1\left(\frac{1}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2} (1-i a x)\right)\right)}{72 a^4 \sqrt[4]{1+i a x}}","-\frac{i (1-i a x)^{5/4} (1+i a x)^{3/4} (452 a x+521 i)}{96 a^4}+\frac{475 \sqrt[4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac{475 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{128 \sqrt{2} a^4}-\frac{475 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{128 \sqrt{2} a^4}+\frac{475 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{64 \sqrt{2} a^4}-\frac{475 \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{64 \sqrt{2} a^4}-\frac{17 x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^2}+\frac{4 i x^3 (1-i a x)^{5/4}}{a \sqrt[4]{1+i a x}}",1,"-1/72*((1 - I*a*x)^(1/4)*(I + a*x)^2*(3*(59 + (5*I)*a*x + 6*a^2*x^2) - 95*2^(3/4)*(1 + I*a*x)^(1/4)*Hypergeometric2F1[1/4, 9/4, 13/4, (1 - I*a*x)/2]))/(a^4*(1 + I*a*x)^(1/4))","C",0
107,1,91,371,0.0325227,"\int e^{-\frac{5}{2} i \tan ^{-1}(a x)} x^2 \, dx","Integrate[x^2/E^(((5*I)/2)*ArcTan[a*x]),x]","-\frac{\sqrt[4]{1-i a x} (a x+i)^2 \left(11 i 2^{3/4} \sqrt[4]{1+i a x} \, _2F_1\left(\frac{1}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2} (1-i a x)\right)+3 a x-21 i\right)}{9 a^3 \sqrt[4]{1+i a x}}","-\frac{i (1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a^3}-\frac{2 i (1-i a x)^{9/4}}{a^3 \sqrt[4]{1+i a x}}-\frac{11 i (1+i a x)^{3/4} (1-i a x)^{5/4}}{4 a^3}-\frac{55 i (1+i a x)^{3/4} \sqrt[4]{1-i a x}}{8 a^3}-\frac{55 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{16 \sqrt{2} a^3}+\frac{55 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{16 \sqrt{2} a^3}-\frac{55 i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{8 \sqrt{2} a^3}+\frac{55 i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{8 \sqrt{2} a^3}",1,"-1/9*((1 - I*a*x)^(1/4)*(I + a*x)^2*(-21*I + 3*a*x + (11*I)*2^(3/4)*(1 + I*a*x)^(1/4)*Hypergeometric2F1[1/4, 9/4, 13/4, (1 - I*a*x)/2]))/(a^3*(1 + I*a*x)^(1/4))","C",0
108,1,63,324,0.0305636,"\int e^{-\frac{5}{2} i \tan ^{-1}(a x)} x \, dx","Integrate[x/E^(((5*I)/2)*ArcTan[a*x]),x]","\frac{2 (1-i a x)^{9/4} \left(5\ 2^{3/4} \, _2F_1\left(\frac{1}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2} (1-i a x)\right)-\frac{9}{\sqrt[4]{1+i a x}}\right)}{9 a^2}","-\frac{2 (1-i a x)^{9/4}}{a^2 \sqrt[4]{1+i a x}}-\frac{5 (1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}-\frac{25 (1+i a x)^{3/4} \sqrt[4]{1-i a x}}{4 a^2}-\frac{25 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{8 \sqrt{2} a^2}+\frac{25 \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{8 \sqrt{2} a^2}-\frac{25 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{4 \sqrt{2} a^2}+\frac{25 \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{4 \sqrt{2} a^2}",1,"(2*(1 - I*a*x)^(9/4)*(-9/(1 + I*a*x)^(1/4) + 5*2^(3/4)*Hypergeometric2F1[1/4, 9/4, 13/4, (1 - I*a*x)/2]))/(9*a^2)","C",0
109,1,39,299,0.0500893,"\int e^{-\frac{5}{2} i \tan ^{-1}(a x)} \, dx","Integrate[E^(((-5*I)/2)*ArcTan[a*x]),x]","\frac{8 i e^{-\frac{1}{2} i \tan ^{-1}(a x)} \, _2F_1\left(-\frac{1}{4},2;\frac{3}{4};-e^{2 i \tan ^{-1}(a x)}\right)}{a}","\frac{4 i (1-i a x)^{5/4}}{a \sqrt[4]{1+i a x}}+\frac{5 i (1+i a x)^{3/4} \sqrt[4]{1-i a x}}{a}+\frac{5 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{2 \sqrt{2} a}-\frac{5 i \log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{2 \sqrt{2} a}+\frac{5 i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{\sqrt{2} a}-\frac{5 i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)}{\sqrt{2} a}",1,"((8*I)*Hypergeometric2F1[-1/4, 2, 3/4, -E^((2*I)*ArcTan[a*x])])/(a*E^((I/2)*ArcTan[a*x]))","C",0
110,1,106,293,0.0685183,"\int \frac{e^{-\frac{5}{2} i \tan ^{-1}(a x)}}{x} \, dx","Integrate[1/(E^(((5*I)/2)*ArcTan[a*x])*x),x]","\frac{\sqrt[4]{1-i a x} \left(-20 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a x+i}{i-a x}\right)+2^{3/4} (1-i a x) \sqrt[4]{1+i a x} \, _2F_1\left(\frac{5}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-i a x)\right)+20\right)}{5 \sqrt[4]{1+i a x}}","\frac{8 \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{\sqrt{2}}-\frac{\log \left(\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+1\right)}{\sqrt{2}}+2 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)-\sqrt{2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}\right)-2 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)",1,"((1 - I*a*x)^(1/4)*(20 - 20*Hypergeometric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)] + 2^(3/4)*(1 - I*a*x)*(1 + I*a*x)^(1/4)*Hypergeometric2F1[5/4, 5/4, 9/4, (1 - I*a*x)/2]))/(5*(1 + I*a*x)^(1/4))","C",0
111,1,69,121,0.0189805,"\int \frac{e^{-\frac{5}{2} i \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[1/(E^(((5*I)/2)*ArcTan[a*x])*x^2),x]","\frac{i \sqrt[4]{1-i a x} \left(10 a x \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a x+i}{i-a x}\right)-9 a x+i\right)}{x \sqrt[4]{1+i a x}}","-\frac{(1-i a x)^{5/4}}{x \sqrt[4]{1+i a x}}-\frac{10 i a \sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}-5 i a \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+5 i a \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)",1,"(I*(1 - I*a*x)^(1/4)*(I - 9*a*x + 10*a*x*Hypergeometric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)]))/(x*(1 + I*a*x)^(1/4))","C",0
112,1,81,163,0.0202971,"\int \frac{e^{-\frac{5}{2} i \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[1/(E^(((5*I)/2)*ArcTan[a*x])*x^3),x]","\frac{\sqrt[4]{1-i a x} \left(50 a^2 x^2 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a x+i}{i-a x}\right)-43 a^2 x^2+9 i a x-2\right)}{4 x^2 \sqrt[4]{1+i a x}}","-\frac{25 a^2 \sqrt[4]{1-i a x}}{2 \sqrt[4]{1+i a x}}-\frac{25}{4} a^2 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{25}{4} a^2 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{(1-i a x)^{9/4}}{2 x^2 \sqrt[4]{1+i a x}}+\frac{5 i a (1-i a x)^{5/4}}{4 x \sqrt[4]{1+i a x}}",1,"((1 - I*a*x)^(1/4)*(-2 + (9*I)*a*x - 43*a^2*x^2 + 50*a^2*x^2*Hypergeometric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)]))/(4*x^2*(1 + I*a*x)^(1/4))","C",0
113,1,93,203,0.0273962,"\int \frac{e^{-\frac{5}{2} i \tan ^{-1}(a x)}}{x^4} \, dx","Integrate[1/(E^(((5*I)/2)*ArcTan[a*x])*x^4),x]","\frac{\sqrt[4]{1-i a x} \left(-330 i a^3 x^3 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a x+i}{i-a x}\right)+287 i a^3 x^3+61 a^2 x^2+26 i a x-8\right)}{24 x^3 \sqrt[4]{1+i a x}}","\frac{287 i a^3 \sqrt[4]{1-i a x}}{24 \sqrt[4]{1+i a x}}+\frac{55}{8} i a^3 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{55}{8} i a^3 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)+\frac{61 a^2 \sqrt[4]{1-i a x}}{24 x \sqrt[4]{1+i a x}}-\frac{\sqrt[4]{1-i a x}}{3 x^3 \sqrt[4]{1+i a x}}+\frac{13 i a \sqrt[4]{1-i a x}}{12 x^2 \sqrt[4]{1+i a x}}",1,"((1 - I*a*x)^(1/4)*(-8 + (26*I)*a*x + 61*a^2*x^2 + (287*I)*a^3*x^3 - (330*I)*a^3*x^3*Hypergeometric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)]))/(24*x^3*(1 + I*a*x)^(1/4))","C",0
114,1,99,233,0.0365038,"\int \frac{e^{-\frac{5}{2} i \tan ^{-1}(a x)}}{x^5} \, dx","Integrate[1/(E^(((5*I)/2)*ArcTan[a*x])*x^5),x]","\frac{\sqrt[4]{1-i a x} \left(-2850 a^4 x^4 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a x+i}{i-a x}\right)+2467 a^4 x^4-521 i a^3 x^3+226 a^2 x^2+136 i a x-48\right)}{192 x^4 \sqrt[4]{1+i a x}}","\frac{2467 a^4 \sqrt[4]{1-i a x}}{192 \sqrt[4]{1+i a x}}+\frac{475}{64} a^4 \tan ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{475}{64} a^4 \tanh ^{-1}\left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}\right)-\frac{521 i a^3 \sqrt[4]{1-i a x}}{192 x \sqrt[4]{1+i a x}}+\frac{113 a^2 \sqrt[4]{1-i a x}}{96 x^2 \sqrt[4]{1+i a x}}-\frac{\sqrt[4]{1-i a x}}{4 x^4 \sqrt[4]{1+i a x}}+\frac{17 i a \sqrt[4]{1-i a x}}{24 x^3 \sqrt[4]{1+i a x}}",1,"((1 - I*a*x)^(1/4)*(-48 + (136*I)*a*x + 226*a^2*x^2 - (521*I)*a^3*x^3 + 2467*a^4*x^4 - 2850*a^4*x^4*Hypergeometric2F1[1/4, 1, 5/4, (I + a*x)/(I - a*x)]))/(192*x^4*(1 + I*a*x)^(1/4))","C",0
115,1,73,319,0.0360994,"\int e^{\frac{1}{3} i \tan ^{-1}(x)} x^2 \, dx","Integrate[E^((I/3)*ArcTan[x])*x^2,x]","\frac{1}{90} (1-i x)^{5/6} \left(5 \sqrt[6]{1+i x} \left(6 i x^2+7 x-i\right)-38 i \sqrt[6]{2} \, _2F_1\left(-\frac{1}{6},\frac{5}{6};\frac{11}{6};\frac{1}{2}-\frac{i x}{2}\right)\right)","\frac{1}{3} (1-i x)^{5/6} x (1+i x)^{7/6}-\frac{1}{18} i (1-i x)^{5/6} (1+i x)^{7/6}-\frac{19}{54} i (1-i x)^{5/6} \sqrt[6]{1+i x}-\frac{19 i \log \left(\frac{\sqrt[3]{1-i x}}{\sqrt[3]{1+i x}}-\frac{\sqrt{3} \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}+1\right)}{108 \sqrt{3}}+\frac{19 i \log \left(\frac{\sqrt[3]{1-i x}}{\sqrt[3]{1+i x}}+\frac{\sqrt{3} \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}+1\right)}{108 \sqrt{3}}+\frac{19}{162} i \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}\right)-\frac{19}{162} i \tan ^{-1}\left(\sqrt{3}+\frac{2 \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}\right)-\frac{19}{81} i \tan ^{-1}\left(\frac{\sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}\right)",1,"((1 - I*x)^(5/6)*(5*(1 + I*x)^(1/6)*(-I + 7*x + (6*I)*x^2) - (38*I)*2^(1/6)*Hypergeometric2F1[-1/6, 5/6, 11/6, 1/2 - (I/2)*x]))/90","C",0
116,1,57,278,0.0238242,"\int e^{\frac{1}{3} i \tan ^{-1}(x)} x \, dx","Integrate[E^((I/3)*ArcTan[x])*x,x]","\frac{1}{10} (1-i x)^{5/6} \left(2 \sqrt[6]{2} \, _2F_1\left(-\frac{1}{6},\frac{5}{6};\frac{11}{6};\frac{1}{2}-\frac{i x}{2}\right)+5 (1+i x)^{7/6}\right)","\frac{1}{2} (1-i x)^{5/6} (1+i x)^{7/6}+\frac{1}{6} (1-i x)^{5/6} \sqrt[6]{1+i x}+\frac{\log \left(\frac{\sqrt[3]{1-i x}}{\sqrt[3]{1+i x}}-\frac{\sqrt{3} \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}+1\right)}{12 \sqrt{3}}-\frac{\log \left(\frac{\sqrt[3]{1-i x}}{\sqrt[3]{1+i x}}+\frac{\sqrt{3} \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}+1\right)}{12 \sqrt{3}}-\frac{1}{18} \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}\right)+\frac{1}{18} \tan ^{-1}\left(\sqrt{3}+\frac{2 \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}\right)+\frac{1}{9} \tan ^{-1}\left(\frac{\sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}\right)",1,"((1 - I*x)^(5/6)*(5*(1 + I*x)^(7/6) + 2*2^(1/6)*Hypergeometric2F1[-1/6, 5/6, 11/6, 1/2 - (I/2)*x]))/10","C",0
117,1,34,262,0.0179018,"\int e^{\frac{1}{3} i \tan ^{-1}(x)} \, dx","Integrate[E^((I/3)*ArcTan[x]),x]","-\frac{12}{7} i e^{\frac{7}{3} i \tan ^{-1}(x)} \, _2F_1\left(\frac{7}{6},2;\frac{13}{6};-e^{2 i \tan ^{-1}(x)}\right)","i (1-i x)^{5/6} \sqrt[6]{1+i x}+\frac{i \log \left(\frac{\sqrt[3]{1-i x}}{\sqrt[3]{1+i x}}-\frac{\sqrt{3} \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}+1\right)}{2 \sqrt{3}}-\frac{i \log \left(\frac{\sqrt[3]{1-i x}}{\sqrt[3]{1+i x}}+\frac{\sqrt{3} \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}+1\right)}{2 \sqrt{3}}-\frac{1}{3} i \tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}\right)+\frac{1}{3} i \tan ^{-1}\left(\sqrt{3}+\frac{2 \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}\right)+\frac{2}{3} i \tan ^{-1}\left(\frac{\sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}\right)",1,"((-12*I)/7)*E^(((7*I)/3)*ArcTan[x])*Hypergeometric2F1[7/6, 2, 13/6, -E^((2*I)*ArcTan[x])]","C",0
118,1,90,430,0.0309675,"\int \frac{e^{\frac{1}{3} i \tan ^{-1}(x)}}{x} \, dx","Integrate[E^((I/3)*ArcTan[x])/x,x]","-\frac{3 (1-i x)^{5/6} \left(\sqrt[6]{2} (1+i x)^{5/6} \, _2F_1\left(\frac{5}{6},\frac{5}{6};\frac{11}{6};\frac{1}{2}-\frac{i x}{2}\right)+2 \, _2F_1\left(\frac{5}{6},1;\frac{11}{6};\frac{x+i}{i-x}\right)\right)}{5 (1+i x)^{5/6}}","-\frac{1}{2} \sqrt{3} \log \left(\frac{\sqrt[3]{1-i x}}{\sqrt[3]{1+i x}}-\frac{\sqrt{3} \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}+1\right)+\frac{1}{2} \sqrt{3} \log \left(\frac{\sqrt[3]{1-i x}}{\sqrt[3]{1+i x}}+\frac{\sqrt{3} \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}+1\right)+\frac{1}{2} \log \left(\frac{\sqrt[3]{1+i x}}{\sqrt[3]{1-i x}}-\frac{\sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}+1\right)-\frac{1}{2} \log \left(\frac{\sqrt[3]{1+i x}}{\sqrt[3]{1-i x}}+\frac{\sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}+1\right)+\tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}\right)-\tan ^{-1}\left(\sqrt{3}+\frac{2 \sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}\right)+\sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}}{\sqrt{3}}\right)-\sqrt{3} \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}}{\sqrt{3}}\right)-2 \tan ^{-1}\left(\frac{\sqrt[6]{1-i x}}{\sqrt[6]{1+i x}}\right)-2 \tanh ^{-1}\left(\frac{\sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}\right)",1,"(-3*(1 - I*x)^(5/6)*(2^(1/6)*(1 + I*x)^(5/6)*Hypergeometric2F1[5/6, 5/6, 11/6, 1/2 - (I/2)*x] + 2*Hypergeometric2F1[5/6, 1, 11/6, (I + x)/(I - x)]))/(5*(1 + I*x)^(5/6))","C",0
119,1,64,253,0.0159701,"\int \frac{e^{\frac{1}{3} i \tan ^{-1}(x)}}{x^2} \, dx","Integrate[E^((I/3)*ArcTan[x])/x^2,x]","-\frac{i (1-i x)^{5/6} \left(2 x \, _2F_1\left(\frac{5}{6},1;\frac{11}{6};\frac{x+i}{i-x}\right)+5 x-5 i\right)}{5 (1+i x)^{5/6} x}","-\frac{(1-i x)^{5/6} \sqrt[6]{1+i x}}{x}+\frac{1}{6} i \log \left(\frac{\sqrt[3]{1+i x}}{\sqrt[3]{1-i x}}-\frac{\sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}+1\right)-\frac{1}{6} i \log \left(\frac{\sqrt[3]{1+i x}}{\sqrt[3]{1-i x}}+\frac{\sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}+1\right)+\frac{i \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{i \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}}{\sqrt{3}}\right)}{\sqrt{3}}-\frac{2}{3} i \tanh ^{-1}\left(\frac{\sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}\right)",1,"((-1/5*I)*(1 - I*x)^(5/6)*(-5*I + 5*x + 2*x*Hypergeometric2F1[5/6, 1, 11/6, (I + x)/(I - x)]))/((1 + I*x)^(5/6)*x)","C",0
120,1,72,280,0.0175788,"\int \frac{e^{\frac{1}{3} i \tan ^{-1}(x)}}{x^3} \, dx","Integrate[E^((I/3)*ArcTan[x])/x^3,x]","\frac{(1-i x)^{5/6} \left(2 x^2 \, _2F_1\left(\frac{5}{6},1;\frac{11}{6};\frac{x+i}{i-x}\right)+5 \left(4 x^2-7 i x-3\right)\right)}{30 (1+i x)^{5/6} x^2}","-\frac{(1-i x)^{5/6} (1+i x)^{7/6}}{2 x^2}-\frac{i (1-i x)^{5/6} \sqrt[6]{1+i x}}{6 x}-\frac{1}{36} \log \left(\frac{\sqrt[3]{1+i x}}{\sqrt[3]{1-i x}}-\frac{\sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}+1\right)+\frac{1}{36} \log \left(\frac{\sqrt[3]{1+i x}}{\sqrt[3]{1-i x}}+\frac{\sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}+1\right)-\frac{\tan ^{-1}\left(\frac{1-\frac{2 \sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}}{\sqrt{3}}\right)}{6 \sqrt{3}}+\frac{\tan ^{-1}\left(\frac{1+\frac{2 \sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}}{\sqrt{3}}\right)}{6 \sqrt{3}}+\frac{1}{9} \tanh ^{-1}\left(\frac{\sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}\right)",1,"((1 - I*x)^(5/6)*(5*(-3 - (7*I)*x + 4*x^2) + 2*x^2*Hypergeometric2F1[5/6, 1, 11/6, (I + x)/(I - x)]))/(30*(1 + I*x)^(5/6)*x^2)","C",0
121,1,81,319,0.0207491,"\int \frac{e^{\frac{1}{3} i \tan ^{-1}(x)}}{x^4} \, dx","Integrate[E^((I/3)*ArcTan[x])/x^4,x]","\frac{(1-i x)^{5/6} \left(38 i x^3 \, _2F_1\left(\frac{5}{6},1;\frac{11}{6};\frac{x+i}{i-x}\right)+5 \left(22 i x^3+43 x^2-39 i x-18\right)\right)}{270 (1+i x)^{5/6} x^3}","-\frac{(1-i x)^{5/6} \sqrt[6]{1+i x}}{3 x^3}-\frac{7 i (1-i x)^{5/6} \sqrt[6]{1+i x}}{18 x^2}+\frac{11 (1-i x)^{5/6} \sqrt[6]{1+i x}}{27 x}-\frac{19}{324} i \log \left(\frac{\sqrt[3]{1+i x}}{\sqrt[3]{1-i x}}-\frac{\sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}+1\right)+\frac{19}{324} i \log \left(\frac{\sqrt[3]{1+i x}}{\sqrt[3]{1-i x}}+\frac{\sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}+1\right)-\frac{19 i \tan ^{-1}\left(\frac{1-\frac{2 \sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}}{\sqrt{3}}\right)}{54 \sqrt{3}}+\frac{19 i \tan ^{-1}\left(\frac{1+\frac{2 \sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}}{\sqrt{3}}\right)}{54 \sqrt{3}}+\frac{19}{81} i \tanh ^{-1}\left(\frac{\sqrt[6]{1+i x}}{\sqrt[6]{1-i x}}\right)",1,"((1 - I*x)^(5/6)*(5*(-18 - (39*I)*x + 43*x^2 + (22*I)*x^3) + (38*I)*x^3*Hypergeometric2F1[5/6, 1, 11/6, (I + x)/(I - x)]))/(270*(1 + I*x)^(5/6)*x^3)","C",0
122,1,73,177,0.0356146,"\int e^{\frac{2}{3} i \tan ^{-1}(x)} x^2 \, dx","Integrate[E^(((2*I)/3)*ArcTan[x])*x^2,x]","\frac{1}{18} (1-i x)^{2/3} \left(2 \sqrt[3]{1+i x} \left(3 i x^2+4 x-i\right)-11 i \sqrt[3]{2} \, _2F_1\left(-\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{1}{2}-\frac{i x}{2}\right)\right)","\frac{1}{3} (1-i x)^{2/3} x (1+i x)^{4/3}-\frac{1}{9} i (1-i x)^{2/3} (1+i x)^{4/3}-\frac{11}{27} i (1-i x)^{2/3} \sqrt[3]{1+i x}+\frac{11}{27} i \log \left(1+\frac{\sqrt[3]{1-i x}}{\sqrt[3]{1+i x}}\right)+\frac{11}{81} i \log (1+i x)+\frac{22 i \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{1-i x}}{\sqrt{3} \sqrt[3]{1+i x}}\right)}{27 \sqrt{3}}",1,"((1 - I*x)^(2/3)*(2*(1 + I*x)^(1/3)*(-I + 4*x + (3*I)*x^2) - (11*I)*2^(1/3)*Hypergeometric2F1[-1/3, 2/3, 5/3, 1/2 - (I/2)*x]))/18","C",0
123,1,54,140,0.0218901,"\int e^{\frac{2}{3} i \tan ^{-1}(x)} x \, dx","Integrate[E^(((2*I)/3)*ArcTan[x])*x,x]","\frac{1}{2} (1-i x)^{2/3} \left(\sqrt[3]{2} \, _2F_1\left(-\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{1}{2}-\frac{i x}{2}\right)+(1+i x)^{4/3}\right)","\frac{1}{2} (1-i x)^{2/3} (1+i x)^{4/3}+\frac{1}{3} (1-i x)^{2/3} \sqrt[3]{1+i x}-\frac{1}{3} \log \left(1+\frac{\sqrt[3]{1-i x}}{\sqrt[3]{1+i x}}\right)-\frac{1}{9} \log (1+i x)-\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{1-i x}}{\sqrt{3} \sqrt[3]{1+i x}}\right)}{3 \sqrt{3}}",1,"((1 - I*x)^(2/3)*((1 + I*x)^(4/3) + 2^(1/3)*Hypergeometric2F1[-1/3, 2/3, 5/3, 1/2 - (I/2)*x]))/2","C",0
124,1,34,116,0.0188485,"\int e^{\frac{2}{3} i \tan ^{-1}(x)} \, dx","Integrate[E^(((2*I)/3)*ArcTan[x]),x]","-\frac{3}{2} i e^{\frac{8}{3} i \tan ^{-1}(x)} \, _2F_1\left(\frac{4}{3},2;\frac{7}{3};-e^{2 i \tan ^{-1}(x)}\right)","i (1-i x)^{2/3} \sqrt[3]{1+i x}-i \log \left(1+\frac{\sqrt[3]{1-i x}}{\sqrt[3]{1+i x}}\right)-\frac{1}{3} i \log (1+i x)-\frac{2 i \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{1-i x}}{\sqrt{3} \sqrt[3]{1+i x}}\right)}{\sqrt{3}}",1,"((-3*I)/2)*E^(((8*I)/3)*ArcTan[x])*Hypergeometric2F1[4/3, 2, 7/3, -E^((2*I)*ArcTan[x])]","C",0
125,1,90,163,0.0274586,"\int \frac{e^{\frac{2}{3} i \tan ^{-1}(x)}}{x} \, dx","Integrate[E^(((2*I)/3)*ArcTan[x])/x,x]","-\frac{3 (1-i x)^{2/3} \left(\sqrt[3]{2} (1+i x)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{2}-\frac{i x}{2}\right)+2 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{x+i}{i-x}\right)\right)}{4 (1+i x)^{2/3}}","\frac{3}{2} \log \left(1+\frac{\sqrt[3]{1-i x}}{\sqrt[3]{1+i x}}\right)+\frac{3}{2} \log \left(\sqrt[3]{1-i x}-\sqrt[3]{1+i x}\right)+\frac{1}{2} \log (1+i x)-\frac{\log (x)}{2}+\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2 \sqrt[3]{1-i x}}{\sqrt{3} \sqrt[3]{1+i x}}\right)+\sqrt{3} \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1-i x}}{\sqrt{3} \sqrt[3]{1+i x}}\right)",1,"(-3*(1 - I*x)^(2/3)*(2^(1/3)*(1 + I*x)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, 1/2 - (I/2)*x] + 2*Hypergeometric2F1[2/3, 1, 5/3, (I + x)/(I - x)]))/(4*(1 + I*x)^(2/3))","C",0
126,1,59,111,0.013233,"\int \frac{e^{\frac{2}{3} i \tan ^{-1}(x)}}{x^2} \, dx","Integrate[E^(((2*I)/3)*ArcTan[x])/x^2,x]","-\frac{i (1-i x)^{2/3} \left(x \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{x+i}{i-x}\right)+x-i\right)}{(1+i x)^{2/3} x}","-\frac{(1-i x)^{2/3} \sqrt[3]{1+i x}}{x}+i \log \left(\sqrt[3]{1-i x}-\sqrt[3]{1+i x}\right)-\frac{1}{3} i \log (x)+\frac{2 i \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1-i x}}{\sqrt{3} \sqrt[3]{1+i x}}\right)}{\sqrt{3}}",1,"((-I)*(1 - I*x)^(2/3)*(-I + x + x*Hypergeometric2F1[2/3, 1, 5/3, (I + x)/(I - x)]))/((1 + I*x)^(2/3)*x)","C",0
127,1,69,142,0.016441,"\int \frac{e^{\frac{2}{3} i \tan ^{-1}(x)}}{x^3} \, dx","Integrate[E^(((2*I)/3)*ArcTan[x])/x^3,x]","\frac{(1-i x)^{2/3} \left(2 x^2 \, _2F_1\left(\frac{2}{3},1;\frac{5}{3};\frac{x+i}{i-x}\right)+5 x^2-8 i x-3\right)}{6 (1+i x)^{2/3} x^2}","-\frac{(1-i x)^{2/3} (1+i x)^{4/3}}{2 x^2}-\frac{i (1-i x)^{2/3} \sqrt[3]{1+i x}}{3 x}-\frac{1}{3} \log \left(\sqrt[3]{1-i x}-\sqrt[3]{1+i x}\right)+\frac{\log (x)}{9}-\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{3}}+\frac{2 \sqrt[3]{1-i x}}{\sqrt{3} \sqrt[3]{1+i x}}\right)}{3 \sqrt{3}}",1,"((1 - I*x)^(2/3)*(-3 - (8*I)*x + 5*x^2 + 2*x^2*Hypergeometric2F1[2/3, 1, 5/3, (I + x)/(I - x)]))/(6*(1 + I*x)^(2/3)*x^2)","C",0
128,1,83,741,0.0408256,"\int e^{\frac{1}{4} i \tan ^{-1}(a x)} x^2 \, dx","Integrate[E^((I/4)*ArcTan[a*x])*x^2,x]","\frac{(1-i a x)^{7/8} \left(7 \sqrt[8]{1+i a x} \left(8 i a^2 x^2+9 a x-i\right)-66 i \sqrt[8]{2} \, _2F_1\left(-\frac{1}{8},\frac{7}{8};\frac{15}{8};\frac{1}{2} (1-i a x)\right)\right)}{168 a^3}","-\frac{i (1-i a x)^{7/8} (1+i a x)^{9/8}}{24 a^3}-\frac{11 i (1-i a x)^{7/8} \sqrt[8]{1+i a x}}{32 a^3}-\frac{11 i \sqrt{2-\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}-\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}+1\right)}{256 a^3}+\frac{11 i \sqrt{2-\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}+1\right)}{256 a^3}-\frac{11 i \sqrt{2+\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}-\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}+1\right)}{256 a^3}+\frac{11 i \sqrt{2+\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}+1\right)}{256 a^3}+\frac{11 i \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}}{\sqrt{2+\sqrt{2}}}\right)}{128 a^3}+\frac{11 i \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}}{\sqrt{2-\sqrt{2}}}\right)}{128 a^3}-\frac{11 i \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}+\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}}{\sqrt{2+\sqrt{2}}}\right)}{128 a^3}-\frac{11 i \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}+\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}}{\sqrt{2-\sqrt{2}}}\right)}{128 a^3}+\frac{x (1-i a x)^{7/8} (1+i a x)^{9/8}}{3 a^2}",1,"((1 - I*a*x)^(7/8)*(7*(1 + I*a*x)^(1/8)*(-I + 9*a*x + (8*I)*a^2*x^2) - (66*I)*2^(1/8)*Hypergeometric2F1[-1/8, 7/8, 15/8, (1 - I*a*x)/2]))/(168*a^3)","C",0
129,1,63,689,0.025338,"\int e^{\frac{1}{4} i \tan ^{-1}(a x)} x \, dx","Integrate[E^((I/4)*ArcTan[a*x])*x,x]","\frac{(1-i a x)^{7/8} \left(2 \sqrt[8]{2} \, _2F_1\left(-\frac{1}{8},\frac{7}{8};\frac{15}{8};\frac{1}{2} (1-i a x)\right)+7 (1+i a x)^{9/8}\right)}{14 a^2}","\frac{(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac{(1-i a x)^{7/8} \sqrt[8]{1+i a x}}{8 a^2}+\frac{\sqrt{2-\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}-\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}+1\right)}{64 a^2}-\frac{\sqrt{2-\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}+1\right)}{64 a^2}+\frac{\sqrt{2+\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}-\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}+1\right)}{64 a^2}-\frac{\sqrt{2+\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}+1\right)}{64 a^2}-\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}}{\sqrt{2+\sqrt{2}}}\right)}{32 a^2}-\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}}{\sqrt{2-\sqrt{2}}}\right)}{32 a^2}+\frac{\sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}+\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}}{\sqrt{2+\sqrt{2}}}\right)}{32 a^2}+\frac{\sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}+\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}}{\sqrt{2-\sqrt{2}}}\right)}{32 a^2}",1,"((1 - I*a*x)^(7/8)*(7*(1 + I*a*x)^(9/8) + 2*2^(1/8)*Hypergeometric2F1[-1/8, 7/8, 15/8, (1 - I*a*x)/2]))/(14*a^2)","C",0
130,1,41,674,0.0248265,"\int e^{\frac{1}{4} i \tan ^{-1}(a x)} \, dx","Integrate[E^((I/4)*ArcTan[a*x]),x]","-\frac{16 i e^{\frac{9}{4} i \tan ^{-1}(a x)} \, _2F_1\left(\frac{9}{8},2;\frac{17}{8};-e^{2 i \tan ^{-1}(a x)}\right)}{9 a}","\frac{i (1-i a x)^{7/8} \sqrt[8]{1+i a x}}{a}+\frac{i \sqrt{2-\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}-\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}+1\right)}{8 a}-\frac{i \sqrt{2-\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}+1\right)}{8 a}+\frac{i \sqrt{2+\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}-\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}+1\right)}{8 a}-\frac{i \sqrt{2+\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{1+i a x}}+\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}+1\right)}{8 a}-\frac{i \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}}{\sqrt{2+\sqrt{2}}}\right)}{4 a}-\frac{i \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}}{\sqrt{2-\sqrt{2}}}\right)}{4 a}+\frac{i \sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}+\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}}{\sqrt{2+\sqrt{2}}}\right)}{4 a}+\frac{i \sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}+\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{1+i a x}}}{\sqrt{2-\sqrt{2}}}\right)}{4 a}",1,"(((-16*I)/9)*E^(((9*I)/4)*ArcTan[a*x])*Hypergeometric2F1[9/8, 2, 17/8, -E^((2*I)*ArcTan[a*x])])/a","C",0
