1,1,117,0,0.1033169,"\int x^3 (d+i c d x) \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^3*(d + I*c*d*x)*(a + b*ArcTan[c*x]),x]","\frac{1}{5} i c d x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} d x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{i b d x^2}{10 c^2}-\frac{i b d \log \left(c^2 x^2+1\right)}{10 c^4}+\frac{b d x}{4 c^3}-\frac{b d \tan ^{-1}(c x)}{4 c^4}-\frac{b d x^3}{12 c}-\frac{1}{20} i b d x^4","\frac{1}{5} i c d x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} d x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{i b d x^2}{10 c^2}-\frac{i b d \log \left(c^2 x^2+1\right)}{10 c^4}+\frac{b d x}{4 c^3}-\frac{b d \tan ^{-1}(c x)}{4 c^4}-\frac{b d x^3}{12 c}-\frac{1}{20} i b d x^4",1,"(b*d*x)/(4*c^3) + ((I/10)*b*d*x^2)/c^2 - (b*d*x^3)/(12*c) - (I/20)*b*d*x^4 - (b*d*ArcTan[c*x])/(4*c^4) + (d*x^4*(a + b*ArcTan[c*x]))/4 + (I/5)*c*d*x^5*(a + b*ArcTan[c*x]) - ((I/10)*b*d*Log[1 + c^2*x^2])/c^4","A",7,7,21,0.3333,1,"{43, 4872, 12, 801, 635, 203, 260}"
2,1,105,0,0.0942691,"\int x^2 (d+i c d x) \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^2*(d + I*c*d*x)*(a + b*ArcTan[c*x]),x]","\frac{1}{4} i c d x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} d x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{b d \log \left(c^2 x^2+1\right)}{6 c^3}+\frac{i b d x}{4 c^2}-\frac{i b d \tan ^{-1}(c x)}{4 c^3}-\frac{b d x^2}{6 c}-\frac{1}{12} i b d x^3","\frac{1}{4} i c d x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} d x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{b d \log \left(c^2 x^2+1\right)}{6 c^3}+\frac{i b d x}{4 c^2}-\frac{i b d \tan ^{-1}(c x)}{4 c^3}-\frac{b d x^2}{6 c}-\frac{1}{12} i b d x^3",1,"((I/4)*b*d*x)/c^2 - (b*d*x^2)/(6*c) - (I/12)*b*d*x^3 - ((I/4)*b*d*ArcTan[c*x])/c^3 + (d*x^3*(a + b*ArcTan[c*x]))/3 + (I/4)*c*d*x^4*(a + b*ArcTan[c*x]) + (b*d*Log[1 + c^2*x^2])/(6*c^3)","A",7,7,21,0.3333,1,"{43, 4872, 12, 801, 635, 203, 260}"
3,1,91,0,0.0773123,"\int x (d+i c d x) \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x*(d + I*c*d*x)*(a + b*ArcTan[c*x]),x]","\frac{1}{3} i c d x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{2} d x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{i b d \log \left(c^2 x^2+1\right)}{6 c^2}+\frac{b d \tan ^{-1}(c x)}{2 c^2}-\frac{b d x}{2 c}-\frac{1}{6} i b d x^2","\frac{1}{3} i c d x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{2} d x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{i b d \log \left(c^2 x^2+1\right)}{6 c^2}+\frac{b d \tan ^{-1}(c x)}{2 c^2}-\frac{b d x}{2 c}-\frac{1}{6} i b d x^2",1,"-(b*d*x)/(2*c) - (I/6)*b*d*x^2 + (b*d*ArcTan[c*x])/(2*c^2) + (d*x^2*(a + b*ArcTan[c*x]))/2 + (I/3)*c*d*x^3*(a + b*ArcTan[c*x]) + ((I/6)*b*d*Log[1 + c^2*x^2])/c^2","A",7,7,19,0.3684,1,"{43, 4872, 12, 801, 635, 203, 260}"
4,1,53,0,0.0312274,"\int (d+i c d x) \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[(d + I*c*d*x)*(a + b*ArcTan[c*x]),x]","-\frac{i d (1+i c x)^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c}-\frac{b d \log (c x+i)}{c}-\frac{1}{2} i b d x","-\frac{i d (1+i c x)^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c}-\frac{b d \log (c x+i)}{c}-\frac{1}{2} i b d x",1,"(-I/2)*b*d*x - ((I/2)*d*(1 + I*c*x)^2*(a + b*ArcTan[c*x]))/c - (b*d*Log[I + c*x])/c","A",4,3,18,0.1667,1,"{4862, 627, 43}"
5,1,76,0,0.0858881,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Int[((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x,x]","\frac{1}{2} i b d \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d \text{PolyLog}(2,i c x)+i a c d x+a d \log (x)-\frac{1}{2} i b d \log \left(c^2 x^2+1\right)+i b c d x \tan ^{-1}(c x)","\frac{1}{2} i b d \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d \text{PolyLog}(2,i c x)+i a c d x+a d \log (x)-\frac{1}{2} i b d \log \left(c^2 x^2+1\right)+i b c d x \tan ^{-1}(c x)",1,"I*a*c*d*x + I*b*c*d*x*ArcTan[c*x] + a*d*Log[x] - (I/2)*b*d*Log[1 + c^2*x^2] + (I/2)*b*d*PolyLog[2, (-I)*c*x] - (I/2)*b*d*PolyLog[2, I*c*x]","A",8,5,21,0.2381,1,"{4876, 4846, 260, 4848, 2391}"
6,1,77,0,0.1002277,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Int[((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x^2,x]","-\frac{1}{2} b c d \text{PolyLog}(2,-i c x)+\frac{1}{2} b c d \text{PolyLog}(2,i c x)-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{x}+i a c d \log (x)-\frac{1}{2} b c d \log \left(c^2 x^2+1\right)+b c d \log (x)","-\frac{1}{2} b c d \text{PolyLog}(2,-i c x)+\frac{1}{2} b c d \text{PolyLog}(2,i c x)-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{x}+i a c d \log (x)-\frac{1}{2} b c d \log \left(c^2 x^2+1\right)+b c d \log (x)",1,"-((d*(a + b*ArcTan[c*x]))/x) + I*a*c*d*Log[x] + b*c*d*Log[x] - (b*c*d*Log[1 + c^2*x^2])/2 - (b*c*d*PolyLog[2, (-I)*c*x])/2 + (b*c*d*PolyLog[2, I*c*x])/2","A",10,8,21,0.3810,1,"{4876, 4852, 266, 36, 29, 31, 4848, 2391}"
7,1,65,0,0.0552368,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Int[((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x^3,x]","-\frac{d (1+i c x)^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}+i b c^2 d \log (x)-i b c^2 d \log (c x+i)-\frac{b c d}{2 x}","-\frac{d (1+i c x)^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}+i b c^2 d \log (x)-i b c^2 d \log (c x+i)-\frac{b c d}{2 x}",1,"-(b*c*d)/(2*x) - (d*(1 + I*c*x)^2*(a + b*ArcTan[c*x]))/(2*x^2) + I*b*c^2*d*Log[x] - I*b*c^2*d*Log[I + c*x]","A",4,4,21,0.1905,1,"{37, 4872, 12, 77}"
8,1,106,0,0.0908906,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Int[((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x^4,x]","-\frac{i c d \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{i b c^2 d}{2 x}-\frac{1}{3} b c^3 d \log (x)-\frac{1}{12} b c^3 d \log (-c x+i)+\frac{5}{12} b c^3 d \log (c x+i)-\frac{b c d}{6 x^2}","-\frac{i c d \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{i b c^2 d}{2 x}-\frac{1}{3} b c^3 d \log (x)-\frac{1}{12} b c^3 d \log (-c x+i)+\frac{5}{12} b c^3 d \log (c x+i)-\frac{b c d}{6 x^2}",1,"-(b*c*d)/(6*x^2) - ((I/2)*b*c^2*d)/x - (d*(a + b*ArcTan[c*x]))/(3*x^3) - ((I/2)*c*d*(a + b*ArcTan[c*x]))/x^2 - (b*c^3*d*Log[x])/3 - (b*c^3*d*Log[I - c*x])/12 + (5*b*c^3*d*Log[I + c*x])/12","A",4,4,21,0.1905,1,"{43, 4872, 12, 801}"
9,1,124,0,0.0964044,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Int[((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x^5,x]","-\frac{i c d \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{i b c^2 d}{6 x^2}+\frac{b c^3 d}{4 x}-\frac{1}{3} i b c^4 d \log (x)+\frac{1}{24} i b c^4 d \log (-c x+i)+\frac{7}{24} i b c^4 d \log (c x+i)-\frac{b c d}{12 x^3}","-\frac{i c d \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{i b c^2 d}{6 x^2}+\frac{b c^3 d}{4 x}-\frac{1}{3} i b c^4 d \log (x)+\frac{1}{24} i b c^4 d \log (-c x+i)+\frac{7}{24} i b c^4 d \log (c x+i)-\frac{b c d}{12 x^3}",1,"-(b*c*d)/(12*x^3) - ((I/6)*b*c^2*d)/x^2 + (b*c^3*d)/(4*x) - (d*(a + b*ArcTan[c*x]))/(4*x^4) - ((I/3)*c*d*(a + b*ArcTan[c*x]))/x^3 - (I/3)*b*c^4*d*Log[x] + (I/24)*b*c^4*d*Log[I - c*x] + ((7*I)/24)*b*c^4*d*Log[I + c*x]","A",4,4,21,0.1905,1,"{43, 4872, 12, 801}"
10,1,166,0,0.1633698,"\int x^3 (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^3*(d + I*c*d*x)^2*(a + b*ArcTan[c*x]),x]","-\frac{1}{6} c^2 d^2 x^6 \left(a+b \tan ^{-1}(c x)\right)+\frac{2}{5} i c d^2 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{i b d^2 x^2}{5 c^2}-\frac{i b d^2 \log \left(c^2 x^2+1\right)}{5 c^4}+\frac{5 b d^2 x}{12 c^3}-\frac{5 b d^2 \tan ^{-1}(c x)}{12 c^4}+\frac{1}{30} b c d^2 x^5-\frac{5 b d^2 x^3}{36 c}-\frac{1}{10} i b d^2 x^4","-\frac{1}{6} c^2 d^2 x^6 \left(a+b \tan ^{-1}(c x)\right)+\frac{2}{5} i c d^2 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{i b d^2 x^2}{5 c^2}-\frac{i b d^2 \log \left(c^2 x^2+1\right)}{5 c^4}+\frac{5 b d^2 x}{12 c^3}-\frac{5 b d^2 \tan ^{-1}(c x)}{12 c^4}+\frac{1}{30} b c d^2 x^5-\frac{5 b d^2 x^3}{36 c}-\frac{1}{10} i b d^2 x^4",1,"(5*b*d^2*x)/(12*c^3) + ((I/5)*b*d^2*x^2)/c^2 - (5*b*d^2*x^3)/(36*c) - (I/10)*b*d^2*x^4 + (b*c*d^2*x^5)/30 - (5*b*d^2*ArcTan[c*x])/(12*c^4) + (d^2*x^4*(a + b*ArcTan[c*x]))/4 + ((2*I)/5)*c*d^2*x^5*(a + b*ArcTan[c*x]) - (c^2*d^2*x^6*(a + b*ArcTan[c*x]))/6 - ((I/5)*b*d^2*Log[1 + c^2*x^2])/c^4","A",7,7,23,0.3043,1,"{43, 4872, 12, 1802, 635, 203, 260}"
11,1,152,0,0.1504379,"\int x^2 (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^2*(d + I*c*d*x)^2*(a + b*ArcTan[c*x]),x]","-\frac{1}{5} c^2 d^2 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{2} i c d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{4 b d^2 \log \left(c^2 x^2+1\right)}{15 c^3}+\frac{i b d^2 x}{2 c^2}-\frac{i b d^2 \tan ^{-1}(c x)}{2 c^3}+\frac{1}{20} b c d^2 x^4-\frac{4 b d^2 x^2}{15 c}-\frac{1}{6} i b d^2 x^3","-\frac{1}{5} c^2 d^2 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{2} i c d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{4 b d^2 \log \left(c^2 x^2+1\right)}{15 c^3}+\frac{i b d^2 x}{2 c^2}-\frac{i b d^2 \tan ^{-1}(c x)}{2 c^3}+\frac{1}{20} b c d^2 x^4-\frac{4 b d^2 x^2}{15 c}-\frac{1}{6} i b d^2 x^3",1,"((I/2)*b*d^2*x)/c^2 - (4*b*d^2*x^2)/(15*c) - (I/6)*b*d^2*x^3 + (b*c*d^2*x^4)/20 - ((I/2)*b*d^2*ArcTan[c*x])/c^3 + (d^2*x^3*(a + b*ArcTan[c*x]))/3 + (I/2)*c*d^2*x^4*(a + b*ArcTan[c*x]) - (c^2*d^2*x^5*(a + b*ArcTan[c*x]))/5 + (4*b*d^2*Log[1 + c^2*x^2])/(15*c^3)","A",7,7,23,0.3043,1,"{43, 4872, 12, 1802, 635, 203, 260}"
12,1,136,0,0.1270636,"\int x (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x*(d + I*c*d*x)^2*(a + b*ArcTan[c*x]),x]","-\frac{1}{4} c^2 d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{2}{3} i c d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{2} d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{i b d^2 \log \left(c^2 x^2+1\right)}{3 c^2}+\frac{3 b d^2 \tan ^{-1}(c x)}{4 c^2}+\frac{1}{12} b c d^2 x^3-\frac{3 b d^2 x}{4 c}-\frac{1}{3} i b d^2 x^2","-\frac{1}{4} c^2 d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{2}{3} i c d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{2} d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{i b d^2 \log \left(c^2 x^2+1\right)}{3 c^2}+\frac{3 b d^2 \tan ^{-1}(c x)}{4 c^2}+\frac{1}{12} b c d^2 x^3-\frac{3 b d^2 x}{4 c}-\frac{1}{3} i b d^2 x^2",1,"(-3*b*d^2*x)/(4*c) - (I/3)*b*d^2*x^2 + (b*c*d^2*x^3)/12 + (3*b*d^2*ArcTan[c*x])/(4*c^2) + (d^2*x^2*(a + b*ArcTan[c*x]))/2 + ((2*I)/3)*c*d^2*x^3*(a + b*ArcTan[c*x]) - (c^2*d^2*x^4*(a + b*ArcTan[c*x]))/4 + ((I/3)*b*d^2*Log[1 + c^2*x^2])/c^2","A",7,7,21,0.3333,1,"{43, 4872, 12, 1802, 635, 203, 260}"
13,1,83,0,0.0456393,"\int (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[(d + I*c*d*x)^2*(a + b*ArcTan[c*x]),x]","-\frac{i d^2 (1+i c x)^3 \left(a+b \tan ^{-1}(c x)\right)}{3 c}-\frac{b d^2 (1+i c x)^2}{6 c}-\frac{4 b d^2 \log (1-i c x)}{3 c}-\frac{2}{3} i b d^2 x","-\frac{i d^2 (1+i c x)^3 \left(a+b \tan ^{-1}(c x)\right)}{3 c}-\frac{b d^2 (1+i c x)^2}{6 c}-\frac{4 b d^2 \log (1-i c x)}{3 c}-\frac{2}{3} i b d^2 x",1,"((-2*I)/3)*b*d^2*x - (b*d^2*(1 + I*c*x)^2)/(6*c) - ((I/3)*d^2*(1 + I*c*x)^3*(a + b*ArcTan[c*x]))/c - (4*b*d^2*Log[1 - I*c*x])/(3*c)","A",4,3,20,0.1500,1,"{4862, 627, 43}"
14,1,129,0,0.1268597,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Int[((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x,x]","\frac{1}{2} i b d^2 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^2 \text{PolyLog}(2,i c x)-\frac{1}{2} c^2 d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)+2 i a c d^2 x+a d^2 \log (x)-i b d^2 \log \left(c^2 x^2+1\right)+\frac{1}{2} b c d^2 x-\frac{1}{2} b d^2 \tan ^{-1}(c x)+2 i b c d^2 x \tan ^{-1}(c x)","\frac{1}{2} i b d^2 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^2 \text{PolyLog}(2,i c x)-\frac{1}{2} c^2 d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)+2 i a c d^2 x+a d^2 \log (x)-i b d^2 \log \left(c^2 x^2+1\right)+\frac{1}{2} b c d^2 x-\frac{1}{2} b d^2 \tan ^{-1}(c x)+2 i b c d^2 x \tan ^{-1}(c x)",1,"(2*I)*a*c*d^2*x + (b*c*d^2*x)/2 - (b*d^2*ArcTan[c*x])/2 + (2*I)*b*c*d^2*x*ArcTan[c*x] - (c^2*d^2*x^2*(a + b*ArcTan[c*x]))/2 + a*d^2*Log[x] - I*b*d^2*Log[1 + c^2*x^2] + (I/2)*b*d^2*PolyLog[2, (-I)*c*x] - (I/2)*b*d^2*PolyLog[2, I*c*x]","A",11,8,23,0.3478,1,"{4876, 4846, 260, 4848, 2391, 4852, 321, 203}"
15,1,89,0,0.1375756,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Int[((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^2,x]","-b c d^2 \text{PolyLog}(2,-i c x)+b c d^2 \text{PolyLog}(2,i c x)-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{x}-a c^2 d^2 x+2 i a c d^2 \log (x)-b c^2 d^2 x \tan ^{-1}(c x)+b c d^2 \log (x)","-b c d^2 \text{PolyLog}(2,-i c x)+b c d^2 \text{PolyLog}(2,i c x)-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{x}-a c^2 d^2 x+2 i a c d^2 \log (x)-b c^2 d^2 x \tan ^{-1}(c x)+b c d^2 \log (x)",1,"-(a*c^2*d^2*x) - b*c^2*d^2*x*ArcTan[c*x] - (d^2*(a + b*ArcTan[c*x]))/x + (2*I)*a*c*d^2*Log[x] + b*c*d^2*Log[x] - b*c*d^2*PolyLog[2, (-I)*c*x] + b*c*d^2*PolyLog[2, I*c*x]","A",13,10,23,0.4348,1,"{4876, 4846, 260, 4852, 266, 36, 29, 31, 4848, 2391}"
16,1,152,0,0.1535779,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Int[((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^3,x]","-\frac{1}{2} i b c^2 d^2 \text{PolyLog}(2,-i c x)+\frac{1}{2} i b c^2 d^2 \text{PolyLog}(2,i c x)-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{2 i c d^2 \left(a+b \tan ^{-1}(c x)\right)}{x}-a c^2 d^2 \log (x)-i b c^2 d^2 \log \left(c^2 x^2+1\right)+2 i b c^2 d^2 \log (x)-\frac{1}{2} b c^2 d^2 \tan ^{-1}(c x)-\frac{b c d^2}{2 x}","-\frac{1}{2} i b c^2 d^2 \text{PolyLog}(2,-i c x)+\frac{1}{2} i b c^2 d^2 \text{PolyLog}(2,i c x)-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{2 i c d^2 \left(a+b \tan ^{-1}(c x)\right)}{x}-a c^2 d^2 \log (x)-i b c^2 d^2 \log \left(c^2 x^2+1\right)+2 i b c^2 d^2 \log (x)-\frac{1}{2} b c^2 d^2 \tan ^{-1}(c x)-\frac{b c d^2}{2 x}",1,"-(b*c*d^2)/(2*x) - (b*c^2*d^2*ArcTan[c*x])/2 - (d^2*(a + b*ArcTan[c*x]))/(2*x^2) - ((2*I)*c*d^2*(a + b*ArcTan[c*x]))/x - a*c^2*d^2*Log[x] + (2*I)*b*c^2*d^2*Log[x] - I*b*c^2*d^2*Log[1 + c^2*x^2] - (I/2)*b*c^2*d^2*PolyLog[2, (-I)*c*x] + (I/2)*b*c^2*d^2*PolyLog[2, I*c*x]","A",13,10,23,0.4348,1,"{4876, 4852, 325, 203, 266, 36, 29, 31, 4848, 2391}"
17,1,87,0,0.0818295,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Int[((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^4,x]","-\frac{d^2 (1+i c x)^3 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{i b c^2 d^2}{x}-\frac{4}{3} b c^3 d^2 \log (x)+\frac{4}{3} b c^3 d^2 \log (c x+i)-\frac{b c d^2}{6 x^2}","-\frac{d^2 (1+i c x)^3 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{i b c^2 d^2}{x}-\frac{4}{3} b c^3 d^2 \log (x)+\frac{4}{3} b c^3 d^2 \log (c x+i)-\frac{b c d^2}{6 x^2}",1,"-(b*c*d^2)/(6*x^2) - (I*b*c^2*d^2)/x - (d^2*(1 + I*c*x)^3*(a + b*ArcTan[c*x]))/(3*x^3) - (4*b*c^3*d^2*Log[x])/3 + (4*b*c^3*d^2*Log[I + c*x])/3","A",4,4,23,0.1739,1,"{37, 4872, 12, 88}"
18,1,161,0,0.1498807,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Int[((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^5,x]","\frac{c^2 d^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{2 i c d^2 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{i b c^2 d^2}{3 x^2}+\frac{3 b c^3 d^2}{4 x}-\frac{2}{3} i b c^4 d^2 \log (x)-\frac{1}{24} i b c^4 d^2 \log (-c x+i)+\frac{17}{24} i b c^4 d^2 \log (c x+i)-\frac{b c d^2}{12 x^3}","\frac{c^2 d^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{2 i c d^2 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{i b c^2 d^2}{3 x^2}+\frac{3 b c^3 d^2}{4 x}-\frac{2}{3} i b c^4 d^2 \log (x)-\frac{1}{24} i b c^4 d^2 \log (-c x+i)+\frac{17}{24} i b c^4 d^2 \log (c x+i)-\frac{b c d^2}{12 x^3}",1,"-(b*c*d^2)/(12*x^3) - ((I/3)*b*c^2*d^2)/x^2 + (3*b*c^3*d^2)/(4*x) - (d^2*(a + b*ArcTan[c*x]))/(4*x^4) - (((2*I)/3)*c*d^2*(a + b*ArcTan[c*x]))/x^3 + (c^2*d^2*(a + b*ArcTan[c*x]))/(2*x^2) - ((2*I)/3)*b*c^4*d^2*Log[x] - (I/24)*b*c^4*d^2*Log[I - c*x] + ((17*I)/24)*b*c^4*d^2*Log[I + c*x]","A",4,4,23,0.1739,1,"{43, 4872, 12, 1802}"
19,1,171,0,0.1580009,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Int[((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^6,x]","\frac{c^2 d^2 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{i c d^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^4}-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}+\frac{4 b c^3 d^2}{15 x^2}-\frac{i b c^2 d^2}{6 x^3}+\frac{i b c^4 d^2}{2 x}+\frac{8}{15} b c^5 d^2 \log (x)-\frac{1}{60} b c^5 d^2 \log (-c x+i)-\frac{31}{60} b c^5 d^2 \log (c x+i)-\frac{b c d^2}{20 x^4}","\frac{c^2 d^2 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{i c d^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^4}-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}+\frac{4 b c^3 d^2}{15 x^2}-\frac{i b c^2 d^2}{6 x^3}+\frac{i b c^4 d^2}{2 x}+\frac{8}{15} b c^5 d^2 \log (x)-\frac{1}{60} b c^5 d^2 \log (-c x+i)-\frac{31}{60} b c^5 d^2 \log (c x+i)-\frac{b c d^2}{20 x^4}",1,"-(b*c*d^2)/(20*x^4) - ((I/6)*b*c^2*d^2)/x^3 + (4*b*c^3*d^2)/(15*x^2) + ((I/2)*b*c^4*d^2)/x - (d^2*(a + b*ArcTan[c*x]))/(5*x^5) - ((I/2)*c*d^2*(a + b*ArcTan[c*x]))/x^4 + (c^2*d^2*(a + b*ArcTan[c*x]))/(3*x^3) + (8*b*c^5*d^2*Log[x])/15 - (b*c^5*d^2*Log[I - c*x])/60 - (31*b*c^5*d^2*Log[I + c*x])/60","A",4,4,23,0.1739,1,"{43, 4872, 12, 1802}"
20,1,205,0,0.1838641,"\int x^3 (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^3*(d + I*c*d*x)^3*(a + b*ArcTan[c*x]),x]","-\frac{1}{7} i c^3 d^3 x^7 \left(a+b \tan ^{-1}(c x)\right)-\frac{1}{2} c^2 d^3 x^6 \left(a+b \tan ^{-1}(c x)\right)+\frac{3}{5} i c d^3 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} d^3 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{42} i b c^2 d^3 x^6+\frac{13 i b d^3 x^2}{35 c^2}-\frac{13 i b d^3 \log \left(c^2 x^2+1\right)}{35 c^4}+\frac{3 b d^3 x}{4 c^3}-\frac{3 b d^3 \tan ^{-1}(c x)}{4 c^4}+\frac{1}{10} b c d^3 x^5-\frac{b d^3 x^3}{4 c}-\frac{13}{70} i b d^3 x^4","-\frac{1}{7} i c^3 d^3 x^7 \left(a+b \tan ^{-1}(c x)\right)-\frac{1}{2} c^2 d^3 x^6 \left(a+b \tan ^{-1}(c x)\right)+\frac{3}{5} i c d^3 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} d^3 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{42} i b c^2 d^3 x^6+\frac{13 i b d^3 x^2}{35 c^2}-\frac{13 i b d^3 \log \left(c^2 x^2+1\right)}{35 c^4}+\frac{3 b d^3 x}{4 c^3}-\frac{3 b d^3 \tan ^{-1}(c x)}{4 c^4}+\frac{1}{10} b c d^3 x^5-\frac{b d^3 x^3}{4 c}-\frac{13}{70} i b d^3 x^4",1,"(3*b*d^3*x)/(4*c^3) + (((13*I)/35)*b*d^3*x^2)/c^2 - (b*d^3*x^3)/(4*c) - ((13*I)/70)*b*d^3*x^4 + (b*c*d^3*x^5)/10 + (I/42)*b*c^2*d^3*x^6 - (3*b*d^3*ArcTan[c*x])/(4*c^4) + (d^3*x^4*(a + b*ArcTan[c*x]))/4 + ((3*I)/5)*c*d^3*x^5*(a + b*ArcTan[c*x]) - (c^2*d^3*x^6*(a + b*ArcTan[c*x]))/2 - (I/7)*c^3*d^3*x^7*(a + b*ArcTan[c*x]) - (((13*I)/35)*b*d^3*Log[1 + c^2*x^2])/c^4","A",7,7,23,0.3043,1,"{43, 4872, 12, 1802, 635, 203, 260}"
21,1,191,0,0.171228,"\int x^2 (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^2*(d + I*c*d*x)^3*(a + b*ArcTan[c*x]),x]","-\frac{1}{6} i c^3 d^3 x^6 \left(a+b \tan ^{-1}(c x)\right)-\frac{3}{5} c^2 d^3 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{3}{4} i c d^3 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{30} i b c^2 d^3 x^5+\frac{7 b d^3 \log \left(c^2 x^2+1\right)}{15 c^3}+\frac{11 i b d^3 x}{12 c^2}-\frac{11 i b d^3 \tan ^{-1}(c x)}{12 c^3}+\frac{3}{20} b c d^3 x^4-\frac{7 b d^3 x^2}{15 c}-\frac{11}{36} i b d^3 x^3","-\frac{1}{6} i c^3 d^3 x^6 \left(a+b \tan ^{-1}(c x)\right)-\frac{3}{5} c^2 d^3 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{3}{4} i c d^3 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{30} i b c^2 d^3 x^5+\frac{7 b d^3 \log \left(c^2 x^2+1\right)}{15 c^3}+\frac{11 i b d^3 x}{12 c^2}-\frac{11 i b d^3 \tan ^{-1}(c x)}{12 c^3}+\frac{3}{20} b c d^3 x^4-\frac{7 b d^3 x^2}{15 c}-\frac{11}{36} i b d^3 x^3",1,"(((11*I)/12)*b*d^3*x)/c^2 - (7*b*d^3*x^2)/(15*c) - ((11*I)/36)*b*d^3*x^3 + (3*b*c*d^3*x^4)/20 + (I/30)*b*c^2*d^3*x^5 - (((11*I)/12)*b*d^3*ArcTan[c*x])/c^3 + (d^3*x^3*(a + b*ArcTan[c*x]))/3 + ((3*I)/4)*c*d^3*x^4*(a + b*ArcTan[c*x]) - (3*c^2*d^3*x^5*(a + b*ArcTan[c*x]))/5 - (I/6)*c^3*d^3*x^6*(a + b*ArcTan[c*x]) + (7*b*d^3*Log[1 + c^2*x^2])/(15*c^3)","A",7,7,23,0.3043,1,"{43, 4872, 12, 1802, 635, 203, 260}"
22,1,157,0,0.0968673,"\int x (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x*(d + I*c*d*x)^3*(a + b*ArcTan[c*x]),x]","-\frac{d^3 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{5 c^2}+\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)}{4 c^2}+\frac{i b d^3 (-c x+i)^4}{20 c^2}-\frac{b d^3 (-c x+i)^3}{20 c^2}-\frac{3 i b d^3 (-c x+i)^2}{20 c^2}+\frac{6 i b d^3 \log (c x+i)}{5 c^2}-\frac{3 b d^3 x}{5 c}","-\frac{d^3 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{5 c^2}+\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)}{4 c^2}+\frac{i b d^3 (-c x+i)^4}{20 c^2}-\frac{b d^3 (-c x+i)^3}{20 c^2}-\frac{3 i b d^3 (-c x+i)^2}{20 c^2}+\frac{6 i b d^3 \log (c x+i)}{5 c^2}-\frac{3 b d^3 x}{5 c}",1,"(-3*b*d^3*x)/(5*c) - (((3*I)/20)*b*d^3*(I - c*x)^2)/c^2 - (b*d^3*(I - c*x)^3)/(20*c^2) + ((I/20)*b*d^3*(I - c*x)^4)/c^2 + (d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x]))/(4*c^2) - (d^3*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/(5*c^2) + (((6*I)/5)*b*d^3*Log[I + c*x])/c^2","A",4,4,21,0.1905,1,"{43, 4872, 12, 77}"
23,1,100,0,0.0537698,"\int (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[(d + I*c*d*x)^3*(a + b*ArcTan[c*x]),x]","-\frac{i d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)}{4 c}-\frac{b d^3 (1+i c x)^3}{12 c}-\frac{b d^3 (1+i c x)^2}{4 c}-\frac{2 b d^3 \log (1-i c x)}{c}-i b d^3 x","-\frac{i d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)}{4 c}-\frac{b d^3 (1+i c x)^3}{12 c}-\frac{b d^3 (1+i c x)^2}{4 c}-\frac{2 b d^3 \log (1-i c x)}{c}-i b d^3 x",1,"(-I)*b*d^3*x - (b*d^3*(1 + I*c*x)^2)/(4*c) - (b*d^3*(1 + I*c*x)^3)/(12*c) - ((I/4)*d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x]))/c - (2*b*d^3*Log[1 - I*c*x])/c","A",4,3,20,0.1500,1,"{4862, 627, 43}"
24,1,170,0,0.1745104,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x,x]","\frac{1}{2} i b d^3 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^3 \text{PolyLog}(2,i c x)-\frac{1}{3} i c^3 d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{3}{2} c^2 d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)+3 i a c d^3 x+a d^3 \log (x)+\frac{1}{6} i b c^2 d^3 x^2-\frac{5}{3} i b d^3 \log \left(c^2 x^2+1\right)+\frac{3}{2} b c d^3 x-\frac{3}{2} b d^3 \tan ^{-1}(c x)+3 i b c d^3 x \tan ^{-1}(c x)","\frac{1}{2} i b d^3 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^3 \text{PolyLog}(2,i c x)-\frac{1}{3} i c^3 d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{3}{2} c^2 d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)+3 i a c d^3 x+a d^3 \log (x)+\frac{1}{6} i b c^2 d^3 x^2-\frac{5}{3} i b d^3 \log \left(c^2 x^2+1\right)+\frac{3}{2} b c d^3 x-\frac{3}{2} b d^3 \tan ^{-1}(c x)+3 i b c d^3 x \tan ^{-1}(c x)",1,"(3*I)*a*c*d^3*x + (3*b*c*d^3*x)/2 + (I/6)*b*c^2*d^3*x^2 - (3*b*d^3*ArcTan[c*x])/2 + (3*I)*b*c*d^3*x*ArcTan[c*x] - (3*c^2*d^3*x^2*(a + b*ArcTan[c*x]))/2 - (I/3)*c^3*d^3*x^3*(a + b*ArcTan[c*x]) + a*d^3*Log[x] - ((5*I)/3)*b*d^3*Log[1 + c^2*x^2] + (I/2)*b*d^3*PolyLog[2, (-I)*c*x] - (I/2)*b*d^3*PolyLog[2, I*c*x]","A",15,10,23,0.4348,1,"{4876, 4846, 260, 4848, 2391, 4852, 321, 203, 266, 43}"
25,1,162,0,0.1734821,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^2,x]","-\frac{3}{2} b c d^3 \text{PolyLog}(2,-i c x)+\frac{3}{2} b c d^3 \text{PolyLog}(2,i c x)-\frac{1}{2} i c^3 d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{x}-3 a c^2 d^3 x+3 i a c d^3 \log (x)+b c d^3 \log \left(c^2 x^2+1\right)+\frac{1}{2} i b c^2 d^3 x-3 b c^2 d^3 x \tan ^{-1}(c x)+b c d^3 \log (x)-\frac{1}{2} i b c d^3 \tan ^{-1}(c x)","-\frac{3}{2} b c d^3 \text{PolyLog}(2,-i c x)+\frac{3}{2} b c d^3 \text{PolyLog}(2,i c x)-\frac{1}{2} i c^3 d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{x}-3 a c^2 d^3 x+3 i a c d^3 \log (x)+b c d^3 \log \left(c^2 x^2+1\right)+\frac{1}{2} i b c^2 d^3 x-3 b c^2 d^3 x \tan ^{-1}(c x)+b c d^3 \log (x)-\frac{1}{2} i b c d^3 \tan ^{-1}(c x)",1,"-3*a*c^2*d^3*x + (I/2)*b*c^2*d^3*x - (I/2)*b*c*d^3*ArcTan[c*x] - 3*b*c^2*d^3*x*ArcTan[c*x] - (d^3*(a + b*ArcTan[c*x]))/x - (I/2)*c^3*d^3*x^2*(a + b*ArcTan[c*x]) + (3*I)*a*c*d^3*Log[x] + b*c*d^3*Log[x] + b*c*d^3*Log[1 + c^2*x^2] - (3*b*c*d^3*PolyLog[2, (-I)*c*x])/2 + (3*b*c*d^3*PolyLog[2, I*c*x])/2","A",16,12,23,0.5217,1,"{4876, 4846, 260, 4852, 266, 36, 29, 31, 4848, 2391, 321, 203}"
26,1,180,0,0.1780905,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^3,x]","-\frac{3}{2} i b c^2 d^3 \text{PolyLog}(2,-i c x)+\frac{3}{2} i b c^2 d^3 \text{PolyLog}(2,i c x)-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)}{x}-i a c^3 d^3 x-3 a c^2 d^3 \log (x)-i b c^2 d^3 \log \left(c^2 x^2+1\right)+3 i b c^2 d^3 \log (x)-\frac{1}{2} b c^2 d^3 \tan ^{-1}(c x)-i b c^3 d^3 x \tan ^{-1}(c x)-\frac{b c d^3}{2 x}","-\frac{3}{2} i b c^2 d^3 \text{PolyLog}(2,-i c x)+\frac{3}{2} i b c^2 d^3 \text{PolyLog}(2,i c x)-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)}{x}-i a c^3 d^3 x-3 a c^2 d^3 \log (x)-i b c^2 d^3 \log \left(c^2 x^2+1\right)+3 i b c^2 d^3 \log (x)-\frac{1}{2} b c^2 d^3 \tan ^{-1}(c x)-i b c^3 d^3 x \tan ^{-1}(c x)-\frac{b c d^3}{2 x}",1,"-(b*c*d^3)/(2*x) - I*a*c^3*d^3*x - (b*c^2*d^3*ArcTan[c*x])/2 - I*b*c^3*d^3*x*ArcTan[c*x] - (d^3*(a + b*ArcTan[c*x]))/(2*x^2) - ((3*I)*c*d^3*(a + b*ArcTan[c*x]))/x - 3*a*c^2*d^3*Log[x] + (3*I)*b*c^2*d^3*Log[x] - I*b*c^2*d^3*Log[1 + c^2*x^2] - ((3*I)/2)*b*c^2*d^3*PolyLog[2, (-I)*c*x] + ((3*I)/2)*b*c^2*d^3*PolyLog[2, I*c*x]","A",16,12,23,0.5217,1,"{4876, 4846, 260, 4852, 325, 203, 266, 36, 29, 31, 4848, 2391}"
27,1,189,0,0.2032952,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^4,x]","\frac{1}{2} b c^3 d^3 \text{PolyLog}(2,-i c x)-\frac{1}{2} b c^3 d^3 \text{PolyLog}(2,i c x)+\frac{3 c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-i a c^3 d^3 \log (x)+\frac{5}{3} b c^3 d^3 \log \left(c^2 x^2+1\right)-\frac{3 i b c^2 d^3}{2 x}-\frac{10}{3} b c^3 d^3 \log (x)-\frac{3}{2} i b c^3 d^3 \tan ^{-1}(c x)-\frac{b c d^3}{6 x^2}","\frac{1}{2} b c^3 d^3 \text{PolyLog}(2,-i c x)-\frac{1}{2} b c^3 d^3 \text{PolyLog}(2,i c x)+\frac{3 c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-i a c^3 d^3 \log (x)+\frac{5}{3} b c^3 d^3 \log \left(c^2 x^2+1\right)-\frac{3 i b c^2 d^3}{2 x}-\frac{10}{3} b c^3 d^3 \log (x)-\frac{3}{2} i b c^3 d^3 \tan ^{-1}(c x)-\frac{b c d^3}{6 x^2}",1,"-(b*c*d^3)/(6*x^2) - (((3*I)/2)*b*c^2*d^3)/x - ((3*I)/2)*b*c^3*d^3*ArcTan[c*x] - (d^3*(a + b*ArcTan[c*x]))/(3*x^3) - (((3*I)/2)*c*d^3*(a + b*ArcTan[c*x]))/x^2 + (3*c^2*d^3*(a + b*ArcTan[c*x]))/x - I*a*c^3*d^3*Log[x] - (10*b*c^3*d^3*Log[x])/3 + (5*b*c^3*d^3*Log[1 + c^2*x^2])/3 + (b*c^3*d^3*PolyLog[2, (-I)*c*x])/2 - (b*c^3*d^3*PolyLog[2, I*c*x])/2","A",17,11,23,0.4783,1,"{4876, 4852, 266, 44, 325, 203, 36, 29, 31, 4848, 2391}"
28,1,103,0,0.0906254,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^5,x]","-\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{i b c^2 d^3}{2 x^2}+\frac{7 b c^3 d^3}{4 x}-2 i b c^4 d^3 \log (x)+2 i b c^4 d^3 \log (c x+i)-\frac{b c d^3}{12 x^3}","-\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{i b c^2 d^3}{2 x^2}+\frac{7 b c^3 d^3}{4 x}-2 i b c^4 d^3 \log (x)+2 i b c^4 d^3 \log (c x+i)-\frac{b c d^3}{12 x^3}",1,"-(b*c*d^3)/(12*x^3) - ((I/2)*b*c^2*d^3)/x^2 + (7*b*c^3*d^3)/(4*x) - (d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x]))/(4*x^4) - (2*I)*b*c^4*d^3*Log[x] + (2*I)*b*c^4*d^3*Log[I + c*x]","A",4,4,23,0.1739,1,"{37, 4872, 12, 88}"
29,1,150,0,0.1067871,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^6,x]","\frac{i c d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)}{20 x^4}-\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}+\frac{3 b c^3 d^3}{5 x^2}-\frac{i b c^2 d^3}{4 x^3}+\frac{5 i b c^4 d^3}{4 x}+\frac{6}{5} b c^5 d^3 \log (x)-\frac{6}{5} b c^5 d^3 \log (c x+i)-\frac{b c d^3}{20 x^4}","\frac{i c d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)}{20 x^4}-\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}+\frac{3 b c^3 d^3}{5 x^2}-\frac{i b c^2 d^3}{4 x^3}+\frac{5 i b c^4 d^3}{4 x}+\frac{6}{5} b c^5 d^3 \log (x)-\frac{6}{5} b c^5 d^3 \log (c x+i)-\frac{b c d^3}{20 x^4}",1,"-(b*c*d^3)/(20*x^4) - ((I/4)*b*c^2*d^3)/x^3 + (3*b*c^3*d^3)/(5*x^2) + (((5*I)/4)*b*c^4*d^3)/x - (d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x]))/(5*x^5) + ((I/20)*c*d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x]))/x^4 + (6*b*c^5*d^3*Log[x])/5 - (6*b*c^5*d^3*Log[I + c*x])/5","A",4,5,23,0.2174,1,"{45, 37, 4872, 12, 148}"
30,1,214,0,0.1784933,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^7} \, dx","Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^7,x]","\frac{i c^3 d^3 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}+\frac{3 c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{6 x^6}+\frac{7 i b c^4 d^3}{15 x^2}+\frac{11 b c^3 d^3}{36 x^3}-\frac{3 i b c^2 d^3}{20 x^4}-\frac{11 b c^5 d^3}{12 x}+\frac{14}{15} i b c^6 d^3 \log (x)-\frac{1}{120} i b c^6 d^3 \log (-c x+i)-\frac{37}{40} i b c^6 d^3 \log (c x+i)-\frac{b c d^3}{30 x^5}","\frac{i c^3 d^3 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}+\frac{3 c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{6 x^6}+\frac{7 i b c^4 d^3}{15 x^2}+\frac{11 b c^3 d^3}{36 x^3}-\frac{3 i b c^2 d^3}{20 x^4}-\frac{11 b c^5 d^3}{12 x}+\frac{14}{15} i b c^6 d^3 \log (x)-\frac{1}{120} i b c^6 d^3 \log (-c x+i)-\frac{37}{40} i b c^6 d^3 \log (c x+i)-\frac{b c d^3}{30 x^5}",1,"-(b*c*d^3)/(30*x^5) - (((3*I)/20)*b*c^2*d^3)/x^4 + (11*b*c^3*d^3)/(36*x^3) + (((7*I)/15)*b*c^4*d^3)/x^2 - (11*b*c^5*d^3)/(12*x) - (d^3*(a + b*ArcTan[c*x]))/(6*x^6) - (((3*I)/5)*c*d^3*(a + b*ArcTan[c*x]))/x^5 + (3*c^2*d^3*(a + b*ArcTan[c*x]))/(4*x^4) + ((I/3)*c^3*d^3*(a + b*ArcTan[c*x]))/x^3 + ((14*I)/15)*b*c^6*d^3*Log[x] - (I/120)*b*c^6*d^3*Log[I - c*x] - ((37*I)/40)*b*c^6*d^3*Log[I + c*x]","A",4,4,23,0.1739,1,"{43, 4872, 12, 1802}"
31,1,238,0,0.2142273,"\int x^3 (d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^3*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]),x]","\frac{1}{8} c^4 d^4 x^8 \left(a+b \tan ^{-1}(c x)\right)-\frac{4}{7} i c^3 d^4 x^7 \left(a+b \tan ^{-1}(c x)\right)-c^2 d^4 x^6 \left(a+b \tan ^{-1}(c x)\right)+\frac{4}{5} i c d^4 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} d^4 x^4 \left(a+b \tan ^{-1}(c x)\right)-\frac{1}{56} b c^3 d^4 x^7+\frac{2}{21} i b c^2 d^4 x^6+\frac{24 i b d^4 x^2}{35 c^2}-\frac{24 i b d^4 \log \left(c^2 x^2+1\right)}{35 c^4}+\frac{11 b d^4 x}{8 c^3}-\frac{11 b d^4 \tan ^{-1}(c x)}{8 c^4}+\frac{9}{40} b c d^4 x^5-\frac{11 b d^4 x^3}{24 c}-\frac{12}{35} i b d^4 x^4","\frac{1}{8} c^4 d^4 x^8 \left(a+b \tan ^{-1}(c x)\right)-\frac{4}{7} i c^3 d^4 x^7 \left(a+b \tan ^{-1}(c x)\right)-c^2 d^4 x^6 \left(a+b \tan ^{-1}(c x)\right)+\frac{4}{5} i c d^4 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} d^4 x^4 \left(a+b \tan ^{-1}(c x)\right)-\frac{1}{56} b c^3 d^4 x^7+\frac{2}{21} i b c^2 d^4 x^6+\frac{24 i b d^4 x^2}{35 c^2}-\frac{24 i b d^4 \log \left(c^2 x^2+1\right)}{35 c^4}+\frac{11 b d^4 x}{8 c^3}-\frac{11 b d^4 \tan ^{-1}(c x)}{8 c^4}+\frac{9}{40} b c d^4 x^5-\frac{11 b d^4 x^3}{24 c}-\frac{12}{35} i b d^4 x^4",1,"(11*b*d^4*x)/(8*c^3) + (((24*I)/35)*b*d^4*x^2)/c^2 - (11*b*d^4*x^3)/(24*c) - ((12*I)/35)*b*d^4*x^4 + (9*b*c*d^4*x^5)/40 + ((2*I)/21)*b*c^2*d^4*x^6 - (b*c^3*d^4*x^7)/56 - (11*b*d^4*ArcTan[c*x])/(8*c^4) + (d^4*x^4*(a + b*ArcTan[c*x]))/4 + ((4*I)/5)*c*d^4*x^5*(a + b*ArcTan[c*x]) - c^2*d^4*x^6*(a + b*ArcTan[c*x]) - ((4*I)/7)*c^3*d^4*x^7*(a + b*ArcTan[c*x]) + (c^4*d^4*x^8*(a + b*ArcTan[c*x]))/8 - (((24*I)/35)*b*d^4*Log[1 + c^2*x^2])/c^4","A",7,7,23,0.3043,1,"{43, 4872, 12, 1802, 635, 203, 260}"
32,1,193,0,0.1667175,"\int x^2 (d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^2*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]),x]","\frac{i d^4 (1+i c x)^7 \left(a+b \tan ^{-1}(c x)\right)}{7 c^3}-\frac{i d^4 (1+i c x)^6 \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}+\frac{i d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{5 c^3}-\frac{1}{42} b c^3 d^4 x^6+\frac{2}{15} i b c^2 d^4 x^5+\frac{5 i b d^4 x}{3 c^2}+\frac{176 b d^4 \log (c x+i)}{105 c^3}+\frac{47}{140} b c d^4 x^4-\frac{88 b d^4 x^2}{105 c}-\frac{5}{9} i b d^4 x^3","\frac{i d^4 (1+i c x)^7 \left(a+b \tan ^{-1}(c x)\right)}{7 c^3}-\frac{i d^4 (1+i c x)^6 \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}+\frac{i d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{5 c^3}-\frac{1}{42} b c^3 d^4 x^6+\frac{2}{15} i b c^2 d^4 x^5+\frac{5 i b d^4 x}{3 c^2}+\frac{176 b d^4 \log (c x+i)}{105 c^3}+\frac{47}{140} b c d^4 x^4-\frac{88 b d^4 x^2}{105 c}-\frac{5}{9} i b d^4 x^3",1,"(((5*I)/3)*b*d^4*x)/c^2 - (88*b*d^4*x^2)/(105*c) - ((5*I)/9)*b*d^4*x^3 + (47*b*c*d^4*x^4)/140 + ((2*I)/15)*b*c^2*d^4*x^5 - (b*c^3*d^4*x^6)/42 + ((I/5)*d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/c^3 - ((I/3)*d^4*(1 + I*c*x)^6*(a + b*ArcTan[c*x]))/c^3 + ((I/7)*d^4*(1 + I*c*x)^7*(a + b*ArcTan[c*x]))/c^3 + (176*b*d^4*Log[I + c*x])/(105*c^3)","A",4,4,23,0.1739,1,"{43, 4872, 12, 893}"
33,1,178,0,0.1125499,"\int x (d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]),x]","-\frac{d^4 (1+i c x)^6 \left(a+b \tan ^{-1}(c x)\right)}{6 c^2}+\frac{d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{5 c^2}+\frac{b d^4 (-c x+i)^5}{30 c^2}+\frac{i b d^4 (-c x+i)^4}{30 c^2}-\frac{4 b d^4 (-c x+i)^3}{45 c^2}-\frac{4 i b d^4 (-c x+i)^2}{15 c^2}+\frac{32 i b d^4 \log (c x+i)}{15 c^2}-\frac{16 b d^4 x}{15 c}","-\frac{d^4 (1+i c x)^6 \left(a+b \tan ^{-1}(c x)\right)}{6 c^2}+\frac{d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{5 c^2}+\frac{b d^4 (-c x+i)^5}{30 c^2}+\frac{i b d^4 (-c x+i)^4}{30 c^2}-\frac{4 b d^4 (-c x+i)^3}{45 c^2}-\frac{4 i b d^4 (-c x+i)^2}{15 c^2}+\frac{32 i b d^4 \log (c x+i)}{15 c^2}-\frac{16 b d^4 x}{15 c}",1,"(-16*b*d^4*x)/(15*c) - (((4*I)/15)*b*d^4*(I - c*x)^2)/c^2 - (4*b*d^4*(I - c*x)^3)/(45*c^2) + ((I/30)*b*d^4*(I - c*x)^4)/c^2 + (b*d^4*(I - c*x)^5)/(30*c^2) + (d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/(5*c^2) - (d^4*(1 + I*c*x)^6*(a + b*ArcTan[c*x]))/(6*c^2) + (((32*I)/15)*b*d^4*Log[I + c*x])/c^2","A",4,4,21,0.1905,1,"{43, 4872, 12, 77}"
34,1,125,0,0.063127,"\int (d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[(d + I*c*d*x)^4*(a + b*ArcTan[c*x]),x]","-\frac{i d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{5 c}-\frac{b d^4 (1+i c x)^4}{20 c}-\frac{2 b d^4 (1+i c x)^3}{15 c}-\frac{2 b d^4 (1+i c x)^2}{5 c}-\frac{16 b d^4 \log (1-i c x)}{5 c}-\frac{8}{5} i b d^4 x","-\frac{i d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{5 c}-\frac{b d^4 (1+i c x)^4}{20 c}-\frac{2 b d^4 (1+i c x)^3}{15 c}-\frac{2 b d^4 (1+i c x)^2}{5 c}-\frac{16 b d^4 \log (1-i c x)}{5 c}-\frac{8}{5} i b d^4 x",1,"((-8*I)/5)*b*d^4*x - (2*b*d^4*(1 + I*c*x)^2)/(5*c) - (2*b*d^4*(1 + I*c*x)^3)/(15*c) - (b*d^4*(1 + I*c*x)^4)/(20*c) - ((I/5)*d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/c - (16*b*d^4*Log[1 - I*c*x])/(5*c)","A",4,3,20,0.1500,1,"{4862, 627, 43}"
35,1,203,0,0.2107049,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Int[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x,x]","\frac{1}{2} i b d^4 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^4 \text{PolyLog}(2,i c x)+\frac{1}{4} c^4 d^4 x^4 \left(a+b \tan ^{-1}(c x)\right)-\frac{4}{3} i c^3 d^4 x^3 \left(a+b \tan ^{-1}(c x)\right)-3 c^2 d^4 x^2 \left(a+b \tan ^{-1}(c x)\right)+4 i a c d^4 x+a d^4 \log (x)-\frac{1}{12} b c^3 d^4 x^3+\frac{2}{3} i b c^2 d^4 x^2-\frac{8}{3} i b d^4 \log \left(c^2 x^2+1\right)+\frac{13}{4} b c d^4 x-\frac{13}{4} b d^4 \tan ^{-1}(c x)+4 i b c d^4 x \tan ^{-1}(c x)","\frac{1}{2} i b d^4 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^4 \text{PolyLog}(2,i c x)+\frac{1}{4} c^4 d^4 x^4 \left(a+b \tan ^{-1}(c x)\right)-\frac{4}{3} i c^3 d^4 x^3 \left(a+b \tan ^{-1}(c x)\right)-3 c^2 d^4 x^2 \left(a+b \tan ^{-1}(c x)\right)+4 i a c d^4 x+a d^4 \log (x)-\frac{1}{12} b c^3 d^4 x^3+\frac{2}{3} i b c^2 d^4 x^2-\frac{8}{3} i b d^4 \log \left(c^2 x^2+1\right)+\frac{13}{4} b c d^4 x-\frac{13}{4} b d^4 \tan ^{-1}(c x)+4 i b c d^4 x \tan ^{-1}(c x)",1,"(4*I)*a*c*d^4*x + (13*b*c*d^4*x)/4 + ((2*I)/3)*b*c^2*d^4*x^2 - (b*c^3*d^4*x^3)/12 - (13*b*d^4*ArcTan[c*x])/4 + (4*I)*b*c*d^4*x*ArcTan[c*x] - 3*c^2*d^4*x^2*(a + b*ArcTan[c*x]) - ((4*I)/3)*c^3*d^4*x^3*(a + b*ArcTan[c*x]) + (c^4*d^4*x^4*(a + b*ArcTan[c*x]))/4 + a*d^4*Log[x] - ((8*I)/3)*b*d^4*Log[1 + c^2*x^2] + (I/2)*b*d^4*PolyLog[2, (-I)*c*x] - (I/2)*b*d^4*PolyLog[2, I*c*x]","A",19,11,23,0.4783,1,"{4876, 4846, 260, 4848, 2391, 4852, 321, 203, 266, 43, 302}"
36,1,190,0,0.2065585,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Int[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^2,x]","-2 b c d^4 \text{PolyLog}(2,-i c x)+2 b c d^4 \text{PolyLog}(2,i c x)+\frac{1}{3} c^4 d^4 x^3 \left(a+b \tan ^{-1}(c x)\right)-2 i c^3 d^4 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{d^4 \left(a+b \tan ^{-1}(c x)\right)}{x}-6 a c^2 d^4 x+4 i a c d^4 \log (x)-\frac{1}{6} b c^3 d^4 x^2+\frac{8}{3} b c d^4 \log \left(c^2 x^2+1\right)+2 i b c^2 d^4 x-6 b c^2 d^4 x \tan ^{-1}(c x)+b c d^4 \log (x)-2 i b c d^4 \tan ^{-1}(c x)","-2 b c d^4 \text{PolyLog}(2,-i c x)+2 b c d^4 \text{PolyLog}(2,i c x)+\frac{1}{3} c^4 d^4 x^3 \left(a+b \tan ^{-1}(c x)\right)-2 i c^3 d^4 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{d^4 \left(a+b \tan ^{-1}(c x)\right)}{x}-6 a c^2 d^4 x+4 i a c d^4 \log (x)-\frac{1}{6} b c^3 d^4 x^2+\frac{8}{3} b c d^4 \log \left(c^2 x^2+1\right)+2 i b c^2 d^4 x-6 b c^2 d^4 x \tan ^{-1}(c x)+b c d^4 \log (x)-2 i b c d^4 \tan ^{-1}(c x)",1,"-6*a*c^2*d^4*x + (2*I)*b*c^2*d^4*x - (b*c^3*d^4*x^2)/6 - (2*I)*b*c*d^4*ArcTan[c*x] - 6*b*c^2*d^4*x*ArcTan[c*x] - (d^4*(a + b*ArcTan[c*x]))/x - (2*I)*c^3*d^4*x^2*(a + b*ArcTan[c*x]) + (c^4*d^4*x^3*(a + b*ArcTan[c*x]))/3 + (4*I)*a*c*d^4*Log[x] + b*c*d^4*Log[x] + (8*b*c*d^4*Log[1 + c^2*x^2])/3 - 2*b*c*d^4*PolyLog[2, (-I)*c*x] + 2*b*c*d^4*PolyLog[2, I*c*x]","A",20,13,23,0.5652,1,"{4876, 4846, 260, 4852, 266, 36, 29, 31, 4848, 2391, 321, 203, 43}"
37,1,173,0,0.1994523,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Int[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^3,x]","-3 i b c^2 d^4 \text{PolyLog}(2,-i c x)+3 i b c^2 d^4 \text{PolyLog}(2,i c x)+\frac{1}{2} c^4 d^4 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{d^4 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{4 i c d^4 \left(a+b \tan ^{-1}(c x)\right)}{x}-4 i a c^3 d^4 x-6 a c^2 d^4 \log (x)-\frac{1}{2} b c^3 d^4 x+4 i b c^2 d^4 \log (x)-4 i b c^3 d^4 x \tan ^{-1}(c x)-\frac{b c d^4}{2 x}","-3 i b c^2 d^4 \text{PolyLog}(2,-i c x)+3 i b c^2 d^4 \text{PolyLog}(2,i c x)+\frac{1}{2} c^4 d^4 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{d^4 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{4 i c d^4 \left(a+b \tan ^{-1}(c x)\right)}{x}-4 i a c^3 d^4 x-6 a c^2 d^4 \log (x)-\frac{1}{2} b c^3 d^4 x+4 i b c^2 d^4 \log (x)-4 i b c^3 d^4 x \tan ^{-1}(c x)-\frac{b c d^4}{2 x}",1,"-(b*c*d^4)/(2*x) - (4*I)*a*c^3*d^4*x - (b*c^3*d^4*x)/2 - (4*I)*b*c^3*d^4*x*ArcTan[c*x] - (d^4*(a + b*ArcTan[c*x]))/(2*x^2) - ((4*I)*c*d^4*(a + b*ArcTan[c*x]))/x + (c^4*d^4*x^2*(a + b*ArcTan[c*x]))/2 - 6*a*c^2*d^4*Log[x] + (4*I)*b*c^2*d^4*Log[x] - (3*I)*b*c^2*d^4*PolyLog[2, (-I)*c*x] + (3*I)*b*c^2*d^4*PolyLog[2, I*c*x]","A",19,13,23,0.5652,1,"{4876, 4846, 260, 4852, 325, 203, 266, 36, 29, 31, 4848, 2391, 321}"
38,1,201,0,0.2176668,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Int[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^4,x]","2 b c^3 d^4 \text{PolyLog}(2,-i c x)-2 b c^3 d^4 \text{PolyLog}(2,i c x)+\frac{6 c^2 d^4 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{2 i c d^4 \left(a+b \tan ^{-1}(c x)\right)}{x^2}-\frac{d^4 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}+a c^4 d^4 x-4 i a c^3 d^4 \log (x)+\frac{8}{3} b c^3 d^4 \log \left(c^2 x^2+1\right)-\frac{2 i b c^2 d^4}{x}-\frac{19}{3} b c^3 d^4 \log (x)-2 i b c^3 d^4 \tan ^{-1}(c x)+b c^4 d^4 x \tan ^{-1}(c x)-\frac{b c d^4}{6 x^2}","2 b c^3 d^4 \text{PolyLog}(2,-i c x)-2 b c^3 d^4 \text{PolyLog}(2,i c x)+\frac{6 c^2 d^4 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{2 i c d^4 \left(a+b \tan ^{-1}(c x)\right)}{x^2}-\frac{d^4 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}+a c^4 d^4 x-4 i a c^3 d^4 \log (x)+\frac{8}{3} b c^3 d^4 \log \left(c^2 x^2+1\right)-\frac{2 i b c^2 d^4}{x}-\frac{19}{3} b c^3 d^4 \log (x)-2 i b c^3 d^4 \tan ^{-1}(c x)+b c^4 d^4 x \tan ^{-1}(c x)-\frac{b c d^4}{6 x^2}",1,"-(b*c*d^4)/(6*x^2) - ((2*I)*b*c^2*d^4)/x + a*c^4*d^4*x - (2*I)*b*c^3*d^4*ArcTan[c*x] + b*c^4*d^4*x*ArcTan[c*x] - (d^4*(a + b*ArcTan[c*x]))/(3*x^3) - ((2*I)*c*d^4*(a + b*ArcTan[c*x]))/x^2 + (6*c^2*d^4*(a + b*ArcTan[c*x]))/x - (4*I)*a*c^3*d^4*Log[x] - (19*b*c^3*d^4*Log[x])/3 + (8*b*c^3*d^4*Log[1 + c^2*x^2])/3 + 2*b*c^3*d^4*PolyLog[2, (-I)*c*x] - 2*b*c^3*d^4*PolyLog[2, I*c*x]","A",20,13,23,0.5652,1,"{4876, 4846, 260, 4852, 266, 44, 325, 203, 36, 29, 31, 4848, 2391}"
39,1,227,0,0.2296859,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Int[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^5,x]","\frac{1}{2} i b c^4 d^4 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b c^4 d^4 \text{PolyLog}(2,i c x)+\frac{3 c^2 d^4 \left(a+b \tan ^{-1}(c x)\right)}{x^2}+\frac{4 i c^3 d^4 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{4 i c d^4 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{d^4 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}+a c^4 d^4 \log (x)-\frac{2 i b c^2 d^4}{3 x^2}+\frac{8}{3} i b c^4 d^4 \log \left(c^2 x^2+1\right)+\frac{13 b c^3 d^4}{4 x}-\frac{16}{3} i b c^4 d^4 \log (x)+\frac{13}{4} b c^4 d^4 \tan ^{-1}(c x)-\frac{b c d^4}{12 x^3}","\frac{1}{2} i b c^4 d^4 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b c^4 d^4 \text{PolyLog}(2,i c x)+\frac{3 c^2 d^4 \left(a+b \tan ^{-1}(c x)\right)}{x^2}+\frac{4 i c^3 d^4 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{4 i c d^4 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{d^4 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}+a c^4 d^4 \log (x)-\frac{2 i b c^2 d^4}{3 x^2}+\frac{8}{3} i b c^4 d^4 \log \left(c^2 x^2+1\right)+\frac{13 b c^3 d^4}{4 x}-\frac{16}{3} i b c^4 d^4 \log (x)+\frac{13}{4} b c^4 d^4 \tan ^{-1}(c x)-\frac{b c d^4}{12 x^3}",1,"-(b*c*d^4)/(12*x^3) - (((2*I)/3)*b*c^2*d^4)/x^2 + (13*b*c^3*d^4)/(4*x) + (13*b*c^4*d^4*ArcTan[c*x])/4 - (d^4*(a + b*ArcTan[c*x]))/(4*x^4) - (((4*I)/3)*c*d^4*(a + b*ArcTan[c*x]))/x^3 + (3*c^2*d^4*(a + b*ArcTan[c*x]))/x^2 + ((4*I)*c^3*d^4*(a + b*ArcTan[c*x]))/x + a*c^4*d^4*Log[x] - ((16*I)/3)*b*c^4*d^4*Log[x] + ((8*I)/3)*b*c^4*d^4*Log[1 + c^2*x^2] + (I/2)*b*c^4*d^4*PolyLog[2, (-I)*c*x] - (I/2)*b*c^4*d^4*PolyLog[2, I*c*x]","A",21,11,23,0.4783,1,"{4876, 4852, 325, 203, 266, 44, 36, 29, 31, 4848, 2391}"
40,1,117,0,0.0972312,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Int[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^6,x]","-\frac{d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}+\frac{11 b c^3 d^4}{10 x^2}-\frac{i b c^2 d^4}{3 x^3}+\frac{3 i b c^4 d^4}{x}+\frac{16}{5} b c^5 d^4 \log (x)-\frac{16}{5} b c^5 d^4 \log (c x+i)-\frac{b c d^4}{20 x^4}","-\frac{d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}+\frac{11 b c^3 d^4}{10 x^2}-\frac{i b c^2 d^4}{3 x^3}+\frac{3 i b c^4 d^4}{x}+\frac{16}{5} b c^5 d^4 \log (x)-\frac{16}{5} b c^5 d^4 \log (c x+i)-\frac{b c d^4}{20 x^4}",1,"-(b*c*d^4)/(20*x^4) - ((I/3)*b*c^2*d^4)/x^3 + (11*b*c^3*d^4)/(10*x^2) + ((3*I)*b*c^4*d^4)/x - (d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/(5*x^5) + (16*b*c^5*d^4*Log[x])/5 - (16*b*c^5*d^4*Log[I + c*x])/5","A",4,4,23,0.1739,1,"{37, 4872, 12, 88}"
41,1,168,0,0.1137689,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x^7} \, dx","Int[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^7,x]","\frac{i c d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{30 x^5}-\frac{d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{6 x^6}+\frac{16 i b c^4 d^4}{15 x^2}+\frac{5 b c^3 d^4}{9 x^3}-\frac{i b c^2 d^4}{5 x^4}-\frac{13 b c^5 d^4}{6 x}+\frac{32}{15} i b c^6 d^4 \log (x)-\frac{32}{15} i b c^6 d^4 \log (c x+i)-\frac{b c d^4}{30 x^5}","\frac{i c d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{30 x^5}-\frac{d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{6 x^6}+\frac{16 i b c^4 d^4}{15 x^2}+\frac{5 b c^3 d^4}{9 x^3}-\frac{i b c^2 d^4}{5 x^4}-\frac{13 b c^5 d^4}{6 x}+\frac{32}{15} i b c^6 d^4 \log (x)-\frac{32}{15} i b c^6 d^4 \log (c x+i)-\frac{b c d^4}{30 x^5}",1,"-(b*c*d^4)/(30*x^5) - ((I/5)*b*c^2*d^4)/x^4 + (5*b*c^3*d^4)/(9*x^3) + (((16*I)/15)*b*c^4*d^4)/x^2 - (13*b*c^5*d^4)/(6*x) - (d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/(6*x^6) + ((I/30)*c*d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/x^5 + ((32*I)/15)*b*c^6*d^4*Log[x] - ((32*I)/15)*b*c^6*d^4*Log[I + c*x]","A",4,5,23,0.2174,1,"{45, 37, 4872, 12, 148}"
42,1,243,0,0.1957785,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x^8} \, dx","Int[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^8,x]","-\frac{c^4 d^4 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}+\frac{i c^3 d^4 \left(a+b \tan ^{-1}(c x)\right)}{x^4}+\frac{6 c^2 d^4 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{2 i c d^4 \left(a+b \tan ^{-1}(c x)\right)}{3 x^6}-\frac{d^4 \left(a+b \tan ^{-1}(c x)\right)}{7 x^7}-\frac{88 b c^5 d^4}{105 x^2}+\frac{5 i b c^4 d^4}{9 x^3}+\frac{47 b c^3 d^4}{140 x^4}-\frac{2 i b c^2 d^4}{15 x^5}-\frac{5 i b c^6 d^4}{3 x}-\frac{176}{105} b c^7 d^4 \log (x)+\frac{1}{210} b c^7 d^4 \log (-c x+i)+\frac{117}{70} b c^7 d^4 \log (c x+i)-\frac{b c d^4}{42 x^6}","-\frac{c^4 d^4 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}+\frac{i c^3 d^4 \left(a+b \tan ^{-1}(c x)\right)}{x^4}+\frac{6 c^2 d^4 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{2 i c d^4 \left(a+b \tan ^{-1}(c x)\right)}{3 x^6}-\frac{d^4 \left(a+b \tan ^{-1}(c x)\right)}{7 x^7}-\frac{88 b c^5 d^4}{105 x^2}+\frac{5 i b c^4 d^4}{9 x^3}+\frac{47 b c^3 d^4}{140 x^4}-\frac{2 i b c^2 d^4}{15 x^5}-\frac{5 i b c^6 d^4}{3 x}-\frac{176}{105} b c^7 d^4 \log (x)+\frac{1}{210} b c^7 d^4 \log (-c x+i)+\frac{117}{70} b c^7 d^4 \log (c x+i)-\frac{b c d^4}{42 x^6}",1,"-(b*c*d^4)/(42*x^6) - (((2*I)/15)*b*c^2*d^4)/x^5 + (47*b*c^3*d^4)/(140*x^4) + (((5*I)/9)*b*c^4*d^4)/x^3 - (88*b*c^5*d^4)/(105*x^2) - (((5*I)/3)*b*c^6*d^4)/x - (d^4*(a + b*ArcTan[c*x]))/(7*x^7) - (((2*I)/3)*c*d^4*(a + b*ArcTan[c*x]))/x^6 + (6*c^2*d^4*(a + b*ArcTan[c*x]))/(5*x^5) + (I*c^3*d^4*(a + b*ArcTan[c*x]))/x^4 - (c^4*d^4*(a + b*ArcTan[c*x]))/(3*x^3) - (176*b*c^7*d^4*Log[x])/105 + (b*c^7*d^4*Log[I - c*x])/210 + (117*b*c^7*d^4*Log[I + c*x])/70","A",4,4,23,0.1739,1,"{43, 4872, 12, 1802}"
43,1,196,0,0.2861522,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{d+i c d x} \, dx","Int[(x^3*(a + b*ArcTan[c*x]))/(d + I*c*d*x),x]","\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^4 d}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c^2 d}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d}-\frac{i x^3 \left(a+b \tan ^{-1}(c x)\right)}{3 c d}+\frac{i a x}{c^3 d}+\frac{i b x^2}{6 c^2 d}-\frac{2 i b \log \left(c^2 x^2+1\right)}{3 c^4 d}-\frac{b x}{2 c^3 d}+\frac{i b x \tan ^{-1}(c x)}{c^3 d}+\frac{b \tan ^{-1}(c x)}{2 c^4 d}","\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^4 d}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c^2 d}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d}-\frac{i x^3 \left(a+b \tan ^{-1}(c x)\right)}{3 c d}+\frac{i a x}{c^3 d}+\frac{i b x^2}{6 c^2 d}-\frac{2 i b \log \left(c^2 x^2+1\right)}{3 c^4 d}-\frac{b x}{2 c^3 d}+\frac{i b x \tan ^{-1}(c x)}{c^3 d}+\frac{b \tan ^{-1}(c x)}{2 c^4 d}",1,"(I*a*x)/(c^3*d) - (b*x)/(2*c^3*d) + ((I/6)*b*x^2)/(c^2*d) + (b*ArcTan[c*x])/(2*c^4*d) + (I*b*x*ArcTan[c*x])/(c^3*d) + (x^2*(a + b*ArcTan[c*x]))/(2*c^2*d) - ((I/3)*x^3*(a + b*ArcTan[c*x]))/(c*d) + ((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^4*d) - (((2*I)/3)*b*Log[1 + c^2*x^2])/(c^4*d) + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d)","A",16,11,23,0.4783,1,"{4866, 4852, 266, 43, 321, 203, 4846, 260, 4854, 2402, 2315}"
44,1,156,0,0.1810024,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{d+i c d x} \, dx","Int[(x^2*(a + b*ArcTan[c*x]))/(d + I*c*d*x),x]","\frac{b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^3 d}-\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d}-\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c d}+\frac{a x}{c^2 d}-\frac{b \log \left(c^2 x^2+1\right)}{2 c^3 d}+\frac{i b x}{2 c^2 d}+\frac{b x \tan ^{-1}(c x)}{c^2 d}-\frac{i b \tan ^{-1}(c x)}{2 c^3 d}","\frac{b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^3 d}-\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d}-\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c d}+\frac{a x}{c^2 d}-\frac{b \log \left(c^2 x^2+1\right)}{2 c^3 d}+\frac{i b x}{2 c^2 d}+\frac{b x \tan ^{-1}(c x)}{c^2 d}-\frac{i b \tan ^{-1}(c x)}{2 c^3 d}",1,"(a*x)/(c^2*d) + ((I/2)*b*x)/(c^2*d) - ((I/2)*b*ArcTan[c*x])/(c^3*d) + (b*x*ArcTan[c*x])/(c^2*d) - ((I/2)*x^2*(a + b*ArcTan[c*x]))/(c*d) - (I*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^3*d) - (b*Log[1 + c^2*x^2])/(2*c^3*d) + (b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c^3*d)","A",11,9,23,0.3913,1,"{4866, 4852, 321, 203, 4846, 260, 4854, 2402, 2315}"
45,1,110,0,0.1038564,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{d+i c d x} \, dx","Int[(x*(a + b*ArcTan[c*x]))/(d + I*c*d*x),x]","-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^2 d}-\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d}-\frac{i a x}{c d}+\frac{i b \log \left(c^2 x^2+1\right)}{2 c^2 d}-\frac{i b x \tan ^{-1}(c x)}{c d}","-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^2 d}-\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d}-\frac{i a x}{c d}+\frac{i b \log \left(c^2 x^2+1\right)}{2 c^2 d}-\frac{i b x \tan ^{-1}(c x)}{c d}",1,"((-I)*a*x)/(c*d) - (I*b*x*ArcTan[c*x])/(c*d) - ((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^2*d) + ((I/2)*b*Log[1 + c^2*x^2])/(c^2*d) - ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d)","A",7,6,21,0.2857,1,"{4866, 4846, 260, 4854, 2402, 2315}"
46,1,59,0,0.047172,"\int \frac{a+b \tan ^{-1}(c x)}{d+i c d x} \, dx","Int[(a + b*ArcTan[c*x])/(d + I*c*d*x),x]","\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c d}-\frac{b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c d}","\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c d}-\frac{b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c d}",1,"(I*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c*d) - (b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c*d)","A",3,3,20,0.1500,1,"{4854, 2402, 2315}"
47,1,54,0,0.0711642,"\int \frac{a+b \tan ^{-1}(c x)}{x (d+i c d x)} \, dx","Int[(a + b*ArcTan[c*x])/(x*(d + I*c*d*x)),x]","\frac{i b \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right)}{2 d}+\frac{\log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}","\frac{i b \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right)}{2 d}+\frac{\log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}",1,"((a + b*ArcTan[c*x])*Log[2 - 2/(1 + I*c*x)])/d + ((I/2)*b*PolyLog[2, -1 + 2/(1 + I*c*x)])/d","A",2,2,23,0.08696,1,"{4868, 2447}"
48,1,100,0,0.1546699,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 (d+i c d x)} \, dx","Int[(a + b*ArcTan[c*x])/(x^2*(d + I*c*d*x)),x]","\frac{b c \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right)}{2 d}-\frac{a+b \tan ^{-1}(c x)}{d x}-\frac{i c \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{b c \log \left(c^2 x^2+1\right)}{2 d}+\frac{b c \log (x)}{d}","\frac{b c \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right)}{2 d}-\frac{a+b \tan ^{-1}(c x)}{d x}-\frac{i c \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{b c \log \left(c^2 x^2+1\right)}{2 d}+\frac{b c \log (x)}{d}",1,"-((a + b*ArcTan[c*x])/(d*x)) + (b*c*Log[x])/d - (b*c*Log[1 + c^2*x^2])/(2*d) - (I*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 + I*c*x)])/d + (b*c*PolyLog[2, -1 + 2/(1 + I*c*x)])/(2*d)","A",8,8,23,0.3478,1,"{4870, 4852, 266, 36, 29, 31, 4868, 2447}"
49,1,161,0,0.2357187,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 (d+i c d x)} \, dx","Int[(a + b*ArcTan[c*x])/(x^3*(d + I*c*d*x)),x]","-\frac{i b c^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right)}{2 d}-\frac{c^2 \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{a+b \tan ^{-1}(c x)}{2 d x^2}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)}{d x}+\frac{i b c^2 \log \left(c^2 x^2+1\right)}{2 d}-\frac{i b c^2 \log (x)}{d}-\frac{b c^2 \tan ^{-1}(c x)}{2 d}-\frac{b c}{2 d x}","-\frac{i b c^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right)}{2 d}-\frac{c^2 \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{a+b \tan ^{-1}(c x)}{2 d x^2}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)}{d x}+\frac{i b c^2 \log \left(c^2 x^2+1\right)}{2 d}-\frac{i b c^2 \log (x)}{d}-\frac{b c^2 \tan ^{-1}(c x)}{2 d}-\frac{b c}{2 d x}",1,"-(b*c)/(2*d*x) - (b*c^2*ArcTan[c*x])/(2*d) - (a + b*ArcTan[c*x])/(2*d*x^2) + (I*c*(a + b*ArcTan[c*x]))/(d*x) - (I*b*c^2*Log[x])/d + ((I/2)*b*c^2*Log[1 + c^2*x^2])/d - (c^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 + I*c*x)])/d - ((I/2)*b*c^2*PolyLog[2, -1 + 2/(1 + I*c*x)])/d","A",12,10,23,0.4348,1,"{4870, 4852, 325, 203, 266, 36, 29, 31, 4868, 2447}"
50,1,197,0,0.3392075,"\int \frac{a+b \tan ^{-1}(c x)}{x^4 (d+i c d x)} \, dx","Int[(a + b*ArcTan[c*x])/(x^4*(d + I*c*d*x)),x]","-\frac{b c^3 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right)}{2 d}+\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)}{d x}+\frac{i c^3 \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)}{2 d x^2}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3}+\frac{2 b c^3 \log \left(c^2 x^2+1\right)}{3 d}+\frac{i b c^2}{2 d x}-\frac{4 b c^3 \log (x)}{3 d}+\frac{i b c^3 \tan ^{-1}(c x)}{2 d}-\frac{b c}{6 d x^2}","-\frac{b c^3 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right)}{2 d}+\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)}{d x}+\frac{i c^3 \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)}{2 d x^2}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3}+\frac{2 b c^3 \log \left(c^2 x^2+1\right)}{3 d}+\frac{i b c^2}{2 d x}-\frac{4 b c^3 \log (x)}{3 d}+\frac{i b c^3 \tan ^{-1}(c x)}{2 d}-\frac{b c}{6 d x^2}",1,"-(b*c)/(6*d*x^2) + ((I/2)*b*c^2)/(d*x) + ((I/2)*b*c^3*ArcTan[c*x])/d - (a + b*ArcTan[c*x])/(3*d*x^3) + ((I/2)*c*(a + b*ArcTan[c*x]))/(d*x^2) + (c^2*(a + b*ArcTan[c*x]))/(d*x) - (4*b*c^3*Log[x])/(3*d) + (2*b*c^3*Log[1 + c^2*x^2])/(3*d) + (I*c^3*(a + b*ArcTan[c*x])*Log[2 - 2/(1 + I*c*x)])/d - (b*c^3*PolyLog[2, -1 + 2/(1 + I*c*x)])/(2*d)","A",17,11,23,0.4783,1,"{4870, 4852, 266, 44, 325, 203, 36, 29, 31, 4868, 2447}"
51,1,203,0,0.2219159,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{(d+i c d x)^2} \, dx","Int[(x^3*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^2,x]","-\frac{3 i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^4 d^2}-\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c^2 d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^2 (-c x+i)}-\frac{3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^2}-\frac{2 i a x}{c^3 d^2}+\frac{i b \log \left(c^2 x^2+1\right)}{c^4 d^2}+\frac{b x}{2 c^3 d^2}+\frac{b}{2 c^4 d^2 (-c x+i)}-\frac{2 i b x \tan ^{-1}(c x)}{c^3 d^2}-\frac{b \tan ^{-1}(c x)}{c^4 d^2}","-\frac{3 i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^4 d^2}-\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c^2 d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^2 (-c x+i)}-\frac{3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^2}-\frac{2 i a x}{c^3 d^2}+\frac{i b \log \left(c^2 x^2+1\right)}{c^4 d^2}+\frac{b x}{2 c^3 d^2}+\frac{b}{2 c^4 d^2 (-c x+i)}-\frac{2 i b x \tan ^{-1}(c x)}{c^3 d^2}-\frac{b \tan ^{-1}(c x)}{c^4 d^2}",1,"((-2*I)*a*x)/(c^3*d^2) + (b*x)/(2*c^3*d^2) + b/(2*c^4*d^2*(I - c*x)) - (b*ArcTan[c*x])/(c^4*d^2) - ((2*I)*b*x*ArcTan[c*x])/(c^3*d^2) - (x^2*(a + b*ArcTan[c*x]))/(2*c^2*d^2) + (I*(a + b*ArcTan[c*x]))/(c^4*d^2*(I - c*x)) - (3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^4*d^2) + (I*b*Log[1 + c^2*x^2])/(c^4*d^2) - (((3*I)/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d^2)","A",16,12,23,0.5217,1,"{4876, 4846, 260, 4852, 321, 203, 4862, 627, 44, 4854, 2402, 2315}"
52,1,167,0,0.1902928,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{(d+i c d x)^2} \, dx","Int[(x^2*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^2,x]","-\frac{b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^3 d^2}+\frac{a+b \tan ^{-1}(c x)}{c^3 d^2 (-c x+i)}+\frac{2 i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d^2}-\frac{a x}{c^2 d^2}+\frac{b \log \left(c^2 x^2+1\right)}{2 c^3 d^2}-\frac{i b}{2 c^3 d^2 (-c x+i)}-\frac{b x \tan ^{-1}(c x)}{c^2 d^2}+\frac{i b \tan ^{-1}(c x)}{2 c^3 d^2}","-\frac{b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^3 d^2}+\frac{a+b \tan ^{-1}(c x)}{c^3 d^2 (-c x+i)}+\frac{2 i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d^2}-\frac{a x}{c^2 d^2}+\frac{b \log \left(c^2 x^2+1\right)}{2 c^3 d^2}-\frac{i b}{2 c^3 d^2 (-c x+i)}-\frac{b x \tan ^{-1}(c x)}{c^2 d^2}+\frac{i b \tan ^{-1}(c x)}{2 c^3 d^2}",1,"-((a*x)/(c^2*d^2)) - ((I/2)*b)/(c^3*d^2*(I - c*x)) + ((I/2)*b*ArcTan[c*x])/(c^3*d^2) - (b*x*ArcTan[c*x])/(c^2*d^2) + (a + b*ArcTan[c*x])/(c^3*d^2*(I - c*x)) + ((2*I)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^3*d^2) + (b*Log[1 + c^2*x^2])/(2*c^3*d^2) - (b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d^2)","A",13,10,23,0.4348,1,"{4876, 4846, 260, 4862, 627, 44, 203, 4854, 2402, 2315}"
53,1,122,0,0.1468091,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{(d+i c d x)^2} \, dx","Int[(x*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^2,x]","\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^2 d^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right)}{c^2 d^2 (-c x+i)}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d^2}-\frac{b}{2 c^2 d^2 (-c x+i)}+\frac{b \tan ^{-1}(c x)}{2 c^2 d^2}","\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^2 d^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right)}{c^2 d^2 (-c x+i)}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d^2}-\frac{b}{2 c^2 d^2 (-c x+i)}+\frac{b \tan ^{-1}(c x)}{2 c^2 d^2}",1,"-b/(2*c^2*d^2*(I - c*x)) + (b*ArcTan[c*x])/(2*c^2*d^2) - (I*(a + b*ArcTan[c*x]))/(c^2*d^2*(I - c*x)) + ((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^2*d^2) + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d^2)","A",10,8,21,0.3810,1,"{4876, 4862, 627, 44, 203, 4854, 2402, 2315}"
54,1,69,0,0.0472076,"\int \frac{a+b \tan ^{-1}(c x)}{(d+i c d x)^2} \, dx","Int[(a + b*ArcTan[c*x])/(d + I*c*d*x)^2,x]","\frac{i \left(a+b \tan ^{-1}(c x)\right)}{c d^2 (1+i c x)}+\frac{i b}{2 c d^2 (-c x+i)}-\frac{i b \tan ^{-1}(c x)}{2 c d^2}","\frac{i \left(a+b \tan ^{-1}(c x)\right)}{c d^2 (1+i c x)}+\frac{i b}{2 c d^2 (-c x+i)}-\frac{i b \tan ^{-1}(c x)}{2 c d^2}",1,"((I/2)*b)/(c*d^2*(I - c*x)) - ((I/2)*b*ArcTan[c*x])/(c*d^2) + (I*(a + b*ArcTan[c*x]))/(c*d^2*(1 + I*c*x))","A",5,4,20,0.2000,1,"{4862, 627, 44, 203}"
55,1,150,0,0.193568,"\int \frac{a+b \tan ^{-1}(c x)}{x (d+i c d x)^2} \, dx","Int[(a + b*ArcTan[c*x])/(x*(d + I*c*d*x)^2),x]","\frac{i b \text{PolyLog}(2,-i c x)}{2 d^2}-\frac{i b \text{PolyLog}(2,i c x)}{2 d^2}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right)}{d^2 (-c x+i)}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{a \log (x)}{d^2}+\frac{b}{2 d^2 (-c x+i)}-\frac{b \tan ^{-1}(c x)}{2 d^2}","\frac{i b \text{PolyLog}(2,-i c x)}{2 d^2}-\frac{i b \text{PolyLog}(2,i c x)}{2 d^2}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right)}{d^2 (-c x+i)}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{a \log (x)}{d^2}+\frac{b}{2 d^2 (-c x+i)}-\frac{b \tan ^{-1}(c x)}{2 d^2}",1,"b/(2*d^2*(I - c*x)) - (b*ArcTan[c*x])/(2*d^2) + (I*(a + b*ArcTan[c*x]))/(d^2*(I - c*x)) + (a*Log[x])/d^2 + ((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/d^2 + ((I/2)*b*PolyLog[2, (-I)*c*x])/d^2 - ((I/2)*b*PolyLog[2, I*c*x])/d^2 + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^2","A",13,10,23,0.4348,1,"{4876, 4848, 2391, 4862, 627, 44, 203, 4854, 2402, 2315}"
56,1,194,0,0.2411605,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 (d+i c d x)^2} \, dx","Int[(a + b*ArcTan[c*x])/(x^2*(d + I*c*d*x)^2),x]","\frac{b c \text{PolyLog}(2,-i c x)}{d^2}-\frac{b c \text{PolyLog}(2,i c x)}{d^2}+\frac{b c \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{d^2}+\frac{c \left(a+b \tan ^{-1}(c x)\right)}{d^2 (-c x+i)}-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{2 i c \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{2 i a c \log (x)}{d^2}-\frac{b c \log \left(c^2 x^2+1\right)}{2 d^2}-\frac{i b c}{2 d^2 (-c x+i)}+\frac{b c \log (x)}{d^2}+\frac{i b c \tan ^{-1}(c x)}{2 d^2}","\frac{b c \text{PolyLog}(2,-i c x)}{d^2}-\frac{b c \text{PolyLog}(2,i c x)}{d^2}+\frac{b c \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{d^2}+\frac{c \left(a+b \tan ^{-1}(c x)\right)}{d^2 (-c x+i)}-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{2 i c \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{2 i a c \log (x)}{d^2}-\frac{b c \log \left(c^2 x^2+1\right)}{2 d^2}-\frac{i b c}{2 d^2 (-c x+i)}+\frac{b c \log (x)}{d^2}+\frac{i b c \tan ^{-1}(c x)}{2 d^2}",1,"((-I/2)*b*c)/(d^2*(I - c*x)) + ((I/2)*b*c*ArcTan[c*x])/d^2 - (a + b*ArcTan[c*x])/(d^2*x) + (c*(a + b*ArcTan[c*x]))/(d^2*(I - c*x)) - ((2*I)*a*c*Log[x])/d^2 + (b*c*Log[x])/d^2 - ((2*I)*c*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/d^2 - (b*c*Log[1 + c^2*x^2])/(2*d^2) + (b*c*PolyLog[2, (-I)*c*x])/d^2 - (b*c*PolyLog[2, I*c*x])/d^2 + (b*c*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^2","A",18,15,23,0.6522,1,"{4876, 4852, 266, 36, 29, 31, 4848, 2391, 4862, 627, 44, 203, 4854, 2402, 2315}"
57,1,244,0,0.2675441,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 (d+i c d x)^2} \, dx","Int[(a + b*ArcTan[c*x])/(x^3*(d + I*c*d*x)^2),x]","-\frac{3 i b c^2 \text{PolyLog}(2,-i c x)}{2 d^2}+\frac{3 i b c^2 \text{PolyLog}(2,i c x)}{2 d^2}-\frac{3 i b c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 d^2}-\frac{i c^2 \left(a+b \tan ^{-1}(c x)\right)}{d^2 (-c x+i)}-\frac{3 c^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{a+b \tan ^{-1}(c x)}{2 d^2 x^2}+\frac{2 i c \left(a+b \tan ^{-1}(c x)\right)}{d^2 x}-\frac{3 a c^2 \log (x)}{d^2}+\frac{i b c^2 \log \left(c^2 x^2+1\right)}{d^2}-\frac{b c^2}{2 d^2 (-c x+i)}-\frac{2 i b c^2 \log (x)}{d^2}-\frac{b c}{2 d^2 x}","-\frac{3 i b c^2 \text{PolyLog}(2,-i c x)}{2 d^2}+\frac{3 i b c^2 \text{PolyLog}(2,i c x)}{2 d^2}-\frac{3 i b c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 d^2}-\frac{i c^2 \left(a+b \tan ^{-1}(c x)\right)}{d^2 (-c x+i)}-\frac{3 c^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{a+b \tan ^{-1}(c x)}{2 d^2 x^2}+\frac{2 i c \left(a+b \tan ^{-1}(c x)\right)}{d^2 x}-\frac{3 a c^2 \log (x)}{d^2}+\frac{i b c^2 \log \left(c^2 x^2+1\right)}{d^2}-\frac{b c^2}{2 d^2 (-c x+i)}-\frac{2 i b c^2 \log (x)}{d^2}-\frac{b c}{2 d^2 x}",1,"-(b*c)/(2*d^2*x) - (b*c^2)/(2*d^2*(I - c*x)) - (a + b*ArcTan[c*x])/(2*d^2*x^2) + ((2*I)*c*(a + b*ArcTan[c*x]))/(d^2*x) - (I*c^2*(a + b*ArcTan[c*x]))/(d^2*(I - c*x)) - (3*a*c^2*Log[x])/d^2 - ((2*I)*b*c^2*Log[x])/d^2 - (3*c^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/d^2 + (I*b*c^2*Log[1 + c^2*x^2])/d^2 - (((3*I)/2)*b*c^2*PolyLog[2, (-I)*c*x])/d^2 + (((3*I)/2)*b*c^2*PolyLog[2, I*c*x])/d^2 - (((3*I)/2)*b*c^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^2","A",21,16,23,0.6957,1,"{4876, 4852, 325, 203, 266, 36, 29, 31, 4848, 2391, 4862, 627, 44, 4854, 2402, 2315}"
58,1,256,0,0.284669,"\int \frac{x^4 \left(a+b \tan ^{-1}(c x)\right)}{(d+i c d x)^3} \, dx","Int[(x^4*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^3,x]","-\frac{3 b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^5 d^3}+\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c^3 d^3}+\frac{4 \left(a+b \tan ^{-1}(c x)\right)}{c^5 d^3 (-c x+i)}-\frac{i \left(a+b \tan ^{-1}(c x)\right)}{2 c^5 d^3 (-c x+i)^2}+\frac{6 i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^5 d^3}-\frac{3 a x}{c^4 d^3}+\frac{3 b \log \left(c^2 x^2+1\right)}{2 c^5 d^3}-\frac{i b x}{2 c^4 d^3}-\frac{15 i b}{8 c^5 d^3 (-c x+i)}-\frac{b}{8 c^5 d^3 (-c x+i)^2}-\frac{3 b x \tan ^{-1}(c x)}{c^4 d^3}+\frac{19 i b \tan ^{-1}(c x)}{8 c^5 d^3}","-\frac{3 b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^5 d^3}+\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c^3 d^3}+\frac{4 \left(a+b \tan ^{-1}(c x)\right)}{c^5 d^3 (-c x+i)}-\frac{i \left(a+b \tan ^{-1}(c x)\right)}{2 c^5 d^3 (-c x+i)^2}+\frac{6 i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^5 d^3}-\frac{3 a x}{c^4 d^3}+\frac{3 b \log \left(c^2 x^2+1\right)}{2 c^5 d^3}-\frac{i b x}{2 c^4 d^3}-\frac{15 i b}{8 c^5 d^3 (-c x+i)}-\frac{b}{8 c^5 d^3 (-c x+i)^2}-\frac{3 b x \tan ^{-1}(c x)}{c^4 d^3}+\frac{19 i b \tan ^{-1}(c x)}{8 c^5 d^3}",1,"(-3*a*x)/(c^4*d^3) - ((I/2)*b*x)/(c^4*d^3) - b/(8*c^5*d^3*(I - c*x)^2) - (((15*I)/8)*b)/(c^5*d^3*(I - c*x)) + (((19*I)/8)*b*ArcTan[c*x])/(c^5*d^3) - (3*b*x*ArcTan[c*x])/(c^4*d^3) + ((I/2)*x^2*(a + b*ArcTan[c*x]))/(c^3*d^3) - ((I/2)*(a + b*ArcTan[c*x]))/(c^5*d^3*(I - c*x)^2) + (4*(a + b*ArcTan[c*x]))/(c^5*d^3*(I - c*x)) + ((6*I)*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^5*d^3) + (3*b*Log[1 + c^2*x^2])/(2*c^5*d^3) - (3*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^3)","A",21,12,23,0.5217,1,"{4876, 4846, 260, 4852, 321, 203, 4862, 627, 44, 4854, 2402, 2315}"
59,1,225,0,0.2427804,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{(d+i c d x)^3} \, dx","Int[(x^3*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^3,x]","\frac{3 i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^4 d^3}-\frac{3 i \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^3 (-c x+i)}-\frac{a+b \tan ^{-1}(c x)}{2 c^4 d^3 (-c x+i)^2}+\frac{3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^3}+\frac{i a x}{c^3 d^3}-\frac{i b \log \left(c^2 x^2+1\right)}{2 c^4 d^3}-\frac{11 b}{8 c^4 d^3 (-c x+i)}+\frac{i b}{8 c^4 d^3 (-c x+i)^2}+\frac{i b x \tan ^{-1}(c x)}{c^3 d^3}+\frac{11 b \tan ^{-1}(c x)}{8 c^4 d^3}","\frac{3 i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^4 d^3}-\frac{3 i \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^3 (-c x+i)}-\frac{a+b \tan ^{-1}(c x)}{2 c^4 d^3 (-c x+i)^2}+\frac{3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^3}+\frac{i a x}{c^3 d^3}-\frac{i b \log \left(c^2 x^2+1\right)}{2 c^4 d^3}-\frac{11 b}{8 c^4 d^3 (-c x+i)}+\frac{i b}{8 c^4 d^3 (-c x+i)^2}+\frac{i b x \tan ^{-1}(c x)}{c^3 d^3}+\frac{11 b \tan ^{-1}(c x)}{8 c^4 d^3}",1,"(I*a*x)/(c^3*d^3) + ((I/8)*b)/(c^4*d^3*(I - c*x)^2) - (11*b)/(8*c^4*d^3*(I - c*x)) + (11*b*ArcTan[c*x])/(8*c^4*d^3) + (I*b*x*ArcTan[c*x])/(c^3*d^3) - (a + b*ArcTan[c*x])/(2*c^4*d^3*(I - c*x)^2) - ((3*I)*(a + b*ArcTan[c*x]))/(c^4*d^3*(I - c*x)) + (3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^4*d^3) - ((I/2)*b*Log[1 + c^2*x^2])/(c^4*d^3) + (((3*I)/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d^3)","A",18,10,23,0.4348,1,"{4876, 4846, 260, 4862, 627, 44, 203, 4854, 2402, 2315}"
60,1,176,0,0.2172102,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{(d+i c d x)^3} \, dx","Int[(x^2*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^3,x]","\frac{b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^3 d^3}-\frac{2 \left(a+b \tan ^{-1}(c x)\right)}{c^3 d^3 (-c x+i)}+\frac{i \left(a+b \tan ^{-1}(c x)\right)}{2 c^3 d^3 (-c x+i)^2}-\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d^3}+\frac{7 i b}{8 c^3 d^3 (-c x+i)}+\frac{b}{8 c^3 d^3 (-c x+i)^2}-\frac{7 i b \tan ^{-1}(c x)}{8 c^3 d^3}","\frac{b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^3 d^3}-\frac{2 \left(a+b \tan ^{-1}(c x)\right)}{c^3 d^3 (-c x+i)}+\frac{i \left(a+b \tan ^{-1}(c x)\right)}{2 c^3 d^3 (-c x+i)^2}-\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d^3}+\frac{7 i b}{8 c^3 d^3 (-c x+i)}+\frac{b}{8 c^3 d^3 (-c x+i)^2}-\frac{7 i b \tan ^{-1}(c x)}{8 c^3 d^3}",1,"b/(8*c^3*d^3*(I - c*x)^2) + (((7*I)/8)*b)/(c^3*d^3*(I - c*x)) - (((7*I)/8)*b*ArcTan[c*x])/(c^3*d^3) + ((I/2)*(a + b*ArcTan[c*x]))/(c^3*d^3*(I - c*x)^2) - (2*(a + b*ArcTan[c*x]))/(c^3*d^3*(I - c*x)) - (I*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^3*d^3) + (b*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c^3*d^3)","A",15,8,23,0.3478,1,"{4876, 4862, 627, 44, 203, 4854, 2402, 2315}"
61,1,88,0,0.0767148,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{(d+i c d x)^3} \, dx","Int[(x*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^3,x]","\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 d^3 (1+i c x)^2}+\frac{3 b}{8 c^2 d^3 (-c x+i)}-\frac{i b}{8 c^2 d^3 (-c x+i)^2}+\frac{b \tan ^{-1}(c x)}{8 c^2 d^3}","\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 d^3 (1+i c x)^2}+\frac{3 b}{8 c^2 d^3 (-c x+i)}-\frac{i b}{8 c^2 d^3 (-c x+i)^2}+\frac{b \tan ^{-1}(c x)}{8 c^2 d^3}",1,"((-I/8)*b)/(c^2*d^3*(I - c*x)^2) + (3*b)/(8*c^2*d^3*(I - c*x)) + (b*ArcTan[c*x])/(8*c^2*d^3) + (x^2*(a + b*ArcTan[c*x]))/(2*d^3*(1 + I*c*x)^2)","A",5,5,21,0.2381,1,"{37, 4872, 12, 88, 203}"
62,1,92,0,0.0549023,"\int \frac{a+b \tan ^{-1}(c x)}{(d+i c d x)^3} \, dx","Int[(a + b*ArcTan[c*x])/(d + I*c*d*x)^3,x]","\frac{i \left(a+b \tan ^{-1}(c x)\right)}{2 c d^3 (1+i c x)^2}+\frac{i b}{8 c d^3 (-c x+i)}-\frac{b}{8 c d^3 (-c x+i)^2}-\frac{i b \tan ^{-1}(c x)}{8 c d^3}","\frac{i \left(a+b \tan ^{-1}(c x)\right)}{2 c d^3 (1+i c x)^2}+\frac{i b}{8 c d^3 (-c x+i)}-\frac{b}{8 c d^3 (-c x+i)^2}-\frac{i b \tan ^{-1}(c x)}{8 c d^3}",1,"-b/(8*c*d^3*(I - c*x)^2) + ((I/8)*b)/(c*d^3*(I - c*x)) - ((I/8)*b*ArcTan[c*x])/(c*d^3) + ((I/2)*(a + b*ArcTan[c*x]))/(c*d^3*(1 + I*c*x)^2)","A",5,4,20,0.2000,1,"{4862, 627, 44, 203}"
63,1,195,0,0.2416644,"\int \frac{a+b \tan ^{-1}(c x)}{x (d+i c d x)^3} \, dx","Int[(a + b*ArcTan[c*x])/(x*(d + I*c*d*x)^3),x]","\frac{i b \text{PolyLog}(2,-i c x)}{2 d^3}-\frac{i b \text{PolyLog}(2,i c x)}{2 d^3}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 d^3}+\frac{i \left(a+b \tan ^{-1}(c x)\right)}{d^3 (-c x+i)}-\frac{a+b \tan ^{-1}(c x)}{2 d^3 (-c x+i)^2}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}+\frac{a \log (x)}{d^3}+\frac{5 b}{8 d^3 (-c x+i)}+\frac{i b}{8 d^3 (-c x+i)^2}-\frac{5 b \tan ^{-1}(c x)}{8 d^3}","\frac{i b \text{PolyLog}(2,-i c x)}{2 d^3}-\frac{i b \text{PolyLog}(2,i c x)}{2 d^3}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 d^3}+\frac{i \left(a+b \tan ^{-1}(c x)\right)}{d^3 (-c x+i)}-\frac{a+b \tan ^{-1}(c x)}{2 d^3 (-c x+i)^2}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}+\frac{a \log (x)}{d^3}+\frac{5 b}{8 d^3 (-c x+i)}+\frac{i b}{8 d^3 (-c x+i)^2}-\frac{5 b \tan ^{-1}(c x)}{8 d^3}",1,"((I/8)*b)/(d^3*(I - c*x)^2) + (5*b)/(8*d^3*(I - c*x)) - (5*b*ArcTan[c*x])/(8*d^3) - (a + b*ArcTan[c*x])/(2*d^3*(I - c*x)^2) + (I*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)) + (a*Log[x])/d^3 + ((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/d^3 + ((I/2)*b*PolyLog[2, (-I)*c*x])/d^3 - ((I/2)*b*PolyLog[2, I*c*x])/d^3 + ((I/2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^3","A",18,10,23,0.4348,1,"{4876, 4848, 2391, 4862, 627, 44, 203, 4854, 2402, 2315}"
64,1,250,0,0.2878269,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 (d+i c d x)^3} \, dx","Int[(a + b*ArcTan[c*x])/(x^2*(d + I*c*d*x)^3),x]","\frac{3 b c \text{PolyLog}(2,-i c x)}{2 d^3}-\frac{3 b c \text{PolyLog}(2,i c x)}{2 d^3}+\frac{3 b c \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 d^3}+\frac{2 c \left(a+b \tan ^{-1}(c x)\right)}{d^3 (-c x+i)}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)}{2 d^3 (-c x+i)^2}-\frac{a+b \tan ^{-1}(c x)}{d^3 x}-\frac{3 i c \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}-\frac{3 i a c \log (x)}{d^3}-\frac{b c \log \left(c^2 x^2+1\right)}{2 d^3}-\frac{9 i b c}{8 d^3 (-c x+i)}+\frac{b c}{8 d^3 (-c x+i)^2}+\frac{b c \log (x)}{d^3}+\frac{9 i b c \tan ^{-1}(c x)}{8 d^3}","\frac{3 b c \text{PolyLog}(2,-i c x)}{2 d^3}-\frac{3 b c \text{PolyLog}(2,i c x)}{2 d^3}+\frac{3 b c \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 d^3}+\frac{2 c \left(a+b \tan ^{-1}(c x)\right)}{d^3 (-c x+i)}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)}{2 d^3 (-c x+i)^2}-\frac{a+b \tan ^{-1}(c x)}{d^3 x}-\frac{3 i c \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}-\frac{3 i a c \log (x)}{d^3}-\frac{b c \log \left(c^2 x^2+1\right)}{2 d^3}-\frac{9 i b c}{8 d^3 (-c x+i)}+\frac{b c}{8 d^3 (-c x+i)^2}+\frac{b c \log (x)}{d^3}+\frac{9 i b c \tan ^{-1}(c x)}{8 d^3}",1,"(b*c)/(8*d^3*(I - c*x)^2) - (((9*I)/8)*b*c)/(d^3*(I - c*x)) + (((9*I)/8)*b*c*ArcTan[c*x])/d^3 - (a + b*ArcTan[c*x])/(d^3*x) + ((I/2)*c*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)^2) + (2*c*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)) - ((3*I)*a*c*Log[x])/d^3 + (b*c*Log[x])/d^3 - ((3*I)*c*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/d^3 - (b*c*Log[1 + c^2*x^2])/(2*d^3) + (3*b*c*PolyLog[2, (-I)*c*x])/(2*d^3) - (3*b*c*PolyLog[2, I*c*x])/(2*d^3) + (3*b*c*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*d^3)","A",23,15,23,0.6522,1,"{4876, 4852, 266, 36, 29, 31, 4848, 2391, 4862, 627, 44, 203, 4854, 2402, 2315}"
65,1,306,0,0.3201789,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 (d+i c d x)^3} \, dx","Int[(a + b*ArcTan[c*x])/(x^3*(d + I*c*d*x)^3),x]","-\frac{3 i b c^2 \text{PolyLog}(2,-i c x)}{d^3}+\frac{3 i b c^2 \text{PolyLog}(2,i c x)}{d^3}-\frac{3 i b c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{d^3}-\frac{3 i c^2 \left(a+b \tan ^{-1}(c x)\right)}{d^3 (-c x+i)}+\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)}{2 d^3 (-c x+i)^2}-\frac{6 c^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}-\frac{a+b \tan ^{-1}(c x)}{2 d^3 x^2}+\frac{3 i c \left(a+b \tan ^{-1}(c x)\right)}{d^3 x}-\frac{6 a c^2 \log (x)}{d^3}+\frac{3 i b c^2 \log \left(c^2 x^2+1\right)}{2 d^3}-\frac{13 b c^2}{8 d^3 (-c x+i)}-\frac{i b c^2}{8 d^3 (-c x+i)^2}-\frac{3 i b c^2 \log (x)}{d^3}+\frac{9 b c^2 \tan ^{-1}(c x)}{8 d^3}-\frac{b c}{2 d^3 x}","-\frac{3 i b c^2 \text{PolyLog}(2,-i c x)}{d^3}+\frac{3 i b c^2 \text{PolyLog}(2,i c x)}{d^3}-\frac{3 i b c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{d^3}-\frac{3 i c^2 \left(a+b \tan ^{-1}(c x)\right)}{d^3 (-c x+i)}+\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)}{2 d^3 (-c x+i)^2}-\frac{6 c^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}-\frac{a+b \tan ^{-1}(c x)}{2 d^3 x^2}+\frac{3 i c \left(a+b \tan ^{-1}(c x)\right)}{d^3 x}-\frac{6 a c^2 \log (x)}{d^3}+\frac{3 i b c^2 \log \left(c^2 x^2+1\right)}{2 d^3}-\frac{13 b c^2}{8 d^3 (-c x+i)}-\frac{i b c^2}{8 d^3 (-c x+i)^2}-\frac{3 i b c^2 \log (x)}{d^3}+\frac{9 b c^2 \tan ^{-1}(c x)}{8 d^3}-\frac{b c}{2 d^3 x}",1,"-(b*c)/(2*d^3*x) - ((I/8)*b*c^2)/(d^3*(I - c*x)^2) - (13*b*c^2)/(8*d^3*(I - c*x)) + (9*b*c^2*ArcTan[c*x])/(8*d^3) - (a + b*ArcTan[c*x])/(2*d^3*x^2) + ((3*I)*c*(a + b*ArcTan[c*x]))/(d^3*x) + (c^2*(a + b*ArcTan[c*x]))/(2*d^3*(I - c*x)^2) - ((3*I)*c^2*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)) - (6*a*c^2*Log[x])/d^3 - ((3*I)*b*c^2*Log[x])/d^3 - (6*c^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/d^3 + (((3*I)/2)*b*c^2*Log[1 + c^2*x^2])/d^3 - ((3*I)*b*c^2*PolyLog[2, (-I)*c*x])/d^3 + ((3*I)*b*c^2*PolyLog[2, I*c*x])/d^3 - ((3*I)*b*c^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^3","A",26,16,23,0.6957,1,"{4876, 4852, 325, 203, 266, 36, 29, 31, 4848, 2391, 4862, 627, 44, 4854, 2402, 2315}"
66,1,100,0,0.0512029,"\int \frac{a+b \tan ^{-1}(c x)}{(1+i c x)^4} \, dx","Int[(a + b*ArcTan[c*x])/(1 + I*c*x)^4,x]","\frac{i \left(a+b \tan ^{-1}(c x)\right)}{3 c (1+i c x)^3}+\frac{i b}{24 c (-c x+i)}-\frac{b}{24 c (-c x+i)^2}-\frac{i b}{18 c (-c x+i)^3}-\frac{i b \tan ^{-1}(c x)}{24 c}","\frac{i \left(a+b \tan ^{-1}(c x)\right)}{3 c (1+i c x)^3}+\frac{i b}{24 c (-c x+i)}-\frac{b}{24 c (-c x+i)^2}-\frac{i b}{18 c (-c x+i)^3}-\frac{i b \tan ^{-1}(c x)}{24 c}",1,"((-I/18)*b)/(c*(I - c*x)^3) - b/(24*c*(I - c*x)^2) + ((I/24)*b)/(c*(I - c*x)) - ((I/24)*b*ArcTan[c*x])/c + ((I/3)*(a + b*ArcTan[c*x]))/(c*(1 + I*c*x)^3)","A",5,4,19,0.2105,1,"{4862, 627, 44, 203}"
67,1,49,0,0.0636605,"\int \frac{\tan ^{-1}(a x)}{c x+i a c x^2} \, dx","Int[ArcTan[a*x]/(c*x + I*a*c*x^2),x]","\frac{i \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)}{2 c}+\frac{\log \left(2-\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{c}","\frac{i \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)}{2 c}+\frac{\log \left(2-\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{c}",1,"(ArcTan[a*x]*Log[2 - 2/(1 + I*a*x)])/c + ((I/2)*PolyLog[2, -1 + 2/(1 + I*a*x)])/c","A",3,3,20,0.1500,1,"{1593, 4868, 2447}"
68,1,287,0,0.5963833,"\int x^3 (d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x^3*(d + I*c*d*x)*(a + b*ArcTan[c*x])^2,x]","-\frac{b^2 d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{5 c^4}+\frac{i b d x^2 \left(a+b \tan ^{-1}(c x)\right)}{5 c^2}+\frac{a b d x}{2 c^3}-\frac{9 d \left(a+b \tan ^{-1}(c x)\right)^2}{20 c^4}+\frac{2 i b d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{5 c^4}+\frac{1}{5} i c d x^5 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{4} d x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{10} i b d x^4 \left(a+b \tan ^{-1}(c x)\right)-\frac{b d x^3 \left(a+b \tan ^{-1}(c x)\right)}{6 c}+\frac{b^2 d x^2}{12 c^2}-\frac{b^2 d \log \left(c^2 x^2+1\right)}{3 c^4}-\frac{3 i b^2 d x}{10 c^3}+\frac{b^2 d x \tan ^{-1}(c x)}{2 c^3}+\frac{3 i b^2 d \tan ^{-1}(c x)}{10 c^4}+\frac{i b^2 d x^3}{30 c}","-\frac{b^2 d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{5 c^4}+\frac{i b d x^2 \left(a+b \tan ^{-1}(c x)\right)}{5 c^2}+\frac{a b d x}{2 c^3}-\frac{9 d \left(a+b \tan ^{-1}(c x)\right)^2}{20 c^4}+\frac{2 i b d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{5 c^4}+\frac{1}{5} i c d x^5 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{4} d x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{10} i b d x^4 \left(a+b \tan ^{-1}(c x)\right)-\frac{b d x^3 \left(a+b \tan ^{-1}(c x)\right)}{6 c}+\frac{b^2 d x^2}{12 c^2}-\frac{b^2 d \log \left(c^2 x^2+1\right)}{3 c^4}-\frac{3 i b^2 d x}{10 c^3}+\frac{b^2 d x \tan ^{-1}(c x)}{2 c^3}+\frac{3 i b^2 d \tan ^{-1}(c x)}{10 c^4}+\frac{i b^2 d x^3}{30 c}",1,"(a*b*d*x)/(2*c^3) - (((3*I)/10)*b^2*d*x)/c^3 + (b^2*d*x^2)/(12*c^2) + ((I/30)*b^2*d*x^3)/c + (((3*I)/10)*b^2*d*ArcTan[c*x])/c^4 + (b^2*d*x*ArcTan[c*x])/(2*c^3) + ((I/5)*b*d*x^2*(a + b*ArcTan[c*x]))/c^2 - (b*d*x^3*(a + b*ArcTan[c*x]))/(6*c) - (I/10)*b*d*x^4*(a + b*ArcTan[c*x]) - (9*d*(a + b*ArcTan[c*x])^2)/(20*c^4) + (d*x^4*(a + b*ArcTan[c*x])^2)/4 + (I/5)*c*d*x^5*(a + b*ArcTan[c*x])^2 + (((2*I)/5)*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^4 - (b^2*d*Log[1 + c^2*x^2])/(3*c^4) - (b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/(5*c^4)","A",27,15,23,0.6522,1,"{4876, 4852, 4916, 266, 43, 4846, 260, 4884, 302, 203, 321, 4920, 4854, 2402, 2315}"