131,1,97,859,0.0361696,"\int \frac{e^{\frac{1}{4} i \tan ^{-1}(a x)}}{x} \, dx","Integrate[E^((I/4)*ArcTan[a*x])/x,x]","-\frac{4 (1-i a x)^{7/8} \left(\sqrt[8]{2} (1+i a x)^{7/8} \, _2F_1\left(\frac{7}{8},\frac{7}{8};\frac{15}{8};\frac{1}{2} (1-i a x)\right)+2 \, _2F_1\left(\frac{7}{8},1;\frac{15}{8};\frac{a x+i}{i-a x}\right)\right)}{7 (1+i a x)^{7/8}}","-2 \tan ^{-1}\left(\frac{\sqrt[8]{i a x+1}}{\sqrt[8]{1-i a x}}\right)+\sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{i a x+1}}}{\sqrt{2+\sqrt{2}}}\right)+\sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{i a x+1}}}{\sqrt{2-\sqrt{2}}}\right)-\sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{i a x+1}}+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right)-\sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\frac{2 \sqrt[8]{1-i a x}}{\sqrt[8]{i a x+1}}+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right)+\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[8]{i a x+1}}{\sqrt[8]{1-i a x}}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[8]{i a x+1}}{\sqrt[8]{1-i a x}}+1\right)-2 \tanh ^{-1}\left(\frac{\sqrt[8]{i a x+1}}{\sqrt[8]{1-i a x}}\right)-\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{i a x+1}}-\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{i a x+1}}+1\right)+\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{i a x+1}}+\frac{\sqrt{2-\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{i a x+1}}+1\right)-\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{i a x+1}}-\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{i a x+1}}+1\right)+\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left(\frac{\sqrt[4]{1-i a x}}{\sqrt[4]{i a x+1}}+\frac{\sqrt{2+\sqrt{2}} \sqrt[8]{1-i a x}}{\sqrt[8]{i a x+1}}+1\right)+\frac{\log \left(\frac{\sqrt[4]{i a x+1}}{\sqrt[4]{1-i a x}}-\frac{\sqrt{2} \sqrt[8]{i a x+1}}{\sqrt[8]{1-i a x}}+1\right)}{\sqrt{2}}-\frac{\log \left(\frac{\sqrt[4]{i a x+1}}{\sqrt[4]{1-i a x}}+\frac{\sqrt{2} \sqrt[8]{i a x+1}}{\sqrt[8]{1-i a x}}+1\right)}{\sqrt{2}}",1,"(-4*(1 - I*a*x)^(7/8)*(2^(1/8)*(1 + I*a*x)^(7/8)*Hypergeometric2F1[7/8, 7/8, 15/8, (1 - I*a*x)/2] + 2*Hypergeometric2F1[7/8, 1, 15/8, (I + a*x)/(I - a*x)]))/(7*(1 + I*a*x)^(7/8))","C",0
132,1,71,328,0.0173693,"\int \frac{e^{\frac{1}{4} i \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[E^((I/4)*ArcTan[a*x])/x^2,x]","-\frac{i (1-i a x)^{7/8} \left(2 a x \, _2F_1\left(\frac{7}{8},1;\frac{15}{8};\frac{a x+i}{i-a x}\right)+7 a x-7 i\right)}{7 x (1+i a x)^{7/8}}","-\frac{(1-i a x)^{7/8} \sqrt[8]{1+i a x}}{x}+\frac{i a \log \left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}-\frac{\sqrt{2} \sqrt[8]{1+i a x}}{\sqrt[8]{1-i a x}}+1\right)}{4 \sqrt{2}}-\frac{i a \log \left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}+\frac{\sqrt{2} \sqrt[8]{1+i a x}}{\sqrt[8]{1-i a x}}+1\right)}{4 \sqrt{2}}-\frac{1}{2} i a \tan ^{-1}\left(\frac{\sqrt[8]{1+i a x}}{\sqrt[8]{1-i a x}}\right)+\frac{i a \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[8]{1+i a x}}{\sqrt[8]{1-i a x}}\right)}{2 \sqrt{2}}-\frac{i a \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[8]{1+i a x}}{\sqrt[8]{1-i a x}}\right)}{2 \sqrt{2}}-\frac{1}{2} i a \tanh ^{-1}\left(\frac{\sqrt[8]{1+i a x}}{\sqrt[8]{1-i a x}}\right)",1,"((-1/7*I)*(1 - I*a*x)^(7/8)*(-7*I + 7*a*x + 2*a*x*Hypergeometric2F1[7/8, 1, 15/8, (I + a*x)/(I - a*x)]))/(x*(1 + I*a*x)^(7/8))","C",0
133,1,84,364,0.0226518,"\int \frac{e^{\frac{1}{4} i \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[E^((I/4)*ArcTan[a*x])/x^3,x]","\frac{(1-i a x)^{7/8} \left(2 a^2 x^2 \, _2F_1\left(\frac{7}{8},1;\frac{15}{8};\frac{a x+i}{i-a x}\right)+7 \left(5 a^2 x^2-9 i a x-4\right)\right)}{56 x^2 (1+i a x)^{7/8}}","-\frac{a^2 \log \left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}-\frac{\sqrt{2} \sqrt[8]{1+i a x}}{\sqrt[8]{1-i a x}}+1\right)}{32 \sqrt{2}}+\frac{a^2 \log \left(\frac{\sqrt[4]{1+i a x}}{\sqrt[4]{1-i a x}}+\frac{\sqrt{2} \sqrt[8]{1+i a x}}{\sqrt[8]{1-i a x}}+1\right)}{32 \sqrt{2}}+\frac{1}{16} a^2 \tan ^{-1}\left(\frac{\sqrt[8]{1+i a x}}{\sqrt[8]{1-i a x}}\right)-\frac{a^2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[8]{1+i a x}}{\sqrt[8]{1-i a x}}\right)}{16 \sqrt{2}}+\frac{a^2 \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[8]{1+i a x}}{\sqrt[8]{1-i a x}}\right)}{16 \sqrt{2}}+\frac{1}{16} a^2 \tanh ^{-1}\left(\frac{\sqrt[8]{1+i a x}}{\sqrt[8]{1-i a x}}\right)-\frac{(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 x^2}-\frac{i a (1-i a x)^{7/8} \sqrt[8]{1+i a x}}{8 x}",1,"((1 - I*a*x)^(7/8)*(7*(-4 - (9*I)*a*x + 5*a^2*x^2) + 2*a^2*x^2*Hypergeometric2F1[7/8, 1, 15/8, (I + a*x)/(I - a*x)]))/(56*x^2*(1 + I*a*x)^(7/8))","C",0
134,1,94,114,0.0453827,"\int e^{6 i \tan ^{-1}(a x)} x^m \, dx","Integrate[E^((6*I)*ArcTan[a*x])*x^m,x]","\frac{x^{m+1} \left(-a^2 x^2+2 \left(2 m^2+4 m+3\right) (a x+i)^2 \, _2F_1(1,m+1;m+2;i a x)+m^2 (4-4 i a x)+4 m (2-3 i a x)-10 i a x+5\right)}{(m+1) (a x+i)^2}","\frac{2 \left(2 m^2+4 m+3\right) x^{m+1} \, _2F_1(1,m+1;m+2;i a x)}{m+1}+\frac{4 i x^{m+1} \left(a \left(m^2+3 m+3\right) x+i (m+1)^2\right)}{(m+1) (1-i a x)^2}-\frac{(1+i a x)^2 x^{m+1}}{(m+1) (1-i a x)^2}",1,"(x^(1 + m)*(5 - (10*I)*a*x - a^2*x^2 + 4*m*(2 - (3*I)*a*x) + m^2*(4 - (4*I)*a*x) + 2*(3 + 4*m + 2*m^2)*(I + a*x)^2*Hypergeometric2F1[1, 1 + m, 2 + m, I*a*x]))/((1 + m)*(I + a*x)^2)","A",1
135,1,58,50,0.0237557,"\int e^{4 i \tan ^{-1}(a x)} x^m \, dx","Integrate[E^((4*I)*ArcTan[a*x])*x^m,x]","\frac{x^{m+1} (-4 (m+1) (a x+i) \, _2F_1(1,m+1;m+2;i a x)+a x+4 i m+5 i)}{(m+1) (a x+i)}","-4 x^{m+1} \, _2F_1(1,m+1;m+2;i a x)+\frac{4 x^{m+1}}{1-i a x}+\frac{x^{m+1}}{m+1}",1,"(x^(1 + m)*(5*I + (4*I)*m + a*x - 4*(1 + m)*(I + a*x)*Hypergeometric2F1[1, 1 + m, 2 + m, I*a*x]))/((1 + m)*(I + a*x))","A",1
136,1,29,39,0.0071043,"\int e^{2 i \tan ^{-1}(a x)} x^m \, dx","Integrate[E^((2*I)*ArcTan[a*x])*x^m,x]","\frac{x^{m+1} (-1+2 \, _2F_1(1,m+1;m+2;i a x))}{m+1}","-\frac{x^{m+1}}{m+1}+\frac{2 x^{m+1} \, _2F_1(1,m+1;m+2;i a x)}{m+1}",1,"(x^(1 + m)*(-1 + 2*Hypergeometric2F1[1, 1 + m, 2 + m, I*a*x]))/(1 + m)","A",1
137,1,29,39,0.0090297,"\int e^{-2 i \tan ^{-1}(a x)} x^m \, dx","Integrate[x^m/E^((2*I)*ArcTan[a*x]),x]","\frac{x^{m+1} (-1+2 \, _2F_1(1,m+1;m+2;-i a x))}{m+1}","-\frac{x^{m+1}}{m+1}+\frac{2 x^{m+1} \, _2F_1(1,m+1;m+2;-i a x)}{m+1}",1,"(x^(1 + m)*(-1 + 2*Hypergeometric2F1[1, 1 + m, 2 + m, (-I)*a*x]))/(1 + m)","A",1
138,1,58,50,0.022098,"\int e^{-4 i \tan ^{-1}(a x)} x^m \, dx","Integrate[x^m/E^((4*I)*ArcTan[a*x]),x]","\frac{x^{m+1} (-4 (m+1) (a x-i) \, _2F_1(1,m+1;m+2;-i a x)+a x-4 i m-5 i)}{(m+1) (a x-i)}","-4 x^{m+1} \, _2F_1(1,m+1;m+2;-i a x)+\frac{4 x^{m+1}}{1+i a x}+\frac{x^{m+1}}{m+1}",1,"(x^(1 + m)*(-5*I - (4*I)*m + a*x - 4*(1 + m)*(-I + a*x)*Hypergeometric2F1[1, 1 + m, 2 + m, (-I)*a*x]))/((1 + m)*(-I + a*x))","A",1
139,1,94,115,0.0426046,"\int e^{-6 i \tan ^{-1}(a x)} x^m \, dx","Integrate[x^m/E^((6*I)*ArcTan[a*x]),x]","\frac{x^{m+1} \left(-a^2 x^2+2 \left(2 m^2+4 m+3\right) (a x-i)^2 \, _2F_1(1,m+1;m+2;-i a x)+m^2 (4+4 i a x)+4 m (2+3 i a x)+10 i a x+5\right)}{(m+1) (a x-i)^2}","\frac{2 \left(2 m^2+4 m+3\right) x^{m+1} \, _2F_1(1,m+1;m+2;-i a x)}{m+1}+\frac{4 i x^{m+1} \left(-a \left(m^2+3 m+3\right) x+i (m+1)^2\right)}{(m+1) (1+i a x)^2}-\frac{(1-i a x)^2 x^{m+1}}{(m+1) (1+i a x)^2}",1,"(x^(1 + m)*(5 + (10*I)*a*x - a^2*x^2 + 4*m*(2 + (3*I)*a*x) + m^2*(4 + (4*I)*a*x) + 2*(3 + 4*m + 2*m^2)*(-I + a*x)^2*Hypergeometric2F1[1, 1 + m, 2 + m, (-I)*a*x]))/((1 + m)*(-I + a*x)^2)","A",1
140,1,113,159,0.0847726,"\int e^{3 i \tan ^{-1}(a x)} x^m \, dx","Integrate[E^((3*I)*ArcTan[a*x])*x^m,x]","-\frac{i \sqrt{1-i a x} \sqrt{a x-i} x^{m+1} \left(F_1\left(m+1;-\frac{1}{2},\frac{1}{2};m+2;-i a x,i a x\right)-2 F_1\left(m+1;-\frac{1}{2},\frac{3}{2};m+2;-i a x,i a x\right)\right)}{(m+1) \sqrt{1+i a x} \sqrt{a x+i}}","-\frac{3 x^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-a^2 x^2\right)}{m+1}+\frac{4 x^{m+1} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};-a^2 x^2\right)}{m+1}-\frac{i a x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m+2}+\frac{4 i a x^{m+2} \, _2F_1\left(\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m+2}",1,"((-I)*x^(1 + m)*Sqrt[1 - I*a*x]*Sqrt[-I + a*x]*(AppellF1[1 + m, -1/2, 1/2, 2 + m, (-I)*a*x, I*a*x] - 2*AppellF1[1 + m, -1/2, 3/2, 2 + m, (-I)*a*x, I*a*x]))/((1 + m)*Sqrt[1 + I*a*x]*Sqrt[I + a*x])","C",0
141,1,85,79,0.0366724,"\int e^{i \tan ^{-1}(a x)} x^m \, dx","Integrate[E^(I*ArcTan[a*x])*x^m,x]","\frac{i \sqrt{1-i a x} \sqrt{a x-i} x^{m+1} F_1\left(m+1;-\frac{1}{2},\frac{1}{2};m+2;-i a x,i a x\right)}{(m+1) \sqrt{1+i a x} \sqrt{a x+i}}","\frac{x^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-a^2 x^2\right)}{m+1}+\frac{i a x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m+2}",1,"(I*x^(1 + m)*Sqrt[1 - I*a*x]*Sqrt[-I + a*x]*AppellF1[1 + m, -1/2, 1/2, 2 + m, (-I)*a*x, I*a*x])/((1 + m)*Sqrt[1 + I*a*x]*Sqrt[I + a*x])","C",0
142,1,85,79,0.0423178,"\int e^{-i \tan ^{-1}(a x)} x^m \, dx","Integrate[x^m/E^(I*ArcTan[a*x]),x]","-\frac{i \sqrt{1+i a x} \sqrt{a x+i} x^{m+1} F_1\left(m+1;\frac{1}{2},-\frac{1}{2};m+2;-i a x,i a x\right)}{(m+1) \sqrt{1-i a x} \sqrt{a x-i}}","\frac{x^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-a^2 x^2\right)}{m+1}-\frac{i a x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m+2}",1,"((-I)*x^(1 + m)*Sqrt[1 + I*a*x]*Sqrt[I + a*x]*AppellF1[1 + m, 1/2, -1/2, 2 + m, (-I)*a*x, I*a*x])/((1 + m)*Sqrt[1 - I*a*x]*Sqrt[-I + a*x])","C",0
143,1,113,159,0.0832006,"\int e^{-3 i \tan ^{-1}(a x)} x^m \, dx","Integrate[x^m/E^((3*I)*ArcTan[a*x]),x]","\frac{i \sqrt{1+i a x} \sqrt{a x+i} x^{m+1} \left(F_1\left(m+1;\frac{1}{2},-\frac{1}{2};m+2;-i a x,i a x\right)-2 F_1\left(m+1;\frac{3}{2},-\frac{1}{2};m+2;-i a x,i a x\right)\right)}{(m+1) \sqrt{1-i a x} \sqrt{a x-i}}","-\frac{3 x^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-a^2 x^2\right)}{m+1}+\frac{4 x^{m+1} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};-a^2 x^2\right)}{m+1}+\frac{i a x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m+2}-\frac{4 i a x^{m+2} \, _2F_1\left(\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m+2}",1,"(I*x^(1 + m)*Sqrt[1 + I*a*x]*Sqrt[I + a*x]*(AppellF1[1 + m, 1/2, -1/2, 2 + m, (-I)*a*x, I*a*x] - 2*AppellF1[1 + m, 3/2, -1/2, 2 + m, (-I)*a*x, I*a*x]))/((1 + m)*Sqrt[1 - I*a*x]*Sqrt[-I + a*x])","C",0
144,0,0,36,0.3086566,"\int e^{\frac{5}{2} i \tan ^{-1}(a x)} x^m \, dx","Integrate[E^(((5*I)/2)*ArcTan[a*x])*x^m,x]","\int e^{\frac{5}{2} i \tan ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(m+1;\frac{5}{4},-\frac{5}{4};m+2;i a x,-i a x\right)}{m+1}",1,"Integrate[E^(((5*I)/2)*ArcTan[a*x])*x^m, x]","F",-1
145,0,0,36,0.2645419,"\int e^{\frac{3}{2} i \tan ^{-1}(a x)} x^m \, dx","Integrate[E^(((3*I)/2)*ArcTan[a*x])*x^m,x]","\int e^{\frac{3}{2} i \tan ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(m+1;\frac{3}{4},-\frac{3}{4};m+2;i a x,-i a x\right)}{m+1}",1,"Integrate[E^(((3*I)/2)*ArcTan[a*x])*x^m, x]","F",-1
146,0,0,36,0.2444247,"\int e^{\frac{1}{2} i \tan ^{-1}(a x)} x^m \, dx","Integrate[E^((I/2)*ArcTan[a*x])*x^m,x]","\int e^{\frac{1}{2} i \tan ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(m+1;\frac{1}{4},-\frac{1}{4};m+2;i a x,-i a x\right)}{m+1}",1,"Integrate[E^((I/2)*ArcTan[a*x])*x^m, x]","F",-1
147,0,0,36,0.2683693,"\int e^{-\frac{1}{2} i \tan ^{-1}(a x)} x^m \, dx","Integrate[x^m/E^((I/2)*ArcTan[a*x]),x]","\int e^{-\frac{1}{2} i \tan ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(m+1;-\frac{1}{4},\frac{1}{4};m+2;i a x,-i a x\right)}{m+1}",1,"Integrate[x^m/E^((I/2)*ArcTan[a*x]), x]","F",-1
148,0,0,36,0.2784454,"\int e^{-\frac{3}{2} i \tan ^{-1}(a x)} x^m \, dx","Integrate[x^m/E^(((3*I)/2)*ArcTan[a*x]),x]","\int e^{-\frac{3}{2} i \tan ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(m+1;-\frac{3}{4},\frac{3}{4};m+2;i a x,-i a x\right)}{m+1}",1,"Integrate[x^m/E^(((3*I)/2)*ArcTan[a*x]), x]","F",-1
149,0,0,36,0.3391639,"\int e^{-\frac{5}{2} i \tan ^{-1}(a x)} x^m \, dx","Integrate[x^m/E^(((5*I)/2)*ArcTan[a*x]),x]","\int e^{-\frac{5}{2} i \tan ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(m+1;-\frac{5}{4},\frac{5}{4};m+2;i a x,-i a x\right)}{m+1}",1,"Integrate[x^m/E^(((5*I)/2)*ArcTan[a*x]), x]","F",-1
150,0,0,38,0.2158846,"\int e^{\frac{2}{3} \tan ^{-1}(x)} x^m \, dx","Integrate[E^((2*ArcTan[x])/3)*x^m,x]","\int e^{\frac{2}{3} \tan ^{-1}(x)} x^m \, dx","\frac{x^{m+1} F_1\left(m+1;-\frac{i}{3},\frac{i}{3};m+2;i x,-i x\right)}{m+1}",1,"Integrate[E^((2*ArcTan[x])/3)*x^m, x]","F",-1
151,0,0,38,0.2089887,"\int e^{\frac{1}{3} \tan ^{-1}(x)} x^m \, dx","Integrate[E^(ArcTan[x]/3)*x^m,x]","\int e^{\frac{1}{3} \tan ^{-1}(x)} x^m \, dx","\frac{x^{m+1} F_1\left(m+1;-\frac{i}{6},\frac{i}{6};m+2;i x,-i x\right)}{m+1}",1,"Integrate[E^(ArcTan[x]/3)*x^m, x]","F",-1
152,0,0,36,0.2445633,"\int e^{\frac{1}{4} i \tan ^{-1}(a x)} x^m \, dx","Integrate[E^((I/4)*ArcTan[a*x])*x^m,x]","\int e^{\frac{1}{4} i \tan ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(m+1;\frac{1}{8},-\frac{1}{8};m+2;i a x,-i a x\right)}{m+1}",1,"Integrate[E^((I/4)*ArcTan[a*x])*x^m, x]","F",-1
153,0,0,40,0.2493187,"\int e^{i n \tan ^{-1}(a x)} x^m \, dx","Integrate[E^(I*n*ArcTan[a*x])*x^m,x]","\int e^{i n \tan ^{-1}(a x)} x^m \, dx","\frac{x^{m+1} F_1\left(m+1;\frac{n}{2},-\frac{n}{2};m+2;i a x,-i a x\right)}{m+1}",1,"Integrate[E^(I*n*ArcTan[a*x])*x^m, x]","F",-1
154,1,210,171,0.198137,"\int e^{i n \tan ^{-1}(a x)} x^3 \, dx","Integrate[E^(I*n*ArcTan[a*x])*x^3,x]","\frac{(a x+i) (1-i a x)^{-n/2} \left((n-2) \left(a^2 x^2 (a x-i) (1+i a x)^{n/2}-i 2^{\frac{n}{2}+1} \, _2F_1\left(1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right)\right)-i 2^{\frac{n}{2}+3} n \, _2F_1\left(-\frac{n}{2}-2,1-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right)+i 2^{\frac{n}{2}+3} (n-1) \, _2F_1\left(-\frac{n}{2}-1,1-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right)\right)}{4 a^4 (n-2)}","-\frac{2^{\frac{n}{2}-2} n \left(n^2+8\right) (1-i a x)^{1-\frac{n}{2}} \, _2F_1\left(1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right)}{3 a^4 (2-n)}-\frac{(1+i a x)^{\frac{n+2}{2}} \left(2 i a n x+n^2+6\right) (1-i a x)^{1-\frac{n}{2}}}{24 a^4}+\frac{x^2 (1+i a x)^{\frac{n+2}{2}} (1-i a x)^{1-\frac{n}{2}}}{4 a^2}",1,"((I + a*x)*((-I)*2^(3 + n/2)*n*Hypergeometric2F1[-2 - n/2, 1 - n/2, 2 - n/2, (1 - I*a*x)/2] + I*2^(3 + n/2)*(-1 + n)*Hypergeometric2F1[-1 - n/2, 1 - n/2, 2 - n/2, (1 - I*a*x)/2] + (-2 + n)*(a^2*x^2*(1 + I*a*x)^(n/2)*(-I + a*x) - I*2^(1 + n/2)*Hypergeometric2F1[1 - n/2, -1/2*n, 2 - n/2, (1 - I*a*x)/2])))/(4*a^4*(-2 + n)*(1 - I*a*x)^(n/2))","A",0
155,1,116,159,0.053589,"\int e^{i n \tan ^{-1}(a x)} x^2 \, dx","Integrate[E^(I*n*ArcTan[a*x])*x^2,x]","\frac{(a x+i) (1-i a x)^{-n/2} \left(2^{\frac{n}{2}+1} \left(n^2+2\right) \, _2F_1\left(1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right)+(n-2) (a x-i) (2 a x-i n) (1+i a x)^{n/2}\right)}{6 a^3 (n-2)}","-\frac{i 2^{n/2} \left(n^2+2\right) (1-i a x)^{1-\frac{n}{2}} \, _2F_1\left(1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right)}{3 a^3 (2-n)}-\frac{i n (1+i a x)^{\frac{n+2}{2}} (1-i a x)^{1-\frac{n}{2}}}{6 a^3}+\frac{x (1+i a x)^{\frac{n+2}{2}} (1-i a x)^{1-\frac{n}{2}}}{3 a^2}",1,"((I + a*x)*((-2 + n)*(1 + I*a*x)^(n/2)*(-I + a*x)*((-I)*n + 2*a*x) + 2^(1 + n/2)*(2 + n^2)*Hypergeometric2F1[1 - n/2, -1/2*n, 2 - n/2, (1 - I*a*x)/2]))/(6*a^3*(-2 + n)*(1 - I*a*x)^(n/2))","A",1
156,1,105,107,0.0285973,"\int e^{i n \tan ^{-1}(a x)} x \, dx","Integrate[E^(I*n*ArcTan[a*x])*x,x]","\frac{(a x+i) (1-i a x)^{-n/2} \left(i 2^{\frac{n}{2}+1} n \, _2F_1\left(1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right)+(n-2) (a x-i) (1+i a x)^{n/2}\right)}{2 a^2 (n-2)}","\frac{2^{n/2} n (1-i a x)^{1-\frac{n}{2}} \, _2F_1\left(1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right)}{a^2 (2-n)}+\frac{(1+i a x)^{\frac{n+2}{2}} (1-i a x)^{1-\frac{n}{2}}}{2 a^2}",1,"((I + a*x)*((-2 + n)*(1 + I*a*x)^(n/2)*(-I + a*x) + I*2^(1 + n/2)*n*Hypergeometric2F1[1 - n/2, -1/2*n, 2 - n/2, (1 - I*a*x)/2]))/(2*a^2*(-2 + n)*(1 - I*a*x)^(n/2))","A",1
157,1,53,71,0.0341771,"\int e^{i n \tan ^{-1}(a x)} \, dx","Integrate[E^(I*n*ArcTan[a*x]),x]","-\frac{4 i e^{i (n+2) \tan ^{-1}(a x)} \, _2F_1\left(2,\frac{n}{2}+1;\frac{n}{2}+2;-e^{2 i \tan ^{-1}(a x)}\right)}{a (n+2)}","\frac{i 2^{\frac{n}{2}+1} (1-i a x)^{1-\frac{n}{2}} \, _2F_1\left(1-\frac{n}{2},-\frac{n}{2};2-\frac{n}{2};\frac{1}{2} (1-i a x)\right)}{a (2-n)}",1,"((-4*I)*E^(I*(2 + n)*ArcTan[a*x])*Hypergeometric2F1[2, 1 + n/2, 2 + n/2, -E^((2*I)*ArcTan[a*x])])/(a*(2 + n))","A",0
158,1,106,125,0.0293882,"\int \frac{e^{i n \tan ^{-1}(a x)}}{x} \, dx","Integrate[E^(I*n*ArcTan[a*x])/x,x]","\frac{2 (1-i a x)^{-n/2} \left((1+i a x)^{n/2} \, _2F_1\left(1,-\frac{n}{2};1-\frac{n}{2};\frac{a x+i}{i-a x}\right)-2^{n/2} \, _2F_1\left(-\frac{n}{2},-\frac{n}{2};1-\frac{n}{2};\frac{1}{2} (1-i a x)\right)\right)}{n}","\frac{2 (1-i a x)^{-n/2} (1+i a x)^{n/2} \, _2F_1\left(1,-\frac{n}{2};1-\frac{n}{2};\frac{1-i a x}{i a x+1}\right)}{n}-\frac{2^{\frac{n}{2}+1} (1-i a x)^{-n/2} \, _2F_1\left(-\frac{n}{2},-\frac{n}{2};1-\frac{n}{2};\frac{1}{2} (1-i a x)\right)}{n}",1,"(2*((1 + I*a*x)^(n/2)*Hypergeometric2F1[1, -1/2*n, 1 - n/2, (I + a*x)/(I - a*x)] - 2^(n/2)*Hypergeometric2F1[-1/2*n, -1/2*n, 1 - n/2, (1 - I*a*x)/2]))/(n*(1 - I*a*x)^(n/2))","A",1
159,1,82,79,0.0156381,"\int \frac{e^{i n \tan ^{-1}(a x)}}{x^2} \, dx","Integrate[E^(I*n*ArcTan[a*x])/x^2,x]","-\frac{2 i a (1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{n}{2}-1} \, _2F_1\left(2,1-\frac{n}{2};2-\frac{n}{2};-\frac{1-i a x}{-i a x-1}\right)}{1-\frac{n}{2}}","-\frac{4 i a (1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{n-2}{2}} \, _2F_1\left(2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-i a x}{i a x+1}\right)}{2-n}",1,"((-2*I)*a*(1 - I*a*x)^(1 - n/2)*(1 + I*a*x)^(-1 + n/2)*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, -((1 - I*a*x)/(-1 - I*a*x))])/(1 - n/2)","A",1
160,1,114,120,0.0342748,"\int \frac{e^{i n \tan ^{-1}(a x)}}{x^3} \, dx","Integrate[E^(I*n*ArcTan[a*x])/x^3,x]","\frac{(a x+i) (1-i a x)^{-n/2} (1+i a x)^{n/2} \left(4 a^2 n x^2 \, _2F_1\left(2,1-\frac{n}{2};2-\frac{n}{2};\frac{a x+i}{i-a x}\right)-(n-2) (a x-i)^2\right)}{2 (n-2) x^2 (a x-i)}","\frac{2 a^2 n (1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{n-2}{2}} \, _2F_1\left(2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-i a x}{i a x+1}\right)}{2-n}-\frac{(1-i a x)^{1-\frac{n}{2}} (1+i a x)^{\frac{n+2}{2}}}{2 x^2}",1,"((1 + I*a*x)^(n/2)*(I + a*x)*(-((-2 + n)*(-I + a*x)^2) + 4*a^2*n*x^2*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (I + a*x)/(I - a*x)]))/(2*(-2 + n)*x^2*(1 - I*a*x)^(n/2)*(-I + a*x))","A",1
161,1,119,171,0.0643681,"\int \frac{e^{i n \tan ^{-1}(a x)}}{x^4} \, dx","Integrate[E^(I*n*ArcTan[a*x])/x^4,x]","-\frac{(a x+i) (1-i a x)^{-n/2} (1+i a x)^{\frac{n-2}{2}} \left(4 a^3 \left(n^2+2\right) x^3 \, _2F_1\left(2,1-\frac{n}{2};2-\frac{n}{2};\frac{a x+i}{i-a x}\right)-(n-2) (a x-i)^2 (a n x-2 i)\right)}{6 (n-2) x^3}","\frac{2 i a^3 \left(n^2+2\right) (1+i a x)^{\frac{n-2}{2}} (1-i a x)^{1-\frac{n}{2}} \, _2F_1\left(2,1-\frac{n}{2};2-\frac{n}{2};\frac{1-i a x}{i a x+1}\right)}{3 (2-n)}-\frac{(1+i a x)^{\frac{n+2}{2}} (1-i a x)^{1-\frac{n}{2}}}{3 x^3}-\frac{i a n (1+i a x)^{\frac{n+2}{2}} (1-i a x)^{1-\frac{n}{2}}}{6 x^2}",1,"-1/6*((1 + I*a*x)^((-2 + n)/2)*(I + a*x)*(-((-2 + n)*(-I + a*x)^2*(-2*I + a*n*x)) + 4*a^3*(2 + n^2)*x^3*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (I + a*x)/(I - a*x)]))/((-2 + n)*x^3*(1 - I*a*x)^(n/2))","A",1
162,1,217,276,0.526339,"\int e^{i \tan ^{-1}(a+b x)} x^4 \, dx","Integrate[E^(I*ArcTan[a + b*x])*x^4,x]","\frac{\sqrt[4]{-1} \left(8 a^4+16 i a^3-24 a^2-12 i a+3\right) \sqrt{-i b} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{4 b^{11/2}}+\frac{i \sqrt{a^2+2 a b x+b^2 x^2+1} \left(24 a^4+a^3 (-24 b x+250 i)+2 a^2 \left(12 b^2 x^2-65 i b x-166\right)+a \left(-24 b^3 x^3+70 i b^2 x^2+116 b x-275 i\right)+24 b^4 x^4-30 i b^3 x^3-32 b^2 x^2+45 i b x+64\right)}{120 b^5}","\frac{\sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2} \left(-96 a^3-2 \left(-36 a^2-14 i a+13\right) b x-86 i a^2+114 a+19 i\right)}{120 b^5}+\frac{\left(8 i a^4-16 a^3-24 i a^2+12 a+3 i\right) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^5}+\frac{\left(8 a^4+16 i a^3-24 a^2-12 i a+3\right) \sinh ^{-1}(a+b x)}{8 b^5}-\frac{(8 a+i) x^2 \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{20 b^3}+\frac{x^3 \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{5 b^2}",1,"((I/120)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(64 + 24*a^4 + (45*I)*b*x - 32*b^2*x^2 - (30*I)*b^3*x^3 + 24*b^4*x^4 + a^3*(250*I - 24*b*x) + 2*a^2*(-166 - (65*I)*b*x + 12*b^2*x^2) + a*(-275*I + 116*b*x + (70*I)*b^2*x^2 - 24*b^3*x^3)))/b^5 + ((-1)^(1/4)*(3 - (12*I)*a - 24*a^2 + (16*I)*a^3 + 8*a^4)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(4*b^(11/2))","A",0
163,1,176,201,0.3076666,"\int e^{i \tan ^{-1}(a+b x)} x^3 \, dx","Integrate[E^(I*ArcTan[a + b*x])*x^3,x]","\frac{\sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2+1} \left(-6 i a^3+a^2 (44+6 i b x)+a \left(-6 i b^2 x^2-20 b x+39 i\right)+6 i b^3 x^3+8 b^2 x^2-9 i b x-16\right)-6 \sqrt[4]{-1} \left(8 a^3+12 i a^2-12 a-3 i\right) \sqrt{-i b} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{24 b^{9/2}}","-\frac{\sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2} \left(-18 a^2+2 (6 a+i) b x-10 i a+7\right)}{24 b^4}-\frac{\left(8 i a^3-12 a^2-12 i a+3\right) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^4}+\frac{\left(-8 a^3-12 i a^2+12 a+3 i\right) \sinh ^{-1}(a+b x)}{8 b^4}+\frac{x^2 \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b^2}",1,"(Sqrt[b]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(-16 - (6*I)*a^3 - (9*I)*b*x + 8*b^2*x^2 + (6*I)*b^3*x^3 + a^2*(44 + (6*I)*b*x) + a*(39*I - 20*b*x - (6*I)*b^2*x^2)) - 6*(-1)^(1/4)*(-3*I - 12*a + (12*I)*a^2 + 8*a^3)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(24*b^(9/2))","A",0
164,1,135,171,0.143443,"\int e^{i \tan ^{-1}(a+b x)} x^2 \, dx","Integrate[E^(I*ArcTan[a + b*x])*x^2,x]","\frac{\sqrt[4]{-1} \left(2 a^2+2 i a-1\right) \sqrt{-i b} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{b^{7/2}}+\frac{\sqrt{a^2+2 a b x+b^2 x^2+1} \left(2 i a^2+a (-9-2 i b x)+2 i b^2 x^2+3 b x-4 i\right)}{6 b^3}","-\frac{\left(-2 i a^2+2 a+i\right) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{2 b^3}-\frac{\left(-2 a^2-2 i a+1\right) \sinh ^{-1}(a+b x)}{2 b^3}-\frac{(4 a+i) \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{6 b^3}+\frac{x \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{3 b^2}",1,"(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(-4*I + (2*I)*a^2 + 3*b*x + (2*I)*b^2*x^2 + a*(-9 - (2*I)*b*x)))/(6*b^3) + ((-1)^(1/4)*(-1 + (2*I)*a + 2*a^2)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/b^(7/2)","A",0
165,1,108,110,0.1224462,"\int e^{i \tan ^{-1}(a+b x)} x \, dx","Integrate[E^(I*ArcTan[a + b*x])*x,x]","\frac{\sqrt{a^2+2 a b x+b^2 x^2+1} (-i a+i b x+2)}{2 b^2}+\frac{(-1)^{3/4} (2 a+i) \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{\sqrt{-i b} b^{3/2}}","\frac{\sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{2 b^2}+\frac{(1-2 i a) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{2 b^2}-\frac{(2 a+i) \sinh ^{-1}(a+b x)}{2 b^2}",1,"((2 - I*a + I*b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/(2*b^2) + ((-1)^(3/4)*(I + 2*a)*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(Sqrt[(-I)*b]*b^(3/2))","A",0
166,1,28,52,0.020517,"\int e^{i \tan ^{-1}(a+b x)} \, dx","Integrate[E^(I*ArcTan[a + b*x]),x]","\frac{\sinh ^{-1}(a+b x)+i \sqrt{(a+b x)^2+1}}{b}","\frac{\sinh ^{-1}(a+b x)}{b}+\frac{i \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{b}",1,"(I*Sqrt[1 + (a + b*x)^2] + ArcSinh[a + b*x])/b","A",1
167,1,142,89,0.0800708,"\int \frac{e^{i \tan ^{-1}(a+b x)}}{x} \, dx","Integrate[E^(I*ArcTan[a + b*x])/x,x]","\frac{2 (-1)^{3/4} \sqrt{-i b} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{\sqrt{b}}-\frac{2 \sqrt{-1-i a} \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)}{\sqrt{-1+i a}}","i \sinh ^{-1}(a+b x)-\frac{2 \sqrt{-a+i} \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{\sqrt{a+i}}",1,"(2*(-1)^(3/4)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/Sqrt[b] - (2*Sqrt[-1 - I*a]*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/Sqrt[-1 + I*a]","A",0
168,1,120,130,0.0867043,"\int \frac{e^{i \tan ^{-1}(a+b x)}}{x^2} \, dx","Integrate[E^(I*ArcTan[a + b*x])/x^2,x]","-i \left(\frac{\sqrt{a^2+2 a b x+b^2 x^2+1}}{a x+i x}+\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)}{\sqrt{-1-i a} (-1+i a)^{3/2}}\right)","\frac{2 i b \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{\sqrt{-a+i} (a+i)^{3/2}}-\frac{\sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{(1-i a) x}",1,"(-I)*(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]/(I*x + a*x) + (2*b*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/(Sqrt[-1 - I*a]*(-1 + I*a)^(3/2)))","A",0
169,1,154,201,0.1357045,"\int \frac{e^{i \tan ^{-1}(a+b x)}}{x^3} \, dx","Integrate[E^(I*ArcTan[a + b*x])/x^3,x]","\frac{\frac{2 (2 a-i) b^2 \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)}{\sqrt{-1-i a} \sqrt{-1+i a}}-\frac{i \left(a^2-a b x+2 i b x+1\right) \sqrt{a^2+2 a b x+b^2 x^2+1}}{x^2}}{2 (a-i) (a+i)^2}","-\frac{\sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{2 \left(a^2+1\right) x^2}+\frac{(1+2 i a) b^2 \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{(-a+i)^{3/2} (a+i)^{5/2}}-\frac{(1+2 i a) b \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{2 (-a+i) (a+i)^2 x}",1,"(((-I)*(1 + a^2 + (2*I)*b*x - a*b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/x^2 + (2*(-I + 2*a)*b^2*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/(Sqrt[-1 - I*a]*Sqrt[-1 + I*a]))/(2*(-I + a)*(I + a)^2)","A",0
170,1,247,283,0.2864942,"\int \frac{e^{i \tan ^{-1}(a+b x)}}{x^4} \, dx","Integrate[E^(I*ArcTan[a + b*x])/x^4,x]","\frac{(1+4 i a) b x (a+b x-i) \sqrt{a^2+2 a b x+b^2 x^2+1}+2 (1-i a) (a-i) (a+b x-i) \sqrt{a^2+2 a b x+b^2 x^2+1}+\frac{3 \left(2 a^2-2 i a-1\right) b^2 x^2 \left(\sqrt{-1-i a} \sqrt{-1+i a} \sqrt{a^2+2 a b x+b^2 x^2+1}-2 i b x \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)\right)}{\sqrt{-1-i a} (-1+i a)^{3/2}}}{6 \left(a^2+1\right)^2 x^3}","\frac{\left(2 a-i \left(1-2 a^2\right)\right) b^3 \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{(-a+i)^{5/2} (a+i)^{7/2}}+\frac{\left(-2 a^2+9 i a+4\right) b^2 \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{6 (1-i a) \left(a^2+1\right)^2 x}-\frac{(-2 a+3 i) b \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{6 (1-i a) \left(a^2+1\right) x^2}-\frac{\sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{3 (1-i a) x^3}",1,"(2*(1 - I*a)*(-I + a)*(-I + a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (1 + (4*I)*a)*b*x*(-I + a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (3*(-1 - (2*I)*a + 2*a^2)*b^2*x^2*(Sqrt[-1 - I*a]*Sqrt[-1 + I*a]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] - (2*I)*b*x*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])]))/(Sqrt[-1 - I*a]*(-1 + I*a)^(3/2)))/(6*(1 + a^2)^2*x^3)","A",0
171,1,92,92,0.0784705,"\int e^{2 i \tan ^{-1}(a+b x)} x^4 \, dx","Integrate[E^((2*I)*ArcTan[a + b*x])*x^4,x]","\frac{2 i (a+i)^4 \log (a+b x+i)}{b^5}-\frac{2 (1-i a)^3 x}{b^4}+\frac{i (a+i)^2 x^2}{b^3}+\frac{2 (1-i a) x^3}{3 b^2}+\frac{i x^4}{2 b}-\frac{x^5}{5}","\frac{2 i (a+i)^4 \log (a+b x+i)}{b^5}-\frac{2 (1-i a)^3 x}{b^4}+\frac{i (a+i)^2 x^2}{b^3}+\frac{2 (1-i a) x^3}{3 b^2}+\frac{i x^4}{2 b}-\frac{x^5}{5}",1,"(-2*(1 - I*a)^3*x)/b^4 + (I*(I + a)^2*x^2)/b^3 + (2*(1 - I*a)*x^3)/(3*b^2) + ((I/2)*x^4)/b - x^5/5 + ((2*I)*(I + a)^4*Log[I + a + b*x])/b^5","A",1
172,1,72,72,0.071138,"\int e^{2 i \tan ^{-1}(a+b x)} x^3 \, dx","Integrate[E^((2*I)*ArcTan[a + b*x])*x^3,x]","-\frac{2 (1-i a)^3 \log (a+b x+i)}{b^4}+\frac{2 i (a+i)^2 x}{b^3}+\frac{(1-i a) x^2}{b^2}+\frac{2 i x^3}{3 b}-\frac{x^4}{4}","-\frac{2 (1-i a)^3 \log (a+b x+i)}{b^4}+\frac{2 i (a+i)^2 x}{b^3}+\frac{(1-i a) x^2}{b^2}+\frac{2 i x^3}{3 b}-\frac{x^4}{4}",1,"((2*I)*(I + a)^2*x)/b^3 + ((1 - I*a)*x^2)/b^2 + (((2*I)/3)*x^3)/b - x^4/4 - (2*(1 - I*a)^3*Log[I + a + b*x])/b^4","A",1
173,1,54,54,0.035101,"\int e^{2 i \tan ^{-1}(a+b x)} x^2 \, dx","Integrate[E^((2*I)*ArcTan[a + b*x])*x^2,x]","\frac{2 i (a+i)^2 \log (a+b x+i)}{b^3}+\frac{2 (1-i a) x}{b^2}+\frac{i x^2}{b}-\frac{x^3}{3}","\frac{2 i (a+i)^2 \log (a+b x+i)}{b^3}+\frac{2 (1-i a) x}{b^2}+\frac{i x^2}{b}-\frac{x^3}{3}",1,"(2*(1 - I*a)*x)/b^2 + (I*x^2)/b - x^3/3 + ((2*I)*(I + a)^2*Log[I + a + b*x])/b^3","A",1
174,1,37,37,0.0234892,"\int e^{2 i \tan ^{-1}(a+b x)} x \, dx","Integrate[E^((2*I)*ArcTan[a + b*x])*x,x]","\frac{2 (1-i a) \log (a+b x+i)}{b^2}+\frac{2 i x}{b}-\frac{x^2}{2}","\frac{2 (1-i a) \log (a+b x+i)}{b^2}+\frac{2 i x}{b}-\frac{x^2}{2}",1,"((2*I)*x)/b - x^2/2 + (2*(1 - I*a)*Log[I + a + b*x])/b^2","A",1
175,1,32,20,0.0124602,"\int e^{2 i \tan ^{-1}(a+b x)} \, dx","Integrate[E^((2*I)*ArcTan[a + b*x]),x]","\frac{i \log \left((a+b x)^2+1\right)}{b}+\frac{2 \tan ^{-1}(a+b x)}{b}-x","-x+\frac{2 i \log (a+b x+i)}{b}",1,"-x + (2*ArcTan[a + b*x])/b + (I*Log[1 + (a + b*x)^2])/b","A",0
176,1,31,38,0.020643,"\int \frac{e^{2 i \tan ^{-1}(a+b x)}}{x} \, dx","Integrate[E^((2*I)*ArcTan[a + b*x])/x,x]","-\frac{2 i \log (a+b x+i)+(a-i) \log (x)}{a+i}","\frac{(-a+i) \log (x)}{a+i}-\frac{2 \log (a+b x+i)}{1-i a}",1,"-(((-I + a)*Log[x] + (2*I)*Log[I + a + b*x])/(I + a))","A",1
177,1,39,55,0.0286639,"\int \frac{e^{2 i \tan ^{-1}(a+b x)}}{x^2} \, dx","Integrate[E^((2*I)*ArcTan[a + b*x])/x^2,x]","\frac{a^2+2 i b x \log (a+b x+i)-2 i b x \log (x)+1}{(a+i)^2 x}","-\frac{2 i b \log (x)}{(a+i)^2}+\frac{2 i b \log (a+b x+i)}{(a+i)^2}-\frac{-a+i}{(a+i) x}",1,"(1 + a^2 - (2*I)*b*x*Log[x] + (2*I)*b*x*Log[I + a + b*x])/((I + a)^2*x)","A",1
178,1,63,76,0.0381129,"\int \frac{e^{2 i \tan ^{-1}(a+b x)}}{x^3} \, dx","Integrate[E^((2*I)*ArcTan[a + b*x])/x^3,x]","\frac{(a+i) \left(a^2+4 i b x+1\right)-4 i b^2 x^2 \log (a+b x+i)+4 i b^2 x^2 \log (x)}{2 (a+i)^3 x^2}","-\frac{2 b^2 \log (x)}{(1-i a)^3}+\frac{2 b^2 \log (a+b x+i)}{(1-i a)^3}+\frac{2 i b}{(a+i)^2 x}-\frac{-a+i}{2 (a+i) x^2}",1,"((I + a)*(1 + a^2 + (4*I)*b*x) + (4*I)*b^2*x^2*Log[x] - (4*I)*b^2*x^2*Log[I + a + b*x])/(2*(I + a)^3*x^2)","A",1
179,1,88,93,0.052057,"\int \frac{e^{2 i \tan ^{-1}(a+b x)}}{x^4} \, dx","Integrate[E^((2*I)*ArcTan[a + b*x])/x^4,x]","\frac{(a+i) \left(a^3+i a^2+3 i a b x+a-6 i b^2 x^2-3 b x+i\right)+6 i b^3 x^3 \log (a+b x+i)-6 i b^3 x^3 \log (x)}{3 (a+i)^4 x^3}","-\frac{2 i b^3 \log (x)}{(a+i)^4}+\frac{2 i b^3 \log (a+b x+i)}{(a+i)^4}+\frac{2 b^2}{(1-i a)^3 x}+\frac{i b}{(a+i)^2 x^2}-\frac{-a+i}{3 (a+i) x^3}",1,"((I + a)*(I + a + I*a^2 + a^3 - 3*b*x + (3*I)*a*b*x - (6*I)*b^2*x^2) - (6*I)*b^3*x^3*Log[x] + (6*I)*b^3*x^3*Log[I + a + b*x])/(3*(I + a)^4*x^3)","A",1
180,1,249,324,0.4804465,"\int e^{3 i \tan ^{-1}(a+b x)} x^4 \, dx","Integrate[E^((3*I)*ArcTan[a + b*x])*x^4,x]","\frac{3 (-1)^{3/4} \left(8 a^4+48 i a^3-88 a^2-68 i a+19\right) \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{4 \sqrt{-i b} b^{9/2}}-\frac{\sqrt{i a+i b x+1} \left(8 a^5+418 i a^4+14 i a^3 (8 b x+121 i)-i a^2 \left(52 b^2 x^2-422 i b x+2599\right)+a \left(32 i b^3 x^3+118 b^2 x^2-458 i b x+1763\right)+8 b^5 x^5-22 i b^4 x^4-34 b^3 x^3+61 i b^2 x^2+163 b x+448 i\right)}{40 b^5 \sqrt{-i (a+b x+i)}}","-\frac{i \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2} \left(112 i a^3+2 \left(-52 i a^2+118 a+61 i\right) b x-422 a^2-458 i a+163\right)}{40 b^5}-\frac{3 \left(8 i a^4-48 a^3-88 i a^2+68 a+19 i\right) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^5}-\frac{3 \left(8 a^4+48 i a^3-88 a^2-68 i a+19\right) \sinh ^{-1}(a+b x)}{8 b^5}+\frac{3 (16 a+17 i) x^2 \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{20 b^3}-\frac{11 x^3 \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{5 b^2}-\frac{2 i x^4 (i a+i b x+1)^{3/2}}{b \sqrt{-i a-i b x+1}}",1,"-1/40*(Sqrt[1 + I*a + I*b*x]*(448*I + (418*I)*a^4 + 8*a^5 + 163*b*x + (61*I)*b^2*x^2 - 34*b^3*x^3 - (22*I)*b^4*x^4 + 8*b^5*x^5 + (14*I)*a^3*(121*I + 8*b*x) - I*a^2*(2599 - (422*I)*b*x + 52*b^2*x^2) + a*(1763 - (458*I)*b*x + 118*b^2*x^2 + (32*I)*b^3*x^3)))/(b^5*Sqrt[(-I)*(I + a + b*x)]) + (3*(-1)^(3/4)*(19 - (68*I)*a - 88*a^2 + (48*I)*a^3 + 8*a^4)*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(4*Sqrt[(-I)*b]*b^(9/2))","A",0
181,1,201,249,0.278826,"\int e^{3 i \tan ^{-1}(a+b x)} x^3 \, dx","Integrate[E^((3*I)*ArcTan[a + b*x])*x^3,x]","\frac{3 \sqrt[4]{-1} \left(8 a^3+36 i a^2-44 a-17 i\right) \sqrt{-i b} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{4 b^{9/2}}+\frac{\sqrt{i a+i b x+1} \left(2 a^4+78 i a^3+a^2 (-233+22 i b x)-i a \left(10 b^2 x^2-54 i b x+237\right)-2 b^4 x^4+6 i b^3 x^3+11 b^2 x^2-29 i b x+80\right)}{8 b^4 \sqrt{-i (a+b x+i)}}","-\frac{i \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2} \left(-22 i a^2-2 (11-10 i a) b x+54 a+29 i\right)}{8 b^4}+\frac{3 \left(8 i a^3-36 a^2-44 i a+17\right) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^4}-\frac{3 \left(-8 a^3-36 i a^2+44 a+17 i\right) \sinh ^{-1}(a+b x)}{8 b^4}-\frac{9 x^2 \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b^2}-\frac{2 i x^3 (i a+i b x+1)^{3/2}}{b \sqrt{-i a-i b x+1}}",1,"(Sqrt[1 + I*a + I*b*x]*(80 + (78*I)*a^3 + 2*a^4 - (29*I)*b*x + 11*b^2*x^2 + (6*I)*b^3*x^3 - 2*b^4*x^4 + a^2*(-233 + (22*I)*b*x) - I*a*(237 - (54*I)*b*x + 10*b^2*x^2)))/(8*b^4*Sqrt[(-I)*(I + a + b*x)]) + (3*(-1)^(1/4)*(-17*I - 44*a + (36*I)*a^2 + 8*a^3)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(4*b^(9/2))","A",0
182,1,160,227,0.2392147,"\int e^{3 i \tan ^{-1}(a+b x)} x^2 \, dx","Integrate[E^((3*I)*ArcTan[a + b*x])*x^2,x]","\frac{(-1)^{3/4} \left(6 a^2+18 i a-11\right) \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{\sqrt{-i b} b^{5/2}}+\frac{\sqrt{i a+i b x+1} \left(-2 a^3-53 i a^2+a (103-16 i b x)-2 b^3 x^3+7 i b^2 x^2+19 b x+52 i\right)}{6 b^3 \sqrt{-i (a+b x+i)}}","\frac{\left(-6 i a^2+18 a+11 i\right) \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{6 b^3}+\frac{\left(-6 i a^2+18 a+11 i\right) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{2 b^3}+\frac{\left(-6 a^2-18 i a+11\right) \sinh ^{-1}(a+b x)}{2 b^3}+\frac{i \sqrt{-i a-i b x+1} (i a+i b x+1)^{5/2}}{3 b^3}-\frac{i (a+i)^2 (i a+i b x+1)^{5/2}}{b^3 \sqrt{-i a-i b x+1}}",1,"(Sqrt[1 + I*a + I*b*x]*(52*I - (53*I)*a^2 - 2*a^3 + 19*b*x + (7*I)*b^2*x^2 - 2*b^3*x^3 + a*(103 - (16*I)*b*x)))/(6*b^3*Sqrt[(-I)*(I + a + b*x)]) + ((-1)^(3/4)*(-11 + (18*I)*a + 6*a^2)*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(Sqrt[(-I)*b]*b^(5/2))","A",0
183,1,132,163,0.1815139,"\int e^{3 i \tan ^{-1}(a+b x)} x \, dx","Integrate[E^((3*I)*ArcTan[a + b*x])*x,x]","\frac{\sqrt{i a+i b x+1} \left(a^2+15 i a-b^2 x^2+5 i b x-14\right)}{2 b^2 \sqrt{-i (a+b x+i)}}+\frac{3 \sqrt[4]{-1} (2 a+3 i) \sqrt{-i b} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{b^{5/2}}","-\frac{(1-i a) (i a+i b x+1)^{5/2}}{b^2 \sqrt{-i a-i b x+1}}-\frac{(3-2 i a) \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{2 b^2}-\frac{3 (3-2 i a) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{2 b^2}+\frac{3 (2 a+3 i) \sinh ^{-1}(a+b x)}{2 b^2}",1,"(Sqrt[1 + I*a + I*b*x]*(-14 + (15*I)*a + a^2 + (5*I)*b*x - b^2*x^2))/(2*b^2*Sqrt[(-I)*(I + a + b*x)]) + (3*(-1)^(1/4)*(3*I + 2*a)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/b^(5/2)","A",0
184,1,45,94,0.0452508,"\int e^{3 i \tan ^{-1}(a+b x)} \, dx","Integrate[E^((3*I)*ArcTan[a + b*x]),x]","-\frac{3 \sinh ^{-1}(a+b x)}{b}+\frac{\sqrt{(a+b x)^2+1} \left(\frac{4}{a+b x+i}-i\right)}{b}","-\frac{2 i (i a+i b x+1)^{3/2}}{b \sqrt{-i a-i b x+1}}-\frac{3 i \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{b}-\frac{3 \sinh ^{-1}(a+b x)}{b}",1,"(Sqrt[1 + (a + b*x)^2]*(-I + 4/(I + a + b*x)))/b - (3*ArcSinh[a + b*x])/b","A",0
185,1,196,134,0.7997031,"\int \frac{e^{3 i \tan ^{-1}(a+b x)}}{x} \, dx","Integrate[E^((3*I)*ArcTan[a + b*x])/x,x]","\frac{2 \left(\frac{\sqrt[4]{-1} (a+i) (-i b)^{3/2} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{b^{3/2}}+\frac{2 i \sqrt{i a+i b x+1}}{\sqrt{-i (a+b x+i)}}+\frac{\sqrt{-1-i a} (a-i) \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)}{\sqrt{-1+i a}}\right)}{a+i}","\frac{4 \sqrt{i a+i b x+1}}{(1-i a) \sqrt{-i a-i b x+1}}-i \sinh ^{-1}(a+b x)-\frac{2 (-a+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{(a+i)^{3/2}}",1,"(2*(((2*I)*Sqrt[1 + I*a + I*b*x])/Sqrt[(-I)*(I + a + b*x)] + ((-1)^(1/4)*(I + a)*((-I)*b)^(3/2)*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/b^(3/2) + (Sqrt[-1 - I*a]*(-I + a)*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/Sqrt[-1 + I*a]))/(I + a)","A",0
186,1,145,176,0.1688502,"\int \frac{e^{3 i \tan ^{-1}(a+b x)}}{x^2} \, dx","Integrate[E^((3*I)*ArcTan[a + b*x])/x^2,x]","\frac{\frac{\sqrt{i a+i b x+1} \left(a^2+a b x-5 i b x+1\right)}{x \sqrt{-i (a+b x+i)}}+\frac{6 i \sqrt{-1-i a} b \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)}{\sqrt{-1+i a}}}{(a+i)^2}","-\frac{(i a+i b x+1)^{3/2}}{(1-i a) x \sqrt{-i a-i b x+1}}-\frac{6 i b \sqrt{i a+i b x+1}}{(a+i)^2 \sqrt{-i a-i b x+1}}+\frac{6 i \sqrt{-a+i} b \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{(a+i)^{5/2}}",1,"((Sqrt[1 + I*a + I*b*x]*(1 + a^2 - (5*I)*b*x + a*b*x))/(x*Sqrt[(-I)*(I + a + b*x)]) + ((6*I)*Sqrt[-1 - I*a]*b*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/Sqrt[-1 + I*a])/(I + a)^2","A",0
187,1,194,264,0.2425158,"\int \frac{e^{3 i \tan ^{-1}(a+b x)}}{x^3} \, dx","Integrate[E^((3*I)*ArcTan[a + b*x])/x^3,x]","\frac{\frac{\sqrt{i a+i b x+1} \left(a^3+i a^2-a b^2 x^2+5 i a b x+a+14 i b^2 x^2-5 b x+i\right)}{x^2 \sqrt{-i (a+b x+i)}}-\frac{6 i \sqrt{-1-i a} (2 a-3 i) b^2 \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)}{\sqrt{-1+i a} (a-i)}}{2 (a+i)^3}","-\frac{(i a+i b x+1)^{5/2}}{2 \left(a^2+1\right) x^2 \sqrt{-i a-i b x+1}}+\frac{3 (-2 a+3 i) b^2 \sqrt{i a+i b x+1}}{(1+i a) (a+i)^3 \sqrt{-i a-i b x+1}}+\frac{3 (3+2 i a) b^2 \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{\sqrt{-a+i} (a+i)^{7/2}}+\frac{(-2 a+3 i) b (i a+i b x+1)^{3/2}}{2 (1+i a) (a+i)^2 x \sqrt{-i a-i b x+1}}",1,"((Sqrt[1 + I*a + I*b*x]*(I + a + I*a^2 + a^3 - 5*b*x + (5*I)*a*b*x + (14*I)*b^2*x^2 - a*b^2*x^2))/(x^2*Sqrt[(-I)*(I + a + b*x)]) - ((6*I)*Sqrt[-1 - I*a]*(-3*I + 2*a)*b^2*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/(Sqrt[-1 + I*a]*(-I + a)))/(2*(I + a)^3)","A",0
188,1,282,338,0.4259416,"\int \frac{e^{3 i \tan ^{-1}(a+b x)}}{x^4} \, dx","Integrate[E^((3*I)*ArcTan[a + b*x])/x^4,x]","-\frac{-i \left(6 a^2-18 i a-11\right) b^2 x^2 \left(i \sqrt{-1+i a} \sqrt{i a+i b x+1} \left(a^2+a b x-5 i b x+1\right)-6 \sqrt{-1-i a} b x \sqrt{-i (a+b x+i)} \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)\right)+2 (-1+i a)^{3/2} (1+i a) (a+i)^2 (i a+i b x+1)^{5/2}+(-4 a+3 i) (-1+i a)^{5/2} b x (i a+i b x+1)^{5/2}}{6 (-1+i a)^{5/2} \left(a^2+1\right)^2 x^3 \sqrt{-i (a+b x+i)}}","\frac{\left(-2 a^2+51 i a+52\right) b^3 \sqrt{i a+i b x+1}}{6 (-a+i) (a+i)^4 \sqrt{-i a-i b x+1}}-\frac{\left(-6 i a^2-18 a+11 i\right) b^3 \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{(-a+i)^{3/2} (a+i)^{9/2}}+\frac{(19+16 i a) b^2 \sqrt{i a+i b x+1}}{6 (-a+i) (a+i)^3 x \sqrt{-i a-i b x+1}}-\frac{(-a+i) \sqrt{i a+i b x+1}}{3 (a+i) x^3 \sqrt{-i a-i b x+1}}+\frac{7 i b \sqrt{i a+i b x+1}}{6 (a+i)^2 x^2 \sqrt{-i a-i b x+1}}",1,"-1/6*(2*(-1 + I*a)^(3/2)*(1 + I*a)*(I + a)^2*(1 + I*a + I*b*x)^(5/2) + (3*I - 4*a)*(-1 + I*a)^(5/2)*b*x*(1 + I*a + I*b*x)^(5/2) - I*(-11 - (18*I)*a + 6*a^2)*b^2*x^2*(I*Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x]*(1 + a^2 - (5*I)*b*x + a*b*x) - 6*Sqrt[-1 - I*a]*b*x*Sqrt[(-I)*(I + a + b*x)]*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])]))/((-1 + I*a)^(5/2)*(1 + a^2)^2*x^3*Sqrt[(-I)*(I + a + b*x)])","A",0
189,1,248,276,0.7179672,"\int e^{-i \tan ^{-1}(a+b x)} x^4 \, dx","Integrate[x^4/E^(I*ArcTan[a + b*x]),x]","\frac{\sqrt[4]{-1} \left(-8 i a^4-16 a^3+24 i a^2+12 a-3 i\right) \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{4 \sqrt{-i b} b^{9/2}}+\frac{i \sqrt{i a+i b x+1} \left(24 i a^5+226 a^4+2 a^3 (72 b x-41 i)+a^2 \left(-84 b^2 x^2-346 i b x+57\right)+a \left(64 b^3 x^3+154 i b^2 x^2-346 b x-211 i\right)+24 i b^5 x^5-54 b^4 x^4-62 i b^3 x^3+77 b^2 x^2+109 i b x-64\right)}{120 b^5 \sqrt{-i (a+b x+i)}}","-\frac{(-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1} \left(96 a^3+2 \left(-36 a^2+14 i a+13\right) b x-86 i a^2-114 a+19 i\right)}{120 b^5}-\frac{\left(8 i a^4+16 a^3-24 i a^2-12 a+3 i\right) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^5}+\frac{\left(8 a^4-16 i a^3-24 a^2+12 i a+3\right) \sinh ^{-1}(a+b x)}{8 b^5}+\frac{(-8 a+i) x^2 (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1}}{20 b^3}+\frac{x^3 (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1}}{5 b^2}",1,"((I/120)*Sqrt[1 + I*a + I*b*x]*(-64 + 226*a^4 + (24*I)*a^5 + (109*I)*b*x + 77*b^2*x^2 - (62*I)*b^3*x^3 - 54*b^4*x^4 + (24*I)*b^5*x^5 + 2*a^3*(-41*I + 72*b*x) + a^2*(57 - (346*I)*b*x - 84*b^2*x^2) + a*(-211*I - 346*b*x + (154*I)*b^2*x^2 + 64*b^3*x^3)))/(b^5*Sqrt[(-I)*(I + a + b*x)]) + ((-1)^(1/4)*(-3*I + 12*a + (24*I)*a^2 - 16*a^3 - (8*I)*a^4)*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(4*Sqrt[(-I)*b]*b^(9/2))","A",0
190,1,202,201,0.5161277,"\int e^{-i \tan ^{-1}(a+b x)} x^3 \, dx","Integrate[x^3/E^(I*ArcTan[a + b*x]),x]","\frac{(-1)^{3/4} \left(8 i a^3+12 a^2-12 i a-3\right) \sqrt{-i b} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{4 b^{9/2}}+\frac{\sqrt{i a+i b x+1} \left(6 a^4-38 i a^3+5 a^2 (1-6 i b x)+i a \left(18 b^2 x^2+50 i b x-23\right)-6 b^4 x^4-14 i b^3 x^3+17 b^2 x^2+25 i b x-16\right)}{24 b^4 \sqrt{-i (a+b x+i)}}","-\frac{(-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1} \left(-18 a^2-2 (-6 a+i) b x+10 i a+7\right)}{24 b^4}-\frac{\left(-8 i a^3-12 a^2+12 i a+3\right) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^4}-\frac{\left(8 a^3-12 i a^2-12 a+3 i\right) \sinh ^{-1}(a+b x)}{8 b^4}+\frac{x^2 (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1}}{4 b^2}",1,"(Sqrt[1 + I*a + I*b*x]*(-16 - (38*I)*a^3 + 6*a^4 + (25*I)*b*x + 17*b^2*x^2 - (14*I)*b^3*x^3 - 6*b^4*x^4 + 5*a^2*(1 - (6*I)*b*x) + I*a*(-23 + (50*I)*b*x + 18*b^2*x^2)))/(24*b^4*Sqrt[(-I)*(I + a + b*x)]) + ((-1)^(3/4)*(-3 - (12*I)*a + 12*a^2 + (8*I)*a^3)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(4*b^(9/2))","A",0
191,1,162,171,0.2994351,"\int e^{-i \tan ^{-1}(a+b x)} x^2 \, dx","Integrate[x^2/E^(I*ArcTan[a + b*x]),x]","\frac{\sqrt[4]{-1} \left(2 a^2-2 i a-1\right) \sqrt{-i b} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{b^{7/2}}+\frac{i \sqrt{i a+i b x+1} \left(2 i a^3+7 a^2+a (8 b x+5 i)+2 i b^3 x^3-5 b^2 x^2-7 i b x+4\right)}{6 b^3 \sqrt{-i (a+b x+i)}}","\frac{\left(-2 i a^2-2 a+i\right) \sqrt{i a+i b x+1} \sqrt{-i a-i b x+1}}{2 b^3}-\frac{\left(-2 a^2+2 i a+1\right) \sinh ^{-1}(a+b x)}{2 b^3}+\frac{(-4 a+i) \sqrt{i a+i b x+1} (-i a-i b x+1)^{3/2}}{6 b^3}+\frac{x \sqrt{i a+i b x+1} (-i a-i b x+1)^{3/2}}{3 b^2}",1,"((I/6)*Sqrt[1 + I*a + I*b*x]*(4 + 7*a^2 + (2*I)*a^3 - (7*I)*b*x - 5*b^2*x^2 + (2*I)*b^3*x^3 + a*(5*I + 8*b*x)))/(b^3*Sqrt[(-I)*(I + a + b*x)]) + ((-1)^(1/4)*(-1 - (2*I)*a + 2*a^2)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/b^(7/2)","A",0
192,1,131,110,0.1372969,"\int e^{-i \tan ^{-1}(a+b x)} x \, dx","Integrate[x/E^(I*ArcTan[a + b*x]),x]","\frac{\sqrt{i a+i b x+1} \left(a^2-i a-b^2 x^2-3 i b x+2\right)}{2 b^2 \sqrt{-i (a+b x+i)}}+\frac{(-1)^{3/4} (1+2 i a) \sqrt{-i b} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{b^{5/2}}","\frac{\sqrt{i a+i b x+1} (-i a-i b x+1)^{3/2}}{2 b^2}+\frac{(1+2 i a) \sqrt{i a+i b x+1} \sqrt{-i a-i b x+1}}{2 b^2}+\frac{(-2 a+i) \sinh ^{-1}(a+b x)}{2 b^2}",1,"(Sqrt[1 + I*a + I*b*x]*(2 - I*a + a^2 - (3*I)*b*x - b^2*x^2))/(2*b^2*Sqrt[(-I)*(I + a + b*x)]) + ((-1)^(3/4)*(1 + (2*I)*a)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/b^(5/2)","A",0
193,1,28,52,0.0204978,"\int e^{-i \tan ^{-1}(a+b x)} \, dx","Integrate[E^((-I)*ArcTan[a + b*x]),x]","\frac{\sinh ^{-1}(a+b x)-i \sqrt{(a+b x)^2+1}}{b}","\frac{\sinh ^{-1}(a+b x)}{b}-\frac{i \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{b}",1,"((-I)*Sqrt[1 + (a + b*x)^2] + ArcSinh[a + b*x])/b","A",1
194,1,142,89,0.0769629,"\int \frac{e^{-i \tan ^{-1}(a+b x)}}{x} \, dx","Integrate[1/(E^(I*ArcTan[a + b*x])*x),x]","\frac{2 \sqrt[4]{-1} (-i b)^{3/2} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{b^{3/2}}-\frac{2 \sqrt{-1+i a} \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)}{\sqrt{-1-i a}}","-i \sinh ^{-1}(a+b x)-\frac{2 \sqrt{a+i} \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{\sqrt{-a+i}}",1,"(2*(-1)^(1/4)*((-I)*b)^(3/2)*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/b^(3/2) - (2*Sqrt[-1 + I*a]*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/Sqrt[-1 - I*a]","A",0
195,1,119,130,0.0834942,"\int \frac{e^{-i \tan ^{-1}(a+b x)}}{x^2} \, dx","Integrate[1/(E^(I*ArcTan[a + b*x])*x^2),x]","i \left(\frac{\sqrt{a^2+2 a b x+b^2 x^2+1}}{(a-i) x}+\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)}{(-1-i a)^{3/2} \sqrt{-1+i a}}\right)","-\frac{\sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{(1+i a) x}-\frac{2 i b \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{(-a+i)^{3/2} \sqrt{a+i}}",1,"I*(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]/((-I + a)*x) + (2*b*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/((-1 - I*a)^(3/2)*Sqrt[-1 + I*a]))","A",0
196,1,154,201,0.1489461,"\int \frac{e^{-i \tan ^{-1}(a+b x)}}{x^3} \, dx","Integrate[1/(E^(I*ArcTan[a + b*x])*x^3),x]","\frac{\frac{i \left(a^2-a b x-2 i b x+1\right) \sqrt{a^2+2 a b x+b^2 x^2+1}}{x^2}+\frac{2 (2 a+i) b^2 \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)}{\sqrt{-1-i a} \sqrt{-1+i a}}}{2 (a-i)^2 (a+i)}","-\frac{(-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1}}{2 \left(a^2+1\right) x^2}+\frac{(1-2 i a) b^2 \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{(-a+i)^{5/2} (a+i)^{3/2}}+\frac{(1-2 i a) b \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{2 (-a+i)^2 (a+i) x}",1,"((I*(1 + a^2 - (2*I)*b*x - a*b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/x^2 + (2*(I + 2*a)*b^2*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/(Sqrt[-1 - I*a]*Sqrt[-1 + I*a]))/(2*(-I + a)^2*(I + a))","A",0
197,1,247,283,0.2986283,"\int \frac{e^{-i \tan ^{-1}(a+b x)}}{x^4} \, dx","Integrate[1/(E^(I*ArcTan[a + b*x])*x^4),x]","\frac{(1-4 i a) b x (a+b x+i) \sqrt{a^2+2 a b x+b^2 x^2+1}+2 (1+i a) (a+i) (a+b x+i) \sqrt{a^2+2 a b x+b^2 x^2+1}+\frac{3 \left(2 a^2+2 i a-1\right) b^2 x^2 \left(\sqrt{-1-i a} \sqrt{-1+i a} \sqrt{a^2+2 a b x+b^2 x^2+1}+2 i b x \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)\right)}{(-1-i a)^{3/2} \sqrt{-1+i a}}}{6 \left(a^2+1\right)^2 x^3}","\frac{\left(2 a+i \left(1-2 a^2\right)\right) b^3 \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{(-a+i)^{7/2} (a+i)^{5/2}}+\frac{\left(-2 a^2-9 i a+4\right) b^2 \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{6 (1+i a) \left(a^2+1\right)^2 x}-\frac{\sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{3 (1+i a) x^3}+\frac{(3-2 i a) b \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{6 (-a+i)^2 (a+i) x^2}",1,"(2*(1 + I*a)*(I + a)*(I + a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (1 - (4*I)*a)*b*x*(I + a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (3*(-1 + (2*I)*a + 2*a^2)*b^2*x^2*(Sqrt[-1 - I*a]*Sqrt[-1 + I*a]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (2*I)*b*x*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])]))/((-1 - I*a)^(3/2)*Sqrt[-1 + I*a]))/(6*(1 + a^2)^2*x^3)","A",0
198,1,95,99,0.0744215,"\int e^{-2 i \tan ^{-1}(a+b x)} x^4 \, dx","Integrate[x^4/E^((2*I)*ArcTan[a + b*x]),x]","-\frac{2 i (a-i)^4 \log (-a-b x+i)}{b^5}-\frac{2 (1+i a)^3 x}{b^4}-\frac{i (a-i)^2 x^2}{b^3}+\frac{2 (1+i a) x^3}{3 b^2}-\frac{i x^4}{2 b}-\frac{x^5}{5}","-\frac{2 i (-a+i)^4 \log (-a-b x+i)}{b^5}-\frac{2 (1+i a)^3 x}{b^4}-\frac{i (-a+i)^2 x^2}{b^3}+\frac{2 (1+i a) x^3}{3 b^2}-\frac{i x^4}{2 b}-\frac{x^5}{5}",1,"(-2*(1 + I*a)^3*x)/b^4 - (I*(-I + a)^2*x^2)/b^3 + (2*(1 + I*a)*x^3)/(3*b^2) - ((I/2)*x^4)/b - x^5/5 - ((2*I)*(-I + a)^4*Log[I - a - b*x])/b^5","A",1
199,1,77,77,0.063226,"\int e^{-2 i \tan ^{-1}(a+b x)} x^3 \, dx","Integrate[x^3/E^((2*I)*ArcTan[a + b*x]),x]","-\frac{2 (1+i a)^3 \log (-a-b x+i)}{b^4}-\frac{2 i (-a+i)^2 x}{b^3}+\frac{(1+i a) x^2}{b^2}-\frac{2 i x^3}{3 b}-\frac{x^4}{4}","-\frac{2 (1+i a)^3 \log (-a-b x+i)}{b^4}-\frac{2 i (-a+i)^2 x}{b^3}+\frac{(1+i a) x^2}{b^2}-\frac{2 i x^3}{3 b}-\frac{x^4}{4}",1,"((-2*I)*(I - a)^2*x)/b^3 + ((1 + I*a)*x^2)/b^2 - (((2*I)/3)*x^3)/b - x^4/4 - (2*(1 + I*a)^3*Log[I - a - b*x])/b^4","A",1
200,1,55,59,0.0339239,"\int e^{-2 i \tan ^{-1}(a+b x)} x^2 \, dx","Integrate[x^2/E^((2*I)*ArcTan[a + b*x]),x]","\frac{b x \left(6 i a-b^2 x^2-3 i b x+6\right)-6 i (a-i)^2 \log (-a-b x+i)}{3 b^3}","-\frac{2 i (-a+i)^2 \log (-a-b x+i)}{b^3}+\frac{2 (1+i a) x}{b^2}-\frac{i x^2}{b}-\frac{x^3}{3}",1,"(b*x*(6 + (6*I)*a - (3*I)*b*x - b^2*x^2) - (6*I)*(-I + a)^2*Log[I - a - b*x])/(3*b^3)","A",1
201,1,40,40,0.022775,"\int e^{-2 i \tan ^{-1}(a+b x)} x \, dx","Integrate[x/E^((2*I)*ArcTan[a + b*x]),x]","\frac{2 (1+i a) \log (-a-b x+i)}{b^2}-\frac{2 i x}{b}-\frac{x^2}{2}","\frac{2 (1+i a) \log (-a-b x+i)}{b^2}-\frac{2 i x}{b}-\frac{x^2}{2}",1,"((-2*I)*x)/b - x^2/2 + (2*(1 + I*a)*Log[I - a - b*x])/b^2","A",1
202,1,32,23,0.0115567,"\int e^{-2 i \tan ^{-1}(a+b x)} \, dx","Integrate[E^((-2*I)*ArcTan[a + b*x]),x]","-\frac{i \log \left((a+b x)^2+1\right)}{b}+\frac{2 \tan ^{-1}(a+b x)}{b}-x","-x-\frac{2 i \log (-a-b x+i)}{b}",1,"-x + (2*ArcTan[a + b*x])/b - (I*Log[1 + (a + b*x)^2])/b","A",0
203,1,34,41,0.021285,"\int \frac{e^{-2 i \tan ^{-1}(a+b x)}}{x} \, dx","Integrate[1/(E^((2*I)*ArcTan[a + b*x])*x),x]","\frac{2 i \log (-a-b x+i)-(a+i) \log (x)}{a-i}","\frac{(a+i) \log (x)}{-a+i}-\frac{2 \log (-a-b x+i)}{1+i a}",1,"(-((I + a)*Log[x]) + (2*I)*Log[I - a - b*x])/(-I + a)","A",1