69,1,255,0,0.4909996,"\int x^2 (d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x^2*(d + I*c*d*x)*(a + b*ArcTan[c*x])^2,x]","-\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^3}+\frac{i a b d x}{2 c^2}-\frac{7 i d \left(a+b \tan ^{-1}(c x)\right)^2}{12 c^3}-\frac{2 b d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}+\frac{1}{4} i c d x^4 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{3} d x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{6} i b d x^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{b d x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c}-\frac{i b^2 d \log \left(c^2 x^2+1\right)}{3 c^3}+\frac{b^2 d x}{3 c^2}+\frac{i b^2 d x \tan ^{-1}(c x)}{2 c^2}-\frac{b^2 d \tan ^{-1}(c x)}{3 c^3}+\frac{i b^2 d x^2}{12 c}","-\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^3}+\frac{i a b d x}{2 c^2}-\frac{7 i d \left(a+b \tan ^{-1}(c x)\right)^2}{12 c^3}-\frac{2 b d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}+\frac{1}{4} i c d x^4 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{3} d x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{6} i b d x^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{b d x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c}-\frac{i b^2 d \log \left(c^2 x^2+1\right)}{3 c^3}+\frac{b^2 d x}{3 c^2}+\frac{i b^2 d x \tan ^{-1}(c x)}{2 c^2}-\frac{b^2 d \tan ^{-1}(c x)}{3 c^3}+\frac{i b^2 d x^2}{12 c}",1,"((I/2)*a*b*d*x)/c^2 + (b^2*d*x)/(3*c^2) + ((I/12)*b^2*d*x^2)/c - (b^2*d*ArcTan[c*x])/(3*c^3) + ((I/2)*b^2*d*x*ArcTan[c*x])/c^2 - (b*d*x^2*(a + b*ArcTan[c*x]))/(3*c) - (I/6)*b*d*x^3*(a + b*ArcTan[c*x]) - (((7*I)/12)*d*(a + b*ArcTan[c*x])^2)/c^3 + (d*x^3*(a + b*ArcTan[c*x])^2)/3 + (I/4)*c*d*x^4*(a + b*ArcTan[c*x])^2 - (2*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) - ((I/3)*b^2*d*Log[1 + c^2*x^2])/c^3 - ((I/3)*b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3","A",22,14,23,0.6087,1,"{4876, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 266, 43, 4846, 260, 4884}"
70,1,211,0,0.3580161,"\int x (d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x*(d + I*c*d*x)*(a + b*ArcTan[c*x])^2,x]","\frac{b^2 d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^2}+\frac{5 d \left(a+b \tan ^{-1}(c x)\right)^2}{6 c^2}-\frac{2 i b d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^2}+\frac{1}{3} i c d x^3 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{2} d x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{3} i b d x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{a b d x}{c}+\frac{b^2 d \log \left(c^2 x^2+1\right)}{2 c^2}-\frac{i b^2 d \tan ^{-1}(c x)}{3 c^2}+\frac{i b^2 d x}{3 c}-\frac{b^2 d x \tan ^{-1}(c x)}{c}","\frac{b^2 d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^2}+\frac{5 d \left(a+b \tan ^{-1}(c x)\right)^2}{6 c^2}-\frac{2 i b d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^2}+\frac{1}{3} i c d x^3 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{2} d x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{3} i b d x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{a b d x}{c}+\frac{b^2 d \log \left(c^2 x^2+1\right)}{2 c^2}-\frac{i b^2 d \tan ^{-1}(c x)}{3 c^2}+\frac{i b^2 d x}{3 c}-\frac{b^2 d x \tan ^{-1}(c x)}{c}",1,"-((a*b*d*x)/c) + ((I/3)*b^2*d*x)/c - ((I/3)*b^2*d*ArcTan[c*x])/c^2 - (b^2*d*x*ArcTan[c*x])/c - (I/3)*b*d*x^2*(a + b*ArcTan[c*x]) + (5*d*(a + b*ArcTan[c*x])^2)/(6*c^2) + (d*x^2*(a + b*ArcTan[c*x])^2)/2 + (I/3)*c*d*x^3*(a + b*ArcTan[c*x])^2 - (((2*I)/3)*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^2 + (b^2*d*Log[1 + c^2*x^2])/(2*c^2) + (b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^2)","A",17,12,21,0.5714,1,"{4876, 4852, 4916, 4846, 260, 4884, 321, 203, 4920, 4854, 2402, 2315}"
71,1,130,0,0.1211625,"\int (d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[(d + I*c*d*x)*(a + b*ArcTan[c*x])^2,x]","-\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{c}-\frac{i d (1+i c x)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c}+\frac{2 b d \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}-i a b d x+\frac{i b^2 d \log \left(c^2 x^2+1\right)}{2 c}-i b^2 d x \tan ^{-1}(c x)","-\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{c}-\frac{i d (1+i c x)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c}+\frac{2 b d \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}-i a b d x+\frac{i b^2 d \log \left(c^2 x^2+1\right)}{2 c}-i b^2 d x \tan ^{-1}(c x)",1,"(-I)*a*b*d*x - I*b^2*d*x*ArcTan[c*x] - ((I/2)*d*(1 + I*c*x)^2*(a + b*ArcTan[c*x])^2)/c + (2*b*d*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/c + ((I/2)*b^2*d*Log[1 + c^2*x^2])/c - (I*b^2*d*PolyLog[2, 1 - 2/(1 - I*c*x)])/c","A",9,7,20,0.3500,1,"{4864, 4846, 260, 1586, 4854, 2402, 2315}"
72,1,216,0,0.4208071,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2}{x} \, dx","Int[((d + I*c*d*x)*(a + b*ArcTan[c*x])^2)/x,x]","-i b d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+b^2 (-d) \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)-\frac{1}{2} b^2 d \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{1}{2} b^2 d \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-d \left(a+b \tan ^{-1}(c x)\right)^2+i c d x \left(a+b \tan ^{-1}(c x)\right)^2+2 i b d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 d \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2","-i b d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+b^2 (-d) \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)-\frac{1}{2} b^2 d \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{1}{2} b^2 d \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-d \left(a+b \tan ^{-1}(c x)\right)^2+i c d x \left(a+b \tan ^{-1}(c x)\right)^2+2 i b d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 d \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2",1,"-(d*(a + b*ArcTan[c*x])^2) + I*c*d*x*(a + b*ArcTan[c*x])^2 + 2*d*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + (2*I)*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)] - I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (b^2*d*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*d*PolyLog[3, -1 + 2/(1 + I*c*x)])/2","A",13,11,23,0.4783,1,"{4876, 4846, 4920, 4854, 2402, 2315, 4850, 4988, 4884, 4994, 6610}"
73,1,228,0,0.4662549,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2}{x^2} \, dx","Int[((d + I*c*d*x)*(a + b*ArcTan[c*x])^2)/x^2,x]","b c d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-b c d \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-i b^2 c d \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)-\frac{1}{2} i b^2 c d \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{1}{2} i b^2 c d \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-i c d \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{x}+2 b c d \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 i c d \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2","b c d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-b c d \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-i b^2 c d \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)-\frac{1}{2} i b^2 c d \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{1}{2} i b^2 c d \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-i c d \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{x}+2 b c d \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 i c d \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2",1,"(-I)*c*d*(a + b*ArcTan[c*x])^2 - (d*(a + b*ArcTan[c*x])^2)/x + (2*I)*c*d*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + 2*b*c*d*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - I*b^2*c*d*PolyLog[2, -1 + 2/(1 - I*c*x)] + b*c*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] - b*c*d*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (I/2)*b^2*c*d*PolyLog[3, 1 - 2/(1 + I*c*x)] + (I/2)*b^2*c*d*PolyLog[3, -1 + 2/(1 + I*c*x)]","A",12,10,23,0.4348,1,"{4876, 4852, 4924, 4868, 2447, 4850, 4988, 4884, 4994, 6610}"
74,1,159,0,0.3392144,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2}{x^3} \, dx","Int[((d + I*c*d*x)*(a + b*ArcTan[c*x])^2)/x^3,x]","b^2 c^2 d \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)+\frac{1}{2} c^2 d \left(a+b \tan ^{-1}(c x)\right)^2+2 i b c^2 d \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{i c d \left(a+b \tan ^{-1}(c x)\right)^2}{x}-\frac{b c d \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{1}{2} b^2 c^2 d \log \left(c^2 x^2+1\right)+b^2 c^2 d \log (x)","b^2 c^2 d \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)+\frac{1}{2} c^2 d \left(a+b \tan ^{-1}(c x)\right)^2+2 i b c^2 d \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{i c d \left(a+b \tan ^{-1}(c x)\right)^2}{x}-\frac{b c d \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{1}{2} b^2 c^2 d \log \left(c^2 x^2+1\right)+b^2 c^2 d \log (x)",1,"-((b*c*d*(a + b*ArcTan[c*x]))/x) + (c^2*d*(a + b*ArcTan[c*x])^2)/2 - (d*(a + b*ArcTan[c*x])^2)/(2*x^2) - (I*c*d*(a + b*ArcTan[c*x])^2)/x + b^2*c^2*d*Log[x] - (b^2*c^2*d*Log[1 + c^2*x^2])/2 + (2*I)*b*c^2*d*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] + b^2*c^2*d*PolyLog[2, -1 + 2/(1 - I*c*x)]","A",14,11,23,0.4783,1,"{4876, 4852, 4918, 266, 36, 29, 31, 4884, 4924, 4868, 2447}"
75,1,224,0,0.4324266,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2}{x^4} \, dx","Int[((d + I*c*d*x)*(a + b*ArcTan[c*x])^2)/x^4,x]","\frac{1}{3} i b^2 c^3 d \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)-\frac{1}{6} i c^3 d \left(a+b \tan ^{-1}(c x)\right)^2-\frac{i b c^2 d \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{2}{3} b c^3 d \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{i c d \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{b c d \left(a+b \tan ^{-1}(c x)\right)}{3 x^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{3 x^3}-\frac{1}{2} i b^2 c^3 d \log \left(c^2 x^2+1\right)-\frac{b^2 c^2 d}{3 x}+i b^2 c^3 d \log (x)-\frac{1}{3} b^2 c^3 d \tan ^{-1}(c x)","\frac{1}{3} i b^2 c^3 d \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)-\frac{1}{6} i c^3 d \left(a+b \tan ^{-1}(c x)\right)^2-\frac{i b c^2 d \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{2}{3} b c^3 d \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{i c d \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{b c d \left(a+b \tan ^{-1}(c x)\right)}{3 x^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{3 x^3}-\frac{1}{2} i b^2 c^3 d \log \left(c^2 x^2+1\right)-\frac{b^2 c^2 d}{3 x}+i b^2 c^3 d \log (x)-\frac{1}{3} b^2 c^3 d \tan ^{-1}(c x)",1,"-(b^2*c^2*d)/(3*x) - (b^2*c^3*d*ArcTan[c*x])/3 - (b*c*d*(a + b*ArcTan[c*x]))/(3*x^2) - (I*b*c^2*d*(a + b*ArcTan[c*x]))/x - (I/6)*c^3*d*(a + b*ArcTan[c*x])^2 - (d*(a + b*ArcTan[c*x])^2)/(3*x^3) - ((I/2)*c*d*(a + b*ArcTan[c*x])^2)/x^2 + I*b^2*c^3*d*Log[x] - (I/2)*b^2*c^3*d*Log[1 + c^2*x^2] - (2*b*c^3*d*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/3 + (I/3)*b^2*c^3*d*PolyLog[2, -1 + 2/(1 - I*c*x)]","A",18,13,23,0.5652,1,"{4876, 4852, 4918, 325, 203, 4924, 4868, 2447, 266, 36, 29, 31, 4884}"
76,1,373,0,0.9733518,"\int x^3 (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x^3*(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2,x]","-\frac{2 b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{5 c^4}-\frac{1}{6} c^2 d^2 x^6 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{2 i b d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{5 c^2}+\frac{5 a b d^2 x}{6 c^3}-\frac{49 d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{60 c^4}+\frac{4 i b d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{5 c^4}+\frac{2}{5} i c d^2 x^5 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{15} b c d^2 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{5} i b d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)-\frac{5 b d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)}{18 c}+\frac{31 b^2 d^2 x^2}{180 c^2}-\frac{53 b^2 d^2 \log \left(c^2 x^2+1\right)}{90 c^4}-\frac{3 i b^2 d^2 x}{5 c^3}+\frac{5 b^2 d^2 x \tan ^{-1}(c x)}{6 c^3}+\frac{3 i b^2 d^2 \tan ^{-1}(c x)}{5 c^4}+\frac{i b^2 d^2 x^3}{15 c}-\frac{1}{60} b^2 d^2 x^4","-\frac{2 b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{5 c^4}-\frac{1}{6} c^2 d^2 x^6 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{2 i b d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{5 c^2}+\frac{5 a b d^2 x}{6 c^3}-\frac{49 d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{60 c^4}+\frac{4 i b d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{5 c^4}+\frac{2}{5} i c d^2 x^5 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{15} b c d^2 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{5} i b d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)-\frac{5 b d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)}{18 c}+\frac{31 b^2 d^2 x^2}{180 c^2}-\frac{53 b^2 d^2 \log \left(c^2 x^2+1\right)}{90 c^4}-\frac{3 i b^2 d^2 x}{5 c^3}+\frac{5 b^2 d^2 x \tan ^{-1}(c x)}{6 c^3}+\frac{3 i b^2 d^2 \tan ^{-1}(c x)}{5 c^4}+\frac{i b^2 d^2 x^3}{15 c}-\frac{1}{60} b^2 d^2 x^4",1,"(5*a*b*d^2*x)/(6*c^3) - (((3*I)/5)*b^2*d^2*x)/c^3 + (31*b^2*d^2*x^2)/(180*c^2) + ((I/15)*b^2*d^2*x^3)/c - (b^2*d^2*x^4)/60 + (((3*I)/5)*b^2*d^2*ArcTan[c*x])/c^4 + (5*b^2*d^2*x*ArcTan[c*x])/(6*c^3) + (((2*I)/5)*b*d^2*x^2*(a + b*ArcTan[c*x]))/c^2 - (5*b*d^2*x^3*(a + b*ArcTan[c*x]))/(18*c) - (I/5)*b*d^2*x^4*(a + b*ArcTan[c*x]) + (b*c*d^2*x^5*(a + b*ArcTan[c*x]))/15 - (49*d^2*(a + b*ArcTan[c*x])^2)/(60*c^4) + (d^2*x^4*(a + b*ArcTan[c*x])^2)/4 + ((2*I)/5)*c*d^2*x^5*(a + b*ArcTan[c*x])^2 - (c^2*d^2*x^6*(a + b*ArcTan[c*x])^2)/6 + (((4*I)/5)*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^4 - (53*b^2*d^2*Log[1 + c^2*x^2])/(90*c^4) - (2*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(5*c^4)","A",43,15,25,0.6000,1,"{4876, 4852, 4916, 266, 43, 4846, 260, 4884, 302, 203, 321, 4920, 4854, 2402, 2315}"
77,1,333,0,0.8566619,"\int x^2 (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x^2*(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2,x]","-\frac{8 i b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{15 c^3}-\frac{1}{5} c^2 d^2 x^5 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{i a b d^2 x}{c^2}-\frac{31 i d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{30 c^3}-\frac{16 b d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{15 c^3}+\frac{1}{2} i c d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{10} b c d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{3} i b d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{8 b d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{15 c}-\frac{2 i b^2 d^2 \log \left(c^2 x^2+1\right)}{3 c^3}+\frac{19 b^2 d^2 x}{30 c^2}+\frac{i b^2 d^2 x \tan ^{-1}(c x)}{c^2}-\frac{19 b^2 d^2 \tan ^{-1}(c x)}{30 c^3}+\frac{i b^2 d^2 x^2}{6 c}-\frac{1}{30} b^2 d^2 x^3","-\frac{8 i b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{15 c^3}-\frac{1}{5} c^2 d^2 x^5 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{i a b d^2 x}{c^2}-\frac{31 i d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{30 c^3}-\frac{16 b d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{15 c^3}+\frac{1}{2} i c d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{10} b c d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{3} i b d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{8 b d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{15 c}-\frac{2 i b^2 d^2 \log \left(c^2 x^2+1\right)}{3 c^3}+\frac{19 b^2 d^2 x}{30 c^2}+\frac{i b^2 d^2 x \tan ^{-1}(c x)}{c^2}-\frac{19 b^2 d^2 \tan ^{-1}(c x)}{30 c^3}+\frac{i b^2 d^2 x^2}{6 c}-\frac{1}{30} b^2 d^2 x^3",1,"(I*a*b*d^2*x)/c^2 + (19*b^2*d^2*x)/(30*c^2) + ((I/6)*b^2*d^2*x^2)/c - (b^2*d^2*x^3)/30 - (19*b^2*d^2*ArcTan[c*x])/(30*c^3) + (I*b^2*d^2*x*ArcTan[c*x])/c^2 - (8*b*d^2*x^2*(a + b*ArcTan[c*x]))/(15*c) - (I/3)*b*d^2*x^3*(a + b*ArcTan[c*x]) + (b*c*d^2*x^4*(a + b*ArcTan[c*x]))/10 - (((31*I)/30)*d^2*(a + b*ArcTan[c*x])^2)/c^3 + (d^2*x^3*(a + b*ArcTan[c*x])^2)/3 + (I/2)*c*d^2*x^4*(a + b*ArcTan[c*x])^2 - (c^2*d^2*x^5*(a + b*ArcTan[c*x])^2)/5 - (16*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(15*c^3) - (((2*I)/3)*b^2*d^2*Log[1 + c^2*x^2])/c^3 - (((8*I)/15)*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3","A",36,15,25,0.6000,1,"{4876, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 266, 43, 4846, 260, 4884, 302}"
78,1,293,0,0.6213981,"\int x (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x*(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2,x]","\frac{2 b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^2}-\frac{1}{4} c^2 d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{17 d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{12 c^2}-\frac{4 i b d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^2}+\frac{2}{3} i c d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{6} b c d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{2} d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{2}{3} i b d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{3 a b d^2 x}{2 c}+\frac{5 b^2 d^2 \log \left(c^2 x^2+1\right)}{6 c^2}-\frac{2 i b^2 d^2 \tan ^{-1}(c x)}{3 c^2}+\frac{2 i b^2 d^2 x}{3 c}-\frac{3 b^2 d^2 x \tan ^{-1}(c x)}{2 c}-\frac{1}{12} b^2 d^2 x^2","\frac{2 b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^2}-\frac{1}{4} c^2 d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{17 d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{12 c^2}-\frac{4 i b d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^2}+\frac{2}{3} i c d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{6} b c d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{2} d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{2}{3} i b d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{3 a b d^2 x}{2 c}+\frac{5 b^2 d^2 \log \left(c^2 x^2+1\right)}{6 c^2}-\frac{2 i b^2 d^2 \tan ^{-1}(c x)}{3 c^2}+\frac{2 i b^2 d^2 x}{3 c}-\frac{3 b^2 d^2 x \tan ^{-1}(c x)}{2 c}-\frac{1}{12} b^2 d^2 x^2",1,"(-3*a*b*d^2*x)/(2*c) + (((2*I)/3)*b^2*d^2*x)/c - (b^2*d^2*x^2)/12 - (((2*I)/3)*b^2*d^2*ArcTan[c*x])/c^2 - (3*b^2*d^2*x*ArcTan[c*x])/(2*c) - ((2*I)/3)*b*d^2*x^2*(a + b*ArcTan[c*x]) + (b*c*d^2*x^3*(a + b*ArcTan[c*x]))/6 + (17*d^2*(a + b*ArcTan[c*x])^2)/(12*c^2) + (d^2*x^2*(a + b*ArcTan[c*x])^2)/2 + ((2*I)/3)*c*d^2*x^3*(a + b*ArcTan[c*x])^2 - (c^2*d^2*x^4*(a + b*ArcTan[c*x])^2)/4 - (((4*I)/3)*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^2 + (5*b^2*d^2*Log[1 + c^2*x^2])/(6*c^2) + (2*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^2)","A",28,14,23,0.6087,1,"{4876, 4852, 4916, 4846, 260, 4884, 321, 203, 4920, 4854, 2402, 2315, 266, 43}"
79,1,192,0,0.196765,"\int (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2,x]","-\frac{4 i b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{3 c}+\frac{1}{3} b c d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{i d^2 (1+i c x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 c}+\frac{8 b d^2 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c}-2 i a b d^2 x+\frac{i b^2 d^2 \log \left(c^2 x^2+1\right)}{c}+\frac{b^2 d^2 \tan ^{-1}(c x)}{3 c}-2 i b^2 d^2 x \tan ^{-1}(c x)-\frac{1}{3} b^2 d^2 x","-\frac{4 i b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{3 c}+\frac{1}{3} b c d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{i d^2 (1+i c x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 c}+\frac{8 b d^2 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c}-2 i a b d^2 x+\frac{i b^2 d^2 \log \left(c^2 x^2+1\right)}{c}+\frac{b^2 d^2 \tan ^{-1}(c x)}{3 c}-2 i b^2 d^2 x \tan ^{-1}(c x)-\frac{1}{3} b^2 d^2 x",1,"(-2*I)*a*b*d^2*x - (b^2*d^2*x)/3 + (b^2*d^2*ArcTan[c*x])/(3*c) - (2*I)*b^2*d^2*x*ArcTan[c*x] + (b*c*d^2*x^2*(a + b*ArcTan[c*x]))/3 - ((I/3)*d^2*(1 + I*c*x)^3*(a + b*ArcTan[c*x])^2)/c + (8*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/(3*c) + (I*b^2*d^2*Log[1 + c^2*x^2])/c - (((4*I)/3)*b^2*d^2*PolyLog[2, 1 - 2/(1 - I*c*x)])/c","A",12,10,22,0.4545,1,"{4864, 4846, 260, 4852, 321, 203, 1586, 4854, 2402, 2315}"
80,1,300,0,0.5781195,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x} \, dx","Int[((d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2)/x,x]","-i b d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-2 b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)-\frac{1}{2} b^2 d^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{1}{2} b^2 d^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-\frac{1}{2} c^2 d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)^2+a b c d^2 x+2 i c d^2 x \left(a+b \tan ^{-1}(c x)\right)^2-\frac{5}{2} d^2 \left(a+b \tan ^{-1}(c x)\right)^2+4 i b d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 d^2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{2} b^2 d^2 \log \left(c^2 x^2+1\right)+b^2 c d^2 x \tan ^{-1}(c x)","-i b d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-2 b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)-\frac{1}{2} b^2 d^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{1}{2} b^2 d^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-\frac{1}{2} c^2 d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)^2+a b c d^2 x+2 i c d^2 x \left(a+b \tan ^{-1}(c x)\right)^2-\frac{5}{2} d^2 \left(a+b \tan ^{-1}(c x)\right)^2+4 i b d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 d^2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{2} b^2 d^2 \log \left(c^2 x^2+1\right)+b^2 c d^2 x \tan ^{-1}(c x)",1,"a*b*c*d^2*x + b^2*c*d^2*x*ArcTan[c*x] - (5*d^2*(a + b*ArcTan[c*x])^2)/2 + (2*I)*c*d^2*x*(a + b*ArcTan[c*x])^2 - (c^2*d^2*x^2*(a + b*ArcTan[c*x])^2)/2 + 2*d^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + (4*I)*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (b^2*d^2*Log[1 + c^2*x^2])/2 - 2*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)] - I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (b^2*d^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*d^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/2","A",19,14,25,0.5600,1,"{4876, 4846, 4920, 4854, 2402, 2315, 4850, 4988, 4884, 4994, 6610, 4852, 4916, 260}"
81,1,317,0,0.6196685,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x^2} \, dx","Int[((d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2)/x^2,x]","2 b c d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-2 b c d^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-i b^2 c d^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)-i b^2 c d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)-i b^2 c d^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+i b^2 c d^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-c^2 d^2 x \left(a+b \tan ^{-1}(c x)\right)^2-2 i c d^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x}-2 b c d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 b c d^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+4 i c d^2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2","2 b c d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-2 b c d^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-i b^2 c d^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)-i b^2 c d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)-i b^2 c d^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+i b^2 c d^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-c^2 d^2 x \left(a+b \tan ^{-1}(c x)\right)^2-2 i c d^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x}-2 b c d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 b c d^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+4 i c d^2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2",1,"(-2*I)*c*d^2*(a + b*ArcTan[c*x])^2 - (d^2*(a + b*ArcTan[c*x])^2)/x - c^2*d^2*x*(a + b*ArcTan[c*x])^2 + (4*I)*c*d^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] - 2*b*c*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] + 2*b*c*d^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - I*b^2*c*d^2*PolyLog[2, -1 + 2/(1 - I*c*x)] - I*b^2*c*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)] + 2*b*c*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] - 2*b*c*d^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - I*b^2*c*d^2*PolyLog[3, 1 - 2/(1 + I*c*x)] + I*b^2*c*d^2*PolyLog[3, -1 + 2/(1 + I*c*x)]","A",17,15,25,0.6000,1,"{4876, 4846, 4920, 4854, 2402, 2315, 4852, 4924, 4868, 2447, 4850, 4988, 4884, 4994, 6610}"
82,1,337,0,0.6454572,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x^3} \, dx","Int[((d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2)/x^3,x]","i b c^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-i b c^2 d^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 b^2 c^2 d^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)+\frac{1}{2} b^2 c^2 d^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)-\frac{1}{2} b^2 c^2 d^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)+\frac{3}{2} c^2 d^2 \left(a+b \tan ^{-1}(c x)\right)^2+4 i b c^2 d^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-2 c^2 d^2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{2 i c d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x}-\frac{b c d^2 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{1}{2} b^2 c^2 d^2 \log \left(c^2 x^2+1\right)+b^2 c^2 d^2 \log (x)","i b c^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-i b c^2 d^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 b^2 c^2 d^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)+\frac{1}{2} b^2 c^2 d^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)-\frac{1}{2} b^2 c^2 d^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)+\frac{3}{2} c^2 d^2 \left(a+b \tan ^{-1}(c x)\right)^2+4 i b c^2 d^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-2 c^2 d^2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{2 i c d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x}-\frac{b c d^2 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{1}{2} b^2 c^2 d^2 \log \left(c^2 x^2+1\right)+b^2 c^2 d^2 \log (x)",1,"-((b*c*d^2*(a + b*ArcTan[c*x]))/x) + (3*c^2*d^2*(a + b*ArcTan[c*x])^2)/2 - (d^2*(a + b*ArcTan[c*x])^2)/(2*x^2) - ((2*I)*c*d^2*(a + b*ArcTan[c*x])^2)/x - 2*c^2*d^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + b^2*c^2*d^2*Log[x] - (b^2*c^2*d^2*Log[1 + c^2*x^2])/2 + (4*I)*b*c^2*d^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] + 2*b^2*c^2*d^2*PolyLog[2, -1 + 2/(1 - I*c*x)] + I*b*c^2*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] - I*b*c^2*d^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] + (b^2*c^2*d^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 - (b^2*c^2*d^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/2","A",20,15,25,0.6000,1,"{4876, 4852, 4918, 266, 36, 29, 31, 4884, 4924, 4868, 2447, 4850, 4988, 4994, 6610}"
83,1,267,0,0.2683089,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x^4} \, dx","Int[((d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2)/x^4,x]","-\frac{4}{3} i b^2 c^3 d^2 \text{PolyLog}(2,-i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{PolyLog}(2,i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)-\frac{8}{3} a b c^3 d^2 \log (x)-\frac{2 i b c^2 d^2 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{8}{3} b c^3 d^2 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{b c d^2 \left(a+b \tan ^{-1}(c x)\right)}{3 x^2}-\frac{d^2 (1+i c x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 x^3}-i b^2 c^3 d^2 \log \left(c^2 x^2+1\right)-\frac{b^2 c^2 d^2}{3 x}+2 i b^2 c^3 d^2 \log (x)-\frac{1}{3} b^2 c^3 d^2 \tan ^{-1}(c x)","-\frac{4}{3} i b^2 c^3 d^2 \text{PolyLog}(2,-i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{PolyLog}(2,i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)-\frac{8}{3} a b c^3 d^2 \log (x)-\frac{2 i b c^2 d^2 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{8}{3} b c^3 d^2 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{b c d^2 \left(a+b \tan ^{-1}(c x)\right)}{3 x^2}-\frac{d^2 (1+i c x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 x^3}-i b^2 c^3 d^2 \log \left(c^2 x^2+1\right)-\frac{b^2 c^2 d^2}{3 x}+2 i b^2 c^3 d^2 \log (x)-\frac{1}{3} b^2 c^3 d^2 \tan ^{-1}(c x)",1,"-(b^2*c^2*d^2)/(3*x) - (b^2*c^3*d^2*ArcTan[c*x])/3 - (b*c*d^2*(a + b*ArcTan[c*x]))/(3*x^2) - ((2*I)*b*c^2*d^2*(a + b*ArcTan[c*x]))/x - (d^2*(1 + I*c*x)^3*(a + b*ArcTan[c*x])^2)/(3*x^3) - (8*a*b*c^3*d^2*Log[x])/3 + (2*I)*b^2*c^3*d^2*Log[x] - (8*b*c^3*d^2*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/3 - I*b^2*c^3*d^2*Log[1 + c^2*x^2] - ((4*I)/3)*b^2*c^3*d^2*PolyLog[2, (-I)*c*x] + ((4*I)/3)*b^2*c^3*d^2*PolyLog[2, I*c*x] + ((4*I)/3)*b^2*c^3*d^2*PolyLog[2, 1 - 2/(1 - I*c*x)]","A",16,14,25,0.5600,1,"{37, 4874, 4852, 325, 203, 266, 36, 29, 31, 4848, 2391, 4854, 2402, 2315}"
84,1,438,0,1.3656151,"\int x^3 (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x^3*(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2,x]","-\frac{26 b^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{35 c^4}-\frac{1}{7} i c^3 d^3 x^7 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{2} c^2 d^3 x^6 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{21} i b c^2 d^3 x^6 \left(a+b \tan ^{-1}(c x)\right)+\frac{26 i b d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)}{35 c^2}+\frac{3 a b d^3 x}{2 c^3}-\frac{209 d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{140 c^4}+\frac{52 i b d^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{35 c^4}+\frac{3}{5} i c d^3 x^5 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{5} b c d^3 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} d^3 x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{13}{35} i b d^3 x^4 \left(a+b \tan ^{-1}(c x)\right)-\frac{b d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)}{2 c}+\frac{7 b^2 d^3 x^2}{20 c^2}-\frac{11 b^2 d^3 \log \left(c^2 x^2+1\right)}{10 c^4}-\frac{122 i b^2 d^3 x}{105 c^3}+\frac{3 b^2 d^3 x \tan ^{-1}(c x)}{2 c^3}+\frac{122 i b^2 d^3 \tan ^{-1}(c x)}{105 c^4}-\frac{1}{105} i b^2 c d^3 x^5+\frac{44 i b^2 d^3 x^3}{315 c}-\frac{1}{20} b^2 d^3 x^4","-\frac{26 b^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{35 c^4}-\frac{1}{7} i c^3 d^3 x^7 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{2} c^2 d^3 x^6 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{21} i b c^2 d^3 x^6 \left(a+b \tan ^{-1}(c x)\right)+\frac{26 i b d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)}{35 c^2}+\frac{3 a b d^3 x}{2 c^3}-\frac{209 d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{140 c^4}+\frac{52 i b d^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{35 c^4}+\frac{3}{5} i c d^3 x^5 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{5} b c d^3 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} d^3 x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{13}{35} i b d^3 x^4 \left(a+b \tan ^{-1}(c x)\right)-\frac{b d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)}{2 c}+\frac{7 b^2 d^3 x^2}{20 c^2}-\frac{11 b^2 d^3 \log \left(c^2 x^2+1\right)}{10 c^4}-\frac{122 i b^2 d^3 x}{105 c^3}+\frac{3 b^2 d^3 x \tan ^{-1}(c x)}{2 c^3}+\frac{122 i b^2 d^3 \tan ^{-1}(c x)}{105 c^4}-\frac{1}{105} i b^2 c d^3 x^5+\frac{44 i b^2 d^3 x^3}{315 c}-\frac{1}{20} b^2 d^3 x^4",1,"(3*a*b*d^3*x)/(2*c^3) - (((122*I)/105)*b^2*d^3*x)/c^3 + (7*b^2*d^3*x^2)/(20*c^2) + (((44*I)/315)*b^2*d^3*x^3)/c - (b^2*d^3*x^4)/20 - (I/105)*b^2*c*d^3*x^5 + (((122*I)/105)*b^2*d^3*ArcTan[c*x])/c^4 + (3*b^2*d^3*x*ArcTan[c*x])/(2*c^3) + (((26*I)/35)*b*d^3*x^2*(a + b*ArcTan[c*x]))/c^2 - (b*d^3*x^3*(a + b*ArcTan[c*x]))/(2*c) - ((13*I)/35)*b*d^3*x^4*(a + b*ArcTan[c*x]) + (b*c*d^3*x^5*(a + b*ArcTan[c*x]))/5 + (I/21)*b*c^2*d^3*x^6*(a + b*ArcTan[c*x]) - (209*d^3*(a + b*ArcTan[c*x])^2)/(140*c^4) + (d^3*x^4*(a + b*ArcTan[c*x])^2)/4 + ((3*I)/5)*c*d^3*x^5*(a + b*ArcTan[c*x])^2 - (c^2*d^3*x^6*(a + b*ArcTan[c*x])^2)/2 - (I/7)*c^3*d^3*x^7*(a + b*ArcTan[c*x])^2 + (((52*I)/35)*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^4 - (11*b^2*d^3*Log[1 + c^2*x^2])/(10*c^4) - (26*b^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/(35*c^4)","A",62,15,25,0.6000,1,"{4876, 4852, 4916, 266, 43, 4846, 260, 4884, 302, 203, 321, 4920, 4854, 2402, 2315}"
85,1,402,0,1.1961694,"\int x^2 (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x^2*(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2,x]","-\frac{14 i b^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{15 c^3}-\frac{1}{6} i c^3 d^3 x^6 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{3}{5} c^2 d^3 x^5 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{15} i b c^2 d^3 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{11 i a b d^3 x}{6 c^2}-\frac{37 i d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{20 c^3}-\frac{28 b d^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{15 c^3}+\frac{3}{4} i c d^3 x^4 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{3}{10} b c d^3 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{11}{18} i b d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{14 b d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)}{15 c}-\frac{113 i b^2 d^3 \log \left(c^2 x^2+1\right)}{90 c^3}+\frac{37 b^2 d^3 x}{30 c^2}+\frac{11 i b^2 d^3 x \tan ^{-1}(c x)}{6 c^2}-\frac{37 b^2 d^3 \tan ^{-1}(c x)}{30 c^3}-\frac{1}{60} i b^2 c d^3 x^4+\frac{61 i b^2 d^3 x^2}{180 c}-\frac{1}{10} b^2 d^3 x^3","-\frac{14 i b^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{15 c^3}-\frac{1}{6} i c^3 d^3 x^6 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{3}{5} c^2 d^3 x^5 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{15} i b c^2 d^3 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{11 i a b d^3 x}{6 c^2}-\frac{37 i d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{20 c^3}-\frac{28 b d^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{15 c^3}+\frac{3}{4} i c d^3 x^4 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{3}{10} b c d^3 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{11}{18} i b d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{14 b d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)}{15 c}-\frac{113 i b^2 d^3 \log \left(c^2 x^2+1\right)}{90 c^3}+\frac{37 b^2 d^3 x}{30 c^2}+\frac{11 i b^2 d^3 x \tan ^{-1}(c x)}{6 c^2}-\frac{37 b^2 d^3 \tan ^{-1}(c x)}{30 c^3}-\frac{1}{60} i b^2 c d^3 x^4+\frac{61 i b^2 d^3 x^2}{180 c}-\frac{1}{10} b^2 d^3 x^3",1,"(((11*I)/6)*a*b*d^3*x)/c^2 + (37*b^2*d^3*x)/(30*c^2) + (((61*I)/180)*b^2*d^3*x^2)/c - (b^2*d^3*x^3)/10 - (I/60)*b^2*c*d^3*x^4 - (37*b^2*d^3*ArcTan[c*x])/(30*c^3) + (((11*I)/6)*b^2*d^3*x*ArcTan[c*x])/c^2 - (14*b*d^3*x^2*(a + b*ArcTan[c*x]))/(15*c) - ((11*I)/18)*b*d^3*x^3*(a + b*ArcTan[c*x]) + (3*b*c*d^3*x^4*(a + b*ArcTan[c*x]))/10 + (I/15)*b*c^2*d^3*x^5*(a + b*ArcTan[c*x]) - (((37*I)/20)*d^3*(a + b*ArcTan[c*x])^2)/c^3 + (d^3*x^3*(a + b*ArcTan[c*x])^2)/3 + ((3*I)/4)*c*d^3*x^4*(a + b*ArcTan[c*x])^2 - (3*c^2*d^3*x^5*(a + b*ArcTan[c*x])^2)/5 - (I/6)*c^3*d^3*x^6*(a + b*ArcTan[c*x])^2 - (28*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(15*c^3) - (((113*I)/90)*b^2*d^3*Log[1 + c^2*x^2])/c^3 - (((14*I)/15)*b^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3","A",52,15,25,0.6000,1,"{4876, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 266, 43, 4846, 260, 4884, 302}"
86,1,307,0,0.6136522,"\int x (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x*(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2,x]","-\frac{6 b^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{5 c^2}+\frac{1}{10} i b c^2 d^3 x^4 \left(a+b \tan ^{-1}(c x)\right)-\frac{d^3 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)^2}{5 c^2}+\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^2}-\frac{12 i b d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{5 c^2}+\frac{1}{2} b c d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{6}{5} i b d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{5 a b d^3 x}{2 c}+\frac{3 b^2 d^3 \log \left(c^2 x^2+1\right)}{2 c^2}-\frac{13 i b^2 d^3 \tan ^{-1}(c x)}{10 c^2}-\frac{1}{30} i b^2 c d^3 x^3+\frac{13 i b^2 d^3 x}{10 c}-\frac{5 b^2 d^3 x \tan ^{-1}(c x)}{2 c}-\frac{1}{4} b^2 d^3 x^2","-\frac{6 b^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{5 c^2}+\frac{1}{10} i b c^2 d^3 x^4 \left(a+b \tan ^{-1}(c x)\right)-\frac{d^3 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)^2}{5 c^2}+\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^2}-\frac{12 i b d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{5 c^2}+\frac{1}{2} b c d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{6}{5} i b d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{5 a b d^3 x}{2 c}+\frac{3 b^2 d^3 \log \left(c^2 x^2+1\right)}{2 c^2}-\frac{13 i b^2 d^3 \tan ^{-1}(c x)}{10 c^2}-\frac{1}{30} i b^2 c d^3 x^3+\frac{13 i b^2 d^3 x}{10 c}-\frac{5 b^2 d^3 x \tan ^{-1}(c x)}{2 c}-\frac{1}{4} b^2 d^3 x^2",1,"(-5*a*b*d^3*x)/(2*c) + (((13*I)/10)*b^2*d^3*x)/c - (b^2*d^3*x^2)/4 - (I/30)*b^2*c*d^3*x^3 - (((13*I)/10)*b^2*d^3*ArcTan[c*x])/c^2 - (5*b^2*d^3*x*ArcTan[c*x])/(2*c) - ((6*I)/5)*b*d^3*x^2*(a + b*ArcTan[c*x]) + (b*c*d^3*x^3*(a + b*ArcTan[c*x]))/2 + (I/10)*b*c^2*d^3*x^4*(a + b*ArcTan[c*x]) + (d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^2)/(4*c^2) - (d^3*(1 + I*c*x)^5*(a + b*ArcTan[c*x])^2)/(5*c^2) - (((12*I)/5)*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/c^2 + (3*b^2*d^3*Log[1 + c^2*x^2])/(2*c^2) - (6*b^2*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)])/(5*c^2)","A",38,14,23,0.6087,1,"{4876, 4864, 4846, 260, 4852, 321, 203, 266, 43, 1586, 4854, 2402, 2315, 302}"
87,1,226,0,0.2064898,"\int (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2,x]","-\frac{2 i b^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{c}+\frac{1}{6} i b c^2 d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)+b c d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{i d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)^2}{4 c}+\frac{4 b d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}-\frac{7}{2} i a b d^3 x+\frac{11 i b^2 d^3 \log \left(c^2 x^2+1\right)}{6 c}-\frac{1}{12} i b^2 c d^3 x^2+\frac{b^2 d^3 \tan ^{-1}(c x)}{c}-\frac{7}{2} i b^2 d^3 x \tan ^{-1}(c x)-b^2 d^3 x","-\frac{2 i b^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{c}+\frac{1}{6} i b c^2 d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)+b c d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{i d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)^2}{4 c}+\frac{4 b d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}-\frac{7}{2} i a b d^3 x+\frac{11 i b^2 d^3 \log \left(c^2 x^2+1\right)}{6 c}-\frac{1}{12} i b^2 c d^3 x^2+\frac{b^2 d^3 \tan ^{-1}(c x)}{c}-\frac{7}{2} i b^2 d^3 x \tan ^{-1}(c x)-b^2 d^3 x",1,"((-7*I)/2)*a*b*d^3*x - b^2*d^3*x - (I/12)*b^2*c*d^3*x^2 + (b^2*d^3*ArcTan[c*x])/c - ((7*I)/2)*b^2*d^3*x*ArcTan[c*x] + b*c*d^3*x^2*(a + b*ArcTan[c*x]) + (I/6)*b*c^2*d^3*x^3*(a + b*ArcTan[c*x]) - ((I/4)*d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^2)/c + (4*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/c + (((11*I)/6)*b^2*d^3*Log[1 + c^2*x^2])/c - ((2*I)*b^2*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)])/c","A",16,12,22,0.5455,1,"{4864, 4846, 260, 4852, 321, 203, 266, 43, 1586, 4854, 2402, 2315}"
88,1,385,0,0.7694113,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x} \, dx","Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x,x]","-i b d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d^3 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{10}{3} b^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)-\frac{1}{2} b^2 d^3 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{1}{2} b^2 d^3 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-\frac{1}{3} i c^3 d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{3}{2} c^2 d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{3} i b c^2 d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)+3 a b c d^3 x+3 i c d^3 x \left(a+b \tan ^{-1}(c x)\right)^2-\frac{29}{6} d^3 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{20}{3} i b d^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 d^3 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{3}{2} b^2 d^3 \log \left(c^2 x^2+1\right)-\frac{1}{3} i b^2 c d^3 x+\frac{1}{3} i b^2 d^3 \tan ^{-1}(c x)+3 b^2 c d^3 x \tan ^{-1}(c x)","-i b d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d^3 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{10}{3} b^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)-\frac{1}{2} b^2 d^3 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{1}{2} b^2 d^3 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-\frac{1}{3} i c^3 d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{3}{2} c^2 d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{3} i b c^2 d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)+3 a b c d^3 x+3 i c d^3 x \left(a+b \tan ^{-1}(c x)\right)^2-\frac{29}{6} d^3 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{20}{3} i b d^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 d^3 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{3}{2} b^2 d^3 \log \left(c^2 x^2+1\right)-\frac{1}{3} i b^2 c d^3 x+\frac{1}{3} i b^2 d^3 \tan ^{-1}(c x)+3 b^2 c d^3 x \tan ^{-1}(c x)",1,"3*a*b*c*d^3*x - (I/3)*b^2*c*d^3*x + (I/3)*b^2*d^3*ArcTan[c*x] + 3*b^2*c*d^3*x*ArcTan[c*x] + (I/3)*b*c^2*d^3*x^2*(a + b*ArcTan[c*x]) - (29*d^3*(a + b*ArcTan[c*x])^2)/6 + (3*I)*c*d^3*x*(a + b*ArcTan[c*x])^2 - (3*c^2*d^3*x^2*(a + b*ArcTan[c*x])^2)/2 - (I/3)*c^3*d^3*x^3*(a + b*ArcTan[c*x])^2 + 2*d^3*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + ((20*I)/3)*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (3*b^2*d^3*Log[1 + c^2*x^2])/2 - (10*b^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/3 - I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (b^2*d^3*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*d^3*PolyLog[3, -1 + 2/(1 + I*c*x)])/2","A",28,16,25,0.6400,1,"{4876, 4846, 4920, 4854, 2402, 2315, 4850, 4988, 4884, 4994, 6610, 4852, 4916, 260, 321, 203}"
89,1,402,0,0.7362185,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x^2} \, dx","Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^2,x]","3 b c d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-3 b c d^3 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-i b^2 c d^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)-3 i b^2 c d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)-\frac{3}{2} i b^2 c d^3 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{3}{2} i b^2 c d^3 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-\frac{1}{2} i c^3 d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)^2+i a b c^2 d^3 x-3 c^2 d^3 x \left(a+b \tan ^{-1}(c x)\right)^2-\frac{9}{2} i c d^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x}-6 b c d^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 b c d^3 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+6 i c d^3 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{2} i b^2 c d^3 \log \left(c^2 x^2+1\right)+i b^2 c^2 d^3 x \tan ^{-1}(c x)","3 b c d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-3 b c d^3 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-i b^2 c d^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)-3 i b^2 c d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)-\frac{3}{2} i b^2 c d^3 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{3}{2} i b^2 c d^3 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-\frac{1}{2} i c^3 d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)^2+i a b c^2 d^3 x-3 c^2 d^3 x \left(a+b \tan ^{-1}(c x)\right)^2-\frac{9}{2} i c d^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x}-6 b c d^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 b c d^3 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+6 i c d^3 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{2} i b^2 c d^3 \log \left(c^2 x^2+1\right)+i b^2 c^2 d^3 x \tan ^{-1}(c x)",1,"I*a*b*c^2*d^3*x + I*b^2*c^2*d^3*x*ArcTan[c*x] - ((9*I)/2)*c*d^3*(a + b*ArcTan[c*x])^2 - (d^3*(a + b*ArcTan[c*x])^2)/x - 3*c^2*d^3*x*(a + b*ArcTan[c*x])^2 - (I/2)*c^3*d^3*x^2*(a + b*ArcTan[c*x])^2 + (6*I)*c*d^3*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] - 6*b*c*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (I/2)*b^2*c*d^3*Log[1 + c^2*x^2] + 2*b*c*d^3*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - I*b^2*c*d^3*PolyLog[2, -1 + 2/(1 - I*c*x)] - (3*I)*b^2*c*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)] + 3*b*c*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] - 3*b*c*d^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - ((3*I)/2)*b^2*c*d^3*PolyLog[3, 1 - 2/(1 + I*c*x)] + ((3*I)/2)*b^2*c*d^3*PolyLog[3, -1 + 2/(1 + I*c*x)]","A",23,17,25,0.6800,1,"{4876, 4846, 4920, 4854, 2402, 2315, 4852, 4924, 4868, 2447, 4850, 4988, 4884, 4994, 6610, 4916, 260}"
90,1,416,0,0.7527467,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x^3} \, dx","Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^3,x]","3 i b c^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-3 i b c^2 d^3 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+3 b^2 c^2 d^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)+b^2 c^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)+\frac{3}{2} b^2 c^2 d^3 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)-\frac{3}{2} b^2 c^2 d^3 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)+\frac{7}{2} c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)^2-i c^3 d^3 x \left(a+b \tan ^{-1}(c x)\right)^2-2 i b c^2 d^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+6 i b c^2 d^3 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-6 c^2 d^3 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x}-\frac{b c d^3 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{1}{2} b^2 c^2 d^3 \log \left(c^2 x^2+1\right)+b^2 c^2 d^3 \log (x)","3 i b c^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-3 i b c^2 d^3 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+3 b^2 c^2 d^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)+b^2 c^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)+\frac{3}{2} b^2 c^2 d^3 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)-\frac{3}{2} b^2 c^2 d^3 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)+\frac{7}{2} c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)^2-i c^3 d^3 x \left(a+b \tan ^{-1}(c x)\right)^2-2 i b c^2 d^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+6 i b c^2 d^3 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-6 c^2 d^3 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x}-\frac{b c d^3 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{1}{2} b^2 c^2 d^3 \log \left(c^2 x^2+1\right)+b^2 c^2 d^3 \log (x)",1,"-((b*c*d^3*(a + b*ArcTan[c*x]))/x) + (7*c^2*d^3*(a + b*ArcTan[c*x])^2)/2 - (d^3*(a + b*ArcTan[c*x])^2)/(2*x^2) - ((3*I)*c*d^3*(a + b*ArcTan[c*x])^2)/x - I*c^3*d^3*x*(a + b*ArcTan[c*x])^2 - 6*c^2*d^3*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + b^2*c^2*d^3*Log[x] - (2*I)*b*c^2*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (b^2*c^2*d^3*Log[1 + c^2*x^2])/2 + (6*I)*b*c^2*d^3*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] + 3*b^2*c^2*d^3*PolyLog[2, -1 + 2/(1 - I*c*x)] + b^2*c^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)] + (3*I)*b*c^2*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] - (3*I)*b*c^2*d^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] + (3*b^2*c^2*d^3*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 - (3*b^2*c^2*d^3*PolyLog[3, -1 + 2/(1 + I*c*x)])/2","A",25,20,25,0.8000,1,"{4876, 4846, 4920, 4854, 2402, 2315, 4852, 4918, 266, 36, 29, 31, 4884, 4924, 4868, 2447, 4850, 4988, 4994, 6610}"
91,1,429,0,0.8902533,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x^4} \, dx","Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^4,x]","-b c^3 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+b c^3 d^3 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+\frac{10}{3} i b^2 c^3 d^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)+\frac{1}{2} i b^2 c^3 d^3 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)-\frac{1}{2} i b^2 c^3 d^3 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)+\frac{11}{6} i c^3 d^3 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{3 c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x}-\frac{3 i b c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{20}{3} b c^3 d^3 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-2 i c^3 d^3 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 x^3}-\frac{b c d^3 \left(a+b \tan ^{-1}(c x)\right)}{3 x^2}-\frac{3}{2} i b^2 c^3 d^3 \log \left(c^2 x^2+1\right)-\frac{b^2 c^2 d^3}{3 x}+3 i b^2 c^3 d^3 \log (x)-\frac{1}{3} b^2 c^3 d^3 \tan ^{-1}(c x)","-b c^3 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+b c^3 d^3 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+\frac{10}{3} i b^2 c^3 d^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)+\frac{1}{2} i b^2 c^3 d^3 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)-\frac{1}{2} i b^2 c^3 d^3 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)+\frac{11}{6} i c^3 d^3 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{3 c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x}-\frac{3 i b c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{20}{3} b c^3 d^3 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-2 i c^3 d^3 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 x^3}-\frac{b c d^3 \left(a+b \tan ^{-1}(c x)\right)}{3 x^2}-\frac{3}{2} i b^2 c^3 d^3 \log \left(c^2 x^2+1\right)-\frac{b^2 c^2 d^3}{3 x}+3 i b^2 c^3 d^3 \log (x)-\frac{1}{3} b^2 c^3 d^3 \tan ^{-1}(c x)",1,"-(b^2*c^2*d^3)/(3*x) - (b^2*c^3*d^3*ArcTan[c*x])/3 - (b*c*d^3*(a + b*ArcTan[c*x]))/(3*x^2) - ((3*I)*b*c^2*d^3*(a + b*ArcTan[c*x]))/x + ((11*I)/6)*c^3*d^3*(a + b*ArcTan[c*x])^2 - (d^3*(a + b*ArcTan[c*x])^2)/(3*x^3) - (((3*I)/2)*c*d^3*(a + b*ArcTan[c*x])^2)/x^2 + (3*c^2*d^3*(a + b*ArcTan[c*x])^2)/x - (2*I)*c^3*d^3*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + (3*I)*b^2*c^3*d^3*Log[x] - ((3*I)/2)*b^2*c^3*d^3*Log[1 + c^2*x^2] - (20*b*c^3*d^3*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/3 + ((10*I)/3)*b^2*c^3*d^3*PolyLog[2, -1 + 2/(1 - I*c*x)] - b*c^3*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + b*c^3*d^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] + (I/2)*b^2*c^3*d^3*PolyLog[3, 1 - 2/(1 + I*c*x)] - (I/2)*b^2*c^3*d^3*PolyLog[3, -1 + 2/(1 + I*c*x)]","A",28,17,25,0.6800,1,"{4876, 4852, 4918, 325, 203, 4924, 4868, 2447, 266, 36, 29, 31, 4884, 4850, 4988, 4994, 6610}"
92,1,293,0,0.3219665,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x^5} \, dx","Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^5,x]","2 b^2 c^4 d^3 \text{PolyLog}(2,-i c x)-2 b^2 c^4 d^3 \text{PolyLog}(2,i c x)-2 b^2 c^4 d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)-\frac{i b c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)}{x^2}-4 i a b c^4 d^3 \log (x)+\frac{7 b c^3 d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x}-4 i b c^4 d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{b c d^3 \left(a+b \tan ^{-1}(c x)\right)}{6 x^3}-\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)^2}{4 x^4}-\frac{b^2 c^2 d^3}{12 x^2}+\frac{11}{6} b^2 c^4 d^3 \log \left(c^2 x^2+1\right)-\frac{i b^2 c^3 d^3}{x}-\frac{11}{3} b^2 c^4 d^3 \log (x)-i b^2 c^4 d^3 \tan ^{-1}(c x)","2 b^2 c^4 d^3 \text{PolyLog}(2,-i c x)-2 b^2 c^4 d^3 \text{PolyLog}(2,i c x)-2 b^2 c^4 d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)-\frac{i b c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)}{x^2}-4 i a b c^4 d^3 \log (x)+\frac{7 b c^3 d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x}-4 i b c^4 d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{b c d^3 \left(a+b \tan ^{-1}(c x)\right)}{6 x^3}-\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)^2}{4 x^4}-\frac{b^2 c^2 d^3}{12 x^2}+\frac{11}{6} b^2 c^4 d^3 \log \left(c^2 x^2+1\right)-\frac{i b^2 c^3 d^3}{x}-\frac{11}{3} b^2 c^4 d^3 \log (x)-i b^2 c^4 d^3 \tan ^{-1}(c x)",1,"-(b^2*c^2*d^3)/(12*x^2) - (I*b^2*c^3*d^3)/x - I*b^2*c^4*d^3*ArcTan[c*x] - (b*c*d^3*(a + b*ArcTan[c*x]))/(6*x^3) - (I*b*c^2*d^3*(a + b*ArcTan[c*x]))/x^2 + (7*b*c^3*d^3*(a + b*ArcTan[c*x]))/(2*x) - (d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^2)/(4*x^4) - (4*I)*a*b*c^4*d^3*Log[x] - (11*b^2*c^4*d^3*Log[x])/3 - (4*I)*b*c^4*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)] + (11*b^2*c^4*d^3*Log[1 + c^2*x^2])/6 + 2*b^2*c^4*d^3*PolyLog[2, (-I)*c*x] - 2*b^2*c^4*d^3*PolyLog[2, I*c*x] - 2*b^2*c^4*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)]","A",20,15,25,0.6000,1,"{37, 4874, 4852, 266, 44, 325, 203, 36, 29, 31, 4848, 2391, 4854, 2402, 2315}"
93,1,384,0,0.3666113,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x^6} \, dx","Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^6,x]","\frac{6}{5} i b^2 c^5 d^3 \text{PolyLog}(2,-i c x)-\frac{6}{5} i b^2 c^5 d^3 \text{PolyLog}(2,i c x)-\frac{6}{5} i b^2 c^5 d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)+\frac{6 b c^3 d^3 \left(a+b \tan ^{-1}(c x)\right)}{5 x^2}-\frac{i b c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x^3}+\frac{12}{5} a b c^5 d^3 \log (x)+\frac{5 i b c^4 d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x}+\frac{12}{5} b c^5 d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+\frac{i c d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)^2}{20 x^4}-\frac{b c d^3 \left(a+b \tan ^{-1}(c x)\right)}{10 x^4}-\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)^2}{5 x^5}-\frac{i b^2 c^3 d^3}{4 x^2}-\frac{b^2 c^2 d^3}{30 x^3}+\frac{3}{2} i b^2 c^5 d^3 \log \left(c^2 x^2+1\right)+\frac{13 b^2 c^4 d^3}{10 x}-3 i b^2 c^5 d^3 \log (x)+\frac{13}{10} b^2 c^5 d^3 \tan ^{-1}(c x)","\frac{6}{5} i b^2 c^5 d^3 \text{PolyLog}(2,-i c x)-\frac{6}{5} i b^2 c^5 d^3 \text{PolyLog}(2,i c x)-\frac{6}{5} i b^2 c^5 d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)+\frac{6 b c^3 d^3 \left(a+b \tan ^{-1}(c x)\right)}{5 x^2}-\frac{i b c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x^3}+\frac{12}{5} a b c^5 d^3 \log (x)+\frac{5 i b c^4 d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x}+\frac{12}{5} b c^5 d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+\frac{i c d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)^2}{20 x^4}-\frac{b c d^3 \left(a+b \tan ^{-1}(c x)\right)}{10 x^4}-\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)^2}{5 x^5}-\frac{i b^2 c^3 d^3}{4 x^2}-\frac{b^2 c^2 d^3}{30 x^3}+\frac{3}{2} i b^2 c^5 d^3 \log \left(c^2 x^2+1\right)+\frac{13 b^2 c^4 d^3}{10 x}-3 i b^2 c^5 d^3 \log (x)+\frac{13}{10} b^2 c^5 d^3 \tan ^{-1}(c x)",1,"-(b^2*c^2*d^3)/(30*x^3) - ((I/4)*b^2*c^3*d^3)/x^2 + (13*b^2*c^4*d^3)/(10*x) + (13*b^2*c^5*d^3*ArcTan[c*x])/10 - (b*c*d^3*(a + b*ArcTan[c*x]))/(10*x^4) - ((I/2)*b*c^2*d^3*(a + b*ArcTan[c*x]))/x^3 + (6*b*c^3*d^3*(a + b*ArcTan[c*x]))/(5*x^2) + (((5*I)/2)*b*c^4*d^3*(a + b*ArcTan[c*x]))/x - (d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^2)/(5*x^5) + ((I/20)*c*d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^2)/x^4 + (12*a*b*c^5*d^3*Log[x])/5 - (3*I)*b^2*c^5*d^3*Log[x] + (12*b*c^5*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/5 + ((3*I)/2)*b^2*c^5*d^3*Log[1 + c^2*x^2] + ((6*I)/5)*b^2*c^5*d^3*PolyLog[2, (-I)*c*x] - ((6*I)/5)*b^2*c^5*d^3*PolyLog[2, I*c*x] - ((6*I)/5)*b^2*c^5*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)]","A",24,16,25,0.6400,1,"{45, 37, 4874, 4852, 325, 203, 266, 44, 36, 29, 31, 4848, 2391, 4854, 2402, 2315}"
94,1,513,0,0.5168082,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x^7} \, dx","Int[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^7,x]","-\frac{14}{15} b^2 c^6 d^3 \text{PolyLog}(2,-i c x)+\frac{14}{15} b^2 c^6 d^3 \text{PolyLog}(2,i c x)+\frac{37}{40} b^2 c^6 d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)-\frac{1}{120} b^2 c^6 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)+\frac{14 i b c^4 d^3 \left(a+b \tan ^{-1}(c x)\right)}{15 x^2}+\frac{i c^3 d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 x^3}+\frac{11 b c^3 d^3 \left(a+b \tan ^{-1}(c x)\right)}{18 x^3}+\frac{3 c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{4 x^4}-\frac{3 i b c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)}{10 x^4}+\frac{28}{15} i a b c^6 d^3 \log (x)-\frac{11 b c^5 d^3 \left(a+b \tan ^{-1}(c x)\right)}{6 x}+\frac{37}{20} i b c^6 d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{60} i b c^6 d^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{5 x^5}-\frac{b c d^3 \left(a+b \tan ^{-1}(c x)\right)}{15 x^5}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{6 x^6}+\frac{61 b^2 c^4 d^3}{180 x^2}-\frac{i b^2 c^3 d^3}{10 x^3}-\frac{b^2 c^2 d^3}{60 x^4}-\frac{113}{90} b^2 c^6 d^3 \log \left(c^2 x^2+1\right)+\frac{37 i b^2 c^5 d^3}{30 x}+\frac{113}{45} b^2 c^6 d^3 \log (x)+\frac{37}{30} i b^2 c^6 d^3 \tan ^{-1}(c x)","-\frac{14}{15} b^2 c^6 d^3 \text{PolyLog}(2,-i c x)+\frac{14}{15} b^2 c^6 d^3 \text{PolyLog}(2,i c x)+\frac{37}{40} b^2 c^6 d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)-\frac{1}{120} b^2 c^6 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)+\frac{14 i b c^4 d^3 \left(a+b \tan ^{-1}(c x)\right)}{15 x^2}+\frac{i c^3 d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 x^3}+\frac{11 b c^3 d^3 \left(a+b \tan ^{-1}(c x)\right)}{18 x^3}+\frac{3 c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{4 x^4}-\frac{3 i b c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)}{10 x^4}+\frac{28}{15} i a b c^6 d^3 \log (x)-\frac{11 b c^5 d^3 \left(a+b \tan ^{-1}(c x)\right)}{6 x}+\frac{37}{20} i b c^6 d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{60} i b c^6 d^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{5 x^5}-\frac{b c d^3 \left(a+b \tan ^{-1}(c x)\right)}{15 x^5}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{6 x^6}+\frac{61 b^2 c^4 d^3}{180 x^2}-\frac{i b^2 c^3 d^3}{10 x^3}-\frac{b^2 c^2 d^3}{60 x^4}-\frac{113}{90} b^2 c^6 d^3 \log \left(c^2 x^2+1\right)+\frac{37 i b^2 c^5 d^3}{30 x}+\frac{113}{45} b^2 c^6 d^3 \log (x)+\frac{37}{30} i b^2 c^6 d^3 \tan ^{-1}(c x)",1,"-(b^2*c^2*d^3)/(60*x^4) - ((I/10)*b^2*c^3*d^3)/x^3 + (61*b^2*c^4*d^3)/(180*x^2) + (((37*I)/30)*b^2*c^5*d^3)/x + ((37*I)/30)*b^2*c^6*d^3*ArcTan[c*x] - (b*c*d^3*(a + b*ArcTan[c*x]))/(15*x^5) - (((3*I)/10)*b*c^2*d^3*(a + b*ArcTan[c*x]))/x^4 + (11*b*c^3*d^3*(a + b*ArcTan[c*x]))/(18*x^3) + (((14*I)/15)*b*c^4*d^3*(a + b*ArcTan[c*x]))/x^2 - (11*b*c^5*d^3*(a + b*ArcTan[c*x]))/(6*x) - (d^3*(a + b*ArcTan[c*x])^2)/(6*x^6) - (((3*I)/5)*c*d^3*(a + b*ArcTan[c*x])^2)/x^5 + (3*c^2*d^3*(a + b*ArcTan[c*x])^2)/(4*x^4) + ((I/3)*c^3*d^3*(a + b*ArcTan[c*x])^2)/x^3 + ((28*I)/15)*a*b*c^6*d^3*Log[x] + (113*b^2*c^6*d^3*Log[x])/45 + ((37*I)/20)*b*c^6*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)] + (I/60)*b*c^6*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)] - (113*b^2*c^6*d^3*Log[1 + c^2*x^2])/90 - (14*b^2*c^6*d^3*PolyLog[2, (-I)*c*x])/15 + (14*b^2*c^6*d^3*PolyLog[2, I*c*x])/15 + (37*b^2*c^6*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)])/40 - (b^2*c^6*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/120","A",31,15,25,0.6000,1,"{43, 4874, 4852, 266, 44, 325, 203, 36, 29, 31, 4848, 2391, 4854, 2402, 2315}"
95,1,356,0,0.8231591,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{d+i c d x} \, dx","Int[(x^3*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x),x]","\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d}-\frac{4 b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^4 d}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^4 d}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 d}+\frac{i b x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c^2 d}-\frac{a b x}{c^3 d}+\frac{i x \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d}-\frac{5 \left(a+b \tan ^{-1}(c x)\right)^2}{6 c^4 d}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d}+\frac{8 i b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^4 d}-\frac{i x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 c d}+\frac{b^2 \log \left(c^2 x^2+1\right)}{2 c^4 d}-\frac{i b^2 x}{3 c^3 d}-\frac{b^2 x \tan ^{-1}(c x)}{c^3 d}+\frac{i b^2 \tan ^{-1}(c x)}{3 c^4 d}","\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d}-\frac{4 b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^4 d}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^4 d}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 d}+\frac{i b x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c^2 d}-\frac{a b x}{c^3 d}+\frac{i x \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d}-\frac{5 \left(a+b \tan ^{-1}(c x)\right)^2}{6 c^4 d}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d}+\frac{8 i b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^4 d}-\frac{i x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 c d}+\frac{b^2 \log \left(c^2 x^2+1\right)}{2 c^4 d}-\frac{i b^2 x}{3 c^3 d}-\frac{b^2 x \tan ^{-1}(c x)}{c^3 d}+\frac{i b^2 \tan ^{-1}(c x)}{3 c^4 d}",1,"-((a*b*x)/(c^3*d)) - ((I/3)*b^2*x)/(c^3*d) + ((I/3)*b^2*ArcTan[c*x])/(c^4*d) - (b^2*x*ArcTan[c*x])/(c^3*d) + ((I/3)*b*x^2*(a + b*ArcTan[c*x]))/(c^2*d) - (5*(a + b*ArcTan[c*x])^2)/(6*c^4*d) + (I*x*(a + b*ArcTan[c*x])^2)/(c^3*d) + (x^2*(a + b*ArcTan[c*x])^2)/(2*c^2*d) - ((I/3)*x^3*(a + b*ArcTan[c*x])^2)/(c*d) + (((8*I)/3)*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^4*d) + ((a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^4*d) + (b^2*Log[1 + c^2*x^2])/(2*c^4*d) - (4*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^4*d) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d) + (b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^4*d)","A",26,14,25,0.5600,1,"{4866, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 4846, 260, 4884, 4994, 6610}"
96,1,277,0,0.5130289,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{d+i c d x} \, dx","Int[(x^2*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x),x]","\frac{b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d}+\frac{i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^3 d}-\frac{i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^3 d}+\frac{i a b x}{c^2 d}+\frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 d}+\frac{2 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d}-\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d}-\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c d}-\frac{i b^2 \log \left(c^2 x^2+1\right)}{2 c^3 d}+\frac{i b^2 x \tan ^{-1}(c x)}{c^2 d}","\frac{b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d}+\frac{i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^3 d}-\frac{i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^3 d}+\frac{i a b x}{c^2 d}+\frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 d}+\frac{2 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d}-\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d}-\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c d}-\frac{i b^2 \log \left(c^2 x^2+1\right)}{2 c^3 d}+\frac{i b^2 x \tan ^{-1}(c x)}{c^2 d}",1,"(I*a*b*x)/(c^2*d) + (I*b^2*x*ArcTan[c*x])/(c^2*d) + ((I/2)*(a + b*ArcTan[c*x])^2)/(c^3*d) + (x*(a + b*ArcTan[c*x])^2)/(c^2*d) - ((I/2)*x^2*(a + b*ArcTan[c*x])^2)/(c*d) + (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^3*d) - (I*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^3*d) - ((I/2)*b^2*Log[1 + c^2*x^2])/(c^3*d) + (I*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d) + (b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d) - ((I/2)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^3*d)","A",16,12,25,0.4800,1,"{4866, 4852, 4916, 4846, 260, 4884, 4920, 4854, 2402, 2315, 4994, 6610}"
97,1,192,0,0.2925605,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{d+i c d x} \, dx","Int[(x*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x),x]","-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d}+\frac{b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^2 d}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^2 d}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d}-\frac{2 i b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d}-\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d}-\frac{i x \left(a+b \tan ^{-1}(c x)\right)^2}{c d}","-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d}+\frac{b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^2 d}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^2 d}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d}-\frac{2 i b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d}-\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d}-\frac{i x \left(a+b \tan ^{-1}(c x)\right)^2}{c d}",1,"(a + b*ArcTan[c*x])^2/(c^2*d) - (I*x*(a + b*ArcTan[c*x])^2)/(c*d) - ((2*I)*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^2*d) - ((a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^2*d) + (b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d) - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d) - (b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^2*d)","A",9,9,23,0.3913,1,"{4866, 4846, 4920, 4854, 2402, 2315, 4884, 4994, 6610}"
98,1,98,0,0.1334522,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d+i c d x} \, dx","Int[(a + b*ArcTan[c*x])^2/(d + I*c*d*x),x]","-\frac{b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c d}+\frac{i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c d}+\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c d}","-\frac{b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c d}+\frac{i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c d}+\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c d}",1,"(I*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c*d) - (b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c*d) + ((I/2)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c*d)","A",3,4,22,0.1818,1,"{4854, 4884, 4994, 6610}"
99,1,88,0,0.1606816,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x (d+i c d x)} \, dx","Int[(a + b*ArcTan[c*x])^2/(x*(d + I*c*d*x)),x]","\frac{i b \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{b^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d}+\frac{\log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}","\frac{i b \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{b^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d}+\frac{\log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}",1,"((a + b*ArcTan[c*x])^2*Log[2 - 2/(1 + I*c*x)])/d + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d)","A",3,4,25,0.1600,1,"{4868, 4884, 4994, 6610}"
100,1,186,0,0.3981141,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^2 (d+i c d x)} \, dx","Int[(a + b*ArcTan[c*x])^2/(x^2*(d + I*c*d*x)),x]","\frac{b c \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{i b^2 c \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d}-\frac{i b^2 c \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d}-\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d x}+\frac{2 b c \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{i c \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}","\frac{b c \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{i b^2 c \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d}-\frac{i b^2 c \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d}-\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d x}+\frac{2 b c \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{i c \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}",1,"((-I)*c*(a + b*ArcTan[c*x])^2)/d - (a + b*ArcTan[c*x])^2/(d*x) + (2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d - (I*c*(a + b*ArcTan[c*x])^2*Log[2 - 2/(1 + I*c*x)])/d - (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)])/d + (b*c*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d - ((I/2)*b^2*c*PolyLog[3, -1 + 2/(1 + I*c*x)])/d","A",8,8,25,0.3200,1,"{4870, 4852, 4924, 4868, 2447, 4884, 4994, 6610}"
101,1,273,0,0.6243446,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^3 (d+i c d x)} \, dx","Int[(a + b*ArcTan[c*x])^2/(x^3*(d + I*c*d*x)),x]","-\frac{i b c^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{b^2 c^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d}-\frac{b^2 c^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d}-\frac{3 c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}-\frac{2 i b c^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{c^2 \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 d x^2}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{d x}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{d x}-\frac{b^2 c^2 \log \left(c^2 x^2+1\right)}{2 d}+\frac{b^2 c^2 \log (x)}{d}","-\frac{i b c^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{b^2 c^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d}-\frac{b^2 c^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d}-\frac{3 c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}-\frac{2 i b c^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{c^2 \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 d x^2}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{d x}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{d x}-\frac{b^2 c^2 \log \left(c^2 x^2+1\right)}{2 d}+\frac{b^2 c^2 \log (x)}{d}",1,"-((b*c*(a + b*ArcTan[c*x]))/(d*x)) - (3*c^2*(a + b*ArcTan[c*x])^2)/(2*d) - (a + b*ArcTan[c*x])^2/(2*d*x^2) + (I*c*(a + b*ArcTan[c*x])^2)/(d*x) + (b^2*c^2*Log[x])/d - (b^2*c^2*Log[1 + c^2*x^2])/(2*d) - ((2*I)*b*c^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d - (c^2*(a + b*ArcTan[c*x])^2*Log[2 - 2/(1 + I*c*x)])/d - (b^2*c^2*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - (I*b*c^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d - (b^2*c^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d)","A",17,13,25,0.5200,1,"{4870, 4852, 4918, 266, 36, 29, 31, 4884, 4924, 4868, 2447, 4994, 6610}"
102,1,365,0,0.9748988,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^4 (d+i c d x)} \, dx","Int[(a + b*ArcTan[c*x])^2/(x^4*(d + I*c*d*x)),x]","-\frac{b c^3 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{4 i b^2 c^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{3 d}+\frac{i b^2 c^3 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d}+\frac{11 i c^3 \left(a+b \tan ^{-1}(c x)\right)^2}{6 d}+\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{d x}+\frac{i b c^2 \left(a+b \tan ^{-1}(c x)\right)}{d x}-\frac{8 b c^3 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 d}+\frac{i c^3 \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{2 d x^2}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{3 d x^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{3 d x^3}+\frac{i b^2 c^3 \log \left(c^2 x^2+1\right)}{2 d}-\frac{b^2 c^2}{3 d x}-\frac{i b^2 c^3 \log (x)}{d}-\frac{b^2 c^3 \tan ^{-1}(c x)}{3 d}","-\frac{b c^3 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{4 i b^2 c^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{3 d}+\frac{i b^2 c^3 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d}+\frac{11 i c^3 \left(a+b \tan ^{-1}(c x)\right)^2}{6 d}+\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{d x}+\frac{i b c^2 \left(a+b \tan ^{-1}(c x)\right)}{d x}-\frac{8 b c^3 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 d}+\frac{i c^3 \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{2 d x^2}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{3 d x^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{3 d x^3}+\frac{i b^2 c^3 \log \left(c^2 x^2+1\right)}{2 d}-\frac{b^2 c^2}{3 d x}-\frac{i b^2 c^3 \log (x)}{d}-\frac{b^2 c^3 \tan ^{-1}(c x)}{3 d}",1,"-(b^2*c^2)/(3*d*x) - (b^2*c^3*ArcTan[c*x])/(3*d) - (b*c*(a + b*ArcTan[c*x]))/(3*d*x^2) + (I*b*c^2*(a + b*ArcTan[c*x]))/(d*x) + (((11*I)/6)*c^3*(a + b*ArcTan[c*x])^2)/d - (a + b*ArcTan[c*x])^2/(3*d*x^3) + ((I/2)*c*(a + b*ArcTan[c*x])^2)/(d*x^2) + (c^2*(a + b*ArcTan[c*x])^2)/(d*x) - (I*b^2*c^3*Log[x])/d + ((I/2)*b^2*c^3*Log[1 + c^2*x^2])/d - (8*b*c^3*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/(3*d) + (I*c^3*(a + b*ArcTan[c*x])^2*Log[2 - 2/(1 + I*c*x)])/d + (((4*I)/3)*b^2*c^3*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - (b*c^3*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d + ((I/2)*b^2*c^3*PolyLog[3, -1 + 2/(1 + I*c*x)])/d","A",26,15,25,0.6000,1,"{4870, 4852, 4918, 325, 203, 4924, 4868, 2447, 266, 36, 29, 31, 4884, 4994, 6610}"
103,1,433,0,0.8271702,"\int \frac{x^4 \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^2} \, dx","Int[(x^4*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^2,x]","\frac{4 b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^5 d^2}+\frac{10 i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^5 d^2}-\frac{2 i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{c^5 d^2}-\frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^2 d^2}-\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^2}+\frac{b x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c^3 d^2}+\frac{2 i a b x}{c^4 d^2}+\frac{3 x \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{c^5 d^2 (-c x+i)}+\frac{11 i \left(a+b \tan ^{-1}(c x)\right)^2}{6 c^5 d^2}+\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{c^5 d^2 (-c x+i)}-\frac{4 i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^5 d^2}+\frac{20 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^5 d^2}-\frac{i b^2 \log \left(c^2 x^2+1\right)}{c^5 d^2}-\frac{b^2 x}{3 c^4 d^2}+\frac{b^2}{2 c^5 d^2 (-c x+i)}+\frac{2 i b^2 x \tan ^{-1}(c x)}{c^4 d^2}-\frac{b^2 \tan ^{-1}(c x)}{6 c^5 d^2}","\frac{4 b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^5 d^2}+\frac{10 i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^5 d^2}-\frac{2 i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{c^5 d^2}-\frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^2 d^2}-\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^2}+\frac{b x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c^3 d^2}+\frac{2 i a b x}{c^4 d^2}+\frac{3 x \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{c^5 d^2 (-c x+i)}+\frac{11 i \left(a+b \tan ^{-1}(c x)\right)^2}{6 c^5 d^2}+\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{c^5 d^2 (-c x+i)}-\frac{4 i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^5 d^2}+\frac{20 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^5 d^2}-\frac{i b^2 \log \left(c^2 x^2+1\right)}{c^5 d^2}-\frac{b^2 x}{3 c^4 d^2}+\frac{b^2}{2 c^5 d^2 (-c x+i)}+\frac{2 i b^2 x \tan ^{-1}(c x)}{c^4 d^2}-\frac{b^2 \tan ^{-1}(c x)}{6 c^5 d^2}",1,"((2*I)*a*b*x)/(c^4*d^2) - (b^2*x)/(3*c^4*d^2) + b^2/(2*c^5*d^2*(I - c*x)) - (b^2*ArcTan[c*x])/(6*c^5*d^2) + ((2*I)*b^2*x*ArcTan[c*x])/(c^4*d^2) + (b*x^2*(a + b*ArcTan[c*x]))/(3*c^3*d^2) + (I*b*(a + b*ArcTan[c*x]))/(c^5*d^2*(I - c*x)) + (((11*I)/6)*(a + b*ArcTan[c*x])^2)/(c^5*d^2) + (3*x*(a + b*ArcTan[c*x])^2)/(c^4*d^2) - (I*x^2*(a + b*ArcTan[c*x])^2)/(c^3*d^2) - (x^3*(a + b*ArcTan[c*x])^2)/(3*c^2*d^2) - (a + b*ArcTan[c*x])^2/(c^5*d^2*(I - c*x)) + (20*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^5*d^2) - ((4*I)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^5*d^2) - (I*b^2*Log[1 + c^2*x^2])/(c^5*d^2) + (((10*I)/3)*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^2) + (4*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^2) - ((2*I)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^5*d^2)","A",33,18,25,0.7200,1,"{4876, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 321, 203, 4864, 4862, 627, 44, 4994, 6610}"
104,1,364,0,0.6141991,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^2} \, dx","Int[(x^3*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^2,x]","-\frac{3 i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^2}+\frac{2 b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^4 d^2}-\frac{3 b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^4 d^2}-\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 d^2}+\frac{a b x}{c^3 d^2}+\frac{b \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^2 (-c x+i)}-\frac{2 i x \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^2 (-c x+i)}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^2}-\frac{4 i b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^2}-\frac{3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^2}-\frac{b^2 \log \left(c^2 x^2+1\right)}{2 c^4 d^2}-\frac{i b^2}{2 c^4 d^2 (-c x+i)}+\frac{b^2 x \tan ^{-1}(c x)}{c^3 d^2}+\frac{i b^2 \tan ^{-1}(c x)}{2 c^4 d^2}","-\frac{3 i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^2}+\frac{2 b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^4 d^2}-\frac{3 b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^4 d^2}-\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 d^2}+\frac{a b x}{c^3 d^2}+\frac{b \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^2 (-c x+i)}-\frac{2 i x \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^2 (-c x+i)}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^2}-\frac{4 i b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^2}-\frac{3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^2}-\frac{b^2 \log \left(c^2 x^2+1\right)}{2 c^4 d^2}-\frac{i b^2}{2 c^4 d^2 (-c x+i)}+\frac{b^2 x \tan ^{-1}(c x)}{c^3 d^2}+\frac{i b^2 \tan ^{-1}(c x)}{2 c^4 d^2}",1,"(a*b*x)/(c^3*d^2) - ((I/2)*b^2)/(c^4*d^2*(I - c*x)) + ((I/2)*b^2*ArcTan[c*x])/(c^4*d^2) + (b^2*x*ArcTan[c*x])/(c^3*d^2) + (b*(a + b*ArcTan[c*x]))/(c^4*d^2*(I - c*x)) + (a + b*ArcTan[c*x])^2/(c^4*d^2) - ((2*I)*x*(a + b*ArcTan[c*x])^2)/(c^3*d^2) - (x^2*(a + b*ArcTan[c*x])^2)/(2*c^2*d^2) + (I*(a + b*ArcTan[c*x])^2)/(c^4*d^2*(I - c*x)) - ((4*I)*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^4*d^2) - (3*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^4*d^2) - (b^2*Log[1 + c^2*x^2])/(2*c^4*d^2) + (2*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d^2) - ((3*I)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d^2) - (3*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^4*d^2)","A",24,17,25,0.6800,1,"{4876, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 4864, 4862, 627, 44, 203, 4994, 6610}"
105,1,292,0,0.4946623,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^2} \, dx","Int[(x^2*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^2,x]","-\frac{2 b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d^2}-\frac{i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^3 d^2}+\frac{i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{c^3 d^2}-\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{c^3 d^2 (-c x+i)}-\frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^2 (-c x+i)}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 d^2}-\frac{2 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d^2}+\frac{2 i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^2}-\frac{b^2}{2 c^3 d^2 (-c x+i)}+\frac{b^2 \tan ^{-1}(c x)}{2 c^3 d^2}","-\frac{2 b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d^2}-\frac{i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^3 d^2}+\frac{i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{c^3 d^2}-\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{c^3 d^2 (-c x+i)}-\frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^2 (-c x+i)}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 d^2}-\frac{2 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d^2}+\frac{2 i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^2}-\frac{b^2}{2 c^3 d^2 (-c x+i)}+\frac{b^2 \tan ^{-1}(c x)}{2 c^3 d^2}",1,"-b^2/(2*c^3*d^2*(I - c*x)) + (b^2*ArcTan[c*x])/(2*c^3*d^2) - (I*b*(a + b*ArcTan[c*x]))/(c^3*d^2*(I - c*x)) - ((I/2)*(a + b*ArcTan[c*x])^2)/(c^3*d^2) - (x*(a + b*ArcTan[c*x])^2)/(c^2*d^2) + (a + b*ArcTan[c*x])^2/(c^3*d^2*(I - c*x)) - (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^3*d^2) + ((2*I)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^3*d^2) - (I*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d^2) - (2*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d^2) + (I*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^3*d^2)","A",18,14,25,0.5600,1,"{4876, 4846, 4920, 4854, 2402, 2315, 4864, 4862, 627, 44, 203, 4884, 4994, 6610}"
106,1,216,0,0.3424016,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^2} \, dx","Int[(x*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^2,x]","\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d^2}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^2 d^2}-\frac{b \left(a+b \tan ^{-1}(c x)\right)}{c^2 d^2 (-c x+i)}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d^2 (-c x+i)}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 d^2}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d^2}+\frac{i b^2}{2 c^2 d^2 (-c x+i)}-\frac{i b^2 \tan ^{-1}(c x)}{2 c^2 d^2}","\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d^2}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^2 d^2}-\frac{b \left(a+b \tan ^{-1}(c x)\right)}{c^2 d^2 (-c x+i)}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d^2 (-c x+i)}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 d^2}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d^2}+\frac{i b^2}{2 c^2 d^2 (-c x+i)}-\frac{i b^2 \tan ^{-1}(c x)}{2 c^2 d^2}",1,"((I/2)*b^2)/(c^2*d^2*(I - c*x)) - ((I/2)*b^2*ArcTan[c*x])/(c^2*d^2) - (b*(a + b*ArcTan[c*x]))/(c^2*d^2*(I - c*x)) + (a + b*ArcTan[c*x])^2/(2*c^2*d^2) - (I*(a + b*ArcTan[c*x])^2)/(c^2*d^2*(I - c*x)) + ((a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^2*d^2) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d^2) + (b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^2*d^2)","A",13,10,23,0.4348,1,"{4876, 4864, 4862, 627, 44, 203, 4884, 4854, 4994, 6610}"