204,1,42,62,0.0265557,"\int \frac{e^{-2 i \tan ^{-1}(a+b x)}}{x^2} \, dx","Integrate[1/(E^((2*I)*ArcTan[a + b*x])*x^2),x]","\frac{a^2-2 i b x \log (-a-b x+i)+2 i b x \log (x)+1}{(a-i)^2 x}","\frac{2 i b \log (x)}{(-a+i)^2}-\frac{2 i b \log (-a-b x+i)}{(-a+i)^2}-\frac{a+i}{(-a+i) x}",1,"(1 + a^2 + (2*I)*b*x*Log[x] - (2*I)*b*x*Log[I - a - b*x])/((-I + a)^2*x)","A",1
205,1,66,83,0.0400147,"\int \frac{e^{-2 i \tan ^{-1}(a+b x)}}{x^3} \, dx","Integrate[1/(E^((2*I)*ArcTan[a + b*x])*x^3),x]","\frac{(a-i) \left(a^2-4 i b x+1\right)+4 i b^2 x^2 \log (-a-b x+i)-4 i b^2 x^2 \log (x)}{2 (a-i)^3 x^2}","-\frac{2 b^2 \log (x)}{(1+i a)^3}+\frac{2 b^2 \log (-a-b x+i)}{(1+i a)^3}-\frac{2 i b}{(-a+i)^2 x}+\frac{-a-i}{2 (-a+i) x^2}",1,"((-I + a)*(1 + a^2 - (4*I)*b*x) - (4*I)*b^2*x^2*Log[x] + (4*I)*b^2*x^2*Log[I - a - b*x])/(2*(-I + a)^3*x^2)","A",1
206,1,91,104,0.0504307,"\int \frac{e^{-2 i \tan ^{-1}(a+b x)}}{x^4} \, dx","Integrate[1/(E^((2*I)*ArcTan[a + b*x])*x^4),x]","\frac{(a-i) \left(a^3-i a^2-3 i a b x+a+6 i b^2 x^2-3 b x-i\right)-6 i b^3 x^3 \log (-a-b x+i)+6 i b^3 x^3 \log (x)}{3 (a-i)^4 x^3}","\frac{2 i b^3 \log (x)}{(-a+i)^4}-\frac{2 i b^3 \log (-a-b x+i)}{(-a+i)^4}+\frac{2 b^2}{(1+i a)^3 x}-\frac{i b}{(-a+i)^2 x^2}+\frac{-a-i}{3 (-a+i) x^3}",1,"((-I + a)*(-I + a - I*a^2 + a^3 - 3*b*x - (3*I)*a*b*x + (6*I)*b^2*x^2) + (6*I)*b^3*x^3*Log[x] - (6*I)*b^3*x^3*Log[I - a - b*x])/(3*(-I + a)^4*x^3)","A",1
207,1,299,324,0.523655,"\int e^{-3 i \tan ^{-1}(a+b x)} x^4 \, dx","Integrate[x^4/E^((3*I)*ArcTan[a + b*x]),x]","\frac{\frac{30 \sqrt[4]{-1} \left(8 i a^4+48 a^3-88 i a^2-68 a+19 i\right) \sqrt{b} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{\sqrt{-i b}}+\frac{8 i a^6+a^5 (410+8 i b x)+2 a^4 (265 b x-638 i)+a^3 \left(60 b^2 x^2-2004 i b x-905\right)-a^2 \left(20 b^3 x^3+356 i b^2 x^2+2635 b x+836 i\right)+a \left(8 i b^5 x^5+10 b^4 x^4+116 i b^3 x^3-515 b^2 x^2+1468 i b x-1315\right)+8 i b^6 x^6-30 b^5 x^5-56 i b^4 x^4+95 b^3 x^3+224 i b^2 x^2+285 b x+448 i}{\sqrt{a^2+2 a b x+b^2 x^2+1}}}{40 b^5}","\frac{i (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1} \left(-112 i a^3-2 \left(-52 i a^2-118 a+61 i\right) b x-422 a^2+458 i a+163\right)}{40 b^5}+\frac{3 \left(8 i a^4+48 a^3-88 i a^2-68 a+19 i\right) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^5}-\frac{3 \left(8 a^4-48 i a^3-88 a^2+68 i a+19\right) \sinh ^{-1}(a+b x)}{8 b^5}-\frac{3 (-16 a+17 i) x^2 (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1}}{20 b^3}-\frac{11 x^3 (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1}}{5 b^2}+\frac{2 i x^4 (-i a-i b x+1)^{3/2}}{b \sqrt{i a+i b x+1}}",1,"((448*I + (8*I)*a^6 + 285*b*x + (224*I)*b^2*x^2 + 95*b^3*x^3 - (56*I)*b^4*x^4 - 30*b^5*x^5 + (8*I)*b^6*x^6 + a^5*(410 + (8*I)*b*x) + 2*a^4*(-638*I + 265*b*x) + a^3*(-905 - (2004*I)*b*x + 60*b^2*x^2) - a^2*(836*I + 2635*b*x + (356*I)*b^2*x^2 + 20*b^3*x^3) + a*(-1315 + (1468*I)*b*x - 515*b^2*x^2 + (116*I)*b^3*x^3 + 10*b^4*x^4 + (8*I)*b^5*x^5))/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (30*(-1)^(1/4)*(19*I - 68*a - (88*I)*a^2 + 48*a^3 + (8*I)*a^4)*Sqrt[b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/Sqrt[(-I)*b])/(40*b^5)","A",0
208,1,244,249,0.3897468,"\int e^{-3 i \tan ^{-1}(a+b x)} x^3 \, dx","Integrate[x^3/E^((3*I)*ArcTan[a + b*x]),x]","\frac{3 \sqrt[4]{-1} \left(8 a^3-36 i a^2-44 a+17 i\right) \sqrt{-i b} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{4 b^{9/2}}+\frac{-2 i a^5+a^4 (-76-2 i b x)-5 a^3 (20 b x-31 i)+a^2 \left(-12 b^2 x^2+265 i b x+4\right)+a \left(2 i b^4 x^4+4 b^3 x^3+53 i b^2 x^2+212 b x+157 i\right)+2 i b^5 x^5-8 b^4 x^4-17 i b^3 x^3+40 b^2 x^2-51 i b x+80}{8 b^4 \sqrt{a^2+2 a b x+b^2 x^2+1}}","-\frac{i (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1} \left(-22 i a^2+2 (11+10 i a) b x-54 a+29 i\right)}{8 b^4}+\frac{3 \left(-8 i a^3-36 a^2+44 i a+17\right) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^4}+\frac{3 \left(8 a^3-36 i a^2-44 a+17 i\right) \sinh ^{-1}(a+b x)}{8 b^4}-\frac{9 x^2 (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1}}{4 b^2}+\frac{2 i x^3 (-i a-i b x+1)^{3/2}}{b \sqrt{i a+i b x+1}}",1,"(80 - (2*I)*a^5 - (51*I)*b*x + 40*b^2*x^2 - (17*I)*b^3*x^3 - 8*b^4*x^4 + (2*I)*b^5*x^5 + a^4*(-76 - (2*I)*b*x) - 5*a^3*(-31*I + 20*b*x) + a^2*(4 + (265*I)*b*x - 12*b^2*x^2) + a*(157*I + 212*b*x + (53*I)*b^2*x^2 + 4*b^3*x^3 + (2*I)*b^4*x^4))/(8*b^4*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]) + (3*(-1)^(1/4)*(17*I - 44*a - (36*I)*a^2 + 8*a^3)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(4*b^(9/2))","A",0
209,1,198,229,0.3222918,"\int e^{-3 i \tan ^{-1}(a+b x)} x^2 \, dx","Integrate[x^2/E^((3*I)*ArcTan[a + b*x]),x]","\frac{\sqrt[4]{-1} \left(-6 a^2+18 i a+11\right) \sqrt{-i b} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{b^{7/2}}+\frac{2 i a^4+a^3 (51+2 i b x)+a^2 (69 b x-50 i)+a \left(2 i b^3 x^3+9 b^2 x^2-106 i b x+51\right)+i \left(2 b^4 x^4+9 i b^3 x^3-26 b^2 x^2+33 i b x-52\right)}{6 b^3 \sqrt{a^2+2 a b x+b^2 x^2+1}}","-\frac{\left(-6 i a^2-18 a+11 i\right) \sqrt{i a+i b x+1} (-i a-i b x+1)^{3/2}}{6 b^3}-\frac{\left(-6 i a^2-18 a+11 i\right) \sqrt{i a+i b x+1} \sqrt{-i a-i b x+1}}{2 b^3}+\frac{\left(-6 a^2+18 i a+11\right) \sinh ^{-1}(a+b x)}{2 b^3}-\frac{i \sqrt{i a+i b x+1} (-i a-i b x+1)^{5/2}}{3 b^3}+\frac{i (-a+i)^2 (-i a-i b x+1)^{5/2}}{b^3 \sqrt{i a+i b x+1}}",1,"((2*I)*a^4 + a^3*(51 + (2*I)*b*x) + a^2*(-50*I + 69*b*x) + a*(51 - (106*I)*b*x + 9*b^2*x^2 + (2*I)*b^3*x^3) + I*(-52 + (33*I)*b*x - 26*b^2*x^2 + (9*I)*b^3*x^3 + 2*b^4*x^4))/(6*b^3*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]) + ((-1)^(1/4)*(11 + (18*I)*a - 6*a^2)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/b^(7/2)","A",0
210,1,157,163,0.334909,"\int e^{-3 i \tan ^{-1}(a+b x)} x \, dx","Integrate[x/E^((3*I)*ArcTan[a + b*x]),x]","\frac{i \left(-a^3+a^2 (-b x+14 i)+a \left(b^2 x^2+20 i b x-1\right)+b^3 x^3+6 i b^2 x^2+9 b x+14 i\right)}{2 b^2 \sqrt{a^2+2 a b x+b^2 x^2+1}}+\frac{3 \sqrt[4]{-1} (2 a-3 i) \sqrt{-i b} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{b^{5/2}}","-\frac{(1+i a) (-i a-i b x+1)^{5/2}}{b^2 \sqrt{i a+i b x+1}}-\frac{(3+2 i a) \sqrt{i a+i b x+1} (-i a-i b x+1)^{3/2}}{2 b^2}-\frac{3 (3+2 i a) \sqrt{i a+i b x+1} \sqrt{-i a-i b x+1}}{2 b^2}-\frac{3 (-2 a+3 i) \sinh ^{-1}(a+b x)}{2 b^2}",1,"((I/2)*(14*I - a^3 + 9*b*x + (6*I)*b^2*x^2 + b^3*x^3 + a^2*(14*I - b*x) + a*(-1 + (20*I)*b*x + b^2*x^2)))/(b^2*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]) + (3*(-1)^(1/4)*(-3*I + 2*a)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/b^(5/2)","A",0
211,1,45,94,0.0430374,"\int e^{-3 i \tan ^{-1}(a+b x)} \, dx","Integrate[E^((-3*I)*ArcTan[a + b*x]),x]","-\frac{3 \sinh ^{-1}(a+b x)}{b}+\frac{\sqrt{(a+b x)^2+1} \left(\frac{4}{a+b x-i}+i\right)}{b}","\frac{2 i (-i a-i b x+1)^{3/2}}{b \sqrt{i a+i b x+1}}+\frac{3 i \sqrt{i a+i b x+1} \sqrt{-i a-i b x+1}}{b}-\frac{3 \sinh ^{-1}(a+b x)}{b}",1,"(Sqrt[1 + (a + b*x)^2]*(I + 4/(-I + a + b*x)))/b - (3*ArcSinh[a + b*x])/b","A",0
212,1,189,134,0.6576641,"\int \frac{e^{-3 i \tan ^{-1}(a+b x)}}{x} \, dx","Integrate[1/(E^((3*I)*ArcTan[a + b*x])*x),x]","\frac{2 \left(\frac{\sqrt{-1+i a} (a+i) \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)}{\sqrt{-1-i a}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2+1}}{a+b x-i}\right)}{a-i}+\frac{2 (-1)^{3/4} \sqrt{-i b} \sinh ^{-1}\left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right)}{\sqrt{b}}","\frac{4 \sqrt{-i a-i b x+1}}{(1+i a) \sqrt{i a+i b x+1}}+i \sinh ^{-1}(a+b x)-\frac{2 (a+i)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{(-a+i)^{3/2}}",1,"(2*(-1)^(3/4)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/Sqrt[b] + (2*((-2*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/(-I + a + b*x) + (Sqrt[-1 + I*a]*(I + a)*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/Sqrt[-1 - I*a]))/(-I + a)","A",0
213,1,145,178,0.1741686,"\int \frac{e^{-3 i \tan ^{-1}(a+b x)}}{x^2} \, dx","Integrate[1/(E^((3*I)*ArcTan[a + b*x])*x^2),x]","\frac{\frac{\sqrt{-i (a+b x+i)} \left(a^2+a b x+5 i b x+1\right)}{x \sqrt{i a+i b x+1}}-\frac{6 i \sqrt{-1+i a} b \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)}{\sqrt{-1-i a}}}{(a-i)^2}","-\frac{(-i a-i b x+1)^{3/2}}{(1+i a) x \sqrt{i a+i b x+1}}+\frac{6 i b \sqrt{-i a-i b x+1}}{(-a+i)^2 \sqrt{i a+i b x+1}}-\frac{6 i \sqrt{a+i} b \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{(-a+i)^{5/2}}",1,"((Sqrt[(-I)*(I + a + b*x)]*(1 + a^2 + (5*I)*b*x + a*b*x))/(x*Sqrt[1 + I*a + I*b*x]) - ((6*I)*Sqrt[-1 + I*a]*b*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/Sqrt[-1 - I*a])/(-I + a)^2","A",0
214,1,194,264,0.2687214,"\int \frac{e^{-3 i \tan ^{-1}(a+b x)}}{x^3} \, dx","Integrate[1/(E^((3*I)*ArcTan[a + b*x])*x^3),x]","\frac{\frac{\sqrt{-i (a+b x+i)} \left(a^3-i a^2-a b^2 x^2-5 i a b x+a-14 i b^2 x^2-5 b x-i\right)}{x^2 \sqrt{i a+i b x+1}}+\frac{6 i \sqrt{-1+i a} (2 a+3 i) b^2 \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)}{\sqrt{-1-i a} (a+i)}}{2 (a-i)^3}","-\frac{(-i a-i b x+1)^{5/2}}{2 \left(a^2+1\right) x^2 \sqrt{i a+i b x+1}}-\frac{3 (2 a+3 i) b^2 \sqrt{-i a-i b x+1}}{(1+i a)^3 (a+i) \sqrt{i a+i b x+1}}+\frac{3 (3-2 i a) b^2 \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{(-a+i)^{7/2} \sqrt{a+i}}+\frac{(3-2 i a) b (-i a-i b x+1)^{3/2}}{2 (-a+i)^2 (a+i) x \sqrt{i a+i b x+1}}",1,"((Sqrt[(-I)*(I + a + b*x)]*(-I + a - I*a^2 + a^3 - 5*b*x - (5*I)*a*b*x - (14*I)*b^2*x^2 - a*b^2*x^2))/(x^2*Sqrt[1 + I*a + I*b*x]) + ((6*I)*Sqrt[-1 + I*a]*(3*I + 2*a)*b^2*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/(Sqrt[-1 - I*a]*(I + a)))/(2*(-I + a)^3)","A",0
215,1,275,339,0.424023,"\int \frac{e^{-3 i \tan ^{-1}(a+b x)}}{x^4} \, dx","Integrate[1/(E^((3*I)*ArcTan[a + b*x])*x^4),x]","-\frac{i \left(6 a^2+18 i a-11\right) b^2 x^2 \left(-i \sqrt{-1-i a} \sqrt{-i (a+b x+i)} \left(a^2+a b x+5 i b x+1\right)-6 \sqrt{-1+i a} b x \sqrt{i a+i b x+1} \tanh ^{-1}\left(\frac{\sqrt{-1-i a} \sqrt{-i (a+b x+i)}}{\sqrt{-1+i a} \sqrt{i a+i b x+1}}\right)\right)-2 (1-i a) (-1-i a)^{7/2} (-i (a+b x+i))^{5/2}-(4 a+3 i) (-1-i a)^{5/2} b x (-i (a+b x+i))^{5/2}}{6 (-1-i a)^{5/2} \left(a^2+1\right)^2 x^3 \sqrt{i a+i b x+1}}","-\frac{\left(-2 a^2-51 i a+52\right) b^3 \sqrt{-i a-i b x+1}}{6 (-a+i)^4 (a+i) \sqrt{i a+i b x+1}}+\frac{\left(-6 i a^2+18 a+11 i\right) b^3 \tanh ^{-1}\left(\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right)}{(-a+i)^{9/2} (a+i)^{3/2}}+\frac{(19-16 i a) b^2 \sqrt{-i a-i b x+1}}{6 (-a+i)^3 (a+i) x \sqrt{i a+i b x+1}}-\frac{(a+i) \sqrt{-i a-i b x+1}}{3 (-a+i) x^3 \sqrt{i a+i b x+1}}-\frac{7 i b \sqrt{-i a-i b x+1}}{6 (-a+i)^2 x^2 \sqrt{i a+i b x+1}}",1,"-1/6*(-2*(-1 - I*a)^(7/2)*(1 - I*a)*((-I)*(I + a + b*x))^(5/2) - (-1 - I*a)^(5/2)*(3*I + 4*a)*b*x*((-I)*(I + a + b*x))^(5/2) + I*(-11 + (18*I)*a + 6*a^2)*b^2*x^2*((-I)*Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)]*(1 + a^2 + (5*I)*b*x + a*b*x) - 6*Sqrt[-1 + I*a]*b*x*Sqrt[1 + I*a + I*b*x]*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])]))/((-1 - I*a)^(5/2)*(1 + a^2)^2*x^3*Sqrt[1 + I*a + I*b*x])","A",0
216,1,121,494,0.0947501,"\int e^{\frac{1}{2} i \tan ^{-1}(a+b x)} x^2 \, dx","Integrate[E^((I/2)*ArcTan[a + b*x])*x^2,x]","\frac{(-i (a+b x+i))^{3/4} \left(2 i \sqrt[4]{2} \left(8 a^2+4 i a-3\right) \, _2F_1\left(-\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{1}{2} i (a+b x+i)\right)-i \sqrt[4]{i a+i b x+1} \left(8 a^2+a (4 b x-7 i)-4 b^2 x^2+5 i b x+1\right)\right)}{12 b^3}","-\frac{\left(-8 i a^2+4 a+3 i\right) (-i a-i b x+1)^{3/4} \sqrt[4]{i a+i b x+1}}{8 b^3}-\frac{\left(-8 i a^2+4 a+3 i\right) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{16 \sqrt{2} b^3}+\frac{\left(-8 i a^2+4 a+3 i\right) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{16 \sqrt{2} b^3}+\frac{\left(-8 i a^2+4 a+3 i\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{8 \sqrt{2} b^3}-\frac{\left(-8 i a^2+4 a+3 i\right) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{8 \sqrt{2} b^3}-\frac{(8 a+i) (-i a-i b x+1)^{3/4} (i a+i b x+1)^{5/4}}{12 b^3}+\frac{x (-i a-i b x+1)^{3/4} (i a+i b x+1)^{5/4}}{3 b^2}",1,"(((-I)*(I + a + b*x))^(3/4)*((-I)*(1 + I*a + I*b*x)^(1/4)*(1 + 8*a^2 + (5*I)*b*x - 4*b^2*x^2 + a*(-7*I + 4*b*x)) + (2*I)*2^(1/4)*(-3 + (4*I)*a + 8*a^2)*Hypergeometric2F1[-1/4, 3/4, 7/4, (-1/2*I)*(I + a + b*x)]))/(12*b^3)","C",0
217,1,81,410,0.0483235,"\int e^{\frac{1}{2} i \tan ^{-1}(a+b x)} x \, dx","Integrate[E^((I/2)*ArcTan[a + b*x])*x,x]","\frac{(-i (a+b x+i))^{3/4} \left(2 \sqrt[4]{2} (1-4 i a) \, _2F_1\left(-\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{1}{2} i (a+b x+i)\right)+3 (i a+i b x+1)^{5/4}\right)}{6 b^2}","\frac{(-i a-i b x+1)^{3/4} (i a+i b x+1)^{5/4}}{2 b^2}+\frac{(1-4 i a) (-i a-i b x+1)^{3/4} \sqrt[4]{i a+i b x+1}}{4 b^2}+\frac{(1-4 i a) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{8 \sqrt{2} b^2}-\frac{(1-4 i a) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{8 \sqrt{2} b^2}-\frac{(1-4 i a) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{4 \sqrt{2} b^2}+\frac{(1-4 i a) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{4 \sqrt{2} b^2}",1,"(((-I)*(I + a + b*x))^(3/4)*(3*(1 + I*a + I*b*x)^(5/4) + 2*2^(1/4)*(1 - (4*I)*a)*Hypergeometric2F1[-1/4, 3/4, 7/4, (-1/2*I)*(I + a + b*x)]))/(6*b^2)","C",0
218,1,45,338,0.0171578,"\int e^{\frac{1}{2} i \tan ^{-1}(a+b x)} \, dx","Integrate[E^((I/2)*ArcTan[a + b*x]),x]","-\frac{8 i e^{\frac{5}{2} i \tan ^{-1}(a+b x)} \, _2F_1\left(\frac{5}{4},2;\frac{9}{4};-e^{2 i \tan ^{-1}(a+b x)}\right)}{5 b}","\frac{i (-i a-i b x+1)^{3/4} \sqrt[4]{i a+i b x+1}}{b}+\frac{i \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{2 \sqrt{2} b}-\frac{i \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{2 \sqrt{2} b}-\frac{i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{\sqrt{2} b}+\frac{i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{\sqrt{2} b}",1,"(((-8*I)/5)*E^(((5*I)/2)*ArcTan[a + b*x])*Hypergeometric2F1[5/4, 2, 9/4, -E^((2*I)*ArcTan[a + b*x])])/b","C",0
219,1,124,395,0.1020851,"\int \frac{e^{\frac{1}{2} i \tan ^{-1}(a+b x)}}{x} \, dx","Integrate[E^((I/2)*ArcTan[a + b*x])/x,x]","\frac{2}{3} (-i (a+b x+i))^{3/4} \left(\frac{2 (a-i) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right)}{(a+i) (i a+i b x+1)^{3/4}}-\sqrt[4]{2} \, _2F_1\left(\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{1}{2} i (a+b x+i)\right)\right)","-\frac{\log \left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}-\frac{\sqrt{2} \sqrt[4]{1+i (a+b x)}}{\sqrt[4]{1-i (a+b x)}}+1\right)}{\sqrt{2}}+\frac{\log \left(\frac{\sqrt{1+i (a+b x)}}{\sqrt{1-i (a+b x)}}+\frac{\sqrt{2} \sqrt[4]{1+i (a+b x)}}{\sqrt[4]{1-i (a+b x)}}+1\right)}{\sqrt{2}}-\frac{2 \sqrt[4]{-a+i} \tan ^{-1}\left(\frac{\sqrt[4]{a+i} \sqrt[4]{1+i (a+b x)}}{\sqrt[4]{-a+i} \sqrt[4]{1-i (a+b x)}}\right)}{\sqrt[4]{a+i}}-\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1+i (a+b x)}}{\sqrt[4]{1-i (a+b x)}}\right)+\sqrt{2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1+i (a+b x)}}{\sqrt[4]{1-i (a+b x)}}\right)-\frac{2 \sqrt[4]{-a+i} \tanh ^{-1}\left(\frac{\sqrt[4]{a+i} \sqrt[4]{1+i (a+b x)}}{\sqrt[4]{-a+i} \sqrt[4]{1-i (a+b x)}}\right)}{\sqrt[4]{a+i}}",1,"(2*((-I)*(I + a + b*x))^(3/4)*(-(2^(1/4)*Hypergeometric2F1[3/4, 3/4, 7/4, (-1/2*I)*(I + a + b*x)]) + (2*(-I + a)*Hypergeometric2F1[3/4, 1, 7/4, (1 + a^2 - I*b*x + a*b*x)/(1 + a^2 + I*b*x + a*b*x)])/((I + a)*(1 + I*a + I*b*x)^(3/4))))/3","C",0
220,1,110,205,0.0338607,"\int \frac{e^{\frac{1}{2} i \tan ^{-1}(a+b x)}}{x^2} \, dx","Integrate[E^((I/2)*ArcTan[a + b*x])/x^2,x]","\frac{(-i (a+b x+i))^{3/4} \left(2 i b x \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right)+3 (a+i) (a+b x-i)\right)}{3 (a+i)^2 x (i a+i b x+1)^{3/4}}","-\frac{\sqrt[4]{1+i (a+b x)} (a+b x+i)}{(a+i) x \sqrt[4]{1-i (a+b x)}}+\frac{i b \tan ^{-1}\left(\frac{\sqrt[4]{a+i} \sqrt[4]{1+i (a+b x)}}{\sqrt[4]{-a+i} \sqrt[4]{1-i (a+b x)}}\right)}{(-a+i)^{3/4} (a+i)^{5/4}}+\frac{i b \tanh ^{-1}\left(\frac{\sqrt[4]{a+i} \sqrt[4]{1+i (a+b x)}}{\sqrt[4]{-a+i} \sqrt[4]{1-i (a+b x)}}\right)}{(-a+i)^{3/4} (a+i)^{5/4}}",1,"(((-I)*(I + a + b*x))^(3/4)*(3*(I + a)*(-I + a + b*x) + (2*I)*b*x*Hypergeometric2F1[3/4, 1, 7/4, (1 + a^2 - I*b*x + a*b*x)/(1 + a^2 + I*b*x + a*b*x)]))/(3*(I + a)^2*x*(1 + I*a + I*b*x)^(3/4))","C",0
221,1,121,494,0.0929362,"\int e^{\frac{3}{2} i \tan ^{-1}(a+b x)} x^2 \, dx","Integrate[E^(((3*I)/2)*ArcTan[a + b*x])*x^2,x]","\frac{\sqrt[4]{-i (a+b x+i)} \left(2 i 2^{3/4} \left(24 a^2+36 i a-17\right) \, _2F_1\left(-\frac{3}{4},\frac{1}{4};\frac{5}{4};-\frac{1}{2} i (a+b x+i)\right)-i (i a+i b x+1)^{3/4} \left(8 a^2+a (4 b x-5 i)-4 b^2 x^2+7 i b x+3\right)\right)}{12 b^3}","-\frac{\left(-24 i a^2+36 a+17 i\right) \sqrt[4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{24 b^3}+\frac{\left(-24 i a^2+36 a+17 i\right) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{16 \sqrt{2} b^3}-\frac{\left(-24 i a^2+36 a+17 i\right) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{16 \sqrt{2} b^3}+\frac{\left(-24 i a^2+36 a+17 i\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{8 \sqrt{2} b^3}-\frac{\left(-24 i a^2+36 a+17 i\right) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{8 \sqrt{2} b^3}-\frac{(8 a+3 i) \sqrt[4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{12 b^3}+\frac{x \sqrt[4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}",1,"(((-I)*(I + a + b*x))^(1/4)*((-I)*(1 + I*a + I*b*x)^(3/4)*(3 + 8*a^2 + (7*I)*b*x - 4*b^2*x^2 + a*(-5*I + 4*b*x)) + (2*I)*2^(3/4)*(-17 + (36*I)*a + 24*a^2)*Hypergeometric2F1[-3/4, 1/4, 5/4, (-1/2*I)*(I + a + b*x)]))/(12*b^3)","C",0
222,1,79,410,0.0471128,"\int e^{\frac{3}{2} i \tan ^{-1}(a+b x)} x \, dx","Integrate[E^(((3*I)/2)*ArcTan[a + b*x])*x,x]","\frac{\sqrt[4]{-i (a+b x+i)} \left(2\ 2^{3/4} (3-4 i a) \, _2F_1\left(-\frac{3}{4},\frac{1}{4};\frac{5}{4};-\frac{1}{2} i (a+b x+i)\right)+(i a+i b x+1)^{7/4}\right)}{2 b^2}","\frac{\sqrt[4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b^2}+\frac{(3-4 i a) \sqrt[4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{4 b^2}-\frac{3 (3-4 i a) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{8 \sqrt{2} b^2}+\frac{3 (3-4 i a) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{8 \sqrt{2} b^2}-\frac{3 (3-4 i a) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{4 \sqrt{2} b^2}+\frac{3 (3-4 i a) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{4 \sqrt{2} b^2}",1,"(((-I)*(I + a + b*x))^(1/4)*((1 + I*a + I*b*x)^(7/4) + 2*2^(3/4)*(3 - (4*I)*a)*Hypergeometric2F1[-3/4, 1/4, 5/4, (-1/2*I)*(I + a + b*x)]))/(2*b^2)","C",0
223,1,45,338,0.0194627,"\int e^{\frac{3}{2} i \tan ^{-1}(a+b x)} \, dx","Integrate[E^(((3*I)/2)*ArcTan[a + b*x]),x]","-\frac{8 i e^{\frac{7}{2} i \tan ^{-1}(a+b x)} \, _2F_1\left(\frac{7}{4},2;\frac{11}{4};-e^{2 i \tan ^{-1}(a+b x)}\right)}{7 b}","\frac{i \sqrt[4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{b}-\frac{3 i \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{2 \sqrt{2} b}+\frac{3 i \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{2 \sqrt{2} b}-\frac{3 i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{\sqrt{2} b}+\frac{3 i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{\sqrt{2} b}",1,"(((-8*I)/7)*E^(((7*I)/2)*ArcTan[a + b*x])*Hypergeometric2F1[7/4, 2, 11/4, -E^((2*I)*ArcTan[a + b*x])])/b","C",0
224,1,122,427,0.0955521,"\int \frac{e^{\frac{3}{2} i \tan ^{-1}(a+b x)}}{x} \, dx","Integrate[E^(((3*I)/2)*ArcTan[a + b*x])/x,x]","2 \sqrt[4]{-i (a+b x+i)} \left(\frac{2 (a-i) \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right)}{(a+i) \sqrt[4]{i a+i b x+1}}-2^{3/4} \, _2F_1\left(\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{1}{2} i (a+b x+i)\right)\right)","\frac{\log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{\sqrt{2}}-\frac{\log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{\sqrt{2}}+\frac{2 (-a+i)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a+i} \sqrt[4]{i a+i b x+1}}{\sqrt[4]{-a+i} \sqrt[4]{-i a-i b x+1}}\right)}{(a+i)^{3/4}}+\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)-\sqrt{2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)-\frac{2 (-a+i)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a+i} \sqrt[4]{i a+i b x+1}}{\sqrt[4]{-a+i} \sqrt[4]{-i a-i b x+1}}\right)}{(a+i)^{3/4}}",1,"2*((-I)*(I + a + b*x))^(1/4)*(-(2^(3/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (-1/2*I)*(I + a + b*x)]) + (2*(-I + a)*Hypergeometric2F1[1/4, 1, 5/4, (1 + a^2 - I*b*x + a*b*x)/(1 + a^2 + I*b*x + a*b*x)])/((I + a)*(1 + I*a + I*b*x)^(1/4)))","C",0
225,1,106,211,0.0280142,"\int \frac{e^{\frac{3}{2} i \tan ^{-1}(a+b x)}}{x^2} \, dx","Integrate[E^(((3*I)/2)*ArcTan[a + b*x])/x^2,x]","\frac{\sqrt[4]{-i (a+b x+i)} \left(6 i b x \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right)+a^2+a b x+i b x+1\right)}{(a+i)^2 x \sqrt[4]{i a+i b x+1}}","-\frac{\sqrt[4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{(1-i a) x}-\frac{3 i b \tan ^{-1}\left(\frac{\sqrt[4]{a+i} \sqrt[4]{i a+i b x+1}}{\sqrt[4]{-a+i} \sqrt[4]{-i a-i b x+1}}\right)}{\sqrt[4]{-a+i} (a+i)^{7/4}}+\frac{3 i b \tanh ^{-1}\left(\frac{\sqrt[4]{a+i} \sqrt[4]{i a+i b x+1}}{\sqrt[4]{-a+i} \sqrt[4]{-i a-i b x+1}}\right)}{\sqrt[4]{-a+i} (a+i)^{7/4}}",1,"(((-I)*(I + a + b*x))^(1/4)*(1 + a^2 + I*b*x + a*b*x + (6*I)*b*x*Hypergeometric2F1[1/4, 1, 5/4, (1 + a^2 - I*b*x + a*b*x)/(1 + a^2 + I*b*x + a*b*x)]))/((I + a)^2*x*(1 + I*a + I*b*x)^(1/4))","C",0
226,1,99,494,0.0971239,"\int e^{-\frac{1}{2} i \tan ^{-1}(a+b x)} x^2 \, dx","Integrate[x^2/E^((I/2)*ArcTan[a + b*x]),x]","\frac{(-i (a+b x+i))^{5/4} \left(3\ 2^{3/4} \left(8 i a^2+4 a-3 i\right) \, _2F_1\left(\frac{1}{4},\frac{5}{4};\frac{9}{4};-\frac{1}{2} i (a+b x+i)\right)+5 (i a+i b x+1)^{3/4} (-8 a+4 b x+i)\right)}{60 b^3}","\frac{\left(-8 i a^2-4 a+3 i\right) (i a+i b x+1)^{3/4} \sqrt[4]{-i a-i b x+1}}{8 b^3}+\frac{\left(-8 i a^2-4 a+3 i\right) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{16 \sqrt{2} b^3}-\frac{\left(-8 i a^2-4 a+3 i\right) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{16 \sqrt{2} b^3}+\frac{\left(-8 i a^2-4 a+3 i\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{8 \sqrt{2} b^3}-\frac{\left(-8 i a^2-4 a+3 i\right) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{8 \sqrt{2} b^3}+\frac{(-8 a+i) (i a+i b x+1)^{3/4} (-i a-i b x+1)^{5/4}}{12 b^3}+\frac{x (i a+i b x+1)^{3/4} (-i a-i b x+1)^{5/4}}{3 b^2}",1,"(((-I)*(I + a + b*x))^(5/4)*(5*(1 + I*a + I*b*x)^(3/4)*(I - 8*a + 4*b*x) + 3*2^(3/4)*(-3*I + 4*a + (8*I)*a^2)*Hypergeometric2F1[1/4, 5/4, 9/4, (-1/2*I)*(I + a + b*x)]))/(60*b^3)","C",0
227,1,84,410,0.0639833,"\int e^{-\frac{1}{2} i \tan ^{-1}(a+b x)} x \, dx","Integrate[x/E^((I/2)*ArcTan[a + b*x]),x]","-\frac{i (-i (a+b x+i))^{5/4} \left(2^{3/4} (4 a-i) \, _2F_1\left(\frac{1}{4},\frac{5}{4};\frac{9}{4};-\frac{1}{2} i (a+b x+i)\right)+5 i (i a+i b x+1)^{3/4}\right)}{10 b^2}","\frac{(i a+i b x+1)^{3/4} (-i a-i b x+1)^{5/4}}{2 b^2}+\frac{(1+4 i a) (i a+i b x+1)^{3/4} \sqrt[4]{-i a-i b x+1}}{4 b^2}+\frac{(1+4 i a) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{8 \sqrt{2} b^2}-\frac{(1+4 i a) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{8 \sqrt{2} b^2}+\frac{(1+4 i a) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{4 \sqrt{2} b^2}-\frac{(1+4 i a) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{4 \sqrt{2} b^2}",1,"((-1/10*I)*((-I)*(I + a + b*x))^(5/4)*((5*I)*(1 + I*a + I*b*x)^(3/4) + 2^(3/4)*(-I + 4*a)*Hypergeometric2F1[1/4, 5/4, 9/4, (-1/2*I)*(I + a + b*x)]))/b^2","C",0
228,1,45,338,0.0230933,"\int e^{-\frac{1}{2} i \tan ^{-1}(a+b x)} \, dx","Integrate[E^((-1/2*I)*ArcTan[a + b*x]),x]","-\frac{8 i e^{\frac{3}{2} i \tan ^{-1}(a+b x)} \, _2F_1\left(\frac{3}{4},2;\frac{7}{4};-e^{2 i \tan ^{-1}(a+b x)}\right)}{3 b}","-\frac{i \sqrt[4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{b}-\frac{i \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{2 \sqrt{2} b}+\frac{i \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{2 \sqrt{2} b}-\frac{i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{\sqrt{2} b}+\frac{i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{\sqrt{2} b}",1,"(((-8*I)/3)*E^(((3*I)/2)*ArcTan[a + b*x])*Hypergeometric2F1[3/4, 2, 7/4, -E^((2*I)*ArcTan[a + b*x])])/b","C",0
229,1,126,395,0.0399832,"\int \frac{e^{-\frac{1}{2} i \tan ^{-1}(a+b x)}}{x} \, dx","Integrate[1/(E^((I/2)*ArcTan[a + b*x])*x),x]","\frac{2 \sqrt[4]{-i (a+b x+i)} \left(2^{3/4} \sqrt[4]{i a+i b x+1} \, _2F_1\left(\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{1}{2} i (a+b x+i)\right)-2 \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right)\right)}{\sqrt[4]{i a+i b x+1}}","-\frac{\log \left(\frac{\sqrt{1-i (a+b x)}}{\sqrt{1+i (a+b x)}}-\frac{\sqrt{2} \sqrt[4]{1-i (a+b x)}}{\sqrt[4]{1+i (a+b x)}}+1\right)}{\sqrt{2}}+\frac{\log \left(\frac{\sqrt{1-i (a+b x)}}{\sqrt{1+i (a+b x)}}+\frac{\sqrt{2} \sqrt[4]{1-i (a+b x)}}{\sqrt[4]{1+i (a+b x)}}+1\right)}{\sqrt{2}}-\frac{2 \sqrt[4]{a+i} \tan ^{-1}\left(\frac{\sqrt[4]{-a+i} \sqrt[4]{1-i (a+b x)}}{\sqrt[4]{a+i} \sqrt[4]{1+i (a+b x)}}\right)}{\sqrt[4]{-a+i}}-\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{1-i (a+b x)}}{\sqrt[4]{1+i (a+b x)}}\right)+\sqrt{2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{1-i (a+b x)}}{\sqrt[4]{1+i (a+b x)}}\right)-\frac{2 \sqrt[4]{a+i} \tanh ^{-1}\left(\frac{\sqrt[4]{-a+i} \sqrt[4]{1-i (a+b x)}}{\sqrt[4]{a+i} \sqrt[4]{1+i (a+b x)}}\right)}{\sqrt[4]{-a+i}}",1,"(2*((-I)*(I + a + b*x))^(1/4)*(2^(3/4)*(1 + I*a + I*b*x)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (-1/2*I)*(I + a + b*x)] - 2*Hypergeometric2F1[1/4, 1, 5/4, (1 + a^2 - I*b*x + a*b*x)/(1 + a^2 + I*b*x + a*b*x)]))/(1 + I*a + I*b*x)^(1/4)","C",0
230,1,107,210,0.0315289,"\int \frac{e^{-\frac{1}{2} i \tan ^{-1}(a+b x)}}{x^2} \, dx","Integrate[1/(E^((I/2)*ArcTan[a + b*x])*x^2),x]","-\frac{\sqrt[4]{-i (a+b x+i)} \left(-2 i b x \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};\frac{a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right)+a^2+a b x+i b x+1\right)}{\left(a^2+1\right) x \sqrt[4]{i a+i b x+1}}","-\frac{\sqrt[4]{1-i (a+b x)} (-a-b x+i)}{(-a+i) x \sqrt[4]{1+i (a+b x)}}-\frac{i b \tan ^{-1}\left(\frac{\sqrt[4]{-a+i} \sqrt[4]{1-i (a+b x)}}{\sqrt[4]{a+i} \sqrt[4]{1+i (a+b x)}}\right)}{(-a+i)^{5/4} (a+i)^{3/4}}-\frac{i b \tanh ^{-1}\left(\frac{\sqrt[4]{-a+i} \sqrt[4]{1-i (a+b x)}}{\sqrt[4]{a+i} \sqrt[4]{1+i (a+b x)}}\right)}{(-a+i)^{5/4} (a+i)^{3/4}}",1,"-((((-I)*(I + a + b*x))^(1/4)*(1 + a^2 + I*b*x + a*b*x - (2*I)*b*x*Hypergeometric2F1[1/4, 1, 5/4, (1 + a^2 - I*b*x + a*b*x)/(1 + a^2 + I*b*x + a*b*x)]))/((1 + a^2)*x*(1 + I*a + I*b*x)^(1/4)))","C",0
231,1,98,494,0.0933496,"\int e^{-\frac{3}{2} i \tan ^{-1}(a+b x)} x^2 \, dx","Integrate[x^2/E^(((3*I)/2)*ArcTan[a + b*x]),x]","\frac{(-i (a+b x+i))^{7/4} \left(\sqrt[4]{2} \left(24 i a^2+36 a-17 i\right) \, _2F_1\left(\frac{3}{4},\frac{7}{4};\frac{11}{4};-\frac{1}{2} i (a+b x+i)\right)+7 \sqrt[4]{i a+i b x+1} (-8 a+4 b x+3 i)\right)}{84 b^3}","\frac{\left(-24 i a^2-36 a+17 i\right) \sqrt[4]{i a+i b x+1} (-i a-i b x+1)^{3/4}}{24 b^3}-\frac{\left(-24 i a^2-36 a+17 i\right) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{16 \sqrt{2} b^3}+\frac{\left(-24 i a^2-36 a+17 i\right) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{16 \sqrt{2} b^3}+\frac{\left(-24 i a^2-36 a+17 i\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{8 \sqrt{2} b^3}-\frac{\left(-24 i a^2-36 a+17 i\right) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{8 \sqrt{2} b^3}+\frac{(-8 a+3 i) \sqrt[4]{i a+i b x+1} (-i a-i b x+1)^{7/4}}{12 b^3}+\frac{x \sqrt[4]{i a+i b x+1} (-i a-i b x+1)^{7/4}}{3 b^2}",1,"(((-I)*(I + a + b*x))^(7/4)*(7*(1 + I*a + I*b*x)^(1/4)*(3*I - 8*a + 4*b*x) + 2^(1/4)*(-17*I + 36*a + (24*I)*a^2)*Hypergeometric2F1[3/4, 7/4, 11/4, (-1/2*I)*(I + a + b*x)]))/(84*b^3)","C",0
232,1,84,410,0.044722,"\int e^{-\frac{3}{2} i \tan ^{-1}(a+b x)} x \, dx","Integrate[x/E^(((3*I)/2)*ArcTan[a + b*x]),x]","-\frac{i (-i (a+b x+i))^{7/4} \left(\sqrt[4]{2} (4 a-3 i) \, _2F_1\left(\frac{3}{4},\frac{7}{4};\frac{11}{4};-\frac{1}{2} i (a+b x+i)\right)+7 i \sqrt[4]{i a+i b x+1}\right)}{14 b^2}","\frac{\sqrt[4]{i a+i b x+1} (-i a-i b x+1)^{7/4}}{2 b^2}+\frac{(3+4 i a) \sqrt[4]{i a+i b x+1} (-i a-i b x+1)^{3/4}}{4 b^2}-\frac{3 (3+4 i a) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{8 \sqrt{2} b^2}+\frac{3 (3+4 i a) \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{8 \sqrt{2} b^2}+\frac{3 (3+4 i a) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{4 \sqrt{2} b^2}-\frac{3 (3+4 i a) \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{4 \sqrt{2} b^2}",1,"((-1/14*I)*((-I)*(I + a + b*x))^(7/4)*((7*I)*(1 + I*a + I*b*x)^(1/4) + 2^(1/4)*(-3*I + 4*a)*Hypergeometric2F1[3/4, 7/4, 11/4, (-1/2*I)*(I + a + b*x)]))/b^2","C",0
233,1,43,338,0.0203817,"\int e^{-\frac{3}{2} i \tan ^{-1}(a+b x)} \, dx","Integrate[E^(((-3*I)/2)*ArcTan[a + b*x]),x]","-\frac{8 i e^{\frac{1}{2} i \tan ^{-1}(a+b x)} \, _2F_1\left(\frac{1}{4},2;\frac{5}{4};-e^{2 i \tan ^{-1}(a+b x)}\right)}{b}","-\frac{i (-i a-i b x+1)^{3/4} \sqrt[4]{i a+i b x+1}}{b}+\frac{3 i \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{2 \sqrt{2} b}-\frac{3 i \log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{2 \sqrt{2} b}-\frac{3 i \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{\sqrt{2} b}+\frac{3 i \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)}{\sqrt{2} b}",1,"((-8*I)*E^((I/2)*ArcTan[a + b*x])*Hypergeometric2F1[1/4, 2, 5/4, -E^((2*I)*ArcTan[a + b*x])])/b","C",0
234,1,128,427,0.0432205,"\int \frac{e^{-\frac{3}{2} i \tan ^{-1}(a+b x)}}{x} \, dx","Integrate[1/(E^(((3*I)/2)*ArcTan[a + b*x])*x),x]","\frac{2 (-i (a+b x+i))^{3/4} \left(\sqrt[4]{2} (i a+i b x+1)^{3/4} \, _2F_1\left(\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{1}{2} i (a+b x+i)\right)-2 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right)\right)}{3 (i a+i b x+1)^{3/4}}","\frac{\log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{\sqrt{2}}-\frac{\log \left(\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}+1\right)}{\sqrt{2}}-\frac{2 (a+i)^{3/4} \tan ^{-1}\left(\frac{\sqrt[4]{a+i} \sqrt[4]{i a+i b x+1}}{\sqrt[4]{-a+i} \sqrt[4]{-i a-i b x+1}}\right)}{(-a+i)^{3/4}}-\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)+\sqrt{2} \tan ^{-1}\left(1+\frac{\sqrt{2} \sqrt[4]{-i a-i b x+1}}{\sqrt[4]{i a+i b x+1}}\right)-\frac{2 (a+i)^{3/4} \tanh ^{-1}\left(\frac{\sqrt[4]{a+i} \sqrt[4]{i a+i b x+1}}{\sqrt[4]{-a+i} \sqrt[4]{-i a-i b x+1}}\right)}{(-a+i)^{3/4}}",1,"(2*((-I)*(I + a + b*x))^(3/4)*(2^(1/4)*(1 + I*a + I*b*x)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, (-1/2*I)*(I + a + b*x)] - 2*Hypergeometric2F1[3/4, 1, 7/4, (1 + a^2 - I*b*x + a*b*x)/(1 + a^2 + I*b*x + a*b*x)]))/(3*(1 + I*a + I*b*x)^(3/4))","C",0
235,1,107,211,0.0342757,"\int \frac{e^{-\frac{3}{2} i \tan ^{-1}(a+b x)}}{x^2} \, dx","Integrate[1/(E^(((3*I)/2)*ArcTan[a + b*x])*x^2),x]","-\frac{(-i (a+b x+i))^{3/4} \left(-2 i b x \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};\frac{a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right)+a^2+a b x+i b x+1\right)}{\left(a^2+1\right) x (i a+i b x+1)^{3/4}}","-\frac{(-i a-i b x+1)^{3/4} \sqrt[4]{i a+i b x+1}}{(1+i a) x}-\frac{3 i b \tan ^{-1}\left(\frac{\sqrt[4]{a+i} \sqrt[4]{i a+i b x+1}}{\sqrt[4]{-a+i} \sqrt[4]{-i a-i b x+1}}\right)}{(-a+i)^{7/4} \sqrt[4]{a+i}}-\frac{3 i b \tanh ^{-1}\left(\frac{\sqrt[4]{a+i} \sqrt[4]{i a+i b x+1}}{\sqrt[4]{-a+i} \sqrt[4]{-i a-i b x+1}}\right)}{(-a+i)^{7/4} \sqrt[4]{a+i}}",1,"-((((-I)*(I + a + b*x))^(3/4)*(1 + a^2 + I*b*x + a*b*x - (2*I)*b*x*Hypergeometric2F1[3/4, 1, 7/4, (1 + a^2 - I*b*x + a*b*x)/(1 + a^2 + I*b*x + a*b*x)]))/((1 + a^2)*x*(1 + I*a + I*b*x)^(3/4)))","C",0
236,0,0,140,0.9424453,"\int e^{n \tan ^{-1}(a+b x)} x^m \, dx","Integrate[E^(n*ArcTan[a + b*x])*x^m,x]","\int e^{n \tan ^{-1}(a+b x)} x^m \, dx","\frac{x^{m+1} (-i a-i b x+1)^{\frac{i n}{2}} (i a+i b x+1)^{-\frac{i n}{2}} \left(1-\frac{b x}{-a+i}\right)^{\frac{i n}{2}} \left(1+\frac{b x}{a+i}\right)^{-\frac{i n}{2}} F_1\left(m+1;-\frac{i n}{2},\frac{i n}{2};m+2;-\frac{b x}{a+i},\frac{b x}{i-a}\right)}{m+1}",1,"Integrate[E^(n*ArcTan[a + b*x])*x^m, x]","F",-1
237,1,272,260,0.3846032,"\int e^{n \tan ^{-1}(a+b x)} x^3 \, dx","Integrate[E^(n*ArcTan[a + b*x])*x^3,x]","\frac{(-i (a+b x+i))^{1+\frac{i n}{2}} \left(b^2 (-n+2 i) x^2 (i a+i b x+1)^{1-\frac{i n}{2}}-2^{3-\frac{i n}{2}} (6 a+n) \, _2F_1\left(\frac{i n}{2}-2,\frac{i n}{2}+1;\frac{i n}{2}+2;-\frac{1}{2} i (a+b x+i)\right)+(1+i a) 2^{3-\frac{i n}{2}} (5 a+n-i) \, _2F_1\left(\frac{i n}{2}-1,\frac{i n}{2}+1;\frac{i n}{2}+2;-\frac{1}{2} i (a+b x+i)\right)+(a-i)^2 2^{1-\frac{i n}{2}} (4 a+n-2 i) \, _2F_1\left(\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;-\frac{1}{2} i (a+b x+i)\right)\right)}{4 b^4 (-n+2 i)}","-\frac{(-i a-i b x+1)^{1+\frac{i n}{2}} \left(-18 a^2+2 b x (6 a+n)-10 a n-n^2+6\right) (i a+i b x+1)^{1-\frac{i n}{2}}}{24 b^4}+\frac{2^{-2-\frac{i n}{2}} \left(24 a^3+36 a^2 n-12 a \left(2-n^2\right)-n \left(8-n^2\right)\right) (-i a-i b x+1)^{1+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;\frac{1}{2} (-i a-i b x+1)\right)}{3 b^4 (-n+2 i)}+\frac{x^2 (-i a-i b x+1)^{1+\frac{i n}{2}} (i a+i b x+1)^{1-\frac{i n}{2}}}{4 b^2}",1,"(((-I)*(I + a + b*x))^(1 + (I/2)*n)*(b^2*(2*I - n)*x^2*(1 + I*a + I*b*x)^(1 - (I/2)*n) - 2^(3 - (I/2)*n)*(6*a + n)*Hypergeometric2F1[-2 + (I/2)*n, 1 + (I/2)*n, 2 + (I/2)*n, (-1/2*I)*(I + a + b*x)] + 2^(3 - (I/2)*n)*(1 + I*a)*(-I + 5*a + n)*Hypergeometric2F1[-1 + (I/2)*n, 1 + (I/2)*n, 2 + (I/2)*n, (-1/2*I)*(I + a + b*x)] + 2^(1 - (I/2)*n)*(-I + a)^2*(-2*I + 4*a + n)*Hypergeometric2F1[1 + (I/2)*n, (I/2)*n, 2 + (I/2)*n, (-1/2*I)*(I + a + b*x)]))/(4*b^4*(2*I - n))","A",0
238,1,160,220,0.1646585,"\int e^{n \tan ^{-1}(a+b x)} x^2 \, dx","Integrate[E^(n*ArcTan[a + b*x])*x^2,x]","\frac{(-i (a+b x+i))^{1+\frac{i n}{2}} \left(\frac{2^{1-\frac{i n}{2}} \left(6 a^2+6 a n+n^2-2\right) \, _2F_1\left(\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;-\frac{1}{2} i (a+b x+i)\right)}{n-2 i}-\left((4 a+n) (i a+i b x+1)^{1-\frac{i n}{2}}\right)+2 b x (i a+i b x+1)^{1-\frac{i n}{2}}\right)}{6 b^3}","\frac{2^{-\frac{i n}{2}} \left(-6 a^2-6 a n-n^2+2\right) (-i a-i b x+1)^{1+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;\frac{1}{2} (-i a-i b x+1)\right)}{3 b^3 (-n+2 i)}-\frac{(4 a+n) (-i a-i b x+1)^{1+\frac{i n}{2}} (i a+i b x+1)^{1-\frac{i n}{2}}}{6 b^3}+\frac{x (-i a-i b x+1)^{1+\frac{i n}{2}} (i a+i b x+1)^{1-\frac{i n}{2}}}{3 b^2}",1,"(((-I)*(I + a + b*x))^(1 + (I/2)*n)*(-((4*a + n)*(1 + I*a + I*b*x)^(1 - (I/2)*n)) + 2*b*x*(1 + I*a + I*b*x)^(1 - (I/2)*n) + (2^(1 - (I/2)*n)*(-2 + 6*a^2 + 6*a*n + n^2)*Hypergeometric2F1[1 + (I/2)*n, (I/2)*n, 2 + (I/2)*n, (-1/2*I)*(I + a + b*x)])/(-2*I + n)))/(6*b^3)","A",1
239,1,128,147,0.1318675,"\int e^{n \tan ^{-1}(a+b x)} x \, dx","Integrate[E^(n*ArcTan[a + b*x])*x,x]","\frac{i (-i (a+b x+i))^{1+\frac{i n}{2}} \left(\frac{2^{1-\frac{i n}{2}} (2 a+n) \, _2F_1\left(\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;-\frac{1}{2} i (a+b x+i)\right)}{-2-i n}+(a+b x-i) (i a+i b x+1)^{-\frac{i n}{2}}\right)}{2 b^2}","\frac{2^{-\frac{i n}{2}} (2 a+n) (-i a-i b x+1)^{1+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;\frac{1}{2} (-i a-i b x+1)\right)}{b^2 (-n+2 i)}+\frac{(-i a-i b x+1)^{1+\frac{i n}{2}} (i a+i b x+1)^{1-\frac{i n}{2}}}{2 b^2}",1,"((I/2)*((-I)*(I + a + b*x))^(1 + (I/2)*n)*((-I + a + b*x)/(1 + I*a + I*b*x)^((I/2)*n) + (2^(1 - (I/2)*n)*(2*a + n)*Hypergeometric2F1[1 + (I/2)*n, (I/2)*n, 2 + (I/2)*n, (-1/2*I)*(I + a + b*x)])/(-2 - I*n)))/b^2","A",1
240,1,60,91,0.0327549,"\int e^{n \tan ^{-1}(a+b x)} \, dx","Integrate[E^(n*ArcTan[a + b*x]),x]","\frac{4 e^{(n+2 i) \tan ^{-1}(a+b x)} \, _2F_1\left(2,1-\frac{i n}{2};2-\frac{i n}{2};-e^{2 i \tan ^{-1}(a+b x)}\right)}{b (n+2 i)}","-\frac{2^{1-\frac{i n}{2}} (-i a-i b x+1)^{1+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;\frac{1}{2} (-i a-i b x+1)\right)}{b (-n+2 i)}",1,"(4*E^((2*I + n)*ArcTan[a + b*x])*Hypergeometric2F1[2, 1 - (I/2)*n, 2 - (I/2)*n, -E^((2*I)*ArcTan[a + b*x])])/(b*(2*I + n))","A",0
241,1,170,191,0.0474254,"\int \frac{e^{n \tan ^{-1}(a+b x)}}{x} \, dx","Integrate[E^(n*ArcTan[a + b*x])/x,x]","\frac{2 i (i a+i b x+1)^{-\frac{i n}{2}} (-i (a+b x+i))^{\frac{i n}{2}} \left(\, _2F_1\left(1,\frac{i n}{2};\frac{i n}{2}+1;\frac{a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right)-2^{-\frac{i n}{2}} (i a+i b x+1)^{\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2},\frac{i n}{2};\frac{i n}{2}+1;-\frac{1}{2} i (a+b x+i)\right)\right)}{n}","\frac{2 i (-i a-i b x+1)^{\frac{i n}{2}} (i a+i b x+1)^{-\frac{i n}{2}} \, _2F_1\left(1,\frac{i n}{2};\frac{i n}{2}+1;\frac{(i-a) (-i a-i b x+1)}{(a+i) (i a+i b x+1)}\right)}{n}-\frac{i 2^{1-\frac{i n}{2}} (-i a-i b x+1)^{\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2},\frac{i n}{2};\frac{i n}{2}+1;\frac{1}{2} (-i a-i b x+1)\right)}{n}",1,"((2*I)*((-I)*(I + a + b*x))^((I/2)*n)*(Hypergeometric2F1[1, (I/2)*n, 1 + (I/2)*n, (1 + a^2 - I*b*x + a*b*x)/(1 + a^2 + I*b*x + a*b*x)] - ((1 + I*a + I*b*x)^((I/2)*n)*Hypergeometric2F1[(I/2)*n, (I/2)*n, 1 + (I/2)*n, (-1/2*I)*(I + a + b*x)])/2^((I/2)*n)))/(n*(1 + I*a + I*b*x)^((I/2)*n))","A",1
242,1,125,128,0.0262631,"\int \frac{e^{n \tan ^{-1}(a+b x)}}{x^2} \, dx","Integrate[E^(n*ArcTan[a + b*x])/x^2,x]","-\frac{4 i b (i a+i b x+1)^{-\frac{i n}{2}} (-i (a+b x+i))^{1+\frac{i n}{2}} \, _2F_1\left(2,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right)}{(a+i)^2 (n-2 i) (a+b x-i)}","-\frac{4 b (-i a-i b x+1)^{1+\frac{i n}{2}} (i a+i b x+1)^{-1-\frac{i n}{2}} \, _2F_1\left(2,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{(i-a) (-i a-i b x+1)}{(a+i) (i a+i b x+1)}\right)}{(a+i)^2 (-n+2 i)}",1,"((-4*I)*b*((-I)*(I + a + b*x))^(1 + (I/2)*n)*Hypergeometric2F1[2, 1 + (I/2)*n, 2 + (I/2)*n, (1 + a^2 - I*b*x + a*b*x)/(1 + a^2 + I*b*x + a*b*x)])/((I + a)^2*(-2*I + n)*(1 + I*a + I*b*x)^((I/2)*n)*(-I + a + b*x))","A",1
243,1,173,207,0.0811455,"\int \frac{e^{n \tan ^{-1}(a+b x)}}{x^3} \, dx","Integrate[E^(n*ArcTan[a + b*x])/x^3,x]","-\frac{i (i a+i b x+1)^{-\frac{i n}{2}} (-i (a+b x+i))^{1+\frac{i n}{2}} \left(4 b^2 x^2 (n-2 a) \, _2F_1\left(2,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right)+(a+i)^2 (n-2 i) (a+b x-i)^2\right)}{2 (a-i) (a+i)^3 (n-2 i) x^2 (a+b x-i)}","-\frac{(-i a-i b x+1)^{1+\frac{i n}{2}} (i a+i b x+1)^{1-\frac{i n}{2}}}{2 \left(a^2+1\right) x^2}-\frac{2 b^2 (2 a-n) (-i a-i b x+1)^{1+\frac{i n}{2}} (i a+i b x+1)^{-1-\frac{i n}{2}} \, _2F_1\left(2,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{(i-a) (-i a-i b x+1)}{(a+i) (i a+i b x+1)}\right)}{(-a+i) (a+i)^3 (-n+2 i)}",1,"((-1/2*I)*((-I)*(I + a + b*x))^(1 + (I/2)*n)*((I + a)^2*(-2*I + n)*(-I + a + b*x)^2 + 4*b^2*(-2*a + n)*x^2*Hypergeometric2F1[2, 1 + (I/2)*n, 2 + (I/2)*n, (1 + a^2 - I*b*x + a*b*x)/(1 + a^2 + I*b*x + a*b*x)]))/((-I + a)*(I + a)^3*(-2*I + n)*x^2*(1 + I*a + I*b*x)^((I/2)*n)*(-I + a + b*x))","A",1
244,1,102,102,0.0318613,"\int e^{\tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^p \, dx","Integrate[E^ArcTan[a*x]*(c + a^2*c*x^2)^p,x]","\frac{i 2^{p-\frac{i}{2}} (1-i a x)^{p+\left(1+\frac{i}{2}\right)} \left(a^2 x^2+1\right)^{-p} \left(a^2 c x^2+c\right)^p \, _2F_1\left(\frac{i}{2}-p,p+\left(1+\frac{i}{2}\right);p+\left(2+\frac{i}{2}\right);\frac{1}{2} (1-i a x)\right)}{a \left(p+\left(1+\frac{i}{2}\right)\right)}","\frac{i 2^{p+\left(1-\frac{i}{2}\right)} (1-i a x)^{p+\left(1+\frac{i}{2}\right)} \left(a^2 x^2+1\right)^{-p} \left(a^2 c x^2+c\right)^p \, _2F_1\left(\frac{i}{2}-p,p+\left(1+\frac{i}{2}\right);p+\left(2+\frac{i}{2}\right);\frac{1}{2} (1-i a x)\right)}{a (2 p+(2+i))}",1,"(I*2^(-1/2*I + p)*(1 - I*a*x)^((1 + I/2) + p)*(c + a^2*c*x^2)^p*Hypergeometric2F1[I/2 - p, (1 + I/2) + p, (2 + I/2) + p, (1 - I*a*x)/2])/(a*((1 + I/2) + p)*(1 + a^2*x^2)^p)","A",1
245,1,63,63,0.0153521,"\int e^{\tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^2 \, dx","Integrate[E^ArcTan[a*x]*(c + a^2*c*x^2)^2,x]","\frac{\left(\frac{1}{37}+\frac{6 i}{37}\right) 2^{3-\frac{i}{2}} c^2 (1-i a x)^{3+\frac{i}{2}} \, _2F_1\left(-2+\frac{i}{2},3+\frac{i}{2};4+\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a}","\frac{\left(\frac{1}{37}+\frac{6 i}{37}\right) 2^{3-\frac{i}{2}} c^2 (1-i a x)^{3+\frac{i}{2}} \, _2F_1\left(-2+\frac{i}{2},3+\frac{i}{2};4+\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a}",1,"((1/37 + (6*I)/37)*2^(3 - I/2)*c^2*(1 - I*a*x)^(3 + I/2)*Hypergeometric2F1[-2 + I/2, 3 + I/2, 4 + I/2, (1 - I*a*x)/2])/a","A",1
246,1,61,61,0.0126908,"\int e^{\tan ^{-1}(a x)} \left(c+a^2 c x^2\right) \, dx","Integrate[E^ArcTan[a*x]*(c + a^2*c*x^2),x]","\frac{\left(\frac{1}{17}+\frac{4 i}{17}\right) 2^{2-\frac{i}{2}} c (1-i a x)^{2+\frac{i}{2}} \, _2F_1\left(-1+\frac{i}{2},2+\frac{i}{2};3+\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a}","\frac{\left(\frac{1}{17}+\frac{4 i}{17}\right) 2^{2-\frac{i}{2}} c (1-i a x)^{2+\frac{i}{2}} \, _2F_1\left(-1+\frac{i}{2},2+\frac{i}{2};3+\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a}",1,"((1/17 + (4*I)/17)*2^(2 - I/2)*c*(1 - I*a*x)^(2 + I/2)*Hypergeometric2F1[-1 + I/2, 2 + I/2, 3 + I/2, (1 - I*a*x)/2])/a","A",1
247,1,45,60,0.0185967,"\int e^{\tan ^{-1}(a x)} \, dx","Integrate[E^ArcTan[a*x],x]","\frac{\left(\frac{4}{5}-\frac{8 i}{5}\right) e^{(1+2 i) \tan ^{-1}(a x)} \, _2F_1\left(1-\frac{i}{2},2;2-\frac{i}{2};-e^{2 i \tan ^{-1}(a x)}\right)}{a}","\frac{\left(\frac{1}{5}+\frac{2 i}{5}\right) 2^{1-\frac{i}{2}} (1-i a x)^{1+\frac{i}{2}} \, _2F_1\left(\frac{i}{2},1+\frac{i}{2};2+\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a}",1,"((4/5 - (8*I)/5)*E^((1 + 2*I)*ArcTan[a*x])*Hypergeometric2F1[1 - I/2, 2, 2 - I/2, -E^((2*I)*ArcTan[a*x])])/a","A",0
248,1,35,13,0.0076094,"\int \frac{e^{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Integrate[E^ArcTan[a*x]/(c + a^2*c*x^2),x]","\frac{(1-i a x)^{\frac{i}{2}} (1+i a x)^{-\frac{i}{2}}}{a c}","\frac{e^{\tan ^{-1}(a x)}}{a c}",1,"(1 - I*a*x)^(I/2)/(a*c*(1 + I*a*x)^(I/2))","C",0
249,1,60,50,0.0243622,"\int \frac{e^{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[E^ArcTan[a*x]/(c + a^2*c*x^2)^2,x]","\frac{(1-i a x)^{\frac{i}{2}} (1+i a x)^{-\frac{i}{2}} \left(2 a^2 x^2+2 a x+3\right)}{5 c^2 \left(a^3 x^2+a\right)}","\frac{(2 a x+1) e^{\tan ^{-1}(a x)}}{5 a c^2 \left(a^2 x^2+1\right)}+\frac{2 e^{\tan ^{-1}(a x)}}{5 a c^2}",1,"((1 - I*a*x)^(I/2)*(3 + 2*a*x + 2*a^2*x^2))/(5*c^2*(1 + I*a*x)^(I/2)*(a + a^3*x^2))","C",0
250,1,114,83,0.2851088,"\int \frac{e^{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[E^ArcTan[a*x]/(c + a^2*c*x^2)^3,x]","\frac{\frac{5 (4 a x+1) e^{\tan ^{-1}(a x)}}{\left(a^2 x^2+1\right)^2}+\frac{24 (1-i a x)^{\frac{i}{2}} (1+i a x)^{-\frac{i}{2}} (a x+(1-i))}{a x-i}+(12-24 i) (1-i a x)^{-1+\frac{i}{2}} (1+i a x)^{-1-\frac{i}{2}}}{85 a c^3}","\frac{12 (2 a x+1) e^{\tan ^{-1}(a x)}}{85 a c^3 \left(a^2 x^2+1\right)}+\frac{(4 a x+1) e^{\tan ^{-1}(a x)}}{17 a c^3 \left(a^2 x^2+1\right)^2}+\frac{24 e^{\tan ^{-1}(a x)}}{85 a c^3}",1,"((12 - 24*I)/((1 - I*a*x)^(1 - I/2)*(1 + I*a*x)^(1 + I/2)) + (24*(1 - I*a*x)^(I/2)*((1 - I) + a*x))/((1 + I*a*x)^(I/2)*(-I + a*x)) + (5*E^ArcTan[a*x]*(1 + 4*a*x))/(1 + a^2*x^2)^2)/(85*a*c^3)","C",0
251,1,123,116,0.3032097,"\int \frac{e^{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^4} \, dx","Integrate[E^ArcTan[a*x]/(c + a^2*c*x^2)^4,x]","\frac{17 c (6 a x+1) e^{\tan ^{-1}(a x)}+6 \left(a^2 c x^2+c\right) \left(5 (4 a x+1) e^{\tan ^{-1}(a x)}+12 (1-i a x)^{\frac{i}{2}} (1+i a x)^{-\frac{i}{2}} (a x-i) (a x+i) \left(2 a^2 x^2+2 a x+3\right)\right)}{629 a c^2 \left(a^2 c x^2+c\right)^3}","\frac{72 (2 a x+1) e^{\tan ^{-1}(a x)}}{629 a c^4 \left(a^2 x^2+1\right)}+\frac{30 (4 a x+1) e^{\tan ^{-1}(a x)}}{629 a c^4 \left(a^2 x^2+1\right)^2}+\frac{(6 a x+1) e^{\tan ^{-1}(a x)}}{37 a c^4 \left(a^2 x^2+1\right)^3}+\frac{144 e^{\tan ^{-1}(a x)}}{629 a c^4}",1,"(17*c*E^ArcTan[a*x]*(1 + 6*a*x) + 6*(c + a^2*c*x^2)*(5*E^ArcTan[a*x]*(1 + 4*a*x) + (12*(1 - I*a*x)^(I/2)*(-I + a*x)*(I + a*x)*(3 + 2*a*x + 2*a^2*x^2))/(1 + I*a*x)^(I/2)))/(629*a*c^2*(c + a^2*c*x^2)^3)","C",0
252,1,153,149,0.3236954,"\int \frac{e^{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^5} \, dx","Integrate[E^ArcTan[a*x]/(c + a^2*c*x^2)^5,x]","\frac{629 (8 a x+1) e^{\tan ^{-1}(a x)}+\frac{56 \left(a^2 c x^2+c\right) \left(17 c (6 a x+1) e^{\tan ^{-1}(a x)}+6 \left(a^2 c x^2+c\right) \left(5 (4 a x+1) e^{\tan ^{-1}(a x)}+12 (1-i a x)^{\frac{i}{2}} (1+i a x)^{-\frac{i}{2}} (a x-i) (a x+i) \left(2 a^2 x^2+2 a x+3\right)\right)\right)}{c^2}}{40885 a c \left(a^2 c x^2+c\right)^4}","\frac{4032 (2 a x+1) e^{\tan ^{-1}(a x)}}{40885 a c^5 \left(a^2 x^2+1\right)}+\frac{336 (4 a x+1) e^{\tan ^{-1}(a x)}}{8177 a c^5 \left(a^2 x^2+1\right)^2}+\frac{56 (6 a x+1) e^{\tan ^{-1}(a x)}}{2405 a c^5 \left(a^2 x^2+1\right)^3}+\frac{(8 a x+1) e^{\tan ^{-1}(a x)}}{65 a c^5 \left(a^2 x^2+1\right)^4}+\frac{8064 e^{\tan ^{-1}(a x)}}{40885 a c^5}",1,"(629*E^ArcTan[a*x]*(1 + 8*a*x) + (56*(c + a^2*c*x^2)*(17*c*E^ArcTan[a*x]*(1 + 6*a*x) + 6*(c + a^2*c*x^2)*(5*E^ArcTan[a*x]*(1 + 4*a*x) + (12*(1 - I*a*x)^(I/2)*(-I + a*x)*(I + a*x)*(3 + 2*a*x + 2*a^2*x^2))/(1 + I*a*x)^(I/2))))/c^2)/(40885*a*c*(c + a^2*c*x^2)^4)","C",0
253,1,98,98,0.0240928,"\int e^{\tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^{3/2} \, dx","Integrate[E^ArcTan[a*x]*(c + a^2*c*x^2)^(3/2),x]","\frac{\left(\frac{1}{13}+\frac{5 i}{13}\right) 2^{\frac{3}{2}-\frac{i}{2}} c (1-i a x)^{\frac{5}{2}+\frac{i}{2}} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{3}{2}+\frac{i}{2},\frac{5}{2}+\frac{i}{2};\frac{7}{2}+\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}","\frac{\left(\frac{1}{13}+\frac{5 i}{13}\right) 2^{\frac{3}{2}-\frac{i}{2}} c (1-i a x)^{\frac{5}{2}+\frac{i}{2}} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{3}{2}+\frac{i}{2},\frac{5}{2}+\frac{i}{2};\frac{7}{2}+\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}",1,"((1/13 + (5*I)/13)*2^(3/2 - I/2)*c*(1 - I*a*x)^(5/2 + I/2)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-3/2 + I/2, 5/2 + I/2, 7/2 + I/2, (1 - I*a*x)/2])/(a*Sqrt[1 + a^2*x^2])","A",1
254,1,97,97,0.0201,"\int e^{\tan ^{-1}(a x)} \sqrt{c+a^2 c x^2} \, dx","Integrate[E^ArcTan[a*x]*Sqrt[c + a^2*c*x^2],x]","\frac{\left(\frac{1}{5}+\frac{3 i}{5}\right) 2^{\frac{1}{2}-\frac{i}{2}} (1-i a x)^{\frac{3}{2}+\frac{i}{2}} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{1}{2}+\frac{i}{2},\frac{3}{2}+\frac{i}{2};\frac{5}{2}+\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}","\frac{\left(\frac{1}{5}+\frac{3 i}{5}\right) 2^{\frac{1}{2}-\frac{i}{2}} (1-i a x)^{\frac{3}{2}+\frac{i}{2}} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{1}{2}+\frac{i}{2},\frac{3}{2}+\frac{i}{2};\frac{5}{2}+\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}",1,"((1/5 + (3*I)/5)*2^(1/2 - I/2)*(1 - I*a*x)^(3/2 + I/2)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-1/2 + I/2, 3/2 + I/2, 5/2 + I/2, (1 - I*a*x)/2])/(a*Sqrt[1 + a^2*x^2])","A",1
255,1,93,93,0.0206043,"\int \frac{e^{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[E^ArcTan[a*x]/Sqrt[c + a^2*c*x^2],x]","\frac{(1+i) 2^{-\frac{1}{2}-\frac{i}{2}} (1-i a x)^{\frac{1}{2}+\frac{i}{2}} \sqrt{a^2 x^2+1} \, _2F_1\left(\frac{1}{2}+\frac{i}{2},\frac{1}{2}+\frac{i}{2};\frac{3}{2}+\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 c x^2+c}}","\frac{(1+i) 2^{-\frac{1}{2}-\frac{i}{2}} (1-i a x)^{\frac{1}{2}+\frac{i}{2}} \sqrt{a^2 x^2+1} \, _2F_1\left(\frac{1}{2}+\frac{i}{2},\frac{1}{2}+\frac{i}{2};\frac{3}{2}+\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 c x^2+c}}",1,"((1 + I)*(1 - I*a*x)^(1/2 + I/2)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2 + I/2, 1/2 + I/2, 3/2 + I/2, (1 - I*a*x)/2])/(2^(1/2 + I/2)*a*Sqrt[c + a^2*c*x^2])","A",1
256,1,35,35,0.0188068,"\int \frac{e^{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^ArcTan[a*x]/(c + a^2*c*x^2)^(3/2),x]","\frac{(a x+1) e^{\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}}","\frac{(a x+1) e^{\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}}",1,"(E^ArcTan[a*x]*(1 + a*x))/(2*a*c*Sqrt[c + a^2*c*x^2])","A",1
257,1,60,72,0.0379154,"\int \frac{e^{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[E^ArcTan[a*x]/(c + a^2*c*x^2)^(5/2),x]","\frac{\left(3 a^3 x^3+3 a^2 x^2+6 a x+4\right) e^{\tan ^{-1}(a x)}}{10 c^2 \left(a^3 x^2+a\right) \sqrt{a^2 c x^2+c}}","\frac{3 (a x+1) e^{\tan ^{-1}(a x)}}{10 a c^2 \sqrt{a^2 c x^2+c}}+\frac{(3 a x+1) e^{\tan ^{-1}(a x)}}{10 a c \left(a^2 c x^2+c\right)^{3/2}}",1,"(E^ArcTan[a*x]*(4 + 6*a*x + 3*a^2*x^2 + 3*a^3*x^3))/(10*c^2*(a + a^3*x^2)*Sqrt[c + a^2*c*x^2])","A",1
258,1,79,108,0.0398711,"\int \frac{e^{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{7/2}} \, dx","Integrate[E^ArcTan[a*x]/(c + a^2*c*x^2)^(7/2),x]","\frac{\left(6 a^5 x^5+6 a^4 x^4+18 a^3 x^3+14 a^2 x^2+17 a x+9\right) e^{\tan ^{-1}(a x)}}{26 a c^3 \left(a^2 x^2+1\right)^2 \sqrt{a^2 c x^2+c}}","\frac{3 (a x+1) e^{\tan ^{-1}(a x)}}{13 a c^3 \sqrt{a^2 c x^2+c}}+\frac{(3 a x+1) e^{\tan ^{-1}(a x)}}{13 a c^2 \left(a^2 c x^2+c\right)^{3/2}}+\frac{(5 a x+1) e^{\tan ^{-1}(a x)}}{26 a c \left(a^2 c x^2+c\right)^{5/2}}",1,"(E^ArcTan[a*x]*(9 + 17*a*x + 14*a^2*x^2 + 18*a^3*x^3 + 6*a^4*x^4 + 6*a^5*x^5))/(26*a*c^3*(1 + a^2*x^2)^2*Sqrt[c + a^2*c*x^2])","A",1
259,1,90,90,0.0291866,"\int e^{2 \tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^p \, dx","Integrate[E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^p,x]","\frac{i 2^{p-i} (1-i a x)^{p+(1+i)} \left(a^2 x^2+1\right)^{-p} \left(a^2 c x^2+c\right)^p \, _2F_1\left(i-p,p+(1+i);p+(2+i);\frac{1}{2} (1-i a x)\right)}{a (p+(1+i))}","\frac{i 2^{p-i} (1-i a x)^{p+(1+i)} \left(a^2 x^2+1\right)^{-p} \left(a^2 c x^2+c\right)^p \, _2F_1\left(i-p,p+(1+i);p+(2+i);\frac{1}{2} (1-i a x)\right)}{a (p+(1+i))}",1,"(I*2^(-I + p)*(1 - I*a*x)^((1 + I) + p)*(c + a^2*c*x^2)^p*Hypergeometric2F1[I - p, (1 + I) + p, (2 + I) + p, (1 - I*a*x)/2])/(a*((1 + I) + p)*(1 + a^2*x^2)^p)","A",1
260,1,53,53,0.0162182,"\int e^{2 \tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^2 \, dx","Integrate[E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^2,x]","\frac{\left(\frac{1}{5}+\frac{3 i}{5}\right) 2^{1-i} c^2 (1-i a x)^{3+i} \, _2F_1\left(-2+i,3+i;4+i;\frac{1}{2} (1-i a x)\right)}{a}","\frac{\left(\frac{1}{5}+\frac{3 i}{5}\right) 2^{1-i} c^2 (1-i a x)^{3+i} \, _2F_1\left(-2+i,3+i;4+i;\frac{1}{2} (1-i a x)\right)}{a}",1,"((1/5 + (3*I)/5)*2^(1 - I)*c^2*(1 - I*a*x)^(3 + I)*Hypergeometric2F1[-2 + I, 3 + I, 4 + I, (1 - I*a*x)/2])/a","A",1