107,1,122,0,0.1215824,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^2} \, dx","Int[(a + b*ArcTan[c*x])^2/(d + I*c*d*x)^2,x]","\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{c d^2 (-c x+i)}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{c d^2 (1+i c x)}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{2 c d^2}+\frac{b^2}{2 c d^2 (-c x+i)}-\frac{b^2 \tan ^{-1}(c x)}{2 c d^2}","\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{c d^2 (-c x+i)}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{c d^2 (1+i c x)}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{2 c d^2}+\frac{b^2}{2 c d^2 (-c x+i)}-\frac{b^2 \tan ^{-1}(c x)}{2 c d^2}",1,"b^2/(2*c*d^2*(I - c*x)) - (b^2*ArcTan[c*x])/(2*c*d^2) + (I*b*(a + b*ArcTan[c*x]))/(c*d^2*(I - c*x)) - ((I/2)*(a + b*ArcTan[c*x])^2)/(c*d^2) + (I*(a + b*ArcTan[c*x])^2)/(c*d^2*(1 + I*c*x))","A",8,6,22,0.2727,1,"{4864, 4862, 627, 44, 203, 4884}"
108,1,221,0,0.6161169,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x (d+i c d x)^2} \, dx","Int[(a + b*ArcTan[c*x])^2/(x*(d + I*c*d*x)^2),x]","\frac{i b \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{b^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d^2}+\frac{b \left(a+b \tan ^{-1}(c x)\right)}{d^2 (-c x+i)}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{d^2 (-c x+i)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 d^2}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{i b^2}{2 d^2 (-c x+i)}+\frac{i b^2 \tan ^{-1}(c x)}{2 d^2}","\frac{i b \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{b^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d^2}+\frac{b \left(a+b \tan ^{-1}(c x)\right)}{d^2 (-c x+i)}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{d^2 (-c x+i)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 d^2}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{i b^2}{2 d^2 (-c x+i)}+\frac{i b^2 \tan ^{-1}(c x)}{2 d^2}",1,"((-I/2)*b^2)/(d^2*(I - c*x)) + ((I/2)*b^2*ArcTan[c*x])/d^2 + (b*(a + b*ArcTan[c*x]))/(d^2*(I - c*x)) - (a + b*ArcTan[c*x])^2/(2*d^2) + (I*(a + b*ArcTan[c*x])^2)/(d^2*(I - c*x)) + (2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^2 + ((a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^2 + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^2 + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^2)","A",19,12,25,0.4800,1,"{4876, 4850, 4988, 4884, 4994, 6610, 4864, 4862, 627, 44, 203, 4854}"
109,1,306,0,0.7670476,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^2 (d+i c d x)^2} \, dx","Int[(a + b*ArcTan[c*x])^2/(x^2*(d + I*c*d*x)^2),x]","\frac{2 b c \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{i b^2 c \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d^2}-\frac{i b^2 c \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{d^2}-\frac{i b c \left(a+b \tan ^{-1}(c x)\right)}{d^2 (-c x+i)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d^2 x}+\frac{c \left(a+b \tan ^{-1}(c x)\right)^2}{d^2 (-c x+i)}-\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{2 d^2}+\frac{2 b c \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{2 i c \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{4 i c \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{b^2 c}{2 d^2 (-c x+i)}+\frac{b^2 c \tan ^{-1}(c x)}{2 d^2}","\frac{2 b c \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{i b^2 c \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d^2}-\frac{i b^2 c \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{d^2}-\frac{i b c \left(a+b \tan ^{-1}(c x)\right)}{d^2 (-c x+i)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d^2 x}+\frac{c \left(a+b \tan ^{-1}(c x)\right)^2}{d^2 (-c x+i)}-\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{2 d^2}+\frac{2 b c \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{2 i c \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{4 i c \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{b^2 c}{2 d^2 (-c x+i)}+\frac{b^2 c \tan ^{-1}(c x)}{2 d^2}",1,"-(b^2*c)/(2*d^2*(I - c*x)) + (b^2*c*ArcTan[c*x])/(2*d^2) - (I*b*c*(a + b*ArcTan[c*x]))/(d^2*(I - c*x)) - ((I/2)*c*(a + b*ArcTan[c*x])^2)/d^2 - (a + b*ArcTan[c*x])^2/(d^2*x) + (c*(a + b*ArcTan[c*x])^2)/(d^2*(I - c*x)) - ((4*I)*c*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^2 - ((2*I)*c*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^2 + (2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d^2 - (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)])/d^2 + (2*b*c*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^2 - (I*b^2*c*PolyLog[3, -1 + 2/(1 + I*c*x)])/d^2","A",23,16,25,0.6400,1,"{4876, 4852, 4924, 4868, 2447, 4850, 4988, 4884, 4994, 6610, 4864, 4862, 627, 44, 203, 4854}"
110,1,403,0,0.9338969,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^3 (d+i c d x)^2} \, dx","Int[(a + b*ArcTan[c*x])^2/(x^3*(d + I*c*d*x)^2),x]","-\frac{3 i b c^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{2 b^2 c^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d^2}-\frac{3 b^2 c^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d^2}-\frac{i c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{d^2 (-c x+i)}-\frac{2 c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{b c^2 \left(a+b \tan ^{-1}(c x)\right)}{d^2 (-c x+i)}-\frac{3 c^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{4 i b c^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{6 c^2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 d^2 x^2}+\frac{2 i c \left(a+b \tan ^{-1}(c x)\right)^2}{d^2 x}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{d^2 x}-\frac{b^2 c^2 \log \left(c^2 x^2+1\right)}{2 d^2}+\frac{i b^2 c^2}{2 d^2 (-c x+i)}+\frac{b^2 c^2 \log (x)}{d^2}-\frac{i b^2 c^2 \tan ^{-1}(c x)}{2 d^2}","-\frac{3 i b c^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{2 b^2 c^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d^2}-\frac{3 b^2 c^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d^2}-\frac{i c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{d^2 (-c x+i)}-\frac{2 c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{b c^2 \left(a+b \tan ^{-1}(c x)\right)}{d^2 (-c x+i)}-\frac{3 c^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{4 i b c^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{6 c^2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 d^2 x^2}+\frac{2 i c \left(a+b \tan ^{-1}(c x)\right)^2}{d^2 x}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{d^2 x}-\frac{b^2 c^2 \log \left(c^2 x^2+1\right)}{2 d^2}+\frac{i b^2 c^2}{2 d^2 (-c x+i)}+\frac{b^2 c^2 \log (x)}{d^2}-\frac{i b^2 c^2 \tan ^{-1}(c x)}{2 d^2}",1,"((I/2)*b^2*c^2)/(d^2*(I - c*x)) - ((I/2)*b^2*c^2*ArcTan[c*x])/d^2 - (b*c*(a + b*ArcTan[c*x]))/(d^2*x) - (b*c^2*(a + b*ArcTan[c*x]))/(d^2*(I - c*x)) - (2*c^2*(a + b*ArcTan[c*x])^2)/d^2 - (a + b*ArcTan[c*x])^2/(2*d^2*x^2) + ((2*I)*c*(a + b*ArcTan[c*x])^2)/(d^2*x) - (I*c^2*(a + b*ArcTan[c*x])^2)/(d^2*(I - c*x)) - (6*c^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^2 + (b^2*c^2*Log[x])/d^2 - (3*c^2*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^2 - (b^2*c^2*Log[1 + c^2*x^2])/(2*d^2) - ((4*I)*b*c^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d^2 - (2*b^2*c^2*PolyLog[2, -1 + 2/(1 - I*c*x)])/d^2 - ((3*I)*b*c^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^2 - (3*b^2*c^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^2)","A",31,21,25,0.8400,1,"{4876, 4852, 4918, 266, 36, 29, 31, 4884, 4924, 4868, 2447, 4850, 4988, 4994, 6610, 4864, 4862, 627, 44, 203, 4854}"
111,1,462,0,0.8329219,"\int \frac{x^4 \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^3} \, dx","Int[(x^4*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3,x]","-\frac{6 b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^5 d^3}-\frac{3 i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^5 d^3}+\frac{3 i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{c^5 d^3}+\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 d^3}-\frac{i a b x}{c^4 d^3}-\frac{15 i b \left(a+b \tan ^{-1}(c x)\right)}{4 c^5 d^3 (-c x+i)}-\frac{b \left(a+b \tan ^{-1}(c x)\right)}{4 c^5 d^3 (-c x+i)^2}-\frac{3 x \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^3}+\frac{4 \left(a+b \tan ^{-1}(c x)\right)^2}{c^5 d^3 (-c x+i)}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^5 d^3 (-c x+i)^2}-\frac{5 i \left(a+b \tan ^{-1}(c x)\right)^2}{8 c^5 d^3}-\frac{6 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^5 d^3}+\frac{6 i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^5 d^3}+\frac{i b^2 \log \left(c^2 x^2+1\right)}{2 c^5 d^3}-\frac{29 b^2}{16 c^5 d^3 (-c x+i)}+\frac{i b^2}{16 c^5 d^3 (-c x+i)^2}-\frac{i b^2 x \tan ^{-1}(c x)}{c^4 d^3}+\frac{29 b^2 \tan ^{-1}(c x)}{16 c^5 d^3}","-\frac{6 b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^5 d^3}-\frac{3 i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^5 d^3}+\frac{3 i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{c^5 d^3}+\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 d^3}-\frac{i a b x}{c^4 d^3}-\frac{15 i b \left(a+b \tan ^{-1}(c x)\right)}{4 c^5 d^3 (-c x+i)}-\frac{b \left(a+b \tan ^{-1}(c x)\right)}{4 c^5 d^3 (-c x+i)^2}-\frac{3 x \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^3}+\frac{4 \left(a+b \tan ^{-1}(c x)\right)^2}{c^5 d^3 (-c x+i)}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^5 d^3 (-c x+i)^2}-\frac{5 i \left(a+b \tan ^{-1}(c x)\right)^2}{8 c^5 d^3}-\frac{6 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^5 d^3}+\frac{6 i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^5 d^3}+\frac{i b^2 \log \left(c^2 x^2+1\right)}{2 c^5 d^3}-\frac{29 b^2}{16 c^5 d^3 (-c x+i)}+\frac{i b^2}{16 c^5 d^3 (-c x+i)^2}-\frac{i b^2 x \tan ^{-1}(c x)}{c^4 d^3}+\frac{29 b^2 \tan ^{-1}(c x)}{16 c^5 d^3}",1,"((-I)*a*b*x)/(c^4*d^3) + ((I/16)*b^2)/(c^5*d^3*(I - c*x)^2) - (29*b^2)/(16*c^5*d^3*(I - c*x)) + (29*b^2*ArcTan[c*x])/(16*c^5*d^3) - (I*b^2*x*ArcTan[c*x])/(c^4*d^3) - (b*(a + b*ArcTan[c*x]))/(4*c^5*d^3*(I - c*x)^2) - (((15*I)/4)*b*(a + b*ArcTan[c*x]))/(c^5*d^3*(I - c*x)) - (((5*I)/8)*(a + b*ArcTan[c*x])^2)/(c^5*d^3) - (3*x*(a + b*ArcTan[c*x])^2)/(c^4*d^3) + ((I/2)*x^2*(a + b*ArcTan[c*x])^2)/(c^3*d^3) - ((I/2)*(a + b*ArcTan[c*x])^2)/(c^5*d^3*(I - c*x)^2) + (4*(a + b*ArcTan[c*x])^2)/(c^5*d^3*(I - c*x)) - (6*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^5*d^3) + ((6*I)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^5*d^3) + ((I/2)*b^2*Log[1 + c^2*x^2])/(c^5*d^3) - ((3*I)*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^3) - (6*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^5*d^3) + ((3*I)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^5*d^3)","A",37,17,25,0.6800,1,"{4876, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 4864, 4862, 627, 44, 203, 4994, 6610}"
112,1,383,0,0.6746299,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^3} \, dx","Int[(x^3*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3,x]","\frac{3 i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^3}-\frac{b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^4 d^3}+\frac{3 b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^4 d^3}-\frac{11 b \left(a+b \tan ^{-1}(c x)\right)}{4 c^4 d^3 (-c x+i)}+\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{4 c^4 d^3 (-c x+i)^2}+\frac{i x \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^3}-\frac{3 i \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^3 (-c x+i)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 c^4 d^3 (-c x+i)^2}+\frac{3 \left(a+b \tan ^{-1}(c x)\right)^2}{8 c^4 d^3}+\frac{2 i b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^3}+\frac{3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^3}+\frac{21 i b^2}{16 c^4 d^3 (-c x+i)}+\frac{b^2}{16 c^4 d^3 (-c x+i)^2}-\frac{21 i b^2 \tan ^{-1}(c x)}{16 c^4 d^3}","\frac{3 i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^3}-\frac{b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c^4 d^3}+\frac{3 b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^4 d^3}-\frac{11 b \left(a+b \tan ^{-1}(c x)\right)}{4 c^4 d^3 (-c x+i)}+\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{4 c^4 d^3 (-c x+i)^2}+\frac{i x \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^3}-\frac{3 i \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^3 (-c x+i)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 c^4 d^3 (-c x+i)^2}+\frac{3 \left(a+b \tan ^{-1}(c x)\right)^2}{8 c^4 d^3}+\frac{2 i b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^3}+\frac{3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^3}+\frac{21 i b^2}{16 c^4 d^3 (-c x+i)}+\frac{b^2}{16 c^4 d^3 (-c x+i)^2}-\frac{21 i b^2 \tan ^{-1}(c x)}{16 c^4 d^3}",1,"b^2/(16*c^4*d^3*(I - c*x)^2) + (((21*I)/16)*b^2)/(c^4*d^3*(I - c*x)) - (((21*I)/16)*b^2*ArcTan[c*x])/(c^4*d^3) + ((I/4)*b*(a + b*ArcTan[c*x]))/(c^4*d^3*(I - c*x)^2) - (11*b*(a + b*ArcTan[c*x]))/(4*c^4*d^3*(I - c*x)) + (3*(a + b*ArcTan[c*x])^2)/(8*c^4*d^3) + (I*x*(a + b*ArcTan[c*x])^2)/(c^3*d^3) - (a + b*ArcTan[c*x])^2/(2*c^4*d^3*(I - c*x)^2) - ((3*I)*(a + b*ArcTan[c*x])^2)/(c^4*d^3*(I - c*x)) + ((2*I)*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^4*d^3) + (3*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^4*d^3) - (b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d^3) + ((3*I)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^4*d^3) + (3*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^4*d^3)","A",31,14,25,0.5600,1,"{4876, 4846, 4920, 4854, 2402, 2315, 4864, 4862, 627, 44, 203, 4884, 4994, 6610}"
113,1,304,0,0.5604275,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^3} \, dx","Int[(x^2*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3,x]","\frac{b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d^3}-\frac{i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^3 d^3}+\frac{7 i b \left(a+b \tan ^{-1}(c x)\right)}{4 c^3 d^3 (-c x+i)}+\frac{b \left(a+b \tan ^{-1}(c x)\right)}{4 c^3 d^3 (-c x+i)^2}-\frac{2 \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^3 (-c x+i)}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 d^3 (-c x+i)^2}-\frac{7 i \left(a+b \tan ^{-1}(c x)\right)^2}{8 c^3 d^3}-\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^3}+\frac{13 b^2}{16 c^3 d^3 (-c x+i)}-\frac{i b^2}{16 c^3 d^3 (-c x+i)^2}-\frac{13 b^2 \tan ^{-1}(c x)}{16 c^3 d^3}","\frac{b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d^3}-\frac{i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^3 d^3}+\frac{7 i b \left(a+b \tan ^{-1}(c x)\right)}{4 c^3 d^3 (-c x+i)}+\frac{b \left(a+b \tan ^{-1}(c x)\right)}{4 c^3 d^3 (-c x+i)^2}-\frac{2 \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^3 (-c x+i)}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 d^3 (-c x+i)^2}-\frac{7 i \left(a+b \tan ^{-1}(c x)\right)^2}{8 c^3 d^3}-\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^3}+\frac{13 b^2}{16 c^3 d^3 (-c x+i)}-\frac{i b^2}{16 c^3 d^3 (-c x+i)^2}-\frac{13 b^2 \tan ^{-1}(c x)}{16 c^3 d^3}",1,"((-I/16)*b^2)/(c^3*d^3*(I - c*x)^2) + (13*b^2)/(16*c^3*d^3*(I - c*x)) - (13*b^2*ArcTan[c*x])/(16*c^3*d^3) + (b*(a + b*ArcTan[c*x]))/(4*c^3*d^3*(I - c*x)^2) + (((7*I)/4)*b*(a + b*ArcTan[c*x]))/(c^3*d^3*(I - c*x)) - (((7*I)/8)*(a + b*ArcTan[c*x])^2)/(c^3*d^3) + ((I/2)*(a + b*ArcTan[c*x])^2)/(c^3*d^3*(I - c*x)^2) - (2*(a + b*ArcTan[c*x])^2)/(c^3*d^3*(I - c*x)) - (I*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^3*d^3) + (b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d^3) - ((I/2)*b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^3*d^3)","A",26,10,25,0.4000,1,"{4876, 4864, 4862, 627, 44, 203, 4884, 4854, 4994, 6610}"
114,1,178,0,0.2167779,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^3} \, dx","Int[(x*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3,x]","\frac{3 b \left(a+b \tan ^{-1}(c x)\right)}{4 c^2 d^3 (-c x+i)}-\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{4 c^2 d^3 (-c x+i)^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{8 c^2 d^3}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d^3 (1+i c x)^2}-\frac{5 i b^2}{16 c^2 d^3 (-c x+i)}-\frac{b^2}{16 c^2 d^3 (-c x+i)^2}+\frac{5 i b^2 \tan ^{-1}(c x)}{16 c^2 d^3}","\frac{3 b \left(a+b \tan ^{-1}(c x)\right)}{4 c^2 d^3 (-c x+i)}-\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{4 c^2 d^3 (-c x+i)^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{8 c^2 d^3}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d^3 (1+i c x)^2}-\frac{5 i b^2}{16 c^2 d^3 (-c x+i)}-\frac{b^2}{16 c^2 d^3 (-c x+i)^2}+\frac{5 i b^2 \tan ^{-1}(c x)}{16 c^2 d^3}",1,"-b^2/(16*c^2*d^3*(I - c*x)^2) - (((5*I)/16)*b^2)/(c^2*d^3*(I - c*x)) + (((5*I)/16)*b^2*ArcTan[c*x])/(c^2*d^3) - ((I/4)*b*(a + b*ArcTan[c*x]))/(c^2*d^3*(I - c*x)^2) + (3*b*(a + b*ArcTan[c*x]))/(4*c^2*d^3*(I - c*x)) + (a + b*ArcTan[c*x])^2/(8*c^2*d^3) + (x^2*(a + b*ArcTan[c*x])^2)/(2*d^3*(1 + I*c*x)^2)","A",13,7,23,0.3043,1,"{37, 4874, 4862, 627, 44, 203, 4884}"
115,1,180,0,0.1795499,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^3} \, dx","Int[(a + b*ArcTan[c*x])^2/(d + I*c*d*x)^3,x]","\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{4 c d^3 (-c x+i)}-\frac{b \left(a+b \tan ^{-1}(c x)\right)}{4 c d^3 (-c x+i)^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{2 c d^3 (1+i c x)^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{8 c d^3}+\frac{3 b^2}{16 c d^3 (-c x+i)}+\frac{i b^2}{16 c d^3 (-c x+i)^2}-\frac{3 b^2 \tan ^{-1}(c x)}{16 c d^3}","\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{4 c d^3 (-c x+i)}-\frac{b \left(a+b \tan ^{-1}(c x)\right)}{4 c d^3 (-c x+i)^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{2 c d^3 (1+i c x)^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{8 c d^3}+\frac{3 b^2}{16 c d^3 (-c x+i)}+\frac{i b^2}{16 c d^3 (-c x+i)^2}-\frac{3 b^2 \tan ^{-1}(c x)}{16 c d^3}",1,"((I/16)*b^2)/(c*d^3*(I - c*x)^2) + (3*b^2)/(16*c*d^3*(I - c*x)) - (3*b^2*ArcTan[c*x])/(16*c*d^3) - (b*(a + b*ArcTan[c*x]))/(4*c*d^3*(I - c*x)^2) + ((I/4)*b*(a + b*ArcTan[c*x]))/(c*d^3*(I - c*x)) - ((I/8)*(a + b*ArcTan[c*x])^2)/(c*d^3) + ((I/2)*(a + b*ArcTan[c*x])^2)/(c*d^3*(1 + I*c*x)^2)","A",13,6,22,0.2727,1,"{4864, 4862, 627, 44, 203, 4884}"
116,1,299,0,0.7947151,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x (d+i c d x)^3} \, dx","Int[(a + b*ArcTan[c*x])^2/(x*(d + I*c*d*x)^3),x]","\frac{i b \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}+\frac{b^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d^3}+\frac{5 b \left(a+b \tan ^{-1}(c x)\right)}{4 d^3 (-c x+i)}+\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{4 d^3 (-c x+i)^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{d^3 (-c x+i)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 d^3 (-c x+i)^2}-\frac{5 \left(a+b \tan ^{-1}(c x)\right)^2}{8 d^3}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^3}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^3}-\frac{11 i b^2}{16 d^3 (-c x+i)}+\frac{b^2}{16 d^3 (-c x+i)^2}+\frac{11 i b^2 \tan ^{-1}(c x)}{16 d^3}","\frac{i b \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}+\frac{b^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d^3}+\frac{5 b \left(a+b \tan ^{-1}(c x)\right)}{4 d^3 (-c x+i)}+\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{4 d^3 (-c x+i)^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{d^3 (-c x+i)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 d^3 (-c x+i)^2}-\frac{5 \left(a+b \tan ^{-1}(c x)\right)^2}{8 d^3}+\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^3}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^3}-\frac{11 i b^2}{16 d^3 (-c x+i)}+\frac{b^2}{16 d^3 (-c x+i)^2}+\frac{11 i b^2 \tan ^{-1}(c x)}{16 d^3}",1,"b^2/(16*d^3*(I - c*x)^2) - (((11*I)/16)*b^2)/(d^3*(I - c*x)) + (((11*I)/16)*b^2*ArcTan[c*x])/d^3 + ((I/4)*b*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)^2) + (5*b*(a + b*ArcTan[c*x]))/(4*d^3*(I - c*x)) - (5*(a + b*ArcTan[c*x])^2)/(8*d^3) - (a + b*ArcTan[c*x])^2/(2*d^3*(I - c*x)^2) + (I*(a + b*ArcTan[c*x])^2)/(d^3*(I - c*x)) + (2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^3 + ((a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^3 + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^3 + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^3)","A",32,12,25,0.4800,1,"{4876, 4850, 4988, 4884, 4994, 6610, 4864, 4862, 627, 44, 203, 4854}"
117,1,391,0,0.9764818,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^2 (d+i c d x)^3} \, dx","Int[(a + b*ArcTan[c*x])^2/(x^2*(d + I*c*d*x)^3),x]","\frac{3 b c \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}-\frac{i b^2 c \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d^3}-\frac{3 i b^2 c \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d^3}-\frac{9 i b c \left(a+b \tan ^{-1}(c x)\right)}{4 d^3 (-c x+i)}+\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{4 d^3 (-c x+i)^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d^3 x}+\frac{2 c \left(a+b \tan ^{-1}(c x)\right)^2}{d^3 (-c x+i)}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{2 d^3 (-c x+i)^2}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{8 d^3}+\frac{2 b c \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}-\frac{3 i c \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^3}-\frac{6 i c \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^3}-\frac{19 b^2 c}{16 d^3 (-c x+i)}-\frac{i b^2 c}{16 d^3 (-c x+i)^2}+\frac{19 b^2 c \tan ^{-1}(c x)}{16 d^3}","\frac{3 b c \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}-\frac{i b^2 c \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d^3}-\frac{3 i b^2 c \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d^3}-\frac{9 i b c \left(a+b \tan ^{-1}(c x)\right)}{4 d^3 (-c x+i)}+\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{4 d^3 (-c x+i)^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d^3 x}+\frac{2 c \left(a+b \tan ^{-1}(c x)\right)^2}{d^3 (-c x+i)}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{2 d^3 (-c x+i)^2}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{8 d^3}+\frac{2 b c \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}-\frac{3 i c \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^3}-\frac{6 i c \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^3}-\frac{19 b^2 c}{16 d^3 (-c x+i)}-\frac{i b^2 c}{16 d^3 (-c x+i)^2}+\frac{19 b^2 c \tan ^{-1}(c x)}{16 d^3}",1,"((-I/16)*b^2*c)/(d^3*(I - c*x)^2) - (19*b^2*c)/(16*d^3*(I - c*x)) + (19*b^2*c*ArcTan[c*x])/(16*d^3) + (b*c*(a + b*ArcTan[c*x]))/(4*d^3*(I - c*x)^2) - (((9*I)/4)*b*c*(a + b*ArcTan[c*x]))/(d^3*(I - c*x)) + ((I/8)*c*(a + b*ArcTan[c*x])^2)/d^3 - (a + b*ArcTan[c*x])^2/(d^3*x) + ((I/2)*c*(a + b*ArcTan[c*x])^2)/(d^3*(I - c*x)^2) + (2*c*(a + b*ArcTan[c*x])^2)/(d^3*(I - c*x)) - ((6*I)*c*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^3 - ((3*I)*c*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/d^3 + (2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d^3 - (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)])/d^3 + (3*b*c*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^3 - (((3*I)/2)*b^2*c*PolyLog[3, -1 + 2/(1 + I*c*x)])/d^3","A",36,16,25,0.6400,1,"{4876, 4852, 4924, 4868, 2447, 4850, 4988, 4884, 4994, 6610, 4864, 4862, 627, 44, 203, 4854}"
118,1,207,0,0.222115,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{(1+i c x)^4} \, dx","Int[(a + b*ArcTan[c*x])^2/(1 + I*c*x)^4,x]","\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{12 c (-c x+i)}-\frac{b \left(a+b \tan ^{-1}(c x)\right)}{12 c (-c x+i)^2}-\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{9 c (-c x+i)^3}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{24 c}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{3 c (1+i c x)^3}+\frac{11 b^2}{144 c (-c x+i)}+\frac{5 i b^2}{144 c (-c x+i)^2}-\frac{b^2}{54 c (-c x+i)^3}-\frac{11 b^2 \tan ^{-1}(c x)}{144 c}","\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{12 c (-c x+i)}-\frac{b \left(a+b \tan ^{-1}(c x)\right)}{12 c (-c x+i)^2}-\frac{i b \left(a+b \tan ^{-1}(c x)\right)}{9 c (-c x+i)^3}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{24 c}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{3 c (1+i c x)^3}+\frac{11 b^2}{144 c (-c x+i)}+\frac{5 i b^2}{144 c (-c x+i)^2}-\frac{b^2}{54 c (-c x+i)^3}-\frac{11 b^2 \tan ^{-1}(c x)}{144 c}",1,"-b^2/(54*c*(I - c*x)^3) + (((5*I)/144)*b^2)/(c*(I - c*x)^2) + (11*b^2)/(144*c*(I - c*x)) - (11*b^2*ArcTan[c*x])/(144*c) - ((I/9)*b*(a + b*ArcTan[c*x]))/(c*(I - c*x)^3) - (b*(a + b*ArcTan[c*x]))/(12*c*(I - c*x)^2) + ((I/12)*b*(a + b*ArcTan[c*x]))/(c*(I - c*x)) - ((I/24)*(a + b*ArcTan[c*x])^2)/c + ((I/3)*(a + b*ArcTan[c*x])^2)/(c*(1 + I*c*x)^3)","A",18,6,21,0.2857,1,"{4864, 4862, 627, 44, 203, 4884}"
119,1,76,0,0.1376443,"\int \frac{\tan ^{-1}(a x)^2}{c x-i a c x^2} \, dx","Int[ArcTan[a*x]^2/(c*x - I*a*c*x^2),x]","\frac{\text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c}","\frac{\text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c}",1,"(ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c - (I*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c + PolyLog[3, -1 + 2/(1 - I*a*x)]/(2*c)","A",4,5,22,0.2273,1,"{1593, 4868, 4884, 4992, 6610}"
120,1,382,0,0.7069872,"\int (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Int[(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^3,x]","-\frac{6 i b^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{11 b^3 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c}+\frac{3 b^3 d^3 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{c}-\frac{1}{4} i b^2 c d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{11 i b^2 d^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}-3 a b^2 d^3 x+\frac{1}{4} i b c^2 d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{3}{2} b c d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{i d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)^3}{4 c}+\frac{7 b d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{c}-\frac{21}{4} i b d^3 x \left(a+b \tan ^{-1}(c x)\right)^2+\frac{6 b d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c}+\frac{3 b^3 d^3 \log \left(c^2 x^2+1\right)}{2 c}-\frac{i b^3 d^3 \tan ^{-1}(c x)}{4 c}-3 b^3 d^3 x \tan ^{-1}(c x)+\frac{1}{4} i b^3 d^3 x","-\frac{6 i b^2 d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{11 b^3 d^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c}+\frac{3 b^3 d^3 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{c}-\frac{1}{4} i b^2 c d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{11 i b^2 d^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}-3 a b^2 d^3 x+\frac{1}{4} i b c^2 d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{3}{2} b c d^3 x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{i d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)^3}{4 c}+\frac{7 b d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{c}-\frac{21}{4} i b d^3 x \left(a+b \tan ^{-1}(c x)\right)^2+\frac{6 b d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c}+\frac{3 b^3 d^3 \log \left(c^2 x^2+1\right)}{2 c}-\frac{i b^3 d^3 \tan ^{-1}(c x)}{4 c}-3 b^3 d^3 x \tan ^{-1}(c x)+\frac{1}{4} i b^3 d^3 x",1,"-3*a*b^2*d^3*x + (I/4)*b^3*d^3*x - ((I/4)*b^3*d^3*ArcTan[c*x])/c - 3*b^3*d^3*x*ArcTan[c*x] - (I/4)*b^2*c*d^3*x^2*(a + b*ArcTan[c*x]) + (7*b*d^3*(a + b*ArcTan[c*x])^2)/c - ((21*I)/4)*b*d^3*x*(a + b*ArcTan[c*x])^2 + (3*b*c*d^3*x^2*(a + b*ArcTan[c*x])^2)/2 + (I/4)*b*c^2*d^3*x^3*(a + b*ArcTan[c*x])^2 - ((I/4)*d^3*(1 + I*c*x)^4*(a + b*ArcTan[c*x])^3)/c + (6*b*d^3*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/c - ((11*I)*b^2*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c + (3*b^3*d^3*Log[1 + c^2*x^2])/(2*c) - ((6*I)*b^2*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/c + (11*b^3*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c) + (3*b^3*d^3*PolyLog[3, 1 - 2/(1 - I*c*x)])/c","A",26,15,22,0.6818,1,"{4864, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 321, 203, 1586, 4992, 6610}"
121,1,298,0,0.4781835,"\int (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Int[(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^3,x]","-\frac{4 i b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{3 b^3 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c}+\frac{2 b^3 d^2 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{c}-\frac{6 i b^2 d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}-a b^2 d^2 x+\frac{1}{2} b c d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{7 b d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c}-3 i b d^2 x \left(a+b \tan ^{-1}(c x)\right)^2-\frac{i d^2 (1+i c x)^3 \left(a+b \tan ^{-1}(c x)\right)^3}{3 c}+\frac{4 b d^2 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c}+\frac{b^3 d^2 \log \left(c^2 x^2+1\right)}{2 c}+b^3 \left(-d^2\right) x \tan ^{-1}(c x)","-\frac{4 i b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{3 b^3 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c}+\frac{2 b^3 d^2 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{c}-\frac{6 i b^2 d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}-a b^2 d^2 x+\frac{1}{2} b c d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{7 b d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c}-3 i b d^2 x \left(a+b \tan ^{-1}(c x)\right)^2-\frac{i d^2 (1+i c x)^3 \left(a+b \tan ^{-1}(c x)\right)^3}{3 c}+\frac{4 b d^2 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c}+\frac{b^3 d^2 \log \left(c^2 x^2+1\right)}{2 c}+b^3 \left(-d^2\right) x \tan ^{-1}(c x)",1,"-(a*b^2*d^2*x) - b^3*d^2*x*ArcTan[c*x] + (7*b*d^2*(a + b*ArcTan[c*x])^2)/(2*c) - (3*I)*b*d^2*x*(a + b*ArcTan[c*x])^2 + (b*c*d^2*x^2*(a + b*ArcTan[c*x])^2)/2 - ((I/3)*d^2*(1 + I*c*x)^3*(a + b*ArcTan[c*x])^3)/c + (4*b*d^2*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/c - ((6*I)*b^2*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c + (b^3*d^2*Log[1 + c^2*x^2])/(2*c) - ((4*I)*b^2*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/c + (3*b^3*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c + (2*b^3*d^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/c","A",17,13,22,0.5909,1,"{4864, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 1586, 4992, 6610}"
122,1,220,0,0.339436,"\int (d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Int[(d + I*c*d*x)*(a + b*ArcTan[c*x])^3,x]","-\frac{3 i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{3 b^3 d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c}+\frac{3 b^3 d \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 c}-\frac{3 i b^2 d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{3 b d \left(a+b \tan ^{-1}(c x)\right)^2}{2 c}-\frac{3}{2} i b d x \left(a+b \tan ^{-1}(c x)\right)^2-\frac{i d (1+i c x)^2 \left(a+b \tan ^{-1}(c x)\right)^3}{2 c}+\frac{3 b d \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c}","-\frac{3 i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{3 b^3 d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c}+\frac{3 b^3 d \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 c}-\frac{3 i b^2 d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{3 b d \left(a+b \tan ^{-1}(c x)\right)^2}{2 c}-\frac{3}{2} i b d x \left(a+b \tan ^{-1}(c x)\right)^2-\frac{i d (1+i c x)^2 \left(a+b \tan ^{-1}(c x)\right)^3}{2 c}+\frac{3 b d \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c}",1,"(3*b*d*(a + b*ArcTan[c*x])^2)/(2*c) - ((3*I)/2)*b*d*x*(a + b*ArcTan[c*x])^2 - ((I/2)*d*(1 + I*c*x)^2*(a + b*ArcTan[c*x])^3)/c + (3*b*d*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/c - ((3*I)*b^2*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c - ((3*I)*b^2*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/c + (3*b^3*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c) + (3*b^3*d*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*c)","A",11,10,20,0.5000,1,"{4864, 4846, 4920, 4854, 2402, 2315, 1586, 4884, 4992, 6610}"
123,1,139,0,0.2299562,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{d+i c d x} \, dx","Int[(a + b*ArcTan[c*x])^3/(d + I*c*d*x),x]","\frac{3 i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c d}-\frac{3 b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 c d}+\frac{3 b^3 \text{PolyLog}\left(4,1-\frac{2}{1+i c x}\right)}{4 c d}+\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{c d}","\frac{3 i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c d}-\frac{3 b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 c d}+\frac{3 b^3 \text{PolyLog}\left(4,1-\frac{2}{1+i c x}\right)}{4 c d}+\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{c d}",1,"(I*(a + b*ArcTan[c*x])^3*Log[2/(1 + I*c*x)])/(c*d) - (3*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c*d) + (((3*I)/2)*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c*d) + (3*b^3*PolyLog[4, 1 - 2/(1 + I*c*x)])/(4*c*d)","A",4,5,22,0.2273,1,"{4854, 4884, 4994, 4998, 6610}"
124,1,182,0,0.2198026,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{(d+i c d x)^2} \, dx","Int[(a + b*ArcTan[c*x])^3/(d + I*c*d*x)^2,x]","\frac{3 b^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c d^2 (-c x+i)}+\frac{3 i b \left(a+b \tan ^{-1}(c x)\right)^2}{2 c d^2 (-c x+i)}-\frac{3 b \left(a+b \tan ^{-1}(c x)\right)^2}{4 c d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{c d^2 (1+i c x)}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{2 c d^2}-\frac{3 i b^3}{4 c d^2 (-c x+i)}+\frac{3 i b^3 \tan ^{-1}(c x)}{4 c d^2}","\frac{3 b^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c d^2 (-c x+i)}+\frac{3 i b \left(a+b \tan ^{-1}(c x)\right)^2}{2 c d^2 (-c x+i)}-\frac{3 b \left(a+b \tan ^{-1}(c x)\right)^2}{4 c d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{c d^2 (1+i c x)}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{2 c d^2}-\frac{3 i b^3}{4 c d^2 (-c x+i)}+\frac{3 i b^3 \tan ^{-1}(c x)}{4 c d^2}",1,"(((-3*I)/4)*b^3)/(c*d^2*(I - c*x)) + (((3*I)/4)*b^3*ArcTan[c*x])/(c*d^2) + (3*b^2*(a + b*ArcTan[c*x]))/(2*c*d^2*(I - c*x)) - (3*b*(a + b*ArcTan[c*x])^2)/(4*c*d^2) + (((3*I)/2)*b*(a + b*ArcTan[c*x])^2)/(c*d^2*(I - c*x)) - ((I/2)*(a + b*ArcTan[c*x])^3)/(c*d^2) + (I*(a + b*ArcTan[c*x])^3)/(c*d^2*(1 + I*c*x))","A",11,6,22,0.2727,1,"{4864, 4862, 627, 44, 203, 4884}"
125,1,271,0,0.4030005,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{(d+i c d x)^3} \, dx","Int[(a + b*ArcTan[c*x])^3/(d + I*c*d*x)^3,x]","\frac{9 b^2 \left(a+b \tan ^{-1}(c x)\right)}{16 c d^3 (-c x+i)}+\frac{3 i b^2 \left(a+b \tan ^{-1}(c x)\right)}{16 c d^3 (-c x+i)^2}+\frac{3 i b \left(a+b \tan ^{-1}(c x)\right)^2}{8 c d^3 (-c x+i)}-\frac{3 b \left(a+b \tan ^{-1}(c x)\right)^2}{8 c d^3 (-c x+i)^2}-\frac{9 b \left(a+b \tan ^{-1}(c x)\right)^2}{32 c d^3}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{2 c d^3 (1+i c x)^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{8 c d^3}-\frac{21 i b^3}{64 c d^3 (-c x+i)}+\frac{3 b^3}{64 c d^3 (-c x+i)^2}+\frac{21 i b^3 \tan ^{-1}(c x)}{64 c d^3}","\frac{9 b^2 \left(a+b \tan ^{-1}(c x)\right)}{16 c d^3 (-c x+i)}+\frac{3 i b^2 \left(a+b \tan ^{-1}(c x)\right)}{16 c d^3 (-c x+i)^2}+\frac{3 i b \left(a+b \tan ^{-1}(c x)\right)^2}{8 c d^3 (-c x+i)}-\frac{3 b \left(a+b \tan ^{-1}(c x)\right)^2}{8 c d^3 (-c x+i)^2}-\frac{9 b \left(a+b \tan ^{-1}(c x)\right)^2}{32 c d^3}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{2 c d^3 (1+i c x)^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{8 c d^3}-\frac{21 i b^3}{64 c d^3 (-c x+i)}+\frac{3 b^3}{64 c d^3 (-c x+i)^2}+\frac{21 i b^3 \tan ^{-1}(c x)}{64 c d^3}",1,"(3*b^3)/(64*c*d^3*(I - c*x)^2) - (((21*I)/64)*b^3)/(c*d^3*(I - c*x)) + (((21*I)/64)*b^3*ArcTan[c*x])/(c*d^3) + (((3*I)/16)*b^2*(a + b*ArcTan[c*x]))/(c*d^3*(I - c*x)^2) + (9*b^2*(a + b*ArcTan[c*x]))/(16*c*d^3*(I - c*x)) - (9*b*(a + b*ArcTan[c*x])^2)/(32*c*d^3) - (3*b*(a + b*ArcTan[c*x])^2)/(8*c*d^3*(I - c*x)^2) + (((3*I)/8)*b*(a + b*ArcTan[c*x])^2)/(c*d^3*(I - c*x)) - ((I/8)*(a + b*ArcTan[c*x])^3)/(c*d^3) + ((I/2)*(a + b*ArcTan[c*x])^3)/(c*d^3*(1 + I*c*x)^2)","A",24,6,22,0.2727,1,"{4864, 4862, 627, 44, 203, 4884}"
126,1,360,0,0.6735461,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{(d+i c d x)^4} \, dx","Int[(a + b*ArcTan[c*x])^3/(d + I*c*d*x)^4,x]","\frac{11 b^2 \left(a+b \tan ^{-1}(c x)\right)}{48 c d^4 (-c x+i)}+\frac{5 i b^2 \left(a+b \tan ^{-1}(c x)\right)}{48 c d^4 (-c x+i)^2}-\frac{b^2 \left(a+b \tan ^{-1}(c x)\right)}{18 c d^4 (-c x+i)^3}+\frac{i b \left(a+b \tan ^{-1}(c x)\right)^2}{8 c d^4 (-c x+i)}-\frac{b \left(a+b \tan ^{-1}(c x)\right)^2}{8 c d^4 (-c x+i)^2}-\frac{i b \left(a+b \tan ^{-1}(c x)\right)^2}{6 c d^4 (-c x+i)^3}-\frac{11 b \left(a+b \tan ^{-1}(c x)\right)^2}{96 c d^4}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{3 c d^4 (1+i c x)^3}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{24 c d^4}-\frac{85 i b^3}{576 c d^4 (-c x+i)}+\frac{19 b^3}{576 c d^4 (-c x+i)^2}+\frac{i b^3}{108 c d^4 (-c x+i)^3}+\frac{85 i b^3 \tan ^{-1}(c x)}{576 c d^4}","\frac{11 b^2 \left(a+b \tan ^{-1}(c x)\right)}{48 c d^4 (-c x+i)}+\frac{5 i b^2 \left(a+b \tan ^{-1}(c x)\right)}{48 c d^4 (-c x+i)^2}-\frac{b^2 \left(a+b \tan ^{-1}(c x)\right)}{18 c d^4 (-c x+i)^3}+\frac{i b \left(a+b \tan ^{-1}(c x)\right)^2}{8 c d^4 (-c x+i)}-\frac{b \left(a+b \tan ^{-1}(c x)\right)^2}{8 c d^4 (-c x+i)^2}-\frac{i b \left(a+b \tan ^{-1}(c x)\right)^2}{6 c d^4 (-c x+i)^3}-\frac{11 b \left(a+b \tan ^{-1}(c x)\right)^2}{96 c d^4}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{3 c d^4 (1+i c x)^3}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{24 c d^4}-\frac{85 i b^3}{576 c d^4 (-c x+i)}+\frac{19 b^3}{576 c d^4 (-c x+i)^2}+\frac{i b^3}{108 c d^4 (-c x+i)^3}+\frac{85 i b^3 \tan ^{-1}(c x)}{576 c d^4}",1,"((I/108)*b^3)/(c*d^4*(I - c*x)^3) + (19*b^3)/(576*c*d^4*(I - c*x)^2) - (((85*I)/576)*b^3)/(c*d^4*(I - c*x)) + (((85*I)/576)*b^3*ArcTan[c*x])/(c*d^4) - (b^2*(a + b*ArcTan[c*x]))/(18*c*d^4*(I - c*x)^3) + (((5*I)/48)*b^2*(a + b*ArcTan[c*x]))/(c*d^4*(I - c*x)^2) + (11*b^2*(a + b*ArcTan[c*x]))/(48*c*d^4*(I - c*x)) - (11*b*(a + b*ArcTan[c*x])^2)/(96*c*d^4) - ((I/6)*b*(a + b*ArcTan[c*x])^2)/(c*d^4*(I - c*x)^3) - (b*(a + b*ArcTan[c*x])^2)/(8*c*d^4*(I - c*x)^2) + ((I/8)*b*(a + b*ArcTan[c*x])^2)/(c*d^4*(I - c*x)) - ((I/24)*(a + b*ArcTan[c*x])^3)/(c*d^4) + ((I/3)*(a + b*ArcTan[c*x])^3)/(c*d^4*(1 + I*c*x)^3)","A",42,6,22,0.2727,1,"{4864, 4862, 627, 44, 203, 4884}"
127,1,410,0,0.8618982,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^3}{d+i c d x} \, dx","Int[(x^2*(a + b*ArcTan[c*x])^3)/(d + I*c*d*x),x]","\frac{3 i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d}-\frac{3 i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c^3 d}+\frac{3 b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 d}-\frac{3 b^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^3 d}+\frac{3 b^3 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^3 d}-\frac{3 b^3 \text{PolyLog}\left(4,1-\frac{2}{1+i c x}\right)}{4 c^3 d}+\frac{3 i b^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d}+\frac{3 i b x \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 d}-\frac{3 b \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 d}+\frac{x \left(a+b \tan ^{-1}(c x)\right)^3}{c^2 d}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{2 c^3 d}+\frac{3 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d}-\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{c^3 d}-\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)^3}{2 c d}","\frac{3 i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d}-\frac{3 i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c^3 d}+\frac{3 b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 d}-\frac{3 b^3 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{2 c^3 d}+\frac{3 b^3 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^3 d}-\frac{3 b^3 \text{PolyLog}\left(4,1-\frac{2}{1+i c x}\right)}{4 c^3 d}+\frac{3 i b^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d}+\frac{3 i b x \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 d}-\frac{3 b \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 d}+\frac{x \left(a+b \tan ^{-1}(c x)\right)^3}{c^2 d}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^3}{2 c^3 d}+\frac{3 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d}-\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{c^3 d}-\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)^3}{2 c d}",1,"(-3*b*(a + b*ArcTan[c*x])^2)/(2*c^3*d) + (((3*I)/2)*b*x*(a + b*ArcTan[c*x])^2)/(c^2*d) + ((I/2)*(a + b*ArcTan[c*x])^3)/(c^3*d) + (x*(a + b*ArcTan[c*x])^3)/(c^2*d) - ((I/2)*x^2*(a + b*ArcTan[c*x])^3)/(c*d) + ((3*I)*b^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c^3*d) + (3*b*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^3*d) - (I*(a + b*ArcTan[c*x])^3*Log[2/(1 + I*c*x)])/(c^3*d) - (3*b^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c^3*d) + ((3*I)*b^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*d) + (3*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c^3*d) + (3*b^3*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^3*d) - (((3*I)/2)*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^3*d) - (3*b^3*PolyLog[4, 1 - 2/(1 + I*c*x)])/(4*c^3*d)","A",19,12,25,0.4800,1,"{4866, 4852, 4916, 4846, 4920, 4854, 2402, 2315, 4884, 4994, 6610, 4998}"
128,1,277,0,0.5034948,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)^3}{d+i c d x} \, dx","Int[(x*(a + b*ArcTan[c*x])^3)/(d + I*c*d*x),x]","\frac{3 b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d}-\frac{3 b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c^2 d}-\frac{3 i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 d}-\frac{3 i b^3 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^2 d}+\frac{3 i b^3 \text{PolyLog}\left(4,1-\frac{2}{1+i c x}\right)}{4 c^2 d}+\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{c^2 d}-\frac{3 i b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d}-\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{c^2 d}-\frac{i x \left(a+b \tan ^{-1}(c x)\right)^3}{c d}","\frac{3 b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d}-\frac{3 b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c^2 d}-\frac{3 i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 d}-\frac{3 i b^3 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 c^2 d}+\frac{3 i b^3 \text{PolyLog}\left(4,1-\frac{2}{1+i c x}\right)}{4 c^2 d}+\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{c^2 d}-\frac{3 i b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d}-\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{c^2 d}-\frac{i x \left(a+b \tan ^{-1}(c x)\right)^3}{c d}",1,"(a + b*ArcTan[c*x])^3/(c^2*d) - (I*x*(a + b*ArcTan[c*x])^3)/(c*d) - ((3*I)*b*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/(c^2*d) - ((a + b*ArcTan[c*x])^3*Log[2/(1 + I*c*x)])/(c^2*d) + (3*b^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d) - (((3*I)/2)*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^2*d) - (((3*I)/2)*b^3*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c^2*d) - (3*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^2*d) + (((3*I)/4)*b^3*PolyLog[4, 1 - 2/(1 + I*c*x)])/(c^2*d)","A",10,8,23,0.3478,1,"{4866, 4846, 4920, 4854, 4884, 4994, 6610, 4998}"
129,1,139,0,0.2151991,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{d+i c d x} \, dx","Int[(a + b*ArcTan[c*x])^3/(d + I*c*d*x),x]","\frac{3 i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c d}-\frac{3 b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 c d}+\frac{3 b^3 \text{PolyLog}\left(4,1-\frac{2}{1+i c x}\right)}{4 c d}+\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{c d}","\frac{3 i b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c d}-\frac{3 b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 c d}+\frac{3 b^3 \text{PolyLog}\left(4,1-\frac{2}{1+i c x}\right)}{4 c d}+\frac{i \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{c d}",1,"(I*(a + b*ArcTan[c*x])^3*Log[2/(1 + I*c*x)])/(c*d) - (3*b*(a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(2*c*d) + (((3*I)/2)*b^2*(a + b*ArcTan[c*x])*PolyLog[3, 1 - 2/(1 + I*c*x)])/(c*d) + (3*b^3*PolyLog[4, 1 - 2/(1 + I*c*x)])/(4*c*d)","A",4,5,22,0.2273,1,"{4854, 4884, 4994, 4998, 6610}"
130,1,128,0,0.2320612,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{x (d+i c d x)} \, dx","Int[(a + b*ArcTan[c*x])^3/(x*(d + I*c*d*x)),x]","\frac{3 b^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 d}+\frac{3 i b \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}-\frac{3 i b^3 \text{PolyLog}\left(4,-1+\frac{2}{1+i c x}\right)}{4 d}+\frac{\log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{d}","\frac{3 b^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 d}+\frac{3 i b \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}-\frac{3 i b^3 \text{PolyLog}\left(4,-1+\frac{2}{1+i c x}\right)}{4 d}+\frac{\log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{d}",1,"((a + b*ArcTan[c*x])^3*Log[2 - 2/(1 + I*c*x)])/d + (((3*I)/2)*b*(a + b*ArcTan[c*x])^2*PolyLog[2, -1 + 2/(1 + I*c*x)])/d + (3*b^2*(a + b*ArcTan[c*x])*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d) - (((3*I)/4)*b^3*PolyLog[4, -1 + 2/(1 + I*c*x)])/d","A",4,5,25,0.2000,1,"{4868, 4884, 4994, 4998, 6610}"
131,1,263,0,0.6001295,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{x^2 (d+i c d x)} \, dx","Int[(a + b*ArcTan[c*x])^3/(x^2*(d + I*c*d*x)),x]","-\frac{3 i b^2 c \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{3 i b^2 c \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 d}+\frac{3 b c \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}+\frac{3 b^3 c \text{PolyLog}\left(3,-1+\frac{2}{1-i c x}\right)}{2 d}-\frac{3 b^3 c \text{PolyLog}\left(4,-1+\frac{2}{1+i c x}\right)}{4 d}-\frac{i c \left(a+b \tan ^{-1}(c x)\right)^3}{d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{d x}+\frac{3 b c \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}-\frac{i c \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{d}","-\frac{3 i b^2 c \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{3 i b^2 c \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 d}+\frac{3 b c \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}+\frac{3 b^3 c \text{PolyLog}\left(3,-1+\frac{2}{1-i c x}\right)}{2 d}-\frac{3 b^3 c \text{PolyLog}\left(4,-1+\frac{2}{1+i c x}\right)}{4 d}-\frac{i c \left(a+b \tan ^{-1}(c x)\right)^3}{d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{d x}+\frac{3 b c \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}-\frac{i c \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{d}",1,"((-I)*c*(a + b*ArcTan[c*x])^3)/d - (a + b*ArcTan[c*x])^3/(d*x) + (3*b*c*(a + b*ArcTan[c*x])^2*Log[2 - 2/(1 - I*c*x)])/d - (I*c*(a + b*ArcTan[c*x])^3*Log[2 - 2/(1 + I*c*x)])/d - ((3*I)*b^2*c*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 - I*c*x)])/d + (3*b*c*(a + b*ArcTan[c*x])^2*PolyLog[2, -1 + 2/(1 + I*c*x)])/(2*d) + (3*b^3*c*PolyLog[3, -1 + 2/(1 - I*c*x)])/(2*d) - (((3*I)/2)*b^2*c*(a + b*ArcTan[c*x])*PolyLog[3, -1 + 2/(1 + I*c*x)])/d - (3*b^3*c*PolyLog[4, -1 + 2/(1 + I*c*x)])/(4*d)","A",10,9,25,0.3600,1,"{4870, 4852, 4924, 4868, 4884, 4992, 6610, 4994, 4998}"
132,1,414,0,1.0195377,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{x^3 (d+i c d x)} \, dx","Int[(a + b*ArcTan[c*x])^3/(x^3*(d + I*c*d*x)),x]","-\frac{3 b^2 c^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{3 b^2 c^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 d}-\frac{3 i b c^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}-\frac{3 i b^3 c^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{2 d}-\frac{3 i b^3 c^2 \text{PolyLog}\left(3,-1+\frac{2}{1-i c x}\right)}{2 d}+\frac{3 i b^3 c^2 \text{PolyLog}\left(4,-1+\frac{2}{1+i c x}\right)}{4 d}+\frac{3 b^2 c^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{3 i b c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}-\frac{3 c^2 \left(a+b \tan ^{-1}(c x)\right)^3}{2 d}-\frac{3 i b c^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}-\frac{c^2 \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{2 d x^2}-\frac{3 b c \left(a+b \tan ^{-1}(c x)\right)^2}{2 d x}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)^3}{d x}","-\frac{3 b^2 c^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{3 b^2 c^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 d}-\frac{3 i b c^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}-\frac{3 i b^3 c^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{2 d}-\frac{3 i b^3 c^2 \text{PolyLog}\left(3,-1+\frac{2}{1-i c x}\right)}{2 d}+\frac{3 i b^3 c^2 \text{PolyLog}\left(4,-1+\frac{2}{1+i c x}\right)}{4 d}+\frac{3 b^2 c^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{3 i b c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}-\frac{3 c^2 \left(a+b \tan ^{-1}(c x)\right)^3}{2 d}-\frac{3 i b c^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}-\frac{c^2 \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{2 d x^2}-\frac{3 b c \left(a+b \tan ^{-1}(c x)\right)^2}{2 d x}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)^3}{d x}",1,"(((-3*I)/2)*b*c^2*(a + b*ArcTan[c*x])^2)/d - (3*b*c*(a + b*ArcTan[c*x])^2)/(2*d*x) - (3*c^2*(a + b*ArcTan[c*x])^3)/(2*d) - (a + b*ArcTan[c*x])^3/(2*d*x^2) + (I*c*(a + b*ArcTan[c*x])^3)/(d*x) + (3*b^2*c^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d - ((3*I)*b*c^2*(a + b*ArcTan[c*x])^2*Log[2 - 2/(1 - I*c*x)])/d - (c^2*(a + b*ArcTan[c*x])^3*Log[2 - 2/(1 + I*c*x)])/d - (((3*I)/2)*b^3*c^2*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - (3*b^2*c^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - (((3*I)/2)*b*c^2*(a + b*ArcTan[c*x])^2*PolyLog[2, -1 + 2/(1 + I*c*x)])/d - (((3*I)/2)*b^3*c^2*PolyLog[3, -1 + 2/(1 - I*c*x)])/d - (3*b^2*c^2*(a + b*ArcTan[c*x])*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d) + (((3*I)/4)*b^3*c^2*PolyLog[4, -1 + 2/(1 + I*c*x)])/d","A",18,11,25,0.4400,1,"{4870, 4852, 4918, 4924, 4868, 2447, 4884, 4992, 6610, 4994, 4998}"
133,0,0,0,0.0373987,"\int \frac{1}{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)} \, dx","Int[1/((d + I*c*d*x)*(a + b*ArcTan[c*x])),x]","\int \frac{1}{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)},x\right)",0,"Defer[Int][1/((d + I*c*d*x)*(a + b*ArcTan[c*x])), x]","A",0,0,0,0,-1,"{}"
134,1,297,0,0.2701357,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{d+e x} \, dx","Int[(x^3*(a + b*ArcTan[c*x]))/(d + e*x),x]","-\frac{i b d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e^4}+\frac{i b d^3 \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e^4}+\frac{d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^4}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^4}-\frac{d x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 e^2}+\frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{3 e}+\frac{a d^2 x}{e^3}-\frac{b d^2 \log \left(c^2 x^2+1\right)}{2 c e^3}-\frac{b d \tan ^{-1}(c x)}{2 c^2 e^2}+\frac{b \log \left(c^2 x^2+1\right)}{6 c^3 e}+\frac{b d^2 x \tan ^{-1}(c x)}{e^3}+\frac{b d x}{2 c e^2}-\frac{b x^2}{6 c e}","-\frac{i b d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e^4}+\frac{i b d^3 \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e^4}+\frac{d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^4}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^4}-\frac{d x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 e^2}+\frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{3 e}+\frac{a d^2 x}{e^3}-\frac{b d^2 \log \left(c^2 x^2+1\right)}{2 c e^3}-\frac{b d \tan ^{-1}(c x)}{2 c^2 e^2}+\frac{b \log \left(c^2 x^2+1\right)}{6 c^3 e}+\frac{b d^2 x \tan ^{-1}(c x)}{e^3}+\frac{b d x}{2 c e^2}-\frac{b x^2}{6 c e}",1,"(a*d^2*x)/e^3 + (b*d*x)/(2*c*e^2) - (b*x^2)/(6*c*e) - (b*d*ArcTan[c*x])/(2*c^2*e^2) + (b*d^2*x*ArcTan[c*x])/e^3 - (d*x^2*(a + b*ArcTan[c*x]))/(2*e^2) + (x^3*(a + b*ArcTan[c*x]))/(3*e) + (d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e^4 - (d^3*(a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^4 - (b*d^2*Log[1 + c^2*x^2])/(2*c*e^3) + (b*Log[1 + c^2*x^2])/(6*c^3*e) - ((I/2)*b*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^4 + ((I/2)*b*d^3*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^4","A",16,12,19,0.6316,1,"{4876, 4846, 260, 4852, 321, 203, 266, 43, 4856, 2402, 2315, 2447}"
135,1,237,0,0.2084987,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{d+e x} \, dx","Int[(x^2*(a + b*ArcTan[c*x]))/(d + e*x),x]","\frac{i b d^2 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e^3}-\frac{i b d^2 \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e^3}-\frac{d^2 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^3}+\frac{d^2 \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^3}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 e}-\frac{a d x}{e^2}+\frac{b d \log \left(c^2 x^2+1\right)}{2 c e^2}+\frac{b \tan ^{-1}(c x)}{2 c^2 e}-\frac{b d x \tan ^{-1}(c x)}{e^2}-\frac{b x}{2 c e}","\frac{i b d^2 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e^3}-\frac{i b d^2 \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e^3}-\frac{d^2 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^3}+\frac{d^2 \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^3}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 e}-\frac{a d x}{e^2}+\frac{b d \log \left(c^2 x^2+1\right)}{2 c e^2}+\frac{b \tan ^{-1}(c x)}{2 c^2 e}-\frac{b d x \tan ^{-1}(c x)}{e^2}-\frac{b x}{2 c e}",1,"-((a*d*x)/e^2) - (b*x)/(2*c*e) + (b*ArcTan[c*x])/(2*c^2*e) - (b*d*x*ArcTan[c*x])/e^2 + (x^2*(a + b*ArcTan[c*x]))/(2*e) - (d^2*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e^3 + (d^2*(a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^3 + (b*d*Log[1 + c^2*x^2])/(2*c*e^2) + ((I/2)*b*d^2*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^3 - ((I/2)*b*d^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^3","A",12,10,19,0.5263,1,"{4876, 4846, 260, 4852, 321, 203, 4856, 2402, 2315, 2447}"
136,1,179,0,0.1585947,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{d+e x} \, dx","Int[(x*(a + b*ArcTan[c*x]))/(d + e*x),x]","-\frac{i b d \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e^2}+\frac{i b d \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e^2}+\frac{d \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^2}+\frac{a x}{e}-\frac{b \log \left(c^2 x^2+1\right)}{2 c e}+\frac{b x \tan ^{-1}(c x)}{e}","-\frac{i b d \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e^2}+\frac{i b d \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e^2}+\frac{d \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^2}+\frac{a x}{e}-\frac{b \log \left(c^2 x^2+1\right)}{2 c e}+\frac{b x \tan ^{-1}(c x)}{e}",1,"(a*x)/e + (b*x*ArcTan[c*x])/e + (d*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e^2 - (d*(a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^2 - (b*Log[1 + c^2*x^2])/(2*c*e) - ((I/2)*b*d*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^2 + ((I/2)*b*d*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^2","A",9,7,17,0.4118,1,"{4876, 4846, 260, 4856, 2402, 2315, 2447}"
137,1,138,0,0.0745539,"\int \frac{a+b \tan ^{-1}(c x)}{d+e x} \, dx","Int[(a + b*ArcTan[c*x])/(d + e*x),x]","-\frac{i b \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e}+\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e}","-\frac{i b \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e}+\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e}",1,"-(((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e) + ((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e + ((I/2)*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/e - ((I/2)*b*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e","A",4,4,16,0.2500,1,"{4856, 2402, 2315, 2447}"
138,1,181,0,0.1861014,"\int \frac{a+b \tan ^{-1}(c x)}{x (d+e x)} \, dx","Int[(a + b*ArcTan[c*x])/(x*(d + e*x)),x]","\frac{i b \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 d}+\frac{i b \text{PolyLog}(2,-i c x)}{2 d}-\frac{i b \text{PolyLog}(2,i c x)}{2 d}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d}-\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d}+\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{a \log (x)}{d}","\frac{i b \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 d}+\frac{i b \text{PolyLog}(2,-i c x)}{2 d}-\frac{i b \text{PolyLog}(2,i c x)}{2 d}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d}-\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d}+\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{a \log (x)}{d}",1,"(a*Log[x])/d + ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d - ((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d + ((I/2)*b*PolyLog[2, (-I)*c*x])/d - ((I/2)*b*PolyLog[2, I*c*x])/d - ((I/2)*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/d + ((I/2)*b*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d","A",9,7,19,0.3684,1,"{4876, 4848, 2391, 4856, 2402, 2315, 2447}"
139,1,232,0,0.2413351,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 (d+e x)} \, dx","Int[(a + b*ArcTan[c*x])/(x^2*(d + e*x)),x]","-\frac{i b e \text{PolyLog}(2,-i c x)}{2 d^2}+\frac{i b e \text{PolyLog}(2,i c x)}{2 d^2}+\frac{i b e \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d^2}-\frac{i b e \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 d^2}-\frac{e \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d^2}-\frac{a+b \tan ^{-1}(c x)}{d x}-\frac{a e \log (x)}{d^2}-\frac{b c \log \left(c^2 x^2+1\right)}{2 d}+\frac{b c \log (x)}{d}","-\frac{i b e \text{PolyLog}(2,-i c x)}{2 d^2}+\frac{i b e \text{PolyLog}(2,i c x)}{2 d^2}+\frac{i b e \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d^2}-\frac{i b e \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 d^2}-\frac{e \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d^2}-\frac{a+b \tan ^{-1}(c x)}{d x}-\frac{a e \log (x)}{d^2}-\frac{b c \log \left(c^2 x^2+1\right)}{2 d}+\frac{b c \log (x)}{d}",1,"-((a + b*ArcTan[c*x])/(d*x)) + (b*c*Log[x])/d - (a*e*Log[x])/d^2 - (e*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d^2 + (e*(a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d^2 - (b*c*Log[1 + c^2*x^2])/(2*d) - ((I/2)*b*e*PolyLog[2, (-I)*c*x])/d^2 + ((I/2)*b*e*PolyLog[2, I*c*x])/d^2 + ((I/2)*b*e*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^2 - ((I/2)*b*e*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d^2","A",14,12,19,0.6316,1,"{4876, 4852, 266, 36, 29, 31, 4848, 2391, 4856, 2402, 2315, 2447}"
140,1,293,0,0.2844294,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 (d+e x)} \, dx","Int[(a + b*ArcTan[c*x])/(x^3*(d + e*x)),x]","\frac{i b e^2 \text{PolyLog}(2,-i c x)}{2 d^3}-\frac{i b e^2 \text{PolyLog}(2,i c x)}{2 d^3}-\frac{i b e^2 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d^3}+\frac{i b e^2 \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 d^3}+\frac{e^2 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}-\frac{e^2 \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d^3}+\frac{e \left(a+b \tan ^{-1}(c x)\right)}{d^2 x}-\frac{a+b \tan ^{-1}(c x)}{2 d x^2}+\frac{a e^2 \log (x)}{d^3}+\frac{b c e \log \left(c^2 x^2+1\right)}{2 d^2}-\frac{b c^2 \tan ^{-1}(c x)}{2 d}-\frac{b c e \log (x)}{d^2}-\frac{b c}{2 d x}","\frac{i b e^2 \text{PolyLog}(2,-i c x)}{2 d^3}-\frac{i b e^2 \text{PolyLog}(2,i c x)}{2 d^3}-\frac{i b e^2 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d^3}+\frac{i b e^2 \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 d^3}+\frac{e^2 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}-\frac{e^2 \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d^3}+\frac{e \left(a+b \tan ^{-1}(c x)\right)}{d^2 x}-\frac{a+b \tan ^{-1}(c x)}{2 d x^2}+\frac{a e^2 \log (x)}{d^3}+\frac{b c e \log \left(c^2 x^2+1\right)}{2 d^2}-\frac{b c^2 \tan ^{-1}(c x)}{2 d}-\frac{b c e \log (x)}{d^2}-\frac{b c}{2 d x}",1,"-(b*c)/(2*d*x) - (b*c^2*ArcTan[c*x])/(2*d) - (a + b*ArcTan[c*x])/(2*d*x^2) + (e*(a + b*ArcTan[c*x]))/(d^2*x) - (b*c*e*Log[x])/d^2 + (a*e^2*Log[x])/d^3 + (e^2*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d^3 - (e^2*(a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d^3 + (b*c*e*Log[1 + c^2*x^2])/(2*d^2) + ((I/2)*b*e^2*PolyLog[2, (-I)*c*x])/d^3 - ((I/2)*b*e^2*PolyLog[2, I*c*x])/d^3 - ((I/2)*b*e^2*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^3 + ((I/2)*b*e^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d^3","A",17,14,19,0.7368,1,"{4876, 4852, 325, 203, 266, 36, 29, 31, 4848, 2391, 4856, 2402, 2315, 2447}"
141,1,598,0,0.671122,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{d+e x} \, dx","Int[(x^3*(a + b*ArcTan[c*x])^2)/(d + e*x),x]","-\frac{i b d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^4}+\frac{i b d^3 \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^4}-\frac{i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^3 e}+\frac{b^2 d^3 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 e^4}-\frac{b^2 d^3 \text{PolyLog}\left(3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e^4}+\frac{i b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c e^3}-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 e^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^3 e}-\frac{2 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3 e}+\frac{d^2 x \left(a+b \tan ^{-1}(c x)\right)^2}{e^3}+\frac{i d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{c e^3}+\frac{d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{e^4}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^4}+\frac{2 b d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c e^3}-\frac{d x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 e^2}+\frac{a b d x}{c e^2}+\frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 e}-\frac{b x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c e}-\frac{b^2 d \log \left(c^2 x^2+1\right)}{2 c^2 e^2}+\frac{b^2 x}{3 c^2 e}-\frac{b^2 \tan ^{-1}(c x)}{3 c^3 e}+\frac{b^2 d x \tan ^{-1}(c x)}{c e^2}","-\frac{i b d^3 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^4}+\frac{i b d^3 \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^4}-\frac{i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^3 e}+\frac{b^2 d^3 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 e^4}-\frac{b^2 d^3 \text{PolyLog}\left(3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e^4}+\frac{i b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c e^3}-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 e^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^3 e}-\frac{2 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3 e}+\frac{d^2 x \left(a+b \tan ^{-1}(c x)\right)^2}{e^3}+\frac{i d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{c e^3}+\frac{d^3 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{e^4}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^4}+\frac{2 b d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c e^3}-\frac{d x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 e^2}+\frac{a b d x}{c e^2}+\frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 e}-\frac{b x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c e}-\frac{b^2 d \log \left(c^2 x^2+1\right)}{2 c^2 e^2}+\frac{b^2 x}{3 c^2 e}-\frac{b^2 \tan ^{-1}(c x)}{3 c^3 e}+\frac{b^2 d x \tan ^{-1}(c x)}{c e^2}",1,"(a*b*d*x)/(c*e^2) + (b^2*x)/(3*c^2*e) - (b^2*ArcTan[c*x])/(3*c^3*e) + (b^2*d*x*ArcTan[c*x])/(c*e^2) - (b*x^2*(a + b*ArcTan[c*x]))/(3*c*e) + (I*d^2*(a + b*ArcTan[c*x])^2)/(c*e^3) - (d*(a + b*ArcTan[c*x])^2)/(2*c^2*e^2) - ((I/3)*(a + b*ArcTan[c*x])^2)/(c^3*e) + (d^2*x*(a + b*ArcTan[c*x])^2)/e^3 - (d*x^2*(a + b*ArcTan[c*x])^2)/(2*e^2) + (x^3*(a + b*ArcTan[c*x])^2)/(3*e) + (d^3*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e^4 + (2*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c*e^3) - (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3*e) - (d^3*(a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^4 - (b^2*d*Log[1 + c^2*x^2])/(2*c^2*e^2) - (I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^4 + (I*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c*e^3) - ((I/3)*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c^3*e) + (I*b*d^3*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^4 + (b^2*d^3*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e^4) - (b^2*d^3*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e^4)","A",23,13,21,0.6190,1,"{4876, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 321, 203, 4858}"
142,1,430,0,0.4245683,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{d+e x} \, dx","Int[(x^2*(a + b*ArcTan[c*x])^2)/(d + e*x),x]","\frac{i b d^2 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^3}-\frac{i b d^2 \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^3}-\frac{b^2 d^2 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 e^3}+\frac{b^2 d^2 \text{PolyLog}\left(3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e^3}-\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c e^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 e}-\frac{d^2 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{e^3}+\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^3}-\frac{d x \left(a+b \tan ^{-1}(c x)\right)^2}{e^2}-\frac{i d \left(a+b \tan ^{-1}(c x)\right)^2}{c e^2}-\frac{2 b d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c e^2}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 e}-\frac{a b x}{c e}+\frac{b^2 \log \left(c^2 x^2+1\right)}{2 c^2 e}-\frac{b^2 x \tan ^{-1}(c x)}{c e}","\frac{i b d^2 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^3}-\frac{i b d^2 \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^3}-\frac{b^2 d^2 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 e^3}+\frac{b^2 d^2 \text{PolyLog}\left(3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e^3}-\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c e^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 e}-\frac{d^2 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{e^3}+\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^3}-\frac{d x \left(a+b \tan ^{-1}(c x)\right)^2}{e^2}-\frac{i d \left(a+b \tan ^{-1}(c x)\right)^2}{c e^2}-\frac{2 b d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c e^2}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 e}-\frac{a b x}{c e}+\frac{b^2 \log \left(c^2 x^2+1\right)}{2 c^2 e}-\frac{b^2 x \tan ^{-1}(c x)}{c e}",1,"-((a*b*x)/(c*e)) - (b^2*x*ArcTan[c*x])/(c*e) - (I*d*(a + b*ArcTan[c*x])^2)/(c*e^2) + (a + b*ArcTan[c*x])^2/(2*c^2*e) - (d*x*(a + b*ArcTan[c*x])^2)/e^2 + (x^2*(a + b*ArcTan[c*x])^2)/(2*e) - (d^2*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e^3 - (2*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c*e^2) + (d^2*(a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^3 + (b^2*Log[1 + c^2*x^2])/(2*c^2*e) + (I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^3 - (I*b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c*e^2) - (I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^3 - (b^2*d^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e^3) + (b^2*d^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e^3)","A",14,11,21,0.5238,1,"{4876, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 4858}"
143,1,323,0,0.2655611,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{d+e x} \, dx","Int[(x*(a + b*ArcTan[c*x])^2)/(d + e*x),x]","-\frac{i b d \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^2}+\frac{i b d \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^2}+\frac{b^2 d \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 e^2}-\frac{b^2 d \text{PolyLog}\left(3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e^2}+\frac{i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c e}+\frac{d \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{e^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^2}+\frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{e}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{c e}+\frac{2 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c e}","-\frac{i b d \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^2}+\frac{i b d \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^2}+\frac{b^2 d \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 e^2}-\frac{b^2 d \text{PolyLog}\left(3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e^2}+\frac{i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c e}+\frac{d \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{e^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e^2}+\frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{e}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{c e}+\frac{2 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c e}",1,"(I*(a + b*ArcTan[c*x])^2)/(c*e) + (x*(a + b*ArcTan[c*x])^2)/e + (d*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e^2 + (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c*e) - (d*(a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^2 - (I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^2 + (I*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c*e) + (I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e^2 + (b^2*d*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e^2) - (b^2*d*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e^2)","A",8,7,19,0.3684,1,"{4876, 4846, 4920, 4854, 2402, 2315, 4858}"
144,1,223,0,0.0476133,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d+e x} \, dx","Int[(a + b*ArcTan[c*x])^2/(d + e*x),x]","-\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 e}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{e}","-\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 e}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 e}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{e}",1,"-(((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/e - (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e) + (b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*e)","A",1,1,18,0.05556,1,"{4858}"
145,1,369,0,0.4330212,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x (d+e x)} \, dx","Int[(a + b*ArcTan[c*x])^2/(x*(d + e*x)),x]","\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{i b \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 d}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 d}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 d}+\frac{b^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d}+\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}","\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{i b \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 d}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 d}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 d}+\frac{b^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d}+\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}",1,"(2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d + ((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d - ((a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d + (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*d) - (b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*d) + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d) - (b^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*d)","A",9,7,21,0.3333,1,"{4876, 4850, 4988, 4884, 4994, 6610, 4858}"
146,1,473,0,0.6035131,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^2 (d+e x)} \, dx","Int[(a + b*ArcTan[c*x])^2/(x^2*(d + e*x)),x]","\frac{i b e \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{i b e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{i b e \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{i b e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d^2}-\frac{b^2 e \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 d^2}+\frac{b^2 e \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 d^2}-\frac{b^2 e \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d^2}+\frac{b^2 e \text{PolyLog}\left(3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 d^2}-\frac{i b^2 c \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d}-\frac{e \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d^2}-\frac{2 e \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d x}+\frac{2 b c \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}","\frac{i b e \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{i b e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{i b e \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{i b e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d^2}-\frac{b^2 e \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 d^2}+\frac{b^2 e \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 d^2}-\frac{b^2 e \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d^2}+\frac{b^2 e \text{PolyLog}\left(3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 d^2}-\frac{i b^2 c \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d}-\frac{e \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d^2}-\frac{2 e \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d x}+\frac{2 b c \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}",1,"((-I)*c*(a + b*ArcTan[c*x])^2)/d - (a + b*ArcTan[c*x])^2/(d*x) - (2*e*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^2 - (e*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d^2 + (e*(a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d^2 + (2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d + (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^2 - (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)])/d + (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^2 - (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^2 - (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d^2 - (b^2*e*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*d^2) + (b^2*e*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*d^2) - (b^2*e*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^2) + (b^2*e*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*d^2)","A",13,11,21,0.5238,1,"{4876, 4852, 4924, 4868, 2447, 4850, 4988, 4884, 4994, 6610, 4858}"
147,1,591,0,0.8418595,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^3 (d+e x)} \, dx","Int[(a + b*ArcTan[c*x])^2/(x^3*(d + e*x)),x]","-\frac{i b e^2 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}-\frac{i b e^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}+\frac{i b e^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}+\frac{i b e^2 \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d^3}+\frac{b^2 e^2 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 d^3}-\frac{b^2 e^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 d^3}+\frac{b^2 e^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d^3}-\frac{b^2 e^2 \text{PolyLog}\left(3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 d^3}+\frac{i b^2 c e \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d^2}-\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}+\frac{e^2 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^3}-\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d^3}+\frac{2 e^2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^3}+\frac{i c e \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{d^2 x}-\frac{2 b c e \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 d x^2}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{d x}-\frac{b^2 c^2 \log \left(c^2 x^2+1\right)}{2 d}+\frac{b^2 c^2 \log (x)}{d}","-\frac{i b e^2 \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}-\frac{i b e^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}+\frac{i b e^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}+\frac{i b e^2 \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d^3}+\frac{b^2 e^2 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 d^3}-\frac{b^2 e^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 d^3}+\frac{b^2 e^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d^3}-\frac{b^2 e^2 \text{PolyLog}\left(3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{2 d^3}+\frac{i b^2 c e \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d^2}-\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}+\frac{e^2 \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^3}-\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{d^3}+\frac{2 e^2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^3}+\frac{i c e \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{d^2 x}-\frac{2 b c e \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 d x^2}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{d x}-\frac{b^2 c^2 \log \left(c^2 x^2+1\right)}{2 d}+\frac{b^2 c^2 \log (x)}{d}",1,"-((b*c*(a + b*ArcTan[c*x]))/(d*x)) - (c^2*(a + b*ArcTan[c*x])^2)/(2*d) + (I*c*e*(a + b*ArcTan[c*x])^2)/d^2 - (a + b*ArcTan[c*x])^2/(2*d*x^2) + (e*(a + b*ArcTan[c*x])^2)/(d^2*x) + (2*e^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^3 + (b^2*c^2*Log[x])/d + (e^2*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d^3 - (e^2*(a + b*ArcTan[c*x])^2*Log[(2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d^3 - (b^2*c^2*Log[1 + c^2*x^2])/(2*d) - (2*b*c*e*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d^2 - (I*b*e^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^3 + (I*b^2*c*e*PolyLog[2, -1 + 2/(1 - I*c*x)])/d^2 - (I*b*e^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^3 + (I*b*e^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^3 + (I*b*e^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/d^3 + (b^2*e^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*d^3) - (b^2*e^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*d^3) + (b^2*e^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^3) - (b^2*e^2*PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + I*e)*(1 - I*c*x))])/(2*d^3)","A",21,16,21,0.7619,1,"{4876, 4852, 4918, 266, 36, 29, 31, 4884, 4924, 4868, 2447, 4850, 4988, 4994, 6610, 4858}"
148,0,0,0,0.0313904,"\int \frac{1}{(d+e x) \left(a+b \tan ^{-1}(c x)\right)} \, dx","Int[1/((d + e*x)*(a + b*ArcTan[c*x])),x]","\int \frac{1}{(d+e x) \left(a+b \tan ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{(d+e x) \left(a+b \tan ^{-1}(c x)\right)},x\right)",0,"Defer[Int][1/((d + e*x)*(a + b*ArcTan[c*x])), x]","A",0,0,0,0,-1,"{}"
149,1,69,0,0.0855232,"\int x^3 \left(c+a^2 c x^2\right) \tan ^{-1}(a x) \, dx","Int[x^3*(c + a^2*c*x^2)*ArcTan[a*x],x]","\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)+\frac{c x}{12 a^3}-\frac{c \tan ^{-1}(a x)}{12 a^4}-\frac{1}{30} a c x^5-\frac{c x^3}{36 a}+\frac{1}{4} c x^4 \tan ^{-1}(a x)","\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)+\frac{c x}{12 a^3}-\frac{c \tan ^{-1}(a x)}{12 a^4}-\frac{1}{30} a c x^5-\frac{c x^3}{36 a}+\frac{1}{4} c x^4 \tan ^{-1}(a x)",1,"(c*x)/(12*a^3) - (c*x^3)/(36*a) - (a*c*x^5)/30 - (c*ArcTan[a*x])/(12*a^4) + (c*x^4*ArcTan[a*x])/4 + (a^2*c*x^6*ArcTan[a*x])/6","A",9,4,18,0.2222,1,"{4950, 4852, 302, 203}"
150,1,66,0,0.0948406,"\int x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x) \, dx","Int[x^2*(c + a^2*c*x^2)*ArcTan[a*x],x]","\frac{c \log \left(a^2 x^2+1\right)}{15 a^3}+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)-\frac{1}{20} a c x^4-\frac{c x^2}{15 a}+\frac{1}{3} c x^3 \tan ^{-1}(a x)","\frac{c \log \left(a^2 x^2+1\right)}{15 a^3}+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)-\frac{1}{20} a c x^4-\frac{c x^2}{15 a}+\frac{1}{3} c x^3 \tan ^{-1}(a x)",1,"-(c*x^2)/(15*a) - (a*c*x^4)/20 + (c*x^3*ArcTan[a*x])/3 + (a^2*c*x^5*ArcTan[a*x])/5 + (c*Log[1 + a^2*x^2])/(15*a^3)","A",9,4,18,0.2222,1,"{4950, 4852, 266, 43}"