261,1,51,51,0.0131416,"\int e^{2 \tan ^{-1}(a x)} \left(c+a^2 c x^2\right) \, dx","Integrate[E^(2*ArcTan[a*x])*(c + a^2*c*x^2),x]","\frac{\left(\frac{1}{5}+\frac{2 i}{5}\right) 2^{1-i} c (1-i a x)^{2+i} \, _2F_1\left(-1+i,2+i;3+i;\frac{1}{2} (1-i a x)\right)}{a}","\frac{\left(\frac{1}{5}+\frac{2 i}{5}\right) 2^{1-i} c (1-i a x)^{2+i} \, _2F_1\left(-1+i,2+i;3+i;\frac{1}{2} (1-i a x)\right)}{a}",1,"((1/5 + (2*I)/5)*2^(1 - I)*c*(1 - I*a*x)^(2 + I)*Hypergeometric2F1[-1 + I, 2 + I, 3 + I, (1 - I*a*x)/2])/a","A",1
262,1,37,46,0.0235335,"\int e^{2 \tan ^{-1}(a x)} \, dx","Integrate[E^(2*ArcTan[a*x]),x]","\frac{(1-i) e^{(2+2 i) \tan ^{-1}(a x)} \, _2F_1\left(1-i,2;2-i;-e^{2 i \tan ^{-1}(a x)}\right)}{a}","\frac{(1+i) 2^{-1-i} (1-i a x)^{1+i} \, _2F_1\left(i,1+i;2+i;\frac{1}{2} (1-i a x)\right)}{a}",1,"((1 - I)*E^((2 + 2*I)*ArcTan[a*x])*Hypergeometric2F1[1 - I, 2, 2 - I, -E^((2*I)*ArcTan[a*x])])/a","A",0
263,1,34,18,0.0081297,"\int \frac{e^{2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Integrate[E^(2*ArcTan[a*x])/(c + a^2*c*x^2),x]","\frac{(1-i a x)^i (1+i a x)^{-i}}{2 a c}","\frac{e^{2 \tan ^{-1}(a x)}}{2 a c}",1,"(1 - I*a*x)^I/(2*a*c*(1 + I*a*x)^I)","C",0
264,1,55,53,0.0224109,"\int \frac{e^{2 \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^2,x]","\frac{(1-i a x)^i (1+i a x)^{-i} \left(a^2 x^2+2 a x+3\right)}{8 c^2 \left(a^3 x^2+a\right)}","\frac{(a x+1) e^{2 \tan ^{-1}(a x)}}{4 a c^2 \left(a^2 x^2+1\right)}+\frac{e^{2 \tan ^{-1}(a x)}}{8 a c^2}",1,"((1 - I*a*x)^I*(3 + 2*a*x + a^2*x^2))/(8*c^2*(1 + I*a*x)^I*(a + a^3*x^2))","C",0
265,1,86,88,0.1368218,"\int \frac{e^{2 \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^3,x]","\frac{4 (2 a x+1) e^{2 \tan ^{-1}(a x)}+3 (1-i a x)^i (1+i a x)^{-i} \left(a^2 x^2+1\right) \left(a^2 x^2+2 a x+3\right)}{40 a c^3 \left(a^2 x^2+1\right)^2}","\frac{3 (a x+1) e^{2 \tan ^{-1}(a x)}}{20 a c^3 \left(a^2 x^2+1\right)}+\frac{(2 a x+1) e^{2 \tan ^{-1}(a x)}}{10 a c^3 \left(a^2 x^2+1\right)^2}+\frac{3 e^{2 \tan ^{-1}(a x)}}{40 a c^3}",1,"(4*E^(2*ArcTan[a*x])*(1 + 2*a*x) + (3*(1 - I*a*x)^I*(1 + a^2*x^2)*(3 + 2*a*x + a^2*x^2))/(1 + I*a*x)^I)/(40*a*c^3*(1 + a^2*x^2)^2)","C",0
266,1,122,123,0.2734657,"\int \frac{e^{2 \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^4} \, dx","Integrate[E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^4,x]","\frac{8 c (3 a x+1) e^{2 \tan ^{-1}(a x)}+3 \left(a^2 c x^2+c\right) \left(4 (2 a x+1) e^{2 \tan ^{-1}(a x)}+3 (1-i a x)^i (1+i a x)^{-i} (a x-i) (a x+i) \left(a^2 x^2+2 a x+3\right)\right)}{160 a c^2 \left(a^2 c x^2+c\right)^3}","\frac{9 (a x+1) e^{2 \tan ^{-1}(a x)}}{80 a c^4 \left(a^2 x^2+1\right)}+\frac{3 (2 a x+1) e^{2 \tan ^{-1}(a x)}}{40 a c^4 \left(a^2 x^2+1\right)^2}+\frac{(3 a x+1) e^{2 \tan ^{-1}(a x)}}{20 a c^4 \left(a^2 x^2+1\right)^3}+\frac{9 e^{2 \tan ^{-1}(a x)}}{160 a c^4}",1,"(8*c*E^(2*ArcTan[a*x])*(1 + 3*a*x) + 3*(c + a^2*c*x^2)*(4*E^(2*ArcTan[a*x])*(1 + 2*a*x) + (3*(1 - I*a*x)^I*(-I + a*x)*(I + a*x)*(3 + 2*a*x + a^2*x^2))/(1 + I*a*x)^I))/(160*a*c^2*(c + a^2*c*x^2)^3)","C",0
267,1,88,88,0.0240685,"\int e^{2 \tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^{3/2} \, dx","Integrate[E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^(3/2),x]","\frac{\left(\frac{2}{29}+\frac{5 i}{29}\right) 2^{\frac{5}{2}-i} c (1-i a x)^{\frac{5}{2}+i} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{3}{2}+i,\frac{5}{2}+i;\frac{7}{2}+i;\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}","\frac{\left(\frac{2}{29}+\frac{5 i}{29}\right) 2^{\frac{5}{2}-i} c (1-i a x)^{\frac{5}{2}+i} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{3}{2}+i,\frac{5}{2}+i;\frac{7}{2}+i;\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}",1,"((2/29 + (5*I)/29)*2^(5/2 - I)*c*(1 - I*a*x)^(5/2 + I)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-3/2 + I, 5/2 + I, 7/2 + I, (1 - I*a*x)/2])/(a*Sqrt[1 + a^2*x^2])","A",1
268,1,87,87,0.0218525,"\int e^{2 \tan ^{-1}(a x)} \sqrt{c+a^2 c x^2} \, dx","Integrate[E^(2*ArcTan[a*x])*Sqrt[c + a^2*c*x^2],x]","\frac{\left(\frac{2}{13}+\frac{3 i}{13}\right) 2^{\frac{3}{2}-i} (1-i a x)^{\frac{3}{2}+i} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{1}{2}+i,\frac{3}{2}+i;\frac{5}{2}+i;\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}","\frac{\left(\frac{2}{13}+\frac{3 i}{13}\right) 2^{\frac{3}{2}-i} (1-i a x)^{\frac{3}{2}+i} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{1}{2}+i,\frac{3}{2}+i;\frac{5}{2}+i;\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}",1,"((2/13 + (3*I)/13)*2^(3/2 - I)*(1 - I*a*x)^(3/2 + I)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-1/2 + I, 3/2 + I, 5/2 + I, (1 - I*a*x)/2])/(a*Sqrt[1 + a^2*x^2])","A",1
269,1,87,87,0.0197108,"\int \frac{e^{2 \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[E^(2*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","\frac{\left(\frac{2}{5}+\frac{i}{5}\right) 2^{\frac{1}{2}-i} (1-i a x)^{\frac{1}{2}+i} \sqrt{a^2 x^2+1} \, _2F_1\left(\frac{1}{2}+i,\frac{1}{2}+i;\frac{3}{2}+i;\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 c x^2+c}}","\frac{\left(\frac{2}{5}+\frac{i}{5}\right) 2^{\frac{1}{2}-i} (1-i a x)^{\frac{1}{2}+i} \sqrt{a^2 x^2+1} \, _2F_1\left(\frac{1}{2}+i,\frac{1}{2}+i;\frac{3}{2}+i;\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 c x^2+c}}",1,"((2/5 + I/5)*2^(1/2 - I)*(1 - I*a*x)^(1/2 + I)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2 + I, 1/2 + I, 3/2 + I, (1 - I*a*x)/2])/(a*Sqrt[c + a^2*c*x^2])","A",1
270,1,37,37,0.0183031,"\int \frac{e^{2 \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]","\frac{(a x+2) e^{2 \tan ^{-1}(a x)}}{5 a c \sqrt{a^2 c x^2+c}}","\frac{(a x+2) e^{2 \tan ^{-1}(a x)}}{5 a c \sqrt{a^2 c x^2+c}}",1,"(E^(2*ArcTan[a*x])*(2 + a*x))/(5*a*c*Sqrt[c + a^2*c*x^2])","A",1
271,1,62,76,0.036976,"\int \frac{e^{2 \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2),x]","\frac{\left(6 a^3 x^3+12 a^2 x^2+21 a x+22\right) e^{2 \tan ^{-1}(a x)}}{65 c^2 \left(a^3 x^2+a\right) \sqrt{a^2 c x^2+c}}","\frac{6 (a x+2) e^{2 \tan ^{-1}(a x)}}{65 a c^2 \sqrt{a^2 c x^2+c}}+\frac{(3 a x+2) e^{2 \tan ^{-1}(a x)}}{13 a c \left(a^2 c x^2+c\right)^{3/2}}",1,"(E^(2*ArcTan[a*x])*(22 + 21*a*x + 12*a^2*x^2 + 6*a^3*x^3))/(65*c^2*(a + a^3*x^2)*Sqrt[c + a^2*c*x^2])","A",1
272,1,81,114,0.0404583,"\int \frac{e^{2 \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{7/2}} \, dx","Integrate[E^(2*ArcTan[a*x])/(c + a^2*c*x^2)^(7/2),x]","\frac{\left(24 a^5 x^5+48 a^4 x^4+108 a^3 x^3+136 a^2 x^2+149 a x+114\right) e^{2 \tan ^{-1}(a x)}}{377 a c^3 \left(a^2 x^2+1\right)^2 \sqrt{a^2 c x^2+c}}","\frac{24 (a x+2) e^{2 \tan ^{-1}(a x)}}{377 a c^3 \sqrt{a^2 c x^2+c}}+\frac{20 (3 a x+2) e^{2 \tan ^{-1}(a x)}}{377 a c^2 \left(a^2 c x^2+c\right)^{3/2}}+\frac{(5 a x+2) e^{2 \tan ^{-1}(a x)}}{29 a c \left(a^2 c x^2+c\right)^{5/2}}",1,"(E^(2*ArcTan[a*x])*(114 + 149*a*x + 136*a^2*x^2 + 108*a^3*x^3 + 48*a^4*x^4 + 24*a^5*x^5))/(377*a*c^3*(1 + a^2*x^2)^2*Sqrt[c + a^2*c*x^2])","A",1
273,1,102,101,0.0280403,"\int e^{-\tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^p \, dx","Integrate[(c + a^2*c*x^2)^p/E^ArcTan[a*x],x]","\frac{i 2^{p+\frac{i}{2}} (1-i a x)^{p+\left(1-\frac{i}{2}\right)} \left(a^2 x^2+1\right)^{-p} \left(a^2 c x^2+c\right)^p \, _2F_1\left(-p-\frac{i}{2},p+\left(1-\frac{i}{2}\right);p+\left(2-\frac{i}{2}\right);\frac{1}{2} (1-i a x)\right)}{a \left(p+\left(1-\frac{i}{2}\right)\right)}","\frac{2^{p+\left(1+\frac{i}{2}\right)} (1-i a x)^{p+\left(1-\frac{i}{2}\right)} \left(a^2 x^2+1\right)^{-p} \left(a^2 c x^2+c\right)^p \, _2F_1\left(-p-\frac{i}{2},p+\left(1-\frac{i}{2}\right);p+\left(2-\frac{i}{2}\right);\frac{1}{2} (1-i a x)\right)}{a (-2 i p-(1+2 i))}",1,"(I*2^(I/2 + p)*(1 - I*a*x)^((1 - I/2) + p)*(c + a^2*c*x^2)^p*Hypergeometric2F1[-1/2*I - p, (1 - I/2) + p, (2 - I/2) + p, (1 - I*a*x)/2])/(a*((1 - I/2) + p)*(1 + a^2*x^2)^p)","A",1
274,1,63,63,0.0156677,"\int e^{-\tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^2 \, dx","Integrate[(c + a^2*c*x^2)^2/E^ArcTan[a*x],x]","-\frac{\left(\frac{1}{37}-\frac{6 i}{37}\right) 2^{3+\frac{i}{2}} c^2 (1-i a x)^{3-\frac{i}{2}} \, _2F_1\left(-2-\frac{i}{2},3-\frac{i}{2};4-\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a}","-\frac{\left(\frac{1}{37}-\frac{6 i}{37}\right) 2^{3+\frac{i}{2}} c^2 (1-i a x)^{3-\frac{i}{2}} \, _2F_1\left(-2-\frac{i}{2},3-\frac{i}{2};4-\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a}",1,"((-1/37 + (6*I)/37)*2^(3 + I/2)*c^2*(1 - I*a*x)^(3 - I/2)*Hypergeometric2F1[-2 - I/2, 3 - I/2, 4 - I/2, (1 - I*a*x)/2])/a","A",1
275,1,61,61,0.0138092,"\int e^{-\tan ^{-1}(a x)} \left(c+a^2 c x^2\right) \, dx","Integrate[(c + a^2*c*x^2)/E^ArcTan[a*x],x]","-\frac{\left(\frac{1}{17}-\frac{4 i}{17}\right) 2^{2+\frac{i}{2}} c (1-i a x)^{2-\frac{i}{2}} \, _2F_1\left(-1-\frac{i}{2},2-\frac{i}{2};3-\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a}","-\frac{\left(\frac{1}{17}-\frac{4 i}{17}\right) 2^{2+\frac{i}{2}} c (1-i a x)^{2-\frac{i}{2}} \, _2F_1\left(-1-\frac{i}{2},2-\frac{i}{2};3-\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a}",1,"((-1/17 + (4*I)/17)*2^(2 + I/2)*c*(1 - I*a*x)^(2 - I/2)*Hypergeometric2F1[-1 - I/2, 2 - I/2, 3 - I/2, (1 - I*a*x)/2])/a","A",1
276,1,45,60,0.0235825,"\int e^{-\tan ^{-1}(a x)} \, dx","Integrate[E^(-ArcTan[a*x]),x]","-\frac{\left(\frac{4}{5}+\frac{8 i}{5}\right) e^{(-1+2 i) \tan ^{-1}(a x)} \, _2F_1\left(1+\frac{i}{2},2;2+\frac{i}{2};-e^{2 i \tan ^{-1}(a x)}\right)}{a}","-\frac{\left(\frac{1}{5}-\frac{2 i}{5}\right) 2^{1+\frac{i}{2}} (1-i a x)^{1-\frac{i}{2}} \, _2F_1\left(-\frac{i}{2},1-\frac{i}{2};2-\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a}",1,"((-4/5 - (8*I)/5)*Hypergeometric2F1[1 + I/2, 2, 2 + I/2, -E^((2*I)*ArcTan[a*x])])/(a*E^((1 - 2*I)*ArcTan[a*x]))","A",0
277,1,36,16,0.0103587,"\int \frac{e^{-\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Integrate[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)),x]","-\frac{(1-i a x)^{-\frac{i}{2}} (1+i a x)^{\frac{i}{2}}}{a c}","-\frac{e^{-\tan ^{-1}(a x)}}{a c}",1,"-((1 + I*a*x)^(I/2)/(a*c*(1 - I*a*x)^(I/2)))","C",0
278,1,60,54,0.0263329,"\int \frac{e^{-\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)^2),x]","-\frac{(1-i a x)^{-\frac{i}{2}} (1+i a x)^{\frac{i}{2}} \left(2 a^2 x^2-2 a x+3\right)}{5 c^2 \left(a^3 x^2+a\right)}","-\frac{(1-2 a x) e^{-\tan ^{-1}(a x)}}{5 a c^2 \left(a^2 x^2+1\right)}-\frac{2 e^{-\tan ^{-1}(a x)}}{5 a c^2}",1,"-1/5*((1 + I*a*x)^(I/2)*(3 - 2*a*x + 2*a^2*x^2))/(c^2*(1 - I*a*x)^(I/2)*(a + a^3*x^2))","C",0
279,1,91,89,0.1759348,"\int \frac{e^{-\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)^3),x]","\frac{5 (4 a x-1) e^{-\tan ^{-1}(a x)}-12 (1-i a x)^{-\frac{i}{2}} (1+i a x)^{\frac{i}{2}} \left(a^2 x^2+1\right) \left(2 a^2 x^2-2 a x+3\right)}{85 a c^3 \left(a^2 x^2+1\right)^2}","-\frac{(1-4 a x) e^{-\tan ^{-1}(a x)}}{17 a c^3 \left(a^2 x^2+1\right)^2}-\frac{12 (1-2 a x) e^{-\tan ^{-1}(a x)}}{85 a c^3 \left(a^2 x^2+1\right)}-\frac{24 e^{-\tan ^{-1}(a x)}}{85 a c^3}",1,"((5*(-1 + 4*a*x))/E^ArcTan[a*x] - (12*(1 + I*a*x)^(I/2)*(1 + a^2*x^2)*(3 - 2*a*x + 2*a^2*x^2))/(1 - I*a*x)^(I/2))/(85*a*c^3*(1 + a^2*x^2)^2)","C",0
280,1,127,124,0.3893425,"\int \frac{e^{-\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^4} \, dx","Integrate[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)^4),x]","\frac{17 c (6 a x-1) e^{-\tan ^{-1}(a x)}-6 \left(a^2 c x^2+c\right) \left(5 (1-4 a x) e^{-\tan ^{-1}(a x)}+12 (1-i a x)^{-\frac{i}{2}} (1+i a x)^{\frac{i}{2}} (a x-i) (a x+i) \left(2 a^2 x^2-2 a x+3\right)\right)}{629 a c^2 \left(a^2 c x^2+c\right)^3}","-\frac{(1-6 a x) e^{-\tan ^{-1}(a x)}}{37 a c^4 \left(a^2 x^2+1\right)^3}-\frac{72 (1-2 a x) e^{-\tan ^{-1}(a x)}}{629 a c^4 \left(a^2 x^2+1\right)}-\frac{30 (1-4 a x) e^{-\tan ^{-1}(a x)}}{629 a c^4 \left(a^2 x^2+1\right)^2}-\frac{144 e^{-\tan ^{-1}(a x)}}{629 a c^4}",1,"((17*c*(-1 + 6*a*x))/E^ArcTan[a*x] - 6*(c + a^2*c*x^2)*((5*(1 - 4*a*x))/E^ArcTan[a*x] + (12*(1 + I*a*x)^(I/2)*(-I + a*x)*(I + a*x)*(3 - 2*a*x + 2*a^2*x^2))/(1 - I*a*x)^(I/2)))/(629*a*c^2*(c + a^2*c*x^2)^3)","C",0
281,1,98,98,0.0263022,"\int e^{-\tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^{3/2} \, dx","Integrate[(c + a^2*c*x^2)^(3/2)/E^ArcTan[a*x],x]","-\frac{\left(\frac{1}{13}-\frac{5 i}{13}\right) 2^{\frac{3}{2}+\frac{i}{2}} c (1-i a x)^{\frac{5}{2}-\frac{i}{2}} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{3}{2}-\frac{i}{2},\frac{5}{2}-\frac{i}{2};\frac{7}{2}-\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}","-\frac{\left(\frac{1}{13}-\frac{5 i}{13}\right) 2^{\frac{3}{2}+\frac{i}{2}} c (1-i a x)^{\frac{5}{2}-\frac{i}{2}} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{3}{2}-\frac{i}{2},\frac{5}{2}-\frac{i}{2};\frac{7}{2}-\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}",1,"((-1/13 + (5*I)/13)*2^(3/2 + I/2)*c*(1 - I*a*x)^(5/2 - I/2)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-3/2 - I/2, 5/2 - I/2, 7/2 - I/2, (1 - I*a*x)/2])/(a*Sqrt[1 + a^2*x^2])","A",1
282,1,97,97,0.0231769,"\int e^{-\tan ^{-1}(a x)} \sqrt{c+a^2 c x^2} \, dx","Integrate[Sqrt[c + a^2*c*x^2]/E^ArcTan[a*x],x]","-\frac{\left(\frac{1}{5}-\frac{3 i}{5}\right) 2^{\frac{1}{2}+\frac{i}{2}} (1-i a x)^{\frac{3}{2}-\frac{i}{2}} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{1}{2}-\frac{i}{2},\frac{3}{2}-\frac{i}{2};\frac{5}{2}-\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}","-\frac{\left(\frac{1}{5}-\frac{3 i}{5}\right) 2^{\frac{1}{2}+\frac{i}{2}} (1-i a x)^{\frac{3}{2}-\frac{i}{2}} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{1}{2}-\frac{i}{2},\frac{3}{2}-\frac{i}{2};\frac{5}{2}-\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}",1,"((-1/5 + (3*I)/5)*2^(1/2 + I/2)*(1 - I*a*x)^(3/2 - I/2)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-1/2 - I/2, 3/2 - I/2, 5/2 - I/2, (1 - I*a*x)/2])/(a*Sqrt[1 + a^2*x^2])","A",1
283,1,93,93,0.0261913,"\int \frac{e^{-\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[1/(E^ArcTan[a*x]*Sqrt[c + a^2*c*x^2]),x]","-\frac{(1-i) 2^{-\frac{1}{2}+\frac{i}{2}} (1-i a x)^{\frac{1}{2}-\frac{i}{2}} \sqrt{a^2 x^2+1} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},\frac{1}{2}-\frac{i}{2};\frac{3}{2}-\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 c x^2+c}}","-\frac{(1-i) 2^{-\frac{1}{2}+\frac{i}{2}} (1-i a x)^{\frac{1}{2}-\frac{i}{2}} \sqrt{a^2 x^2+1} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},\frac{1}{2}-\frac{i}{2};\frac{3}{2}-\frac{i}{2};\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 c x^2+c}}",1,"((-1 + I)*(1 - I*a*x)^(1/2 - I/2)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2 - I/2, 1/2 - I/2, 3/2 - I/2, (1 - I*a*x)/2])/(2^(1/2 - I/2)*a*Sqrt[c + a^2*c*x^2])","A",1
284,1,37,38,0.0247882,"\int \frac{e^{-\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)^(3/2)),x]","\frac{(a x-1) e^{-\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}}","-\frac{(1-a x) e^{-\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}}",1,"(-1 + a*x)/(2*a*c*E^ArcTan[a*x]*Sqrt[c + a^2*c*x^2])","A",1
285,1,62,77,0.0496158,"\int \frac{e^{-\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)^(5/2)),x]","\frac{\left(3 a^3 x^3-3 a^2 x^2+6 a x-4\right) e^{-\tan ^{-1}(a x)}}{10 c^2 \left(a^3 x^2+a\right) \sqrt{a^2 c x^2+c}}","-\frac{3 (1-a x) e^{-\tan ^{-1}(a x)}}{10 a c^2 \sqrt{a^2 c x^2+c}}-\frac{(1-3 a x) e^{-\tan ^{-1}(a x)}}{10 a c \left(a^2 c x^2+c\right)^{3/2}}",1,"(-4 + 6*a*x - 3*a^2*x^2 + 3*a^3*x^3)/(10*c^2*E^ArcTan[a*x]*(a + a^3*x^2)*Sqrt[c + a^2*c*x^2])","A",1
286,1,81,115,0.0512201,"\int \frac{e^{-\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{7/2}} \, dx","Integrate[1/(E^ArcTan[a*x]*(c + a^2*c*x^2)^(7/2)),x]","\frac{\left(6 a^5 x^5-6 a^4 x^4+18 a^3 x^3-14 a^2 x^2+17 a x-9\right) e^{-\tan ^{-1}(a x)}}{26 a c^3 \left(a^2 x^2+1\right)^2 \sqrt{a^2 c x^2+c}}","-\frac{3 (1-a x) e^{-\tan ^{-1}(a x)}}{13 a c^3 \sqrt{a^2 c x^2+c}}-\frac{(1-3 a x) e^{-\tan ^{-1}(a x)}}{13 a c^2 \left(a^2 c x^2+c\right)^{3/2}}-\frac{(1-5 a x) e^{-\tan ^{-1}(a x)}}{26 a c \left(a^2 c x^2+c\right)^{5/2}}",1,"(-9 + 17*a*x - 14*a^2*x^2 + 18*a^3*x^3 - 6*a^4*x^4 + 6*a^5*x^5)/(26*a*c^3*E^ArcTan[a*x]*(1 + a^2*x^2)^2*Sqrt[c + a^2*c*x^2])","A",1
287,1,90,90,0.0271935,"\int e^{-2 \tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^p \, dx","Integrate[(c + a^2*c*x^2)^p/E^(2*ArcTan[a*x]),x]","\frac{i 2^{p+i} (1-i a x)^{p+(1-i)} \left(a^2 x^2+1\right)^{-p} \left(a^2 c x^2+c\right)^p \, _2F_1\left(-p-i,p+(1-i);p+(2-i);\frac{1}{2} (1-i a x)\right)}{a (p+(1-i))}","\frac{i 2^{p+i} (1-i a x)^{p+(1-i)} \left(a^2 x^2+1\right)^{-p} \left(a^2 c x^2+c\right)^p \, _2F_1\left(-p-i,p+(1-i);p+(2-i);\frac{1}{2} (1-i a x)\right)}{a (p+(1-i))}",1,"(I*2^(I + p)*(1 - I*a*x)^((1 - I) + p)*(c + a^2*c*x^2)^p*Hypergeometric2F1[-I - p, (1 - I) + p, (2 - I) + p, (1 - I*a*x)/2])/(a*((1 - I) + p)*(1 + a^2*x^2)^p)","A",1
288,1,53,53,0.0167506,"\int e^{-2 \tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^2 \, dx","Integrate[(c + a^2*c*x^2)^2/E^(2*ArcTan[a*x]),x]","-\frac{\left(\frac{1}{5}-\frac{3 i}{5}\right) 2^{1+i} c^2 (1-i a x)^{3-i} \, _2F_1\left(-2-i,3-i;4-i;\frac{1}{2} (1-i a x)\right)}{a}","-\frac{\left(\frac{1}{5}-\frac{3 i}{5}\right) 2^{1+i} c^2 (1-i a x)^{3-i} \, _2F_1\left(-2-i,3-i;4-i;\frac{1}{2} (1-i a x)\right)}{a}",1,"((-1/5 + (3*I)/5)*2^(1 + I)*c^2*(1 - I*a*x)^(3 - I)*Hypergeometric2F1[-2 - I, 3 - I, 4 - I, (1 - I*a*x)/2])/a","A",1
289,1,51,51,0.0147004,"\int e^{-2 \tan ^{-1}(a x)} \left(c+a^2 c x^2\right) \, dx","Integrate[(c + a^2*c*x^2)/E^(2*ArcTan[a*x]),x]","-\frac{\left(\frac{1}{5}-\frac{2 i}{5}\right) 2^{1+i} c (1-i a x)^{2-i} \, _2F_1\left(-1-i,2-i;3-i;\frac{1}{2} (1-i a x)\right)}{a}","-\frac{\left(\frac{1}{5}-\frac{2 i}{5}\right) 2^{1+i} c (1-i a x)^{2-i} \, _2F_1\left(-1-i,2-i;3-i;\frac{1}{2} (1-i a x)\right)}{a}",1,"((-1/5 + (2*I)/5)*2^(1 + I)*c*(1 - I*a*x)^(2 - I)*Hypergeometric2F1[-1 - I, 2 - I, 3 - I, (1 - I*a*x)/2])/a","A",1
290,1,37,46,0.0234641,"\int e^{-2 \tan ^{-1}(a x)} \, dx","Integrate[E^(-2*ArcTan[a*x]),x]","-\frac{(1+i) e^{(-2+2 i) \tan ^{-1}(a x)} \, _2F_1\left(1+i,2;2+i;-e^{2 i \tan ^{-1}(a x)}\right)}{a}","-\frac{(1-i) 2^{-1+i} (1-i a x)^{1-i} \, _2F_1\left(-i,1-i;2-i;\frac{1}{2} (1-i a x)\right)}{a}",1,"((-1 - I)*Hypergeometric2F1[1 + I, 2, 2 + I, -E^((2*I)*ArcTan[a*x])])/(a*E^((2 - 2*I)*ArcTan[a*x]))","A",0
291,1,34,18,0.0093282,"\int \frac{e^{-2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Integrate[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)),x]","-\frac{(1-i a x)^{-i} (1+i a x)^i}{2 a c}","-\frac{e^{-2 \tan ^{-1}(a x)}}{2 a c}",1,"-1/2*(1 + I*a*x)^I/(a*c*(1 - I*a*x)^I)","C",0
292,1,55,54,0.0249061,"\int \frac{e^{-2 \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^2),x]","-\frac{(1-i a x)^{-i} (1+i a x)^i \left(a^2 x^2-2 a x+3\right)}{8 c^2 \left(a^3 x^2+a\right)}","-\frac{(1-a x) e^{-2 \tan ^{-1}(a x)}}{4 a c^2 \left(a^2 x^2+1\right)}-\frac{e^{-2 \tan ^{-1}(a x)}}{8 a c^2}",1,"-1/8*((1 + I*a*x)^I*(3 - 2*a*x + a^2*x^2))/(c^2*(1 - I*a*x)^I*(a + a^3*x^2))","C",0
293,1,85,89,0.1745469,"\int \frac{e^{-2 \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^3),x]","\frac{(8 a x-4) e^{-2 \tan ^{-1}(a x)}-3 (1-i a x)^{-i} (1+i a x)^i \left(a^2 x^2+1\right) \left(a^2 x^2-2 a x+3\right)}{40 a c^3 \left(a^2 x^2+1\right)^2}","-\frac{(1-2 a x) e^{-2 \tan ^{-1}(a x)}}{10 a c^3 \left(a^2 x^2+1\right)^2}-\frac{3 (1-a x) e^{-2 \tan ^{-1}(a x)}}{20 a c^3 \left(a^2 x^2+1\right)}-\frac{3 e^{-2 \tan ^{-1}(a x)}}{40 a c^3}",1,"((-4 + 8*a*x)/E^(2*ArcTan[a*x]) - (3*(1 + I*a*x)^I*(1 + a^2*x^2)*(3 - 2*a*x + a^2*x^2))/(1 - I*a*x)^I)/(40*a*c^3*(1 + a^2*x^2)^2)","C",0
294,1,121,124,0.3691176,"\int \frac{e^{-2 \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^4} \, dx","Integrate[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^4),x]","\frac{8 c (3 a x-1) e^{-2 \tan ^{-1}(a x)}-3 \left(a^2 c x^2+c\right) \left((4-8 a x) e^{-2 \tan ^{-1}(a x)}+3 (1-i a x)^{-i} (1+i a x)^i (a x-i) (a x+i) \left(a^2 x^2-2 a x+3\right)\right)}{160 a c^2 \left(a^2 c x^2+c\right)^3}","-\frac{(1-3 a x) e^{-2 \tan ^{-1}(a x)}}{20 a c^4 \left(a^2 x^2+1\right)^3}-\frac{9 (1-a x) e^{-2 \tan ^{-1}(a x)}}{80 a c^4 \left(a^2 x^2+1\right)}-\frac{3 (1-2 a x) e^{-2 \tan ^{-1}(a x)}}{40 a c^4 \left(a^2 x^2+1\right)^2}-\frac{9 e^{-2 \tan ^{-1}(a x)}}{160 a c^4}",1,"((8*c*(-1 + 3*a*x))/E^(2*ArcTan[a*x]) - 3*(c + a^2*c*x^2)*((4 - 8*a*x)/E^(2*ArcTan[a*x]) + (3*(1 + I*a*x)^I*(-I + a*x)*(I + a*x)*(3 - 2*a*x + a^2*x^2))/(1 - I*a*x)^I))/(160*a*c^2*(c + a^2*c*x^2)^3)","C",0
295,1,88,88,0.0275024,"\int e^{-2 \tan ^{-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*ArcTan[a*x]),x]","-\frac{\left(\frac{2}{29}-\frac{5 i}{29}\right) 2^{\frac{5}{2}+i} c (1-i a x)^{\frac{5}{2}-i} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{3}{2}-i,\frac{5}{2}-i;\frac{7}{2}-i;\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}","-\frac{\left(\frac{2}{29}-\frac{5 i}{29}\right) 2^{\frac{5}{2}+i} c (1-i a x)^{\frac{5}{2}-i} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{3}{2}-i,\frac{5}{2}-i;\frac{7}{2}-i;\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}",1,"((-2/29 + (5*I)/29)*2^(5/2 + I)*c*(1 - I*a*x)^(5/2 - I)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-3/2 - I, 5/2 - I, 7/2 - I, (1 - I*a*x)/2])/(a*Sqrt[1 + a^2*x^2])","A",1
296,1,87,87,0.0206123,"\int e^{-2 \tan ^{-1}(a x)} \sqrt{c+a^2 c x^2} \, dx","Integrate[Sqrt[c + a^2*c*x^2]/E^(2*ArcTan[a*x]),x]","-\frac{\left(\frac{2}{13}-\frac{3 i}{13}\right) 2^{\frac{3}{2}+i} (1-i a x)^{\frac{3}{2}-i} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{1}{2}-i,\frac{3}{2}-i;\frac{5}{2}-i;\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}","-\frac{\left(\frac{2}{13}-\frac{3 i}{13}\right) 2^{\frac{3}{2}+i} (1-i a x)^{\frac{3}{2}-i} \sqrt{a^2 c x^2+c} \, _2F_1\left(-\frac{1}{2}-i,\frac{3}{2}-i;\frac{5}{2}-i;\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 x^2+1}}",1,"((-2/13 + (3*I)/13)*2^(3/2 + I)*(1 - I*a*x)^(3/2 - I)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[-1/2 - I, 3/2 - I, 5/2 - I, (1 - I*a*x)/2])/(a*Sqrt[1 + a^2*x^2])","A",1
297,1,87,87,0.0217379,"\int \frac{e^{-2 \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[1/(E^(2*ArcTan[a*x])*Sqrt[c + a^2*c*x^2]),x]","-\frac{\left(\frac{2}{5}-\frac{i}{5}\right) 2^{\frac{1}{2}+i} (1-i a x)^{\frac{1}{2}-i} \sqrt{a^2 x^2+1} \, _2F_1\left(\frac{1}{2}-i,\frac{1}{2}-i;\frac{3}{2}-i;\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 c x^2+c}}","-\frac{\left(\frac{2}{5}-\frac{i}{5}\right) 2^{\frac{1}{2}+i} (1-i a x)^{\frac{1}{2}-i} \sqrt{a^2 x^2+1} \, _2F_1\left(\frac{1}{2}-i,\frac{1}{2}-i;\frac{3}{2}-i;\frac{1}{2} (1-i a x)\right)}{a \sqrt{a^2 c x^2+c}}",1,"((-2/5 + I/5)*2^(1/2 + I)*(1 - I*a*x)^(1/2 - I)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2 - I, 1/2 - I, 3/2 - I, (1 - I*a*x)/2])/(a*Sqrt[c + a^2*c*x^2])","A",1
298,1,37,38,0.0241008,"\int \frac{e^{-2 \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^(3/2)),x]","\frac{(a x-2) e^{-2 \tan ^{-1}(a x)}}{5 a c \sqrt{a^2 c x^2+c}}","-\frac{(2-a x) e^{-2 \tan ^{-1}(a x)}}{5 a c \sqrt{a^2 c x^2+c}}",1,"(-2 + a*x)/(5*a*c*E^(2*ArcTan[a*x])*Sqrt[c + a^2*c*x^2])","A",1
299,1,62,77,0.0517427,"\int \frac{e^{-2 \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^(5/2)),x]","\frac{\left(6 a^3 x^3-12 a^2 x^2+21 a x-22\right) e^{-2 \tan ^{-1}(a x)}}{65 c^2 \left(a^3 x^2+a\right) \sqrt{a^2 c x^2+c}}","-\frac{6 (2-a x) e^{-2 \tan ^{-1}(a x)}}{65 a c^2 \sqrt{a^2 c x^2+c}}-\frac{(2-3 a x) e^{-2 \tan ^{-1}(a x)}}{13 a c \left(a^2 c x^2+c\right)^{3/2}}",1,"(-22 + 21*a*x - 12*a^2*x^2 + 6*a^3*x^3)/(65*c^2*E^(2*ArcTan[a*x])*(a + a^3*x^2)*Sqrt[c + a^2*c*x^2])","A",1
300,1,81,115,0.0562229,"\int \frac{e^{-2 \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{7/2}} \, dx","Integrate[1/(E^(2*ArcTan[a*x])*(c + a^2*c*x^2)^(7/2)),x]","\frac{\left(24 a^5 x^5-48 a^4 x^4+108 a^3 x^3-136 a^2 x^2+149 a x-114\right) e^{-2 \tan ^{-1}(a x)}}{377 a c^3 \left(a^2 x^2+1\right)^2 \sqrt{a^2 c x^2+c}}","-\frac{24 (2-a x) e^{-2 \tan ^{-1}(a x)}}{377 a c^3 \sqrt{a^2 c x^2+c}}-\frac{20 (2-3 a x) e^{-2 \tan ^{-1}(a x)}}{377 a c^2 \left(a^2 c x^2+c\right)^{3/2}}-\frac{(2-5 a x) e^{-2 \tan ^{-1}(a x)}}{29 a c \left(a^2 c x^2+c\right)^{5/2}}",1,"(-114 + 149*a*x - 136*a^2*x^2 + 108*a^3*x^3 - 48*a^4*x^4 + 24*a^5*x^5)/(377*a*c^3*E^(2*ArcTan[a*x])*(1 + a^2*x^2)^2*Sqrt[c + a^2*c*x^2])","A",1
301,1,42,50,0.0251829,"\int \frac{e^{5 i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx","Integrate[E^((5*I)*ArcTan[a*x])/Sqrt[1 + a^2*x^2],x]","\frac{i \left(4 i a x+(a x+i)^2 \log (a x+i)-2\right)}{a (a x+i)^2}","\frac{4 i}{a (1-i a x)}-\frac{2 i}{a (1-i a x)^2}+\frac{i \log (a x+i)}{a}",1,"(I*(-2 + (4*I)*a*x + (I + a*x)^2*Log[I + a*x]))/(a*(I + a*x)^2)","A",1