151,1,42,0,0.0249456,"\int x \left(c+a^2 c x^2\right) \tan ^{-1}(a x) \, dx","Int[x*(c + a^2*c*x^2)*ArcTan[a*x],x]","\frac{c \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}{4 a^2}-\frac{1}{12} a c x^3-\frac{c x}{4 a}","\frac{c \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}{4 a^2}-\frac{1}{12} a c x^3-\frac{c x}{4 a}",1,"-(c*x)/(4*a) - (a*c*x^3)/12 + (c*(1 + a^2*x^2)^2*ArcTan[a*x])/(4*a^2)","A",2,1,16,0.06250,1,"{4930}"
152,1,65,0,0.0234937,"\int \left(c+a^2 c x^2\right) \tan ^{-1}(a x) \, dx","Int[(c + a^2*c*x^2)*ArcTan[a*x],x]","-\frac{c \left(a^2 x^2+1\right)}{6 a}-\frac{c \log \left(a^2 x^2+1\right)}{3 a}+\frac{1}{3} c x \left(a^2 x^2+1\right) \tan ^{-1}(a x)+\frac{2}{3} c x \tan ^{-1}(a x)","-\frac{c \log \left(a^2 x^2+1\right)}{3 a}+\frac{1}{3} a^2 c x^3 \tan ^{-1}(a x)-\frac{1}{6} a c x^2+c x \tan ^{-1}(a x)",1,"-(c*(1 + a^2*x^2))/(6*a) + (2*c*x*ArcTan[a*x])/3 + (c*x*(1 + a^2*x^2)*ArcTan[a*x])/3 - (c*Log[1 + a^2*x^2])/(3*a)","A",3,3,15,0.2000,1,"{4878, 4846, 260}"
153,1,62,0,0.0661995,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)}{x} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x])/x,x]","\frac{1}{2} i c \text{PolyLog}(2,-i a x)-\frac{1}{2} i c \text{PolyLog}(2,i a x)+\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)-\frac{a c x}{2}+\frac{1}{2} c \tan ^{-1}(a x)","\frac{1}{2} i c \text{PolyLog}(2,-i a x)-\frac{1}{2} i c \text{PolyLog}(2,i a x)+\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)-\frac{a c x}{2}+\frac{1}{2} c \tan ^{-1}(a x)",1,"-(a*c*x)/2 + (c*ArcTan[a*x])/2 + (a^2*c*x^2*ArcTan[a*x])/2 + (I/2)*c*PolyLog[2, (-I)*a*x] - (I/2)*c*PolyLog[2, I*a*x]","A",7,6,18,0.3333,1,"{4950, 4848, 2391, 4852, 321, 203}"
154,1,40,0,0.0548211,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)}{x^2} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x])/x^2,x]","-a c \log \left(a^2 x^2+1\right)+a^2 c x \tan ^{-1}(a x)+a c \log (x)-\frac{c \tan ^{-1}(a x)}{x}","-a c \log \left(a^2 x^2+1\right)+a^2 c x \tan ^{-1}(a x)+a c \log (x)-\frac{c \tan ^{-1}(a x)}{x}",1,"-((c*ArcTan[a*x])/x) + a^2*c*x*ArcTan[a*x] + a*c*Log[x] - a*c*Log[1 + a^2*x^2]","A",8,8,18,0.4444,1,"{4950, 4852, 266, 36, 29, 31, 4846, 260}"
155,1,70,0,0.0706943,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)}{x^3} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x])/x^3,x]","\frac{1}{2} i a^2 c \text{PolyLog}(2,-i a x)-\frac{1}{2} i a^2 c \text{PolyLog}(2,i a x)-\frac{1}{2} a^2 c \tan ^{-1}(a x)-\frac{c \tan ^{-1}(a x)}{2 x^2}-\frac{a c}{2 x}","\frac{1}{2} i a^2 c \text{PolyLog}(2,-i a x)-\frac{1}{2} i a^2 c \text{PolyLog}(2,i a x)-\frac{1}{2} a^2 c \tan ^{-1}(a x)-\frac{c \tan ^{-1}(a x)}{2 x^2}-\frac{a c}{2 x}",1,"-(a*c)/(2*x) - (a^2*c*ArcTan[a*x])/2 - (c*ArcTan[a*x])/(2*x^2) + (I/2)*a^2*c*PolyLog[2, (-I)*a*x] - (I/2)*a^2*c*PolyLog[2, I*a*x]","A",7,6,18,0.3333,1,"{4950, 4852, 325, 203, 4848, 2391}"
156,1,63,0,0.0822095,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)}{x^4} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x])/x^4,x]","-\frac{1}{3} a^3 c \log \left(a^2 x^2+1\right)+\frac{2}{3} a^3 c \log (x)-\frac{a^2 c \tan ^{-1}(a x)}{x}-\frac{a c}{6 x^2}-\frac{c \tan ^{-1}(a x)}{3 x^3}","-\frac{1}{3} a^3 c \log \left(a^2 x^2+1\right)+\frac{2}{3} a^3 c \log (x)-\frac{a^2 c \tan ^{-1}(a x)}{x}-\frac{a c}{6 x^2}-\frac{c \tan ^{-1}(a x)}{3 x^3}",1,"-(a*c)/(6*x^2) - (c*ArcTan[a*x])/(3*x^3) - (a^2*c*ArcTan[a*x])/x + (2*a^3*c*Log[x])/3 - (a^3*c*Log[1 + a^2*x^2])/3","A",10,7,18,0.3889,1,"{4950, 4852, 266, 44, 36, 29, 31}"
157,1,111,0,0.1556482,"\int x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x) \, dx","Int[x^3*(c + a^2*c*x^2)^2*ArcTan[a*x],x]","-\frac{1}{56} a^3 c^2 x^7+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac{c^2 x}{24 a^3}-\frac{c^2 \tan ^{-1}(a x)}{24 a^4}-\frac{1}{24} a c^2 x^5-\frac{c^2 x^3}{72 a}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)","-\frac{1}{56} a^3 c^2 x^7+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac{c^2 x}{24 a^3}-\frac{c^2 \tan ^{-1}(a x)}{24 a^4}-\frac{1}{24} a c^2 x^5-\frac{c^2 x^3}{72 a}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)",1,"(c^2*x)/(24*a^3) - (c^2*x^3)/(72*a) - (a*c^2*x^5)/24 - (a^3*c^2*x^7)/56 - (c^2*ArcTan[a*x])/(24*a^4) + (c^2*x^4*ArcTan[a*x])/4 + (a^2*c^2*x^6*ArcTan[a*x])/3 + (a^4*c^2*x^8*ArcTan[a*x])/8","A",14,4,20,0.2000,1,"{4948, 4852, 302, 203}"
158,1,106,0,0.1716917,"\int x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x) \, dx","Int[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x],x]","-\frac{1}{42} a^3 c^2 x^6+\frac{4 c^2 \log \left(a^2 x^2+1\right)}{105 a^3}+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)-\frac{9}{140} a c^2 x^4-\frac{4 c^2 x^2}{105 a}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)","-\frac{1}{42} a^3 c^2 x^6+\frac{4 c^2 \log \left(a^2 x^2+1\right)}{105 a^3}+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)-\frac{9}{140} a c^2 x^4-\frac{4 c^2 x^2}{105 a}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)",1,"(-4*c^2*x^2)/(105*a) - (9*a*c^2*x^4)/140 - (a^3*c^2*x^6)/42 + (c^2*x^3*ArcTan[a*x])/3 + (2*a^2*c^2*x^5*ArcTan[a*x])/5 + (a^4*c^2*x^7*ArcTan[a*x])/7 + (4*c^2*Log[1 + a^2*x^2])/(105*a^3)","A",14,4,20,0.2000,1,"{4948, 4852, 266, 43}"
159,1,61,0,0.0425797,"\int x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x) \, dx","Int[x*(c + a^2*c*x^2)^2*ArcTan[a*x],x]","-\frac{1}{30} a^3 c^2 x^5+\frac{c^2 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)}{6 a^2}-\frac{1}{9} a c^2 x^3-\frac{c^2 x}{6 a}","-\frac{1}{30} a^3 c^2 x^5+\frac{c^2 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)}{6 a^2}-\frac{1}{9} a c^2 x^3-\frac{c^2 x}{6 a}",1,"-(c^2*x)/(6*a) - (a*c^2*x^3)/9 - (a^3*c^2*x^5)/30 + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x])/(6*a^2)","A",3,2,18,0.1111,1,"{4930, 194}"
160,1,117,0,0.0454285,"\int \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x) \, dx","Int[(c + a^2*c*x^2)^2*ArcTan[a*x],x]","-\frac{c^2 \left(a^2 x^2+1\right)^2}{20 a}-\frac{2 c^2 \left(a^2 x^2+1\right)}{15 a}-\frac{4 c^2 \log \left(a^2 x^2+1\right)}{15 a}+\frac{1}{5} c^2 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)+\frac{4}{15} c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)+\frac{8}{15} c^2 x \tan ^{-1}(a x)","-\frac{c^2 \left(a^2 x^2+1\right)^2}{20 a}-\frac{2 c^2 \left(a^2 x^2+1\right)}{15 a}-\frac{4 c^2 \log \left(a^2 x^2+1\right)}{15 a}+\frac{1}{5} c^2 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)+\frac{4}{15} c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)+\frac{8}{15} c^2 x \tan ^{-1}(a x)",1,"(-2*c^2*(1 + a^2*x^2))/(15*a) - (c^2*(1 + a^2*x^2)^2)/(20*a) + (8*c^2*x*ArcTan[a*x])/15 + (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x])/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x])/5 - (4*c^2*Log[1 + a^2*x^2])/(15*a)","A",4,3,17,0.1765,1,"{4878, 4846, 260}"
161,1,99,0,0.1191932,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)}{x} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x])/x,x]","\frac{1}{2} i c^2 \text{PolyLog}(2,-i a x)-\frac{1}{2} i c^2 \text{PolyLog}(2,i a x)-\frac{1}{12} a^3 c^2 x^3+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)+a^2 c^2 x^2 \tan ^{-1}(a x)-\frac{3}{4} a c^2 x+\frac{3}{4} c^2 \tan ^{-1}(a x)","\frac{1}{2} i c^2 \text{PolyLog}(2,-i a x)-\frac{1}{2} i c^2 \text{PolyLog}(2,i a x)-\frac{1}{12} a^3 c^2 x^3+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)+a^2 c^2 x^2 \tan ^{-1}(a x)-\frac{3}{4} a c^2 x+\frac{3}{4} c^2 \tan ^{-1}(a x)",1,"(-3*a*c^2*x)/4 - (a^3*c^2*x^3)/12 + (3*c^2*ArcTan[a*x])/4 + a^2*c^2*x^2*ArcTan[a*x] + (a^4*c^2*x^4*ArcTan[a*x])/4 + (I/2)*c^2*PolyLog[2, (-I)*a*x] - (I/2)*c^2*PolyLog[2, I*a*x]","A",12,7,20,0.3500,1,"{4948, 4848, 2391, 4852, 321, 203, 302}"
162,1,81,0,0.1167582,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)}{x^2} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x])/x^2,x]","-\frac{1}{6} a^3 c^2 x^2-\frac{4}{3} a c^2 \log \left(a^2 x^2+1\right)+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)+2 a^2 c^2 x \tan ^{-1}(a x)+a c^2 \log (x)-\frac{c^2 \tan ^{-1}(a x)}{x}","-\frac{1}{6} a^3 c^2 x^2-\frac{4}{3} a c^2 \log \left(a^2 x^2+1\right)+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)+2 a^2 c^2 x \tan ^{-1}(a x)+a c^2 \log (x)-\frac{c^2 \tan ^{-1}(a x)}{x}",1,"-(a^3*c^2*x^2)/6 - (c^2*ArcTan[a*x])/x + 2*a^2*c^2*x*ArcTan[a*x] + (a^4*c^2*x^3*ArcTan[a*x])/3 + a*c^2*Log[x] - (4*a*c^2*Log[1 + a^2*x^2])/3","A",13,9,20,0.4500,1,"{4948, 4846, 260, 4852, 266, 36, 29, 31, 43}"
163,1,90,0,0.1231633,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)}{x^3} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x])/x^3,x]","i a^2 c^2 \text{PolyLog}(2,-i a x)-i a^2 c^2 \text{PolyLog}(2,i a x)+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)-\frac{1}{2} a^3 c^2 x-\frac{c^2 \tan ^{-1}(a x)}{2 x^2}-\frac{a c^2}{2 x}","i a^2 c^2 \text{PolyLog}(2,-i a x)-i a^2 c^2 \text{PolyLog}(2,i a x)+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)-\frac{1}{2} a^3 c^2 x-\frac{c^2 \tan ^{-1}(a x)}{2 x^2}-\frac{a c^2}{2 x}",1,"-(a*c^2)/(2*x) - (a^3*c^2*x)/2 - (c^2*ArcTan[a*x])/(2*x^2) + (a^4*c^2*x^2*ArcTan[a*x])/2 + I*a^2*c^2*PolyLog[2, (-I)*a*x] - I*a^2*c^2*PolyLog[2, I*a*x]","A",11,7,20,0.3500,1,"{4948, 4852, 325, 203, 4848, 2391, 321}"
164,1,85,0,0.1240276,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)}{x^4} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x])/x^4,x]","-\frac{4}{3} a^3 c^2 \log \left(a^2 x^2+1\right)+\frac{5}{3} a^3 c^2 \log (x)+a^4 c^2 x \tan ^{-1}(a x)-\frac{2 a^2 c^2 \tan ^{-1}(a x)}{x}-\frac{a c^2}{6 x^2}-\frac{c^2 \tan ^{-1}(a x)}{3 x^3}","-\frac{4}{3} a^3 c^2 \log \left(a^2 x^2+1\right)+\frac{5}{3} a^3 c^2 \log (x)+a^4 c^2 x \tan ^{-1}(a x)-\frac{2 a^2 c^2 \tan ^{-1}(a x)}{x}-\frac{a c^2}{6 x^2}-\frac{c^2 \tan ^{-1}(a x)}{3 x^3}",1,"-(a*c^2)/(6*x^2) - (c^2*ArcTan[a*x])/(3*x^3) - (2*a^2*c^2*ArcTan[a*x])/x + a^4*c^2*x*ArcTan[a*x] + (5*a^3*c^2*Log[x])/3 - (4*a^3*c^2*Log[1 + a^2*x^2])/3","A",13,9,20,0.4500,1,"{4948, 4846, 260, 4852, 266, 44, 36, 29, 31}"
165,1,141,0,0.2072557,"\int x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x) \, dx","Int[x^3*(c + a^2*c*x^2)^3*ArcTan[a*x],x]","-\frac{1}{90} a^5 c^3 x^9-\frac{11}{280} a^3 c^3 x^7+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)+\frac{c^3 x}{40 a^3}-\frac{c^3 \tan ^{-1}(a x)}{40 a^4}-\frac{9}{200} a c^3 x^5-\frac{c^3 x^3}{120 a}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)","-\frac{1}{90} a^5 c^3 x^9-\frac{11}{280} a^3 c^3 x^7+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)+\frac{c^3 x}{40 a^3}-\frac{c^3 \tan ^{-1}(a x)}{40 a^4}-\frac{9}{200} a c^3 x^5-\frac{c^3 x^3}{120 a}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)",1,"(c^3*x)/(40*a^3) - (c^3*x^3)/(120*a) - (9*a*c^3*x^5)/200 - (11*a^3*c^3*x^7)/280 - (a^5*c^3*x^9)/90 - (c^3*ArcTan[a*x])/(40*a^4) + (c^3*x^4*ArcTan[a*x])/4 + (a^2*c^3*x^6*ArcTan[a*x])/2 + (3*a^4*c^3*x^8*ArcTan[a*x])/8 + (a^6*c^3*x^10*ArcTan[a*x])/10","A",18,4,20,0.2000,1,"{4948, 4852, 302, 203}"
166,1,136,0,0.2333566,"\int x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x) \, dx","Int[x^2*(c + a^2*c*x^2)^3*ArcTan[a*x],x]","-\frac{1}{72} a^5 c^3 x^8-\frac{10}{189} a^3 c^3 x^6+\frac{8 c^3 \log \left(a^2 x^2+1\right)}{315 a^3}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)-\frac{89 a c^3 x^4}{1260}-\frac{8 c^3 x^2}{315 a}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)","-\frac{1}{72} a^5 c^3 x^8-\frac{10}{189} a^3 c^3 x^6+\frac{8 c^3 \log \left(a^2 x^2+1\right)}{315 a^3}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)-\frac{89 a c^3 x^4}{1260}-\frac{8 c^3 x^2}{315 a}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)",1,"(-8*c^3*x^2)/(315*a) - (89*a*c^3*x^4)/1260 - (10*a^3*c^3*x^6)/189 - (a^5*c^3*x^8)/72 + (c^3*x^3*ArcTan[a*x])/3 + (3*a^2*c^3*x^5*ArcTan[a*x])/5 + (3*a^4*c^3*x^7*ArcTan[a*x])/7 + (a^6*c^3*x^9*ArcTan[a*x])/9 + (8*c^3*Log[1 + a^2*x^2])/(315*a^3)","A",18,4,20,0.2000,1,"{4948, 4852, 266, 43}"
167,1,74,0,0.050066,"\int x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x) \, dx","Int[x*(c + a^2*c*x^2)^3*ArcTan[a*x],x]","-\frac{1}{56} a^5 c^3 x^7-\frac{3}{40} a^3 c^3 x^5+\frac{c^3 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x)}{8 a^2}-\frac{1}{8} a c^3 x^3-\frac{c^3 x}{8 a}","-\frac{1}{56} a^5 c^3 x^7-\frac{3}{40} a^3 c^3 x^5+\frac{c^3 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x)}{8 a^2}-\frac{1}{8} a c^3 x^3-\frac{c^3 x}{8 a}",1,"-(c^3*x)/(8*a) - (a*c^3*x^3)/8 - (3*a^3*c^3*x^5)/40 - (a^5*c^3*x^7)/56 + (c^3*(1 + a^2*x^2)^4*ArcTan[a*x])/(8*a^2)","A",3,2,18,0.1111,1,"{4930, 194}"
168,1,161,0,0.0763048,"\int \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x) \, dx","Int[(c + a^2*c*x^2)^3*ArcTan[a*x],x]","-\frac{c^3 \left(a^2 x^2+1\right)^3}{42 a}-\frac{3 c^3 \left(a^2 x^2+1\right)^2}{70 a}-\frac{4 c^3 \left(a^2 x^2+1\right)}{35 a}-\frac{8 c^3 \log \left(a^2 x^2+1\right)}{35 a}+\frac{1}{7} c^3 x \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)+\frac{6}{35} c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)+\frac{8}{35} c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)+\frac{16}{35} c^3 x \tan ^{-1}(a x)","-\frac{c^3 \left(a^2 x^2+1\right)^3}{42 a}-\frac{3 c^3 \left(a^2 x^2+1\right)^2}{70 a}-\frac{4 c^3 \left(a^2 x^2+1\right)}{35 a}-\frac{8 c^3 \log \left(a^2 x^2+1\right)}{35 a}+\frac{1}{7} c^3 x \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)+\frac{6}{35} c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)+\frac{8}{35} c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)+\frac{16}{35} c^3 x \tan ^{-1}(a x)",1,"(-4*c^3*(1 + a^2*x^2))/(35*a) - (3*c^3*(1 + a^2*x^2)^2)/(70*a) - (c^3*(1 + a^2*x^2)^3)/(42*a) + (16*c^3*x*ArcTan[a*x])/35 + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x])/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x])/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x])/7 - (8*c^3*Log[1 + a^2*x^2])/(35*a)","A",5,3,17,0.1765,1,"{4878, 4846, 260}"
169,1,132,0,0.1540439,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)}{x} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x])/x,x]","\frac{1}{2} i c^3 \text{PolyLog}(2,-i a x)-\frac{1}{2} i c^3 \text{PolyLog}(2,i a x)-\frac{1}{30} a^5 c^3 x^5-\frac{7}{36} a^3 c^3 x^3+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)-\frac{11}{12} a c^3 x+\frac{11}{12} c^3 \tan ^{-1}(a x)","\frac{1}{2} i c^3 \text{PolyLog}(2,-i a x)-\frac{1}{2} i c^3 \text{PolyLog}(2,i a x)-\frac{1}{30} a^5 c^3 x^5-\frac{7}{36} a^3 c^3 x^3+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)-\frac{11}{12} a c^3 x+\frac{11}{12} c^3 \tan ^{-1}(a x)",1,"(-11*a*c^3*x)/12 - (7*a^3*c^3*x^3)/36 - (a^5*c^3*x^5)/30 + (11*c^3*ArcTan[a*x])/12 + (3*a^2*c^3*x^2*ArcTan[a*x])/2 + (3*a^4*c^3*x^4*ArcTan[a*x])/4 + (a^6*c^3*x^6*ArcTan[a*x])/6 + (I/2)*c^3*PolyLog[2, (-I)*a*x] - (I/2)*c^3*PolyLog[2, I*a*x]","A",16,7,20,0.3500,1,"{4948, 4848, 2391, 4852, 321, 203, 302}"
170,1,108,0,0.1559429,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)}{x^2} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x])/x^2,x]","-\frac{1}{20} a^5 c^3 x^4-\frac{2}{5} a^3 c^3 x^2-\frac{8}{5} a c^3 \log \left(a^2 x^2+1\right)+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)+a^4 c^3 x^3 \tan ^{-1}(a x)+3 a^2 c^3 x \tan ^{-1}(a x)+a c^3 \log (x)-\frac{c^3 \tan ^{-1}(a x)}{x}","-\frac{1}{20} a^5 c^3 x^4-\frac{2}{5} a^3 c^3 x^2-\frac{8}{5} a c^3 \log \left(a^2 x^2+1\right)+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)+a^4 c^3 x^3 \tan ^{-1}(a x)+3 a^2 c^3 x \tan ^{-1}(a x)+a c^3 \log (x)-\frac{c^3 \tan ^{-1}(a x)}{x}",1,"(-2*a^3*c^3*x^2)/5 - (a^5*c^3*x^4)/20 - (c^3*ArcTan[a*x])/x + 3*a^2*c^3*x*ArcTan[a*x] + a^4*c^3*x^3*ArcTan[a*x] + (a^6*c^3*x^5*ArcTan[a*x])/5 + a*c^3*Log[x] - (8*a*c^3*Log[1 + a^2*x^2])/5","A",17,9,20,0.4500,1,"{4948, 4846, 260, 4852, 266, 36, 29, 31, 43}"
171,1,138,0,0.1525375,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)}{x^3} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x])/x^3,x]","\frac{3}{2} i a^2 c^3 \text{PolyLog}(2,-i a x)-\frac{3}{2} i a^2 c^3 \text{PolyLog}(2,i a x)-\frac{1}{12} a^5 c^3 x^3+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac{5}{4} a^3 c^3 x+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)}{2 x^2}-\frac{a c^3}{2 x}","\frac{3}{2} i a^2 c^3 \text{PolyLog}(2,-i a x)-\frac{3}{2} i a^2 c^3 \text{PolyLog}(2,i a x)-\frac{1}{12} a^5 c^3 x^3+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac{5}{4} a^3 c^3 x+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)}{2 x^2}-\frac{a c^3}{2 x}",1,"-(a*c^3)/(2*x) - (5*a^3*c^3*x)/4 - (a^5*c^3*x^3)/12 + (3*a^2*c^3*ArcTan[a*x])/4 - (c^3*ArcTan[a*x])/(2*x^2) + (3*a^4*c^3*x^2*ArcTan[a*x])/2 + (a^6*c^3*x^4*ArcTan[a*x])/4 + ((3*I)/2)*a^2*c^3*PolyLog[2, (-I)*a*x] - ((3*I)/2)*a^2*c^3*PolyLog[2, I*a*x]","A",15,8,20,0.4000,1,"{4948, 4852, 325, 203, 4848, 2391, 321, 302}"
172,1,116,0,0.1579059,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)}{x^4} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x])/x^4,x]","-\frac{1}{6} a^5 c^3 x^2-\frac{8}{3} a^3 c^3 \log \left(a^2 x^2+1\right)+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)+\frac{8}{3} a^3 c^3 \log (x)+3 a^4 c^3 x \tan ^{-1}(a x)-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}-\frac{a c^3}{6 x^2}-\frac{c^3 \tan ^{-1}(a x)}{3 x^3}","-\frac{1}{6} a^5 c^3 x^2-\frac{8}{3} a^3 c^3 \log \left(a^2 x^2+1\right)+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)+\frac{8}{3} a^3 c^3 \log (x)+3 a^4 c^3 x \tan ^{-1}(a x)-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}-\frac{a c^3}{6 x^2}-\frac{c^3 \tan ^{-1}(a x)}{3 x^3}",1,"-(a*c^3)/(6*x^2) - (a^5*c^3*x^2)/6 - (c^3*ArcTan[a*x])/(3*x^3) - (3*a^2*c^3*ArcTan[a*x])/x + 3*a^4*c^3*x*ArcTan[a*x] + (a^6*c^3*x^3*ArcTan[a*x])/3 + (8*a^3*c^3*Log[x])/3 - (8*a^3*c^3*Log[1 + a^2*x^2])/3","A",17,10,20,0.5000,1,"{4948, 4846, 260, 4852, 266, 44, 36, 29, 31, 43}"
173,1,80,0,0.1547066,"\int \frac{x^4 \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx","Int[(x^4*ArcTan[a*x])/(c + a^2*c*x^2),x]","-\frac{x^2}{6 a^3 c}+\frac{2 \log \left(a^2 x^2+1\right)}{3 a^5 c}+\frac{x^3 \tan ^{-1}(a x)}{3 a^2 c}-\frac{x \tan ^{-1}(a x)}{a^4 c}+\frac{\tan ^{-1}(a x)^2}{2 a^5 c}","-\frac{x^2}{6 a^3 c}+\frac{2 \log \left(a^2 x^2+1\right)}{3 a^5 c}+\frac{x^3 \tan ^{-1}(a x)}{3 a^2 c}-\frac{x \tan ^{-1}(a x)}{a^4 c}+\frac{\tan ^{-1}(a x)^2}{2 a^5 c}",1,"-x^2/(6*a^3*c) - (x*ArcTan[a*x])/(a^4*c) + (x^3*ArcTan[a*x])/(3*a^2*c) + ArcTan[a*x]^2/(2*a^5*c) + (2*Log[1 + a^2*x^2])/(3*a^5*c)","A",9,7,20,0.3500,1,"{4916, 4852, 266, 43, 4846, 260, 4884}"
174,1,113,0,0.142134,"\int \frac{x^3 \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx","Int[(x^3*ArcTan[a*x])/(c + a^2*c*x^2),x]","\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)}{2 a^2 c}-\frac{x}{2 a^3 c}+\frac{i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac{\tan ^{-1}(a x)}{2 a^4 c}+\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{a^4 c}","\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)}{2 a^2 c}-\frac{x}{2 a^3 c}+\frac{i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac{\tan ^{-1}(a x)}{2 a^4 c}+\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{a^4 c}",1,"-x/(2*a^3*c) + ArcTan[a*x]/(2*a^4*c) + (x^2*ArcTan[a*x])/(2*a^2*c) + ((I/2)*ArcTan[a*x]^2)/(a^4*c) + (ArcTan[a*x]*Log[2/(1 + I*a*x)])/(a^4*c) + ((I/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*c)","A",8,8,20,0.4000,1,"{4916, 4852, 321, 203, 4920, 4854, 2402, 2315}"
175,1,49,0,0.072521,"\int \frac{x^2 \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx","Int[(x^2*ArcTan[a*x])/(c + a^2*c*x^2),x]","-\frac{\log \left(a^2 x^2+1\right)}{2 a^3 c}-\frac{\tan ^{-1}(a x)^2}{2 a^3 c}+\frac{x \tan ^{-1}(a x)}{a^2 c}","-\frac{\log \left(a^2 x^2+1\right)}{2 a^3 c}-\frac{\tan ^{-1}(a x)^2}{2 a^3 c}+\frac{x \tan ^{-1}(a x)}{a^2 c}",1,"(x*ArcTan[a*x])/(a^2*c) - ArcTan[a*x]^2/(2*a^3*c) - Log[1 + a^2*x^2]/(2*a^3*c)","A",4,4,20,0.2000,1,"{4916, 4846, 260, 4884}"
176,1,72,0,0.0721384,"\int \frac{x \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx","Int[(x*ArcTan[a*x])/(c + a^2*c*x^2),x]","-\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^2 c}-\frac{i \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{a^2 c}","-\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^2 c}-\frac{i \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{a^2 c}",1,"((-I/2)*ArcTan[a*x]^2)/(a^2*c) - (ArcTan[a*x]*Log[2/(1 + I*a*x)])/(a^2*c) - ((I/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^2*c)","A",4,4,18,0.2222,1,"{4920, 4854, 2402, 2315}"
177,1,16,0,0.0173921,"\int \frac{\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx","Int[ArcTan[a*x]/(c + a^2*c*x^2),x]","\frac{\tan ^{-1}(a x)^2}{2 a c}","\frac{\tan ^{-1}(a x)^2}{2 a c}",1,"ArcTan[a*x]^2/(2*a*c)","A",1,1,17,0.05882,1,"{4884}"
178,1,64,0,0.0996789,"\int \frac{\tan ^{-1}(a x)}{x \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]/(x*(c + a^2*c*x^2)),x]","-\frac{i \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c}-\frac{i \tan ^{-1}(a x)^2}{2 c}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c}","-\frac{i \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c}-\frac{i \tan ^{-1}(a x)^2}{2 c}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c}",1,"((-I/2)*ArcTan[a*x]^2)/c + (ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c - ((I/2)*PolyLog[2, -1 + 2/(1 - I*a*x)])/c","A",3,3,20,0.1500,1,"{4924, 4868, 2447}"
179,1,52,0,0.0865378,"\int \frac{\tan ^{-1}(a x)}{x^2 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]/(x^2*(c + a^2*c*x^2)),x]","-\frac{a \log \left(a^2 x^2+1\right)}{2 c}+\frac{a \log (x)}{c}-\frac{a \tan ^{-1}(a x)^2}{2 c}-\frac{\tan ^{-1}(a x)}{c x}","-\frac{a \log \left(a^2 x^2+1\right)}{2 c}+\frac{a \log (x)}{c}-\frac{a \tan ^{-1}(a x)^2}{2 c}-\frac{\tan ^{-1}(a x)}{c x}",1,"-(ArcTan[a*x]/(c*x)) - (a*ArcTan[a*x]^2)/(2*c) + (a*Log[x])/c - (a*Log[1 + a^2*x^2])/(2*c)","A",7,7,20,0.3500,1,"{4918, 4852, 266, 36, 29, 31, 4884}"
180,1,113,0,0.1623609,"\int \frac{\tan ^{-1}(a x)}{x^3 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]/(x^3*(c + a^2*c*x^2)),x]","\frac{i a^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c}+\frac{i a^2 \tan ^{-1}(a x)^2}{2 c}-\frac{a^2 \tan ^{-1}(a x)}{2 c}-\frac{a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c}-\frac{\tan ^{-1}(a x)}{2 c x^2}-\frac{a}{2 c x}","\frac{i a^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c}+\frac{i a^2 \tan ^{-1}(a x)^2}{2 c}-\frac{a^2 \tan ^{-1}(a x)}{2 c}-\frac{a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c}-\frac{\tan ^{-1}(a x)}{2 c x^2}-\frac{a}{2 c x}",1,"-a/(2*c*x) - (a^2*ArcTan[a*x])/(2*c) - ArcTan[a*x]/(2*c*x^2) + ((I/2)*a^2*ArcTan[a*x]^2)/c - (a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c + ((I/2)*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c","A",7,7,20,0.3500,1,"{4918, 4852, 325, 203, 4924, 4868, 2447}"
181,1,88,0,0.164881,"\int \frac{\tan ^{-1}(a x)}{x^4 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)),x]","\frac{2 a^3 \log \left(a^2 x^2+1\right)}{3 c}-\frac{4 a^3 \log (x)}{3 c}+\frac{a^3 \tan ^{-1}(a x)^2}{2 c}+\frac{a^2 \tan ^{-1}(a x)}{c x}-\frac{a}{6 c x^2}-\frac{\tan ^{-1}(a x)}{3 c x^3}","\frac{2 a^3 \log \left(a^2 x^2+1\right)}{3 c}-\frac{4 a^3 \log (x)}{3 c}+\frac{a^3 \tan ^{-1}(a x)^2}{2 c}+\frac{a^2 \tan ^{-1}(a x)}{c x}-\frac{a}{6 c x^2}-\frac{\tan ^{-1}(a x)}{3 c x^3}",1,"-a/(6*c*x^2) - ArcTan[a*x]/(3*c*x^3) + (a^2*ArcTan[a*x])/(c*x) + (a^3*ArcTan[a*x]^2)/(2*c) - (4*a^3*Log[x])/(3*c) + (2*a^3*Log[1 + a^2*x^2])/(3*c)","A",12,8,20,0.4000,1,"{4918, 4852, 266, 44, 36, 29, 31, 4884}"
182,1,157,0,0.3631895,"\int \frac{x^5 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^5*ArcTan[a*x])/(c + a^2*c*x^2)^2,x]","\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^6 c^2}+\frac{x}{4 a^5 c^2 \left(a^2 x^2+1\right)}+\frac{x^2 \tan ^{-1}(a x)}{2 a^4 c^2}-\frac{\tan ^{-1}(a x)}{2 a^6 c^2 \left(a^2 x^2+1\right)}-\frac{x}{2 a^5 c^2}+\frac{i \tan ^{-1}(a x)^2}{a^6 c^2}+\frac{3 \tan ^{-1}(a x)}{4 a^6 c^2}+\frac{2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{a^6 c^2}","\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^6 c^2}+\frac{x}{4 a^5 c^2 \left(a^2 x^2+1\right)}+\frac{x^2 \tan ^{-1}(a x)}{2 a^4 c^2}-\frac{\tan ^{-1}(a x)}{2 a^6 c^2 \left(a^2 x^2+1\right)}-\frac{x}{2 a^5 c^2}+\frac{i \tan ^{-1}(a x)^2}{a^6 c^2}+\frac{3 \tan ^{-1}(a x)}{4 a^6 c^2}+\frac{2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{a^6 c^2}",1,"-x/(2*a^5*c^2) + x/(4*a^5*c^2*(1 + a^2*x^2)) + (3*ArcTan[a*x])/(4*a^6*c^2) + (x^2*ArcTan[a*x])/(2*a^4*c^2) - ArcTan[a*x]/(2*a^6*c^2*(1 + a^2*x^2)) + (I*ArcTan[a*x]^2)/(a^6*c^2) + (2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(a^6*c^2) + (I*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^6*c^2)","A",17,12,20,0.6000,1,"{4964, 4916, 4852, 321, 203, 4920, 4854, 2402, 2315, 4930, 199, 205}"
183,1,96,0,0.1804871,"\int \frac{x^4 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^4*ArcTan[a*x])/(c + a^2*c*x^2)^2,x]","\frac{1}{4 a^5 c^2 \left(a^2 x^2+1\right)}-\frac{\log \left(a^2 x^2+1\right)}{2 a^5 c^2}+\frac{x \tan ^{-1}(a x)}{2 a^4 c^2 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)^2}{4 a^5 c^2}+\frac{x \tan ^{-1}(a x)}{a^4 c^2}","\frac{1}{4 a^5 c^2 \left(a^2 x^2+1\right)}-\frac{\log \left(a^2 x^2+1\right)}{2 a^5 c^2}+\frac{x \tan ^{-1}(a x)}{2 a^4 c^2 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)^2}{4 a^5 c^2}+\frac{x \tan ^{-1}(a x)}{a^4 c^2}",1,"1/(4*a^5*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(a^4*c^2) + (x*ArcTan[a*x])/(2*a^4*c^2*(1 + a^2*x^2)) - (3*ArcTan[a*x]^2)/(4*a^5*c^2) - Log[1 + a^2*x^2]/(2*a^5*c^2)","A",7,6,20,0.3000,1,"{4964, 4916, 4846, 260, 4884, 4934}"
184,1,133,0,0.1629148,"\int \frac{x^3 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^3*ArcTan[a*x])/(c + a^2*c*x^2)^2,x]","-\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^4 c^2}-\frac{x}{4 a^3 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)}{2 a^4 c^2 \left(a^2 x^2+1\right)}-\frac{i \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac{\tan ^{-1}(a x)}{4 a^4 c^2}-\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{a^4 c^2}","-\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^4 c^2}-\frac{x}{4 a^3 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)}{2 a^4 c^2 \left(a^2 x^2+1\right)}-\frac{i \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac{\tan ^{-1}(a x)}{4 a^4 c^2}-\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{a^4 c^2}",1,"-x/(4*a^3*c^2*(1 + a^2*x^2)) - ArcTan[a*x]/(4*a^4*c^2) + ArcTan[a*x]/(2*a^4*c^2*(1 + a^2*x^2)) - ((I/2)*ArcTan[a*x]^2)/(a^4*c^2) - (ArcTan[a*x]*Log[2/(1 + I*a*x)])/(a^4*c^2) - ((I/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*c^2)","A",8,8,20,0.4000,1,"{4964, 4920, 4854, 2402, 2315, 4930, 199, 205}"
185,1,64,0,0.0643417,"\int \frac{x^2 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^2*ArcTan[a*x])/(c + a^2*c*x^2)^2,x]","-\frac{1}{4 a^3 c^2 \left(a^2 x^2+1\right)}-\frac{x \tan ^{-1}(a x)}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^2}{4 a^3 c^2}","-\frac{1}{4 a^3 c^2 \left(a^2 x^2+1\right)}-\frac{x \tan ^{-1}(a x)}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^2}{4 a^3 c^2}",1,"-1/(4*a^3*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x])/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a^3*c^2)","A",2,2,20,0.1000,1,"{4934, 4884}"
186,1,62,0,0.041147,"\int \frac{x \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x*ArcTan[a*x])/(c + a^2*c*x^2)^2,x]","\frac{x}{4 a c^2 \left(a^2 x^2+1\right)}-\frac{\tan ^{-1}(a x)}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)}{4 a^2 c^2}","\frac{x}{4 a c^2 \left(a^2 x^2+1\right)}-\frac{\tan ^{-1}(a x)}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)}{4 a^2 c^2}",1,"x/(4*a*c^2*(1 + a^2*x^2)) + ArcTan[a*x]/(4*a^2*c^2) - ArcTan[a*x]/(2*a^2*c^2*(1 + a^2*x^2))","A",3,3,18,0.1667,1,"{4930, 199, 205}"
187,1,61,0,0.0258874,"\int \frac{\tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]/(c + a^2*c*x^2)^2,x]","\frac{1}{4 a c^2 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^2}{4 a c^2}","\frac{1}{4 a c^2 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^2}{4 a c^2}",1,"1/(4*a*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a*c^2)","A",2,2,17,0.1176,1,"{4892, 261}"
188,1,117,0,0.1841503,"\int \frac{\tan ^{-1}(a x)}{x \left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]/(x*(c + a^2*c*x^2)^2),x]","-\frac{i \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^2}-\frac{a x}{4 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}-\frac{i \tan ^{-1}(a x)^2}{2 c^2}-\frac{\tan ^{-1}(a x)}{4 c^2}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^2}","-\frac{i \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^2}-\frac{a x}{4 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}-\frac{i \tan ^{-1}(a x)^2}{2 c^2}-\frac{\tan ^{-1}(a x)}{4 c^2}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^2}",1,"-(a*x)/(4*c^2*(1 + a^2*x^2)) - ArcTan[a*x]/(4*c^2) + ArcTan[a*x]/(2*c^2*(1 + a^2*x^2)) - ((I/2)*ArcTan[a*x]^2)/c^2 + (ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^2 - ((I/2)*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2","A",7,7,20,0.3500,1,"{4966, 4924, 4868, 2447, 4930, 199, 205}"
189,1,97,0,0.1619342,"\int \frac{\tan ^{-1}(a x)}{x^2 \left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]/(x^2*(c + a^2*c*x^2)^2),x]","-\frac{a}{4 c^2 \left(a^2 x^2+1\right)}-\frac{a \log \left(a^2 x^2+1\right)}{2 c^2}-\frac{a^2 x \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}+\frac{a \log (x)}{c^2}-\frac{3 a \tan ^{-1}(a x)^2}{4 c^2}-\frac{\tan ^{-1}(a x)}{c^2 x}","-\frac{a}{4 c^2 \left(a^2 x^2+1\right)}-\frac{a \log \left(a^2 x^2+1\right)}{2 c^2}-\frac{a^2 x \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}+\frac{a \log (x)}{c^2}-\frac{3 a \tan ^{-1}(a x)^2}{4 c^2}-\frac{\tan ^{-1}(a x)}{c^2 x}",1,"-a/(4*c^2*(1 + a^2*x^2)) - ArcTan[a*x]/(c^2*x) - (a^2*x*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) - (3*a*ArcTan[a*x]^2)/(4*c^2) + (a*Log[x])/c^2 - (a*Log[1 + a^2*x^2])/(2*c^2)","A",10,10,20,0.5000,1,"{4966, 4918, 4852, 266, 36, 29, 31, 4884, 4892, 261}"
190,1,156,0,0.4059768,"\int \frac{\tan ^{-1}(a x)}{x^3 \left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]/(x^3*(c + a^2*c*x^2)^2),x]","\frac{i a^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^2}+\frac{a^3 x}{4 c^2 \left(a^2 x^2+1\right)}-\frac{a^2 \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}+\frac{i a^2 \tan ^{-1}(a x)^2}{c^2}-\frac{a^2 \tan ^{-1}(a x)}{4 c^2}-\frac{2 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^2}-\frac{\tan ^{-1}(a x)}{2 c^2 x^2}-\frac{a}{2 c^2 x}","\frac{i a^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^2}+\frac{a^3 x}{4 c^2 \left(a^2 x^2+1\right)}-\frac{a^2 \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}+\frac{i a^2 \tan ^{-1}(a x)^2}{c^2}-\frac{a^2 \tan ^{-1}(a x)}{4 c^2}-\frac{2 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^2}-\frac{\tan ^{-1}(a x)}{2 c^2 x^2}-\frac{a}{2 c^2 x}",1,"-a/(2*c^2*x) + (a^3*x)/(4*c^2*(1 + a^2*x^2)) - (a^2*ArcTan[a*x])/(4*c^2) - ArcTan[a*x]/(2*c^2*x^2) - (a^2*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) + (I*a^2*ArcTan[a*x]^2)/c^2 - (2*a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^2 + (I*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2","A",15,11,20,0.5500,1,"{4966, 4918, 4852, 325, 203, 4924, 4868, 2447, 4930, 199, 205}"
191,1,136,0,0.3749027,"\int \frac{\tan ^{-1}(a x)}{x^4 \left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^2),x]","\frac{a^3}{4 c^2 \left(a^2 x^2+1\right)}+\frac{7 a^3 \log \left(a^2 x^2+1\right)}{6 c^2}+\frac{a^4 x \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}-\frac{7 a^3 \log (x)}{3 c^2}+\frac{5 a^3 \tan ^{-1}(a x)^2}{4 c^2}+\frac{2 a^2 \tan ^{-1}(a x)}{c^2 x}-\frac{a}{6 c^2 x^2}-\frac{\tan ^{-1}(a x)}{3 c^2 x^3}","\frac{a^3}{4 c^2 \left(a^2 x^2+1\right)}+\frac{7 a^3 \log \left(a^2 x^2+1\right)}{6 c^2}+\frac{a^4 x \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}-\frac{7 a^3 \log (x)}{3 c^2}+\frac{5 a^3 \tan ^{-1}(a x)^2}{4 c^2}+\frac{2 a^2 \tan ^{-1}(a x)}{c^2 x}-\frac{a}{6 c^2 x^2}-\frac{\tan ^{-1}(a x)}{3 c^2 x^3}",1,"-a/(6*c^2*x^2) + a^3/(4*c^2*(1 + a^2*x^2)) - ArcTan[a*x]/(3*c^2*x^3) + (2*a^2*ArcTan[a*x])/(c^2*x) + (a^4*x*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) + (5*a^3*ArcTan[a*x]^2)/(4*c^2) - (7*a^3*Log[x])/(3*c^2) + (7*a^3*Log[1 + a^2*x^2])/(6*c^2)","A",23,11,20,0.5500,1,"{4966, 4918, 4852, 266, 44, 36, 29, 31, 4884, 4892, 261}"
192,1,86,0,0.0655447,"\int \frac{x^3 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^3*ArcTan[a*x])/(c + a^2*c*x^2)^3,x]","\frac{x^3}{16 a c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x}{32 a^3 c^3 \left(a^2 x^2+1\right)}+\frac{x^4 \tan ^{-1}(a x)}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{3 \tan ^{-1}(a x)}{32 a^4 c^3}","\frac{x^3}{16 a c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x}{32 a^3 c^3 \left(a^2 x^2+1\right)}+\frac{x^4 \tan ^{-1}(a x)}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{3 \tan ^{-1}(a x)}{32 a^4 c^3}",1,"x^3/(16*a*c^3*(1 + a^2*x^2)^2) + (3*x)/(32*a^3*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x])/(32*a^4*c^3) + (x^4*ArcTan[a*x])/(4*c^3*(1 + a^2*x^2)^2)","A",4,3,20,0.1500,1,"{4944, 288, 205}"
193,1,111,0,0.0748615,"\int \frac{x^2 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^2*ArcTan[a*x])/(c + a^2*c*x^2)^3,x]","\frac{1}{16 a^3 c^3 \left(a^2 x^2+1\right)}-\frac{1}{16 a^3 c^3 \left(a^2 x^2+1\right)^2}+\frac{x \tan ^{-1}(a x)}{8 a^2 c^3 \left(a^2 x^2+1\right)}-\frac{x \tan ^{-1}(a x)}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)^2}{16 a^3 c^3}","\frac{1}{16 a^3 c^3 \left(a^2 x^2+1\right)}-\frac{1}{16 a^3 c^3 \left(a^2 x^2+1\right)^2}+\frac{x \tan ^{-1}(a x)}{8 a^2 c^3 \left(a^2 x^2+1\right)}-\frac{x \tan ^{-1}(a x)}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)^2}{16 a^3 c^3}",1,"-1/(16*a^3*c^3*(1 + a^2*x^2)^2) + 1/(16*a^3*c^3*(1 + a^2*x^2)) - (x*ArcTan[a*x])/(4*a^2*c^3*(1 + a^2*x^2)^2) + (x*ArcTan[a*x])/(8*a^2*c^3*(1 + a^2*x^2)) + ArcTan[a*x]^2/(16*a^3*c^3)","A",3,3,20,0.1500,1,"{4934, 4892, 261}"
194,1,84,0,0.0496819,"\int \frac{x \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x*ArcTan[a*x])/(c + a^2*c*x^2)^3,x]","\frac{3 x}{32 a c^3 \left(a^2 x^2+1\right)}+\frac{x}{16 a c^3 \left(a^2 x^2+1\right)^2}-\frac{\tan ^{-1}(a x)}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)}{32 a^2 c^3}","\frac{3 x}{32 a c^3 \left(a^2 x^2+1\right)}+\frac{x}{16 a c^3 \left(a^2 x^2+1\right)^2}-\frac{\tan ^{-1}(a x)}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)}{32 a^2 c^3}",1,"x/(16*a*c^3*(1 + a^2*x^2)^2) + (3*x)/(32*a*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x])/(32*a^2*c^3) - ArcTan[a*x]/(4*a^2*c^3*(1 + a^2*x^2)^2)","A",4,3,18,0.1667,1,"{4930, 199, 205}"
195,1,105,0,0.0461152,"\int \frac{\tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]/(c + a^2*c*x^2)^3,x]","\frac{3}{16 a c^3 \left(a^2 x^2+1\right)}+\frac{1}{16 a c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)^2}{16 a c^3}","\frac{3}{16 a c^3 \left(a^2 x^2+1\right)}+\frac{1}{16 a c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)^2}{16 a c^3}",1,"1/(16*a*c^3*(1 + a^2*x^2)^2) + 3/(16*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(4*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^2)/(16*a*c^3)","A",3,3,17,0.1765,1,"{4896, 4892, 261}"
196,1,159,0,0.2839518,"\int \frac{\tan ^{-1}(a x)}{x \left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]/(x*(c + a^2*c*x^2)^3),x]","-\frac{i \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^3}-\frac{11 a x}{32 c^3 \left(a^2 x^2+1\right)}-\frac{a x}{16 c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)}{2 c^3 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{i \tan ^{-1}(a x)^2}{2 c^3}-\frac{11 \tan ^{-1}(a x)}{32 c^3}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^3}","-\frac{i \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^3}-\frac{11 a x}{32 c^3 \left(a^2 x^2+1\right)}-\frac{a x}{16 c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)}{2 c^3 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{i \tan ^{-1}(a x)^2}{2 c^3}-\frac{11 \tan ^{-1}(a x)}{32 c^3}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^3}",1,"-(a*x)/(16*c^3*(1 + a^2*x^2)^2) - (11*a*x)/(32*c^3*(1 + a^2*x^2)) - (11*ArcTan[a*x])/(32*c^3) + ArcTan[a*x]/(4*c^3*(1 + a^2*x^2)^2) + ArcTan[a*x]/(2*c^3*(1 + a^2*x^2)) - ((I/2)*ArcTan[a*x]^2)/c^3 + (ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^3 - ((I/2)*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3","A",12,7,20,0.3500,1,"{4966, 4924, 4868, 2447, 4930, 199, 205}"
197,1,142,0,0.2625657,"\int \frac{\tan ^{-1}(a x)}{x^2 \left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]/(x^2*(c + a^2*c*x^2)^3),x]","-\frac{7 a}{16 c^3 \left(a^2 x^2+1\right)}-\frac{a}{16 c^3 \left(a^2 x^2+1\right)^2}-\frac{a \log \left(a^2 x^2+1\right)}{2 c^3}-\frac{7 a^2 x \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)}-\frac{a^2 x \tan ^{-1}(a x)}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{a \log (x)}{c^3}-\frac{15 a \tan ^{-1}(a x)^2}{16 c^3}-\frac{\tan ^{-1}(a x)}{c^3 x}","-\frac{7 a}{16 c^3 \left(a^2 x^2+1\right)}-\frac{a}{16 c^3 \left(a^2 x^2+1\right)^2}-\frac{a \log \left(a^2 x^2+1\right)}{2 c^3}-\frac{7 a^2 x \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)}-\frac{a^2 x \tan ^{-1}(a x)}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{a \log (x)}{c^3}-\frac{15 a \tan ^{-1}(a x)^2}{16 c^3}-\frac{\tan ^{-1}(a x)}{c^3 x}",1,"-a/(16*c^3*(1 + a^2*x^2)^2) - (7*a)/(16*c^3*(1 + a^2*x^2)) - ArcTan[a*x]/(c^3*x) - (a^2*x*ArcTan[a*x])/(4*c^3*(1 + a^2*x^2)^2) - (7*a^2*x*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)) - (15*a*ArcTan[a*x]^2)/(16*c^3) + (a*Log[x])/c^3 - (a*Log[1 + a^2*x^2])/(2*c^3)","A",14,11,20,0.5500,1,"{4966, 4918, 4852, 266, 36, 29, 31, 4884, 4892, 261, 4896}"
198,1,205,0,0.7603182,"\int \frac{\tan ^{-1}(a x)}{x^3 \left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]/(x^3*(c + a^2*c*x^2)^3),x]","\frac{3 i a^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^3}+\frac{19 a^3 x}{32 c^3 \left(a^2 x^2+1\right)}+\frac{a^3 x}{16 c^3 \left(a^2 x^2+1\right)^2}-\frac{a^2 \tan ^{-1}(a x)}{c^3 \left(a^2 x^2+1\right)}-\frac{a^2 \tan ^{-1}(a x)}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c^3}+\frac{3 a^2 \tan ^{-1}(a x)}{32 c^3}-\frac{3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^3}-\frac{\tan ^{-1}(a x)}{2 c^3 x^2}-\frac{a}{2 c^3 x}","\frac{3 i a^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^3}+\frac{19 a^3 x}{32 c^3 \left(a^2 x^2+1\right)}+\frac{a^3 x}{16 c^3 \left(a^2 x^2+1\right)^2}-\frac{a^2 \tan ^{-1}(a x)}{c^3 \left(a^2 x^2+1\right)}-\frac{a^2 \tan ^{-1}(a x)}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c^3}+\frac{3 a^2 \tan ^{-1}(a x)}{32 c^3}-\frac{3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^3}-\frac{\tan ^{-1}(a x)}{2 c^3 x^2}-\frac{a}{2 c^3 x}",1,"-a/(2*c^3*x) + (a^3*x)/(16*c^3*(1 + a^2*x^2)^2) + (19*a^3*x)/(32*c^3*(1 + a^2*x^2)) + (3*a^2*ArcTan[a*x])/(32*c^3) - ArcTan[a*x]/(2*c^3*x^2) - (a^2*ArcTan[a*x])/(4*c^3*(1 + a^2*x^2)^2) - (a^2*ArcTan[a*x])/(c^3*(1 + a^2*x^2)) + (((3*I)/2)*a^2*ArcTan[a*x]^2)/c^3 - (3*a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^3 + (((3*I)/2)*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3","A",28,11,20,0.5500,1,"{4966, 4918, 4852, 325, 203, 4924, 4868, 2447, 4930, 199, 205}"
199,1,183,0,0.6899698,"\int \frac{\tan ^{-1}(a x)}{x^4 \left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^3),x]","\frac{11 a^3}{16 c^3 \left(a^2 x^2+1\right)}+\frac{a^3}{16 c^3 \left(a^2 x^2+1\right)^2}+\frac{5 a^3 \log \left(a^2 x^2+1\right)}{3 c^3}+\frac{11 a^4 x \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)}+\frac{a^4 x \tan ^{-1}(a x)}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{10 a^3 \log (x)}{3 c^3}+\frac{35 a^3 \tan ^{-1}(a x)^2}{16 c^3}+\frac{3 a^2 \tan ^{-1}(a x)}{c^3 x}-\frac{a}{6 c^3 x^2}-\frac{\tan ^{-1}(a x)}{3 c^3 x^3}","\frac{11 a^3}{16 c^3 \left(a^2 x^2+1\right)}+\frac{a^3}{16 c^3 \left(a^2 x^2+1\right)^2}+\frac{5 a^3 \log \left(a^2 x^2+1\right)}{3 c^3}+\frac{11 a^4 x \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)}+\frac{a^4 x \tan ^{-1}(a x)}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{10 a^3 \log (x)}{3 c^3}+\frac{35 a^3 \tan ^{-1}(a x)^2}{16 c^3}+\frac{3 a^2 \tan ^{-1}(a x)}{c^3 x}-\frac{a}{6 c^3 x^2}-\frac{\tan ^{-1}(a x)}{3 c^3 x^3}",1,"-a/(6*c^3*x^2) + a^3/(16*c^3*(1 + a^2*x^2)^2) + (11*a^3)/(16*c^3*(1 + a^2*x^2)) - ArcTan[a*x]/(3*c^3*x^3) + (3*a^2*ArcTan[a*x])/(c^3*x) + (a^4*x*ArcTan[a*x])/(4*c^3*(1 + a^2*x^2)^2) + (11*a^4*x*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)) + (35*a^3*ArcTan[a*x]^2)/(16*c^3) - (10*a^3*Log[x])/(3*c^3) + (5*a^3*Log[1 + a^2*x^2])/(3*c^3)","A",38,12,20,0.6000,1,"{4966, 4918, 4852, 266, 44, 36, 29, 31, 4884, 4892, 261, 4896}"
200,1,160,0,0.271102,"\int x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx","Int[x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x],x]","-\frac{x^3 \sqrt{a^2 c x^2+c}}{20 a}+\frac{x \sqrt{a^2 c x^2+c}}{24 a^3}+\frac{1}{5} x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{15 a^2}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{15 a^4}+\frac{11 \sqrt{c} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{120 a^4}","-\frac{x^3 \sqrt{a^2 c x^2+c}}{20 a}+\frac{x \sqrt{a^2 c x^2+c}}{24 a^3}+\frac{1}{5} x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{15 a^2}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{15 a^4}+\frac{11 \sqrt{c} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{120 a^4}",1,"(x*Sqrt[c + a^2*c*x^2])/(24*a^3) - (x^3*Sqrt[c + a^2*c*x^2])/(20*a) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(15*a^4) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(15*a^2) + (x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/5 + (11*Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(120*a^4)","A",12,6,22,0.2727,1,"{4946, 4952, 321, 217, 206, 4930}"
201,1,298,0,0.2696684,"\int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx","Int[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x],x]","-\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{\left(a^2 c x^2+c\right)^{3/2}}{12 a^3 c}+\frac{\sqrt{a^2 c x^2+c}}{8 a^3}+\frac{1}{4} x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{8 a^2}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}","-\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{\left(a^2 c x^2+c\right)^{3/2}}{12 a^3 c}+\frac{\sqrt{a^2 c x^2+c}}{8 a^3}+\frac{1}{4} x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{8 a^2}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}",1,"Sqrt[c + a^2*c*x^2]/(8*a^3) - (c + a^2*c*x^2)^(3/2)/(12*a^3*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(8*a^2) + (x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/4 + ((I/4)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - ((I/8)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + ((I/8)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2])","A",8,7,22,0.3182,1,"{4946, 4952, 261, 4890, 4886, 266, 43}"
202,1,86,0,0.0605705,"\int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx","Int[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x],x]","-\frac{x \sqrt{a^2 c x^2+c}}{6 a}+\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{3 a^2 c}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{6 a^2}","-\frac{x \sqrt{a^2 c x^2+c}}{6 a}+\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{3 a^2 c}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{6 a^2}",1,"-(x*Sqrt[c + a^2*c*x^2])/(6*a) + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(3*a^2*c) - (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(6*a^2)","A",4,4,20,0.2000,1,"{4930, 195, 217, 206}"
203,1,244,0,0.0932314,"\int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx","Int[Sqrt[c + a^2*c*x^2]*ArcTan[a*x],x]","\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a \sqrt{a^2 c x^2+c}}-\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c}}{2 a}-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a \sqrt{a^2 c x^2+c}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)","\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a \sqrt{a^2 c x^2+c}}-\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c}}{2 a}-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a \sqrt{a^2 c x^2+c}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)",1,"-Sqrt[c + a^2*c*x^2]/(2*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/2 - (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + ((I/2)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - ((I/2)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2])","A",3,3,19,0.1579,1,"{4878, 4890, 4886}"
204,1,229,0,0.2209833,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x} \, dx","Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x,x]","\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\sqrt{c} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}","\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\sqrt{c} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}",1,"Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + (I*c*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (I*c*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",5,5,22,0.2273,1,"{4946, 4958, 4954, 217, 206}"
205,1,242,0,0.2269256,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^2} \, dx","Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x^2,x]","\frac{i a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)","\frac{i a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-a \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)",1,"-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) - ((2*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - a*Sqrt[c]*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + (I*a*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (I*a*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",7,7,22,0.3182,1,"{4950, 4944, 266, 63, 208, 4890, 4886}"
206,1,240,0,0.3488075,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^3} \, dx","Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x^3,x]","\frac{i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c}}{2 x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}-\frac{a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}","\frac{i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c}}{2 x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}-\frac{a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}",1,"-(a*Sqrt[c + a^2*c*x^2])/(2*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*x^2) - (a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + ((I/2)*a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((I/2)*a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",6,5,22,0.2273,1,"{4946, 4962, 264, 4958, 4954}"
207,1,84,0,0.1018679,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^4} \, dx","Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x^4,x]","-\frac{a \sqrt{a^2 c x^2+c}}{6 x^2}-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{3 c x^3}-\frac{1}{6} a^3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)","-\frac{a \sqrt{a^2 c x^2+c}}{6 x^2}-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{3 c x^3}-\frac{1}{6} a^3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)",1,"-(a*Sqrt[c + a^2*c*x^2])/(6*x^2) - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(3*c*x^3) - (a^3*Sqrt[c]*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/6","A",5,5,22,0.2273,1,"{4944, 266, 47, 63, 208}"
208,1,217,0,0.7646243,"\int x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x) \, dx","Int[x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x],x]","\frac{17 c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{560 a^4}-\frac{1}{42} a c x^5 \sqrt{a^2 c x^2+c}-\frac{23 c x^3 \sqrt{a^2 c x^2+c}}{840 a}+\frac{3 c x \sqrt{a^2 c x^2+c}}{112 a^3}+\frac{1}{7} a^2 c x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{8}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{35 a^2}-\frac{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{35 a^4}","\frac{17 c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{560 a^4}-\frac{1}{42} a c x^5 \sqrt{a^2 c x^2+c}-\frac{23 c x^3 \sqrt{a^2 c x^2+c}}{840 a}+\frac{3 c x \sqrt{a^2 c x^2+c}}{112 a^3}+\frac{1}{7} a^2 c x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{8}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{35 a^2}-\frac{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{35 a^4}",1,"(3*c*x*Sqrt[c + a^2*c*x^2])/(112*a^3) - (23*c*x^3*Sqrt[c + a^2*c*x^2])/(840*a) - (a*c*x^5*Sqrt[c + a^2*c*x^2])/42 - (2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(35*a^4) + (c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(35*a^2) + (8*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/35 + (a^2*c*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/7 + (17*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(560*a^4)","A",31,7,22,0.3182,1,"{4950, 4946, 4952, 321, 217, 206, 4930}"
209,1,357,0,0.7825416,"\int x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x) \, dx","Int[x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x],x]","-\frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{16 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{16 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{\left(a^2 c x^2+c\right)^{5/2}}{30 a^3 c}+\frac{\left(a^2 c x^2+c\right)^{3/2}}{72 a^3}+\frac{c \sqrt{a^2 c x^2+c}}{16 a^3}+\frac{1}{6} a^2 c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{7}{24} c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{16 a^2}","-\frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{16 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{16 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{\left(a^2 c x^2+c\right)^{5/2}}{30 a^3 c}+\frac{\left(a^2 c x^2+c\right)^{3/2}}{72 a^3}+\frac{c \sqrt{a^2 c x^2+c}}{16 a^3}+\frac{1}{6} a^2 c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{7}{24} c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{16 a^2}",1,"(c*Sqrt[c + a^2*c*x^2])/(16*a^3) + (c + a^2*c*x^2)^(3/2)/(72*a^3) - (c + a^2*c*x^2)^(5/2)/(30*a^3*c) + (c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(16*a^2) + (7*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/24 + (a^2*c*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/6 + ((I/8)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - ((I/16)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + ((I/16)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2])","A",21,8,22,0.3636,1,"{4950, 4946, 4952, 261, 4890, 4886, 266, 43}"
210,1,109,0,0.0746509,"\int x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x) \, dx","Int[x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x],x]","-\frac{3 c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{40 a^2}-\frac{x \left(a^2 c x^2+c\right)^{3/2}}{20 a}-\frac{3 c x \sqrt{a^2 c x^2+c}}{40 a}+\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)}{5 a^2 c}","-\frac{3 c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{40 a^2}-\frac{x \left(a^2 c x^2+c\right)^{3/2}}{20 a}-\frac{3 c x \sqrt{a^2 c x^2+c}}{40 a}+\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)}{5 a^2 c}",1,"(-3*c*x*Sqrt[c + a^2*c*x^2])/(40*a) - (x*(c + a^2*c*x^2)^(3/2))/(20*a) + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(5*a^2*c) - (3*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(40*a^2)","A",5,4,20,0.2000,1,"{4930, 195, 217, 206}"
211,1,298,0,0.1382013,"\int \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x) \, dx","Int[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x],x]","\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 c \sqrt{a^2 c x^2+c}}{8 a}-\frac{\left(a^2 c x^2+c\right)^{3/2}}{12 a}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{1}{4} x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)","\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 c \sqrt{a^2 c x^2+c}}{8 a}-\frac{\left(a^2 c x^2+c\right)^{3/2}}{12 a}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{1}{4} x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)",1,"(-3*c*Sqrt[c + a^2*c*x^2])/(8*a) - (c + a^2*c*x^2)^(3/2)/(12*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/4 - (((3*I)/4)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (((3*I)/8)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (((3*I)/8)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2])","A",4,3,19,0.1579,1,"{4878, 4890, 4886}"
212,1,281,0,0.3760183,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)}{x} \, dx","Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x,x]","\frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{7}{6} c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{1}{6} a c x \sqrt{a^2 c x^2+c}+c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{1}{3} \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)","\frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{7}{6} c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{1}{6} a c x \sqrt{a^2 c x^2+c}+c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{1}{3} \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)",1,"-(a*c*x*Sqrt[c + a^2*c*x^2])/6 + c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/3 - (2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (7*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/6 + (I*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (I*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",10,8,22,0.3636,1,"{4950, 4946, 4958, 4954, 217, 206, 4930, 195}"
213,1,300,0,0.4220008,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)}{x^2} \, dx","Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x^2,x]","\frac{3 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{3 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{3 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-a c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{1}{2} a c \sqrt{a^2 c x^2+c}+\frac{1}{2} a^2 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}","\frac{3 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{3 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{3 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-a c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{1}{2} a c \sqrt{a^2 c x^2+c}+\frac{1}{2} a^2 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}",1,"-(a*c*Sqrt[c + a^2*c*x^2])/2 - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x + (a^2*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/2 - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - a*c^(3/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + (((3*I)/2)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (((3*I)/2)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",11,8,22,0.3636,1,"{4950, 4944, 266, 63, 208, 4890, 4886, 4878}"
214,1,304,0,0.6417784,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)}{x^3} \, dx","Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x^3,x]","\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-a^2 c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a c \sqrt{a^2 c x^2+c}}{2 x}+a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}","\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-a^2 c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a c \sqrt{a^2 c x^2+c}}{2 x}+a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}",1,"-(a*c*Sqrt[c + a^2*c*x^2])/(2*x) + a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*x^2) - (3*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - a^2*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + (((3*I)/2)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (((3*I)/2)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",12,8,22,0.3636,1,"{4950, 4946, 4962, 264, 4958, 4954, 217, 206}"
215,1,310,0,0.434311,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)}{x^4} \, dx","Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x^4,x]","\frac{i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{7}{6} a^3 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{a c \sqrt{a^2 c x^2+c}}{6 x^2}-\frac{a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{3 x^3}","\frac{i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{7}{6} a^3 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{a c \sqrt{a^2 c x^2+c}}{6 x^2}-\frac{a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{3 x^3}",1,"-(a*c*Sqrt[c + a^2*c*x^2])/(6*x^2) - (a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(3*x^3) - ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (7*a^3*c^(3/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/6 + (I*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (I*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",13,8,22,0.3636,1,"{4950, 4944, 266, 47, 63, 208, 4890, 4886}"
216,1,289,0,1.9768125,"\int x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x) \, dx","Int[x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x],x]","-\frac{1}{72} a^3 c^2 x^7 \sqrt{a^2 c x^2+c}-\frac{103 a c^2 x^5 \sqrt{a^2 c x^2+c}}{3024}-\frac{205 c^2 x^3 \sqrt{a^2 c x^2+c}}{12096 a}+\frac{47 c^2 x \sqrt{a^2 c x^2+c}}{2688 a^3}+\frac{1}{9} a^4 c^2 x^8 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{19}{63} a^2 c^2 x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{5}{21} c^2 x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{c^2 x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{63 a^2}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{63 a^4}+\frac{115 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{8064 a^4}","-\frac{1}{72} a^3 c^2 x^7 \sqrt{a^2 c x^2+c}-\frac{103 a c^2 x^5 \sqrt{a^2 c x^2+c}}{3024}-\frac{205 c^2 x^3 \sqrt{a^2 c x^2+c}}{12096 a}+\frac{47 c^2 x \sqrt{a^2 c x^2+c}}{2688 a^3}+\frac{1}{9} a^4 c^2 x^8 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{19}{63} a^2 c^2 x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{5}{21} c^2 x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{c^2 x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{63 a^2}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{63 a^4}+\frac{115 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{8064 a^4}",1,"(47*c^2*x*Sqrt[c + a^2*c*x^2])/(2688*a^3) - (205*c^2*x^3*Sqrt[c + a^2*c*x^2])/(12096*a) - (103*a*c^2*x^5*Sqrt[c + a^2*c*x^2])/3024 - (a^3*c^2*x^7*Sqrt[c + a^2*c*x^2])/72 - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(63*a^4) + (c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(63*a^2) + (5*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/21 + (19*a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/63 + (a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/9 + (115*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(8064*a^4)","A",76,7,22,0.3182,1,"{4950, 4946, 4952, 321, 217, 206, 4930}"
217,1,418,0,2.031113,"\int x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x) \, dx","Int[x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x],x]","-\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{128 a^3 \sqrt{a^2 c x^2+c}}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{128 a^3 \sqrt{a^2 c x^2+c}}+\frac{5 c^2 \sqrt{a^2 c x^2+c}}{128 a^3}+\frac{1}{8} a^4 c^2 x^7 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{17}{48} a^2 c^2 x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{59}{192} c^2 x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{5 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{128 a^2}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}-\frac{\left(a^2 c x^2+c\right)^{7/2}}{56 a^3 c}+\frac{\left(a^2 c x^2+c\right)^{5/2}}{240 a^3}+\frac{5 c \left(a^2 c x^2+c\right)^{3/2}}{576 a^3}","-\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{128 a^3 \sqrt{a^2 c x^2+c}}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{128 a^3 \sqrt{a^2 c x^2+c}}+\frac{5 c^2 \sqrt{a^2 c x^2+c}}{128 a^3}+\frac{1}{8} a^4 c^2 x^7 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{17}{48} a^2 c^2 x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{59}{192} c^2 x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{5 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{128 a^2}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}-\frac{\left(a^2 c x^2+c\right)^{7/2}}{56 a^3 c}+\frac{\left(a^2 c x^2+c\right)^{5/2}}{240 a^3}+\frac{5 c \left(a^2 c x^2+c\right)^{3/2}}{576 a^3}",1,"(5*c^2*Sqrt[c + a^2*c*x^2])/(128*a^3) + (5*c*(c + a^2*c*x^2)^(3/2))/(576*a^3) + (c + a^2*c*x^2)^(5/2)/(240*a^3) - (c + a^2*c*x^2)^(7/2)/(56*a^3*c) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(128*a^2) + (59*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/192 + (17*a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/48 + (a^4*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/8 + (((5*I)/64)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - (((5*I)/128)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (((5*I)/128)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2])","A",51,8,22,0.3636,1,"{4950, 4946, 4952, 261, 4890, 4886, 266, 43}"
218,1,134,0,0.0853006,"\int x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x) \, dx","Int[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x],x]","-\frac{5 c^2 x \sqrt{a^2 c x^2+c}}{112 a}-\frac{5 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{112 a^2}-\frac{x \left(a^2 c x^2+c\right)^{5/2}}{42 a}-\frac{5 c x \left(a^2 c x^2+c\right)^{3/2}}{168 a}+\frac{\left(a^2 c x^2+c\right)^{7/2} \tan ^{-1}(a x)}{7 a^2 c}","-\frac{5 c^2 x \sqrt{a^2 c x^2+c}}{112 a}-\frac{5 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{112 a^2}-\frac{x \left(a^2 c x^2+c\right)^{5/2}}{42 a}-\frac{5 c x \left(a^2 c x^2+c\right)^{3/2}}{168 a}+\frac{\left(a^2 c x^2+c\right)^{7/2} \tan ^{-1}(a x)}{7 a^2 c}",1,"(-5*c^2*x*Sqrt[c + a^2*c*x^2])/(112*a) - (5*c*x*(c + a^2*c*x^2)^(3/2))/(168*a) - (x*(c + a^2*c*x^2)^(5/2))/(42*a) + ((c + a^2*c*x^2)^(7/2)*ArcTan[a*x])/(7*a^2*c) - (5*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(112*a^2)","A",6,4,20,0.2000,1,"{4930, 195, 217, 206}"
219,1,348,0,0.1942205,"\int \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x) \, dx","Int[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x],x]","\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{16 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{16 a \sqrt{a^2 c x^2+c}}-\frac{5 c^2 \sqrt{a^2 c x^2+c}}{16 a}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 a \sqrt{a^2 c x^2+c}}+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{5 c \left(a^2 c x^2+c\right)^{3/2}}{72 a}-\frac{\left(a^2 c x^2+c\right)^{5/2}}{30 a}+\frac{5}{24} c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)+\frac{1}{6} x \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)","\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{16 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{16 a \sqrt{a^2 c x^2+c}}-\frac{5 c^2 \sqrt{a^2 c x^2+c}}{16 a}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 a \sqrt{a^2 c x^2+c}}+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{5 c \left(a^2 c x^2+c\right)^{3/2}}{72 a}-\frac{\left(a^2 c x^2+c\right)^{5/2}}{30 a}+\frac{5}{24} c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)+\frac{1}{6} x \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)",1,"(-5*c^2*Sqrt[c + a^2*c*x^2])/(16*a) - (5*c*(c + a^2*c*x^2)^(3/2))/(72*a) - (c + a^2*c*x^2)^(5/2)/(30*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/6 - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (((5*I)/16)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (((5*I)/16)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2])","A",5,3,19,0.1579,1,"{4878, 4890, 4886}"
220,1,329,0,0.550225,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)}{x} \, dx","Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x,x]","\frac{i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{29}{120} a c^2 x \sqrt{a^2 c x^2+c}+c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{149}{120} c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{1}{20} a c x \left(a^2 c x^2+c\right)^{3/2}+\frac{1}{3} c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)+\frac{1}{5} \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)","\frac{i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{29}{120} a c^2 x \sqrt{a^2 c x^2+c}+c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{149}{120} c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{1}{20} a c x \left(a^2 c x^2+c\right)^{3/2}+\frac{1}{3} c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)+\frac{1}{5} \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)",1,"(-29*a*c^2*x*Sqrt[c + a^2*c*x^2])/120 - (a*c*x*(c + a^2*c*x^2)^(3/2))/20 + c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/3 + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/5 - (2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (149*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/120 + (I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (I*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",16,8,22,0.3636,1,"{4950, 4946, 4958, 4954, 217, 206, 4930, 195}"
221,1,355,0,0.7726989,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)}{x^2} \, dx","Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x^2,x]","\frac{15 i a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 \sqrt{a^2 c x^2+c}}-\frac{15 i a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 \sqrt{a^2 c x^2+c}}-\frac{7}{8} a c^2 \sqrt{a^2 c x^2+c}-\frac{15 i a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{4 \sqrt{a^2 c x^2+c}}+\frac{7}{8} a^2 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-a c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{1}{12} a c \left(a^2 c x^2+c\right)^{3/2}+\frac{1}{4} a^2 c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)","\frac{15 i a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 \sqrt{a^2 c x^2+c}}-\frac{15 i a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{8 \sqrt{a^2 c x^2+c}}-\frac{7}{8} a c^2 \sqrt{a^2 c x^2+c}-\frac{15 i a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{4 \sqrt{a^2 c x^2+c}}+\frac{7}{8} a^2 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-a c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{1}{12} a c \left(a^2 c x^2+c\right)^{3/2}+\frac{1}{4} a^2 c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)",1,"(-7*a*c^2*Sqrt[c + a^2*c*x^2])/8 - (a*c*(c + a^2*c*x^2)^(3/2))/12 - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x + (7*a^2*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/8 + (a^2*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/4 - (((15*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - a*c^(5/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + (((15*I)/8)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (((15*I)/8)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",16,8,22,0.3636,1,"{4950, 4944, 266, 63, 208, 4890, 4886, 4878}"
222,1,364,0,1.1440202,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)}{x^3} \, dx","Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x^3,x]","\frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{1}{6} a^3 c^2 x \sqrt{a^2 c x^2+c}-\frac{a c^2 \sqrt{a^2 c x^2+c}}{2 x}+2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}-\frac{13}{6} a^2 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{5 a^2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} a^2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)","\frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{1}{6} a^3 c^2 x \sqrt{a^2 c x^2+c}-\frac{a c^2 \sqrt{a^2 c x^2+c}}{2 x}+2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}-\frac{13}{6} a^2 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{5 a^2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} a^2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)",1,"-(a*c^2*Sqrt[c + a^2*c*x^2])/(2*x) - (a^3*c^2*x*Sqrt[c + a^2*c*x^2])/6 + 2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*x^2) + (a^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/3 - (5*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (13*a^2*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/6 + (((5*I)/2)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (((5*I)/2)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",23,10,22,0.4545,1,"{4950, 4946, 4962, 264, 4958, 4954, 217, 206, 4930, 195}"
223,1,372,0,0.9751088,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)}{x^4} \, dx","Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x^4,x]","\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{1}{2} a^3 c^2 \sqrt{a^2 c x^2+c}-\frac{a c^2 \sqrt{a^2 c x^2+c}}{6 x^2}+\frac{1}{2} a^4 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{13}{6} a^3 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{3 x^3}","\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{1}{2} a^3 c^2 \sqrt{a^2 c x^2+c}-\frac{a c^2 \sqrt{a^2 c x^2+c}}{6 x^2}+\frac{1}{2} a^4 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{13}{6} a^3 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{3 x^3}",1,"-(a^3*c^2*Sqrt[c + a^2*c*x^2])/2 - (a*c^2*Sqrt[c + a^2*c*x^2])/(6*x^2) - (2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x + (a^4*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/2 - (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(3*x^3) - ((5*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (13*a^3*c^(5/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/6 + (((5*I)/2)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (((5*I)/2)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",25,9,22,0.4091,1,"{4950, 4944, 266, 47, 63, 208, 4890, 4886, 4878}"
224,1,120,0,0.1526621,"\int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^3*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","-\frac{x \sqrt{a^2 c x^2+c}}{6 a^3 c}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 a^2 c}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 a^4 c}+\frac{5 \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{6 a^4 \sqrt{c}}","-\frac{x \sqrt{a^2 c x^2+c}}{6 a^3 c}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 a^2 c}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 a^4 c}+\frac{5 \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{6 a^4 \sqrt{c}}",1,"-(x*Sqrt[c + a^2*c*x^2])/(6*a^3*c) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*a^2*c) + (5*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(6*a^4*Sqrt[c])","A",7,5,22,0.2273,1,"{4952, 321, 217, 206, 4930}"
225,1,250,0,0.1465249,"\int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^2*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a^3 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c}}{2 a^3 c}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 a^2 c}","-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a^3 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c}}{2 a^3 c}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 a^2 c}",1,"-Sqrt[c + a^2*c*x^2]/(2*a^3*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*a^2*c) + (I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - ((I/2)*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + ((I/2)*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2])","A",4,4,22,0.1818,1,"{4952, 261, 4890, 4886}"
226,1,59,0,0.0576372,"\int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^2 c}-\frac{\tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a^2 \sqrt{c}}","\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^2 c}-\frac{\tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a^2 \sqrt{c}}",1,"(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^2*c) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^2*Sqrt[c])","A",3,3,20,0.1500,1,"{4930, 217, 206}"
227,1,193,0,0.0572581,"\int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]/Sqrt[c + a^2*c*x^2],x]","\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a \sqrt{a^2 c x^2+c}}","\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a \sqrt{a^2 c x^2+c}}",1,"((-2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2])","A",2,2,19,0.1053,1,"{4890, 4886}"
228,1,177,0,0.1341268,"\int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]/(x*Sqrt[c + a^2*c*x^2]),x]","\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}","\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}",1,"(-2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",2,2,22,0.09091,1,"{4958, 4954}"
229,1,56,0,0.0926184,"\int \frac{\tan ^{-1}(a x)}{x^2 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]/(x^2*Sqrt[c + a^2*c*x^2]),x]","-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c x}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{\sqrt{c}}","-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c x}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{\sqrt{c}}",1,"-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c*x)) - (a*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/Sqrt[c]","A",4,4,22,0.1818,1,"{4944, 266, 63, 208}"
230,1,242,0,0.2206124,"\int \frac{\tan ^{-1}(a x)}{x^3 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]/(x^3*Sqrt[c + a^2*c*x^2]),x]","-\frac{i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}+\frac{i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c}}{2 c x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 c x^2}+\frac{a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}","-\frac{i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}+\frac{i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c}}{2 c x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 c x^2}+\frac{a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}",1,"-(a*Sqrt[c + a^2*c*x^2])/(2*c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*c*x^2) + (a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - ((I/2)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] + ((I/2)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",4,4,22,0.1818,1,"{4962, 264, 4958, 4954}"
231,1,118,0,0.2003554,"\int \frac{\tan ^{-1}(a x)}{x^4 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]/(x^4*Sqrt[c + a^2*c*x^2]),x]","-\frac{a \sqrt{a^2 c x^2+c}}{6 c x^2}+\frac{2 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c x^3}+\frac{5 a^3 \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{6 \sqrt{c}}","-\frac{a \sqrt{a^2 c x^2+c}}{6 c x^2}+\frac{2 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c x^3}+\frac{5 a^3 \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{6 \sqrt{c}}",1,"-(a*Sqrt[c + a^2*c*x^2])/(6*c*x^2) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c*x) + (5*a^3*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/(6*Sqrt[c])","A",9,6,22,0.2727,1,"{4962, 266, 51, 63, 208, 4944}"
232,1,107,0,0.2020652,"\int \frac{x^3 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^3*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]","\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^4 c^2}-\frac{\tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a^4 c^{3/2}}-\frac{x}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)}{a^4 c \sqrt{a^2 c x^2+c}}","\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^4 c^2}-\frac{\tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a^4 c^{3/2}}-\frac{x}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)}{a^4 c \sqrt{a^2 c x^2+c}}",1,"-(x/(a^3*c*Sqrt[c + a^2*c*x^2])) + ArcTan[a*x]/(a^4*c*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^4*c^2) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^4*c^(3/2))","A",6,5,22,0.2273,1,"{4964, 4930, 217, 206, 191}"
233,1,251,0,0.1586159,"\int \frac{x^2 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^2*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]","\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{1}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}}","\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{1}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}}",1,"-(1/(a^3*c*Sqrt[c + a^2*c*x^2])) - (x*ArcTan[a*x])/(a^2*c*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*c*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*c*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*c*Sqrt[c + a^2*c*x^2])","A",3,3,22,0.1364,1,"{4934, 4890, 4886}"
234,1,49,0,0.0553146,"\int \frac{x \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]","\frac{x}{a c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}}","\frac{x}{a c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}}",1,"x/(a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]/(a^2*c*Sqrt[c + a^2*c*x^2])","A",2,2,20,0.1000,1,"{4930, 191}"
235,1,45,0,0.0248004,"\int \frac{\tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]/(c + a^2*c*x^2)^(3/2),x]","\frac{1}{a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}","\frac{1}{a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}",1,"1/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2])","A",1,1,19,0.05263,1,"{4894}"
236,1,229,0,0.2799531,"\int \frac{\tan ^{-1}(a x)}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]/(x*(c + a^2*c*x^2)^(3/2)),x]","\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{a x}{c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}","\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{a x}{c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}",1,"-((a*x)/(c*Sqrt[c + a^2*c*x^2])) + ArcTan[a*x]/(c*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/(c*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c*Sqrt[c + a^2*c*x^2])","A",5,5,22,0.2273,1,"{4966, 4958, 4954, 4930, 191}"
237,1,103,0,0.2018895,"\int \frac{\tan ^{-1}(a x)}{x^2 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]/(x^2*(c + a^2*c*x^2)^(3/2)),x]","-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c^2 x}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{c^{3/2}}-\frac{a}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}","-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c^2 x}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{c^{3/2}}-\frac{a}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}",1,"-(a/(c*Sqrt[c + a^2*c*x^2])) - (a^2*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c^2*x) - (a*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/c^(3/2)","A",6,6,22,0.2727,1,"{4966, 4944, 266, 63, 208, 4894}"
238,1,300,0,0.6142565,"\int \frac{\tan ^{-1}(a x)}{x^3 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]/(x^3*(c + a^2*c*x^2)^(3/2)),x]","-\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 c \sqrt{a^2 c x^2+c}}+\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 c \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c}}{2 c^2 x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 c^2 x^2}+\frac{a^3 x}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}+\frac{3 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}","-\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 c \sqrt{a^2 c x^2+c}}+\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 c \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c}}{2 c^2 x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 c^2 x^2}+\frac{a^3 x}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}+\frac{3 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}",1,"(a^3*x)/(c*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[c + a^2*c*x^2])/(2*c^2*x) - (a^2*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*c^2*x^2) + (3*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c*Sqrt[c + a^2*c*x^2]) - (((3*I)/2)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/(c*Sqrt[c + a^2*c*x^2]) + (((3*I)/2)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c*Sqrt[c + a^2*c*x^2])","A",10,7,22,0.3182,1,"{4966, 4962, 264, 4958, 4954, 4930, 191}"
239,1,165,0,0.4946927,"\int \frac{\tan ^{-1}(a x)}{x^4 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^(3/2)),x]","-\frac{a \sqrt{a^2 c x^2+c}}{6 c^2 x^2}+\frac{5 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c^2 x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c^2 x^3}+\frac{11 a^3 \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{6 c^{3/2}}+\frac{a^3}{c \sqrt{a^2 c x^2+c}}+\frac{a^4 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}","-\frac{a \sqrt{a^2 c x^2+c}}{6 c^2 x^2}+\frac{5 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c^2 x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c^2 x^3}+\frac{11 a^3 \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{6 c^{3/2}}+\frac{a^3}{c \sqrt{a^2 c x^2+c}}+\frac{a^4 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}",1,"a^3/(c*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[c + a^2*c*x^2])/(6*c^2*x^2) + (a^4*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c^2*x^3) + (5*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c^2*x) + (11*a^3*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/(6*c^(3/2))","A",16,8,22,0.3636,1,"{4966, 4962, 266, 51, 63, 208, 4944, 4894}"
240,1,170,0,0.4331264,"\int \frac{x^5 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^5*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2),x]","-\frac{5 x}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^6 c^3}+\frac{5 \tan ^{-1}(a x)}{3 a^6 c^2 \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^6 c^{5/2}}-\frac{x^3}{9 a^3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^2 \tan ^{-1}(a x)}{3 a^4 c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{5 x}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^6 c^3}+\frac{5 \tan ^{-1}(a x)}{3 a^6 c^2 \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^6 c^{5/2}}-\frac{x^3}{9 a^3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^2 \tan ^{-1}(a x)}{3 a^4 c \left(a^2 c x^2+c\right)^{3/2}}",1,"-x^3/(9*a^3*c*(c + a^2*c*x^2)^(3/2)) - (5*x)/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x])/(3*a^4*c*(c + a^2*c*x^2)^(3/2)) + (5*ArcTan[a*x])/(3*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^6*c^3) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^6*c^(5/2))","A",10,6,22,0.2727,1,"{4964, 4930, 217, 206, 191, 4938}"
241,1,308,0,0.3699599,"\int \frac{x^4 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^4*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2),x]","\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{4}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)}{a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{1}{9 a^5 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^3 \tan ^{-1}(a x)}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}","\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{4}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)}{a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{1}{9 a^5 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^3 \tan ^{-1}(a x)}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}",1,"1/(9*a^5*c*(c + a^2*c*x^2)^(3/2)) - 4/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (x^3*ArcTan[a*x])/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (x*ArcTan[a*x])/(a^4*c^2*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^5*c^2*Sqrt[c + a^2*c*x^2])","A",8,7,22,0.3182,1,"{4964, 4934, 4890, 4886, 4944, 266, 43}"
242,1,112,0,0.1371876,"\int \frac{x^3 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^3*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2),x]","\frac{2 x}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^3}{9 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^2 \tan ^{-1}(a x)}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}","\frac{2 x}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^3}{9 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^2 \tan ^{-1}(a x)}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}",1,"x^3/(9*a*c*(c + a^2*c*x^2)^(3/2)) + (2*x)/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (x^2*ArcTan[a*x])/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (2*ArcTan[a*x])/(3*a^4*c^2*Sqrt[c + a^2*c*x^2])","A",3,3,22,0.1364,1,"{4938, 4930, 191}"
243,1,77,0,0.1128828,"\int \frac{x^2 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^2*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2),x]","\frac{1}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{1}{9 a^3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^3 \tan ^{-1}(a x)}{3 c \left(a^2 c x^2+c\right)^{3/2}}","\frac{1}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{1}{9 a^3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^3 \tan ^{-1}(a x)}{3 c \left(a^2 c x^2+c\right)^{3/2}}",1,"-1/(9*a^3*c*(c + a^2*c*x^2)^(3/2)) + 1/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (x^3*ArcTan[a*x])/(3*c*(c + a^2*c*x^2)^(3/2))","A",4,3,22,0.1364,1,"{4944, 266, 43}"
244,1,79,0,0.0606544,"\int \frac{x \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2),x]","\frac{2 x}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{x}{9 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{\tan ^{-1}(a x)}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}","\frac{2 x}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{x}{9 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{\tan ^{-1}(a x)}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}",1,"x/(9*a*c*(c + a^2*c*x^2)^(3/2)) + (2*x)/(9*a*c^2*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]/(3*a^2*c*(c + a^2*c*x^2)^(3/2))","A",3,3,20,0.1500,1,"{4930, 192, 191}"
245,1,101,0,0.0539432,"\int \frac{\tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[ArcTan[a*x]/(c + a^2*c*x^2)^(5/2),x]","\frac{2}{3 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{1}{9 a c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x \tan ^{-1}(a x)}{3 c \left(a^2 c x^2+c\right)^{3/2}}","\frac{2}{3 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{1}{9 a c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x \tan ^{-1}(a x)}{3 c \left(a^2 c x^2+c\right)^{3/2}}",1,"1/(9*a*c*(c + a^2*c*x^2)^(3/2)) + 2/(3*a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x])/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x])/(3*c^2*Sqrt[c + a^2*c*x^2])","A",2,2,19,0.1053,1,"{4896, 4894}"
246,1,279,0,0.4301694,"\int \frac{\tan ^{-1}(a x)}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[ArcTan[a*x]/(x*(c + a^2*c*x^2)^(5/2)),x]","\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{11 a x}{9 c^2 \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{a x}{9 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{\tan ^{-1}(a x)}{3 c \left(a^2 c x^2+c\right)^{3/2}}","\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{11 a x}{9 c^2 \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{a x}{9 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{\tan ^{-1}(a x)}{3 c \left(a^2 c x^2+c\right)^{3/2}}",1,"-(a*x)/(9*c*(c + a^2*c*x^2)^(3/2)) - (11*a*x)/(9*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]/(3*c*(c + a^2*c*x^2)^(3/2)) + ArcTan[a*x]/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c^2*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - (I*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c^2*Sqrt[c + a^2*c*x^2])","A",9,6,22,0.2727,1,"{4966, 4958, 4954, 4930, 191, 192}"
247,1,158,0,0.3374802,"\int \frac{\tan ^{-1}(a x)}{x^2 \left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[ArcTan[a*x]/(x^2*(c + a^2*c*x^2)^(5/2)),x]","-\frac{5 a}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 a^2 x \tan ^{-1}(a x)}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c^3 x}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{c^{5/2}}-\frac{a}{9 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{a^2 x \tan ^{-1}(a x)}{3 c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{5 a}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 a^2 x \tan ^{-1}(a x)}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c^3 x}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{c^{5/2}}-\frac{a}{9 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{a^2 x \tan ^{-1}(a x)}{3 c \left(a^2 c x^2+c\right)^{3/2}}",1,"-a/(9*c*(c + a^2*c*x^2)^(3/2)) - (5*a)/(3*c^2*Sqrt[c + a^2*c*x^2]) - (a^2*x*ArcTan[a*x])/(3*c*(c + a^2*c*x^2)^(3/2)) - (5*a^2*x*ArcTan[a*x])/(3*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c^3*x) - (a*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/c^(5/2)","A",9,7,22,0.3182,1,"{4966, 4944, 266, 63, 208, 4894, 4896}"
248,1,270,0,0.2252012,"\int x^m \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x) \, dx","Int[x^m*(c + a^2*c*x^2)^3*ArcTan[a*x],x]","-\frac{a c^3 x^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m^2+3 m+2}-\frac{3 a^3 c^3 x^{m+4} \, _2F_1\left(1,\frac{m+4}{2};\frac{m+6}{2};-a^2 x^2\right)}{m^2+7 m+12}-\frac{3 a^5 c^3 x^{m+6} \, _2F_1\left(1,\frac{m+6}{2};\frac{m+8}{2};-a^2 x^2\right)}{(m+5) (m+6)}-\frac{a^7 c^3 x^{m+8} \, _2F_1\left(1,\frac{m+8}{2};\frac{m+10}{2};-a^2 x^2\right)}{(m+7) (m+8)}+\frac{3 a^2 c^3 x^{m+3} \tan ^{-1}(a x)}{m+3}+\frac{3 a^4 c^3 x^{m+5} \tan ^{-1}(a x)}{m+5}+\frac{a^6 c^3 x^{m+7} \tan ^{-1}(a x)}{m+7}+\frac{c^3 x^{m+1} \tan ^{-1}(a x)}{m+1}","-\frac{a c^3 x^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m^2+3 m+2}-\frac{3 a^3 c^3 x^{m+4} \, _2F_1\left(1,\frac{m+4}{2};\frac{m+6}{2};-a^2 x^2\right)}{m^2+7 m+12}-\frac{3 a^5 c^3 x^{m+6} \, _2F_1\left(1,\frac{m+6}{2};\frac{m+8}{2};-a^2 x^2\right)}{(m+5) (m+6)}-\frac{a^7 c^3 x^{m+8} \, _2F_1\left(1,\frac{m+8}{2};\frac{m+10}{2};-a^2 x^2\right)}{(m+7) (m+8)}+\frac{3 a^2 c^3 x^{m+3} \tan ^{-1}(a x)}{m+3}+\frac{3 a^4 c^3 x^{m+5} \tan ^{-1}(a x)}{m+5}+\frac{a^6 c^3 x^{m+7} \tan ^{-1}(a x)}{m+7}+\frac{c^3 x^{m+1} \tan ^{-1}(a x)}{m+1}",1,"(c^3*x^(1 + m)*ArcTan[a*x])/(1 + m) + (3*a^2*c^3*x^(3 + m)*ArcTan[a*x])/(3 + m) + (3*a^4*c^3*x^(5 + m)*ArcTan[a*x])/(5 + m) + (a^6*c^3*x^(7 + m)*ArcTan[a*x])/(7 + m) - (a*c^3*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + 3*m + m^2) - (3*a^3*c^3*x^(4 + m)*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, -(a^2*x^2)])/(12 + 7*m + m^2) - (3*a^5*c^3*x^(6 + m)*Hypergeometric2F1[1, (6 + m)/2, (8 + m)/2, -(a^2*x^2)])/((5 + m)*(6 + m)) - (a^7*c^3*x^(8 + m)*Hypergeometric2F1[1, (8 + m)/2, (10 + m)/2, -(a^2*x^2)])/((7 + m)*(8 + m))","A",10,3,20,0.1500,1,"{4948, 4852, 364}"
249,1,201,0,0.1569964,"\int x^m \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x) \, dx","Int[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x],x]","-\frac{a c^2 x^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m^2+3 m+2}-\frac{2 a^3 c^2 x^{m+4} \, _2F_1\left(1,\frac{m+4}{2};\frac{m+6}{2};-a^2 x^2\right)}{m^2+7 m+12}-\frac{a^5 c^2 x^{m+6} \, _2F_1\left(1,\frac{m+6}{2};\frac{m+8}{2};-a^2 x^2\right)}{(m+5) (m+6)}+\frac{2 a^2 c^2 x^{m+3} \tan ^{-1}(a x)}{m+3}+\frac{a^4 c^2 x^{m+5} \tan ^{-1}(a x)}{m+5}+\frac{c^2 x^{m+1} \tan ^{-1}(a x)}{m+1}","-\frac{a c^2 x^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m^2+3 m+2}-\frac{2 a^3 c^2 x^{m+4} \, _2F_1\left(1,\frac{m+4}{2};\frac{m+6}{2};-a^2 x^2\right)}{m^2+7 m+12}-\frac{a^5 c^2 x^{m+6} \, _2F_1\left(1,\frac{m+6}{2};\frac{m+8}{2};-a^2 x^2\right)}{(m+5) (m+6)}+\frac{2 a^2 c^2 x^{m+3} \tan ^{-1}(a x)}{m+3}+\frac{a^4 c^2 x^{m+5} \tan ^{-1}(a x)}{m+5}+\frac{c^2 x^{m+1} \tan ^{-1}(a x)}{m+1}",1,"(c^2*x^(1 + m)*ArcTan[a*x])/(1 + m) + (2*a^2*c^2*x^(3 + m)*ArcTan[a*x])/(3 + m) + (a^4*c^2*x^(5 + m)*ArcTan[a*x])/(5 + m) - (a*c^2*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + 3*m + m^2) - (2*a^3*c^2*x^(4 + m)*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, -(a^2*x^2)])/(12 + 7*m + m^2) - (a^5*c^2*x^(6 + m)*Hypergeometric2F1[1, (6 + m)/2, (8 + m)/2, -(a^2*x^2)])/((5 + m)*(6 + m))","A",8,3,20,0.1500,1,"{4948, 4852, 364}"
250,1,124,0,0.0758609,"\int x^m \left(c+a^2 c x^2\right) \tan ^{-1}(a x) \, dx","Int[x^m*(c + a^2*c*x^2)*ArcTan[a*x],x]","-\frac{a c x^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m^2+3 m+2}-\frac{a^3 c x^{m+4} \, _2F_1\left(1,\frac{m+4}{2};\frac{m+6}{2};-a^2 x^2\right)}{m^2+7 m+12}+\frac{a^2 c x^{m+3} \tan ^{-1}(a x)}{m+3}+\frac{c x^{m+1} \tan ^{-1}(a x)}{m+1}","-\frac{a c x^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m^2+3 m+2}-\frac{a^3 c x^{m+4} \, _2F_1\left(1,\frac{m+4}{2};\frac{m+6}{2};-a^2 x^2\right)}{m^2+7 m+12}+\frac{a^2 c x^{m+3} \tan ^{-1}(a x)}{m+3}+\frac{c x^{m+1} \tan ^{-1}(a x)}{m+1}",1,"(c*x^(1 + m)*ArcTan[a*x])/(1 + m) + (a^2*c*x^(3 + m)*ArcTan[a*x])/(3 + m) - (a*c*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + 3*m + m^2) - (a^3*c*x^(4 + m)*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, -(a^2*x^2)])/(12 + 7*m + m^2)","A",5,3,18,0.1667,1,"{4950, 4852, 364}"
251,0,0,0,0.0479666,"\int \frac{x^m \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx","Int[(x^m*ArcTan[a*x])/(c + a^2*c*x^2),x]","\int \frac{x^m \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)}{a^2 c x^2+c},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x])/(c + a^2*c*x^2), x]","A",0,0,0,0,-1,"{}"
252,0,0,0,0.0498953,"\int \frac{x^m \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^m*ArcTan[a*x])/(c + a^2*c*x^2)^2,x]","\int \frac{x^m \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)}{\left(a^2 c x^2+c\right)^2},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x])/(c + a^2*c*x^2)^2, x]","A",0,0,0,0,-1,"{}"
253,0,0,0,0.0855001,"\int x^m \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x) \, dx","Int[x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x],x]","\int x^m \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x) \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x),x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
254,0,0,0,0.0812673,"\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x) \, dx","Int[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x],x]","\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x) \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x),x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
255,0,0,0,0.1703203,"\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx","Int[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x],x]","\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx","\frac{c \text{Int}\left(\frac{x^m \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}},x\right)}{m+2}-\frac{a x^{m+2} \sqrt{a^2 c x^2+c} \, _2F_1\left(1,\frac{m+3}{2};\frac{m+4}{2};-a^2 x^2\right)}{(m+2)^2}+\frac{x^{m+1} \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{m+2}",0,"(x^(1 + m)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2 + m) - (a*c*x^(2 + m)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/((2 + m)^2*Sqrt[c + a^2*c*x^2]) + (c*Defer[Int][(x^m*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x])/(2 + m)","A",0,0,0,0,-1,"{}"
256,0,0,0,0.0726201,"\int \frac{x^m \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^m*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","\int \frac{x^m \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x]","A",0,0,0,0,-1,"{}"
257,0,0,0,0.0843549,"\int \frac{x^m \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^m*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]","\int \frac{x^m \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)}{\left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x]","A",0,0,0,0,-1,"{}"
258,1,124,0,0.4251792,"\int x^3 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2 \, dx","Int[x^3*(c + a^2*c*x^2)*ArcTan[a*x]^2,x]","-\frac{c x^2}{180 a^2}-\frac{7 c \log \left(a^2 x^2+1\right)}{90 a^4}+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^2+\frac{c x \tan ^{-1}(a x)}{6 a^3}-\frac{c \tan ^{-1}(a x)^2}{12 a^4}-\frac{1}{15} a c x^5 \tan ^{-1}(a x)+\frac{1}{4} c x^4 \tan ^{-1}(a x)^2-\frac{c x^3 \tan ^{-1}(a x)}{18 a}+\frac{c x^4}{60}","-\frac{c x^2}{180 a^2}-\frac{7 c \log \left(a^2 x^2+1\right)}{90 a^4}+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^2+\frac{c x \tan ^{-1}(a x)}{6 a^3}-\frac{c \tan ^{-1}(a x)^2}{12 a^4}-\frac{1}{15} a c x^5 \tan ^{-1}(a x)+\frac{1}{4} c x^4 \tan ^{-1}(a x)^2-\frac{c x^3 \tan ^{-1}(a x)}{18 a}+\frac{c x^4}{60}",1,"-(c*x^2)/(180*a^2) + (c*x^4)/60 + (c*x*ArcTan[a*x])/(6*a^3) - (c*x^3*ArcTan[a*x])/(18*a) - (a*c*x^5*ArcTan[a*x])/15 - (c*ArcTan[a*x]^2)/(12*a^4) + (c*x^4*ArcTan[a*x]^2)/4 + (a^2*c*x^6*ArcTan[a*x]^2)/6 - (7*c*Log[1 + a^2*x^2])/(90*a^4)","A",26,8,20,0.4000,1,"{4950, 4852, 4916, 266, 43, 4846, 260, 4884}"
259,1,156,0,0.4094196,"\int x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2 \, dx","Int[x^2*(c + a^2*c*x^2)*ArcTan[a*x]^2,x]","-\frac{2 i c \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{15 a^3}+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^2+\frac{c x}{30 a^2}-\frac{2 i c \tan ^{-1}(a x)^2}{15 a^3}-\frac{c \tan ^{-1}(a x)}{30 a^3}-\frac{4 c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{15 a^3}-\frac{1}{10} a c x^4 \tan ^{-1}(a x)+\frac{1}{3} c x^3 \tan ^{-1}(a x)^2-\frac{2 c x^2 \tan ^{-1}(a x)}{15 a}+\frac{c x^3}{30}","-\frac{2 i c \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{15 a^3}+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^2+\frac{c x}{30 a^2}-\frac{2 i c \tan ^{-1}(a x)^2}{15 a^3}-\frac{c \tan ^{-1}(a x)}{30 a^3}-\frac{4 c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{15 a^3}-\frac{1}{10} a c x^4 \tan ^{-1}(a x)+\frac{1}{3} c x^3 \tan ^{-1}(a x)^2-\frac{2 c x^2 \tan ^{-1}(a x)}{15 a}+\frac{c x^3}{30}",1,"(c*x)/(30*a^2) + (c*x^3)/30 - (c*ArcTan[a*x])/(30*a^3) - (2*c*x^2*ArcTan[a*x])/(15*a) - (a*c*x^4*ArcTan[a*x])/10 - (((2*I)/15)*c*ArcTan[a*x]^2)/a^3 + (c*x^3*ArcTan[a*x]^2)/3 + (a^2*c*x^5*ArcTan[a*x]^2)/5 - (4*c*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a^3) - (((2*I)/15)*c*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3","A",24,10,20,0.5000,1,"{4950, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 302}"
260,1,96,0,0.0525261,"\int x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2 \, dx","Int[x*(c + a^2*c*x^2)*ArcTan[a*x]^2,x]","\frac{c \left(a^2 x^2+1\right)}{12 a^2}+\frac{c \log \left(a^2 x^2+1\right)}{6 a^2}+\frac{c \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}{4 a^2}-\frac{c x \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{6 a}-\frac{c x \tan ^{-1}(a x)}{3 a}","\frac{c \left(a^2 x^2+1\right)}{12 a^2}+\frac{c \log \left(a^2 x^2+1\right)}{6 a^2}+\frac{c \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}{4 a^2}-\frac{c x \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{6 a}-\frac{c x \tan ^{-1}(a x)}{3 a}",1,"(c*(1 + a^2*x^2))/(12*a^2) - (c*x*ArcTan[a*x])/(3*a) - (c*x*(1 + a^2*x^2)*ArcTan[a*x])/(6*a) + (c*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(4*a^2) + (c*Log[1 + a^2*x^2])/(6*a^2)","A",4,4,18,0.2222,1,"{4930, 4878, 4846, 260}"
261,1,128,0,0.0960557,"\int \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2 \, dx","Int[(c + a^2*c*x^2)*ArcTan[a*x]^2,x]","\frac{2 i c \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{3 a}+\frac{1}{3} c x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2-\frac{c \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{3 a}+\frac{2 i c \tan ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^2+\frac{4 c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{3 a}+\frac{c x}{3}","\frac{2 i c \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{3 a}+\frac{1}{3} c x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2-\frac{c \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{3 a}+\frac{2 i c \tan ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^2+\frac{4 c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{3 a}+\frac{c x}{3}",1,"(c*x)/3 - (c*(1 + a^2*x^2)*ArcTan[a*x])/(3*a) + (((2*I)/3)*c*ArcTan[a*x]^2)/a + (2*c*x*ArcTan[a*x]^2)/3 + (c*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/3 + (4*c*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(3*a) + (((2*I)/3)*c*PolyLog[2, 1 - 2/(1 + I*a*x)])/a","A",7,7,17,0.4118,1,"{4880, 4846, 4920, 4854, 2402, 2315, 8}"
262,1,169,0,0.3112468,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2}{x} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x]^2)/x,x]","-\frac{1}{2} c \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{1}{2} c \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-i c \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+i c \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)+\frac{1}{2} c \log \left(a^2 x^2+1\right)+\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)^2+\frac{1}{2} c \tan ^{-1}(a x)^2-a c x \tan ^{-1}(a x)+2 c \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)","-\frac{1}{2} c \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{1}{2} c \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-i c \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+i c \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)+\frac{1}{2} c \log \left(a^2 x^2+1\right)+\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)^2+\frac{1}{2} c \tan ^{-1}(a x)^2-a c x \tan ^{-1}(a x)+2 c \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)",1,"-(a*c*x*ArcTan[a*x]) + (c*ArcTan[a*x]^2)/2 + (a^2*c*x^2*ArcTan[a*x]^2)/2 + 2*c*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + (c*Log[1 + a^2*x^2])/2 - I*c*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + I*c*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - (c*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (c*PolyLog[3, -1 + 2/(1 + I*a*x)])/2","A",12,10,20,0.5000,1,"{4950, 4850, 4988, 4884, 4994, 6610, 4852, 4916, 4846, 260}"
263,1,113,0,0.2227044,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2}{x^2} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x]^2)/x^2,x]","-i a c \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+i a c \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+a^2 c x \tan ^{-1}(a x)^2-\frac{c \tan ^{-1}(a x)^2}{x}+2 a c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+2 a c \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)","-i a c \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+i a c \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+a^2 c x \tan ^{-1}(a x)^2-\frac{c \tan ^{-1}(a x)^2}{x}+2 a c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+2 a c \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)",1,"-((c*ArcTan[a*x]^2)/x) + a^2*c*x*ArcTan[a*x]^2 + 2*a*c*ArcTan[a*x]*Log[2/(1 + I*a*x)] + 2*a*c*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a*c*PolyLog[2, -1 + 2/(1 - I*a*x)] + I*a*c*PolyLog[2, 1 - 2/(1 + I*a*x)]","A",10,10,20,0.5000,1,"{4950, 4852, 4924, 4868, 2447, 4846, 4920, 4854, 2402, 2315}"
264,1,196,0,0.3253557,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2}{x^3} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x]^2)/x^3,x]","-\frac{1}{2} a^2 c \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{1}{2} a^2 c \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-i a^2 c \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+i a^2 c \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)-\frac{1}{2} a^2 c \log \left(a^2 x^2+1\right)+a^2 c \log (x)-\frac{1}{2} a^2 c \tan ^{-1}(a x)^2+2 a^2 c \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)-\frac{c \tan ^{-1}(a x)^2}{2 x^2}-\frac{a c \tan ^{-1}(a x)}{x}","-\frac{1}{2} a^2 c \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{1}{2} a^2 c \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-i a^2 c \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+i a^2 c \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)-\frac{1}{2} a^2 c \log \left(a^2 x^2+1\right)+a^2 c \log (x)-\frac{1}{2} a^2 c \tan ^{-1}(a x)^2+2 a^2 c \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)-\frac{c \tan ^{-1}(a x)^2}{2 x^2}-\frac{a c \tan ^{-1}(a x)}{x}",1,"-((a*c*ArcTan[a*x])/x) - (a^2*c*ArcTan[a*x]^2)/2 - (c*ArcTan[a*x]^2)/(2*x^2) + 2*a^2*c*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + a^2*c*Log[x] - (a^2*c*Log[1 + a^2*x^2])/2 - I*a^2*c*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + I*a^2*c*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - (a^2*c*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (a^2*c*PolyLog[3, -1 + 2/(1 + I*a*x)])/2","A",15,12,20,0.6000,1,"{4950, 4852, 4918, 266, 36, 29, 31, 4884, 4850, 4988, 4994, 6610}"
265,1,135,0,0.3121382,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2}{x^4} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x]^2)/x^4,x]","-\frac{2}{3} i a^3 c \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-\frac{a^2 c}{3 x}-\frac{2}{3} i a^3 c \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c \tan ^{-1}(a x)-\frac{a^2 c \tan ^{-1}(a x)^2}{x}+\frac{4}{3} a^3 c \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)-\frac{a c \tan ^{-1}(a x)}{3 x^2}-\frac{c \tan ^{-1}(a x)^2}{3 x^3}","-\frac{2}{3} i a^3 c \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-\frac{a^2 c}{3 x}-\frac{2}{3} i a^3 c \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c \tan ^{-1}(a x)-\frac{a^2 c \tan ^{-1}(a x)^2}{x}+\frac{4}{3} a^3 c \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)-\frac{a c \tan ^{-1}(a x)}{3 x^2}-\frac{c \tan ^{-1}(a x)^2}{3 x^3}",1,"-(a^2*c)/(3*x) - (a^3*c*ArcTan[a*x])/3 - (a*c*ArcTan[a*x])/(3*x^2) - ((2*I)/3)*a^3*c*ArcTan[a*x]^2 - (c*ArcTan[a*x]^2)/(3*x^3) - (a^2*c*ArcTan[a*x]^2)/x + (4*a^3*c*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/3 - ((2*I)/3)*a^3*c*PolyLog[2, -1 + 2/(1 - I*a*x)]","A",13,8,20,0.4000,1,"{4950, 4852, 4918, 325, 203, 4924, 4868, 2447}"
266,1,191,0,0.7886428,"\int x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2 \, dx","Int[x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]","\frac{1}{168} a^2 c^2 x^6-\frac{5 c^2 x^2}{504 a^2}-\frac{2 c^2 \log \left(a^2 x^2+1\right)}{63 a^4}+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac{1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac{c^2 x \tan ^{-1}(a x)}{12 a^3}-\frac{c^2 \tan ^{-1}(a x)^2}{24 a^4}-\frac{1}{12} a c^2 x^5 \tan ^{-1}(a x)+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^2-\frac{c^2 x^3 \tan ^{-1}(a x)}{36 a}+\frac{c^2 x^4}{84}","\frac{1}{168} a^2 c^2 x^6-\frac{5 c^2 x^2}{504 a^2}-\frac{2 c^2 \log \left(a^2 x^2+1\right)}{63 a^4}+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac{1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac{c^2 x \tan ^{-1}(a x)}{12 a^3}-\frac{c^2 \tan ^{-1}(a x)^2}{24 a^4}-\frac{1}{12} a c^2 x^5 \tan ^{-1}(a x)+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^2-\frac{c^2 x^3 \tan ^{-1}(a x)}{36 a}+\frac{c^2 x^4}{84}",1,"(-5*c^2*x^2)/(504*a^2) + (c^2*x^4)/84 + (a^2*c^2*x^6)/168 + (c^2*x*ArcTan[a*x])/(12*a^3) - (c^2*x^3*ArcTan[a*x])/(36*a) - (a*c^2*x^5*ArcTan[a*x])/12 - (a^3*c^2*x^7*ArcTan[a*x])/28 - (c^2*ArcTan[a*x]^2)/(24*a^4) + (c^2*x^4*ArcTan[a*x]^2)/4 + (a^2*c^2*x^6*ArcTan[a*x]^2)/3 + (a^4*c^2*x^8*ArcTan[a*x]^2)/8 - (2*c^2*Log[1 + a^2*x^2])/(63*a^4)","A",47,8,22,0.3636,1,"{4948, 4852, 4916, 266, 43, 4846, 260, 4884}"
267,1,225,0,0.7524737,"\int x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2 \, dx","Int[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]","-\frac{8 i c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{105 a^3}+\frac{1}{105} a^2 c^2 x^5+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2-\frac{c^2 x}{210 a^2}-\frac{8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac{c^2 \tan ^{-1}(a x)}{210 a^3}-\frac{16 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{105 a^3}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2-\frac{8 c^2 x^2 \tan ^{-1}(a x)}{105 a}+\frac{17 c^2 x^3}{630}","-\frac{8 i c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{105 a^3}+\frac{1}{105} a^2 c^2 x^5+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2-\frac{c^2 x}{210 a^2}-\frac{8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac{c^2 \tan ^{-1}(a x)}{210 a^3}-\frac{16 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{105 a^3}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2-\frac{8 c^2 x^2 \tan ^{-1}(a x)}{105 a}+\frac{17 c^2 x^3}{630}",1,"-(c^2*x)/(210*a^2) + (17*c^2*x^3)/630 + (a^2*c^2*x^5)/105 + (c^2*ArcTan[a*x])/(210*a^3) - (8*c^2*x^2*ArcTan[a*x])/(105*a) - (9*a*c^2*x^4*ArcTan[a*x])/70 - (a^3*c^2*x^6*ArcTan[a*x])/21 - (((8*I)/105)*c^2*ArcTan[a*x]^2)/a^3 + (c^2*x^3*ArcTan[a*x]^2)/3 + (2*a^2*c^2*x^5*ArcTan[a*x]^2)/5 + (a^4*c^2*x^7*ArcTan[a*x]^2)/7 - (16*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(105*a^3) - (((8*I)/105)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3","A",44,10,22,0.4545,1,"{4948, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 302}"
268,1,153,0,0.0930782,"\int x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2 \, dx","Int[x*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]","\frac{c^2 \left(a^2 x^2+1\right)^2}{60 a^2}+\frac{2 c^2 \left(a^2 x^2+1\right)}{45 a^2}+\frac{4 c^2 \log \left(a^2 x^2+1\right)}{45 a^2}+\frac{c^2 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^2}{6 a^2}-\frac{c^2 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}{15 a}-\frac{4 c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{45 a}-\frac{8 c^2 x \tan ^{-1}(a x)}{45 a}","\frac{c^2 \left(a^2 x^2+1\right)^2}{60 a^2}+\frac{2 c^2 \left(a^2 x^2+1\right)}{45 a^2}+\frac{4 c^2 \log \left(a^2 x^2+1\right)}{45 a^2}+\frac{c^2 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^2}{6 a^2}-\frac{c^2 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}{15 a}-\frac{4 c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{45 a}-\frac{8 c^2 x \tan ^{-1}(a x)}{45 a}",1,"(2*c^2*(1 + a^2*x^2))/(45*a^2) + (c^2*(1 + a^2*x^2)^2)/(60*a^2) - (8*c^2*x*ArcTan[a*x])/(45*a) - (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x])/(45*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x])/(15*a) + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/(6*a^2) + (4*c^2*Log[1 + a^2*x^2])/(45*a^2)","A",5,4,20,0.2000,1,"{4930, 4878, 4846, 260}"
269,1,205,0,0.138938,"\int \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2 \, dx","Int[(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]","\frac{8 i c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{15 a}+\frac{1}{30} a^2 c^2 x^3+\frac{1}{5} c^2 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2+\frac{4}{15} c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2-\frac{c^2 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}{10 a}-\frac{4 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{15 a}+\frac{8}{15} c^2 x \tan ^{-1}(a x)^2+\frac{8 i c^2 \tan ^{-1}(a x)^2}{15 a}+\frac{16 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{15 a}+\frac{11 c^2 x}{30}","\frac{8 i c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{15 a}+\frac{1}{30} a^2 c^2 x^3+\frac{1}{5} c^2 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2+\frac{4}{15} c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2-\frac{c^2 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}{10 a}-\frac{4 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{15 a}+\frac{8}{15} c^2 x \tan ^{-1}(a x)^2+\frac{8 i c^2 \tan ^{-1}(a x)^2}{15 a}+\frac{16 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{15 a}+\frac{11 c^2 x}{30}",1,"(11*c^2*x)/30 + (a^2*c^2*x^3)/30 - (4*c^2*(1 + a^2*x^2)*ArcTan[a*x])/(15*a) - (c^2*(1 + a^2*x^2)^2*ArcTan[a*x])/(10*a) + (((8*I)/15)*c^2*ArcTan[a*x]^2)/a + (8*c^2*x*ArcTan[a*x]^2)/15 + (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/5 + (16*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a) + (((8*I)/15)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a","A",9,7,19,0.3684,1,"{4880, 4846, 4920, 4854, 2402, 2315, 8}"
270,1,235,0,0.5145684,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2}{x} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x]^2)/x,x]","-\frac{1}{2} c^2 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{1}{2} c^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-i c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+i c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)+\frac{1}{12} a^2 c^2 x^2+\frac{2}{3} c^2 \log \left(a^2 x^2+1\right)+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^2-\frac{1}{6} a^3 c^2 x^3 \tan ^{-1}(a x)+a^2 c^2 x^2 \tan ^{-1}(a x)^2-\frac{3}{2} a c^2 x \tan ^{-1}(a x)+\frac{3}{4} c^2 \tan ^{-1}(a x)^2+2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)","-\frac{1}{2} c^2 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{1}{2} c^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-i c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+i c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)+\frac{1}{12} a^2 c^2 x^2+\frac{2}{3} c^2 \log \left(a^2 x^2+1\right)+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^2-\frac{1}{6} a^3 c^2 x^3 \tan ^{-1}(a x)+a^2 c^2 x^2 \tan ^{-1}(a x)^2-\frac{3}{2} a c^2 x \tan ^{-1}(a x)+\frac{3}{4} c^2 \tan ^{-1}(a x)^2+2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)",1,"(a^2*c^2*x^2)/12 - (3*a*c^2*x*ArcTan[a*x])/2 - (a^3*c^2*x^3*ArcTan[a*x])/6 + (3*c^2*ArcTan[a*x]^2)/4 + a^2*c^2*x^2*ArcTan[a*x]^2 + (a^4*c^2*x^4*ArcTan[a*x]^2)/4 + 2*c^2*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + (2*c^2*Log[1 + a^2*x^2])/3 - I*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + I*c^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - (c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (c^2*PolyLog[3, -1 + 2/(1 + I*a*x)])/2","A",23,12,22,0.5455,1,"{4948, 4850, 4988, 4884, 4994, 6610, 4852, 4916, 4846, 260, 266, 43}"
271,1,205,0,0.4230999,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2}{x^2} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x]^2)/x^2,x]","-i a c^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+\frac{5}{3} i a c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c^2 x^2 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x+2 a^2 c^2 x \tan ^{-1}(a x)^2+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^2-\frac{1}{3} a c^2 \tan ^{-1}(a x)-\frac{c^2 \tan ^{-1}(a x)^2}{x}+\frac{10}{3} a c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+2 a c^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)","-i a c^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+\frac{5}{3} i a c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c^2 x^2 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x+2 a^2 c^2 x \tan ^{-1}(a x)^2+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^2-\frac{1}{3} a c^2 \tan ^{-1}(a x)-\frac{c^2 \tan ^{-1}(a x)^2}{x}+\frac{10}{3} a c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+2 a c^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)",1,"(a^2*c^2*x)/3 - (a*c^2*ArcTan[a*x])/3 - (a^3*c^2*x^2*ArcTan[a*x])/3 + ((2*I)/3)*a*c^2*ArcTan[a*x]^2 - (c^2*ArcTan[a*x]^2)/x + 2*a^2*c^2*x*ArcTan[a*x]^2 + (a^4*c^2*x^3*ArcTan[a*x]^2)/3 + (10*a*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/3 + 2*a*c^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a*c^2*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((5*I)/3)*a*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)]","A",20,13,22,0.5909,1,"{4948, 4846, 4920, 4854, 2402, 2315, 4852, 4924, 4868, 2447, 4916, 321, 203}"
272,1,207,0,0.4523623,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2}{x^3} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x]^2)/x^3,x]","-a^2 c^2 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+a^2 c^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-2 i a^2 c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+2 i a^2 c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+a^2 c^2 \log (x)-a^3 c^2 x \tan ^{-1}(a x)+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)-\frac{c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac{a c^2 \tan ^{-1}(a x)}{x}","-a^2 c^2 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+a^2 c^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-2 i a^2 c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+2 i a^2 c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2+a^2 c^2 \log (x)-a^3 c^2 x \tan ^{-1}(a x)+4 a^2 c^2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)-\frac{c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac{a c^2 \tan ^{-1}(a x)}{x}",1,"-((a*c^2*ArcTan[a*x])/x) - a^3*c^2*x*ArcTan[a*x] - (c^2*ArcTan[a*x]^2)/(2*x^2) + (a^4*c^2*x^2*ArcTan[a*x]^2)/2 + 4*a^2*c^2*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + a^2*c^2*Log[x] - (2*I)*a^2*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (2*I)*a^2*c^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - a^2*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)] + a^2*c^2*PolyLog[3, -1 + 2/(1 + I*a*x)]","A",21,15,22,0.6818,1,"{4948, 4852, 4918, 266, 36, 29, 31, 4884, 4850, 4988, 4994, 6610, 4916, 4846, 260}"
273,1,216,0,0.4397258,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2}{x^4} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x]^2)/x^4,x]","-\frac{5}{3} i a^3 c^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+i a^3 c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)-\frac{a^2 c^2}{3 x}+a^4 c^2 x \tan ^{-1}(a x)^2-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c^2 \tan ^{-1}(a x)-\frac{2 a^2 c^2 \tan ^{-1}(a x)^2}{x}+2 a^3 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+\frac{10}{3} a^3 c^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)-\frac{a c^2 \tan ^{-1}(a x)}{3 x^2}-\frac{c^2 \tan ^{-1}(a x)^2}{3 x^3}","-\frac{5}{3} i a^3 c^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+i a^3 c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)-\frac{a^2 c^2}{3 x}+a^4 c^2 x \tan ^{-1}(a x)^2-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c^2 \tan ^{-1}(a x)-\frac{2 a^2 c^2 \tan ^{-1}(a x)^2}{x}+2 a^3 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+\frac{10}{3} a^3 c^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)-\frac{a c^2 \tan ^{-1}(a x)}{3 x^2}-\frac{c^2 \tan ^{-1}(a x)^2}{3 x^3}",1,"-(a^2*c^2)/(3*x) - (a^3*c^2*ArcTan[a*x])/3 - (a*c^2*ArcTan[a*x])/(3*x^2) - ((2*I)/3)*a^3*c^2*ArcTan[a*x]^2 - (c^2*ArcTan[a*x]^2)/(3*x^3) - (2*a^2*c^2*ArcTan[a*x]^2)/x + a^4*c^2*x*ArcTan[a*x]^2 + 2*a^3*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)] + (10*a^3*c^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/3 - ((5*I)/3)*a^3*c^2*PolyLog[2, -1 + 2/(1 - I*a*x)] + I*a^3*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)]","A",19,13,22,0.5909,1,"{4948, 4846, 4920, 4854, 2402, 2315, 4852, 4918, 325, 203, 4924, 4868, 2447}"
274,1,240,0,1.2269462,"\int x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2 \, dx","Int[x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^2,x]","\frac{1}{360} a^4 c^3 x^8+\frac{71 a^2 c^3 x^6}{7560}-\frac{107 c^3 x^2}{12600 a^2}-\frac{26 c^3 \log \left(a^2 x^2+1\right)}{1575 a^4}+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^2-\frac{1}{45} a^5 c^3 x^9 \tan ^{-1}(a x)+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^2-\frac{11}{140} a^3 c^3 x^7 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^2+\frac{c^3 x \tan ^{-1}(a x)}{20 a^3}-\frac{c^3 \tan ^{-1}(a x)^2}{40 a^4}-\frac{9}{100} a c^3 x^5 \tan ^{-1}(a x)+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^2-\frac{c^3 x^3 \tan ^{-1}(a x)}{60 a}+\frac{53 c^3 x^4}{6300}","\frac{1}{360} a^4 c^3 x^8+\frac{71 a^2 c^3 x^6}{7560}-\frac{107 c^3 x^2}{12600 a^2}-\frac{26 c^3 \log \left(a^2 x^2+1\right)}{1575 a^4}+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^2-\frac{1}{45} a^5 c^3 x^9 \tan ^{-1}(a x)+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^2-\frac{11}{140} a^3 c^3 x^7 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^2+\frac{c^3 x \tan ^{-1}(a x)}{20 a^3}-\frac{c^3 \tan ^{-1}(a x)^2}{40 a^4}-\frac{9}{100} a c^3 x^5 \tan ^{-1}(a x)+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^2-\frac{c^3 x^3 \tan ^{-1}(a x)}{60 a}+\frac{53 c^3 x^4}{6300}",1,"(-107*c^3*x^2)/(12600*a^2) + (53*c^3*x^4)/6300 + (71*a^2*c^3*x^6)/7560 + (a^4*c^3*x^8)/360 + (c^3*x*ArcTan[a*x])/(20*a^3) - (c^3*x^3*ArcTan[a*x])/(60*a) - (9*a*c^3*x^5*ArcTan[a*x])/100 - (11*a^3*c^3*x^7*ArcTan[a*x])/140 - (a^5*c^3*x^9*ArcTan[a*x])/45 - (c^3*ArcTan[a*x]^2)/(40*a^4) + (c^3*x^4*ArcTan[a*x]^2)/4 + (a^2*c^3*x^6*ArcTan[a*x]^2)/2 + (3*a^4*c^3*x^8*ArcTan[a*x]^2)/8 + (a^6*c^3*x^10*ArcTan[a*x]^2)/10 - (26*c^3*Log[1 + a^2*x^2])/(1575*a^4)","A",72,8,22,0.3636,1,"{4948, 4852, 4916, 266, 43, 4846, 260, 4884}"
275,1,274,0,1.1539998,"\int x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2 \, dx","Int[x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^2,x]","-\frac{16 i c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{315 a^3}+\frac{1}{252} a^4 c^3 x^7+\frac{59 a^2 c^3 x^5}{3780}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac{1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2-\frac{20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2-\frac{47 c^3 x}{3780 a^2}-\frac{16 i c^3 \tan ^{-1}(a x)^2}{315 a^3}+\frac{47 c^3 \tan ^{-1}(a x)}{3780 a^3}-\frac{32 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{315 a^3}-\frac{89}{630} a c^3 x^4 \tan ^{-1}(a x)+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^2-\frac{16 c^3 x^2 \tan ^{-1}(a x)}{315 a}+\frac{239 c^3 x^3}{11340}","-\frac{16 i c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{315 a^3}+\frac{1}{252} a^4 c^3 x^7+\frac{59 a^2 c^3 x^5}{3780}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac{1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2-\frac{20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2-\frac{47 c^3 x}{3780 a^2}-\frac{16 i c^3 \tan ^{-1}(a x)^2}{315 a^3}+\frac{47 c^3 \tan ^{-1}(a x)}{3780 a^3}-\frac{32 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{315 a^3}-\frac{89}{630} a c^3 x^4 \tan ^{-1}(a x)+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^2-\frac{16 c^3 x^2 \tan ^{-1}(a x)}{315 a}+\frac{239 c^3 x^3}{11340}",1,"(-47*c^3*x)/(3780*a^2) + (239*c^3*x^3)/11340 + (59*a^2*c^3*x^5)/3780 + (a^4*c^3*x^7)/252 + (47*c^3*ArcTan[a*x])/(3780*a^3) - (16*c^3*x^2*ArcTan[a*x])/(315*a) - (89*a*c^3*x^4*ArcTan[a*x])/630 - (20*a^3*c^3*x^6*ArcTan[a*x])/189 - (a^5*c^3*x^8*ArcTan[a*x])/36 - (((16*I)/315)*c^3*ArcTan[a*x]^2)/a^3 + (c^3*x^3*ArcTan[a*x]^2)/3 + (3*a^2*c^3*x^5*ArcTan[a*x]^2)/5 + (3*a^4*c^3*x^7*ArcTan[a*x]^2)/7 + (a^6*c^3*x^9*ArcTan[a*x]^2)/9 - (32*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(315*a^3) - (((16*I)/315)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3","A",68,10,22,0.4545,1,"{4948, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 302}"
276,1,200,0,0.1212944,"\int x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2 \, dx","Int[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^2,x]","\frac{c^3 \left(a^2 x^2+1\right)^3}{168 a^2}+\frac{3 c^3 \left(a^2 x^2+1\right)^2}{280 a^2}+\frac{c^3 \left(a^2 x^2+1\right)}{35 a^2}+\frac{2 c^3 \log \left(a^2 x^2+1\right)}{35 a^2}+\frac{c^3 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x)^2}{8 a^2}-\frac{c^3 x \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)}{28 a}-\frac{3 c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}{70 a}-\frac{2 c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{35 a}-\frac{4 c^3 x \tan ^{-1}(a x)}{35 a}","\frac{c^3 \left(a^2 x^2+1\right)^3}{168 a^2}+\frac{3 c^3 \left(a^2 x^2+1\right)^2}{280 a^2}+\frac{c^3 \left(a^2 x^2+1\right)}{35 a^2}+\frac{2 c^3 \log \left(a^2 x^2+1\right)}{35 a^2}+\frac{c^3 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x)^2}{8 a^2}-\frac{c^3 x \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)}{28 a}-\frac{3 c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}{70 a}-\frac{2 c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{35 a}-\frac{4 c^3 x \tan ^{-1}(a x)}{35 a}",1,"(c^3*(1 + a^2*x^2))/(35*a^2) + (3*c^3*(1 + a^2*x^2)^2)/(280*a^2) + (c^3*(1 + a^2*x^2)^3)/(168*a^2) - (4*c^3*x*ArcTan[a*x])/(35*a) - (2*c^3*x*(1 + a^2*x^2)*ArcTan[a*x])/(35*a) - (3*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x])/(70*a) - (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x])/(28*a) + (c^3*(1 + a^2*x^2)^4*ArcTan[a*x]^2)/(8*a^2) + (2*c^3*Log[1 + a^2*x^2])/(35*a^2)","A",6,4,20,0.2000,1,"{4930, 4878, 4846, 260}"
277,1,268,0,0.1841948,"\int \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2 \, dx","Int[(c + a^2*c*x^2)^3*ArcTan[a*x]^2,x]","\frac{16 i c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{35 a}+\frac{1}{105} a^4 c^3 x^5+\frac{19}{315} a^2 c^3 x^3+\frac{1}{7} c^3 x \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^2+\frac{6}{35} c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2+\frac{8}{35} c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2-\frac{c^3 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)}{21 a}-\frac{3 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}{35 a}-\frac{8 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{35 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)^2+\frac{16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac{32 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{35 a}+\frac{38 c^3 x}{105}","\frac{16 i c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{35 a}+\frac{1}{105} a^4 c^3 x^5+\frac{19}{315} a^2 c^3 x^3+\frac{1}{7} c^3 x \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^2+\frac{6}{35} c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2+\frac{8}{35} c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2-\frac{c^3 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)}{21 a}-\frac{3 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}{35 a}-\frac{8 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{35 a}+\frac{16}{35} c^3 x \tan ^{-1}(a x)^2+\frac{16 i c^3 \tan ^{-1}(a x)^2}{35 a}+\frac{32 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{35 a}+\frac{38 c^3 x}{105}",1,"(38*c^3*x)/105 + (19*a^2*c^3*x^3)/315 + (a^4*c^3*x^5)/105 - (8*c^3*(1 + a^2*x^2)*ArcTan[a*x])/(35*a) - (3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])/(35*a) - (c^3*(1 + a^2*x^2)^3*ArcTan[a*x])/(21*a) + (((16*I)/35)*c^3*ArcTan[a*x]^2)/a + (16*c^3*x*ArcTan[a*x]^2)/35 + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/7 + (32*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(35*a) + (((16*I)/35)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)])/a","A",12,8,19,0.4211,1,"{4880, 4846, 4920, 4854, 2402, 2315, 8, 194}"
278,1,287,0,0.7434138,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2}{x} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x]^2)/x,x]","-\frac{1}{2} c^3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{1}{2} c^3 \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-i c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+i c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)+\frac{1}{60} a^4 c^3 x^4+\frac{29}{180} a^2 c^3 x^2+\frac{34}{45} c^3 \log \left(a^2 x^2+1\right)+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2-\frac{1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2-\frac{7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2-\frac{11}{6} a c^3 x \tan ^{-1}(a x)+\frac{11}{12} c^3 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)","-\frac{1}{2} c^3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{1}{2} c^3 \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-i c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+i c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)+\frac{1}{60} a^4 c^3 x^4+\frac{29}{180} a^2 c^3 x^2+\frac{34}{45} c^3 \log \left(a^2 x^2+1\right)+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2-\frac{1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2-\frac{7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2-\frac{11}{6} a c^3 x \tan ^{-1}(a x)+\frac{11}{12} c^3 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)",1,"(29*a^2*c^3*x^2)/180 + (a^4*c^3*x^4)/60 - (11*a*c^3*x*ArcTan[a*x])/6 - (7*a^3*c^3*x^3*ArcTan[a*x])/18 - (a^5*c^3*x^5*ArcTan[a*x])/15 + (11*c^3*ArcTan[a*x]^2)/12 + (3*a^2*c^3*x^2*ArcTan[a*x]^2)/2 + (3*a^4*c^3*x^4*ArcTan[a*x]^2)/4 + (a^6*c^3*x^6*ArcTan[a*x]^2)/6 + 2*c^3*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + (34*c^3*Log[1 + a^2*x^2])/45 - I*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + I*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - (c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (c^3*PolyLog[3, -1 + 2/(1 + I*a*x)])/2","A",38,12,22,0.5455,1,"{4948, 4850, 4988, 4884, 4994, 6610, 4852, 4916, 4846, 260, 266, 43}"
279,1,251,0,0.6453295,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2}{x^2} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x]^2)/x^2,x]","-i a c^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+\frac{11}{5} i a c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{1}{30} a^4 c^3 x^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^2-\frac{1}{10} a^5 c^3 x^4 \tan ^{-1}(a x)+a^4 c^3 x^3 \tan ^{-1}(a x)^2-\frac{4}{5} a^3 c^3 x^2 \tan ^{-1}(a x)+\frac{7}{10} a^2 c^3 x+3 a^2 c^3 x \tan ^{-1}(a x)^2+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^2-\frac{7}{10} a c^3 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)^2}{x}+\frac{22}{5} a c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+2 a c^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)","-i a c^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+\frac{11}{5} i a c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{1}{30} a^4 c^3 x^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^2-\frac{1}{10} a^5 c^3 x^4 \tan ^{-1}(a x)+a^4 c^3 x^3 \tan ^{-1}(a x)^2-\frac{4}{5} a^3 c^3 x^2 \tan ^{-1}(a x)+\frac{7}{10} a^2 c^3 x+3 a^2 c^3 x \tan ^{-1}(a x)^2+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^2-\frac{7}{10} a c^3 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)^2}{x}+\frac{22}{5} a c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+2 a c^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)",1,"(7*a^2*c^3*x)/10 + (a^4*c^3*x^3)/30 - (7*a*c^3*ArcTan[a*x])/10 - (4*a^3*c^3*x^2*ArcTan[a*x])/5 - (a^5*c^3*x^4*ArcTan[a*x])/10 + ((6*I)/5)*a*c^3*ArcTan[a*x]^2 - (c^3*ArcTan[a*x]^2)/x + 3*a^2*c^3*x*ArcTan[a*x]^2 + a^4*c^3*x^3*ArcTan[a*x]^2 + (a^6*c^3*x^5*ArcTan[a*x]^2)/5 + (22*a*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/5 + 2*a*c^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a*c^3*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((11*I)/5)*a*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)]","A",34,14,22,0.6364,1,"{4948, 4846, 4920, 4854, 2402, 2315, 4852, 4924, 4868, 2447, 4916, 321, 203, 302}"
280,1,299,0,0.6031829,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2}{x^3} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x]^2)/x^3,x]","-\frac{3}{2} a^2 c^3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{3}{2} a^2 c^3 \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-3 i a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+3 i a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)+\frac{1}{12} a^4 c^3 x^2+\frac{2}{3} a^2 c^3 \log \left(a^2 x^2+1\right)+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^2-\frac{1}{6} a^5 c^3 x^3 \tan ^{-1}(a x)+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^2+a^2 c^3 \log (x)-\frac{5}{2} a^3 c^3 x \tan ^{-1}(a x)+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^2+6 a^2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)-\frac{c^3 \tan ^{-1}(a x)^2}{2 x^2}-\frac{a c^3 \tan ^{-1}(a x)}{x}","-\frac{3}{2} a^2 c^3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{3}{2} a^2 c^3 \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-3 i a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+3 i a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)+\frac{1}{12} a^4 c^3 x^2+\frac{2}{3} a^2 c^3 \log \left(a^2 x^2+1\right)+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^2-\frac{1}{6} a^5 c^3 x^3 \tan ^{-1}(a x)+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^2+a^2 c^3 \log (x)-\frac{5}{2} a^3 c^3 x \tan ^{-1}(a x)+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^2+6 a^2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)-\frac{c^3 \tan ^{-1}(a x)^2}{2 x^2}-\frac{a c^3 \tan ^{-1}(a x)}{x}",1,"(a^4*c^3*x^2)/12 - (a*c^3*ArcTan[a*x])/x - (5*a^3*c^3*x*ArcTan[a*x])/2 - (a^5*c^3*x^3*ArcTan[a*x])/6 + (3*a^2*c^3*ArcTan[a*x]^2)/4 - (c^3*ArcTan[a*x]^2)/(2*x^2) + (3*a^4*c^3*x^2*ArcTan[a*x]^2)/2 + (a^6*c^3*x^4*ArcTan[a*x]^2)/4 + 6*a^2*c^3*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + a^2*c^3*Log[x] + (2*a^2*c^3*Log[1 + a^2*x^2])/3 - (3*I)*a^2*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*I)*a^2*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - (3*a^2*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*a^2*c^3*PolyLog[3, -1 + 2/(1 + I*a*x)])/2","A",31,16,22,0.7273,1,"{4948, 4852, 4918, 266, 36, 29, 31, 4884, 4850, 4988, 4994, 6610, 4916, 4846, 260, 43}"
281,1,250,0,0.6083745,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2}{x^4} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x]^2)/x^4,x]","-\frac{8}{3} i a^3 c^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+\frac{8}{3} i a^3 c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)^2-\frac{1}{3} a^5 c^3 x^2 \tan ^{-1}(a x)+\frac{1}{3} a^4 c^3 x-\frac{a^2 c^3}{3 x}+3 a^4 c^3 x \tan ^{-1}(a x)^2-\frac{2}{3} a^3 c^3 \tan ^{-1}(a x)-\frac{3 a^2 c^3 \tan ^{-1}(a x)^2}{x}+\frac{16}{3} a^3 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+\frac{16}{3} a^3 c^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)-\frac{a c^3 \tan ^{-1}(a x)}{3 x^2}-\frac{c^3 \tan ^{-1}(a x)^2}{3 x^3}","-\frac{8}{3} i a^3 c^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+\frac{8}{3} i a^3 c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)^2-\frac{1}{3} a^5 c^3 x^2 \tan ^{-1}(a x)+\frac{1}{3} a^4 c^3 x-\frac{a^2 c^3}{3 x}+3 a^4 c^3 x \tan ^{-1}(a x)^2-\frac{2}{3} a^3 c^3 \tan ^{-1}(a x)-\frac{3 a^2 c^3 \tan ^{-1}(a x)^2}{x}+\frac{16}{3} a^3 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+\frac{16}{3} a^3 c^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)-\frac{a c^3 \tan ^{-1}(a x)}{3 x^2}-\frac{c^3 \tan ^{-1}(a x)^2}{3 x^3}",1,"-(a^2*c^3)/(3*x) + (a^4*c^3*x)/3 - (2*a^3*c^3*ArcTan[a*x])/3 - (a*c^3*ArcTan[a*x])/(3*x^2) - (a^5*c^3*x^2*ArcTan[a*x])/3 - (c^3*ArcTan[a*x]^2)/(3*x^3) - (3*a^2*c^3*ArcTan[a*x]^2)/x + 3*a^4*c^3*x*ArcTan[a*x]^2 + (a^6*c^3*x^3*ArcTan[a*x]^2)/3 + (16*a^3*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/3 + (16*a^3*c^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/3 - ((8*I)/3)*a^3*c^3*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((8*I)/3)*a^3*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)]","A",28,15,22,0.6818,1,"{4948, 4846, 4920, 4854, 2402, 2315, 4852, 4918, 325, 203, 4924, 4868, 2447, 4916, 321}"
282,1,166,0,0.3871747,"\int \frac{x^4 \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx","Int[(x^4*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]","-\frac{4 i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{3 a^5 c}+\frac{x^3 \tan ^{-1}(a x)^2}{3 a^2 c}-\frac{x^2 \tan ^{-1}(a x)}{3 a^3 c}+\frac{x}{3 a^4 c}-\frac{x \tan ^{-1}(a x)^2}{a^4 c}+\frac{\tan ^{-1}(a x)^3}{3 a^5 c}-\frac{4 i \tan ^{-1}(a x)^2}{3 a^5 c}-\frac{\tan ^{-1}(a x)}{3 a^5 c}-\frac{8 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{3 a^5 c}","-\frac{4 i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{3 a^5 c}+\frac{x^3 \tan ^{-1}(a x)^2}{3 a^2 c}-\frac{x^2 \tan ^{-1}(a x)}{3 a^3 c}+\frac{x}{3 a^4 c}-\frac{x \tan ^{-1}(a x)^2}{a^4 c}+\frac{\tan ^{-1}(a x)^3}{3 a^5 c}-\frac{4 i \tan ^{-1}(a x)^2}{3 a^5 c}-\frac{\tan ^{-1}(a x)}{3 a^5 c}-\frac{8 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{3 a^5 c}",1,"x/(3*a^4*c) - ArcTan[a*x]/(3*a^5*c) - (x^2*ArcTan[a*x])/(3*a^3*c) - (((4*I)/3)*ArcTan[a*x]^2)/(a^5*c) - (x*ArcTan[a*x]^2)/(a^4*c) + (x^3*ArcTan[a*x]^2)/(3*a^2*c) + ArcTan[a*x]^3/(3*a^5*c) - (8*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(3*a^5*c) - (((4*I)/3)*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^5*c)","A",17,10,22,0.4545,1,"{4916, 4852, 321, 203, 4920, 4854, 2402, 2315, 4846, 4884}"
283,1,169,0,0.2937059,"\int \frac{x^3 \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx","Int[(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]","\frac{\text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^4 c}+\frac{i \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^4 c}+\frac{\log \left(a^2 x^2+1\right)}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{i \tan ^{-1}(a x)^3}{3 a^4 c}+\frac{\tan ^{-1}(a x)^2}{2 a^4 c}-\frac{x \tan ^{-1}(a x)}{a^3 c}+\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{a^4 c}","\frac{\text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^4 c}+\frac{i \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^4 c}+\frac{\log \left(a^2 x^2+1\right)}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{i \tan ^{-1}(a x)^3}{3 a^4 c}+\frac{\tan ^{-1}(a x)^2}{2 a^4 c}-\frac{x \tan ^{-1}(a x)}{a^3 c}+\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{a^4 c}",1,"-((x*ArcTan[a*x])/(a^3*c)) + ArcTan[a*x]^2/(2*a^4*c) + (x^2*ArcTan[a*x]^2)/(2*a^2*c) + ((I/3)*ArcTan[a*x]^3)/(a^4*c) + (ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(a^4*c) + Log[1 + a^2*x^2]/(2*a^4*c) + (I*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*c) + PolyLog[3, 1 - 2/(1 + I*a*x)]/(2*a^4*c)","A",10,9,22,0.4091,1,"{4916, 4852, 4846, 260, 4884, 4920, 4854, 4994, 6610}"
284,1,98,0,0.1669249,"\int \frac{x^2 \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx","Int[(x^2*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]","\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^3 c}-\frac{\tan ^{-1}(a x)^3}{3 a^3 c}+\frac{x \tan ^{-1}(a x)^2}{a^2 c}+\frac{i \tan ^{-1}(a x)^2}{a^3 c}+\frac{2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{a^3 c}","\frac{i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^3 c}-\frac{\tan ^{-1}(a x)^3}{3 a^3 c}+\frac{x \tan ^{-1}(a x)^2}{a^2 c}+\frac{i \tan ^{-1}(a x)^2}{a^3 c}+\frac{2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{a^3 c}",1,"(I*ArcTan[a*x]^2)/(a^3*c) + (x*ArcTan[a*x]^2)/(a^2*c) - ArcTan[a*x]^3/(3*a^3*c) + (2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(a^3*c) + (I*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^3*c)","A",7,7,22,0.3182,1,"{4916, 4846, 4920, 4854, 2402, 2315, 4884}"
285,1,102,0,0.1460671,"\int \frac{x \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx","Int[(x*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]","-\frac{\text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^2 c}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^2 c}-\frac{i \tan ^{-1}(a x)^3}{3 a^2 c}-\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{a^2 c}","-\frac{\text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^2 c}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^2 c}-\frac{i \tan ^{-1}(a x)^3}{3 a^2 c}-\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{a^2 c}",1,"((-I/3)*ArcTan[a*x]^3)/(a^2*c) - (ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(a^2*c) - (I*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^2*c) - PolyLog[3, 1 - 2/(1 + I*a*x)]/(2*a^2*c)","A",4,5,20,0.2500,1,"{4920, 4854, 4884, 4994, 6610}"
286,1,16,0,0.0235709,"\int \frac{\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx","Int[ArcTan[a*x]^2/(c + a^2*c*x^2),x]","\frac{\tan ^{-1}(a x)^3}{3 a c}","\frac{\tan ^{-1}(a x)^3}{3 a c}",1,"ArcTan[a*x]^3/(3*a*c)","A",1,1,19,0.05263,1,"{4884}"
287,1,91,0,0.1793314,"\int \frac{\tan ^{-1}(a x)^2}{x \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)),x]","\frac{\text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c}-\frac{i \tan ^{-1}(a x)^3}{3 c}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c}","\frac{\text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c}-\frac{i \tan ^{-1}(a x)^3}{3 c}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c}",1,"((-I/3)*ArcTan[a*x]^3)/c + (ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c - (I*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c + PolyLog[3, -1 + 2/(1 - I*a*x)]/(2*c)","A",4,5,22,0.2273,1,"{4924, 4868, 4884, 4992, 6610}"
288,1,92,0,0.195493,"\int \frac{\tan ^{-1}(a x)^2}{x^2 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)),x]","-\frac{i a \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c}-\frac{a \tan ^{-1}(a x)^3}{3 c}-\frac{i a \tan ^{-1}(a x)^2}{c}-\frac{\tan ^{-1}(a x)^2}{c x}+\frac{2 a \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c}","-\frac{i a \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c}-\frac{a \tan ^{-1}(a x)^3}{3 c}-\frac{i a \tan ^{-1}(a x)^2}{c}-\frac{\tan ^{-1}(a x)^2}{c x}+\frac{2 a \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c}",1,"((-I)*a*ArcTan[a*x]^2)/c - ArcTan[a*x]^2/(c*x) - (a*ArcTan[a*x]^3)/(3*c) + (2*a*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c - (I*a*PolyLog[2, -1 + 2/(1 - I*a*x)])/c","A",6,6,22,0.2727,1,"{4918, 4852, 4924, 4868, 2447, 4884}"
289,1,178,0,0.3355152,"\int \frac{\tan ^{-1}(a x)^2}{x^3 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^2/(x^3*(c + a^2*c*x^2)),x]","-\frac{a^2 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c}+\frac{i a^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c}-\frac{a^2 \log \left(a^2 x^2+1\right)}{2 c}+\frac{a^2 \log (x)}{c}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c}-\frac{a^2 \tan ^{-1}(a x)^2}{2 c}-\frac{a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c}-\frac{\tan ^{-1}(a x)^2}{2 c x^2}-\frac{a \tan ^{-1}(a x)}{c x}","-\frac{a^2 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c}+\frac{i a^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c}-\frac{a^2 \log \left(a^2 x^2+1\right)}{2 c}+\frac{a^2 \log (x)}{c}+\frac{i a^2 \tan ^{-1}(a x)^3}{3 c}-\frac{a^2 \tan ^{-1}(a x)^2}{2 c}-\frac{a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c}-\frac{\tan ^{-1}(a x)^2}{2 c x^2}-\frac{a \tan ^{-1}(a x)}{c x}",1,"-((a*ArcTan[a*x])/(c*x)) - (a^2*ArcTan[a*x]^2)/(2*c) - ArcTan[a*x]^2/(2*c*x^2) + ((I/3)*a^2*ArcTan[a*x]^3)/c + (a^2*Log[x])/c - (a^2*Log[1 + a^2*x^2])/(2*c) - (a^2*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c + (I*a^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c - (a^2*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c)","A",13,11,22,0.5000,1,"{4918, 4852, 266, 36, 29, 31, 4884, 4924, 4868, 4992, 6610}"
290,1,166,0,0.4359521,"\int \frac{\tan ^{-1}(a x)^2}{x^4 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)),x]","\frac{4 i a^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{3 c}-\frac{a^2}{3 c x}+\frac{a^3 \tan ^{-1}(a x)^3}{3 c}+\frac{4 i a^3 \tan ^{-1}(a x)^2}{3 c}-\frac{a^3 \tan ^{-1}(a x)}{3 c}+\frac{a^2 \tan ^{-1}(a x)^2}{c x}-\frac{8 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{3 c}-\frac{a \tan ^{-1}(a x)}{3 c x^2}-\frac{\tan ^{-1}(a x)^2}{3 c x^3}","\frac{4 i a^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{3 c}-\frac{a^2}{3 c x}+\frac{a^3 \tan ^{-1}(a x)^3}{3 c}+\frac{4 i a^3 \tan ^{-1}(a x)^2}{3 c}-\frac{a^3 \tan ^{-1}(a x)}{3 c}+\frac{a^2 \tan ^{-1}(a x)^2}{c x}-\frac{8 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{3 c}-\frac{a \tan ^{-1}(a x)}{3 c x^2}-\frac{\tan ^{-1}(a x)^2}{3 c x^3}",1,"-a^2/(3*c*x) - (a^3*ArcTan[a*x])/(3*c) - (a*ArcTan[a*x])/(3*c*x^2) + (((4*I)/3)*a^3*ArcTan[a*x]^2)/c - ArcTan[a*x]^2/(3*c*x^3) + (a^2*ArcTan[a*x]^2)/(c*x) + (a^3*ArcTan[a*x]^3)/(3*c) - (8*a^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/(3*c) + (((4*I)/3)*a^3*PolyLog[2, -1 + 2/(1 - I*a*x)])/c","A",15,8,22,0.3636,1,"{4918, 4852, 325, 203, 4924, 4868, 2447, 4884}"
291,1,192,0,0.2897027,"\int \frac{x^3 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2,x]","-\frac{\text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^4 c^2}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^4 c^2}-\frac{1}{4 a^4 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^2}{2 a^4 c^2 \left(a^2 x^2+1\right)}-\frac{x \tan ^{-1}(a x)}{2 a^3 c^2 \left(a^2 x^2+1\right)}-\frac{i \tan ^{-1}(a x)^3}{3 a^4 c^2}-\frac{\tan ^{-1}(a x)^2}{4 a^4 c^2}-\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{a^4 c^2}","-\frac{\text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^4 c^2}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^4 c^2}-\frac{1}{4 a^4 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^2}{2 a^4 c^2 \left(a^2 x^2+1\right)}-\frac{x \tan ^{-1}(a x)}{2 a^3 c^2 \left(a^2 x^2+1\right)}-\frac{i \tan ^{-1}(a x)^3}{3 a^4 c^2}-\frac{\tan ^{-1}(a x)^2}{4 a^4 c^2}-\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{a^4 c^2}",1,"-1/(4*a^4*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x])/(2*a^3*c^2*(1 + a^2*x^2)) - ArcTan[a*x]^2/(4*a^4*c^2) + ArcTan[a*x]^2/(2*a^4*c^2*(1 + a^2*x^2)) - ((I/3)*ArcTan[a*x]^3)/(a^4*c^2) - (ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(a^4*c^2) - (I*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*c^2) - PolyLog[3, 1 - 2/(1 + I*a*x)]/(2*a^4*c^2)","A",8,9,22,0.4091,1,"{4964, 4920, 4854, 4884, 4994, 6610, 4930, 4892, 261}"
292,1,106,0,0.1100118,"\int \frac{x^2 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^2*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2,x]","\frac{x}{4 a^2 c^2 \left(a^2 x^2+1\right)}-\frac{x \tan ^{-1}(a x)^2}{2 a^2 c^2 \left(a^2 x^2+1\right)}-\frac{\tan ^{-1}(a x)}{2 a^3 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^3}{6 a^3 c^2}+\frac{\tan ^{-1}(a x)}{4 a^3 c^2}","\frac{x}{4 a^2 c^2 \left(a^2 x^2+1\right)}-\frac{x \tan ^{-1}(a x)^2}{2 a^2 c^2 \left(a^2 x^2+1\right)}-\frac{\tan ^{-1}(a x)}{2 a^3 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^3}{6 a^3 c^2}+\frac{\tan ^{-1}(a x)}{4 a^3 c^2}",1,"x/(4*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]/(4*a^3*c^2) - ArcTan[a*x]/(2*a^3*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x]^2)/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^3/(6*a^3*c^2)","A",4,4,22,0.1818,1,"{4936, 4930, 199, 205}"
293,1,91,0,0.069906,"\int \frac{x \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2,x]","\frac{1}{4 a^2 c^2 \left(a^2 x^2+1\right)}-\frac{\tan ^{-1}(a x)^2}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)}{2 a c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^2}{4 a^2 c^2}","\frac{1}{4 a^2 c^2 \left(a^2 x^2+1\right)}-\frac{\tan ^{-1}(a x)^2}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)}{2 a c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^2}{4 a^2 c^2}",1,"1/(4*a^2*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*a*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a^2*c^2) - ArcTan[a*x]^2/(2*a^2*c^2*(1 + a^2*x^2))","A",3,3,20,0.1500,1,"{4930, 4892, 261}"
294,1,100,0,0.0688284,"\int \frac{\tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]^2/(c + a^2*c*x^2)^2,x]","-\frac{x}{4 c^2 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)^2}{2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)}{2 a c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^3}{6 a c^2}-\frac{\tan ^{-1}(a x)}{4 a c^2}","-\frac{x}{4 c^2 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)^2}{2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)}{2 a c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^3}{6 a c^2}-\frac{\tan ^{-1}(a x)}{4 a c^2}",1,"-x/(4*c^2*(1 + a^2*x^2)) - ArcTan[a*x]/(4*a*c^2) + ArcTan[a*x]/(2*a*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x]^2)/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^3/(6*a*c^2)","A",4,4,19,0.2105,1,"{4892, 4930, 199, 205}"
295,1,170,0,0.3112175,"\int \frac{\tan ^{-1}(a x)^2}{x \left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)^2),x]","\frac{\text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^2}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^2}-\frac{1}{4 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^2}{2 c^2 \left(a^2 x^2+1\right)}-\frac{a x \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}-\frac{i \tan ^{-1}(a x)^3}{3 c^2}-\frac{\tan ^{-1}(a x)^2}{4 c^2}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^2}","\frac{\text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^2}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^2}-\frac{1}{4 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^2}{2 c^2 \left(a^2 x^2+1\right)}-\frac{a x \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}-\frac{i \tan ^{-1}(a x)^3}{3 c^2}-\frac{\tan ^{-1}(a x)^2}{4 c^2}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^2}",1,"-1/(4*c^2*(1 + a^2*x^2)) - (a*x*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) - ArcTan[a*x]^2/(4*c^2) + ArcTan[a*x]^2/(2*c^2*(1 + a^2*x^2)) - ((I/3)*ArcTan[a*x]^3)/c^2 + (ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^2 - (I*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2 + PolyLog[3, -1 + 2/(1 - I*a*x)]/(2*c^2)","A",8,9,22,0.4091,1,"{4966, 4924, 4868, 4884, 4992, 6610, 4930, 4892, 261}"
296,1,177,0,0.3399191,"\int \frac{\tan ^{-1}(a x)^2}{x^2 \left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)^2),x]","-\frac{i a \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^2}+\frac{a^2 x}{4 c^2 \left(a^2 x^2+1\right)}-\frac{a^2 x \tan ^{-1}(a x)^2}{2 c^2 \left(a^2 x^2+1\right)}-\frac{a \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}-\frac{a \tan ^{-1}(a x)^3}{2 c^2}-\frac{\tan ^{-1}(a x)^2}{c^2 x}-\frac{i a \tan ^{-1}(a x)^2}{c^2}+\frac{a \tan ^{-1}(a x)}{4 c^2}+\frac{2 a \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^2}","-\frac{i a \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^2}+\frac{a^2 x}{4 c^2 \left(a^2 x^2+1\right)}-\frac{a^2 x \tan ^{-1}(a x)^2}{2 c^2 \left(a^2 x^2+1\right)}-\frac{a \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}-\frac{a \tan ^{-1}(a x)^3}{2 c^2}-\frac{\tan ^{-1}(a x)^2}{c^2 x}-\frac{i a \tan ^{-1}(a x)^2}{c^2}+\frac{a \tan ^{-1}(a x)}{4 c^2}+\frac{2 a \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^2}",1,"(a^2*x)/(4*c^2*(1 + a^2*x^2)) + (a*ArcTan[a*x])/(4*c^2) - (a*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) - (I*a*ArcTan[a*x]^2)/c^2 - ArcTan[a*x]^2/(c^2*x) - (a^2*x*ArcTan[a*x]^2)/(2*c^2*(1 + a^2*x^2)) - (a*ArcTan[a*x]^3)/(2*c^2) + (2*a*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^2 - (I*a*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2","A",11,11,22,0.5000,1,"{4966, 4918, 4852, 4924, 4868, 2447, 4884, 4892, 4930, 199, 205}"
297,1,250,0,0.74027,"\int \frac{\tan ^{-1}(a x)^2}{x^3 \left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]^2/(x^3*(c + a^2*c*x^2)^2),x]","-\frac{a^2 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{c^2}+\frac{2 i a^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^2}+\frac{a^2}{4 c^2 \left(a^2 x^2+1\right)}-\frac{a^2 \log \left(a^2 x^2+1\right)}{2 c^2}+\frac{a^3 x \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}-\frac{a^2 \tan ^{-1}(a x)^2}{2 c^2 \left(a^2 x^2+1\right)}+\frac{a^2 \log (x)}{c^2}+\frac{2 i a^2 \tan ^{-1}(a x)^3}{3 c^2}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^2}-\frac{2 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^2}-\frac{\tan ^{-1}(a x)^2}{2 c^2 x^2}-\frac{a \tan ^{-1}(a x)}{c^2 x}","-\frac{a^2 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{c^2}+\frac{2 i a^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^2}+\frac{a^2}{4 c^2 \left(a^2 x^2+1\right)}-\frac{a^2 \log \left(a^2 x^2+1\right)}{2 c^2}+\frac{a^3 x \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}-\frac{a^2 \tan ^{-1}(a x)^2}{2 c^2 \left(a^2 x^2+1\right)}+\frac{a^2 \log (x)}{c^2}+\frac{2 i a^2 \tan ^{-1}(a x)^3}{3 c^2}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^2}-\frac{2 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^2}-\frac{\tan ^{-1}(a x)^2}{2 c^2 x^2}-\frac{a \tan ^{-1}(a x)}{c^2 x}",1,"a^2/(4*c^2*(1 + a^2*x^2)) - (a*ArcTan[a*x])/(c^2*x) + (a^3*x*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) - (a^2*ArcTan[a*x]^2)/(4*c^2) - ArcTan[a*x]^2/(2*c^2*x^2) - (a^2*ArcTan[a*x]^2)/(2*c^2*(1 + a^2*x^2)) + (((2*I)/3)*a^2*ArcTan[a*x]^3)/c^2 + (a^2*Log[x])/c^2 - (a^2*Log[1 + a^2*x^2])/(2*c^2) - (2*a^2*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^2 + ((2*I)*a^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2 - (a^2*PolyLog[3, -1 + 2/(1 - I*a*x)])/c^2","A",22,15,22,0.6818,1,"{4966, 4918, 4852, 266, 36, 29, 31, 4884, 4924, 4868, 4992, 6610, 4930, 4892, 261}"
298,1,242,0,0.8726295,"\int \frac{\tan ^{-1}(a x)^2}{x^4 \left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)^2),x]","\frac{7 i a^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{3 c^2}-\frac{a^4 x}{4 c^2 \left(a^2 x^2+1\right)}+\frac{a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left(a^2 x^2+1\right)}+\frac{a^3 \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}-\frac{a^2}{3 c^2 x}+\frac{5 a^3 \tan ^{-1}(a x)^3}{6 c^2}+\frac{7 i a^3 \tan ^{-1}(a x)^2}{3 c^2}-\frac{7 a^3 \tan ^{-1}(a x)}{12 c^2}+\frac{2 a^2 \tan ^{-1}(a x)^2}{c^2 x}-\frac{14 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{3 c^2}-\frac{a \tan ^{-1}(a x)}{3 c^2 x^2}-\frac{\tan ^{-1}(a x)^2}{3 c^2 x^3}","\frac{7 i a^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{3 c^2}-\frac{a^4 x}{4 c^2 \left(a^2 x^2+1\right)}+\frac{a^4 x \tan ^{-1}(a x)^2}{2 c^2 \left(a^2 x^2+1\right)}+\frac{a^3 \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}-\frac{a^2}{3 c^2 x}+\frac{5 a^3 \tan ^{-1}(a x)^3}{6 c^2}+\frac{7 i a^3 \tan ^{-1}(a x)^2}{3 c^2}-\frac{7 a^3 \tan ^{-1}(a x)}{12 c^2}+\frac{2 a^2 \tan ^{-1}(a x)^2}{c^2 x}-\frac{14 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{3 c^2}-\frac{a \tan ^{-1}(a x)}{3 c^2 x^2}-\frac{\tan ^{-1}(a x)^2}{3 c^2 x^3}",1,"-a^2/(3*c^2*x) - (a^4*x)/(4*c^2*(1 + a^2*x^2)) - (7*a^3*ArcTan[a*x])/(12*c^2) - (a*ArcTan[a*x])/(3*c^2*x^2) + (a^3*ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) + (((7*I)/3)*a^3*ArcTan[a*x]^2)/c^2 - ArcTan[a*x]^2/(3*c^2*x^3) + (2*a^2*ArcTan[a*x]^2)/(c^2*x) + (a^4*x*ArcTan[a*x]^2)/(2*c^2*(1 + a^2*x^2)) + (5*a^3*ArcTan[a*x]^3)/(6*c^2) - (14*a^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/(3*c^2) + (((7*I)/3)*a^3*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2","A",27,13,22,0.5909,1,"{4966, 4918, 4852, 325, 203, 4924, 4868, 2447, 4884, 4892, 4930, 199, 205}"
299,1,140,0,0.1851927,"\int \frac{x^3 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2)^3,x]","-\frac{x^4}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{3}{32 a^4 c^3 \left(a^2 x^2+1\right)}+\frac{x^4 \tan ^{-1}(a x)^2}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{x^3 \tan ^{-1}(a x)}{8 a c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x \tan ^{-1}(a x)}{16 a^3 c^3 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)^2}{32 a^4 c^3}","-\frac{x^4}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{3}{32 a^4 c^3 \left(a^2 x^2+1\right)}+\frac{x^4 \tan ^{-1}(a x)^2}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{x^3 \tan ^{-1}(a x)}{8 a c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x \tan ^{-1}(a x)}{16 a^3 c^3 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)^2}{32 a^4 c^3}",1,"-x^4/(32*c^3*(1 + a^2*x^2)^2) + 3/(32*a^4*c^3*(1 + a^2*x^2)) + (x^3*ArcTan[a*x])/(8*a*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x])/(16*a^3*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x]^2)/(32*a^4*c^3) + (x^4*ArcTan[a*x]^2)/(4*c^3*(1 + a^2*x^2)^2)","A",4,4,22,0.1818,1,"{4944, 4938, 4934, 4884}"
300,1,181,0,0.2664429,"\int \frac{x^2 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^2*ArcTan[a*x]^2)/(c + a^2*c*x^2)^3,x]","-\frac{x}{64 a^2 c^3 \left(a^2 x^2+1\right)}+\frac{x}{32 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{x \tan ^{-1}(a x)^2}{8 a^2 c^3 \left(a^2 x^2+1\right)}-\frac{x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left(a^2 x^2+1\right)}-\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)^3}{24 a^3 c^3}-\frac{\tan ^{-1}(a x)}{64 a^3 c^3}","-\frac{x}{64 a^2 c^3 \left(a^2 x^2+1\right)}+\frac{x}{32 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{x \tan ^{-1}(a x)^2}{8 a^2 c^3 \left(a^2 x^2+1\right)}-\frac{x \tan ^{-1}(a x)^2}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left(a^2 x^2+1\right)}-\frac{\tan ^{-1}(a x)}{8 a^3 c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)^3}{24 a^3 c^3}-\frac{\tan ^{-1}(a x)}{64 a^3 c^3}",1,"x/(32*a^2*c^3*(1 + a^2*x^2)^2) - x/(64*a^2*c^3*(1 + a^2*x^2)) - ArcTan[a*x]/(64*a^3*c^3) - ArcTan[a*x]/(8*a^3*c^3*(1 + a^2*x^2)^2) + ArcTan[a*x]/(8*a^3*c^3*(1 + a^2*x^2)) - (x*ArcTan[a*x]^2)/(4*a^2*c^3*(1 + a^2*x^2)^2) + (x*ArcTan[a*x]^2)/(8*a^2*c^3*(1 + a^2*x^2)) + ArcTan[a*x]^3/(24*a^3*c^3)","A",13,6,22,0.2727,1,"{4964, 4892, 4930, 199, 205, 4900}"
301,1,138,0,0.0964198,"\int \frac{x \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^3,x]","\frac{3}{32 a^2 c^3 \left(a^2 x^2+1\right)}+\frac{1}{32 a^2 c^3 \left(a^2 x^2+1\right)^2}-\frac{\tan ^{-1}(a x)^2}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x \tan ^{-1}(a x)}{16 a c^3 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)}{8 a c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)^2}{32 a^2 c^3}","\frac{3}{32 a^2 c^3 \left(a^2 x^2+1\right)}+\frac{1}{32 a^2 c^3 \left(a^2 x^2+1\right)^2}-\frac{\tan ^{-1}(a x)^2}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x \tan ^{-1}(a x)}{16 a c^3 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)}{8 a c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)^2}{32 a^2 c^3}",1,"1/(32*a^2*c^3*(1 + a^2*x^2)^2) + 3/(32*a^2*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(8*a*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x])/(16*a*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^2)/(32*a^2*c^3) - ArcTan[a*x]^2/(4*a^2*c^3*(1 + a^2*x^2)^2)","A",4,4,20,0.2000,1,"{4930, 4896, 4892, 261}"
302,1,169,0,0.1186442,"\int \frac{\tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]^2/(c + a^2*c*x^2)^3,x]","-\frac{15 x}{64 c^3 \left(a^2 x^2+1\right)}-\frac{x}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x \tan ^{-1}(a x)^2}{8 c^3 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)^2}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)}{8 a c^3 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)}{8 a c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)^3}{8 a c^3}-\frac{15 \tan ^{-1}(a x)}{64 a c^3}","-\frac{15 x}{64 c^3 \left(a^2 x^2+1\right)}-\frac{x}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x \tan ^{-1}(a x)^2}{8 c^3 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)^2}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)}{8 a c^3 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)}{8 a c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)^3}{8 a c^3}-\frac{15 \tan ^{-1}(a x)}{64 a c^3}",1,"-x/(32*c^3*(1 + a^2*x^2)^2) - (15*x)/(64*c^3*(1 + a^2*x^2)) - (15*ArcTan[a*x])/(64*a*c^3) + ArcTan[a*x]/(8*a*c^3*(1 + a^2*x^2)^2) + (3*ArcTan[a*x])/(8*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x]^2)/(4*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x]^2)/(8*c^3*(1 + a^2*x^2)) + ArcTan[a*x]^3/(8*a*c^3)","A",8,5,19,0.2632,1,"{4900, 4892, 4930, 199, 205}"
303,1,236,0,0.4849137,"\int \frac{\tan ^{-1}(a x)^2}{x \left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)^3),x]","\frac{\text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^3}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^3}-\frac{11}{32 c^3 \left(a^2 x^2+1\right)}-\frac{1}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)^2}{2 c^3 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^2}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{11 a x \tan ^{-1}(a x)}{16 c^3 \left(a^2 x^2+1\right)}-\frac{a x \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)^2}-\frac{i \tan ^{-1}(a x)^3}{3 c^3}-\frac{11 \tan ^{-1}(a x)^2}{32 c^3}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^3}","\frac{\text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^3}-\frac{i \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^3}-\frac{11}{32 c^3 \left(a^2 x^2+1\right)}-\frac{1}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)^2}{2 c^3 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^2}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{11 a x \tan ^{-1}(a x)}{16 c^3 \left(a^2 x^2+1\right)}-\frac{a x \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)^2}-\frac{i \tan ^{-1}(a x)^3}{3 c^3}-\frac{11 \tan ^{-1}(a x)^2}{32 c^3}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^3}",1,"-1/(32*c^3*(1 + a^2*x^2)^2) - 11/(32*c^3*(1 + a^2*x^2)) - (a*x*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)^2) - (11*a*x*ArcTan[a*x])/(16*c^3*(1 + a^2*x^2)) - (11*ArcTan[a*x]^2)/(32*c^3) + ArcTan[a*x]^2/(4*c^3*(1 + a^2*x^2)^2) + ArcTan[a*x]^2/(2*c^3*(1 + a^2*x^2)) - ((I/3)*ArcTan[a*x]^3)/c^3 + (ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^3 - (I*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 + PolyLog[3, -1 + 2/(1 - I*a*x)]/(2*c^3)","A",13,10,22,0.4545,1,"{4966, 4924, 4868, 4884, 4992, 6610, 4930, 4892, 261, 4896}"