302,1,48,73,0.0138901,"\int \frac{e^{4 i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx","Integrate[E^((4*I)*ArcTan[a*x])/Sqrt[1 + a^2*x^2],x]","-\frac{4 i \sqrt{2} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{1}{2} (1-i a x)\right)}{3 a (1-i a x)^{3/2}}","-\frac{2 i (1+i a x)^{3/2}}{3 a (1-i a x)^{3/2}}+\frac{2 i \sqrt{1+i a x}}{a \sqrt{1-i a x}}+\frac{\sinh ^{-1}(a x)}{a}",1,"(((-4*I)/3)*Sqrt[2]*Hypergeometric2F1[-3/2, -3/2, -1/2, (1 - I*a*x)/2])/(a*(1 - I*a*x)^(3/2))","C",0
303,1,30,30,0.0176728,"\int \frac{e^{3 i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx","Integrate[E^((3*I)*ArcTan[a*x])/Sqrt[1 + a^2*x^2],x]","\frac{2}{a (a x+i)}-\frac{i \log (a x+i)}{a}","\frac{2}{a (a x+i)}-\frac{i \log (a x+i)}{a}",1,"2/(a*(I + a*x)) - (I*Log[I + a*x])/a","A",1
304,1,52,41,0.0329792,"\int \frac{e^{2 i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx","Integrate[E^((2*I)*ArcTan[a*x])/Sqrt[1 + a^2*x^2],x]","-\frac{2 i \left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}+\sin ^{-1}\left(\frac{\sqrt{1-i a x}}{\sqrt{2}}\right)\right)}{a}","-\frac{\sinh ^{-1}(a x)}{a}-\frac{2 i \sqrt{1+i a x}}{a \sqrt{1-i a x}}",1,"((-2*I)*(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x] + ArcSin[Sqrt[1 - I*a*x]/Sqrt[2]]))/a","A",0
305,1,15,15,0.0057945,"\int \frac{e^{i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx","Integrate[E^(I*ArcTan[a*x])/Sqrt[1 + a^2*x^2],x]","\frac{i \log (a x+i)}{a}","\frac{i \log (a x+i)}{a}",1,"(I*Log[I + a*x])/a","A",1
306,1,16,16,0.0060593,"\int \frac{e^{-i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx","Integrate[1/(E^(I*ArcTan[a*x])*Sqrt[1 + a^2*x^2]),x]","-\frac{i \log (-a x+i)}{a}","-\frac{i \log (-a x+i)}{a}",1,"((-I)*Log[I - a*x])/a","A",1
307,1,56,41,0.06554,"\int \frac{e^{-2 i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx","Integrate[1/(E^((2*I)*ArcTan[a*x])*Sqrt[1 + a^2*x^2]),x]","\frac{2 \left(\sqrt{a^2 x^2+1}+(-1-i a x) \sin ^{-1}\left(\frac{\sqrt{1-i a x}}{\sqrt{2}}\right)\right)}{a (a x-i)}","-\frac{\sinh ^{-1}(a x)}{a}+\frac{2 i \sqrt{1-i a x}}{a \sqrt{1+i a x}}",1,"(2*(Sqrt[1 + a^2*x^2] + (-1 - I*a*x)*ArcSin[Sqrt[1 - I*a*x]/Sqrt[2]]))/(a*(-I + a*x))","A",0
308,1,32,32,0.0164519,"\int \frac{e^{-3 i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx","Integrate[1/(E^((3*I)*ArcTan[a*x])*Sqrt[1 + a^2*x^2]),x]","\frac{i \log (-a x+i)}{a}-\frac{2}{a (-a x+i)}","\frac{i \log (-a x+i)}{a}-\frac{2}{a (-a x+i)}",1,"-2/(a*(I - a*x)) + (I*Log[I - a*x])/a","A",1
309,1,82,73,0.0850344,"\int \frac{e^{-4 i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx","Integrate[1/(E^((4*I)*ArcTan[a*x])*Sqrt[1 + a^2*x^2]),x]","\frac{2 i \left(\frac{2 \sqrt{1+i a x} \left(2 a^2 x^2+i a x+1\right)}{\sqrt{1-i a x} (a x-i)^2}+3 \sin ^{-1}\left(\frac{\sqrt{1-i a x}}{\sqrt{2}}\right)\right)}{3 a}","\frac{2 i (1-i a x)^{3/2}}{3 a (1+i a x)^{3/2}}-\frac{2 i \sqrt{1-i a x}}{a \sqrt{1+i a x}}+\frac{\sinh ^{-1}(a x)}{a}",1,"(((2*I)/3)*((2*Sqrt[1 + I*a*x]*(1 + I*a*x + 2*a^2*x^2))/(Sqrt[1 - I*a*x]*(-I + a*x)^2) + 3*ArcSin[Sqrt[1 - I*a*x]/Sqrt[2]]))/a","A",0
310,1,69,131,0.0412946,"\int \frac{e^{5 i \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[E^((5*I)*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","\frac{i \sqrt{a^2 x^2+1} \left(4 i a x+(a x+i)^2 \log (a x+i)-2\right)}{a (a x+i)^2 \sqrt{a^2 c x^2+c}}","\frac{4 i \sqrt{a^2 x^2+1}}{a (1-i a x) \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1}}{a (1-i a x)^2 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \log (a x+i)}{a \sqrt{a^2 c x^2+c}}",1,"(I*Sqrt[1 + a^2*x^2]*(-2 + (4*I)*a*x + (I + a*x)^2*Log[I + a*x]))/(a*(I + a*x)^2*Sqrt[c + a^2*c*x^2])","A",1
311,1,71,96,0.0252078,"\int \frac{e^{4 i \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[E^((4*I)*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","-\frac{4 i \sqrt{2 a^2 x^2+2} \, _2F_1\left(-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{1}{2} (1-i a x)\right)}{3 a (1-i a x)^{3/2} \sqrt{a^2 c x^2+c}}","-\frac{2 i c (1+i a x)^3}{3 a \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 i (1+i a x)}{a \sqrt{a^2 c x^2+c}}+\frac{\tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a \sqrt{c}}",1,"(((-4*I)/3)*Sqrt[2 + 2*a^2*x^2]*Hypergeometric2F1[-3/2, -3/2, -1/2, (1 - I*a*x)/2])/(a*(1 - I*a*x)^(3/2)*Sqrt[c + a^2*c*x^2])","C",0
312,1,55,84,0.0311676,"\int \frac{e^{3 i \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[E^((3*I)*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","\frac{\sqrt{a^2 x^2+1} \left(\frac{2}{a x+i}-i \log (a x+i)\right)}{a \sqrt{a^2 c x^2+c}}","\frac{2 \sqrt{a^2 x^2+1}}{a (a x+i) \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \log (a x+i)}{a \sqrt{a^2 c x^2+c}}",1,"(Sqrt[1 + a^2*x^2]*(2/(I + a*x) - I*Log[I + a*x]))/(a*Sqrt[c + a^2*c*x^2])","A",1
313,1,91,63,0.038515,"\int \frac{e^{2 i \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[E^((2*I)*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","-\frac{2 i \sqrt{a^2 x^2+1} \left(\sqrt{1+i a x}+\sqrt{1-i a x} \sin ^{-1}\left(\frac{\sqrt{1-i a x}}{\sqrt{2}}\right)\right)}{a \sqrt{1-i a x} \sqrt{a^2 c x^2+c}}","-\frac{\tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a \sqrt{c}}-\frac{2 i (1+i a x)}{a \sqrt{a^2 c x^2+c}}",1,"((-2*I)*Sqrt[1 + a^2*x^2]*(Sqrt[1 + I*a*x] + Sqrt[1 - I*a*x]*ArcSin[Sqrt[1 - I*a*x]/Sqrt[2]]))/(a*Sqrt[1 - I*a*x]*Sqrt[c + a^2*c*x^2])","A",0
314,1,42,42,0.0156449,"\int \frac{e^{i \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[E^(I*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","\frac{i \sqrt{a^2 x^2+1} \log (a x+i)}{a \sqrt{a^2 c x^2+c}}","\frac{i \sqrt{a^2 x^2+1} \log (a x+i)}{a \sqrt{a^2 c x^2+c}}",1,"(I*Sqrt[1 + a^2*x^2]*Log[I + a*x])/(a*Sqrt[c + a^2*c*x^2])","A",1
315,1,43,43,0.0174479,"\int \frac{e^{-i \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[1/(E^(I*ArcTan[a*x])*Sqrt[c + a^2*c*x^2]),x]","-\frac{i \sqrt{a^2 x^2+1} \log (-a x+i)}{a \sqrt{a^2 c x^2+c}}","-\frac{i \sqrt{a^2 x^2+1} \log (-a x+i)}{a \sqrt{a^2 c x^2+c}}",1,"((-I)*Sqrt[1 + a^2*x^2]*Log[I - a*x])/(a*Sqrt[c + a^2*c*x^2])","A",1
316,1,117,63,0.0649604,"\int \frac{e^{-2 i \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[1/(E^((2*I)*ArcTan[a*x])*Sqrt[c + a^2*c*x^2]),x]","\frac{2 \sqrt{a^2 x^2+1} \left((1-i a x) \sqrt{1+i a x}-i \sqrt{1-i a x} (a x-i) \sin ^{-1}\left(\frac{\sqrt{1-i a x}}{\sqrt{2}}\right)\right)}{a \sqrt{1-i a x} (a x-i) \sqrt{a^2 c x^2+c}}","-\frac{\tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a \sqrt{c}}+\frac{2 i (1-i a x)}{a \sqrt{a^2 c x^2+c}}",1,"(2*Sqrt[1 + a^2*x^2]*((1 - I*a*x)*Sqrt[1 + I*a*x] - I*Sqrt[1 - I*a*x]*(-I + a*x)*ArcSin[Sqrt[1 - I*a*x]/Sqrt[2]]))/(a*Sqrt[1 - I*a*x]*(-I + a*x)*Sqrt[c + a^2*c*x^2])","A",0
317,1,60,86,0.0333849,"\int \frac{e^{-3 i \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[1/(E^((3*I)*ArcTan[a*x])*Sqrt[c + a^2*c*x^2]),x]","\frac{\sqrt{a^2 x^2+1} \left(\frac{i \log (-a x+i)}{a}-\frac{2}{a (-a x+i)}\right)}{\sqrt{a^2 c x^2+c}}","\frac{i \sqrt{a^2 x^2+1} \log (-a x+i)}{a \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1}}{a (-a x+i) \sqrt{a^2 c x^2+c}}",1,"(Sqrt[1 + a^2*x^2]*(-2/(a*(I - a*x)) + (I*Log[I - a*x])/a))/Sqrt[c + a^2*c*x^2]","A",1
318,1,132,96,0.0955889,"\int \frac{e^{-4 i \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[1/(E^((4*I)*ArcTan[a*x])*Sqrt[c + a^2*c*x^2]),x]","\frac{2 \sqrt{a^2 x^2+1} \left(2 i \sqrt{1+i a x} \left(2 a^2 x^2+i a x+1\right)+3 i \sqrt{1-i a x} (a x-i)^2 \sin ^{-1}\left(\frac{\sqrt{1-i a x}}{\sqrt{2}}\right)\right)}{3 a \sqrt{1-i a x} (a x-i)^2 \sqrt{a^2 c x^2+c}}","\frac{2 i c (1-i a x)^3}{3 a \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 i (1-i a x)}{a \sqrt{a^2 c x^2+c}}+\frac{\tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a \sqrt{c}}",1,"(2*Sqrt[1 + a^2*x^2]*((2*I)*Sqrt[1 + I*a*x]*(1 + I*a*x + 2*a^2*x^2) + (3*I)*Sqrt[1 - I*a*x]*(-I + a*x)^2*ArcSin[Sqrt[1 - I*a*x]/Sqrt[2]]))/(3*a*Sqrt[1 - I*a*x]*(-I + a*x)^2*Sqrt[c + a^2*c*x^2])","A",0
319,1,24,35,0.0186085,"\int \frac{e^{5 i \tan ^{-1}(a x)}}{\left(1+a^2 x^2\right)^{3/2}} \, dx","Integrate[E^((5*I)*ArcTan[a*x])/(1 + a^2*x^2)^(3/2),x]","-\frac{1+3 i a x}{6 a (a x+i)^3}","-\frac{i}{2 a (a x+i)^2}-\frac{2}{3 a (a x+i)^3}",1,"-1/6*(1 + (3*I)*a*x)/(a*(I + a*x)^3)","A",1
320,1,47,67,0.017203,"\int \frac{e^{4 i \tan ^{-1}(a x)}}{\left(1+a^2 x^2\right)^{3/2}} \, dx","Integrate[E^((4*I)*ArcTan[a*x])/(1 + a^2*x^2)^(3/2),x]","\frac{(1+i a x)^{3/2} (a x+4 i)}{15 a \sqrt{1-i a x} (a x+i)^2}","-\frac{i (1+i a x)^{3/2}}{15 a (1-i a x)^{3/2}}-\frac{i (1+i a x)^{3/2}}{5 a (1-i a x)^{5/2}}",1,"((1 + I*a*x)^(3/2)*(4*I + a*x))/(15*a*Sqrt[1 - I*a*x]*(I + a*x)^2)","A",1
321,1,18,19,0.0218773,"\int \frac{e^{3 i \tan ^{-1}(a x)}}{\left(1+a^2 x^2\right)^{3/2}} \, dx","Integrate[E^((3*I)*ArcTan[a*x])/(1 + a^2*x^2)^(3/2),x]","\frac{i}{2 a (a x+i)^2}","-\frac{i}{2 a (1-i a x)^2}",1,"(I/2)/(a*(I + a*x)^2)","A",1
322,1,48,67,0.0141742,"\int \frac{e^{2 i \tan ^{-1}(a x)}}{\left(1+a^2 x^2\right)^{3/2}} \, dx","Integrate[E^((2*I)*ArcTan[a*x])/(1 + a^2*x^2)^(3/2),x]","\frac{(2-i a x) \sqrt{1+i a x}}{3 a \sqrt{1-i a x} (a x+i)}","-\frac{i \sqrt{1+i a x}}{3 a \sqrt{1-i a x}}-\frac{i \sqrt{1+i a x}}{3 a (1-i a x)^{3/2}}",1,"((2 - I*a*x)*Sqrt[1 + I*a*x])/(3*a*Sqrt[1 - I*a*x]*(I + a*x))","A",1
323,1,21,28,0.0182099,"\int \frac{e^{i \tan ^{-1}(a x)}}{\left(1+a^2 x^2\right)^{3/2}} \, dx","Integrate[E^(I*ArcTan[a*x])/(1 + a^2*x^2)^(3/2),x]","\frac{\tan ^{-1}(a x)+\frac{1}{a x+i}}{2 a}","\frac{\tan ^{-1}(a x)}{2 a}+\frac{1}{2 a (a x+i)}",1,"((I + a*x)^(-1) + ArcTan[a*x])/(2*a)","A",1
324,1,21,29,0.017249,"\int \frac{e^{-i \tan ^{-1}(a x)}}{\left(1+a^2 x^2\right)^{3/2}} \, dx","Integrate[1/(E^(I*ArcTan[a*x])*(1 + a^2*x^2)^(3/2)),x]","\frac{\tan ^{-1}(a x)+\frac{1}{a x-i}}{2 a}","\frac{\tan ^{-1}(a x)}{2 a}-\frac{1}{2 a (-a x+i)}",1,"((-I + a*x)^(-1) + ArcTan[a*x])/(2*a)","A",1
325,1,48,67,0.0143025,"\int \frac{e^{-2 i \tan ^{-1}(a x)}}{\left(1+a^2 x^2\right)^{3/2}} \, dx","Integrate[1/(E^((2*I)*ArcTan[a*x])*(1 + a^2*x^2)^(3/2)),x]","\frac{\sqrt{1-i a x} (2+i a x)}{3 a \sqrt{1+i a x} (a x-i)}","\frac{i \sqrt{1-i a x}}{3 a \sqrt{1+i a x}}+\frac{i \sqrt{1-i a x}}{3 a (1+i a x)^{3/2}}",1,"(Sqrt[1 - I*a*x]*(2 + I*a*x))/(3*a*Sqrt[1 + I*a*x]*(-I + a*x))","A",1
326,1,18,19,0.0208372,"\int \frac{e^{-3 i \tan ^{-1}(a x)}}{\left(1+a^2 x^2\right)^{3/2}} \, dx","Integrate[1/(E^((3*I)*ArcTan[a*x])*(1 + a^2*x^2)^(3/2)),x]","-\frac{i}{2 a (a x-i)^2}","\frac{i}{2 a (1+i a x)^2}",1,"(-1/2*I)/(a*(-I + a*x)^2)","A",1
327,1,47,67,0.0168815,"\int \frac{e^{-4 i \tan ^{-1}(a x)}}{\left(1+a^2 x^2\right)^{3/2}} \, dx","Integrate[1/(E^((4*I)*ArcTan[a*x])*(1 + a^2*x^2)^(3/2)),x]","\frac{(1-i a x)^{3/2} (a x-4 i)}{15 a \sqrt{1+i a x} (a x-i)^2}","\frac{i (1-i a x)^{3/2}}{15 a (1+i a x)^{3/2}}+\frac{i (1-i a x)^{3/2}}{5 a (1+i a x)^{5/2}}",1,"((1 - I*a*x)^(3/2)*(-4*I + a*x))/(15*a*Sqrt[1 + I*a*x]*(-I + a*x)^2)","A",1
328,1,56,95,0.0332748,"\int \frac{e^{5 i \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^((5*I)*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]","-\frac{i (3 a x-i) \sqrt{a^2 x^2+1}}{6 a c (a x+i)^3 \sqrt{a^2 c x^2+c}}","-\frac{i \sqrt{a^2 x^2+1}}{2 a c (a x+i)^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1}}{3 a c (a x+i)^3 \sqrt{a^2 c x^2+c}}",1,"((-1/6*I)*(-I + 3*a*x)*Sqrt[1 + a^2*x^2])/(a*c*(I + a*x)^3*Sqrt[c + a^2*c*x^2])","A",1
329,1,77,69,0.0377056,"\int \frac{e^{4 i \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^((4*I)*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]","\frac{(1+i a x)^{3/2} (a x+4 i) \sqrt{a^2 x^2+1}}{15 a c \sqrt{1-i a x} (a x+i)^2 \sqrt{a^2 c x^2+c}}","\frac{i c (1+i a x)^5}{15 a \left(a^2 c x^2+c\right)^{5/2}}-\frac{i c (1+i a x)^4}{3 a \left(a^2 c x^2+c\right)^{5/2}}",1,"((1 + I*a*x)^(3/2)*(4*I + a*x)*Sqrt[1 + a^2*x^2])/(15*a*c*Sqrt[1 - I*a*x]*(I + a*x)^2*Sqrt[c + a^2*c*x^2])","A",0
330,1,57,49,0.041366,"\int \frac{e^{3 i \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^((3*I)*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]","\frac{i \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}{2 a c^2 (a x-i) (a x+i)^3}","-\frac{i \sqrt{a^2 x^2+1}}{2 a c (1-i a x)^2 \sqrt{a^2 c x^2+c}}",1,"((I/2)*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2])/(a*c^2*(-I + a*x)*(I + a*x)^3)","A",1
331,1,78,54,0.0324698,"\int \frac{e^{2 i \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^((2*I)*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]","\frac{(2-i a x) \sqrt{1+i a x} \sqrt{a^2 x^2+1}}{3 a c \sqrt{1-i a x} (a x+i) \sqrt{a^2 c x^2+c}}","\frac{x}{3 c \sqrt{a^2 c x^2+c}}-\frac{2 i (1+i a x)}{3 a \left(a^2 c x^2+c\right)^{3/2}}",1,"((2 - I*a*x)*Sqrt[1 + I*a*x]*Sqrt[1 + a^2*x^2])/(3*a*c*Sqrt[1 - I*a*x]*(I + a*x)*Sqrt[c + a^2*c*x^2])","A",0
332,1,51,88,0.0353346,"\int \frac{e^{i \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[E^(I*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]","\frac{\sqrt{a^2 x^2+1} \left(\tan ^{-1}(a x)+\frac{1}{a x+i}\right)}{2 a c \sqrt{a^2 c x^2+c}}","\frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x)}{2 a c \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1}}{2 a c (a x+i) \sqrt{a^2 c x^2+c}}",1,"(Sqrt[1 + a^2*x^2]*((I + a*x)^(-1) + ArcTan[a*x]))/(2*a*c*Sqrt[c + a^2*c*x^2])","A",1
333,1,60,89,0.0406164,"\int \frac{e^{-i \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[1/(E^(I*ArcTan[a*x])*(c + a^2*c*x^2)^(3/2)),x]","\frac{\sqrt{a^2 x^2+1} \left(\frac{\tan ^{-1}(a x)}{2 a}-\frac{1}{2 a (-a x+i)}\right)}{c \sqrt{a^2 c x^2+c}}","\frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x)}{2 a c \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1}}{2 a c (-a x+i) \sqrt{a^2 c x^2+c}}",1,"(Sqrt[1 + a^2*x^2]*(-1/2*1/(a*(I - a*x)) + ArcTan[a*x]/(2*a)))/(c*Sqrt[c + a^2*c*x^2])","A",1
334,1,78,54,0.0367937,"\int \frac{e^{-2 i \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[1/(E^((2*I)*ArcTan[a*x])*(c + a^2*c*x^2)^(3/2)),x]","\frac{\sqrt{1-i a x} (2+i a x) \sqrt{a^2 x^2+1}}{3 a c \sqrt{1+i a x} (a x-i) \sqrt{a^2 c x^2+c}}","\frac{x}{3 c \sqrt{a^2 c x^2+c}}+\frac{2 i (1-i a x)}{3 a \left(a^2 c x^2+c\right)^{3/2}}",1,"(Sqrt[1 - I*a*x]*(2 + I*a*x)*Sqrt[1 + a^2*x^2])/(3*a*c*Sqrt[1 + I*a*x]*(-I + a*x)*Sqrt[c + a^2*c*x^2])","A",0
335,1,57,49,0.0478217,"\int \frac{e^{-3 i \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[1/(E^((3*I)*ArcTan[a*x])*(c + a^2*c*x^2)^(3/2)),x]","-\frac{i \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}{2 a c^2 (a x-i)^3 (a x+i)}","\frac{i \sqrt{a^2 x^2+1}}{2 a c (1+i a x)^2 \sqrt{a^2 c x^2+c}}",1,"((-1/2*I)*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2])/(a*c^2*(-I + a*x)^3*(I + a*x))","A",1
336,1,77,69,0.0462398,"\int \frac{e^{-4 i \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[1/(E^((4*I)*ArcTan[a*x])*(c + a^2*c*x^2)^(3/2)),x]","\frac{(1-i a x)^{3/2} (a x-4 i) \sqrt{a^2 x^2+1}}{15 a c \sqrt{1+i a x} (a x-i)^2 \sqrt{a^2 c x^2+c}}","\frac{i c (1-i a x)^4}{3 a \left(a^2 c x^2+c\right)^{5/2}}-\frac{i c (1-i a x)^5}{15 a \left(a^2 c x^2+c\right)^{5/2}}",1,"((1 - I*a*x)^(3/2)*(-4*I + a*x)*Sqrt[1 + a^2*x^2])/(15*a*c*Sqrt[1 + I*a*x]*(-I + a*x)^2*Sqrt[c + a^2*c*x^2])","A",0
337,1,90,86,0.0244528,"\int e^{n \tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^2 \, dx","Integrate[E^(n*ArcTan[a*x])*(c + a^2*c*x^2)^2,x]","\frac{i c^2 2^{2-\frac{i n}{2}} (1-i a x)^{3+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}-2,\frac{i n}{2}+3;\frac{i n}{2}+4;\frac{1}{2} (1-i a x)\right)}{a \left(3+\frac{i n}{2}\right)}","-\frac{c^2 2^{3-\frac{i n}{2}} (1-i a x)^{3+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}-2,\frac{i n}{2}+3;\frac{i n}{2}+4;\frac{1}{2} (1-i a x)\right)}{a (-n+6 i)}",1,"(I*2^(2 - (I/2)*n)*c^2*(1 - I*a*x)^(3 + (I/2)*n)*Hypergeometric2F1[-2 + (I/2)*n, 3 + (I/2)*n, 4 + (I/2)*n, (1 - I*a*x)/2])/(a*(3 + (I/2)*n))","A",1
338,1,88,84,0.0183542,"\int e^{n \tan ^{-1}(a x)} \left(c+a^2 c x^2\right) \, dx","Integrate[E^(n*ArcTan[a*x])*(c + a^2*c*x^2),x]","\frac{i c 2^{1-\frac{i n}{2}} (1-i a x)^{2+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}-1,\frac{i n}{2}+2;\frac{i n}{2}+3;\frac{1}{2} (1-i a x)\right)}{a \left(2+\frac{i n}{2}\right)}","-\frac{c 2^{2-\frac{i n}{2}} (1-i a x)^{2+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}-1,\frac{i n}{2}+2;\frac{i n}{2}+3;\frac{1}{2} (1-i a x)\right)}{a (-n+4 i)}",1,"(I*2^(1 - (I/2)*n)*c*(1 - I*a*x)^(2 + (I/2)*n)*Hypergeometric2F1[-1 + (I/2)*n, 2 + (I/2)*n, 3 + (I/2)*n, (1 - I*a*x)/2])/(a*(2 + (I/2)*n))","A",1
339,1,56,81,0.0194418,"\int e^{n \tan ^{-1}(a x)} \, dx","Integrate[E^(n*ArcTan[a*x]),x]","\frac{4 e^{(n+2 i) \tan ^{-1}(a x)} \, _2F_1\left(2,1-\frac{i n}{2};2-\frac{i n}{2};-e^{2 i \tan ^{-1}(a x)}\right)}{a (n+2 i)}","-\frac{2^{1-\frac{i n}{2}} (1-i a x)^{1+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;\frac{1}{2} (1-i a x)\right)}{a (-n+2 i)}",1,"(4*E^((2*I + n)*ArcTan[a*x])*Hypergeometric2F1[2, 1 - (I/2)*n, 2 - (I/2)*n, -E^((2*I)*ArcTan[a*x])])/(a*(2*I + n))","A",0
340,1,141,131,0.1287576,"\int \frac{e^{n \tan ^{-1}(a x)} x^3}{c+a^2 c x^2} \, dx","Integrate[(E^(n*ArcTan[a*x])*x^3)/(c + a^2*c*x^2),x]","\frac{(1-i a x)^{\frac{i n}{2}} \left(\frac{\left(a^2 n x^2-\left(n^2 (a x+i)\right)+n+2 i\right) (1+i a x)^{-\frac{i n}{2}}}{n}+\frac{2^{-\frac{i n}{2}} \left(n^2-2\right) (a x+i) \, _2F_1\left(\frac{i n}{2}+1,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{1}{2} (1-i a x)\right)}{n-2 i}\right)}{2 a^4 c}","\frac{i \left(n^2-2\right) e^{n \tan ^{-1}(a x)} \, _2F_1\left(1,-\frac{i n}{2};1-\frac{i n}{2};-e^{2 i \tan ^{-1}(a x)}\right)}{a^4 c n}+\frac{\left(-i n^2+n+2 i\right) e^{n \tan ^{-1}(a x)}}{2 a^4 c n}-\frac{n x e^{n \tan ^{-1}(a x)}}{2 a^3 c}+\frac{x^2 e^{n \tan ^{-1}(a x)}}{2 a^2 c}",1,"((1 - I*a*x)^((I/2)*n)*((2*I + n + a^2*n*x^2 - n^2*(I + a*x))/(n*(1 + I*a*x)^((I/2)*n)) + ((-2 + n^2)*(I + a*x)*Hypergeometric2F1[1 + (I/2)*n, 1 + (I/2)*n, 2 + (I/2)*n, (1 - I*a*x)/2])/(2^((I/2)*n)*(-2*I + n))))/(2*a^4*c)","A",0
341,1,121,164,0.1281193,"\int \frac{e^{n \tan ^{-1}(a x)} x^2}{c+a^2 c x^2} \, dx","Integrate[(E^(n*ArcTan[a*x])*x^2)/(c + a^2*c*x^2),x]","\frac{(1-i a x)^{\frac{i n}{2}} (2+2 i a x)^{-\frac{i n}{2}} \left(2 i n (1+i a x)^{\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2},\frac{i n}{2};\frac{i n}{2}+1;\frac{1}{2} (1-i a x)\right)+2^{\frac{i n}{2}} (-1+n (a x-i))\right)}{a^3 c n}","\frac{i 2^{1-\frac{i n}{2}} (1-i a x)^{\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2},\frac{i n}{2};\frac{i n}{2}+1;\frac{1}{2} (1-i a x)\right)}{a^3 c}-\frac{(1+i n) (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{a^3 c n}+\frac{x (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{a^2 c}",1,"((1 - I*a*x)^((I/2)*n)*(2^((I/2)*n)*(-1 + n*(-I + a*x)) + (2*I)*n*(1 + I*a*x)^((I/2)*n)*Hypergeometric2F1[(I/2)*n, (I/2)*n, 1 + (I/2)*n, (1 - I*a*x)/2]))/(a^3*c*n*(2 + (2*I)*a*x)^((I/2)*n))","A",1
342,1,109,122,0.0642834,"\int \frac{e^{n \tan ^{-1}(a x)} x}{c+a^2 c x^2} \, dx","Integrate[(E^(n*ArcTan[a*x])*x)/(c + a^2*c*x^2),x]","\frac{i (1-i a x)^{\frac{i n}{2}} (2+2 i a x)^{-\frac{i n}{2}} \left(2^{\frac{i n}{2}}-2 (1+i a x)^{\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2},\frac{i n}{2};\frac{i n}{2}+1;\frac{1}{2} (1-i a x)\right)\right)}{a^2 c n}","\frac{i (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{a^2 c n}-\frac{i 2^{1-\frac{i n}{2}} (1-i a x)^{\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2},\frac{i n}{2};\frac{i n}{2}+1;\frac{1}{2} (1-i a x)\right)}{a^2 c n}",1,"(I*(1 - I*a*x)^((I/2)*n)*(2^((I/2)*n) - 2*(1 + I*a*x)^((I/2)*n)*Hypergeometric2F1[(I/2)*n, (I/2)*n, 1 + (I/2)*n, (1 - I*a*x)/2]))/(a^2*c*n*(2 + (2*I)*a*x)^((I/2)*n))","A",1
343,1,42,18,0.0075106,"\int \frac{e^{n \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Integrate[E^(n*ArcTan[a*x])/(c + a^2*c*x^2),x]","\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{a c n}","\frac{e^{n \tan ^{-1}(a x)}}{a c n}",1,"(1 - I*a*x)^((I/2)*n)/(a*c*n*(1 + I*a*x)^((I/2)*n))","C",0
344,1,120,65,0.0422724,"\int \frac{e^{n \tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)} \, dx","Integrate[E^(n*ArcTan[a*x])/(x*(c + a^2*c*x^2)),x]","\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \left(2 (n-i a n x) \, _2F_1\left(1,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{a x+i}{i-a x}\right)+(2+i n) (a x-i)\right)}{c n (n-2 i) (a x-i)}","\frac{i e^{n \tan ^{-1}(a x)}}{c n}-\frac{2 i e^{n \tan ^{-1}(a x)} \, _2F_1\left(1,-\frac{i n}{2};1-\frac{i n}{2};e^{2 i \tan ^{-1}(a x)}\right)}{c n}",1,"((1 - I*a*x)^((I/2)*n)*((2 + I*n)*(-I + a*x) + 2*(n - I*a*n*x)*Hypergeometric2F1[1, 1 + (I/2)*n, 2 + (I/2)*n, (I + a*x)/(I - a*x)]))/(c*n*(-2*I + n)*(1 + I*a*x)^((I/2)*n)*(-I + a*x))","A",0
345,1,142,90,0.055564,"\int \frac{e^{n \tan ^{-1}(a x)}}{x^2 \left(c+a^2 c x^2\right)} \, dx","Integrate[E^(n*ArcTan[a*x])/(x^2*(c + a^2*c*x^2)),x]","\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \left(2 a n^2 x (1-i a x) \, _2F_1\left(1,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{a x+i}{i-a x}\right)+(n-2 i) (1+i a x) (n (a x+i)+i a x)\right)}{c n (n-2 i) x (a x-i)}","-\frac{2 i a \, _2F_1\left(1,-\frac{i n}{2};1-\frac{i n}{2};\frac{2 i}{a x+i}-1\right) e^{n \tan ^{-1}(a x)}}{c}+\frac{i a (n+i) e^{n \tan ^{-1}(a x)}}{c n}-\frac{e^{n \tan ^{-1}(a x)}}{c x}",1,"((1 - I*a*x)^((I/2)*n)*((-2*I + n)*(1 + I*a*x)*(I*a*x + n*(I + a*x)) + 2*a*n^2*x*(1 - I*a*x)*Hypergeometric2F1[1, 1 + (I/2)*n, 2 + (I/2)*n, (I + a*x)/(I - a*x)]))/(c*n*(-2*I + n)*x*(1 + I*a*x)^((I/2)*n)*(-I + a*x))","A",0
346,1,174,126,0.0814013,"\int \frac{e^{n \tan ^{-1}(a x)}}{x^3 \left(c+a^2 c x^2\right)} \, dx","Integrate[E^(n*ArcTan[a*x])/(x^3*(c + a^2*c*x^2)),x]","\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \left(2 a^2 n \left(n^2-2\right) x^2 (1-i a x) \, _2F_1\left(1,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{a x+i}{i-a x}\right)+i (n-2 i) (a x-i) \left(i n \left(a^2 x^2+1\right)-2 a^2 x^2+a n^2 x (a x+i)\right)\right)}{2 c n (n-2 i) x^2 (a x-i)}","-\frac{i a^2 \left(n^2-2\right) e^{n \tan ^{-1}(a x)} \, _2F_1\left(1,-\frac{i n}{2};1-\frac{i n}{2};e^{2 i \tan ^{-1}(a x)}\right)}{c n}+\frac{i a^2 \left(n^2+i n-2\right) e^{n \tan ^{-1}(a x)}}{2 c n}-\frac{e^{n \tan ^{-1}(a x)}}{2 c x^2}-\frac{a n e^{n \tan ^{-1}(a x)}}{2 c x}",1,"((1 - I*a*x)^((I/2)*n)*(I*(-2*I + n)*(-I + a*x)*(-2*a^2*x^2 + a*n^2*x*(I + a*x) + I*n*(1 + a^2*x^2)) + 2*a^2*n*(-2 + n^2)*x^2*(1 - I*a*x)*Hypergeometric2F1[1, 1 + (I/2)*n, 2 + (I/2)*n, (I + a*x)/(I - a*x)]))/(2*c*n*(-2*I + n)*x^2*(1 + I*a*x)^((I/2)*n)*(-I + a*x))","A",0
347,1,165,181,0.5210212,"\int \frac{e^{n \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^4} \, dx","Integrate[E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^4,x]","\frac{(6 a x+n) e^{n \tan ^{-1}(a x)}+\frac{30 \left(a^2 c x^2+c\right) \left(12 (a x-i) (a x+i) (1-i a x)^{\frac{i n}{2}} \left(2 a^2 x^2+2 a n x+n^2+2\right) (1+i a x)^{-\frac{i n}{2}}+n (n-2 i) (n+2 i) (4 a x+n) e^{n \tan ^{-1}(a x)}\right)}{c n \left(n^4+20 n^2+64\right)}}{a c \left(n^2+36\right) \left(a^2 c x^2+c\right)^3}","\frac{360 (2 a x+n) e^{n \tan ^{-1}(a x)}}{a c^4 \left(n^2+4\right) \left(n^2+16\right) \left(n^2+36\right) \left(a^2 x^2+1\right)}+\frac{30 (4 a x+n) e^{n \tan ^{-1}(a x)}}{a c^4 \left(n^2+16\right) \left(n^2+36\right) \left(a^2 x^2+1\right)^2}+\frac{(6 a x+n) e^{n \tan ^{-1}(a x)}}{a c^4 \left(n^2+36\right) \left(a^2 x^2+1\right)^3}+\frac{720 e^{n \tan ^{-1}(a x)}}{a c^4 n \left(n^2+4\right) \left(n^2+16\right) \left(n^2+36\right)}",1,"(E^(n*ArcTan[a*x])*(n + 6*a*x) + (30*(c + a^2*c*x^2)*(E^(n*ArcTan[a*x])*n*(-2*I + n)*(2*I + n)*(n + 4*a*x) + (12*(1 - I*a*x)^((I/2)*n)*(-I + a*x)*(I + a*x)*(2 + n^2 + 2*a*n*x + 2*a^2*x^2))/(1 + I*a*x)^((I/2)*n)))/(c*n*(64 + 20*n^2 + n^4)))/(a*c*(36 + n^2)*(c + a^2*c*x^2)^3)","C",0
348,1,118,121,0.1164016,"\int e^{n \tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^{3/2} \, dx","Integrate[E^(n*ArcTan[a*x])*(c + a^2*c*x^2)^(3/2),x]","\frac{c 2^{\frac{5}{2}-\frac{i n}{2}} \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{5}{2}+\frac{i n}{2}} \, _2F_1\left(\frac{1}{2} (i n+5),\frac{1}{2} i (n+3 i);\frac{1}{2} (i n+7);\frac{1}{2} (1-i a x)\right)}{a (n-5 i) \sqrt{a^2 x^2+1}}","-\frac{c 2^{\frac{5}{2}-\frac{i n}{2}} \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (5+i n)} \, _2F_1\left(\frac{1}{2} (i n-3),\frac{1}{2} (i n+5);\frac{1}{2} (i n+7);\frac{1}{2} (1-i a x)\right)}{a (-n+5 i) \sqrt{a^2 x^2+1}}",1,"(2^(5/2 - (I/2)*n)*c*(1 - I*a*x)^(5/2 + (I/2)*n)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[(5 + I*n)/2, (I/2)*(3*I + n), (7 + I*n)/2, (1 - I*a*x)/2])/(a*(-5*I + n)*Sqrt[1 + a^2*x^2])","A",1
349,1,117,120,0.0568657,"\int e^{n \tan ^{-1}(a x)} \sqrt{c+a^2 c x^2} \, dx","Integrate[E^(n*ArcTan[a*x])*Sqrt[c + a^2*c*x^2],x]","\frac{2^{\frac{3}{2}-\frac{i n}{2}} \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{3}{2}+\frac{i n}{2}} \, _2F_1\left(\frac{1}{2} (i n+3),\frac{1}{2} i (n+i);\frac{1}{2} (i n+5);\frac{1}{2} (1-i a x)\right)}{a (n-3 i) \sqrt{a^2 x^2+1}}","-\frac{2^{\frac{3}{2}-\frac{i n}{2}} \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (3+i n)} \, _2F_1\left(\frac{1}{2} (i n-1),\frac{1}{2} (i n+3);\frac{1}{2} (i n+5);\frac{1}{2} (1-i a x)\right)}{a (-n+3 i) \sqrt{a^2 x^2+1}}",1,"(2^(3/2 - (I/2)*n)*(1 - I*a*x)^(3/2 + (I/2)*n)*Sqrt[c + a^2*c*x^2]*Hypergeometric2F1[(3 + I*n)/2, (I/2)*(I + n), (5 + I*n)/2, (1 - I*a*x)/2])/(a*(-3*I + n)*Sqrt[1 + a^2*x^2])","A",1