304,1,250,0,0.551444,"\int \frac{\tan ^{-1}(a x)^2}{x^2 \left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)^3),x]","-\frac{i a \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^3}+\frac{31 a^2 x}{64 c^3 \left(a^2 x^2+1\right)}+\frac{a^2 x}{32 c^3 \left(a^2 x^2+1\right)^2}-\frac{7 a^2 x \tan ^{-1}(a x)^2}{8 c^3 \left(a^2 x^2+1\right)}-\frac{a^2 x \tan ^{-1}(a x)^2}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{7 a \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)}-\frac{a \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)^2}-\frac{5 a \tan ^{-1}(a x)^3}{8 c^3}-\frac{\tan ^{-1}(a x)^2}{c^3 x}-\frac{i a \tan ^{-1}(a x)^2}{c^3}+\frac{31 a \tan ^{-1}(a x)}{64 c^3}+\frac{2 a \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^3}","-\frac{i a \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^3}+\frac{31 a^2 x}{64 c^3 \left(a^2 x^2+1\right)}+\frac{a^2 x}{32 c^3 \left(a^2 x^2+1\right)^2}-\frac{7 a^2 x \tan ^{-1}(a x)^2}{8 c^3 \left(a^2 x^2+1\right)}-\frac{a^2 x \tan ^{-1}(a x)^2}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{7 a \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)}-\frac{a \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)^2}-\frac{5 a \tan ^{-1}(a x)^3}{8 c^3}-\frac{\tan ^{-1}(a x)^2}{c^3 x}-\frac{i a \tan ^{-1}(a x)^2}{c^3}+\frac{31 a \tan ^{-1}(a x)}{64 c^3}+\frac{2 a \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^3}",1,"(a^2*x)/(32*c^3*(1 + a^2*x^2)^2) + (31*a^2*x)/(64*c^3*(1 + a^2*x^2)) + (31*a*ArcTan[a*x])/(64*c^3) - (a*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)^2) - (7*a*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)) - (I*a*ArcTan[a*x]^2)/c^3 - ArcTan[a*x]^2/(c^3*x) - (a^2*x*ArcTan[a*x]^2)/(4*c^3*(1 + a^2*x^2)^2) - (7*a^2*x*ArcTan[a*x]^2)/(8*c^3*(1 + a^2*x^2)) - (5*a*ArcTan[a*x]^3)/(8*c^3) + (2*a*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^3 - (I*a*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3","A",20,12,22,0.5455,1,"{4966, 4918, 4852, 4924, 4868, 2447, 4884, 4892, 4930, 199, 205, 4900}"
305,1,322,0,1.3334352,"\int \frac{\tan ^{-1}(a x)^2}{x^3 \left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]^2/(x^3*(c + a^2*c*x^2)^3),x]","-\frac{3 a^2 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^3}+\frac{3 i a^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^3}+\frac{19 a^2}{32 c^3 \left(a^2 x^2+1\right)}+\frac{a^2}{32 c^3 \left(a^2 x^2+1\right)^2}-\frac{a^2 \log \left(a^2 x^2+1\right)}{2 c^3}+\frac{19 a^3 x \tan ^{-1}(a x)}{16 c^3 \left(a^2 x^2+1\right)}+\frac{a^3 x \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)^2}-\frac{a^2 \tan ^{-1}(a x)^2}{c^3 \left(a^2 x^2+1\right)}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{a^2 \log (x)}{c^3}+\frac{i a^2 \tan ^{-1}(a x)^3}{c^3}+\frac{3 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac{3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^3}-\frac{\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{a \tan ^{-1}(a x)}{c^3 x}","-\frac{3 a^2 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^3}+\frac{3 i a^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^3}+\frac{19 a^2}{32 c^3 \left(a^2 x^2+1\right)}+\frac{a^2}{32 c^3 \left(a^2 x^2+1\right)^2}-\frac{a^2 \log \left(a^2 x^2+1\right)}{2 c^3}+\frac{19 a^3 x \tan ^{-1}(a x)}{16 c^3 \left(a^2 x^2+1\right)}+\frac{a^3 x \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)^2}-\frac{a^2 \tan ^{-1}(a x)^2}{c^3 \left(a^2 x^2+1\right)}-\frac{a^2 \tan ^{-1}(a x)^2}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{a^2 \log (x)}{c^3}+\frac{i a^2 \tan ^{-1}(a x)^3}{c^3}+\frac{3 a^2 \tan ^{-1}(a x)^2}{32 c^3}-\frac{3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^3}-\frac{\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{a \tan ^{-1}(a x)}{c^3 x}",1,"a^2/(32*c^3*(1 + a^2*x^2)^2) + (19*a^2)/(32*c^3*(1 + a^2*x^2)) - (a*ArcTan[a*x])/(c^3*x) + (a^3*x*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)^2) + (19*a^3*x*ArcTan[a*x])/(16*c^3*(1 + a^2*x^2)) + (3*a^2*ArcTan[a*x]^2)/(32*c^3) - ArcTan[a*x]^2/(2*c^3*x^2) - (a^2*ArcTan[a*x]^2)/(4*c^3*(1 + a^2*x^2)^2) - (a^2*ArcTan[a*x]^2)/(c^3*(1 + a^2*x^2)) + (I*a^2*ArcTan[a*x]^3)/c^3 + (a^2*Log[x])/c^3 - (a^2*Log[1 + a^2*x^2])/(2*c^3) - (3*a^2*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^3 + ((3*I)*a^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 - (3*a^2*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^3)","A",36,16,22,0.7273,1,"{4966, 4918, 4852, 266, 36, 29, 31, 4884, 4924, 4868, 4992, 6610, 4930, 4892, 261, 4896}"
306,1,317,0,1.5257487,"\int \frac{\tan ^{-1}(a x)^2}{x^4 \left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)^3),x]","\frac{10 i a^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{3 c^3}-\frac{47 a^4 x}{64 c^3 \left(a^2 x^2+1\right)}-\frac{a^4 x}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{11 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left(a^2 x^2+1\right)}+\frac{a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{11 a^3 \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)}+\frac{a^3 \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)^2}-\frac{a^2}{3 c^3 x}+\frac{35 a^3 \tan ^{-1}(a x)^3}{24 c^3}+\frac{10 i a^3 \tan ^{-1}(a x)^2}{3 c^3}-\frac{205 a^3 \tan ^{-1}(a x)}{192 c^3}+\frac{3 a^2 \tan ^{-1}(a x)^2}{c^3 x}-\frac{20 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{3 c^3}-\frac{a \tan ^{-1}(a x)}{3 c^3 x^2}-\frac{\tan ^{-1}(a x)^2}{3 c^3 x^3}","\frac{10 i a^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{3 c^3}-\frac{47 a^4 x}{64 c^3 \left(a^2 x^2+1\right)}-\frac{a^4 x}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{11 a^4 x \tan ^{-1}(a x)^2}{8 c^3 \left(a^2 x^2+1\right)}+\frac{a^4 x \tan ^{-1}(a x)^2}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{11 a^3 \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)}+\frac{a^3 \tan ^{-1}(a x)}{8 c^3 \left(a^2 x^2+1\right)^2}-\frac{a^2}{3 c^3 x}+\frac{35 a^3 \tan ^{-1}(a x)^3}{24 c^3}+\frac{10 i a^3 \tan ^{-1}(a x)^2}{3 c^3}-\frac{205 a^3 \tan ^{-1}(a x)}{192 c^3}+\frac{3 a^2 \tan ^{-1}(a x)^2}{c^3 x}-\frac{20 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{3 c^3}-\frac{a \tan ^{-1}(a x)}{3 c^3 x^2}-\frac{\tan ^{-1}(a x)^2}{3 c^3 x^3}",1,"-a^2/(3*c^3*x) - (a^4*x)/(32*c^3*(1 + a^2*x^2)^2) - (47*a^4*x)/(64*c^3*(1 + a^2*x^2)) - (205*a^3*ArcTan[a*x])/(192*c^3) - (a*ArcTan[a*x])/(3*c^3*x^2) + (a^3*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)^2) + (11*a^3*ArcTan[a*x])/(8*c^3*(1 + a^2*x^2)) + (((10*I)/3)*a^3*ArcTan[a*x]^2)/c^3 - ArcTan[a*x]^2/(3*c^3*x^3) + (3*a^2*ArcTan[a*x]^2)/(c^3*x) + (a^4*x*ArcTan[a*x]^2)/(4*c^3*(1 + a^2*x^2)^2) + (11*a^4*x*ArcTan[a*x]^2)/(8*c^3*(1 + a^2*x^2)) + (35*a^3*ArcTan[a*x]^3)/(24*c^3) - (20*a^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/(3*c^3) + (((10*I)/3)*a^3*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3","A",48,14,22,0.6364,1,"{4966, 4918, 4852, 325, 203, 4924, 4868, 2447, 4884, 4892, 4930, 199, 205, 4900}"
307,1,385,0,1.4252549,"\int x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx","Int[x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]","\frac{11 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{60 a^4 \sqrt{a^2 c x^2+c}}-\frac{11 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{60 a^4 \sqrt{a^2 c x^2+c}}+\frac{\left(a^2 c x^2+c\right)^{3/2}}{30 a^4 c}-\frac{11 \sqrt{a^2 c x^2+c}}{60 a^4}+\frac{1}{5} x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{10 a}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{15 a^2}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{12 a^3}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{15 a^4}-\frac{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{30 a^4 \sqrt{a^2 c x^2+c}}","\frac{11 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{60 a^4 \sqrt{a^2 c x^2+c}}-\frac{11 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{60 a^4 \sqrt{a^2 c x^2+c}}+\frac{\left(a^2 c x^2+c\right)^{3/2}}{30 a^4 c}-\frac{11 \sqrt{a^2 c x^2+c}}{60 a^4}+\frac{1}{5} x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{10 a}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{15 a^2}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{12 a^3}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{15 a^4}-\frac{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{30 a^4 \sqrt{a^2 c x^2+c}}",1,"(-11*Sqrt[c + a^2*c*x^2])/(60*a^4) + (c + a^2*c*x^2)^(3/2)/(30*a^4*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(12*a^3) - (x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(10*a) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(15*a^4) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(15*a^2) + (x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/5 - (((11*I)/30)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2]) + (((11*I)/60)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2]) - (((11*I)/60)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2])","A",26,8,24,0.3333,1,"{4950, 4952, 261, 4890, 4886, 4930, 266, 43}"
308,1,436,0,1.1345148,"\int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx","Int[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]","-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c}}{12 a^2}+\frac{1}{4} x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{6 a}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^2}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{12 a^3}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{6 a^3}","-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c}}{12 a^2}+\frac{1}{4} x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{6 a}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^2}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{12 a^3}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{6 a^3}",1,"(x*Sqrt[c + a^2*c*x^2])/(12*a^2) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(12*a^3) - (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(6*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(8*a^2) + (x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/4 + ((I/4)*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^3*Sqrt[c + a^2*c*x^2]) - (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(6*a^3) - ((I/4)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + ((I/4)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (c*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2]) - (c*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2])","A",35,12,24,0.5000,1,"{4950, 4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589, 321}"
309,1,279,0,0.1759361,"\int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx","Int[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]","-\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 a^2 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 a^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c}}{3 a^2}+\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{3 a^2 c}+\frac{2 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{3 a^2 \sqrt{a^2 c x^2+c}}-\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 a}","-\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 a^2 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 a^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c}}{3 a^2}+\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{3 a^2 c}+\frac{2 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{3 a^2 \sqrt{a^2 c x^2+c}}-\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 a}",1,"Sqrt[c + a^2*c*x^2]/(3*a^2) - (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*a) + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(3*a^2*c) + (((2*I)/3)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) - ((I/3)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) + ((I/3)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2])","A",4,4,22,0.1818,1,"{4930, 4878, 4890, 4886}"
310,1,340,0,0.2152145,"\int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx","Int[Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]","\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}+\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a \sqrt{a^2 c x^2+c}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a}","\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}+\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a \sqrt{a^2 c x^2+c}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a}",1,"-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 - (I*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a*Sqrt[c + a^2*c*x^2]) + (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/a + (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (c*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (c*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])","A",12,9,21,0.4286,1,"{4880, 4890, 4888, 4181, 2531, 2282, 6589, 217, 206}"
311,1,439,0,0.5134937,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x} \, dx","Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x,x]","-\frac{2 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+\frac{4 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}","-\frac{2 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+\frac{4 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}",1,"Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + ((4*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + ((2*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*c*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (2*c*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",13,10,24,0.4167,1,"{4950, 4958, 4956, 4183, 2531, 2282, 6589, 4930, 4890, 4886}"
312,1,458,0,0.5293232,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^2} \, dx","Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x^2,x]","\frac{2 i a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 i a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac{4 a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}","\frac{2 i a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 i a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac{4 a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}",1,"-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x) - ((2*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] - (4*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + ((2*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*a*c*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a*c*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*a*c*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (2*a*c*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",13,10,24,0.4167,1,"{4950, 4944, 4958, 4954, 4890, 4888, 4181, 2531, 2282, 6589}"
313,1,328,0,0.8564175,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^3} \, dx","Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x^3,x]","\frac{i a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x^2}-a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}","\frac{i a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x^2}-a^2 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}",1,"-((a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x^2) - (a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - a^2*Sqrt[c]*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + (I*a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (I*a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",24,12,24,0.5000,1,"{4950, 4962, 4944, 266, 63, 208, 4958, 4956, 4183, 2531, 2282, 6589}"
314,1,275,0,0.4269359,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^4} \, dx","Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x^4,x]","\frac{i a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{i a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c}}{3 x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 x^2}-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{3 c x^3}-\frac{2 a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}","\frac{i a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{i a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c}}{3 x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 x^2}-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{3 c x^3}-\frac{2 a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}",1,"-(a^2*Sqrt[c + a^2*c*x^2])/(3*x) - (a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*x^2) - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(3*c*x^3) - (2*a^3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(3*Sqrt[c + a^2*c*x^2]) + ((I/3)*a^3*c*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((I/3)*a^3*c*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",7,6,24,0.2500,1,"{4944, 4946, 4962, 264, 4958, 4954}"
315,1,476,0,4.0727562,"\int x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2 \, dx","Int[x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2,x]","\frac{17 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{17 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{17 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{140 a^4 \sqrt{a^2 c x^2+c}}+\frac{\left(a^2 c x^2+c\right)^{5/2}}{105 a^4 c}-\frac{17 \left(a^2 c x^2+c\right)^{3/2}}{1260 a^4}-\frac{17 c \sqrt{a^2 c x^2+c}}{280 a^4}+\frac{1}{7} a^2 c x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{21} a c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{8}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{23 c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{420 a}+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{35 a^2}+\frac{3 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{56 a^3}-\frac{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{35 a^4}","\frac{17 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{17 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{17 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{140 a^4 \sqrt{a^2 c x^2+c}}+\frac{\left(a^2 c x^2+c\right)^{5/2}}{105 a^4 c}-\frac{17 \left(a^2 c x^2+c\right)^{3/2}}{1260 a^4}-\frac{17 c \sqrt{a^2 c x^2+c}}{280 a^4}+\frac{1}{7} a^2 c x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{21} a c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{8}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{23 c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{420 a}+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{35 a^2}+\frac{3 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{56 a^3}-\frac{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{35 a^4}",1,"(-17*c*Sqrt[c + a^2*c*x^2])/(280*a^4) - (17*(c + a^2*c*x^2)^(3/2))/(1260*a^4) + (c + a^2*c*x^2)^(5/2)/(105*a^4*c) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(56*a^3) - (23*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(420*a) - (a*c*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/21 - (2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(35*a^4) + (c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(35*a^2) + (8*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/35 + (a^2*c*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/7 - (((17*I)/140)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2]) + (((17*I)/280)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2]) - (((17*I)/280)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2])","A",75,8,24,0.3333,1,"{4950, 4952, 261, 4890, 4886, 4930, 266, 43}"
316,1,531,0,3.1863074,"\int x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2 \, dx","Int[x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2,x]","-\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{41 c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{360 a^3}+\frac{1}{60} c x^3 \sqrt{a^2 c x^2+c}+\frac{c x \sqrt{a^2 c x^2+c}}{36 a^2}+\frac{1}{6} a^2 c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{15} a c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{7}{24} c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{19 c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{180 a}+\frac{c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{16 a^2}+\frac{31 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{360 a^3}","-\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{41 c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{360 a^3}+\frac{1}{60} c x^3 \sqrt{a^2 c x^2+c}+\frac{c x \sqrt{a^2 c x^2+c}}{36 a^2}+\frac{1}{6} a^2 c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{15} a c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{7}{24} c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{19 c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{180 a}+\frac{c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{16 a^2}+\frac{31 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{360 a^3}",1,"(c*x*Sqrt[c + a^2*c*x^2])/(36*a^2) + (c*x^3*Sqrt[c + a^2*c*x^2])/60 + (31*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(360*a^3) - (19*c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(180*a) - (a*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/15 + (c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(16*a^2) + (7*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/24 + (a^2*c*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/6 + ((I/8)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^3*Sqrt[c + a^2*c*x^2]) - (41*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(360*a^3) - ((I/8)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + ((I/8)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2]) - (c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2])","A",92,12,24,0.5000,1,"{4950, 4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589, 321}"
317,1,334,0,0.2321959,"\int x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2 \, dx","Int[x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2,x]","-\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{10 a^2 \sqrt{a^2 c x^2+c}}+\frac{\left(a^2 c x^2+c\right)^{3/2}}{30 a^2}+\frac{3 c \sqrt{a^2 c x^2+c}}{20 a^2}+\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}-\frac{x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{10 a}-\frac{3 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{20 a}","-\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{10 a^2 \sqrt{a^2 c x^2+c}}+\frac{\left(a^2 c x^2+c\right)^{3/2}}{30 a^2}+\frac{3 c \sqrt{a^2 c x^2+c}}{20 a^2}+\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^2}{5 a^2 c}-\frac{x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{10 a}-\frac{3 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{20 a}",1,"(3*c*Sqrt[c + a^2*c*x^2])/(20*a^2) + (c + a^2*c*x^2)^(3/2)/(30*a^2) - (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(20*a) - (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(10*a) + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/(5*a^2*c) + (((3*I)/10)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) - (((3*I)/20)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) + (((3*I)/20)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2])","A",5,4,22,0.1818,1,"{4930, 4878, 4890, 4886}"
318,1,438,0,0.3113589,"\int \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2 \, dx","Int[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2,x]","\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}+\frac{3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{4 a \sqrt{a^2 c x^2+c}}+\frac{5 c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{6 a}+\frac{1}{12} c x \sqrt{a^2 c x^2+c}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{3 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{4 a}+\frac{1}{4} x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{6 a}","\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}+\frac{3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{4 a \sqrt{a^2 c x^2+c}}+\frac{5 c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{6 a}+\frac{1}{12} c x \sqrt{a^2 c x^2+c}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{3 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{4 a}+\frac{1}{4} x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{6 a}",1,"(c*x*Sqrt[c + a^2*c*x^2])/12 - (3*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(4*a) - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(6*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/4 - (((3*I)/4)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a*Sqrt[c + a^2*c*x^2]) + (5*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(6*a) + (((3*I)/4)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((3*I)/4)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*a*Sqrt[c + a^2*c*x^2]) + (3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*a*Sqrt[c + a^2*c*x^2])","A",16,10,21,0.4762,1,"{4880, 4890, 4888, 4181, 2531, 2282, 6589, 217, 206, 195}"
319,1,530,0,0.8821953,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2}{x} \, dx","Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/x,x]","-\frac{7 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\frac{7 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\frac{2 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{14 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} c \sqrt{a^2 c x^2+c}+c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{3} a c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{1}{3} \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2","-\frac{7 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\frac{7 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\frac{2 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{14 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} c \sqrt{a^2 c x^2+c}+c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{3} a c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{1}{3} \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2",1,"(c*Sqrt[c + a^2*c*x^2])/3 - (a*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/3 + c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/3 + (((14*I)/3)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((7*I)/3)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (((7*I)/3)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",18,11,24,0.4583,1,"{4950, 4958, 4956, 4183, 2531, 2282, 6589, 4930, 4890, 4886, 4878}"
320,1,556,0,0.9676325,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2}{x^2} \, dx","Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/x^2,x]","\frac{2 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}+a c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{4 a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{2} a^2 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)","\frac{2 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}+a c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{4 a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{2} a^2 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)",1,"-(a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x + (a^2*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] - (4*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + a*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (3*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (3*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",26,13,24,0.5417,1,"{4950, 4944, 4958, 4954, 4890, 4888, 4181, 2531, 2282, 6589, 4880, 217, 206}"
321,1,567,0,1.6545874,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2}{x^3} \, dx","Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/x^3,x]","-\frac{2 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{4 i a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-a^2 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x^2}","-\frac{2 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{4 i a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-a^2 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x^2}",1,"-((a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) + a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x^2) + ((4*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (3*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - a^2*c^(3/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + ((3*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + ((2*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (3*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (3*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",38,15,24,0.6250,1,"{4950, 4962, 4944, 266, 63, 208, 4958, 4956, 4183, 2531, 2282, 6589, 4930, 4890, 4886}"
322,1,579,0,1.1464169,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2}{x^4} \, dx","Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/x^4,x]","\frac{7 i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{7 i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-\frac{14 a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{a^2 c \sqrt{a^2 c x^2+c}}{3 x}-\frac{a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 x^2}-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{3 x^3}","\frac{7 i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{7 i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-\frac{14 a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{a^2 c \sqrt{a^2 c x^2+c}}{3 x}-\frac{a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 x^2}-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{3 x^3}",1,"-(a^2*c*Sqrt[c + a^2*c*x^2])/(3*x) - (a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*x^2) - (a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(3*x^3) - ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] - (14*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(3*Sqrt[c + a^2*c*x^2]) + ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((7*I)/3)*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (((7*I)/3)*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (2*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",21,13,24,0.5417,1,"{4950, 4944, 4946, 4962, 264, 4958, 4954, 4890, 4888, 4181, 2531, 2282, 6589}"
323,1,578,0,10.7012802,"\int x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2 \, dx","Int[x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]","\frac{115 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{4032 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{4032 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 c^2 \sqrt{a^2 c x^2+c}}{4032 a^4}+\frac{1}{9} a^4 c^2 x^8 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{36} a^3 c^2 x^7 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{19}{63} a^2 c^2 x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{103 a c^2 x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{1512}+\frac{5}{21} c^2 x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{205 c^2 x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{6048 a}+\frac{c^2 x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{63 a^2}+\frac{47 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{1344 a^3}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{63 a^4}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2016 a^4 \sqrt{a^2 c x^2+c}}+\frac{\left(a^2 c x^2+c\right)^{7/2}}{252 a^4 c}-\frac{23 \left(a^2 c x^2+c\right)^{5/2}}{7560 a^4}-\frac{115 c \left(a^2 c x^2+c\right)^{3/2}}{18144 a^4}","\frac{115 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{4032 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{4032 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 c^2 \sqrt{a^2 c x^2+c}}{4032 a^4}+\frac{1}{9} a^4 c^2 x^8 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{36} a^3 c^2 x^7 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{19}{63} a^2 c^2 x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{103 a c^2 x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{1512}+\frac{5}{21} c^2 x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{205 c^2 x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{6048 a}+\frac{c^2 x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{63 a^2}+\frac{47 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{1344 a^3}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{63 a^4}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2016 a^4 \sqrt{a^2 c x^2+c}}+\frac{\left(a^2 c x^2+c\right)^{7/2}}{252 a^4 c}-\frac{23 \left(a^2 c x^2+c\right)^{5/2}}{7560 a^4}-\frac{115 c \left(a^2 c x^2+c\right)^{3/2}}{18144 a^4}",1,"(-115*c^2*Sqrt[c + a^2*c*x^2])/(4032*a^4) - (115*c*(c + a^2*c*x^2)^(3/2))/(18144*a^4) - (23*(c + a^2*c*x^2)^(5/2))/(7560*a^4) + (c + a^2*c*x^2)^(7/2)/(252*a^4*c) + (47*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(1344*a^3) - (205*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(6048*a) - (103*a*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/1512 - (a^3*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/36 - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(63*a^4) + (c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(63*a^2) + (5*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/21 + (19*a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/63 + (a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/9 - (((115*I)/2016)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2]) + (((115*I)/4032)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2]) - (((115*I)/4032)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2])","A",203,8,24,0.3333,1,"{4950, 4952, 261, 4890, 4886, 4930, 266, 43}"
324,1,638,0,8.3479946,"\int x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2 \, dx","Int[x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]","-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}+\frac{5 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}-\frac{5 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}+\frac{1}{168} a^2 c^2 x^5 \sqrt{a^2 c x^2+c}+\frac{29 c^2 x^3 \sqrt{a^2 c x^2+c}}{1680}+\frac{43 c^2 x \sqrt{a^2 c x^2+c}}{4032 a^2}+\frac{1}{8} a^4 c^2 x^7 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{28} a^3 c^2 x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{17}{48} a^2 c^2 x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{83}{840} a c^2 x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{59}{192} c^2 x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{737 c^2 x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{10080 a}+\frac{5 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{128 a^2}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{64 a^3 \sqrt{a^2 c x^2+c}}+\frac{1373 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{20160 a^3}-\frac{397 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{5040 a^3}","-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}+\frac{5 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}-\frac{5 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}+\frac{1}{168} a^2 c^2 x^5 \sqrt{a^2 c x^2+c}+\frac{29 c^2 x^3 \sqrt{a^2 c x^2+c}}{1680}+\frac{43 c^2 x \sqrt{a^2 c x^2+c}}{4032 a^2}+\frac{1}{8} a^4 c^2 x^7 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{1}{28} a^3 c^2 x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{17}{48} a^2 c^2 x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{83}{840} a c^2 x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{59}{192} c^2 x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{737 c^2 x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{10080 a}+\frac{5 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{128 a^2}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{64 a^3 \sqrt{a^2 c x^2+c}}+\frac{1373 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{20160 a^3}-\frac{397 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{5040 a^3}",1,"(43*c^2*x*Sqrt[c + a^2*c*x^2])/(4032*a^2) + (29*c^2*x^3*Sqrt[c + a^2*c*x^2])/1680 + (a^2*c^2*x^5*Sqrt[c + a^2*c*x^2])/168 + (1373*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(20160*a^3) - (737*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(10080*a) - (83*a*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/840 - (a^3*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/28 + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(128*a^2) + (59*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/192 + (17*a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/48 + (a^4*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/8 + (((5*I)/64)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^3*Sqrt[c + a^2*c*x^2]) - (397*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(5040*a^3) - (((5*I)/64)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (((5*I)/64)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) - (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2])","A",238,12,24,0.5000,1,"{4950, 4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589, 321}"
325,1,387,0,0.2808124,"\int x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2 \, dx","Int[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]","-\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{56 a^2 \sqrt{a^2 c x^2+c}}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{56 a^2 \sqrt{a^2 c x^2+c}}+\frac{5 c^2 \sqrt{a^2 c x^2+c}}{56 a^2}-\frac{5 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{56 a}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{28 a^2 \sqrt{a^2 c x^2+c}}+\frac{\left(a^2 c x^2+c\right)^{5/2}}{105 a^2}+\frac{5 c \left(a^2 c x^2+c\right)^{3/2}}{252 a^2}+\frac{\left(a^2 c x^2+c\right)^{7/2} \tan ^{-1}(a x)^2}{7 a^2 c}-\frac{x \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)}{21 a}-\frac{5 c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{84 a}","-\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{56 a^2 \sqrt{a^2 c x^2+c}}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{56 a^2 \sqrt{a^2 c x^2+c}}+\frac{5 c^2 \sqrt{a^2 c x^2+c}}{56 a^2}-\frac{5 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{56 a}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{28 a^2 \sqrt{a^2 c x^2+c}}+\frac{\left(a^2 c x^2+c\right)^{5/2}}{105 a^2}+\frac{5 c \left(a^2 c x^2+c\right)^{3/2}}{252 a^2}+\frac{\left(a^2 c x^2+c\right)^{7/2} \tan ^{-1}(a x)^2}{7 a^2 c}-\frac{x \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)}{21 a}-\frac{5 c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{84 a}",1,"(5*c^2*Sqrt[c + a^2*c*x^2])/(56*a^2) + (5*c*(c + a^2*c*x^2)^(3/2))/(252*a^2) + (c + a^2*c*x^2)^(5/2)/(105*a^2) - (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(56*a) - (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(84*a) - (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(21*a) + ((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^2)/(7*a^2*c) + (((5*I)/28)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) - (((5*I)/56)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) + (((5*I)/56)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2])","A",6,4,22,0.1818,1,"{4930, 4878, 4890, 4886}"
326,1,516,0,0.3904802,"\int \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2 \, dx","Int[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]","\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{8 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{8 a \sqrt{a^2 c x^2+c}}-\frac{5 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{8 a \sqrt{a^2 c x^2+c}}+\frac{5 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{8 a \sqrt{a^2 c x^2+c}}+\frac{17}{180} c^2 x \sqrt{a^2 c x^2+c}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{8 a \sqrt{a^2 c x^2+c}}+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{5 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{8 a}+\frac{259 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{360 a}+\frac{1}{60} c x \left(a^2 c x^2+c\right)^{3/2}+\frac{5}{24} c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2-\frac{5 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{36 a}+\frac{1}{6} x \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^2-\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)}{15 a}","\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{8 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{8 a \sqrt{a^2 c x^2+c}}-\frac{5 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{8 a \sqrt{a^2 c x^2+c}}+\frac{5 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{8 a \sqrt{a^2 c x^2+c}}+\frac{17}{180} c^2 x \sqrt{a^2 c x^2+c}-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{8 a \sqrt{a^2 c x^2+c}}+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{5 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{8 a}+\frac{259 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{360 a}+\frac{1}{60} c x \left(a^2 c x^2+c\right)^{3/2}+\frac{5}{24} c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2-\frac{5 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{36 a}+\frac{1}{6} x \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^2-\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)}{15 a}",1,"(17*c^2*x*Sqrt[c + a^2*c*x^2])/180 + (c*x*(c + a^2*c*x^2)^(3/2))/60 - (5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(8*a) - (5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(36*a) - ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(15*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/6 - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a*Sqrt[c + a^2*c*x^2]) + (259*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(360*a) + (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(8*a*Sqrt[c + a^2*c*x^2]) + (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(8*a*Sqrt[c + a^2*c*x^2])","A",21,10,21,0.4762,1,"{4880, 4890, 4888, 4181, 2531, 2282, 6589, 217, 206, 195}"
327,1,605,0,1.2629977,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2}{x} \, dx","Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/x,x]","-\frac{149 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{60 \sqrt{a^2 c x^2+c}}+\frac{149 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{60 \sqrt{a^2 c x^2+c}}+\frac{2 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{29}{60} c^2 \sqrt{a^2 c x^2+c}+\frac{149 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{30 \sqrt{a^2 c x^2+c}}+c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{29}{60} a c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{30} c \left(a^2 c x^2+c\right)^{3/2}+\frac{1}{3} c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2-\frac{1}{10} a c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)+\frac{1}{5} \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^2","-\frac{149 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{60 \sqrt{a^2 c x^2+c}}+\frac{149 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{60 \sqrt{a^2 c x^2+c}}+\frac{2 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{29}{60} c^2 \sqrt{a^2 c x^2+c}+\frac{149 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{30 \sqrt{a^2 c x^2+c}}+c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{29}{60} a c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{30} c \left(a^2 c x^2+c\right)^{3/2}+\frac{1}{3} c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2-\frac{1}{10} a c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)+\frac{1}{5} \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^2",1,"(29*c^2*Sqrt[c + a^2*c*x^2])/60 + (c*(c + a^2*c*x^2)^(3/2))/30 - (29*a*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/60 - (a*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/10 + c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 + (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/3 + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/5 + (((149*I)/30)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((149*I)/60)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (((149*I)/60)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",24,11,24,0.4583,1,"{4950, 4958, 4956, 4183, 2531, 2282, 6589, 4930, 4890, 4886, 4878}"
328,1,655,0,1.4075033,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2}{x^2} \, dx","Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/x^2,x]","\frac{2 i a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{15 i a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{4 \sqrt{a^2 c x^2+c}}-\frac{15 i a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{4 \sqrt{a^2 c x^2+c}}-\frac{15 a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{4 \sqrt{a^2 c x^2+c}}+\frac{15 a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{4 \sqrt{a^2 c x^2+c}}+\frac{1}{12} a^2 c^2 x \sqrt{a^2 c x^2+c}-\frac{15 i a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{4 \sqrt{a^2 c x^2+c}}+\frac{7}{8} a^2 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac{7}{4} a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{11}{6} a c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{4 a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{4} a^2 c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2-\frac{1}{6} a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)","\frac{2 i a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{15 i a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{4 \sqrt{a^2 c x^2+c}}-\frac{15 i a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{4 \sqrt{a^2 c x^2+c}}-\frac{15 a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{4 \sqrt{a^2 c x^2+c}}+\frac{15 a c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{4 \sqrt{a^2 c x^2+c}}+\frac{1}{12} a^2 c^2 x \sqrt{a^2 c x^2+c}-\frac{15 i a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{4 \sqrt{a^2 c x^2+c}}+\frac{7}{8} a^2 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac{7}{4} a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{11}{6} a c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{4 a c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{4} a^2 c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2-\frac{1}{6} a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)",1,"(a^2*c^2*x*Sqrt[c + a^2*c*x^2])/12 - (7*a*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/4 - (a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/6 - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x + (7*a^2*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/8 + (a^2*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/4 - (((15*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] - (4*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (11*a*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/6 + (((15*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((15*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (15*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*Sqrt[c + a^2*c*x^2]) + (15*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*Sqrt[c + a^2*c*x^2])","A",43,14,24,0.5833,1,"{4950, 4944, 4958, 4954, 4890, 4888, 4181, 2531, 2282, 6589, 4880, 217, 206, 195}"
329,1,661,0,2.6210956,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2}{x^3} \, dx","Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/x^3,x]","-\frac{13 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\frac{13 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{5 a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{5 a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} a^2 c^2 \sqrt{a^2 c x^2+c}-\frac{1}{3} a^3 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+\frac{26 i a^2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x^2}-a^2 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{5 a^2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} a^2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2","-\frac{13 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\frac{13 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{5 a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{5 a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} a^2 c^2 \sqrt{a^2 c x^2+c}-\frac{1}{3} a^3 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+\frac{26 i a^2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x^2}-a^2 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{5 a^2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} a^2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2",1,"(a^2*c^2*Sqrt[c + a^2*c*x^2])/3 - (a*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x - (a^3*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/3 + 2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2 - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x^2) + (a^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/3 + (((26*I)/3)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (5*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - a^2*c^(5/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + ((5*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((5*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((13*I)/3)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (((13*I)/3)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (5*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (5*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",57,16,24,0.6667,1,"{4950, 4962, 4944, 266, 63, 208, 4958, 4956, 4183, 2531, 2282, 6589, 4930, 4890, 4886, 4878}"
330,1,675,0,2.308881,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2}{x^4} \, dx","Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/x^4,x]","\frac{13 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{13 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{5 a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{5 a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a^2 c^2 \sqrt{a^2 c x^2+c}}{3 x}+\frac{1}{2} a^4 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac{a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 x^2}+a^3 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{26 a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{3 x^3}","\frac{13 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{13 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{5 a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{5 a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a^2 c^2 \sqrt{a^2 c x^2+c}}{3 x}+\frac{1}{2} a^4 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac{a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 x^2}+a^3 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{26 a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{3 x^3}",1,"-(a^2*c^2*Sqrt[c + a^2*c*x^2])/(3*x) - a^3*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (a*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*x^2) - (2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x + (a^4*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 - (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(3*x^3) - ((5*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] - (26*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(3*Sqrt[c + a^2*c*x^2]) + a^3*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + ((5*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((5*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((13*I)/3)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - (((13*I)/3)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (5*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (5*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",48,16,24,0.6667,1,"{4950, 4944, 4946, 4962, 264, 4958, 4954, 4890, 4888, 4181, 2531, 2282, 6589, 4880, 217, 206}"
331,1,315,0,0.425524,"\int \frac{x^3 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^3*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2],x]","\frac{5 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 a^4 \sqrt{a^2 c x^2+c}}-\frac{5 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 a^4 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c}}{3 a^4 c}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 a^2 c}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 a^4 c}-\frac{10 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{3 a^4 \sqrt{a^2 c x^2+c}}-\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 a^3 c}","\frac{5 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 a^4 \sqrt{a^2 c x^2+c}}-\frac{5 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 a^4 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c}}{3 a^4 c}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 a^2 c}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 a^4 c}-\frac{10 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{3 a^4 \sqrt{a^2 c x^2+c}}-\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 a^3 c}",1,"Sqrt[c + a^2*c*x^2]/(3*a^4*c) - (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*a^3*c) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*a^2*c) - (((10*I)/3)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2]) + (((5*I)/3)*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2]) - (((5*I)/3)*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*Sqrt[c + a^2*c*x^2])","A",8,5,24,0.2083,1,"{4952, 261, 4890, 4886, 4930}"
332,1,344,0,0.3338788,"\int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^2*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2],x]","-\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^3 c}+\frac{\tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a^3 \sqrt{c}}","-\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^3 c}+\frac{\tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a^3 \sqrt{c}}",1,"-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^3*c)) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a^2*c) + (I*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^3*Sqrt[c + a^2*c*x^2]) + ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^3*Sqrt[c]) - (I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (I*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2])","A",13,10,24,0.4167,1,"{4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589}"
333,1,220,0,0.1431219,"\int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2],x]","-\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{a^2 c}+\frac{4 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a^2 \sqrt{a^2 c x^2+c}}","-\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{a^2 c}+\frac{4 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a^2 \sqrt{a^2 c x^2+c}}",1,"(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(a^2*c) + ((4*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2]) + ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^2*Sqrt[c + a^2*c*x^2])","A",3,3,22,0.1364,1,"{4930, 4890, 4886}"
334,1,256,0,0.1538516,"\int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^2/Sqrt[c + a^2*c*x^2],x]","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a \sqrt{a^2 c x^2+c}}","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a \sqrt{a^2 c x^2+c}}",1,"((-2*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a*Sqrt[c + a^2*c*x^2]) + ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])","A",9,6,21,0.2857,1,"{4890, 4888, 4181, 2531, 2282, 6589}"
335,1,227,0,0.2525719,"\int \frac{\tan ^{-1}(a x)^2}{x \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^2/(x*Sqrt[c + a^2*c*x^2]),x]","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}",1,"(-2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",9,6,24,0.2500,1,"{4958, 4956, 4183, 2531, 2282, 6589}"
336,1,208,0,0.251814,"\int \frac{\tan ^{-1}(a x)^2}{x^2 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^2/(x^2*Sqrt[c + a^2*c*x^2]),x]","\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{c x}-\frac{4 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}","\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{c x}-\frac{4 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}",1,"-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(c*x)) - (4*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + ((2*I)*a*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((2*I)*a*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",3,3,24,0.1250,1,"{4944, 4958, 4954}"
337,1,328,0,0.4753943,"\int \frac{\tan ^{-1}(a x)^2}{x^3 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^2/(x^3*Sqrt[c + a^2*c*x^2]),x]","-\frac{i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 c x^2}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{\sqrt{c}}+\frac{a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}","-\frac{i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 c x^2}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{\sqrt{c}}+\frac{a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}",1,"-((a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c*x)) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*c*x^2) + (a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (a^2*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/Sqrt[c] - (I*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (I*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (a^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (a^2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",14,11,24,0.4583,1,"{4962, 4944, 266, 63, 208, 4958, 4956, 4183, 2531, 2282, 6589}"
338,1,311,0,0.6279908,"\int \frac{\tan ^{-1}(a x)^2}{x^4 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^2/(x^4*Sqrt[c + a^2*c*x^2]),x]","-\frac{5 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\frac{5 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c}}{3 c x}+\frac{2 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c x^2}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{10 a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}","-\frac{5 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\frac{5 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c}}{3 c x}+\frac{2 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c x^2}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{10 a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}",1,"-(a^2*Sqrt[c + a^2*c*x^2])/(3*c*x) - (a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c*x^2) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*c*x) + (10*a^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(3*Sqrt[c + a^2*c*x^2]) - (((5*I)/3)*a^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] + (((5*I)/3)*a^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2]","A",8,5,24,0.2083,1,"{4962, 264, 4958, 4954, 4944}"
339,1,305,0,0.3960278,"\int \frac{x^3 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2),x]","-\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{a^4 c^2}-\frac{2}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^2}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{4 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a^4 c \sqrt{a^2 c x^2+c}}-\frac{2 x \tan ^{-1}(a x)}{a^3 c \sqrt{a^2 c x^2+c}}","-\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{a^4 c^2}-\frac{2}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^2}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{4 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a^4 c \sqrt{a^2 c x^2+c}}-\frac{2 x \tan ^{-1}(a x)}{a^3 c \sqrt{a^2 c x^2+c}}",1,"-2/(a^4*c*Sqrt[c + a^2*c*x^2]) - (2*x*ArcTan[a*x])/(a^3*c*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^2/(a^4*c*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(a^4*c^2) + ((4*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^4*c*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*c*Sqrt[c + a^2*c*x^2]) + ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^4*c*Sqrt[c + a^2*c*x^2])","A",6,5,24,0.2083,1,"{4964, 4930, 4890, 4886, 4894}"
340,1,349,0,0.3411215,"\int \frac{x^2 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^2*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2),x]","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{2 x}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)^2}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)}{a^3 c \sqrt{a^2 c x^2+c}}","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{2 x}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)^2}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)}{a^3 c \sqrt{a^2 c x^2+c}}",1,"(2*x)/(a^2*c*Sqrt[c + a^2*c*x^2]) - (2*ArcTan[a*x])/(a^3*c*Sqrt[c + a^2*c*x^2]) - (x*ArcTan[a*x]^2)/(a^2*c*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^3*c*Sqrt[c + a^2*c*x^2]) + ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) + (2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2])","A",12,9,24,0.3750,1,"{4964, 4890, 4888, 4181, 2531, 2282, 6589, 4898, 191}"
341,1,78,0,0.1105774,"\int \frac{x \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2),x]","\frac{2}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^2}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)}{a c \sqrt{a^2 c x^2+c}}","\frac{2}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^2}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)}{a c \sqrt{a^2 c x^2+c}}",1,"2/(a^2*c*Sqrt[c + a^2*c*x^2]) + (2*x*ArcTan[a*x])/(a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^2/(a^2*c*Sqrt[c + a^2*c*x^2])","A",2,2,22,0.09091,1,"{4930, 4894}"
342,1,72,0,0.0455449,"\int \frac{\tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]^2/(c + a^2*c*x^2)^(3/2),x]","-\frac{2 x}{c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}+\frac{2 \tan ^{-1}(a x)}{a c \sqrt{a^2 c x^2+c}}","-\frac{2 x}{c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}+\frac{2 \tan ^{-1}(a x)}{a c \sqrt{a^2 c x^2+c}}",1,"(-2*x)/(c*Sqrt[c + a^2*c*x^2]) + (2*ArcTan[a*x])/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2])","A",2,2,21,0.09524,1,"{4898, 191}"
343,1,310,0,0.5062897,"\int \frac{\tan ^{-1}(a x)^2}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)^(3/2)),x]","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{2}{c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}-\frac{2 a x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{2}{c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}-\frac{2 a x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}",1,"-2/(c*Sqrt[c + a^2*c*x^2]) - (2*a*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^2/(c*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + (2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2])","A",12,9,24,0.3750,1,"{4966, 4958, 4956, 4183, 2531, 2282, 6589, 4930, 4894}"
344,1,293,0,0.4311924,"\int \frac{\tan ^{-1}(a x)^2}{x^2 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)^(3/2)),x]","\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{c^2 x}+\frac{2 a^2 x}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}-\frac{2 a \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{4 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}","\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{c^2 x}+\frac{2 a^2 x}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}-\frac{2 a \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{4 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}",1,"(2*a^2*x)/(c*Sqrt[c + a^2*c*x^2]) - (2*a*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (a^2*x*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(c^2*x) - (4*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c*Sqrt[c + a^2*c*x^2]) + ((2*I)*a*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/(c*Sqrt[c + a^2*c*x^2]) - ((2*I)*a*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c*Sqrt[c + a^2*c*x^2])","A",6,6,24,0.2500,1,"{4966, 4944, 4958, 4954, 4898, 191}"
345,1,422,0,1.1284044,"\int \frac{\tan ^{-1}(a x)^2}{x^3 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]^2/(x^3*(c + a^2*c*x^2)^(3/2)),x]","-\frac{3 i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{3 i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{3 a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{3 a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c^2 x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 c^2 x^2}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{c^{3/2}}+\frac{2 a^2}{c \sqrt{a^2 c x^2+c}}+\frac{2 a^3 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}+\frac{3 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}","-\frac{3 i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{3 i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{3 a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{3 a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c^2 x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 c^2 x^2}-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{c^{3/2}}+\frac{2 a^2}{c \sqrt{a^2 c x^2+c}}+\frac{2 a^3 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}+\frac{3 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}",1,"(2*a^2)/(c*Sqrt[c + a^2*c*x^2]) + (2*a^3*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c^2*x) - (a^2*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*c^2*x^2) + (3*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - (a^2*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/c^(3/2) - ((3*I)*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + ((3*I)*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + (3*a^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - (3*a^2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2])","A",27,14,24,0.5833,1,"{4966, 4962, 4944, 266, 63, 208, 4958, 4956, 4183, 2531, 2282, 6589, 4930, 4894}"
346,1,397,0,1.2016221,"\int \frac{\tan ^{-1}(a x)^2}{x^4 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)^(3/2)),x]","-\frac{11 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 c \sqrt{a^2 c x^2+c}}+\frac{11 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 c \sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c}}{3 c^2 x}+\frac{5 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c^2 x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c^2 x^2}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c^2 x^3}-\frac{2 a^4 x}{c \sqrt{a^2 c x^2+c}}+\frac{a^4 x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}+\frac{2 a^3 \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}+\frac{22 a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 c \sqrt{a^2 c x^2+c}}","-\frac{11 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 c \sqrt{a^2 c x^2+c}}+\frac{11 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 c \sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c}}{3 c^2 x}+\frac{5 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c^2 x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c^2 x^2}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c^2 x^3}-\frac{2 a^4 x}{c \sqrt{a^2 c x^2+c}}+\frac{a^4 x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}+\frac{2 a^3 \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}+\frac{22 a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{3 c \sqrt{a^2 c x^2+c}}",1,"(-2*a^4*x)/(c*Sqrt[c + a^2*c*x^2]) - (a^2*Sqrt[c + a^2*c*x^2])/(3*c^2*x) + (2*a^3*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c^2*x^2) + (a^4*x*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*c^2*x^3) + (5*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*c^2*x) + (22*a^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(3*c*Sqrt[c + a^2*c*x^2]) - (((11*I)/3)*a^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/(c*Sqrt[c + a^2*c*x^2]) + (((11*I)/3)*a^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c*Sqrt[c + a^2*c*x^2])","A",15,8,24,0.3333,1,"{4966, 4962, 264, 4958, 4954, 4944, 4898, 191}"
347,1,400,0,0.8175097,"\int \frac{x^5 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^5*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2),x]","-\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{32}{9 a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{10 x \tan ^{-1}(a x)}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{a^6 c^3}+\frac{5 \tan ^{-1}(a x)^2}{3 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{27 a^6 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 a^3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{32}{9 a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{10 x \tan ^{-1}(a x)}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{a^6 c^3}+\frac{5 \tan ^{-1}(a x)^2}{3 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{27 a^6 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 a^3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 c \left(a^2 c x^2+c\right)^{3/2}}",1,"2/(27*a^6*c*(c + a^2*c*x^2)^(3/2)) - 32/(9*a^6*c^2*Sqrt[c + a^2*c*x^2]) - (2*x^3*ArcTan[a*x])/(9*a^3*c*(c + a^2*c*x^2)^(3/2)) - (10*x*ArcTan[a*x])/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x]^2)/(3*a^4*c*(c + a^2*c*x^2)^(3/2)) + (5*ArcTan[a*x]^2)/(3*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(a^6*c^3) + ((4*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^6*c^2*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^6*c^2*Sqrt[c + a^2*c*x^2]) + ((2*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^6*c^2*Sqrt[c + a^2*c*x^2])","A",13,8,24,0.3333,1,"{4964, 4930, 4890, 4886, 4894, 4940, 266, 43}"
348,1,444,0,0.7690338,"\int \frac{x^4 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^4*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2),x]","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{22 x}{9 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)^2}{a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{22 \tan ^{-1}(a x)}{9 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x^3}{27 a^2 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^3 \tan ^{-1}(a x)^2}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 x^2 \tan ^{-1}(a x)}{9 a^3 c \left(a^2 c x^2+c\right)^{3/2}}","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{22 x}{9 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)^2}{a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{22 \tan ^{-1}(a x)}{9 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x^3}{27 a^2 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^3 \tan ^{-1}(a x)^2}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 x^2 \tan ^{-1}(a x)}{9 a^3 c \left(a^2 c x^2+c\right)^{3/2}}",1,"(2*x^3)/(27*a^2*c*(c + a^2*c*x^2)^(3/2)) + (22*x)/(9*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (2*x^2*ArcTan[a*x])/(9*a^3*c*(c + a^2*c*x^2)^(3/2)) - (22*ArcTan[a*x])/(9*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (x^3*ArcTan[a*x]^2)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (x*ArcTan[a*x]^2)/(a^4*c^2*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + (2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2])","A",17,12,24,0.5000,1,"{4964, 4890, 4888, 4181, 2531, 2282, 6589, 4898, 191, 4944, 4938, 4930}"
349,1,172,0,0.2827452,"\int \frac{x^3 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2),x]","\frac{14}{9 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 x \tan ^{-1}(a x)}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)^2}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 a^4 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}","\frac{14}{9 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 x \tan ^{-1}(a x)}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)^2}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 a^4 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}",1,"-2/(27*a^4*c*(c + a^2*c*x^2)^(3/2)) + 14/(9*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (2*x^3*ArcTan[a*x])/(9*a*c*(c + a^2*c*x^2)^(3/2)) + (4*x*ArcTan[a*x])/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (x^2*ArcTan[a*x]^2)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (2*ArcTan[a*x]^2)/(3*a^4*c^2*Sqrt[c + a^2*c*x^2])","A",6,5,24,0.2083,1,"{4940, 4930, 4894, 266, 43}"
350,1,139,0,0.2673827,"\int \frac{x^2 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^2*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2),x]","-\frac{4 x}{9 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 \tan ^{-1}(a x)}{9 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x^3}{27 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^3 \tan ^{-1}(a x)^2}{3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 x^2 \tan ^{-1}(a x)}{9 a c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{4 x}{9 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 \tan ^{-1}(a x)}{9 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x^3}{27 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^3 \tan ^{-1}(a x)^2}{3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 x^2 \tan ^{-1}(a x)}{9 a c \left(a^2 c x^2+c\right)^{3/2}}",1,"(-2*x^3)/(27*c*(c + a^2*c*x^2)^(3/2)) - (4*x)/(9*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (2*x^2*ArcTan[a*x])/(9*a*c*(c + a^2*c*x^2)^(3/2)) + (4*ArcTan[a*x])/(9*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (x^3*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3/2))","A",4,4,24,0.1667,1,"{4944, 4938, 4930, 191}"
351,1,137,0,0.1418508,"\int \frac{x \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2),x]","\frac{4}{9 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 x \tan ^{-1}(a x)}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{27 a^2 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{\tan ^{-1}(a x)^2}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 x \tan ^{-1}(a x)}{9 a c \left(a^2 c x^2+c\right)^{3/2}}","\frac{4}{9 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 x \tan ^{-1}(a x)}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{27 a^2 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{\tan ^{-1}(a x)^2}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 x \tan ^{-1}(a x)}{9 a c \left(a^2 c x^2+c\right)^{3/2}}",1,"2/(27*a^2*c*(c + a^2*c*x^2)^(3/2)) + 4/(9*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (2*x*ArcTan[a*x])/(9*a*c*(c + a^2*c*x^2)^(3/2)) + (4*x*ArcTan[a*x])/(9*a*c^2*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^2/(3*a^2*c*(c + a^2*c*x^2)^(3/2))","A",3,3,22,0.1364,1,"{4930, 4896, 4894}"
352,1,157,0,0.1024926,"\int \frac{\tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[ArcTan[a*x]^2/(c + a^2*c*x^2)^(5/2),x]","-\frac{40 x}{27 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 \tan ^{-1}(a x)}{3 a c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x}{27 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x \tan ^{-1}(a x)^2}{3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 \tan ^{-1}(a x)}{9 a c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{40 x}{27 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 \tan ^{-1}(a x)}{3 a c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x}{27 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x \tan ^{-1}(a x)^2}{3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 \tan ^{-1}(a x)}{9 a c \left(a^2 c x^2+c\right)^{3/2}}",1,"(-2*x)/(27*c*(c + a^2*c*x^2)^(3/2)) - (40*x)/(27*c^2*Sqrt[c + a^2*c*x^2]) + (2*ArcTan[a*x])/(9*a*c*(c + a^2*c*x^2)^(3/2)) + (4*ArcTan[a*x])/(3*a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^2)/(3*c^2*Sqrt[c + a^2*c*x^2])","A",5,4,21,0.1905,1,"{4900, 4898, 191, 192}"
353,1,389,0,0.7833998,"\int \frac{\tan ^{-1}(a x)^2}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)^(5/2)),x]","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{22}{9 c^2 \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^2}{c^2 \sqrt{a^2 c x^2+c}}-\frac{22 a x \tan ^{-1}(a x)}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{\tan ^{-1}(a x)^2}{3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 a x \tan ^{-1}(a x)}{9 c \left(a^2 c x^2+c\right)^{3/2}}","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{22}{9 c^2 \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^2}{c^2 \sqrt{a^2 c x^2+c}}-\frac{22 a x \tan ^{-1}(a x)}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{\tan ^{-1}(a x)^2}{3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 a x \tan ^{-1}(a x)}{9 c \left(a^2 c x^2+c\right)^{3/2}}",1,"-2/(27*c*(c + a^2*c*x^2)^(3/2)) - 22/(9*c^2*Sqrt[c + a^2*c*x^2]) - (2*a*x*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^(3/2)) - (22*a*x*ArcTan[a*x])/(9*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^2/(3*c*(c + a^2*c*x^2)^(3/2)) + ArcTan[a*x]^2/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + (2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2])","A",16,10,24,0.4167,1,"{4966, 4958, 4956, 4183, 2531, 2282, 6589, 4930, 4894, 4896}"
354,1,381,0,0.6855725,"\int \frac{\tan ^{-1}(a x)^2}{x^2 \left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)^(5/2)),x]","\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{94 a^2 x}{27 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 a^2 x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{10 a \tan ^{-1}(a x)}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{c^3 x}-\frac{4 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 a^2 x}{27 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{a^2 x \tan ^{-1}(a x)^2}{3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 a \tan ^{-1}(a x)}{9 c \left(a^2 c x^2+c\right)^{3/2}}","\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{94 a^2 x}{27 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 a^2 x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{10 a \tan ^{-1}(a x)}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{c^3 x}-\frac{4 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 a^2 x}{27 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{a^2 x \tan ^{-1}(a x)^2}{3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 a \tan ^{-1}(a x)}{9 c \left(a^2 c x^2+c\right)^{3/2}}",1,"(2*a^2*x)/(27*c*(c + a^2*c*x^2)^(3/2)) + (94*a^2*x)/(27*c^2*Sqrt[c + a^2*c*x^2]) - (2*a*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^(3/2)) - (10*a*ArcTan[a*x])/(3*c^2*Sqrt[c + a^2*c*x^2]) - (a^2*x*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3/2)) - (5*a^2*x*ArcTan[a*x]^2)/(3*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(c^3*x) - (4*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c^2*Sqrt[c + a^2*c*x^2]) + ((2*I)*a*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - ((2*I)*a*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(c^2*Sqrt[c + a^2*c*x^2])","A",12,8,24,0.3333,1,"{4966, 4944, 4958, 4954, 4898, 191, 4900, 192}"
355,0,0,0,0.0541744,"\int x^m \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2 \, dx","Int[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]","\int x^m \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2 \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)^2,x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
356,0,0,0,0.0353108,"\int x^m \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2 \, dx","Int[x^m*(c + a^2*c*x^2)*ArcTan[a*x]^2,x]","\int x^m \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2 \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right) \tan ^{-1}(a x)^2,x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)*ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
357,0,0,0,0.0621514,"\int \frac{x^m \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx","Int[(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]","\int \frac{x^m \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^2}{a^2 c x^2+c},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2), x]","A",0,0,0,0,-1,"{}"
358,0,0,0,0.0625476,"\int \frac{x^m \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2,x]","\int \frac{x^m \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^2} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^2}{\left(a^2 c x^2+c\right)^2},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2, x]","A",0,0,0,0,-1,"{}"
359,0,0,0,0.1094105,"\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2 \, dx","Int[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2,x]","\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2 \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2,x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
360,0,0,0,0.0977815,"\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx","Int[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]","\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx","\text{Int}\left(x^m \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2,x\right)",0,"Defer[Int][x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
361,0,0,0,0.1016261,"\int \frac{x^m \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^m*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2],x]","\int \frac{x^m \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2], x]","A",0,0,0,0,-1,"{}"
362,0,0,0,0.1149365,"\int \frac{x^m \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2),x]","\int \frac{x^m \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^2}{\left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2), x]","A",0,0,0,0,-1,"{}"
363,1,219,0,1.1137439,"\int x^3 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3 \, dx","Int[x^3*(c + a^2*c*x^2)*ArcTan[a*x]^3,x]","\frac{7 i c \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{30 a^4}+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^3-\frac{c x^2 \tan ^{-1}(a x)}{60 a^2}+\frac{c x}{15 a^3}+\frac{c x \tan ^{-1}(a x)^2}{4 a^3}-\frac{c \tan ^{-1}(a x)^3}{12 a^4}+\frac{7 i c \tan ^{-1}(a x)^2}{30 a^4}-\frac{c \tan ^{-1}(a x)}{15 a^4}+\frac{7 c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{15 a^4}-\frac{c x^3}{60 a}-\frac{1}{10} a c x^5 \tan ^{-1}(a x)^2+\frac{1}{4} c x^4 \tan ^{-1}(a x)^3+\frac{1}{20} c x^4 \tan ^{-1}(a x)-\frac{c x^3 \tan ^{-1}(a x)^2}{12 a}","\frac{7 i c \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{30 a^4}+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^3-\frac{c x^2 \tan ^{-1}(a x)}{60 a^2}+\frac{c x}{15 a^3}+\frac{c x \tan ^{-1}(a x)^2}{4 a^3}-\frac{c \tan ^{-1}(a x)^3}{12 a^4}+\frac{7 i c \tan ^{-1}(a x)^2}{30 a^4}-\frac{c \tan ^{-1}(a x)}{15 a^4}+\frac{7 c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{15 a^4}-\frac{c x^3}{60 a}-\frac{1}{10} a c x^5 \tan ^{-1}(a x)^2+\frac{1}{4} c x^4 \tan ^{-1}(a x)^3+\frac{1}{20} c x^4 \tan ^{-1}(a x)-\frac{c x^3 \tan ^{-1}(a x)^2}{12 a}",1,"(c*x)/(15*a^3) - (c*x^3)/(60*a) - (c*ArcTan[a*x])/(15*a^4) - (c*x^2*ArcTan[a*x])/(60*a^2) + (c*x^4*ArcTan[a*x])/20 + (((7*I)/30)*c*ArcTan[a*x]^2)/a^4 + (c*x*ArcTan[a*x]^2)/(4*a^3) - (c*x^3*ArcTan[a*x]^2)/(12*a) - (a*c*x^5*ArcTan[a*x]^2)/10 - (c*ArcTan[a*x]^3)/(12*a^4) + (c*x^4*ArcTan[a*x]^3)/4 + (a^2*c*x^6*ArcTan[a*x]^3)/6 + (7*c*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a^4) + (((7*I)/30)*c*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4","A",52,12,20,0.6000,1,"{4950, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 4846, 4884, 302}"
364,1,211,0,0.8819753,"\int x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3 \, dx","Int[x^2*(c + a^2*c*x^2)*ArcTan[a*x]^3,x]","-\frac{c \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{5 a^3}-\frac{2 i c \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{5 a^3}+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^3+\frac{c x \tan ^{-1}(a x)}{10 a^2}-\frac{2 i c \tan ^{-1}(a x)^3}{15 a^3}-\frac{c \tan ^{-1}(a x)^2}{20 a^3}-\frac{2 c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{5 a^3}-\frac{c x^2}{20 a}-\frac{3}{20} a c x^4 \tan ^{-1}(a x)^2+\frac{1}{3} c x^3 \tan ^{-1}(a x)^3+\frac{1}{10} c x^3 \tan ^{-1}(a x)-\frac{c x^2 \tan ^{-1}(a x)^2}{5 a}","-\frac{c \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{5 a^3}-\frac{2 i c \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{5 a^3}+\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)^3+\frac{c x \tan ^{-1}(a x)}{10 a^2}-\frac{2 i c \tan ^{-1}(a x)^3}{15 a^3}-\frac{c \tan ^{-1}(a x)^2}{20 a^3}-\frac{2 c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{5 a^3}-\frac{c x^2}{20 a}-\frac{3}{20} a c x^4 \tan ^{-1}(a x)^2+\frac{1}{3} c x^3 \tan ^{-1}(a x)^3+\frac{1}{10} c x^3 \tan ^{-1}(a x)-\frac{c x^2 \tan ^{-1}(a x)^2}{5 a}",1,"-(c*x^2)/(20*a) + (c*x*ArcTan[a*x])/(10*a^2) + (c*x^3*ArcTan[a*x])/10 - (c*ArcTan[a*x]^2)/(20*a^3) - (c*x^2*ArcTan[a*x]^2)/(5*a) - (3*a*c*x^4*ArcTan[a*x]^2)/20 - (((2*I)/15)*c*ArcTan[a*x]^3)/a^3 + (c*x^3*ArcTan[a*x]^3)/3 + (a^2*c*x^5*ArcTan[a*x]^3)/5 - (2*c*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(5*a^3) - (((2*I)/5)*c*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3 - (c*PolyLog[3, 1 - 2/(1 + I*a*x)])/(5*a^3)","A",34,12,20,0.6000,1,"{4950, 4852, 4916, 4846, 260, 4884, 4920, 4854, 4994, 6610, 266, 43}"
365,1,160,0,0.1290831,"\int x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3 \, dx","Int[x*(c + a^2*c*x^2)*ArcTan[a*x]^3,x]","-\frac{i c \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^2}+\frac{c \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^3}{4 a^2}-\frac{c x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{4 a}+\frac{c \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{4 a^2}-\frac{i c \tan ^{-1}(a x)^2}{2 a^2}-\frac{c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{a^2}-\frac{c x}{4 a}-\frac{c x \tan ^{-1}(a x)^2}{2 a}","-\frac{i c \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^2}+\frac{c \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^3}{4 a^2}-\frac{c x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{4 a}+\frac{c \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{4 a^2}-\frac{i c \tan ^{-1}(a x)^2}{2 a^2}-\frac{c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{a^2}-\frac{c x}{4 a}-\frac{c x \tan ^{-1}(a x)^2}{2 a}",1,"-(c*x)/(4*a) + (c*(1 + a^2*x^2)*ArcTan[a*x])/(4*a^2) - ((I/2)*c*ArcTan[a*x]^2)/a^2 - (c*x*ArcTan[a*x]^2)/(2*a) - (c*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/(4*a) + (c*(1 + a^2*x^2)^2*ArcTan[a*x]^3)/(4*a^2) - (c*ArcTan[a*x]*Log[2/(1 + I*a*x)])/a^2 - ((I/2)*c*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^2","A",8,8,18,0.4444,1,"{4930, 4880, 4846, 4920, 4854, 2402, 2315, 8}"
366,1,172,0,0.1824322,"\int \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3 \, dx","Int[(c + a^2*c*x^2)*ArcTan[a*x]^3,x]","\frac{c \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{a}+\frac{2 i c \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a}-\frac{c \log \left(a^2 x^2+1\right)}{2 a}+\frac{1}{3} c x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3-\frac{c \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{2 a}+\frac{2 i c \tan ^{-1}(a x)^3}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^3+c x \tan ^{-1}(a x)+\frac{2 c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{a}","\frac{c \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{a}+\frac{2 i c \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a}-\frac{c \log \left(a^2 x^2+1\right)}{2 a}+\frac{1}{3} c x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3-\frac{c \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{2 a}+\frac{2 i c \tan ^{-1}(a x)^3}{3 a}+\frac{2}{3} c x \tan ^{-1}(a x)^3+c x \tan ^{-1}(a x)+\frac{2 c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{a}",1,"c*x*ArcTan[a*x] - (c*(1 + a^2*x^2)*ArcTan[a*x]^2)/(2*a) + (((2*I)/3)*c*ArcTan[a*x]^3)/a + (2*c*x*ArcTan[a*x]^3)/3 + (c*x*(1 + a^2*x^2)*ArcTan[a*x]^3)/3 + (2*c*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/a - (c*Log[1 + a^2*x^2])/(2*a) + ((2*I)*c*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a + (c*PolyLog[3, 1 - 2/(1 + I*a*x)])/a","A",8,8,17,0.4706,1,"{4880, 4846, 4920, 4854, 4884, 4994, 6610, 260}"
367,1,276,0,0.5228153,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3}{x} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x,x]","-\frac{3}{2} i c \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{4} i c \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)-\frac{3}{4} i c \text{PolyLog}\left(4,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} c \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{3}{2} c \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)+\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+\frac{1}{2} c \tan ^{-1}(a x)^3-\frac{3}{2} i c \tan ^{-1}(a x)^2-\frac{3}{2} a c x \tan ^{-1}(a x)^2-3 c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)","-\frac{3}{2} i c \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{4} i c \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)-\frac{3}{4} i c \text{PolyLog}\left(4,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{2} i c \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} c \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{3}{2} c \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)+\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)^3+\frac{1}{2} c \tan ^{-1}(a x)^3-\frac{3}{2} i c \tan ^{-1}(a x)^2-\frac{3}{2} a c x \tan ^{-1}(a x)^2-3 c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)",1,"((-3*I)/2)*c*ArcTan[a*x]^2 - (3*a*c*x*ArcTan[a*x]^2)/2 + (c*ArcTan[a*x]^3)/2 + (a^2*c*x^2*ArcTan[a*x]^3)/2 + 2*c*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 3*c*ArcTan[a*x]*Log[2/(1 + I*a*x)] - ((3*I)/2)*c*PolyLog[2, 1 - 2/(1 + I*a*x)] - ((3*I)/2)*c*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((3*I)/2)*c*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (3*c*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*c*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((3*I)/4)*c*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((3*I)/4)*c*PolyLog[4, -1 + 2/(1 + I*a*x)]","A",17,14,20,0.7000,1,"{4950, 4850, 4988, 4884, 4994, 4998, 6610, 4852, 4916, 4846, 4920, 4854, 2402, 2315}"
368,1,169,0,0.39471,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3}{x^2} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^2,x]","\frac{3}{2} a c \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)+\frac{3}{2} a c \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)-3 i a c \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+3 i a c \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+a^2 c x \tan ^{-1}(a x)^3-\frac{c \tan ^{-1}(a x)^3}{x}+3 a c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2+3 a c \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2","\frac{3}{2} a c \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)+\frac{3}{2} a c \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)-3 i a c \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+3 i a c \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+a^2 c x \tan ^{-1}(a x)^3-\frac{c \tan ^{-1}(a x)^3}{x}+3 a c \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2+3 a c \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2",1,"-((c*ArcTan[a*x]^3)/x) + a^2*c*x*ArcTan[a*x]^3 + 3*a*c*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] + 3*a*c*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (3*I)*a*c*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (3*I)*a*c*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*a*c*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (3*a*c*PolyLog[3, 1 - 2/(1 + I*a*x)])/2","A",11,11,20,0.5500,1,"{4950, 4852, 4924, 4868, 4884, 4992, 6610, 4846, 4920, 4854, 4994}"
369,1,310,0,0.5568538,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3}{x^3} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^3,x]","-\frac{3}{2} i a^2 c \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+\frac{3}{4} i a^2 c \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)-\frac{3}{4} i a^2 c \text{PolyLog}\left(4,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-\frac{1}{2} a^2 c \tan ^{-1}(a x)^3-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2+3 a^2 c \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)-\frac{c \tan ^{-1}(a x)^3}{2 x^2}-\frac{3 a c \tan ^{-1}(a x)^2}{2 x}","-\frac{3}{2} i a^2 c \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+\frac{3}{4} i a^2 c \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)-\frac{3}{4} i a^2 c \text{PolyLog}\left(4,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-\frac{1}{2} a^2 c \tan ^{-1}(a x)^3-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2+3 a^2 c \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)-\frac{c \tan ^{-1}(a x)^3}{2 x^2}-\frac{3 a c \tan ^{-1}(a x)^2}{2 x}",1,"((-3*I)/2)*a^2*c*ArcTan[a*x]^2 - (3*a*c*ArcTan[a*x]^2)/(2*x) - (a^2*c*ArcTan[a*x]^3)/2 - (c*ArcTan[a*x]^3)/(2*x^2) + 2*a^2*c*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] + 3*a^2*c*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - ((3*I)/2)*a^2*c*PolyLog[2, -1 + 2/(1 - I*a*x)] - ((3*I)/2)*a^2*c*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((3*I)/2)*a^2*c*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (3*a^2*c*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*a^2*c*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((3*I)/4)*a^2*c*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((3*I)/4)*a^2*c*PolyLog[4, -1 + 2/(1 + I*a*x)]","A",16,12,20,0.6000,1,"{4950, 4852, 4918, 4924, 4868, 2447, 4884, 4850, 4988, 4994, 4998, 6610}"
370,1,189,0,0.5849203,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3}{x^4} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^4,x]","a^3 c \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)-2 i a^3 c \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-\frac{1}{2} a^3 c \log \left(a^2 x^2+1\right)+a^3 c \log (x)-\frac{2}{3} i a^3 c \tan ^{-1}(a x)^3-\frac{1}{2} a^3 c \tan ^{-1}(a x)^2-\frac{a^2 c \tan ^{-1}(a x)^3}{x}-\frac{a^2 c \tan ^{-1}(a x)}{x}+2 a^3 c \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2-\frac{a c \tan ^{-1}(a x)^2}{2 x^2}-\frac{c \tan ^{-1}(a x)^3}{3 x^3}","a^3 c \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)-2 i a^3 c \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-\frac{1}{2} a^3 c \log \left(a^2 x^2+1\right)+a^3 c \log (x)-\frac{2}{3} i a^3 c \tan ^{-1}(a x)^3-\frac{1}{2} a^3 c \tan ^{-1}(a x)^2-\frac{a^2 c \tan ^{-1}(a x)^3}{x}-\frac{a^2 c \tan ^{-1}(a x)}{x}+2 a^3 c \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2-\frac{a c \tan ^{-1}(a x)^2}{2 x^2}-\frac{c \tan ^{-1}(a x)^3}{3 x^3}",1,"-((a^2*c*ArcTan[a*x])/x) - (a^3*c*ArcTan[a*x]^2)/2 - (a*c*ArcTan[a*x]^2)/(2*x^2) - ((2*I)/3)*a^3*c*ArcTan[a*x]^3 - (c*ArcTan[a*x]^3)/(3*x^3) - (a^2*c*ArcTan[a*x]^3)/x + a^3*c*Log[x] - (a^3*c*Log[1 + a^2*x^2])/2 + 2*a^3*c*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (2*I)*a^3*c*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + a^3*c*PolyLog[3, -1 + 2/(1 - I*a*x)]","A",20,12,20,0.6000,1,"{4950, 4852, 4918, 266, 36, 29, 31, 4884, 4924, 4868, 4992, 6610}"
371,1,313,0,2.2829162,"\int x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3 \, dx","Int[x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^3,x]","\frac{2 i c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{21 a^4}+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3-\frac{3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac{1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)-\frac{5 c^2 x^2 \tan ^{-1}(a x)}{168 a^2}+\frac{c^2 x}{21 a^3}+\frac{c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac{c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac{2 i c^2 \tan ^{-1}(a x)^2}{21 a^4}-\frac{c^2 \tan ^{-1}(a x)}{21 a^4}+\frac{4 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{21 a^4}-\frac{1}{280} a c^2 x^5-\frac{c^2 x^3}{168 a}-\frac{1}{8} a c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac{1}{28} c^2 x^4 \tan ^{-1}(a x)-\frac{c^2 x^3 \tan ^{-1}(a x)^2}{24 a}","\frac{2 i c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{21 a^4}+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3-\frac{3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac{1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)-\frac{5 c^2 x^2 \tan ^{-1}(a x)}{168 a^2}+\frac{c^2 x}{21 a^3}+\frac{c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac{c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac{2 i c^2 \tan ^{-1}(a x)^2}{21 a^4}-\frac{c^2 \tan ^{-1}(a x)}{21 a^4}+\frac{4 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{21 a^4}-\frac{1}{280} a c^2 x^5-\frac{c^2 x^3}{168 a}-\frac{1}{8} a c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac{1}{28} c^2 x^4 \tan ^{-1}(a x)-\frac{c^2 x^3 \tan ^{-1}(a x)^2}{24 a}",1,"(c^2*x)/(21*a^3) - (c^2*x^3)/(168*a) - (a*c^2*x^5)/280 - (c^2*ArcTan[a*x])/(21*a^4) - (5*c^2*x^2*ArcTan[a*x])/(168*a^2) + (c^2*x^4*ArcTan[a*x])/28 + (a^2*c^2*x^6*ArcTan[a*x])/56 + (((2*I)/21)*c^2*ArcTan[a*x]^2)/a^4 + (c^2*x*ArcTan[a*x]^2)/(8*a^3) - (c^2*x^3*ArcTan[a*x]^2)/(24*a) - (a*c^2*x^5*ArcTan[a*x]^2)/8 - (3*a^3*c^2*x^7*ArcTan[a*x]^2)/56 - (c^2*ArcTan[a*x]^3)/(24*a^4) + (c^2*x^4*ArcTan[a*x]^3)/4 + (a^2*c^2*x^6*ArcTan[a*x]^3)/3 + (a^4*c^2*x^8*ArcTan[a*x]^3)/8 + (4*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(21*a^4) + (((2*I)/21)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4","A",106,12,22,0.5455,1,"{4948, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 4846, 4884, 302}"
372,1,321,0,1.7953339,"\int x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3 \, dx","Int[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^3,x]","-\frac{4 c^2 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{35 a^3}-\frac{8 i c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{35 a^3}+\frac{c^2 \log \left(a^2 x^2+1\right)}{30 a^3}+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac{1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac{1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)-\frac{c^2 x \tan ^{-1}(a x)}{70 a^2}-\frac{8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac{c^2 \tan ^{-1}(a x)^2}{140 a^3}-\frac{8 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{35 a^3}-\frac{1}{140} a c^2 x^4-\frac{11 c^2 x^2}{420 a}-\frac{27}{140} a c^2 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac{17}{210} c^2 x^3 \tan ^{-1}(a x)-\frac{4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a}","-\frac{4 c^2 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{35 a^3}-\frac{8 i c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{35 a^3}+\frac{c^2 \log \left(a^2 x^2+1\right)}{30 a^3}+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac{1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^3+\frac{1}{35} a^2 c^2 x^5 \tan ^{-1}(a x)-\frac{c^2 x \tan ^{-1}(a x)}{70 a^2}-\frac{8 i c^2 \tan ^{-1}(a x)^3}{105 a^3}+\frac{c^2 \tan ^{-1}(a x)^2}{140 a^3}-\frac{8 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{35 a^3}-\frac{1}{140} a c^2 x^4-\frac{11 c^2 x^2}{420 a}-\frac{27}{140} a c^2 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^3+\frac{17}{210} c^2 x^3 \tan ^{-1}(a x)-\frac{4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a}",1,"(-11*c^2*x^2)/(420*a) - (a*c^2*x^4)/140 - (c^2*x*ArcTan[a*x])/(70*a^2) + (17*c^2*x^3*ArcTan[a*x])/210 + (a^2*c^2*x^5*ArcTan[a*x])/35 + (c^2*ArcTan[a*x]^2)/(140*a^3) - (4*c^2*x^2*ArcTan[a*x]^2)/(35*a) - (27*a*c^2*x^4*ArcTan[a*x]^2)/140 - (a^3*c^2*x^6*ArcTan[a*x]^2)/14 - (((8*I)/105)*c^2*ArcTan[a*x]^3)/a^3 + (c^2*x^3*ArcTan[a*x]^3)/3 + (2*a^2*c^2*x^5*ArcTan[a*x]^3)/5 + (a^4*c^2*x^7*ArcTan[a*x]^3)/7 - (8*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(35*a^3) + (c^2*Log[1 + a^2*x^2])/(30*a^3) - (((8*I)/35)*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3 - (4*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/(35*a^3)","A",73,12,22,0.5455,1,"{4948, 4852, 4916, 4846, 260, 4884, 4920, 4854, 4994, 6610, 266, 43}"
373,1,242,0,0.1880065,"\int x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3 \, dx","Int[x*(c + a^2*c*x^2)^2*ArcTan[a*x]^3,x]","-\frac{4 i c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{15 a^2}-\frac{c^2 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}{10 a}-\frac{2 c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{15 a}+\frac{c^2 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^3}{6 a^2}+\frac{c^2 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}{20 a^2}+\frac{2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{15 a^2}-\frac{4 i c^2 \tan ^{-1}(a x)^2}{15 a^2}-\frac{8 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{15 a^2}-\frac{1}{60} a c^2 x^3-\frac{11 c^2 x}{60 a}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a}","-\frac{4 i c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{15 a^2}-\frac{c^2 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}{10 a}-\frac{2 c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{15 a}+\frac{c^2 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^3}{6 a^2}+\frac{c^2 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}{20 a^2}+\frac{2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{15 a^2}-\frac{4 i c^2 \tan ^{-1}(a x)^2}{15 a^2}-\frac{8 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{15 a^2}-\frac{1}{60} a c^2 x^3-\frac{11 c^2 x}{60 a}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a}",1,"(-11*c^2*x)/(60*a) - (a*c^2*x^3)/60 + (2*c^2*(1 + a^2*x^2)*ArcTan[a*x])/(15*a^2) + (c^2*(1 + a^2*x^2)^2*ArcTan[a*x])/(20*a^2) - (((4*I)/15)*c^2*ArcTan[a*x]^2)/a^2 - (4*c^2*x*ArcTan[a*x]^2)/(15*a) - (2*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/(15*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(10*a) + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x]^3)/(6*a^2) - (8*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a^2) - (((4*I)/15)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^2","A",10,8,20,0.4000,1,"{4930, 4880, 4846, 4920, 4854, 2402, 2315, 8}"
374,1,289,0,0.2452335,"\int \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3 \, dx","Int[(c + a^2*c*x^2)^2*ArcTan[a*x]^3,x]","\frac{4 c^2 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{5 a}+\frac{8 i c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{5 a}-\frac{c^2 \left(a^2 x^2+1\right)}{20 a}-\frac{c^2 \log \left(a^2 x^2+1\right)}{2 a}+\frac{1}{5} c^2 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^3+\frac{4}{15} c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3-\frac{3 c^2 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}{20 a}-\frac{2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{5 a}+\frac{1}{10} c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)+\frac{8 i c^2 \tan ^{-1}(a x)^3}{15 a}+\frac{8}{15} c^2 x \tan ^{-1}(a x)^3+c^2 x \tan ^{-1}(a x)+\frac{8 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{5 a}","\frac{4 c^2 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{5 a}+\frac{8 i c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{5 a}-\frac{c^2 \left(a^2 x^2+1\right)}{20 a}-\frac{c^2 \log \left(a^2 x^2+1\right)}{2 a}+\frac{1}{5} c^2 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^3+\frac{4}{15} c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3-\frac{3 c^2 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}{20 a}-\frac{2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{5 a}+\frac{1}{10} c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)+\frac{8 i c^2 \tan ^{-1}(a x)^3}{15 a}+\frac{8}{15} c^2 x \tan ^{-1}(a x)^3+c^2 x \tan ^{-1}(a x)+\frac{8 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{5 a}",1,"-(c^2*(1 + a^2*x^2))/(20*a) + c^2*x*ArcTan[a*x] + (c^2*x*(1 + a^2*x^2)*ArcTan[a*x])/10 - (2*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2)/(5*a) - (3*c^2*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(20*a) + (((8*I)/15)*c^2*ArcTan[a*x]^3)/a + (8*c^2*x*ArcTan[a*x]^3)/15 + (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^3)/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^3)/5 + (8*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(5*a) - (c^2*Log[1 + a^2*x^2])/(2*a) + (((8*I)/5)*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a + (4*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/(5*a)","A",12,9,19,0.4737,1,"{4880, 4846, 4920, 4854, 4884, 4994, 6610, 260, 4878}"
375,1,370,0,0.9703994,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3}{x} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x]^3)/x,x]","-2 i c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{4} i c^2 \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)-\frac{3}{4} i c^2 \text{PolyLog}\left(4,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} i c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{2} i c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} c^2 \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{3}{2} c^2 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^3-\frac{1}{4} a^3 c^2 x^3 \tan ^{-1}(a x)^2+a^2 c^2 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^2 c^2 x^2 \tan ^{-1}(a x)-\frac{1}{4} a c^2 x-\frac{9}{4} a c^2 x \tan ^{-1}(a x)^2+\frac{3}{4} c^2 \tan ^{-1}(a x)^3-2 i c^2 \tan ^{-1}(a x)^2+\frac{1}{4} c^2 \tan ^{-1}(a x)-4 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)","-2 i c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{4} i c^2 \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)-\frac{3}{4} i c^2 \text{PolyLog}\left(4,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} i c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{2} i c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} c^2 \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{3}{2} c^2 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)+\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)^3-\frac{1}{4} a^3 c^2 x^3 \tan ^{-1}(a x)^2+a^2 c^2 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^2 c^2 x^2 \tan ^{-1}(a x)-\frac{1}{4} a c^2 x-\frac{9}{4} a c^2 x \tan ^{-1}(a x)^2+\frac{3}{4} c^2 \tan ^{-1}(a x)^3-2 i c^2 \tan ^{-1}(a x)^2+\frac{1}{4} c^2 \tan ^{-1}(a x)-4 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)",1,"-(a*c^2*x)/4 + (c^2*ArcTan[a*x])/4 + (a^2*c^2*x^2*ArcTan[a*x])/4 - (2*I)*c^2*ArcTan[a*x]^2 - (9*a*c^2*x*ArcTan[a*x]^2)/4 - (a^3*c^2*x^3*ArcTan[a*x]^2)/4 + (3*c^2*ArcTan[a*x]^3)/4 + a^2*c^2*x^2*ArcTan[a*x]^3 + (a^4*c^2*x^4*ArcTan[a*x]^3)/4 + 2*c^2*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 4*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)] - (2*I)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)] - ((3*I)/2)*c^2*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((3*I)/2)*c^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (3*c^2*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*c^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((3*I)/4)*c^2*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((3*I)/4)*c^2*PolyLog[4, -1 + 2/(1 + I*a*x)]","A",36,16,22,0.7273,1,"{4948, 4850, 4988, 4884, 4994, 4998, 6610, 4852, 4916, 4846, 4920, 4854, 2402, 2315, 321, 203}"
376,1,284,0,0.7512856,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3}{x^2} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x]^3)/x^2,x]","\frac{3}{2} a c^2 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)+\frac{5}{2} a c^2 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)-3 i a c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+5 i a c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)-\frac{1}{2} a c^2 \log \left(a^2 x^2+1\right)+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3-\frac{1}{2} a^3 c^2 x^2 \tan ^{-1}(a x)^2+2 a^2 c^2 x \tan ^{-1}(a x)^3+a^2 c^2 x \tan ^{-1}(a x)+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^3-\frac{1}{2} a c^2 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{x}+5 a c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2+3 a c^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2","\frac{3}{2} a c^2 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)+\frac{5}{2} a c^2 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)-3 i a c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+5 i a c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)-\frac{1}{2} a c^2 \log \left(a^2 x^2+1\right)+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3-\frac{1}{2} a^3 c^2 x^2 \tan ^{-1}(a x)^2+2 a^2 c^2 x \tan ^{-1}(a x)^3+a^2 c^2 x \tan ^{-1}(a x)+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^3-\frac{1}{2} a c^2 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{x}+5 a c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2+3 a c^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2",1,"a^2*c^2*x*ArcTan[a*x] - (a*c^2*ArcTan[a*x]^2)/2 - (a^3*c^2*x^2*ArcTan[a*x]^2)/2 + ((2*I)/3)*a*c^2*ArcTan[a*x]^3 - (c^2*ArcTan[a*x]^3)/x + 2*a^2*c^2*x*ArcTan[a*x]^3 + (a^4*c^2*x^3*ArcTan[a*x]^3)/3 + 5*a*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] - (a*c^2*Log[1 + a^2*x^2])/2 + 3*a*c^2*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (3*I)*a*c^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (5*I)*a*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*a*c^2*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (5*a*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/2","A",23,13,22,0.5909,1,"{4948, 4846, 4920, 4854, 4884, 4994, 6610, 4852, 4924, 4868, 4992, 4916, 260}"
377,1,399,0,0.7962962,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3}{x^3} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x]^3)/x^3,x]","-\frac{3}{2} i a^2 c^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-\frac{3}{2} i a^2 c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{2} i a^2 c^2 \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)-\frac{3}{2} i a^2 c^2 \text{PolyLog}\left(4,-1+\frac{2}{1+i a x}\right)-3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)-3 a^2 c^2 \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+3 a^2 c^2 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^3-\frac{3}{2} a^3 c^2 x \tan ^{-1}(a x)^2-3 i a^2 c^2 \tan ^{-1}(a x)^2-3 a^2 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+3 a^2 c^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)+4 a^2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)-\frac{c^2 \tan ^{-1}(a x)^3}{2 x^2}-\frac{3 a c^2 \tan ^{-1}(a x)^2}{2 x}","-\frac{3}{2} i a^2 c^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-\frac{3}{2} i a^2 c^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{2} i a^2 c^2 \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)-\frac{3}{2} i a^2 c^2 \text{PolyLog}\left(4,-1+\frac{2}{1+i a x}\right)-3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+3 i a^2 c^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)-3 a^2 c^2 \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+3 a^2 c^2 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)+\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^3-\frac{3}{2} a^3 c^2 x \tan ^{-1}(a x)^2-3 i a^2 c^2 \tan ^{-1}(a x)^2-3 a^2 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+3 a^2 c^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)+4 a^2 c^2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)-\frac{c^2 \tan ^{-1}(a x)^3}{2 x^2}-\frac{3 a c^2 \tan ^{-1}(a x)^2}{2 x}",1,"(-3*I)*a^2*c^2*ArcTan[a*x]^2 - (3*a*c^2*ArcTan[a*x]^2)/(2*x) - (3*a^3*c^2*x*ArcTan[a*x]^2)/2 - (c^2*ArcTan[a*x]^3)/(2*x^2) + (a^4*c^2*x^2*ArcTan[a*x]^3)/2 + 4*a^2*c^2*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 3*a^2*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)] + 3*a^2*c^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - ((3*I)/2)*a^2*c^2*PolyLog[2, -1 + 2/(1 - I*a*x)] - ((3*I)/2)*a^2*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)] - (3*I)*a^2*c^2*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*I)*a^2*c^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - 3*a^2*c^2*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)] + 3*a^2*c^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)] + ((3*I)/2)*a^2*c^2*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((3*I)/2)*a^2*c^2*PolyLog[4, -1 + 2/(1 + I*a*x)]","A",25,18,22,0.8182,1,"{4948, 4852, 4918, 4924, 4868, 2447, 4884, 4850, 4988, 4994, 4998, 6610, 4916, 4846, 4920, 4854, 2402, 2315}"
378,1,311,0,0.7891845,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3}{x^4} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x]^3)/x^4,x]","\frac{5}{2} a^3 c^2 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)+\frac{3}{2} a^3 c^2 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)-5 i a^3 c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+3 i a^3 c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)-\frac{1}{2} a^3 c^2 \log \left(a^2 x^2+1\right)+a^3 c^2 \log (x)+a^4 c^2 x \tan ^{-1}(a x)^3-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^3-\frac{1}{2} a^3 c^2 \tan ^{-1}(a x)^2-\frac{2 a^2 c^2 \tan ^{-1}(a x)^3}{x}-\frac{a^2 c^2 \tan ^{-1}(a x)}{x}+3 a^3 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2+5 a^3 c^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2-\frac{a c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac{c^2 \tan ^{-1}(a x)^3}{3 x^3}","\frac{5}{2} a^3 c^2 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)+\frac{3}{2} a^3 c^2 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)-5 i a^3 c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+3 i a^3 c^2 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)-\frac{1}{2} a^3 c^2 \log \left(a^2 x^2+1\right)+a^3 c^2 \log (x)+a^4 c^2 x \tan ^{-1}(a x)^3-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^3-\frac{1}{2} a^3 c^2 \tan ^{-1}(a x)^2-\frac{2 a^2 c^2 \tan ^{-1}(a x)^3}{x}-\frac{a^2 c^2 \tan ^{-1}(a x)}{x}+3 a^3 c^2 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2+5 a^3 c^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2-\frac{a c^2 \tan ^{-1}(a x)^2}{2 x^2}-\frac{c^2 \tan ^{-1}(a x)^3}{3 x^3}",1,"-((a^2*c^2*ArcTan[a*x])/x) - (a^3*c^2*ArcTan[a*x]^2)/2 - (a*c^2*ArcTan[a*x]^2)/(2*x^2) - ((2*I)/3)*a^3*c^2*ArcTan[a*x]^3 - (c^2*ArcTan[a*x]^3)/(3*x^3) - (2*a^2*c^2*ArcTan[a*x]^3)/x + a^4*c^2*x*ArcTan[a*x]^3 + a^3*c^2*Log[x] + 3*a^3*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] - (a^3*c^2*Log[1 + a^2*x^2])/2 + 5*a^3*c^2*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (5*I)*a^3*c^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (3*I)*a^3*c^2*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (5*a^3*c^2*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (3*a^3*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/2","A",26,16,22,0.7273,1,"{4948, 4846, 4920, 4854, 4884, 4994, 6610, 4852, 4918, 266, 36, 29, 31, 4924, 4868, 4992}"
379,1,381,0,3.7253155,"\int x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3 \, dx","Int[x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^3,x]","\frac{26 i c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{525 a^4}-\frac{1}{840} a^3 c^3 x^7+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3-\frac{1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)-\frac{33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}-\frac{107 c^3 x^2 \tan ^{-1}(a x)}{4200 a^2}+\frac{389 c^3 x}{12600 a^3}+\frac{3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac{c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac{26 i c^3 \tan ^{-1}(a x)^2}{525 a^4}-\frac{389 c^3 \tan ^{-1}(a x)}{12600 a^4}+\frac{52 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{525 a^4}-\frac{1}{252} a c^3 x^5-\frac{17 c^3 x^3}{9450 a}-\frac{27}{200} a c^3 x^5 \tan ^{-1}(a x)^2+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{53 c^3 x^4 \tan ^{-1}(a x)}{2100}-\frac{c^3 x^3 \tan ^{-1}(a x)^2}{40 a}","\frac{26 i c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{525 a^4}-\frac{1}{840} a^3 c^3 x^7+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3-\frac{1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac{1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)-\frac{33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac{71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}-\frac{107 c^3 x^2 \tan ^{-1}(a x)}{4200 a^2}+\frac{389 c^3 x}{12600 a^3}+\frac{3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac{c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac{26 i c^3 \tan ^{-1}(a x)^2}{525 a^4}-\frac{389 c^3 \tan ^{-1}(a x)}{12600 a^4}+\frac{52 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{525 a^4}-\frac{1}{252} a c^3 x^5-\frac{17 c^3 x^3}{9450 a}-\frac{27}{200} a c^3 x^5 \tan ^{-1}(a x)^2+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac{53 c^3 x^4 \tan ^{-1}(a x)}{2100}-\frac{c^3 x^3 \tan ^{-1}(a x)^2}{40 a}",1,"(389*c^3*x)/(12600*a^3) - (17*c^3*x^3)/(9450*a) - (a*c^3*x^5)/252 - (a^3*c^3*x^7)/840 - (389*c^3*ArcTan[a*x])/(12600*a^4) - (107*c^3*x^2*ArcTan[a*x])/(4200*a^2) + (53*c^3*x^4*ArcTan[a*x])/2100 + (71*a^2*c^3*x^6*ArcTan[a*x])/2520 + (a^4*c^3*x^8*ArcTan[a*x])/120 + (((26*I)/525)*c^3*ArcTan[a*x]^2)/a^4 + (3*c^3*x*ArcTan[a*x]^2)/(40*a^3) - (c^3*x^3*ArcTan[a*x]^2)/(40*a) - (27*a*c^3*x^5*ArcTan[a*x]^2)/200 - (33*a^3*c^3*x^7*ArcTan[a*x]^2)/280 - (a^5*c^3*x^9*ArcTan[a*x]^2)/30 - (c^3*ArcTan[a*x]^3)/(40*a^4) + (c^3*x^4*ArcTan[a*x]^3)/4 + (a^2*c^3*x^6*ArcTan[a*x]^3)/2 + (3*a^4*c^3*x^8*ArcTan[a*x]^3)/8 + (a^6*c^3*x^10*ArcTan[a*x]^3)/10 + (52*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(525*a^4) + (((26*I)/525)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^4","A",184,12,22,0.5455,1,"{4948, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 4846, 4884, 302}"
380,1,389,0,3.0392298,"\int x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3 \, dx","Int[x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^3,x]","-\frac{8 c^3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{105 a^3}-\frac{16 i c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{105 a^3}-\frac{1}{504} a^3 c^3 x^6+\frac{31 c^3 \log \left(a^2 x^2+1\right)}{945 a^3}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^3-\frac{1}{24} a^5 c^3 x^8 \tan ^{-1}(a x)^2+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^3+\frac{1}{84} a^4 c^3 x^7 \tan ^{-1}(a x)-\frac{10}{63} a^3 c^3 x^6 \tan ^{-1}(a x)^2+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^3+\frac{59 a^2 c^3 x^5 \tan ^{-1}(a x)}{1260}-\frac{47 c^3 x \tan ^{-1}(a x)}{1260 a^2}-\frac{16 i c^3 \tan ^{-1}(a x)^3}{315 a^3}+\frac{47 c^3 \tan ^{-1}(a x)^2}{2520 a^3}-\frac{16 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{105 a^3}-\frac{11 a c^3 x^4}{1260}-\frac{107 c^3 x^2}{7560 a}-\frac{89}{420} a c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^3+\frac{239 c^3 x^3 \tan ^{-1}(a x)}{3780}-\frac{8 c^3 x^2 \tan ^{-1}(a x)^2}{105 a}","-\frac{8 c^3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{105 a^3}-\frac{16 i c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{105 a^3}-\frac{1}{504} a^3 c^3 x^6+\frac{31 c^3 \log \left(a^2 x^2+1\right)}{945 a^3}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^3-\frac{1}{24} a^5 c^3 x^8 \tan ^{-1}(a x)^2+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^3+\frac{1}{84} a^4 c^3 x^7 \tan ^{-1}(a x)-\frac{10}{63} a^3 c^3 x^6 \tan ^{-1}(a x)^2+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^3+\frac{59 a^2 c^3 x^5 \tan ^{-1}(a x)}{1260}-\frac{47 c^3 x \tan ^{-1}(a x)}{1260 a^2}-\frac{16 i c^3 \tan ^{-1}(a x)^3}{315 a^3}+\frac{47 c^3 \tan ^{-1}(a x)^2}{2520 a^3}-\frac{16 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{105 a^3}-\frac{11 a c^3 x^4}{1260}-\frac{107 c^3 x^2}{7560 a}-\frac{89}{420} a c^3 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^3+\frac{239 c^3 x^3 \tan ^{-1}(a x)}{3780}-\frac{8 c^3 x^2 \tan ^{-1}(a x)^2}{105 a}",1,"(-107*c^3*x^2)/(7560*a) - (11*a*c^3*x^4)/1260 - (a^3*c^3*x^6)/504 - (47*c^3*x*ArcTan[a*x])/(1260*a^2) + (239*c^3*x^3*ArcTan[a*x])/3780 + (59*a^2*c^3*x^5*ArcTan[a*x])/1260 + (a^4*c^3*x^7*ArcTan[a*x])/84 + (47*c^3*ArcTan[a*x]^2)/(2520*a^3) - (8*c^3*x^2*ArcTan[a*x]^2)/(105*a) - (89*a*c^3*x^4*ArcTan[a*x]^2)/420 - (10*a^3*c^3*x^6*ArcTan[a*x]^2)/63 - (a^5*c^3*x^8*ArcTan[a*x]^2)/24 - (((16*I)/315)*c^3*ArcTan[a*x]^3)/a^3 + (c^3*x^3*ArcTan[a*x]^3)/3 + (3*a^2*c^3*x^5*ArcTan[a*x]^3)/5 + (3*a^4*c^3*x^7*ArcTan[a*x]^3)/7 + (a^6*c^3*x^9*ArcTan[a*x]^3)/9 - (16*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(105*a^3) + (31*c^3*Log[1 + a^2*x^2])/(945*a^3) - (((16*I)/105)*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3 - (8*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(105*a^3)","A",132,12,22,0.5455,1,"{4948, 4852, 4916, 4846, 260, 4884, 4920, 4854, 4994, 6610, 266, 43}"
381,1,308,0,0.2542165,"\int x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3 \, dx","Int[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^3,x]","-\frac{6 i c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{35 a^2}-\frac{1}{280} a^3 c^3 x^5-\frac{3 c^3 x \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^2}{56 a}-\frac{9 c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}{140 a}-\frac{3 c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{35 a}+\frac{c^3 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x)^3}{8 a^2}+\frac{c^3 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)}{56 a^2}+\frac{9 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}{280 a^2}+\frac{3 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{35 a^2}-\frac{6 i c^3 \tan ^{-1}(a x)^2}{35 a^2}-\frac{12 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{35 a^2}-\frac{19}{840} a c^3 x^3-\frac{19 c^3 x}{140 a}-\frac{6 c^3 x \tan ^{-1}(a x)^2}{35 a}","-\frac{6 i c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{35 a^2}-\frac{1}{280} a^3 c^3 x^5-\frac{3 c^3 x \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^2}{56 a}-\frac{9 c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}{140 a}-\frac{3 c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{35 a}+\frac{c^3 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x)^3}{8 a^2}+\frac{c^3 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)}{56 a^2}+\frac{9 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}{280 a^2}+\frac{3 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{35 a^2}-\frac{6 i c^3 \tan ^{-1}(a x)^2}{35 a^2}-\frac{12 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{35 a^2}-\frac{19}{840} a c^3 x^3-\frac{19 c^3 x}{140 a}-\frac{6 c^3 x \tan ^{-1}(a x)^2}{35 a}",1,"(-19*c^3*x)/(140*a) - (19*a*c^3*x^3)/840 - (a^3*c^3*x^5)/280 + (3*c^3*(1 + a^2*x^2)*ArcTan[a*x])/(35*a^2) + (9*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])/(280*a^2) + (c^3*(1 + a^2*x^2)^3*ArcTan[a*x])/(56*a^2) - (((6*I)/35)*c^3*ArcTan[a*x]^2)/a^2 - (6*c^3*x*ArcTan[a*x]^2)/(35*a) - (3*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^2)/(35*a) - (9*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(140*a) - (3*c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/(56*a) + (c^3*(1 + a^2*x^2)^4*ArcTan[a*x]^3)/(8*a^2) - (12*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(35*a^2) - (((6*I)/35)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^2","A",13,9,20,0.4500,1,"{4930, 4880, 4846, 4920, 4854, 2402, 2315, 8, 194}"
382,1,388,0,0.3404024,"\int \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3 \, dx","Int[(c + a^2*c*x^2)^3*ArcTan[a*x]^3,x]","\frac{24 c^3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{35 a}+\frac{48 i c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{35 a}-\frac{c^3 \left(a^2 x^2+1\right)^2}{140 a}-\frac{13 c^3 \left(a^2 x^2+1\right)}{210 a}-\frac{7 c^3 \log \left(a^2 x^2+1\right)}{15 a}+\frac{1}{7} c^3 x \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^3+\frac{6}{35} c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^3+\frac{8}{35} c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3-\frac{c^3 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^2}{14 a}-\frac{9 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}{70 a}-\frac{12 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{35 a}+\frac{1}{35} c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)+\frac{13}{105} c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)+\frac{16}{35} c^3 x \tan ^{-1}(a x)^3+\frac{16 i c^3 \tan ^{-1}(a x)^3}{35 a}+\frac{14}{15} c^3 x \tan ^{-1}(a x)+\frac{48 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{35 a}","\frac{24 c^3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{35 a}+\frac{48 i c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{35 a}-\frac{c^3 \left(a^2 x^2+1\right)^2}{140 a}-\frac{13 c^3 \left(a^2 x^2+1\right)}{210 a}-\frac{7 c^3 \log \left(a^2 x^2+1\right)}{15 a}+\frac{1}{7} c^3 x \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^3+\frac{6}{35} c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^3+\frac{8}{35} c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3-\frac{c^3 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^2}{14 a}-\frac{9 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}{70 a}-\frac{12 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{35 a}+\frac{1}{35} c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)+\frac{13}{105} c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)+\frac{16}{35} c^3 x \tan ^{-1}(a x)^3+\frac{16 i c^3 \tan ^{-1}(a x)^3}{35 a}+\frac{14}{15} c^3 x \tan ^{-1}(a x)+\frac{48 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{35 a}",1,"(-13*c^3*(1 + a^2*x^2))/(210*a) - (c^3*(1 + a^2*x^2)^2)/(140*a) + (14*c^3*x*ArcTan[a*x])/15 + (13*c^3*x*(1 + a^2*x^2)*ArcTan[a*x])/105 + (c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x])/35 - (12*c^3*(1 + a^2*x^2)*ArcTan[a*x]^2)/(35*a) - (9*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(70*a) - (c^3*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/(14*a) + (((16*I)/35)*c^3*ArcTan[a*x]^3)/a + (16*c^3*x*ArcTan[a*x]^3)/35 + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^3)/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^3)/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^3)/7 + (48*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(35*a) - (7*c^3*Log[1 + a^2*x^2])/(15*a) + (((48*I)/35)*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a + (24*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(35*a)","A",17,9,19,0.4737,1,"{4880, 4846, 4920, 4854, 4884, 4994, 6610, 260, 4878}"
383,1,447,0,1.6554229,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3}{x} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x]^3)/x,x]","-\frac{34}{15} i c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{4} i c^3 \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)-\frac{3}{4} i c^3 \text{PolyLog}\left(4,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} i c^3 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{2} i c^3 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} c^3 \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{3}{2} c^3 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-\frac{1}{60} a^3 c^3 x^3+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^3-\frac{1}{10} a^5 c^3 x^5 \tan ^{-1}(a x)^2+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^3+\frac{1}{20} a^4 c^3 x^4 \tan ^{-1}(a x)-\frac{7}{12} a^3 c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^3+\frac{29}{60} a^2 c^3 x^2 \tan ^{-1}(a x)-\frac{13}{30} a c^3 x-\frac{11}{4} a c^3 x \tan ^{-1}(a x)^2+\frac{11}{12} c^3 \tan ^{-1}(a x)^3-\frac{34}{15} i c^3 \tan ^{-1}(a x)^2+\frac{13}{30} c^3 \tan ^{-1}(a x)-\frac{68}{15} c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)","-\frac{34}{15} i c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{4} i c^3 \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)-\frac{3}{4} i c^3 \text{PolyLog}\left(4,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} i c^3 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{3}{2} i c^3 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)-\frac{3}{2} c^3 \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{3}{2} c^3 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)-\frac{1}{60} a^3 c^3 x^3+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^3-\frac{1}{10} a^5 c^3 x^5 \tan ^{-1}(a x)^2+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^3+\frac{1}{20} a^4 c^3 x^4 \tan ^{-1}(a x)-\frac{7}{12} a^3 c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^3+\frac{29}{60} a^2 c^3 x^2 \tan ^{-1}(a x)-\frac{13}{30} a c^3 x-\frac{11}{4} a c^3 x \tan ^{-1}(a x)^2+\frac{11}{12} c^3 \tan ^{-1}(a x)^3-\frac{34}{15} i c^3 \tan ^{-1}(a x)^2+\frac{13}{30} c^3 \tan ^{-1}(a x)-\frac{68}{15} c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)",1,"(-13*a*c^3*x)/30 - (a^3*c^3*x^3)/60 + (13*c^3*ArcTan[a*x])/30 + (29*a^2*c^3*x^2*ArcTan[a*x])/60 + (a^4*c^3*x^4*ArcTan[a*x])/20 - ((34*I)/15)*c^3*ArcTan[a*x]^2 - (11*a*c^3*x*ArcTan[a*x]^2)/4 - (7*a^3*c^3*x^3*ArcTan[a*x]^2)/12 - (a^5*c^3*x^5*ArcTan[a*x]^2)/10 + (11*c^3*ArcTan[a*x]^3)/12 + (3*a^2*c^3*x^2*ArcTan[a*x]^3)/2 + (3*a^4*c^3*x^4*ArcTan[a*x]^3)/4 + (a^6*c^3*x^6*ArcTan[a*x]^3)/6 + 2*c^3*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - (68*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/15 - ((34*I)/15)*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)] - ((3*I)/2)*c^3*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((3*I)/2)*c^3*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (3*c^3*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (3*c^3*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((3*I)/4)*c^3*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((3*I)/4)*c^3*PolyLog[4, -1 + 2/(1 + I*a*x)]","A",69,17,22,0.7727,1,"{4948, 4850, 4988, 4884, 4994, 4998, 6610, 4852, 4916, 4846, 4920, 4854, 2402, 2315, 321, 203, 302}"
384,1,354,0,1.2802146,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3}{x^2} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x]^3)/x^2,x]","\frac{3}{2} a c^3 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)+\frac{33}{10} a c^3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)-3 i a c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+\frac{33}{5} i a c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)-\frac{1}{20} a^3 c^3 x^2-a c^3 \log \left(a^2 x^2+1\right)+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{10} a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2+3 a^2 c^3 x \tan ^{-1}(a x)^3+\frac{21}{10} a^2 c^3 x \tan ^{-1}(a x)+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{21}{20} a c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^3}{x}+\frac{33}{5} a c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2+3 a c^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2","\frac{3}{2} a c^3 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)+\frac{33}{10} a c^3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)-3 i a c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+\frac{33}{5} i a c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)-\frac{1}{20} a^3 c^3 x^2-a c^3 \log \left(a^2 x^2+1\right)+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{10} a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2+3 a^2 c^3 x \tan ^{-1}(a x)^3+\frac{21}{10} a^2 c^3 x \tan ^{-1}(a x)+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{21}{20} a c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^3}{x}+\frac{33}{5} a c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2+3 a c^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2",1,"-(a^3*c^3*x^2)/20 + (21*a^2*c^3*x*ArcTan[a*x])/10 + (a^4*c^3*x^3*ArcTan[a*x])/10 - (21*a*c^3*ArcTan[a*x]^2)/20 - (6*a^3*c^3*x^2*ArcTan[a*x]^2)/5 - (3*a^5*c^3*x^4*ArcTan[a*x]^2)/20 + ((6*I)/5)*a*c^3*ArcTan[a*x]^3 - (c^3*ArcTan[a*x]^3)/x + 3*a^2*c^3*x*ArcTan[a*x]^3 + a^4*c^3*x^3*ArcTan[a*x]^3 + (a^6*c^3*x^5*ArcTan[a*x]^3)/5 + (33*a*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/5 - a*c^3*Log[1 + a^2*x^2] + 3*a*c^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (3*I)*a*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((33*I)/5)*a*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + (3*a*c^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/2 + (33*a*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)])/10","A",45,15,22,0.6818,1,"{4948, 4846, 4920, 4854, 4884, 4994, 6610, 4852, 4924, 4868, 4992, 4916, 260, 266, 43}"
385,1,503,0,1.1919814,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3}{x^3} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x]^3)/x^3,x]","-\frac{3}{2} i a^2 c^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-\frac{7}{2} i a^2 c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{9}{4} i a^2 c^3 \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)-\frac{9}{4} i a^2 c^3 \text{PolyLog}\left(4,-1+\frac{2}{1+i a x}\right)-\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)-\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3-\frac{1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac{1}{4} a^3 c^3 x-\frac{15}{4} a^3 c^3 x \tan ^{-1}(a x)^2+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^3-5 i a^2 c^3 \tan ^{-1}(a x)^2+\frac{1}{4} a^2 c^3 \tan ^{-1}(a x)-7 a^2 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+3 a^2 c^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)-\frac{c^3 \tan ^{-1}(a x)^3}{2 x^2}-\frac{3 a c^3 \tan ^{-1}(a x)^2}{2 x}","-\frac{3}{2} i a^2 c^3 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)-\frac{7}{2} i a^2 c^3 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{9}{4} i a^2 c^3 \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)-\frac{9}{4} i a^2 c^3 \text{PolyLog}\left(4,-1+\frac{2}{1+i a x}\right)-\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)+\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i a x}\right)-\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)+\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1+i a x}\right)+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3-\frac{1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac{1}{4} a^3 c^3 x-\frac{15}{4} a^3 c^3 x \tan ^{-1}(a x)^2+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^3-5 i a^2 c^3 \tan ^{-1}(a x)^2+\frac{1}{4} a^2 c^3 \tan ^{-1}(a x)-7 a^2 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)+3 a^2 c^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2}{1+i a x}\right)-\frac{c^3 \tan ^{-1}(a x)^3}{2 x^2}-\frac{3 a c^3 \tan ^{-1}(a x)^2}{2 x}",1,"-(a^3*c^3*x)/4 + (a^2*c^3*ArcTan[a*x])/4 + (a^4*c^3*x^2*ArcTan[a*x])/4 - (5*I)*a^2*c^3*ArcTan[a*x]^2 - (3*a*c^3*ArcTan[a*x]^2)/(2*x) - (15*a^3*c^3*x*ArcTan[a*x]^2)/4 - (a^5*c^3*x^3*ArcTan[a*x]^2)/4 + (3*a^2*c^3*ArcTan[a*x]^3)/4 - (c^3*ArcTan[a*x]^3)/(2*x^2) + (3*a^4*c^3*x^2*ArcTan[a*x]^3)/2 + (a^6*c^3*x^4*ArcTan[a*x]^3)/4 + 6*a^2*c^3*ArcTan[a*x]^3*ArcTanh[1 - 2/(1 + I*a*x)] - 7*a^2*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)] + 3*a^2*c^3*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - ((3*I)/2)*a^2*c^3*PolyLog[2, -1 + 2/(1 - I*a*x)] - ((7*I)/2)*a^2*c^3*PolyLog[2, 1 - 2/(1 + I*a*x)] - ((9*I)/2)*a^2*c^3*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)] + ((9*I)/2)*a^2*c^3*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 + I*a*x)] - (9*a^2*c^3*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/2 + (9*a^2*c^3*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x)])/2 + ((9*I)/4)*a^2*c^3*PolyLog[4, 1 - 2/(1 + I*a*x)] - ((9*I)/4)*a^2*c^3*PolyLog[4, -1 + 2/(1 + I*a*x)]","A",43,20,22,0.9091,1,"{4948, 4852, 4918, 4924, 4868, 2447, 4884, 4850, 4988, 4994, 4998, 6610, 4916, 4846, 4920, 4854, 2402, 2315, 321, 203}"
386,1,336,0,1.1085749,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3}{x^4} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x]^3)/x^4,x]","4 a^3 c^3 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)+4 a^3 c^3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)-8 i a^3 c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+8 i a^3 c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)-a^3 c^3 \log \left(a^2 x^2+1\right)+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)^3-\frac{1}{2} a^5 c^3 x^2 \tan ^{-1}(a x)^2+a^3 c^3 \log (x)+3 a^4 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x \tan ^{-1}(a x)-a^3 c^3 \tan ^{-1}(a x)^2-\frac{3 a^2 c^3 \tan ^{-1}(a x)^3}{x}-\frac{a^2 c^3 \tan ^{-1}(a x)}{x}+8 a^3 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2+8 a^3 c^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2-\frac{a c^3 \tan ^{-1}(a x)^2}{2 x^2}-\frac{c^3 \tan ^{-1}(a x)^3}{3 x^3}","4 a^3 c^3 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)+4 a^3 c^3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)-8 i a^3 c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)+8 i a^3 c^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)-a^3 c^3 \log \left(a^2 x^2+1\right)+\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)^3-\frac{1}{2} a^5 c^3 x^2 \tan ^{-1}(a x)^2+a^3 c^3 \log (x)+3 a^4 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x \tan ^{-1}(a x)-a^3 c^3 \tan ^{-1}(a x)^2-\frac{3 a^2 c^3 \tan ^{-1}(a x)^3}{x}-\frac{a^2 c^3 \tan ^{-1}(a x)}{x}+8 a^3 c^3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2+8 a^3 c^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2-\frac{a c^3 \tan ^{-1}(a x)^2}{2 x^2}-\frac{c^3 \tan ^{-1}(a x)^3}{3 x^3}",1,"-((a^2*c^3*ArcTan[a*x])/x) + a^4*c^3*x*ArcTan[a*x] - a^3*c^3*ArcTan[a*x]^2 - (a*c^3*ArcTan[a*x]^2)/(2*x^2) - (a^5*c^3*x^2*ArcTan[a*x]^2)/2 - (c^3*ArcTan[a*x]^3)/(3*x^3) - (3*a^2*c^3*ArcTan[a*x]^3)/x + 3*a^4*c^3*x*ArcTan[a*x]^3 + (a^6*c^3*x^3*ArcTan[a*x]^3)/3 + a^3*c^3*Log[x] + 8*a^3*c^3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)] - a^3*c^3*Log[1 + a^2*x^2] + 8*a^3*c^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (8*I)*a^3*c^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)] + (8*I)*a^3*c^3*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)] + 4*a^3*c^3*PolyLog[3, -1 + 2/(1 - I*a*x)] + 4*a^3*c^3*PolyLog[3, 1 - 2/(1 + I*a*x)]","A",37,18,22,0.8182,1,"{4948, 4846, 4920, 4854, 4884, 4994, 6610, 4852, 4918, 266, 36, 29, 31, 4924, 4868, 4992, 4916, 260}"
387,1,217,0,0.6262977,"\int \frac{x^4 \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx","Int[(x^4*ArcTan[a*x]^3)/(c + a^2*c*x^2),x]","-\frac{2 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{a^5 c}-\frac{4 i \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^5 c}-\frac{\log \left(a^2 x^2+1\right)}{2 a^5 c}+\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c}-\frac{x^2 \tan ^{-1}(a x)^2}{2 a^3 c}+\frac{\tan ^{-1}(a x)^4}{4 a^5 c}-\frac{x \tan ^{-1}(a x)^3}{a^4 c}-\frac{4 i \tan ^{-1}(a x)^3}{3 a^5 c}-\frac{\tan ^{-1}(a x)^2}{2 a^5 c}+\frac{x \tan ^{-1}(a x)}{a^4 c}-\frac{4 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{a^5 c}","-\frac{2 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{a^5 c}-\frac{4 i \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^5 c}-\frac{\log \left(a^2 x^2+1\right)}{2 a^5 c}+\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c}-\frac{x^2 \tan ^{-1}(a x)^2}{2 a^3 c}+\frac{\tan ^{-1}(a x)^4}{4 a^5 c}-\frac{x \tan ^{-1}(a x)^3}{a^4 c}-\frac{4 i \tan ^{-1}(a x)^3}{3 a^5 c}-\frac{\tan ^{-1}(a x)^2}{2 a^5 c}+\frac{x \tan ^{-1}(a x)}{a^4 c}-\frac{4 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{a^5 c}",1,"(x*ArcTan[a*x])/(a^4*c) - ArcTan[a*x]^2/(2*a^5*c) - (x^2*ArcTan[a*x]^2)/(2*a^3*c) - (((4*I)/3)*ArcTan[a*x]^3)/(a^5*c) - (x*ArcTan[a*x]^3)/(a^4*c) + (x^3*ArcTan[a*x]^3)/(3*a^2*c) + ArcTan[a*x]^4/(4*a^5*c) - (4*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(a^5*c) - Log[1 + a^2*x^2]/(2*a^5*c) - ((4*I)*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^5*c) - (2*PolyLog[3, 1 - 2/(1 + I*a*x)])/(a^5*c)","A",19,9,22,0.4091,1,"{4916, 4852, 4846, 260, 4884, 4920, 4854, 4994, 6610}"
388,1,260,0,0.4467511,"\int \frac{x^3 \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx","Int[(x^3*ArcTan[a*x]^3)/(c + a^2*c*x^2),x]","-\frac{3 i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^4 c}-\frac{3 i \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)}{4 a^4 c}+\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^4 c}+\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)^3}{2 a^2 c}+\frac{i \tan ^{-1}(a x)^4}{4 a^4 c}+\frac{\tan ^{-1}(a x)^3}{2 a^4 c}-\frac{3 x \tan ^{-1}(a x)^2}{2 a^3 c}-\frac{3 i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^3}{a^4 c}-\frac{3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{a^4 c}","-\frac{3 i \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^4 c}-\frac{3 i \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)}{4 a^4 c}+\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^4 c}+\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^4 c}+\frac{x^2 \tan ^{-1}(a x)^3}{2 a^2 c}+\frac{i \tan ^{-1}(a x)^4}{4 a^4 c}+\frac{\tan ^{-1}(a x)^3}{2 a^4 c}-\frac{3 x \tan ^{-1}(a x)^2}{2 a^3 c}-\frac{3 i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^3}{a^4 c}-\frac{3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)}{a^4 c}",1,"(((-3*I)/2)*ArcTan[a*x]^2)/(a^4*c) - (3*x*ArcTan[a*x]^2)/(2*a^3*c) + ArcTan[a*x]^3/(2*a^4*c) + (x^2*ArcTan[a*x]^3)/(2*a^2*c) + ((I/4)*ArcTan[a*x]^4)/(a^4*c) - (3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(a^4*c) + (ArcTan[a*x]^3*Log[2/(1 + I*a*x)])/(a^4*c) - (((3*I)/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*c) + (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*c) + (3*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/(2*a^4*c) - (((3*I)/4)*PolyLog[4, 1 - 2/(1 + I*a*x)])/(a^4*c)","A",14,11,22,0.5000,1,"{4916, 4852, 4846, 4920, 4854, 2402, 2315, 4884, 4994, 4998, 6610}"
389,1,130,0,0.2452694,"\int \frac{x^2 \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx","Int[(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2),x]","\frac{3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^3 c}+\frac{3 i \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^3 c}-\frac{\tan ^{-1}(a x)^4}{4 a^3 c}+\frac{x \tan ^{-1}(a x)^3}{a^2 c}+\frac{i \tan ^{-1}(a x)^3}{a^3 c}+\frac{3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{a^3 c}","\frac{3 \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^3 c}+\frac{3 i \tan ^{-1}(a x) \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{a^3 c}-\frac{\tan ^{-1}(a x)^4}{4 a^3 c}+\frac{x \tan ^{-1}(a x)^3}{a^2 c}+\frac{i \tan ^{-1}(a x)^3}{a^3 c}+\frac{3 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{a^3 c}",1,"(I*ArcTan[a*x]^3)/(a^3*c) + (x*ArcTan[a*x]^3)/(a^2*c) - ArcTan[a*x]^4/(4*a^3*c) + (3*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(a^3*c) + ((3*I)*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^3*c) + (3*PolyLog[3, 1 - 2/(1 + I*a*x)])/(2*a^3*c)","A",7,7,22,0.3182,1,"{4916, 4846, 4920, 4854, 4884, 4994, 6610}"
390,1,138,0,0.2177253,"\int \frac{x \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx","Int[(x*ArcTan[a*x]^3)/(c + a^2*c*x^2),x]","\frac{3 i \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)}{4 a^2 c}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^2 c}-\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^2 c}-\frac{i \tan ^{-1}(a x)^4}{4 a^2 c}-\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^3}{a^2 c}","\frac{3 i \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)}{4 a^2 c}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^2 c}-\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^2 c}-\frac{i \tan ^{-1}(a x)^4}{4 a^2 c}-\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^3}{a^2 c}",1,"((-I/4)*ArcTan[a*x]^4)/(a^2*c) - (ArcTan[a*x]^3*Log[2/(1 + I*a*x)])/(a^2*c) - (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^2*c) - (3*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/(2*a^2*c) + (((3*I)/4)*PolyLog[4, 1 - 2/(1 + I*a*x)])/(a^2*c)","A",5,6,20,0.3000,1,"{4920, 4854, 4884, 4994, 4998, 6610}"
391,1,16,0,0.0242376,"\int \frac{\tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx","Int[ArcTan[a*x]^3/(c + a^2*c*x^2),x]","\frac{\tan ^{-1}(a x)^4}{4 a c}","\frac{\tan ^{-1}(a x)^4}{4 a c}",1,"ArcTan[a*x]^4/(4*a*c)","A",1,1,19,0.05263,1,"{4884}"
392,1,124,0,0.2307543,"\int \frac{\tan ^{-1}(a x)^3}{x \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^3/(x*(c + a^2*c*x^2)),x]","\frac{3 i \text{PolyLog}\left(4,-1+\frac{2}{1-i a x}\right)}{4 c}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c}+\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c}-\frac{i \tan ^{-1}(a x)^4}{4 c}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^3}{c}","\frac{3 i \text{PolyLog}\left(4,-1+\frac{2}{1-i a x}\right)}{4 c}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c}+\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c}-\frac{i \tan ^{-1}(a x)^4}{4 c}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^3}{c}",1,"((-I/4)*ArcTan[a*x]^4)/c + (ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)])/c - (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c + (3*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c) + (((3*I)/4)*PolyLog[4, -1 + 2/(1 - I*a*x)])/c","A",5,6,22,0.2727,1,"{4924, 4868, 4884, 4992, 4996, 6610}"
393,1,122,0,0.2859108,"\int \frac{\tan ^{-1}(a x)^3}{x^2 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)),x]","\frac{3 a \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c}-\frac{3 i a \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c}-\frac{a \tan ^{-1}(a x)^4}{4 c}-\frac{i a \tan ^{-1}(a x)^3}{c}-\frac{\tan ^{-1}(a x)^3}{c x}+\frac{3 a \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c}","\frac{3 a \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c}-\frac{3 i a \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c}-\frac{a \tan ^{-1}(a x)^4}{4 c}-\frac{i a \tan ^{-1}(a x)^3}{c}-\frac{\tan ^{-1}(a x)^3}{c x}+\frac{3 a \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c}",1,"((-I)*a*ArcTan[a*x]^3)/c - ArcTan[a*x]^3/(c*x) - (a*ArcTan[a*x]^4)/(4*c) + (3*a*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c - ((3*I)*a*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c + (3*a*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c)","A",7,7,22,0.3182,1,"{4918, 4852, 4924, 4868, 4884, 4992, 6610}"
394,1,262,0,0.5105517,"\int \frac{\tan ^{-1}(a x)^3}{x^3 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^3/(x^3*(c + a^2*c*x^2)),x]","-\frac{3 i a^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c}-\frac{3 i a^2 \text{PolyLog}\left(4,-1+\frac{2}{1-i a x}\right)}{4 c}+\frac{3 i a^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c}-\frac{3 a^2 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c}+\frac{i a^2 \tan ^{-1}(a x)^4}{4 c}-\frac{a^2 \tan ^{-1}(a x)^3}{2 c}-\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c}-\frac{a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^3}{c}+\frac{3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c}-\frac{\tan ^{-1}(a x)^3}{2 c x^2}-\frac{3 a \tan ^{-1}(a x)^2}{2 c x}","-\frac{3 i a^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c}-\frac{3 i a^2 \text{PolyLog}\left(4,-1+\frac{2}{1-i a x}\right)}{4 c}+\frac{3 i a^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c}-\frac{3 a^2 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c}+\frac{i a^2 \tan ^{-1}(a x)^4}{4 c}-\frac{a^2 \tan ^{-1}(a x)^3}{2 c}-\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c}-\frac{a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^3}{c}+\frac{3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c}-\frac{\tan ^{-1}(a x)^3}{2 c x^2}-\frac{3 a \tan ^{-1}(a x)^2}{2 c x}",1,"(((-3*I)/2)*a^2*ArcTan[a*x]^2)/c - (3*a*ArcTan[a*x]^2)/(2*c*x) - (a^2*ArcTan[a*x]^3)/(2*c) - ArcTan[a*x]^3/(2*c*x^2) + ((I/4)*a^2*ArcTan[a*x]^4)/c + (3*a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c - (a^2*ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)])/c - (((3*I)/2)*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c + (((3*I)/2)*a^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c - (3*a^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c) - (((3*I)/4)*a^2*PolyLog[4, -1 + 2/(1 - I*a*x)])/c","A",13,9,22,0.4091,1,"{4918, 4852, 4924, 4868, 2447, 4884, 4992, 4996, 6610}"
395,1,227,0,0.7221259,"\int \frac{\tan ^{-1}(a x)^3}{x^4 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)),x]","-\frac{2 a^3 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{c}+\frac{4 i a^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c}-\frac{a^3 \log \left(a^2 x^2+1\right)}{2 c}+\frac{a^3 \log (x)}{c}+\frac{a^3 \tan ^{-1}(a x)^4}{4 c}+\frac{4 i a^3 \tan ^{-1}(a x)^3}{3 c}+\frac{a^2 \tan ^{-1}(a x)^3}{c x}-\frac{a^3 \tan ^{-1}(a x)^2}{2 c}-\frac{a^2 \tan ^{-1}(a x)}{c x}-\frac{4 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c}-\frac{\tan ^{-1}(a x)^3}{3 c x^3}-\frac{a \tan ^{-1}(a x)^2}{2 c x^2}","-\frac{2 a^3 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{c}+\frac{4 i a^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c}-\frac{a^3 \log \left(a^2 x^2+1\right)}{2 c}+\frac{a^3 \log (x)}{c}+\frac{a^3 \tan ^{-1}(a x)^4}{4 c}+\frac{4 i a^3 \tan ^{-1}(a x)^3}{3 c}+\frac{a^2 \tan ^{-1}(a x)^3}{c x}-\frac{a^3 \tan ^{-1}(a x)^2}{2 c}-\frac{a^2 \tan ^{-1}(a x)}{c x}-\frac{4 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c}-\frac{\tan ^{-1}(a x)^3}{3 c x^3}-\frac{a \tan ^{-1}(a x)^2}{2 c x^2}",1,"-((a^2*ArcTan[a*x])/(c*x)) - (a^3*ArcTan[a*x]^2)/(2*c) - (a*ArcTan[a*x]^2)/(2*c*x^2) + (((4*I)/3)*a^3*ArcTan[a*x]^3)/c - ArcTan[a*x]^3/(3*c*x^3) + (a^2*ArcTan[a*x]^3)/(c*x) + (a^3*ArcTan[a*x]^4)/(4*c) + (a^3*Log[x])/c - (a^3*Log[1 + a^2*x^2])/(2*c) - (4*a^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c + ((4*I)*a^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c - (2*a^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/c","A",22,11,22,0.5000,1,"{4918, 4852, 266, 36, 29, 31, 4884, 4924, 4868, 4992, 6610}"
396,1,270,0,0.4127319,"\int \frac{x^3 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^3*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2,x]","\frac{3 i \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)}{4 a^4 c^2}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^4 c^2}-\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^4 c^2}+\frac{3 x}{8 a^3 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^3}{2 a^4 c^2 \left(a^2 x^2+1\right)}-\frac{3 x \tan ^{-1}(a x)^2}{4 a^3 c^2 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)}{4 a^4 c^2 \left(a^2 x^2+1\right)}-\frac{i \tan ^{-1}(a x)^4}{4 a^4 c^2}-\frac{\tan ^{-1}(a x)^3}{4 a^4 c^2}+\frac{3 \tan ^{-1}(a x)}{8 a^4 c^2}-\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^3}{a^4 c^2}","\frac{3 i \text{PolyLog}\left(4,1-\frac{2}{1+i a x}\right)}{4 a^4 c^2}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,1-\frac{2}{1+i a x}\right)}{2 a^4 c^2}-\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left(3,1-\frac{2}{1+i a x}\right)}{2 a^4 c^2}+\frac{3 x}{8 a^3 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^3}{2 a^4 c^2 \left(a^2 x^2+1\right)}-\frac{3 x \tan ^{-1}(a x)^2}{4 a^3 c^2 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)}{4 a^4 c^2 \left(a^2 x^2+1\right)}-\frac{i \tan ^{-1}(a x)^4}{4 a^4 c^2}-\frac{\tan ^{-1}(a x)^3}{4 a^4 c^2}+\frac{3 \tan ^{-1}(a x)}{8 a^4 c^2}-\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^3}{a^4 c^2}",1,"(3*x)/(8*a^3*c^2*(1 + a^2*x^2)) + (3*ArcTan[a*x])/(8*a^4*c^2) - (3*ArcTan[a*x])/(4*a^4*c^2*(1 + a^2*x^2)) - (3*x*ArcTan[a*x]^2)/(4*a^3*c^2*(1 + a^2*x^2)) - ArcTan[a*x]^3/(4*a^4*c^2) + ArcTan[a*x]^3/(2*a^4*c^2*(1 + a^2*x^2)) - ((I/4)*ArcTan[a*x]^4)/(a^4*c^2) - (ArcTan[a*x]^3*Log[2/(1 + I*a*x)])/(a^4*c^2) - (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*c^2) - (3*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/(2*a^4*c^2) + (((3*I)/4)*PolyLog[4, 1 - 2/(1 + I*a*x)])/(a^4*c^2)","A",11,11,22,0.5000,1,"{4964, 4920, 4854, 4884, 4994, 4998, 6610, 4930, 4892, 199, 205}"
397,1,135,0,0.1437488,"\int \frac{x^2 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2,x]","\frac{3}{8 a^3 c^2 \left(a^2 x^2+1\right)}-\frac{x \tan ^{-1}(a x)^3}{2 a^2 c^2 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)^2}{4 a^3 c^2 \left(a^2 x^2+1\right)}+\frac{3 x \tan ^{-1}(a x)}{4 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^4}{8 a^3 c^2}+\frac{3 \tan ^{-1}(a x)^2}{8 a^3 c^2}","\frac{3}{8 a^3 c^2 \left(a^2 x^2+1\right)}-\frac{x \tan ^{-1}(a x)^3}{2 a^2 c^2 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)^2}{4 a^3 c^2 \left(a^2 x^2+1\right)}+\frac{3 x \tan ^{-1}(a x)}{4 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^4}{8 a^3 c^2}+\frac{3 \tan ^{-1}(a x)^2}{8 a^3 c^2}",1,"3/(8*a^3*c^2*(1 + a^2*x^2)) + (3*x*ArcTan[a*x])/(4*a^2*c^2*(1 + a^2*x^2)) + (3*ArcTan[a*x]^2)/(8*a^3*c^2) - (3*ArcTan[a*x]^2)/(4*a^3*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x]^3)/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^4/(8*a^3*c^2)","A",4,4,22,0.1818,1,"{4936, 4930, 4892, 261}"
398,1,133,0,0.1216098,"\int \frac{x \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2,x]","-\frac{3 x}{8 a c^2 \left(a^2 x^2+1\right)}-\frac{\tan ^{-1}(a x)^3}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{3 x \tan ^{-1}(a x)^2}{4 a c^2 \left(a^2 x^2+1\right)}+\frac{3 \tan ^{-1}(a x)}{4 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^3}{4 a^2 c^2}-\frac{3 \tan ^{-1}(a x)}{8 a^2 c^2}","-\frac{3 x}{8 a c^2 \left(a^2 x^2+1\right)}-\frac{\tan ^{-1}(a x)^3}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{3 x \tan ^{-1}(a x)^2}{4 a c^2 \left(a^2 x^2+1\right)}+\frac{3 \tan ^{-1}(a x)}{4 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^3}{4 a^2 c^2}-\frac{3 \tan ^{-1}(a x)}{8 a^2 c^2}",1,"(-3*x)/(8*a*c^2*(1 + a^2*x^2)) - (3*ArcTan[a*x])/(8*a^2*c^2) + (3*ArcTan[a*x])/(4*a^2*c^2*(1 + a^2*x^2)) + (3*x*ArcTan[a*x]^2)/(4*a*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^3/(4*a^2*c^2) - ArcTan[a*x]^3/(2*a^2*c^2*(1 + a^2*x^2))","A",5,4,20,0.2000,1,"{4930, 4892, 199, 205}"
399,1,129,0,0.1043869,"\int \frac{\tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]^3/(c + a^2*c*x^2)^2,x]","-\frac{3}{8 a c^2 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)^3}{2 c^2 \left(a^2 x^2+1\right)}+\frac{3 \tan ^{-1}(a x)^2}{4 a c^2 \left(a^2 x^2+1\right)}-\frac{3 x \tan ^{-1}(a x)}{4 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^4}{8 a c^2}-\frac{3 \tan ^{-1}(a x)^2}{8 a c^2}","-\frac{3}{8 a c^2 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)^3}{2 c^2 \left(a^2 x^2+1\right)}+\frac{3 \tan ^{-1}(a x)^2}{4 a c^2 \left(a^2 x^2+1\right)}-\frac{3 x \tan ^{-1}(a x)}{4 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^4}{8 a c^2}-\frac{3 \tan ^{-1}(a x)^2}{8 a c^2}",1,"-3/(8*a*c^2*(1 + a^2*x^2)) - (3*x*ArcTan[a*x])/(4*c^2*(1 + a^2*x^2)) - (3*ArcTan[a*x]^2)/(8*a*c^2) + (3*ArcTan[a*x]^2)/(4*a*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x]^3)/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^4/(8*a*c^2)","A",4,3,19,0.1579,1,"{4892, 4930, 261}"
400,1,240,0,0.428387,"\int \frac{\tan ^{-1}(a x)^3}{x \left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]^3/(x*(c + a^2*c*x^2)^2),x]","\frac{3 i \text{PolyLog}\left(4,-1+\frac{2}{1-i a x}\right)}{4 c^2}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^2}+\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^2}+\frac{3 a x}{8 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^3}{2 c^2 \left(a^2 x^2+1\right)}-\frac{3 a x \tan ^{-1}(a x)^2}{4 c^2 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)}{4 c^2 \left(a^2 x^2+1\right)}-\frac{i \tan ^{-1}(a x)^4}{4 c^2}-\frac{\tan ^{-1}(a x)^3}{4 c^2}+\frac{3 \tan ^{-1}(a x)}{8 c^2}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^3}{c^2}","\frac{3 i \text{PolyLog}\left(4,-1+\frac{2}{1-i a x}\right)}{4 c^2}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^2}+\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^2}+\frac{3 a x}{8 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^3}{2 c^2 \left(a^2 x^2+1\right)}-\frac{3 a x \tan ^{-1}(a x)^2}{4 c^2 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)}{4 c^2 \left(a^2 x^2+1\right)}-\frac{i \tan ^{-1}(a x)^4}{4 c^2}-\frac{\tan ^{-1}(a x)^3}{4 c^2}+\frac{3 \tan ^{-1}(a x)}{8 c^2}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^3}{c^2}",1,"(3*a*x)/(8*c^2*(1 + a^2*x^2)) + (3*ArcTan[a*x])/(8*c^2) - (3*ArcTan[a*x])/(4*c^2*(1 + a^2*x^2)) - (3*a*x*ArcTan[a*x]^2)/(4*c^2*(1 + a^2*x^2)) - ArcTan[a*x]^3/(4*c^2) + ArcTan[a*x]^3/(2*c^2*(1 + a^2*x^2)) - ((I/4)*ArcTan[a*x]^4)/c^2 + (ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)])/c^2 - (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2 + (3*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^2) + (((3*I)/4)*PolyLog[4, -1 + 2/(1 - I*a*x)])/c^2","A",11,11,22,0.5000,1,"{4966, 4924, 4868, 4884, 4992, 4996, 6610, 4930, 4892, 199, 205}"
401,1,234,0,0.4681474,"\int \frac{\tan ^{-1}(a x)^3}{x^2 \left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)^2),x]","\frac{3 a \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^2}-\frac{3 i a \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^2}+\frac{3 a}{8 c^2 \left(a^2 x^2+1\right)}-\frac{a^2 x \tan ^{-1}(a x)^3}{2 c^2 \left(a^2 x^2+1\right)}-\frac{3 a \tan ^{-1}(a x)^2}{4 c^2 \left(a^2 x^2+1\right)}+\frac{3 a^2 x \tan ^{-1}(a x)}{4 c^2 \left(a^2 x^2+1\right)}-\frac{3 a \tan ^{-1}(a x)^4}{8 c^2}-\frac{\tan ^{-1}(a x)^3}{c^2 x}-\frac{i a \tan ^{-1}(a x)^3}{c^2}+\frac{3 a \tan ^{-1}(a x)^2}{8 c^2}+\frac{3 a \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^2}","\frac{3 a \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^2}-\frac{3 i a \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^2}+\frac{3 a}{8 c^2 \left(a^2 x^2+1\right)}-\frac{a^2 x \tan ^{-1}(a x)^3}{2 c^2 \left(a^2 x^2+1\right)}-\frac{3 a \tan ^{-1}(a x)^2}{4 c^2 \left(a^2 x^2+1\right)}+\frac{3 a^2 x \tan ^{-1}(a x)}{4 c^2 \left(a^2 x^2+1\right)}-\frac{3 a \tan ^{-1}(a x)^4}{8 c^2}-\frac{\tan ^{-1}(a x)^3}{c^2 x}-\frac{i a \tan ^{-1}(a x)^3}{c^2}+\frac{3 a \tan ^{-1}(a x)^2}{8 c^2}+\frac{3 a \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^2}",1,"(3*a)/(8*c^2*(1 + a^2*x^2)) + (3*a^2*x*ArcTan[a*x])/(4*c^2*(1 + a^2*x^2)) + (3*a*ArcTan[a*x]^2)/(8*c^2) - (3*a*ArcTan[a*x]^2)/(4*c^2*(1 + a^2*x^2)) - (I*a*ArcTan[a*x]^3)/c^2 - ArcTan[a*x]^3/(c^2*x) - (a^2*x*ArcTan[a*x]^3)/(2*c^2*(1 + a^2*x^2)) - (3*a*ArcTan[a*x]^4)/(8*c^2) + (3*a*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^2 - ((3*I)*a*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2 + (3*a*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^2)","A",12,11,22,0.5000,1,"{4966, 4918, 4852, 4924, 4868, 4884, 4992, 6610, 4892, 4930, 261}"
402,1,374,0,1.0262853,"\int \frac{\tan ^{-1}(a x)^3}{x^3 \left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]^3/(x^3*(c + a^2*c*x^2)^2),x]","-\frac{3 i a^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^2}-\frac{3 i a^2 \text{PolyLog}\left(4,-1+\frac{2}{1-i a x}\right)}{2 c^2}+\frac{3 i a^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^2}-\frac{3 a^2 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{c^2}-\frac{3 a^3 x}{8 c^2 \left(a^2 x^2+1\right)}-\frac{a^2 \tan ^{-1}(a x)^3}{2 c^2 \left(a^2 x^2+1\right)}+\frac{3 a^3 x \tan ^{-1}(a x)^2}{4 c^2 \left(a^2 x^2+1\right)}+\frac{3 a^2 \tan ^{-1}(a x)}{4 c^2 \left(a^2 x^2+1\right)}+\frac{i a^2 \tan ^{-1}(a x)^4}{2 c^2}-\frac{a^2 \tan ^{-1}(a x)^3}{4 c^2}-\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c^2}-\frac{3 a^2 \tan ^{-1}(a x)}{8 c^2}-\frac{2 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^3}{c^2}+\frac{3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^2}-\frac{\tan ^{-1}(a x)^3}{2 c^2 x^2}-\frac{3 a \tan ^{-1}(a x)^2}{2 c^2 x}","-\frac{3 i a^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^2}-\frac{3 i a^2 \text{PolyLog}\left(4,-1+\frac{2}{1-i a x}\right)}{2 c^2}+\frac{3 i a^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^2}-\frac{3 a^2 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{c^2}-\frac{3 a^3 x}{8 c^2 \left(a^2 x^2+1\right)}-\frac{a^2 \tan ^{-1}(a x)^3}{2 c^2 \left(a^2 x^2+1\right)}+\frac{3 a^3 x \tan ^{-1}(a x)^2}{4 c^2 \left(a^2 x^2+1\right)}+\frac{3 a^2 \tan ^{-1}(a x)}{4 c^2 \left(a^2 x^2+1\right)}+\frac{i a^2 \tan ^{-1}(a x)^4}{2 c^2}-\frac{a^2 \tan ^{-1}(a x)^3}{4 c^2}-\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c^2}-\frac{3 a^2 \tan ^{-1}(a x)}{8 c^2}-\frac{2 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^3}{c^2}+\frac{3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^2}-\frac{\tan ^{-1}(a x)^3}{2 c^2 x^2}-\frac{3 a \tan ^{-1}(a x)^2}{2 c^2 x}",1,"(-3*a^3*x)/(8*c^2*(1 + a^2*x^2)) - (3*a^2*ArcTan[a*x])/(8*c^2) + (3*a^2*ArcTan[a*x])/(4*c^2*(1 + a^2*x^2)) - (((3*I)/2)*a^2*ArcTan[a*x]^2)/c^2 - (3*a*ArcTan[a*x]^2)/(2*c^2*x) + (3*a^3*x*ArcTan[a*x]^2)/(4*c^2*(1 + a^2*x^2)) - (a^2*ArcTan[a*x]^3)/(4*c^2) - ArcTan[a*x]^3/(2*c^2*x^2) - (a^2*ArcTan[a*x]^3)/(2*c^2*(1 + a^2*x^2)) + ((I/2)*a^2*ArcTan[a*x]^4)/c^2 + (3*a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^2 - (2*a^2*ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)])/c^2 - (((3*I)/2)*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2 + ((3*I)*a^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2 - (3*a^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/c^2 - (((3*I)/2)*a^2*PolyLog[4, -1 + 2/(1 - I*a*x)])/c^2","A",25,14,22,0.6364,1,"{4966, 4918, 4852, 4924, 4868, 2447, 4884, 4992, 4996, 6610, 4930, 4892, 199, 205}"