350,1,117,120,0.0385878,"\int \frac{e^{n \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[E^(n*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","\frac{2^{\frac{1}{2}-\frac{i n}{2}} \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}+\frac{1}{2},\frac{i n}{2}+\frac{1}{2};\frac{i n}{2}+\frac{3}{2};\frac{1}{2}-\frac{i a x}{2}\right)}{a (n-i) \sqrt{a^2 c x^2+c}}","-\frac{2^{\frac{1}{2}-\frac{i n}{2}} \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} \, _2F_1\left(\frac{1}{2} (i n+1),\frac{1}{2} (i n+1);\frac{1}{2} (i n+3);\frac{1}{2} (1-i a x)\right)}{a (-n+i) \sqrt{a^2 c x^2+c}}",1,"(2^(1/2 - (I/2)*n)*(1 - I*a*x)^(1/2 + (I/2)*n)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2 + (I/2)*n, 1/2 + (I/2)*n, 3/2 + (I/2)*n, 1/2 - (I/2)*a*x])/(a*(-I + n)*Sqrt[c + a^2*c*x^2])","A",1
351,1,217,283,0.2747006,"\int e^{n \tan ^{-1}(a x)} x^2 \left(c+a^2 c x^2\right)^{3/2} \, dx","Integrate[E^(n*ArcTan[a*x])*x^2*(c + a^2*c*x^2)^(3/2),x]","\frac{c 2^{-1-\frac{i n}{2}} (a x+i)^2 \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \left(2^{\frac{i n}{2}} (n-5 i) \sqrt{1+i a x} (a x-i)^2 (5 a x-n)-4 \sqrt{2} \left(n^2-5\right) (1+i a x)^{\frac{i n}{2}} \, _2F_1\left(\frac{1}{2} (i n+5),\frac{1}{2} i (n+3 i);\frac{1}{2} (i n+7);\frac{1}{2} (1-i a x)\right)\right)}{15 a^3 (n-5 i) \sqrt{a^2 x^2+1}}","\frac{c x \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (5+i n)} (1+i a x)^{\frac{1}{2} (5-i n)}}{6 a^2 \sqrt{a^2 x^2+1}}+\frac{c 2^{\frac{3}{2}-\frac{i n}{2}} \left(5-n^2\right) \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (5+i n)} \, _2F_1\left(\frac{1}{2} (i n-3),\frac{1}{2} (i n+5);\frac{1}{2} (i n+7);\frac{1}{2} (1-i a x)\right)}{15 a^3 (-n+5 i) \sqrt{a^2 x^2+1}}-\frac{c n \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (5+i n)} (1+i a x)^{\frac{1}{2} (5-i n)}}{30 a^3 \sqrt{a^2 x^2+1}}",1,"(2^(-1 - (I/2)*n)*c*(1 - I*a*x)^(1/2 + (I/2)*n)*(I + a*x)^2*Sqrt[c + a^2*c*x^2]*(2^((I/2)*n)*(-5*I + n)*Sqrt[1 + I*a*x]*(-I + a*x)^2*(-n + 5*a*x) - 4*Sqrt[2]*(-5 + n^2)*(1 + I*a*x)^((I/2)*n)*Hypergeometric2F1[(5 + I*n)/2, (I/2)*(3*I + n), (7 + I*n)/2, (1 - I*a*x)/2]))/(15*a^3*(-5*I + n)*(1 + I*a*x)^((I/2)*n)*Sqrt[1 + a^2*x^2])","A",1
352,1,214,280,0.2182094,"\int e^{n \tan ^{-1}(a x)} x^2 \sqrt{c+a^2 c x^2} \, dx","Integrate[E^(n*ArcTan[a*x])*x^2*Sqrt[c + a^2*c*x^2],x]","\frac{2^{-2-\frac{i n}{2}} (a x+i) \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \left(2^{\frac{i n}{2}} (n-3 i) \sqrt{1+i a x} (a x-i) (3 a x-n)-2 i \sqrt{2} \left(n^2-3\right) (1+i a x)^{\frac{i n}{2}} \, _2F_1\left(\frac{1}{2} (i n+3),\frac{1}{2} i (n+i);\frac{1}{2} (i n+5);\frac{1}{2} (1-i a x)\right)\right)}{3 a^3 (n-3 i) \sqrt{a^2 x^2+1}}","\frac{x \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (3+i n)} (1+i a x)^{\frac{1}{2} (3-i n)}}{4 a^2 \sqrt{a^2 x^2+1}}+\frac{2^{-\frac{1}{2}-\frac{i n}{2}} \left(3-n^2\right) \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (3+i n)} \, _2F_1\left(\frac{1}{2} (i n-1),\frac{1}{2} (i n+3);\frac{1}{2} (i n+5);\frac{1}{2} (1-i a x)\right)}{3 a^3 (-n+3 i) \sqrt{a^2 x^2+1}}-\frac{n \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (3+i n)} (1+i a x)^{\frac{1}{2} (3-i n)}}{12 a^3 \sqrt{a^2 x^2+1}}",1,"(2^(-2 - (I/2)*n)*(1 - I*a*x)^(1/2 + (I/2)*n)*(I + a*x)*Sqrt[c + a^2*c*x^2]*(2^((I/2)*n)*(-3*I + n)*Sqrt[1 + I*a*x]*(-I + a*x)*(-n + 3*a*x) - (2*I)*Sqrt[2]*(-3 + n^2)*(1 + I*a*x)^((I/2)*n)*Hypergeometric2F1[(3 + I*n)/2, (I/2)*(I + n), (5 + I*n)/2, (1 - I*a*x)/2]))/(3*a^3*(-3*I + n)*(1 + I*a*x)^((I/2)*n)*Sqrt[1 + a^2*x^2])","A",1
353,1,248,322,0.3239257,"\int \frac{e^{n \tan ^{-1}(a x)} x^3}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(E^(n*ArcTan[a*x])*x^3)/Sqrt[c + a^2*c*x^2],x]","\frac{2^{-\frac{3}{2}-\frac{i n}{2}} \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \left(2^{\frac{1}{2}+\frac{i n}{2}} (n-3 i) \sqrt{1+i a x} \left(n \left(2 a^2 x^2+i a x+1\right)-2 i \left(a^2 x^2-2\right)-\left(n^2 (a x+i)\right)\right)+2 n \left(n^2-5\right) (a x+i) (1+i a x)^{\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}+\frac{1}{2},\frac{i n}{2}+\frac{3}{2};\frac{i n}{2}+\frac{5}{2};\frac{1}{2}-\frac{i a x}{2}\right)\right)}{3 a^4 \left(n^2-4 i n-3\right) \sqrt{a^2 c x^2+c}}","\frac{x^2 \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)}}{3 a^2 \sqrt{a^2 c x^2+c}}+\frac{2^{-\frac{1}{2}-\frac{i n}{2}} n \left(5-n^2\right) \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (3+i n)} \, _2F_1\left(\frac{1}{2} (i n+1),\frac{1}{2} (i n+3);\frac{1}{2} (i n+5);\frac{1}{2} (1-i a x)\right)}{3 a^4 \left(4 n-i \left(3-n^2\right)\right) \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} \left(a (1+i n) n x-n^2-i n+4\right) (1+i a x)^{\frac{1}{2} (1-i n)}}{6 a^4 (1+i n) \sqrt{a^2 c x^2+c}}",1,"(2^(-3/2 - (I/2)*n)*(1 - I*a*x)^(1/2 + (I/2)*n)*Sqrt[1 + a^2*x^2]*(2^(1/2 + (I/2)*n)*(-3*I + n)*Sqrt[1 + I*a*x]*(-(n^2*(I + a*x)) - (2*I)*(-2 + a^2*x^2) + n*(1 + I*a*x + 2*a^2*x^2)) + 2*n*(-5 + n^2)*(1 + I*a*x)^((I/2)*n)*(I + a*x)*Hypergeometric2F1[1/2 + (I/2)*n, 3/2 + (I/2)*n, 5/2 + (I/2)*n, 1/2 - (I/2)*a*x]))/(3*a^4*(-3 - (4*I)*n + n^2)*(1 + I*a*x)^((I/2)*n)*Sqrt[c + a^2*c*x^2])","A",1
354,1,206,291,0.1846789,"\int \frac{e^{n \tan ^{-1}(a x)} x^2}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(E^(n*ArcTan[a*x])*x^2)/Sqrt[c + a^2*c*x^2],x]","\frac{2^{-1-\frac{i n}{2}} \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \left(2 i \sqrt{2} \left(n^2-1\right) (1+i a x)^{\frac{i n}{2}} \, _2F_1\left(\frac{1}{2} (i n+1),\frac{1}{2} i (n+i);\frac{1}{2} (i n+3);\frac{1}{2} (1-i a x)\right)+2^{\frac{i n}{2}} (n-i) \sqrt{1+i a x} (n (a x-i)+i a x-1)\right)}{a^3 \left(n^2+1\right) \sqrt{a^2 c x^2+c}}","\frac{x \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)}}{2 a^2 \sqrt{a^2 c x^2+c}}-\frac{i 2^{\frac{1}{2}-\frac{i n}{2}} \left(1-n^2\right) \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} \, _2F_1\left(\frac{1}{2} (i n-1),\frac{1}{2} (i n+1);\frac{1}{2} (i n+3);\frac{1}{2} (1-i a x)\right)}{a^3 \left(n^2+1\right) \sqrt{a^2 c x^2+c}}-\frac{(1+i n) \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)}}{2 a^3 (n+i) \sqrt{a^2 c x^2+c}}",1,"(2^(-1 - (I/2)*n)*(1 - I*a*x)^(1/2 + (I/2)*n)*Sqrt[1 + a^2*x^2]*(2^((I/2)*n)*(-I + n)*Sqrt[1 + I*a*x]*(-1 + I*a*x + n*(-I + a*x)) + (2*I)*Sqrt[2]*(-1 + n^2)*(1 + I*a*x)^((I/2)*n)*Hypergeometric2F1[(1 + I*n)/2, (I/2)*(I + n), (3 + I*n)/2, (1 - I*a*x)/2]))/(a^3*(1 + n^2)*(1 + I*a*x)^((I/2)*n)*Sqrt[c + a^2*c*x^2])","A",1
355,1,187,202,0.1470021,"\int \frac{e^{n \tan ^{-1}(a x)} x}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(E^(n*ArcTan[a*x])*x)/Sqrt[c + a^2*c*x^2],x]","\frac{i 2^{-\frac{1}{2}-\frac{i n}{2}} \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \left(2^{\frac{1}{2}+\frac{i n}{2}} (n-i) \sqrt{1+i a x}-4 n (1+i a x)^{\frac{i n}{2}} \, _2F_1\left(\frac{1}{2} (i n+1),\frac{1}{2} i (n+i);\frac{1}{2} (i n+3);\frac{1}{2} (1-i a x)\right)\right)}{a^2 \left(n^2+1\right) \sqrt{a^2 c x^2+c}}","\frac{\sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)}}{a^2 (1-i n) \sqrt{a^2 c x^2+c}}-\frac{i 2^{\frac{3}{2}-\frac{i n}{2}} n \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} \, _2F_1\left(\frac{1}{2} (i n-1),\frac{1}{2} (i n+1);\frac{1}{2} (i n+3);\frac{1}{2} (1-i a x)\right)}{a^2 \left(n^2+1\right) \sqrt{a^2 c x^2+c}}",1,"(I*2^(-1/2 - (I/2)*n)*(1 - I*a*x)^(1/2 + (I/2)*n)*Sqrt[1 + a^2*x^2]*(2^(1/2 + (I/2)*n)*(-I + n)*Sqrt[1 + I*a*x] - 4*n*(1 + I*a*x)^((I/2)*n)*Hypergeometric2F1[(1 + I*n)/2, (I/2)*(I + n), (3 + I*n)/2, (1 - I*a*x)/2]))/(a^2*(1 + n^2)*(1 + I*a*x)^((I/2)*n)*Sqrt[c + a^2*c*x^2])","A",1
356,1,117,120,0.0289693,"\int \frac{e^{n \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[E^(n*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","\frac{2^{\frac{1}{2}-\frac{i n}{2}} \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}+\frac{1}{2},\frac{i n}{2}+\frac{1}{2};\frac{i n}{2}+\frac{3}{2};\frac{1}{2}-\frac{i a x}{2}\right)}{a (n-i) \sqrt{a^2 c x^2+c}}","-\frac{2^{\frac{1}{2}-\frac{i n}{2}} \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} \, _2F_1\left(\frac{1}{2} (i n+1),\frac{1}{2} (i n+1);\frac{1}{2} (i n+3);\frac{1}{2} (1-i a x)\right)}{a (-n+i) \sqrt{a^2 c x^2+c}}",1,"(2^(1/2 - (I/2)*n)*(1 - I*a*x)^(1/2 + (I/2)*n)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2 + (I/2)*n, 1/2 + (I/2)*n, 3/2 + (I/2)*n, 1/2 - (I/2)*a*x])/(a*(-I + n)*Sqrt[c + a^2*c*x^2])","A",1
357,1,120,121,0.0473486,"\int \frac{e^{n \tan ^{-1}(a x)}}{x \sqrt{c+a^2 c x^2}} \, dx","Integrate[E^(n*ArcTan[a*x])/(x*Sqrt[c + a^2*c*x^2]),x]","\frac{2 \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{1}{2}-\frac{i n}{2}} \, _2F_1\left(1,\frac{i n}{2}+\frac{1}{2};\frac{i n}{2}+\frac{3}{2};\frac{a x+i}{i-a x}\right)}{(-1-i n) \sqrt{a^2 c x^2+c}}","-\frac{2 \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (-1-i n)} \, _2F_1\left(1,\frac{1}{2} (i n+1);\frac{1}{2} (i n+3);\frac{1-i a x}{i a x+1}\right)}{(1+i n) \sqrt{a^2 c x^2+c}}",1,"(2*(1 - I*a*x)^(1/2 + (I/2)*n)*(1 + I*a*x)^(-1/2 - (I/2)*n)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1, 1/2 + (I/2)*n, 3/2 + (I/2)*n, (I + a*x)/(I - a*x)])/((-1 - I*n)*Sqrt[c + a^2*c*x^2])","A",1
358,1,142,196,0.0694467,"\int \frac{e^{n \tan ^{-1}(a x)}}{x^2 \sqrt{c+a^2 c x^2}} \, dx","Integrate[E^(n*ArcTan[a*x])/(x^2*Sqrt[c + a^2*c*x^2]),x]","\frac{\sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{1}{2}-\frac{i n}{2}} \left(2 a n x \, _2F_1\left(1,\frac{i n}{2}+\frac{1}{2};\frac{i n}{2}+\frac{3}{2};\frac{a x+i}{i-a x}\right)-(n-i) (a x-i)\right)}{(-1-i n) x \sqrt{a^2 c x^2+c}}","-\frac{2 a n \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (-1-i n)} \, _2F_1\left(1,\frac{1}{2} (i n+1);\frac{1}{2} (i n+3);\frac{1-i a x}{i a x+1}\right)}{(1+i n) \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)}}{x \sqrt{a^2 c x^2+c}}",1,"((1 - I*a*x)^(1/2 + (I/2)*n)*(1 + I*a*x)^(-1/2 - (I/2)*n)*Sqrt[1 + a^2*x^2]*(-((-I + n)*(-I + a*x)) + 2*a*n*x*Hypergeometric2F1[1, 1/2 + (I/2)*n, 3/2 + (I/2)*n, (I + a*x)/(I - a*x)]))/((-1 - I*n)*x*Sqrt[c + a^2*c*x^2])","A",1
359,1,159,281,0.0842987,"\int \frac{e^{n \tan ^{-1}(a x)}}{x^3 \sqrt{c+a^2 c x^2}} \, dx","Integrate[E^(n*ArcTan[a*x])/(x^3*Sqrt[c + a^2*c*x^2]),x]","\frac{i \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{1}{2}-\frac{i n}{2}} \left(2 a^2 \left(n^2-1\right) x^2 \, _2F_1\left(1,\frac{i n}{2}+\frac{1}{2};\frac{i n}{2}+\frac{3}{2};\frac{a x+i}{i-a x}\right)-(n-i) (a x-i) (a n x+1)\right)}{2 (n-i) x^2 \sqrt{a^2 c x^2+c}}","\frac{a^2 \left(1-n^2\right) \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (-1-i n)} \, _2F_1\left(1,\frac{1}{2} (i n+1);\frac{1}{2} (i n+3);\frac{1-i a x}{i a x+1}\right)}{(1+i n) \sqrt{a^2 c x^2+c}}-\frac{a n \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)}}{2 x \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)}}{2 x^2 \sqrt{a^2 c x^2+c}}",1,"((I/2)*(1 - I*a*x)^(1/2 + (I/2)*n)*(1 + I*a*x)^(-1/2 - (I/2)*n)*Sqrt[1 + a^2*x^2]*(-((-I + n)*(-I + a*x)*(1 + a*n*x)) + 2*a^2*(-1 + n^2)*x^2*Hypergeometric2F1[1, 1/2 + (I/2)*n, 3/2 + (I/2)*n, (I + a*x)/(I - a*x)]))/((-I + n)*x^2*Sqrt[c + a^2*c*x^2])","A",1
360,1,120,120,0.0686937,"\int e^{n \tan ^{-1}(a x)} \sqrt[3]{c+a^2 c x^2} \, dx","Integrate[E^(n*ArcTan[a*x])*(c + a^2*c*x^2)^(1/3),x]","\frac{3\ 2^{\frac{4}{3}-\frac{i n}{2}} \sqrt[3]{a^2 c x^2+c} (1-i a x)^{\frac{4}{3}+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}-\frac{1}{3},\frac{i n}{2}+\frac{4}{3};\frac{i n}{2}+\frac{7}{3};\frac{1}{2}-\frac{i a x}{2}\right)}{a (3 n-8 i) \sqrt[3]{a^2 x^2+1}}","-\frac{3\ 2^{\frac{4}{3}-\frac{i n}{2}} \sqrt[3]{a^2 c x^2+c} (1-i a x)^{\frac{1}{6} (8+3 i n)} \, _2F_1\left(\frac{1}{6} (3 i n-2),\frac{1}{6} (3 i n+8);\frac{1}{6} (3 i n+14);\frac{1}{2} (1-i a x)\right)}{a (-3 n+8 i) \sqrt[3]{a^2 x^2+1}}",1,"(3*2^(4/3 - (I/2)*n)*(1 - I*a*x)^(4/3 + (I/2)*n)*(c + a^2*c*x^2)^(1/3)*Hypergeometric2F1[-1/3 + (I/2)*n, 4/3 + (I/2)*n, 7/3 + (I/2)*n, 1/2 - (I/2)*a*x])/(a*(-8*I + 3*n)*(1 + a^2*x^2)^(1/3))","A",1
361,1,120,120,0.0496476,"\int \frac{e^{n \tan ^{-1}(a x)}}{\sqrt[3]{c+a^2 c x^2}} \, dx","Integrate[E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(1/3),x]","\frac{3\ 2^{\frac{2}{3}-\frac{i n}{2}} \sqrt[3]{a^2 x^2+1} (1-i a x)^{\frac{2}{3}+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}+\frac{1}{3},\frac{i n}{2}+\frac{2}{3};\frac{i n}{2}+\frac{5}{3};\frac{1}{2}-\frac{i a x}{2}\right)}{a (3 n-4 i) \sqrt[3]{a^2 c x^2+c}}","-\frac{3\ 2^{\frac{2}{3}-\frac{i n}{2}} \sqrt[3]{a^2 x^2+1} (1-i a x)^{\frac{1}{6} (4+3 i n)} \, _2F_1\left(\frac{1}{6} (3 i n+2),\frac{1}{6} (3 i n+4);\frac{1}{6} (3 i n+10);\frac{1}{2} (1-i a x)\right)}{a (-3 n+4 i) \sqrt[3]{a^2 c x^2+c}}",1,"(3*2^(2/3 - (I/2)*n)*(1 - I*a*x)^(2/3 + (I/2)*n)*(1 + a^2*x^2)^(1/3)*Hypergeometric2F1[1/3 + (I/2)*n, 2/3 + (I/2)*n, 5/3 + (I/2)*n, 1/2 - (I/2)*a*x])/(a*(-4*I + 3*n)*(c + a^2*c*x^2)^(1/3))","A",1
362,1,120,120,0.0478423,"\int \frac{e^{n \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{2/3}} \, dx","Integrate[E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(2/3),x]","\frac{3\ 2^{\frac{1}{3}-\frac{i n}{2}} \left(a^2 x^2+1\right)^{2/3} (1-i a x)^{\frac{1}{3}+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}+\frac{1}{3},\frac{i n}{2}+\frac{2}{3};\frac{i n}{2}+\frac{4}{3};\frac{1}{2}-\frac{i a x}{2}\right)}{a (3 n-2 i) \left(a^2 c x^2+c\right)^{2/3}}","-\frac{3\ 2^{\frac{1}{3}-\frac{i n}{2}} \left(a^2 x^2+1\right)^{2/3} (1-i a x)^{\frac{1}{6} (2+3 i n)} \, _2F_1\left(\frac{1}{6} (3 i n+2),\frac{1}{6} (3 i n+4);\frac{1}{6} (3 i n+8);\frac{1}{2} (1-i a x)\right)}{a (-3 n+2 i) \left(a^2 c x^2+c\right)^{2/3}}",1,"(3*2^(1/3 - (I/2)*n)*(1 - I*a*x)^(1/3 + (I/2)*n)*(1 + a^2*x^2)^(2/3)*Hypergeometric2F1[1/3 + (I/2)*n, 2/3 + (I/2)*n, 4/3 + (I/2)*n, 1/2 - (I/2)*a*x])/(a*(-2*I + 3*n)*(c + a^2*c*x^2)^(2/3))","A",1
363,1,123,123,0.0570282,"\int \frac{e^{n \tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{4/3}} \, dx","Integrate[E^(n*ArcTan[a*x])/(c + a^2*c*x^2)^(4/3),x]","\frac{3\ 2^{-\frac{1}{3}-\frac{i n}{2}} \sqrt[3]{a^2 x^2+1} (1-i a x)^{-\frac{1}{3}+\frac{i n}{2}} \, _2F_1\left(\frac{i n}{2}-\frac{1}{3},\frac{i n}{2}+\frac{4}{3};\frac{i n}{2}+\frac{2}{3};\frac{1}{2}-\frac{i a x}{2}\right)}{a c (3 n+2 i) \sqrt[3]{a^2 c x^2+c}}","\frac{3\ 2^{-\frac{1}{3}-\frac{i n}{2}} \sqrt[3]{a^2 x^2+1} (1-i a x)^{\frac{1}{6} (-2+3 i n)} \, _2F_1\left(\frac{1}{6} (3 i n-2),\frac{1}{6} (3 i n+8);\frac{1}{6} (3 i n+4);\frac{1}{2} (1-i a x)\right)}{a c (3 n+2 i) \sqrt[3]{a^2 c x^2+c}}",1,"(3*2^(-1/3 - (I/2)*n)*(1 - I*a*x)^(-1/3 + (I/2)*n)*(1 + a^2*x^2)^(1/3)*Hypergeometric2F1[-1/3 + (I/2)*n, 4/3 + (I/2)*n, 2/3 + (I/2)*n, 1/2 - (I/2)*a*x])/(a*c*(2*I + 3*n)*(c + a^2*c*x^2)^(1/3))","A",1
364,0,0,49,0.5210118,"\int e^{n \tan ^{-1}(a x)} x^m \left(c+a^2 c x^2\right) \, dx","Integrate[E^(n*ArcTan[a*x])*x^m*(c + a^2*c*x^2),x]","\int e^{n \tan ^{-1}(a x)} x^m \left(c+a^2 c x^2\right) \, dx","\frac{c x^{m+1} F_1\left(m+1;-\frac{i n}{2}-1,\frac{i n}{2}-1;m+2;i a x,-i a x\right)}{m+1}",1,"Integrate[E^(n*ArcTan[a*x])*x^m*(c + a^2*c*x^2), x]","F",-1
365,1,96,51,0.1710563,"\int \frac{e^{n \tan ^{-1}(a x)} x^m}{c+a^2 c x^2} \, dx","Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2),x]","\frac{x^m \left(1-e^{2 i \tan ^{-1}(a x)}\right)^{-m} \left(1+e^{2 i \tan ^{-1}(a x)}\right)^m e^{n \tan ^{-1}(a x)} F_1\left(-\frac{i n}{2};m,-m;1-\frac{i n}{2};-e^{2 i \tan ^{-1}(a x)},e^{2 i \tan ^{-1}(a x)}\right)}{a c n}","\frac{x^{m+1} F_1\left(m+1;1-\frac{i n}{2},\frac{i n}{2}+1;m+2;i a x,-i a x\right)}{c (m+1)}",1,"(E^(n*ArcTan[a*x])*(1 + E^((2*I)*ArcTan[a*x]))^m*x^m*AppellF1[(-1/2*I)*n, m, -m, 1 - (I/2)*n, -E^((2*I)*ArcTan[a*x]), E^((2*I)*ArcTan[a*x])])/(a*c*(1 - E^((2*I)*ArcTan[a*x]))^m*n)","A",0
366,0,0,51,0.4948043,"\int \frac{e^{n \tan ^{-1}(a x)} x^m}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^2,x]","\int \frac{e^{n \tan ^{-1}(a x)} x^m}{\left(c+a^2 c x^2\right)^2} \, dx","\frac{x^{m+1} F_1\left(m+1;2-\frac{i n}{2},\frac{i n}{2}+2;m+2;i a x,-i a x\right)}{c^2 (m+1)}",1,"Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^2, x]","F",-1
367,0,0,51,0.7935395,"\int \frac{e^{n \tan ^{-1}(a x)} x^m}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^3,x]","\int \frac{e^{n \tan ^{-1}(a x)} x^m}{\left(c+a^2 c x^2\right)^3} \, dx","\frac{x^{m+1} F_1\left(m+1;3-\frac{i n}{2},\frac{i n}{2}+3;m+2;i a x,-i a x\right)}{c^3 (m+1)}",1,"Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^3, x]","F",-1
368,0,0,79,0.3719061,"\int \frac{e^{n \tan ^{-1}(a x)} x^m}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(E^(n*ArcTan[a*x])*x^m)/Sqrt[c + a^2*c*x^2],x]","\int \frac{e^{n \tan ^{-1}(a x)} x^m}{\sqrt{c+a^2 c x^2}} \, dx","\frac{\sqrt{a^2 x^2+1} x^{m+1} F_1\left(m+1;\frac{1}{2} (1-i n),\frac{1}{2} (i n+1);m+2;i a x,-i a x\right)}{(m+1) \sqrt{a^2 c x^2+c}}",1,"Integrate[(E^(n*ArcTan[a*x])*x^m)/Sqrt[c + a^2*c*x^2], x]","F",-1
369,0,0,82,0.4834627,"\int \frac{e^{n \tan ^{-1}(a x)} x^m}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^(3/2),x]","\int \frac{e^{n \tan ^{-1}(a x)} x^m}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\frac{\sqrt{a^2 x^2+1} x^{m+1} F_1\left(m+1;\frac{1}{2} (3-i n),\frac{1}{2} (i n+3);m+2;i a x,-i a x\right)}{c (m+1) \sqrt{a^2 c x^2+c}}",1,"Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^(3/2), x]","F",-1
370,0,0,82,0.6286495,"\int \frac{e^{n \tan ^{-1}(a x)} x^m}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^(5/2),x]","\int \frac{e^{n \tan ^{-1}(a x)} x^m}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","\frac{\sqrt{a^2 x^2+1} x^{m+1} F_1\left(m+1;\frac{1}{2} (5-i n),\frac{1}{2} (i n+5);m+2;i a x,-i a x\right)}{c^2 (m+1) \sqrt{a^2 c x^2+c}}",1,"Integrate[(E^(n*ArcTan[a*x])*x^m)/(c + a^2*c*x^2)^(5/2), x]","F",-1
371,1,115,115,0.0359992,"\int e^{n \tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^p \, dx","Integrate[E^(n*ArcTan[a*x])*(c + a^2*c*x^2)^p,x]","\frac{2^{-\frac{i n}{2}+p+1} \left(a^2 x^2+1\right)^{-p} \left(a^2 c x^2+c\right)^p (1-i a x)^{\frac{i n}{2}+p+1} \, _2F_1\left(\frac{i n}{2}-p,\frac{i n}{2}+p+1;\frac{i n}{2}+p+2;\frac{1}{2} (1-i a x)\right)}{a (n-2 i (p+1))}","\frac{2^{-\frac{i n}{2}+p+1} \left(a^2 x^2+1\right)^{-p} \left(a^2 c x^2+c\right)^p (1-i a x)^{\frac{i n}{2}+p+1} \, _2F_1\left(\frac{i n}{2}-p,\frac{i n}{2}+p+1;\frac{i n}{2}+p+2;\frac{1}{2} (1-i a x)\right)}{a (n-2 i (p+1))}",1,"(2^(1 - (I/2)*n + p)*(1 - I*a*x)^(1 + (I/2)*n + p)*(c + a^2*c*x^2)^p*Hypergeometric2F1[(I/2)*n - p, 1 + (I/2)*n + p, 2 + (I/2)*n + p, (1 - I*a*x)/2])/(a*(n - (2*I)*(1 + p))*(1 + a^2*x^2)^p)","A",1
372,1,39,53,0.0321612,"\int e^{-2 i p \tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^p \, dx","Integrate[(c + a^2*c*x^2)^p/E^((2*I)*p*ArcTan[a*x]),x]","\frac{(a x+i) \left(a^2 c x^2+c\right)^p e^{-2 i p \tan ^{-1}(a x)}}{2 a p+a}","\frac{i (1-i a x)^{2 p+1} \left(a^2 x^2+1\right)^{-p} \left(a^2 c x^2+c\right)^p}{a (2 p+1)}",1,"((I + a*x)*(c + a^2*c*x^2)^p)/(E^((2*I)*p*ArcTan[a*x])*(a + 2*a*p))","A",1
373,1,39,53,0.0282673,"\int e^{2 i p \tan ^{-1}(a x)} \left(c+a^2 c x^2\right)^p \, dx","Integrate[E^((2*I)*p*ArcTan[a*x])*(c + a^2*c*x^2)^p,x]","\frac{(a x-i) \left(a^2 c x^2+c\right)^p e^{2 i p \tan ^{-1}(a x)}}{2 a p+a}","-\frac{i (1+i a x)^{2 p+1} \left(a^2 x^2+1\right)^{-p} \left(a^2 c x^2+c\right)^p}{a (2 p+1)}",1,"(E^((2*I)*p*ArcTan[a*x])*(-I + a*x)*(c + a^2*c*x^2)^p)/(a + 2*a*p)","A",1
374,1,55,60,0.0259042,"\int e^{i n \tan ^{-1}(a x)} x^2 \left(c+a^2 c x^2\right)^{-1-\frac{n^2}{2}} \, dx","Integrate[E^(I*n*ArcTan[a*x])*x^2*(c + a^2*c*x^2)^(-1 - n^2/2),x]","-\frac{(a n x+i) \left(a^2 c x^2+c\right)^{-\frac{n^2}{2}} e^{i n \tan ^{-1}(a x)}}{a^3 c n \left(n^2-1\right)}","\frac{i (1-i a n x) \left(a^2 c x^2+c\right)^{-\frac{n^2}{2}} e^{i n \tan ^{-1}(a x)}}{a^3 c n \left(1-n^2\right)}",1,"-((E^(I*n*ArcTan[a*x])*(I + a*n*x))/(a^3*c*n*(-1 + n^2)*(c + a^2*c*x^2)^(n^2/2)))","A",1
375,1,36,38,1.2265815,"\int \frac{e^{6 i \tan ^{-1}(a x)} x^2}{\left(c+a^2 c x^2\right)^{19}} \, dx","Integrate[(E^((6*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^19,x]","\frac{6 a x+i}{210 a^3 c^{19} (a x-i)^{15} (a x+i)^{21}}","-\frac{6 a x+i}{210 a^3 c^{19} (1-i a x)^{21} (1+i a x)^{15}}",1,"(I + 6*a*x)/(210*a^3*c^19*(-I + a*x)^15*(I + a*x)^21)","A",1
376,1,36,38,0.2347042,"\int \frac{e^{4 i \tan ^{-1}(a x)} x^2}{\left(c+a^2 c x^2\right)^9} \, dx","Integrate[(E^((4*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^9,x]","-\frac{4 a x+i}{60 a^3 c^9 (a x-i)^6 (a x+i)^{10}}","-\frac{4 a x+i}{60 a^3 c^9 (1-i a x)^{10} (1+i a x)^6}",1,"-1/60*(I + 4*a*x)/(a^3*c^9*(-I + a*x)^6*(I + a*x)^10)","A",1
377,1,36,38,0.0378818,"\int \frac{e^{2 i \tan ^{-1}(a x)} x^2}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(E^((2*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^3,x]","\frac{2 a x+i}{6 a^3 c^3 (a x-i) (a x+i)^3}","-\frac{2 a x+i}{6 a^3 c^3 (1-i a x)^3 (1+i a x)}",1,"(I + 2*a*x)/(6*a^3*c^3*(-I + a*x)*(I + a*x)^3)","A",1
378,1,36,38,0.0361862,"\int \frac{e^{-2 i \tan ^{-1}(a x)} x^2}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[x^2/(E^((2*I)*ArcTan[a*x])*(c + a^2*c*x^2)^3),x]","\frac{2 a x-i}{6 a^3 c^3 (a x-i)^3 (a x+i)}","\frac{-2 a x+i}{6 a^3 c^3 (1-i a x) (1+i a x)^3}",1,"(-I + 2*a*x)/(6*a^3*c^3*(-I + a*x)^3*(I + a*x))","A",1
379,1,36,38,0.2292494,"\int \frac{e^{-4 i \tan ^{-1}(a x)} x^2}{\left(c+a^2 c x^2\right)^9} \, dx","Integrate[x^2/(E^((4*I)*ArcTan[a*x])*(c + a^2*c*x^2)^9),x]","\frac{-4 a x+i}{60 a^3 c^9 (a x-i)^{10} (a x+i)^6}","\frac{-4 a x+i}{60 a^3 c^9 (1-i a x)^6 (1+i a x)^{10}}",1,"(I - 4*a*x)/(60*a^3*c^9*(-I + a*x)^10*(I + a*x)^6)","A",1
380,1,63,65,0.5679357,"\int \frac{e^{5 i \tan ^{-1}(a x)} x^2}{\left(c+a^2 c x^2\right)^{27/2}} \, dx","Integrate[(E^((5*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^(27/2),x]","\frac{(1-5 i a x) \sqrt{a^2 x^2+1}}{120 a^3 c^{13} (a x-i)^{10} (a x+i)^{15} \sqrt{a^2 c x^2+c}}","-\frac{(5 a x+i) \sqrt{a^2 x^2+1}}{120 a^3 c^{13} (1-i a x)^{15} (1+i a x)^{10} \sqrt{a^2 c x^2+c}}",1,"((1 - (5*I)*a*x)*Sqrt[1 + a^2*x^2])/(120*a^3*c^13*(-I + a*x)^10*(I + a*x)^15*Sqrt[c + a^2*c*x^2])","A",1
381,1,65,65,0.1093551,"\int \frac{e^{3 i \tan ^{-1}(a x)} x^2}{\left(c+a^2 c x^2\right)^{11/2}} \, dx","Integrate[(E^((3*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^(11/2),x]","\frac{i (3 a x+i) \sqrt{a^2 x^2+1}}{24 a^3 c^5 (a x-i)^3 (a x+i)^6 \sqrt{a^2 c x^2+c}}","-\frac{(3 a x+i) \sqrt{a^2 x^2+1}}{24 a^3 c^5 (1-i a x)^6 (1+i a x)^3 \sqrt{a^2 c x^2+c}}",1,"((I/24)*(I + 3*a*x)*Sqrt[1 + a^2*x^2])/(a^3*c^5*(-I + a*x)^3*(I + a*x)^6*Sqrt[c + a^2*c*x^2])","A",1
382,1,74,142,0.0588336,"\int \frac{e^{i \tan ^{-1}(a x)} x^2}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(E^(I*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^(3/2),x]","\frac{\sqrt{a^2 x^2+1} \left(-\frac{2}{a x+i}+i \log (-a x+i)+3 i \log (a x+i)\right)}{4 a^3 c \sqrt{a^2 c x^2+c}}","-\frac{\sqrt{a^2 x^2+1}}{2 a^3 c (a x+i) \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \log (-a x+i)}{4 a^3 c \sqrt{a^2 c x^2+c}}+\frac{3 i \sqrt{a^2 x^2+1} \log (a x+i)}{4 a^3 c \sqrt{a^2 c x^2+c}}",1,"(Sqrt[1 + a^2*x^2]*(-2/(I + a*x) + I*Log[I - a*x] + (3*I)*Log[I + a*x]))/(4*a^3*c*Sqrt[c + a^2*c*x^2])","A",1
383,1,75,143,0.0640863,"\int \frac{e^{-i \tan ^{-1}(a x)} x^2}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[x^2/(E^(I*ArcTan[a*x])*(c + a^2*c*x^2)^(3/2)),x]","\frac{\sqrt{a^2 x^2+1} \left(\frac{2}{-a x+i}-3 i \log (-a x+i)-i \log (a x+i)\right)}{4 a^3 c \sqrt{a^2 c x^2+c}}","\frac{\sqrt{a^2 x^2+1}}{2 a^3 c (-a x+i) \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \log (-a x+i)}{4 a^3 c \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \log (a x+i)}{4 a^3 c \sqrt{a^2 c x^2+c}}",1,"(Sqrt[1 + a^2*x^2]*(2/(I - a*x) - (3*I)*Log[I - a*x] - I*Log[I + a*x]))/(4*a^3*c*Sqrt[c + a^2*c*x^2])","A",1
384,1,65,65,0.1180627,"\int \frac{e^{-3 i \tan ^{-1}(a x)} x^2}{\left(c+a^2 c x^2\right)^{11/2}} \, dx","Integrate[x^2/(E^((3*I)*ArcTan[a*x])*(c + a^2*c*x^2)^(11/2)),x]","-\frac{i (3 a x-i) \sqrt{a^2 x^2+1}}{24 a^3 c^5 (a x-i)^6 (a x+i)^3 \sqrt{a^2 c x^2+c}}","\frac{(-3 a x+i) \sqrt{a^2 x^2+1}}{24 a^3 c^5 (1-i a x)^3 (1+i a x)^6 \sqrt{a^2 c x^2+c}}",1,"((-1/24*I)*(-I + 3*a*x)*Sqrt[1 + a^2*x^2])/(a^3*c^5*(-I + a*x)^6*(I + a*x)^3*Sqrt[c + a^2*c*x^2])","A",1
385,1,63,65,0.601448,"\int \frac{e^{-5 i \tan ^{-1}(a x)} x^2}{\left(c+a^2 c x^2\right)^{27/2}} \, dx","Integrate[x^2/(E^((5*I)*ArcTan[a*x])*(c + a^2*c*x^2)^(27/2)),x]","\frac{(1+5 i a x) \sqrt{a^2 x^2+1}}{120 a^3 c^{13} (a x-i)^{15} (a x+i)^{10} \sqrt{a^2 c x^2+c}}","\frac{(-5 a x+i) \sqrt{a^2 x^2+1}}{120 a^3 c^{13} (1-i a x)^{10} (1+i a x)^{15} \sqrt{a^2 c x^2+c}}",1,"((1 + (5*I)*a*x)*Sqrt[1 + a^2*x^2])/(120*a^3*c^13*(-I + a*x)^15*(I + a*x)^10*Sqrt[c + a^2*c*x^2])","A",1