403,1,332,0,1.2900436,"\int \frac{\tan ^{-1}(a x)^3}{x^4 \left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)^2),x]","-\frac{7 a^3 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^2}+\frac{7 i a^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^2}-\frac{3 a^3}{8 c^2 \left(a^2 x^2+1\right)}-\frac{a^3 \log \left(a^2 x^2+1\right)}{2 c^2}+\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^2 \left(a^2 x^2+1\right)}-\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^2 \left(a^2 x^2+1\right)}+\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^2 \left(a^2 x^2+1\right)}+\frac{a^3 \log (x)}{c^2}+\frac{5 a^3 \tan ^{-1}(a x)^4}{8 c^2}+\frac{7 i a^3 \tan ^{-1}(a x)^3}{3 c^2}-\frac{7 a^3 \tan ^{-1}(a x)^2}{8 c^2}+\frac{2 a^2 \tan ^{-1}(a x)^3}{c^2 x}-\frac{a^2 \tan ^{-1}(a x)}{c^2 x}-\frac{7 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^2}-\frac{a \tan ^{-1}(a x)^2}{2 c^2 x^2}-\frac{\tan ^{-1}(a x)^3}{3 c^2 x^3}","-\frac{7 a^3 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^2}+\frac{7 i a^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^2}-\frac{3 a^3}{8 c^2 \left(a^2 x^2+1\right)}-\frac{a^3 \log \left(a^2 x^2+1\right)}{2 c^2}+\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^2 \left(a^2 x^2+1\right)}-\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^2 \left(a^2 x^2+1\right)}+\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^2 \left(a^2 x^2+1\right)}+\frac{a^3 \log (x)}{c^2}+\frac{5 a^3 \tan ^{-1}(a x)^4}{8 c^2}+\frac{7 i a^3 \tan ^{-1}(a x)^3}{3 c^2}-\frac{7 a^3 \tan ^{-1}(a x)^2}{8 c^2}+\frac{2 a^2 \tan ^{-1}(a x)^3}{c^2 x}-\frac{a^2 \tan ^{-1}(a x)}{c^2 x}-\frac{7 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^2}-\frac{a \tan ^{-1}(a x)^2}{2 c^2 x^2}-\frac{\tan ^{-1}(a x)^3}{3 c^2 x^3}",1,"(-3*a^3)/(8*c^2*(1 + a^2*x^2)) - (a^2*ArcTan[a*x])/(c^2*x) - (3*a^4*x*ArcTan[a*x])/(4*c^2*(1 + a^2*x^2)) - (7*a^3*ArcTan[a*x]^2)/(8*c^2) - (a*ArcTan[a*x]^2)/(2*c^2*x^2) + (3*a^3*ArcTan[a*x]^2)/(4*c^2*(1 + a^2*x^2)) + (((7*I)/3)*a^3*ArcTan[a*x]^3)/c^2 - ArcTan[a*x]^3/(3*c^2*x^3) + (2*a^2*ArcTan[a*x]^3)/(c^2*x) + (a^4*x*ArcTan[a*x]^3)/(2*c^2*(1 + a^2*x^2)) + (5*a^3*ArcTan[a*x]^4)/(8*c^2) + (a^3*Log[x])/c^2 - (a^3*Log[1 + a^2*x^2])/(2*c^2) - (7*a^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^2 + ((7*I)*a^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^2 - (7*a^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^2)","A",35,15,22,0.6818,1,"{4966, 4918, 4852, 266, 36, 29, 31, 4884, 4924, 4868, 4992, 6610, 4892, 4930, 261}"
404,1,212,0,0.2939019,"\int \frac{x^3 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^3*ArcTan[a*x]^3)/(c + a^2*c*x^2)^3,x]","-\frac{3 x^3}{128 a c^3 \left(a^2 x^2+1\right)^2}-\frac{45 x}{256 a^3 c^3 \left(a^2 x^2+1\right)}+\frac{x^4 \tan ^{-1}(a x)^3}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{3 x^4 \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x^3 \tan ^{-1}(a x)^2}{16 a c^3 \left(a^2 x^2+1\right)^2}+\frac{9 x \tan ^{-1}(a x)^2}{32 a^3 c^3 \left(a^2 x^2+1\right)}+\frac{9 \tan ^{-1}(a x)}{32 a^4 c^3 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)^3}{32 a^4 c^3}-\frac{27 \tan ^{-1}(a x)}{256 a^4 c^3}","-\frac{3 x^3}{128 a c^3 \left(a^2 x^2+1\right)^2}-\frac{45 x}{256 a^3 c^3 \left(a^2 x^2+1\right)}+\frac{x^4 \tan ^{-1}(a x)^3}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{3 x^4 \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x^3 \tan ^{-1}(a x)^2}{16 a c^3 \left(a^2 x^2+1\right)^2}+\frac{9 x \tan ^{-1}(a x)^2}{32 a^3 c^3 \left(a^2 x^2+1\right)}+\frac{9 \tan ^{-1}(a x)}{32 a^4 c^3 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)^3}{32 a^4 c^3}-\frac{27 \tan ^{-1}(a x)}{256 a^4 c^3}",1,"(-3*x^3)/(128*a*c^3*(1 + a^2*x^2)^2) - (45*x)/(256*a^3*c^3*(1 + a^2*x^2)) - (27*ArcTan[a*x])/(256*a^4*c^3) - (3*x^4*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) + (9*ArcTan[a*x])/(32*a^4*c^3*(1 + a^2*x^2)) + (3*x^3*ArcTan[a*x]^2)/(16*a*c^3*(1 + a^2*x^2)^2) + (9*x*ArcTan[a*x]^2)/(32*a^3*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x]^3)/(32*a^4*c^3) + (x^4*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2)","A",9,7,22,0.3182,1,"{4944, 4940, 4936, 4930, 199, 205, 288}"
405,1,237,0,0.3885716,"\int \frac{x^2 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^3,x]","-\frac{3}{128 a^3 c^3 \left(a^2 x^2+1\right)}+\frac{3}{128 a^3 c^3 \left(a^2 x^2+1\right)^2}+\frac{x \tan ^{-1}(a x)^3}{8 a^2 c^3 \left(a^2 x^2+1\right)}-\frac{x \tan ^{-1}(a x)^3}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left(a^2 x^2+1\right)^2}-\frac{3 x \tan ^{-1}(a x)}{64 a^2 c^3 \left(a^2 x^2+1\right)}+\frac{3 x \tan ^{-1}(a x)}{32 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)^4}{32 a^3 c^3}-\frac{3 \tan ^{-1}(a x)^2}{128 a^3 c^3}","-\frac{3}{128 a^3 c^3 \left(a^2 x^2+1\right)}+\frac{3}{128 a^3 c^3 \left(a^2 x^2+1\right)^2}+\frac{x \tan ^{-1}(a x)^3}{8 a^2 c^3 \left(a^2 x^2+1\right)}-\frac{x \tan ^{-1}(a x)^3}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left(a^2 x^2+1\right)^2}-\frac{3 x \tan ^{-1}(a x)}{64 a^2 c^3 \left(a^2 x^2+1\right)}+\frac{3 x \tan ^{-1}(a x)}{32 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)^4}{32 a^3 c^3}-\frac{3 \tan ^{-1}(a x)^2}{128 a^3 c^3}",1,"3/(128*a^3*c^3*(1 + a^2*x^2)^2) - 3/(128*a^3*c^3*(1 + a^2*x^2)) + (3*x*ArcTan[a*x])/(32*a^2*c^3*(1 + a^2*x^2)^2) - (3*x*ArcTan[a*x])/(64*a^2*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x]^2)/(128*a^3*c^3) - (3*ArcTan[a*x]^2)/(16*a^3*c^3*(1 + a^2*x^2)^2) + (3*ArcTan[a*x]^2)/(16*a^3*c^3*(1 + a^2*x^2)) - (x*ArcTan[a*x]^3)/(4*a^2*c^3*(1 + a^2*x^2)^2) + (x*ArcTan[a*x]^3)/(8*a^2*c^3*(1 + a^2*x^2)) + ArcTan[a*x]^4/(32*a^3*c^3)","A",13,6,22,0.2727,1,"{4964, 4892, 4930, 261, 4900, 4896}"
406,1,208,0,0.1774071,"\int \frac{x \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x*ArcTan[a*x]^3)/(c + a^2*c*x^2)^3,x]","-\frac{45 x}{256 a c^3 \left(a^2 x^2+1\right)}-\frac{3 x}{128 a c^3 \left(a^2 x^2+1\right)^2}-\frac{\tan ^{-1}(a x)^3}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{9 x \tan ^{-1}(a x)^2}{32 a c^3 \left(a^2 x^2+1\right)}+\frac{3 x \tan ^{-1}(a x)^2}{16 a c^3 \left(a^2 x^2+1\right)^2}+\frac{9 \tan ^{-1}(a x)}{32 a^2 c^3 \left(a^2 x^2+1\right)}+\frac{3 \tan ^{-1}(a x)}{32 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)^3}{32 a^2 c^3}-\frac{45 \tan ^{-1}(a x)}{256 a^2 c^3}","-\frac{45 x}{256 a c^3 \left(a^2 x^2+1\right)}-\frac{3 x}{128 a c^3 \left(a^2 x^2+1\right)^2}-\frac{\tan ^{-1}(a x)^3}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{9 x \tan ^{-1}(a x)^2}{32 a c^3 \left(a^2 x^2+1\right)}+\frac{3 x \tan ^{-1}(a x)^2}{16 a c^3 \left(a^2 x^2+1\right)^2}+\frac{9 \tan ^{-1}(a x)}{32 a^2 c^3 \left(a^2 x^2+1\right)}+\frac{3 \tan ^{-1}(a x)}{32 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)^3}{32 a^2 c^3}-\frac{45 \tan ^{-1}(a x)}{256 a^2 c^3}",1,"(-3*x)/(128*a*c^3*(1 + a^2*x^2)^2) - (45*x)/(256*a*c^3*(1 + a^2*x^2)) - (45*ArcTan[a*x])/(256*a^2*c^3) + (3*ArcTan[a*x])/(32*a^2*c^3*(1 + a^2*x^2)^2) + (9*ArcTan[a*x])/(32*a^2*c^3*(1 + a^2*x^2)) + (3*x*ArcTan[a*x]^2)/(16*a*c^3*(1 + a^2*x^2)^2) + (9*x*ArcTan[a*x]^2)/(32*a*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^3)/(32*a^2*c^3) - ArcTan[a*x]^3/(4*a^2*c^3*(1 + a^2*x^2)^2)","A",9,5,20,0.2500,1,"{4930, 4900, 4892, 199, 205}"
407,1,225,0,0.1964437,"\int \frac{\tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]^3/(c + a^2*c*x^2)^3,x]","-\frac{45}{128 a c^3 \left(a^2 x^2+1\right)}-\frac{3}{128 a c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x \tan ^{-1}(a x)^3}{8 c^3 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)^3}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{9 \tan ^{-1}(a x)^2}{16 a c^3 \left(a^2 x^2+1\right)}+\frac{3 \tan ^{-1}(a x)^2}{16 a c^3 \left(a^2 x^2+1\right)^2}-\frac{45 x \tan ^{-1}(a x)}{64 c^3 \left(a^2 x^2+1\right)}-\frac{3 x \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)^4}{32 a c^3}-\frac{45 \tan ^{-1}(a x)^2}{128 a c^3}","-\frac{45}{128 a c^3 \left(a^2 x^2+1\right)}-\frac{3}{128 a c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x \tan ^{-1}(a x)^3}{8 c^3 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)^3}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{9 \tan ^{-1}(a x)^2}{16 a c^3 \left(a^2 x^2+1\right)}+\frac{3 \tan ^{-1}(a x)^2}{16 a c^3 \left(a^2 x^2+1\right)^2}-\frac{45 x \tan ^{-1}(a x)}{64 c^3 \left(a^2 x^2+1\right)}-\frac{3 x \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)^4}{32 a c^3}-\frac{45 \tan ^{-1}(a x)^2}{128 a c^3}",1,"-3/(128*a*c^3*(1 + a^2*x^2)^2) - 45/(128*a*c^3*(1 + a^2*x^2)) - (3*x*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) - (45*x*ArcTan[a*x])/(64*c^3*(1 + a^2*x^2)) - (45*ArcTan[a*x]^2)/(128*a*c^3) + (3*ArcTan[a*x]^2)/(16*a*c^3*(1 + a^2*x^2)^2) + (9*ArcTan[a*x]^2)/(16*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x]^3)/(8*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^4)/(32*a*c^3)","A",8,5,19,0.2632,1,"{4900, 4892, 4930, 261, 4896}"
408,1,332,0,0.7093666,"\int \frac{\tan ^{-1}(a x)^3}{x \left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]^3/(x*(c + a^2*c*x^2)^3),x]","\frac{3 i \text{PolyLog}\left(4,-1+\frac{2}{1-i a x}\right)}{4 c^3}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^3}+\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^3}+\frac{141 a x}{256 c^3 \left(a^2 x^2+1\right)}+\frac{3 a x}{128 c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)^3}{2 c^3 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^3}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{33 a x \tan ^{-1}(a x)^2}{32 c^3 \left(a^2 x^2+1\right)}-\frac{3 a x \tan ^{-1}(a x)^2}{16 c^3 \left(a^2 x^2+1\right)^2}-\frac{33 \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)^2}-\frac{i \tan ^{-1}(a x)^4}{4 c^3}-\frac{11 \tan ^{-1}(a x)^3}{32 c^3}+\frac{141 \tan ^{-1}(a x)}{256 c^3}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^3}{c^3}","\frac{3 i \text{PolyLog}\left(4,-1+\frac{2}{1-i a x}\right)}{4 c^3}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^3}+\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^3}+\frac{141 a x}{256 c^3 \left(a^2 x^2+1\right)}+\frac{3 a x}{128 c^3 \left(a^2 x^2+1\right)^2}+\frac{\tan ^{-1}(a x)^3}{2 c^3 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^3}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{33 a x \tan ^{-1}(a x)^2}{32 c^3 \left(a^2 x^2+1\right)}-\frac{3 a x \tan ^{-1}(a x)^2}{16 c^3 \left(a^2 x^2+1\right)^2}-\frac{33 \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)^2}-\frac{i \tan ^{-1}(a x)^4}{4 c^3}-\frac{11 \tan ^{-1}(a x)^3}{32 c^3}+\frac{141 \tan ^{-1}(a x)}{256 c^3}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^3}{c^3}",1,"(3*a*x)/(128*c^3*(1 + a^2*x^2)^2) + (141*a*x)/(256*c^3*(1 + a^2*x^2)) + (141*ArcTan[a*x])/(256*c^3) - (3*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) - (33*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)) - (3*a*x*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)^2) - (33*a*x*ArcTan[a*x]^2)/(32*c^3*(1 + a^2*x^2)) - (11*ArcTan[a*x]^3)/(32*c^3) + ArcTan[a*x]^3/(4*c^3*(1 + a^2*x^2)^2) + ArcTan[a*x]^3/(2*c^3*(1 + a^2*x^2)) - ((I/4)*ArcTan[a*x]^4)/c^3 + (ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)])/c^3 - (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 + (3*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^3) + (((3*I)/4)*PolyLog[4, -1 + 2/(1 - I*a*x)])/c^3","A",21,12,22,0.5455,1,"{4966, 4924, 4868, 4884, 4992, 4996, 6610, 4930, 4892, 199, 205, 4900}"
409,1,332,0,0.7542895,"\int \frac{\tan ^{-1}(a x)^3}{x^2 \left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)^3),x]","\frac{3 a \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^3}-\frac{3 i a \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^3}+\frac{93 a}{128 c^3 \left(a^2 x^2+1\right)}+\frac{3 a}{128 c^3 \left(a^2 x^2+1\right)^2}-\frac{7 a^2 x \tan ^{-1}(a x)^3}{8 c^3 \left(a^2 x^2+1\right)}-\frac{a^2 x \tan ^{-1}(a x)^3}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{21 a \tan ^{-1}(a x)^2}{16 c^3 \left(a^2 x^2+1\right)}-\frac{3 a \tan ^{-1}(a x)^2}{16 c^3 \left(a^2 x^2+1\right)^2}+\frac{93 a^2 x \tan ^{-1}(a x)}{64 c^3 \left(a^2 x^2+1\right)}+\frac{3 a^2 x \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)^2}-\frac{15 a \tan ^{-1}(a x)^4}{32 c^3}-\frac{\tan ^{-1}(a x)^3}{c^3 x}-\frac{i a \tan ^{-1}(a x)^3}{c^3}+\frac{93 a \tan ^{-1}(a x)^2}{128 c^3}+\frac{3 a \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^3}","\frac{3 a \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^3}-\frac{3 i a \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^3}+\frac{93 a}{128 c^3 \left(a^2 x^2+1\right)}+\frac{3 a}{128 c^3 \left(a^2 x^2+1\right)^2}-\frac{7 a^2 x \tan ^{-1}(a x)^3}{8 c^3 \left(a^2 x^2+1\right)}-\frac{a^2 x \tan ^{-1}(a x)^3}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{21 a \tan ^{-1}(a x)^2}{16 c^3 \left(a^2 x^2+1\right)}-\frac{3 a \tan ^{-1}(a x)^2}{16 c^3 \left(a^2 x^2+1\right)^2}+\frac{93 a^2 x \tan ^{-1}(a x)}{64 c^3 \left(a^2 x^2+1\right)}+\frac{3 a^2 x \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)^2}-\frac{15 a \tan ^{-1}(a x)^4}{32 c^3}-\frac{\tan ^{-1}(a x)^3}{c^3 x}-\frac{i a \tan ^{-1}(a x)^3}{c^3}+\frac{93 a \tan ^{-1}(a x)^2}{128 c^3}+\frac{3 a \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^3}",1,"(3*a)/(128*c^3*(1 + a^2*x^2)^2) + (93*a)/(128*c^3*(1 + a^2*x^2)) + (3*a^2*x*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) + (93*a^2*x*ArcTan[a*x])/(64*c^3*(1 + a^2*x^2)) + (93*a*ArcTan[a*x]^2)/(128*c^3) - (3*a*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)^2) - (21*a*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)) - (I*a*ArcTan[a*x]^3)/c^3 - ArcTan[a*x]^3/(c^3*x) - (a^2*x*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) - (7*a^2*x*ArcTan[a*x]^3)/(8*c^3*(1 + a^2*x^2)) - (15*a*ArcTan[a*x]^4)/(32*c^3) + (3*a*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^3 - ((3*I)*a*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 + (3*a*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^3)","A",21,13,22,0.5909,1,"{4966, 4918, 4852, 4924, 4868, 4884, 4992, 6610, 4892, 4930, 261, 4900, 4896}"
410,1,478,0,1.8384427,"\int \frac{\tan ^{-1}(a x)^3}{x^3 \left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]^3/(x^3*(c + a^2*c*x^2)^3),x]","-\frac{3 i a^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^3}-\frac{9 i a^2 \text{PolyLog}\left(4,-1+\frac{2}{1-i a x}\right)}{4 c^3}+\frac{9 i a^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^3}-\frac{9 a^2 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^3}-\frac{237 a^3 x}{256 c^3 \left(a^2 x^2+1\right)}-\frac{3 a^3 x}{128 c^3 \left(a^2 x^2+1\right)^2}-\frac{a^2 \tan ^{-1}(a x)^3}{c^3 \left(a^2 x^2+1\right)}-\frac{a^2 \tan ^{-1}(a x)^3}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{57 a^3 x \tan ^{-1}(a x)^2}{32 c^3 \left(a^2 x^2+1\right)}+\frac{3 a^3 x \tan ^{-1}(a x)^2}{16 c^3 \left(a^2 x^2+1\right)^2}+\frac{57 a^2 \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)}+\frac{3 a^2 \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 i a^2 \tan ^{-1}(a x)^4}{4 c^3}+\frac{3 a^2 \tan ^{-1}(a x)^3}{32 c^3}-\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c^3}-\frac{237 a^2 \tan ^{-1}(a x)}{256 c^3}-\frac{3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^3}{c^3}+\frac{3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^3}-\frac{\tan ^{-1}(a x)^3}{2 c^3 x^2}-\frac{3 a \tan ^{-1}(a x)^2}{2 c^3 x}","-\frac{3 i a^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^3}-\frac{9 i a^2 \text{PolyLog}\left(4,-1+\frac{2}{1-i a x}\right)}{4 c^3}+\frac{9 i a^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{2 c^3}-\frac{9 a^2 \tan ^{-1}(a x) \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{2 c^3}-\frac{237 a^3 x}{256 c^3 \left(a^2 x^2+1\right)}-\frac{3 a^3 x}{128 c^3 \left(a^2 x^2+1\right)^2}-\frac{a^2 \tan ^{-1}(a x)^3}{c^3 \left(a^2 x^2+1\right)}-\frac{a^2 \tan ^{-1}(a x)^3}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{57 a^3 x \tan ^{-1}(a x)^2}{32 c^3 \left(a^2 x^2+1\right)}+\frac{3 a^3 x \tan ^{-1}(a x)^2}{16 c^3 \left(a^2 x^2+1\right)^2}+\frac{57 a^2 \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)}+\frac{3 a^2 \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 i a^2 \tan ^{-1}(a x)^4}{4 c^3}+\frac{3 a^2 \tan ^{-1}(a x)^3}{32 c^3}-\frac{3 i a^2 \tan ^{-1}(a x)^2}{2 c^3}-\frac{237 a^2 \tan ^{-1}(a x)}{256 c^3}-\frac{3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^3}{c^3}+\frac{3 a^2 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{c^3}-\frac{\tan ^{-1}(a x)^3}{2 c^3 x^2}-\frac{3 a \tan ^{-1}(a x)^2}{2 c^3 x}",1,"(-3*a^3*x)/(128*c^3*(1 + a^2*x^2)^2) - (237*a^3*x)/(256*c^3*(1 + a^2*x^2)) - (237*a^2*ArcTan[a*x])/(256*c^3) + (3*a^2*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) + (57*a^2*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)) - (((3*I)/2)*a^2*ArcTan[a*x]^2)/c^3 - (3*a*ArcTan[a*x]^2)/(2*c^3*x) + (3*a^3*x*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)^2) + (57*a^3*x*ArcTan[a*x]^2)/(32*c^3*(1 + a^2*x^2)) + (3*a^2*ArcTan[a*x]^3)/(32*c^3) - ArcTan[a*x]^3/(2*c^3*x^2) - (a^2*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) - (a^2*ArcTan[a*x]^3)/(c^3*(1 + a^2*x^2)) + (((3*I)/4)*a^2*ArcTan[a*x]^4)/c^3 + (3*a^2*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)])/c^3 - (3*a^2*ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)])/c^3 - (((3*I)/2)*a^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 + (((9*I)/2)*a^2*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 - (9*a^2*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/(2*c^3) - (((9*I)/4)*a^2*PolyLog[4, -1 + 2/(1 - I*a*x)])/c^3","A",47,15,22,0.6818,1,"{4966, 4918, 4852, 4924, 4868, 2447, 4884, 4992, 4996, 6610, 4930, 4892, 199, 205, 4900}"
411,1,432,0,2.1595208,"\int \frac{\tan ^{-1}(a x)^3}{x^4 \left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)^3),x]","-\frac{5 a^3 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{c^3}+\frac{10 i a^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^3}-\frac{141 a^3}{128 c^3 \left(a^2 x^2+1\right)}-\frac{3 a^3}{128 c^3 \left(a^2 x^2+1\right)^2}-\frac{a^3 \log \left(a^2 x^2+1\right)}{2 c^3}+\frac{11 a^4 x \tan ^{-1}(a x)^3}{8 c^3 \left(a^2 x^2+1\right)}+\frac{a^4 x \tan ^{-1}(a x)^3}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{141 a^4 x \tan ^{-1}(a x)}{64 c^3 \left(a^2 x^2+1\right)}-\frac{3 a^4 x \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{33 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left(a^2 x^2+1\right)}+\frac{3 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left(a^2 x^2+1\right)^2}+\frac{a^3 \log (x)}{c^3}+\frac{35 a^3 \tan ^{-1}(a x)^4}{32 c^3}+\frac{10 i a^3 \tan ^{-1}(a x)^3}{3 c^3}-\frac{205 a^3 \tan ^{-1}(a x)^2}{128 c^3}+\frac{3 a^2 \tan ^{-1}(a x)^3}{c^3 x}-\frac{a^2 \tan ^{-1}(a x)}{c^3 x}-\frac{10 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^3}-\frac{a \tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{\tan ^{-1}(a x)^3}{3 c^3 x^3}","-\frac{5 a^3 \text{PolyLog}\left(3,-1+\frac{2}{1-i a x}\right)}{c^3}+\frac{10 i a^3 \tan ^{-1}(a x) \text{PolyLog}\left(2,-1+\frac{2}{1-i a x}\right)}{c^3}-\frac{141 a^3}{128 c^3 \left(a^2 x^2+1\right)}-\frac{3 a^3}{128 c^3 \left(a^2 x^2+1\right)^2}-\frac{a^3 \log \left(a^2 x^2+1\right)}{2 c^3}+\frac{11 a^4 x \tan ^{-1}(a x)^3}{8 c^3 \left(a^2 x^2+1\right)}+\frac{a^4 x \tan ^{-1}(a x)^3}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{141 a^4 x \tan ^{-1}(a x)}{64 c^3 \left(a^2 x^2+1\right)}-\frac{3 a^4 x \tan ^{-1}(a x)}{32 c^3 \left(a^2 x^2+1\right)^2}+\frac{33 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left(a^2 x^2+1\right)}+\frac{3 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left(a^2 x^2+1\right)^2}+\frac{a^3 \log (x)}{c^3}+\frac{35 a^3 \tan ^{-1}(a x)^4}{32 c^3}+\frac{10 i a^3 \tan ^{-1}(a x)^3}{3 c^3}-\frac{205 a^3 \tan ^{-1}(a x)^2}{128 c^3}+\frac{3 a^2 \tan ^{-1}(a x)^3}{c^3 x}-\frac{a^2 \tan ^{-1}(a x)}{c^3 x}-\frac{10 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c^3}-\frac{a \tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{\tan ^{-1}(a x)^3}{3 c^3 x^3}",1,"(-3*a^3)/(128*c^3*(1 + a^2*x^2)^2) - (141*a^3)/(128*c^3*(1 + a^2*x^2)) - (a^2*ArcTan[a*x])/(c^3*x) - (3*a^4*x*ArcTan[a*x])/(32*c^3*(1 + a^2*x^2)^2) - (141*a^4*x*ArcTan[a*x])/(64*c^3*(1 + a^2*x^2)) - (205*a^3*ArcTan[a*x]^2)/(128*c^3) - (a*ArcTan[a*x]^2)/(2*c^3*x^2) + (3*a^3*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)^2) + (33*a^3*ArcTan[a*x]^2)/(16*c^3*(1 + a^2*x^2)) + (((10*I)/3)*a^3*ArcTan[a*x]^3)/c^3 - ArcTan[a*x]^3/(3*c^3*x^3) + (3*a^2*ArcTan[a*x]^3)/(c^3*x) + (a^4*x*ArcTan[a*x]^3)/(4*c^3*(1 + a^2*x^2)^2) + (11*a^4*x*ArcTan[a*x]^3)/(8*c^3*(1 + a^2*x^2)) + (35*a^3*ArcTan[a*x]^4)/(32*c^3) + (a^3*Log[x])/c^3 - (a^3*Log[1 + a^2*x^2])/(2*c^3) - (10*a^3*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)])/c^3 + ((10*I)*a^3*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/c^3 - (5*a^3*PolyLog[3, -1 + 2/(1 - I*a*x)])/c^3","A",57,17,22,0.7727,1,"{4966, 4918, 4852, 266, 36, 29, 31, 4884, 4924, 4868, 4992, 6610, 4892, 4930, 261, 4900, 4896}"
412,1,523,0,2.4372005,"\int x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx","Int[x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]","\frac{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{11 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{20 a^4 \sqrt{a^2 c x^2+c}}+\frac{11 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{x \sqrt{a^2 c x^2+c}}{20 a^3}+\frac{1}{5} x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{3 x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{20 a}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{15 a^2}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{10 a^2}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^3}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{15 a^4}-\frac{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{9 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{20 a^4}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{2 a^4}","\frac{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{11 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{20 a^4 \sqrt{a^2 c x^2+c}}+\frac{11 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{x \sqrt{a^2 c x^2+c}}{20 a^3}+\frac{1}{5} x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{3 x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{20 a}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{15 a^2}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{10 a^2}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^3}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{15 a^4}-\frac{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{20 a^4 \sqrt{a^2 c x^2+c}}-\frac{9 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{20 a^4}+\frac{\sqrt{c} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{2 a^4}",1,"-(x*Sqrt[c + a^2*c*x^2])/(20*a^3) - (9*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(20*a^4) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(10*a^2) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(8*a^3) - (3*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(20*a) - (((11*I)/20)*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^4*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(15*a^4) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(15*a^2) + (x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/5 + (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(2*a^4) + (((11*I)/20)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) - (((11*I)/20)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) - (11*c*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(20*a^4*Sqrt[c + a^2*c*x^2]) + (11*c*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(20*a^4*Sqrt[c + a^2*c*x^2])","A",71,12,24,0.5000,1,"{4950, 4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589, 321}"
413,1,747,0,1.8462743,"\int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx","Int[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]","-\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c}}{4 a^3}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{1}{4} x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{8 a^2}-\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{4 a}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^3}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{4 a^2}","-\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c}}{4 a^3}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{1}{4} x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{8 a^2}-\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{4 a}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^3}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{4 a^2}",1,"-Sqrt[c + a^2*c*x^2]/(4*a^3) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(4*a^2) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(8*a^3) - (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(4*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(8*a^2) + (x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/4 + ((I/4)*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a^3*Sqrt[c + a^2*c*x^2]) + (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - (((3*I)/8)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (((3*I)/8)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - ((I/2)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + ((I/2)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2]) - (3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2]) + (((3*I)/4)*c*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (((3*I)/4)*c*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2])","A",40,12,24,0.5000,1,"{4950, 4952, 4930, 4890, 4886, 4888, 4181, 2531, 6609, 2282, 6589, 261}"
414,1,373,0,0.355473,"\int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx","Int[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]","-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3}{3 a^2 c}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^2 \sqrt{a^2 c x^2+c}}-\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^2}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a^2}","-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3}{3 a^2 c}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^2 \sqrt{a^2 c x^2+c}}-\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^2}-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a^2}",1,"(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/a^2 - (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a) + (I*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^2*Sqrt[c + a^2*c*x^2]) + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/(3*a^2*c) - (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/a^2 - (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (c*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) - (c*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2])","A",13,10,22,0.4545,1,"{4930, 4880, 4890, 4888, 4181, 2531, 2282, 6589, 217, 206}"
415,1,626,0,0.3614527,"\int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx","Int[Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]","\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{2 a \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{2 a \sqrt{a^2 c x^2+c}}-\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}+\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{a \sqrt{a^2 c x^2+c}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a}-\frac{6 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a \sqrt{a^2 c x^2+c}}","\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{2 a \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{2 a \sqrt{a^2 c x^2+c}}-\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}+\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{a \sqrt{a^2 c x^2+c}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a}-\frac{6 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a \sqrt{a^2 c x^2+c}}",1,"(-3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/2 - (I*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a*Sqrt[c + a^2*c*x^2]) - ((6*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (((3*I)/2)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((3*I)/2)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + ((3*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - ((3*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - ((3*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + ((3*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])","A",14,9,21,0.4286,1,"{4880, 4890, 4888, 4181, 2531, 6609, 2282, 6589, 4886}"
416,1,600,0,0.7271694,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x} \, dx","Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x,x]","\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3+\frac{6 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-\frac{2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}","\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3+\frac{6 i c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-\frac{2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}",1,"((6*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] + Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 - (2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*c*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*c*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*c*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",22,12,24,0.5000,1,"{4950, 4958, 4956, 4183, 2531, 6609, 2282, 6589, 4930, 4890, 4888, 4181}"
417,1,622,0,0.7743423,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x^2} \, dx","Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x^2,x]","\frac{3 i a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{x}-\frac{6 a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}","\frac{3 i a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i a c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a c \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{x}-\frac{6 a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}",1,"-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x) - ((2*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2] - (6*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a*c*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*a*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*a*c*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*c*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",22,12,24,0.5000,1,"{4950, 4944, 4958, 4956, 4183, 2531, 2282, 6589, 4890, 4888, 4181, 6609}"
418,1,602,0,1.2376901,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x^3} \, dx","Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x^3,x]","\frac{3 i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{3 a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 x^2}-\frac{3 a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x}-\frac{a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}","\frac{3 i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{2 \sqrt{a^2 c x^2+c}}-\frac{3 a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^2 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 x^2}-\frac{3 a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x}-\frac{a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}",1,"(-3*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(2*x^2) - (a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (((3*I)/2)*a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((3*I)/2)*a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (3*a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (3*a^2*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^2*c*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",27,11,24,0.4583,1,"{4950, 4962, 4944, 4958, 4954, 4956, 4183, 2531, 6609, 2282, 6589}"
419,1,361,0,1.0162154,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x^4} \, dx","Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x^4,x]","\frac{i a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x^2}-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3}{3 c x^3}-a^3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}","\frac{i a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x^2}-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3}{3 c x^3}-a^3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}",1,"-((a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) - (a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x^2) - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/(3*c*x^3) - (a^3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - a^3*Sqrt[c]*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + (I*a^3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (I*a^3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (a^3*c*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (a^3*c*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",25,12,24,0.5000,1,"{4944, 4950, 4962, 266, 63, 208, 4958, 4956, 4183, 2531, 2282, 6589}"
420,1,652,0,7.3705899,"\int x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3 \, dx","Int[x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3,x]","\frac{51 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{51 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{51 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{280 a^4 \sqrt{a^2 c x^2+c}}+\frac{51 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{51 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{280 a^4 \sqrt{a^2 c x^2+c}}+\frac{23 c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{120 a^4}-\frac{c x^3 \sqrt{a^2 c x^2+c}}{140 a}+\frac{c x \sqrt{a^2 c x^2+c}}{420 a^3}+\frac{1}{7} a^2 c x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{1}{14} a c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+\frac{8}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3+\frac{1}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{23 c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{280 a}+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{35 a^2}+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{60 a^2}+\frac{9 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{112 a^3}-\frac{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{35 a^4}-\frac{163 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{840 a^4}","\frac{51 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{51 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{51 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{280 a^4 \sqrt{a^2 c x^2+c}}+\frac{51 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{51 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{280 a^4 \sqrt{a^2 c x^2+c}}+\frac{23 c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{120 a^4}-\frac{c x^3 \sqrt{a^2 c x^2+c}}{140 a}+\frac{c x \sqrt{a^2 c x^2+c}}{420 a^3}+\frac{1}{7} a^2 c x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{1}{14} a c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+\frac{8}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3+\frac{1}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{23 c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{280 a}+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{35 a^2}+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{60 a^2}+\frac{9 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{112 a^3}-\frac{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{35 a^4}-\frac{163 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{840 a^4}",1,"(c*x*Sqrt[c + a^2*c*x^2])/(420*a^3) - (c*x^3*Sqrt[c + a^2*c*x^2])/(140*a) - (163*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(840*a^4) + (c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(60*a^2) + (c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/35 + (9*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(112*a^3) - (23*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(280*a) - (a*c*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/14 - (((51*I)/280)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^4*Sqrt[c + a^2*c*x^2]) - (2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(35*a^4) + (c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(35*a^2) + (8*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/35 + (a^2*c*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/7 + (23*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(120*a^4) + (((51*I)/280)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) - (((51*I)/280)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) - (51*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(280*a^4*Sqrt[c + a^2*c*x^2]) + (51*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(280*a^4*Sqrt[c + a^2*c*x^2])","A",200,12,24,0.5000,1,"{4950, 4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589, 321}"
421,1,882,0,5.4699137,"\int x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3 \, dx","Int[x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3,x]","\frac{1}{6} a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^5-\frac{1}{10} a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^4+\frac{7}{24} c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^3+\frac{1}{20} c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^3-\frac{19 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^2}{120 a}+\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x}{16 a^2}+\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x}{12 a^2}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{31 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{240 a^3}-\frac{\left(a^2 c x^2+c\right)^{3/2}}{60 a^3}+\frac{41 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{60 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{16 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{16 a^3 \sqrt{a^2 c x^2+c}}-\frac{41 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{120 a^3 \sqrt{a^2 c x^2+c}}+\frac{41 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{120 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 c x^2+c}}{30 a^3}","\frac{1}{6} a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^5-\frac{1}{10} a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^4+\frac{7}{24} c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^3+\frac{1}{20} c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^3-\frac{19 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^2}{120 a}+\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x}{16 a^2}+\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x}{12 a^2}+\frac{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{31 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{240 a^3}-\frac{\left(a^2 c x^2+c\right)^{3/2}}{60 a^3}+\frac{41 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{60 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{16 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{16 a^3 \sqrt{a^2 c x^2+c}}-\frac{41 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{120 a^3 \sqrt{a^2 c x^2+c}}+\frac{41 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{120 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{8 a^3 \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 c x^2+c}}{30 a^3}",1,"-(c*Sqrt[c + a^2*c*x^2])/(30*a^3) - (c + a^2*c*x^2)^(3/2)/(60*a^3) + (c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(12*a^2) + (c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/20 + (31*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(240*a^3) - (19*c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(120*a) - (a*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/10 + (c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(16*a^2) + (7*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/24 + (a^2*c*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/6 + ((I/8)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a^3*Sqrt[c + a^2*c*x^2]) + (((41*I)/60)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - (((3*I)/16)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (((3*I)/16)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (((41*I)/120)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (((41*I)/120)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2]) - (3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2]) + (((3*I)/8)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (((3*I)/8)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2])","A",108,14,24,0.5833,1,"{4950, 4952, 4930, 4890, 4886, 4888, 4181, 2531, 6609, 2282, 6589, 261, 266, 43}"
422,1,477,0,0.4161409,"\int x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3 \, dx","Int[x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3,x]","-\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{9 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{20 a^2 \sqrt{a^2 c x^2+c}}-\frac{9 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{20 a^2 \sqrt{a^2 c x^2+c}}-\frac{c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{2 a^2}-\frac{c x \sqrt{a^2 c x^2+c}}{20 a}+\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^3}{5 a^2 c}-\frac{3 x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{20 a}-\frac{9 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{40 a}+\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{10 a^2}+\frac{9 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{20 a^2}","-\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{9 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{20 a^2 \sqrt{a^2 c x^2+c}}-\frac{9 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{20 a^2 \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{20 a^2 \sqrt{a^2 c x^2+c}}-\frac{c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{2 a^2}-\frac{c x \sqrt{a^2 c x^2+c}}{20 a}+\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^3}{5 a^2 c}-\frac{3 x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{20 a}-\frac{9 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{40 a}+\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{10 a^2}+\frac{9 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{20 a^2}",1,"-(c*x*Sqrt[c + a^2*c*x^2])/(20*a) + (9*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(20*a^2) + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(10*a^2) - (9*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(40*a) - (3*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(20*a) + (((9*I)/20)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^2*Sqrt[c + a^2*c*x^2]) + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/(5*a^2*c) - (c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(2*a^2) - (((9*I)/20)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (((9*I)/20)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (9*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(20*a^2*Sqrt[c + a^2*c*x^2]) - (9*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(20*a^2*Sqrt[c + a^2*c*x^2])","A",17,11,22,0.5000,1,"{4930, 4880, 4890, 4888, 4181, 2531, 2282, 6589, 217, 206, 195}"
423,1,760,0,0.5242363,"\int \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3 \, dx","Int[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3,x]","\frac{5 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{8 a \sqrt{a^2 c x^2+c}}-\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{8 a \sqrt{a^2 c x^2+c}}-\frac{9 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}+\frac{9 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}-\frac{9 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{4 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 c x^2+c}}{4 a}+\frac{1}{4} x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{4 a}-\frac{9 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a}+\frac{1}{4} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)","\frac{5 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{2 a \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{8 a \sqrt{a^2 c x^2+c}}-\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{8 a \sqrt{a^2 c x^2+c}}-\frac{9 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}+\frac{9 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}-\frac{9 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}+\frac{9 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{4 a \sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{4 a \sqrt{a^2 c x^2+c}}-\frac{5 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 c x^2+c}}{4 a}+\frac{1}{4} x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{4 a}-\frac{9 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a}+\frac{1}{4} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)",1,"-(c*Sqrt[c + a^2*c*x^2])/(4*a) + (c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/4 - (9*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(8*a) - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(4*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/4 - (((3*I)/4)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a*Sqrt[c + a^2*c*x^2]) - ((5*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (((9*I)/8)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((9*I)/8)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (((5*I)/2)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (((5*I)/2)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (9*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*a*Sqrt[c + a^2*c*x^2]) + (9*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*a*Sqrt[c + a^2*c*x^2]) - (((9*I)/4)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (((9*I)/4)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])","A",18,10,21,0.4762,1,"{4880, 4890, 4888, 4181, 2531, 6609, 2282, 6589, 4886, 4878}"
424,1,726,0,1.1439169,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3}{x} \, dx","Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x,x]","\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{7 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{7 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{7 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{7 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{7 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3+c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{1}{2} a c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)","\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{7 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{7 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{7 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{7 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{7 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)-\frac{2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3+c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{1}{2} a c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)",1,"c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (a*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 + ((7*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] + c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/3 - (2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + ((3*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((7*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((7*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (7*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (7*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",36,15,24,0.6250,1,"{4950, 4958, 4956, 4183, 2531, 6609, 2282, 6589, 4930, 4890, 4888, 4181, 4880, 217, 206}"
425,1,901,0,1.2372692,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3}{x^2} \, dx","Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^2,x]","-\frac{3 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 c x^2+c}}+\frac{1}{2} a^2 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{x}-\frac{6 a c^2 \sqrt{a^2 x^2+1} \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}+\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{2 \sqrt{a^2 c x^2+c}}-\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{2 \sqrt{a^2 c x^2+c}}-\frac{3}{2} a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{6 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{6 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{6 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{9 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{9 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}","-\frac{3 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 c x^2+c}}+\frac{1}{2} a^2 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{x}-\frac{6 a c^2 \sqrt{a^2 x^2+1} \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}+\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{2 \sqrt{a^2 c x^2+c}}-\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{2 \sqrt{a^2 c x^2+c}}-\frac{3}{2} a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{6 i a c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{6 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{6 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{9 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{9 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{9 i a c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}",1,"(-3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x + (a^2*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/2 - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((9*I)/2)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((9*I)/2)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (9*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (9*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((9*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((9*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",37,14,24,0.5833,1,"{4950, 4944, 4958, 4956, 4183, 2531, 2282, 6589, 4890, 4888, 4181, 6609, 4880, 4886}"
426,1,919,0,2.0157875,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3}{x^3} \, dx","Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^3,x]","-\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 c x^2+c}}+a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 x^2}+\frac{6 i a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}+\frac{9 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{2 \sqrt{a^2 c x^2+c}}-\frac{9 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{2 \sqrt{a^2 c x^2+c}}-\frac{3 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x}-\frac{6 a^2 c^2 \sqrt{a^2 x^2+1} \tanh ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{6 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{6 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{9 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{9 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{9 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{9 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}","-\frac{3 a^2 c^2 \sqrt{a^2 x^2+1} \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 c x^2+c}}+a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 x^2}+\frac{6 i a^2 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}+\frac{9 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{2 \sqrt{a^2 c x^2+c}}-\frac{9 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{2 \sqrt{a^2 c x^2+c}}-\frac{3 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x}-\frac{6 a^2 c^2 \sqrt{a^2 x^2+1} \tanh ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{6 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{6 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}-\frac{9 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{9 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{9 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{9 i a^2 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}",1,"(-3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x) + ((6*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] + a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(2*x^2) - (3*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (((9*I)/2)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((9*I)/2)*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (9*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (9*a^2*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((9*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((9*I)*a^2*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",50,15,24,0.6250,1,"{4950, 4962, 4944, 4958, 4954, 4956, 4183, 2531, 6609, 2282, 6589, 4930, 4890, 4888, 4181}"
427,1,788,0,1.8791338,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3}{x^4} \, dx","Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^4,x]","\frac{7 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{7 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{7 a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{7 a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 c x^2+c}}-a^3 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{7 a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{x}-\frac{a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x^2}-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3}{3 x^3}","\frac{7 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{7 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{7 a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{7 a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i a^3 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 c x^2+c}}-a^3 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{7 a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{x}-\frac{a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 x^2}-\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3}{3 x^3}",1,"-((a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) - (a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x^2) - (a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/(3*x^3) - ((2*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2] - (7*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - a^3*c^(3/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + ((7*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((7*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (7*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (7*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",48,16,24,0.6667,1,"{4950, 4944, 4962, 266, 63, 208, 4958, 4956, 4183, 2531, 2282, 6589, 4890, 4888, 4181, 6609}"
428,1,798,0,19.6635801,"\int x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3 \, dx","Int[x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]","\frac{1}{9} a^4 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^8-\frac{1}{24} a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^7+\frac{19}{63} a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^6+\frac{1}{84} a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^6-\frac{103 a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^5}{1008}-\frac{1}{504} a c^2 \sqrt{a^2 c x^2+c} x^5+\frac{5}{21} c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^4+\frac{67 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^4}{2520}-\frac{205 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^3}{4032 a}-\frac{c^2 \sqrt{a^2 c x^2+c} x^3}{240 a}+\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^2}{63 a^2}-\frac{47 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^2}{30240 a^2}+\frac{47 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x}{896 a^3}+\frac{85 c^2 \sqrt{a^2 c x^2+c} x}{12096 a^3}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{63 a^4}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{1344 a^4 \sqrt{a^2 c x^2+c}}-\frac{6157 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{60480 a^4}+\frac{1433 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{15120 a^4}+\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{1344 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{1344 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{1344 a^4 \sqrt{a^2 c x^2+c}}+\frac{115 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{1344 a^4 \sqrt{a^2 c x^2+c}}","\frac{1}{9} a^4 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^8-\frac{1}{24} a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^7+\frac{19}{63} a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^6+\frac{1}{84} a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^6-\frac{103 a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^5}{1008}-\frac{1}{504} a c^2 \sqrt{a^2 c x^2+c} x^5+\frac{5}{21} c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^4+\frac{67 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^4}{2520}-\frac{205 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^3}{4032 a}-\frac{c^2 \sqrt{a^2 c x^2+c} x^3}{240 a}+\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^2}{63 a^2}-\frac{47 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^2}{30240 a^2}+\frac{47 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x}{896 a^3}+\frac{85 c^2 \sqrt{a^2 c x^2+c} x}{12096 a^3}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{63 a^4}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{1344 a^4 \sqrt{a^2 c x^2+c}}-\frac{6157 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{60480 a^4}+\frac{1433 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{15120 a^4}+\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{1344 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{1344 a^4 \sqrt{a^2 c x^2+c}}-\frac{115 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{1344 a^4 \sqrt{a^2 c x^2+c}}+\frac{115 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{1344 a^4 \sqrt{a^2 c x^2+c}}",1,"(85*c^2*x*Sqrt[c + a^2*c*x^2])/(12096*a^3) - (c^2*x^3*Sqrt[c + a^2*c*x^2])/(240*a) - (a*c^2*x^5*Sqrt[c + a^2*c*x^2])/504 - (6157*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(60480*a^4) - (47*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(30240*a^2) + (67*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/2520 + (a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/84 + (47*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(896*a^3) - (205*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(4032*a) - (103*a*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/1008 - (a^3*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/24 - (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^4*Sqrt[c + a^2*c*x^2]) - (2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(63*a^4) + (c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(63*a^2) + (5*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/21 + (19*a^2*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/63 + (a^4*c^2*x^8*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/9 + (1433*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(15120*a^4) + (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) - (((115*I)/1344)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) - (115*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(1344*a^4*Sqrt[c + a^2*c*x^2]) + (115*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(1344*a^4*Sqrt[c + a^2*c*x^2])","A",547,12,24,0.5000,1,"{4950, 4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589, 321}"
429,1,1019,0,15.4172608,"\int x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3 \, dx","Int[x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]","\frac{1}{8} a^4 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^7-\frac{3}{56} a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^6+\frac{17}{48} a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^5+\frac{1}{56} a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^5-\frac{83}{560} a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^4+\frac{59}{192} c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^3+\frac{29}{560} c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^3-\frac{737 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^2}{6720 a}+\frac{5 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x}{128 a^2}+\frac{43 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x}{1344 a^2}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{64 a^3 \sqrt{a^2 c x^2+c}}-\frac{\left(a^2 c x^2+c\right)^{5/2}}{280 a^3}+\frac{1373 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{13440 a^3}-\frac{3 c \left(a^2 c x^2+c\right)^{3/2}}{560 a^3}+\frac{397 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{840 a^3 \sqrt{a^2 c x^2+c}}-\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{128 a^3 \sqrt{a^2 c x^2+c}}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{128 a^3 \sqrt{a^2 c x^2+c}}-\frac{397 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{1680 a^3 \sqrt{a^2 c x^2+c}}+\frac{397 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{1680 a^3 \sqrt{a^2 c x^2+c}}+\frac{15 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}-\frac{15 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}-\frac{15 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}+\frac{13 c^2 \sqrt{a^2 c x^2+c}}{6720 a^3}","\frac{1}{8} a^4 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^7-\frac{3}{56} a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^6+\frac{17}{48} a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^5+\frac{1}{56} a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^5-\frac{83}{560} a c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^4+\frac{59}{192} c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x^3+\frac{29}{560} c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x^3-\frac{737 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 x^2}{6720 a}+\frac{5 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 x}{128 a^2}+\frac{43 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) x}{1344 a^2}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{64 a^3 \sqrt{a^2 c x^2+c}}-\frac{\left(a^2 c x^2+c\right)^{5/2}}{280 a^3}+\frac{1373 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{13440 a^3}-\frac{3 c \left(a^2 c x^2+c\right)^{3/2}}{560 a^3}+\frac{397 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{840 a^3 \sqrt{a^2 c x^2+c}}-\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{128 a^3 \sqrt{a^2 c x^2+c}}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{128 a^3 \sqrt{a^2 c x^2+c}}-\frac{397 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{1680 a^3 \sqrt{a^2 c x^2+c}}+\frac{397 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{1680 a^3 \sqrt{a^2 c x^2+c}}+\frac{15 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}-\frac{15 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}-\frac{15 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{64 a^3 \sqrt{a^2 c x^2+c}}+\frac{13 c^2 \sqrt{a^2 c x^2+c}}{6720 a^3}",1,"(13*c^2*Sqrt[c + a^2*c*x^2])/(6720*a^3) - (3*c*(c + a^2*c*x^2)^(3/2))/(560*a^3) - (c + a^2*c*x^2)^(5/2)/(280*a^3) + (43*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(1344*a^2) + (29*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/560 + (a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/56 + (1373*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(13440*a^3) - (737*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(6720*a) - (83*a*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/560 - (3*a^3*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/56 + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(128*a^2) + (59*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/192 + (17*a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/48 + (a^4*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/8 + (((5*I)/64)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a^3*Sqrt[c + a^2*c*x^2]) + (((397*I)/840)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - (((15*I)/128)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (((15*I)/128)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (((397*I)/1680)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (((397*I)/1680)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (15*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) - (15*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) + (((15*I)/64)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (((15*I)/64)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2])","A",293,14,24,0.5833,1,"{4950, 4952, 4930, 4890, 4886, 4888, 4181, 2531, 6609, 2282, 6589, 261, 266, 43}"
430,1,561,0,0.5313818,"\int x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3 \, dx","Int[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]","-\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{56 a^2 \sqrt{a^2 c x^2+c}}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{56 a^2 \sqrt{a^2 c x^2+c}}+\frac{15 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{56 a^2 \sqrt{a^2 c x^2+c}}-\frac{15 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{56 a^2 \sqrt{a^2 c x^2+c}}-\frac{17 c^2 x \sqrt{a^2 c x^2+c}}{420 a}-\frac{15 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{112 a}+\frac{15 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{56 a^2}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{56 a^2 \sqrt{a^2 c x^2+c}}-\frac{37 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{120 a^2}-\frac{c x \left(a^2 c x^2+c\right)^{3/2}}{140 a}+\frac{\left(a^2 c x^2+c\right)^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac{x \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac{5 c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{56 a}+\frac{5 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{84 a^2}","-\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{56 a^2 \sqrt{a^2 c x^2+c}}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{56 a^2 \sqrt{a^2 c x^2+c}}+\frac{15 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{56 a^2 \sqrt{a^2 c x^2+c}}-\frac{15 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{56 a^2 \sqrt{a^2 c x^2+c}}-\frac{17 c^2 x \sqrt{a^2 c x^2+c}}{420 a}-\frac{15 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{112 a}+\frac{15 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{56 a^2}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{56 a^2 \sqrt{a^2 c x^2+c}}-\frac{37 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{120 a^2}-\frac{c x \left(a^2 c x^2+c\right)^{3/2}}{140 a}+\frac{\left(a^2 c x^2+c\right)^{7/2} \tan ^{-1}(a x)^3}{7 a^2 c}-\frac{x \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^2}{14 a}+\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)}{35 a^2}-\frac{5 c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{56 a}+\frac{5 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{84 a^2}",1,"(-17*c^2*x*Sqrt[c + a^2*c*x^2])/(420*a) - (c*x*(c + a^2*c*x^2)^(3/2))/(140*a) + (15*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(56*a^2) + (5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(84*a^2) + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/(35*a^2) - (15*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(112*a) - (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(56*a) - (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/(14*a) + (((15*I)/56)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^2*Sqrt[c + a^2*c*x^2]) + ((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^3)/(7*a^2*c) - (37*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(120*a^2) - (((15*I)/56)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (((15*I)/56)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (15*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(56*a^2*Sqrt[c + a^2*c*x^2]) - (15*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(56*a^2*Sqrt[c + a^2*c*x^2])","A",22,11,22,0.5000,1,"{4930, 4880, 4890, 4888, 4181, 2531, 2282, 6589, 217, 206, 195}"
431,1,870,0,0.7920934,"\int \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3 \, dx","Int[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]","-\frac{5 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3 c^3}{8 a \sqrt{a^2 c x^2+c}}-\frac{259 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) c^3}{60 a \sqrt{a^2 c x^2+c}}+\frac{15 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right) c^3}{16 a \sqrt{a^2 c x^2+c}}-\frac{15 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right) c^3}{16 a \sqrt{a^2 c x^2+c}}+\frac{259 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right) c^3}{120 a \sqrt{a^2 c x^2+c}}-\frac{259 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right) c^3}{120 a \sqrt{a^2 c x^2+c}}-\frac{15 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right) c^3}{8 a \sqrt{a^2 c x^2+c}}+\frac{15 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right) c^3}{8 a \sqrt{a^2 c x^2+c}}-\frac{15 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right) c^3}{8 a \sqrt{a^2 c x^2+c}}+\frac{15 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right) c^3}{8 a \sqrt{a^2 c x^2+c}}+\frac{5}{16} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2-\frac{15 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 c^2}{16 a}+\frac{17}{60} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) c^2-\frac{17 \sqrt{a^2 c x^2+c} c^2}{60 a}+\frac{5}{24} x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3 c-\frac{5 \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2 c}{24 a}-\frac{\left(a^2 c x^2+c\right)^{3/2} c}{60 a}+\frac{1}{20} x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x) c+\frac{1}{6} x \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^3-\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^2}{10 a}","-\frac{5 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3 c^3}{8 a \sqrt{a^2 c x^2+c}}-\frac{259 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) c^3}{60 a \sqrt{a^2 c x^2+c}}+\frac{15 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right) c^3}{16 a \sqrt{a^2 c x^2+c}}-\frac{15 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right) c^3}{16 a \sqrt{a^2 c x^2+c}}+\frac{259 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right) c^3}{120 a \sqrt{a^2 c x^2+c}}-\frac{259 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right) c^3}{120 a \sqrt{a^2 c x^2+c}}-\frac{15 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right) c^3}{8 a \sqrt{a^2 c x^2+c}}+\frac{15 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right) c^3}{8 a \sqrt{a^2 c x^2+c}}-\frac{15 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right) c^3}{8 a \sqrt{a^2 c x^2+c}}+\frac{15 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right) c^3}{8 a \sqrt{a^2 c x^2+c}}+\frac{5}{16} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2-\frac{15 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 c^2}{16 a}+\frac{17}{60} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) c^2-\frac{17 \sqrt{a^2 c x^2+c} c^2}{60 a}+\frac{5}{24} x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3 c-\frac{5 \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2 c}{24 a}-\frac{\left(a^2 c x^2+c\right)^{3/2} c}{60 a}+\frac{1}{20} x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x) c+\frac{1}{6} x \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^3-\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^2}{10 a}",1,"(-17*c^2*Sqrt[c + a^2*c*x^2])/(60*a) - (c*(c + a^2*c*x^2)^(3/2))/(60*a) + (17*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/60 + (c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/20 - (15*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(16*a) - (5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(24*a) - ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/(10*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/6 - (((5*I)/8)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a*Sqrt[c + a^2*c*x^2]) - (((259*I)/60)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) + (((15*I)/16)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (((15*I)/16)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (((259*I)/120)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (((259*I)/120)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a*Sqrt[c + a^2*c*x^2]) - (15*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(8*a*Sqrt[c + a^2*c*x^2]) + (15*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(8*a*Sqrt[c + a^2*c*x^2]) - (((15*I)/8)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (((15*I)/8)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])","A",23,10,21,0.4762,1,"{4880, 4890, 4888, 4181, 2531, 6609, 2282, 6589, 4886, 4878}"
432,1,845,0,1.783952,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3}{x} \, dx","Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x,x]","\frac{149 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2 c^3}{20 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}+\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}-\frac{149 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right) c^3}{20 \sqrt{a^2 c x^2+c}}+\frac{149 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right) c^3}{20 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}+\frac{149 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right) c^3}{20 \sqrt{a^2 c x^2+c}}-\frac{149 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right) c^3}{20 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}-\frac{3}{2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right) c^{5/2}+\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2-\frac{29}{40} a x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 c^2+\frac{29}{20} \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) c^2-\frac{1}{20} a x \sqrt{a^2 c x^2+c} c^2+\frac{1}{3} \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3 c-\frac{3}{20} a x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2 c+\frac{1}{10} \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x) c+\frac{1}{5} \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^3","\frac{149 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2 c^3}{20 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}+\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}-\frac{149 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right) c^3}{20 \sqrt{a^2 c x^2+c}}+\frac{149 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right) c^3}{20 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}+\frac{149 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right) c^3}{20 \sqrt{a^2 c x^2+c}}-\frac{149 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right) c^3}{20 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}-\frac{3}{2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right) c^{5/2}+\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2-\frac{29}{40} a x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 c^2+\frac{29}{20} \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) c^2-\frac{1}{20} a x \sqrt{a^2 c x^2+c} c^2+\frac{1}{3} \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3 c-\frac{3}{20} a x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2 c+\frac{1}{10} \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x) c+\frac{1}{5} \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^3",1,"-(a*c^2*x*Sqrt[c + a^2*c*x^2])/20 + (29*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/20 + (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/10 - (29*a*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/40 - (3*a*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/20 + (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] + c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/3 + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/5 - (2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (3*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/2 + ((3*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (149*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(20*Sqrt[c + a^2*c*x^2]) - (149*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(20*Sqrt[c + a^2*c*x^2]) + (6*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",54,16,24,0.6667,1,"{4950, 4958, 4956, 4183, 2531, 6609, 2282, 6589, 4930, 4890, 4888, 4181, 4880, 217, 206, 195}"
433,1,1027,0,2.1112129,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3}{x^2} \, dx","Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x^2,x]","-\frac{15 i a \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3 c^3}{4 \sqrt{a^2 c x^2+c}}-\frac{11 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) c^3}{\sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}+\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}+\frac{45 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right) c^3}{8 \sqrt{a^2 c x^2+c}}-\frac{45 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right) c^3}{8 \sqrt{a^2 c x^2+c}}-\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}+\frac{11 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right) c^3}{2 \sqrt{a^2 c x^2+c}}-\frac{11 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right) c^3}{2 \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}-\frac{45 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right) c^3}{4 \sqrt{a^2 c x^2+c}}+\frac{45 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right) c^3}{4 \sqrt{a^2 c x^2+c}}+\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}-\frac{45 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right) c^3}{4 \sqrt{a^2 c x^2+c}}+\frac{45 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right) c^3}{4 \sqrt{a^2 c x^2+c}}+\frac{7}{8} a^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2}{x}-\frac{21}{8} a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 c^2+\frac{1}{4} a^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) c^2-\frac{1}{4} a \sqrt{a^2 c x^2+c} c^2+\frac{1}{4} a^2 x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3 c-\frac{1}{4} a \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2 c","-\frac{15 i a \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3 c^3}{4 \sqrt{a^2 c x^2+c}}-\frac{11 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) c^3}{\sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}+\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}+\frac{45 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right) c^3}{8 \sqrt{a^2 c x^2+c}}-\frac{45 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right) c^3}{8 \sqrt{a^2 c x^2+c}}-\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}+\frac{11 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right) c^3}{2 \sqrt{a^2 c x^2+c}}-\frac{11 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right) c^3}{2 \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}-\frac{45 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right) c^3}{4 \sqrt{a^2 c x^2+c}}+\frac{45 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right) c^3}{4 \sqrt{a^2 c x^2+c}}+\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right) c^3}{\sqrt{a^2 c x^2+c}}-\frac{45 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right) c^3}{4 \sqrt{a^2 c x^2+c}}+\frac{45 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right) c^3}{4 \sqrt{a^2 c x^2+c}}+\frac{7}{8} a^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2}{x}-\frac{21}{8} a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 c^2+\frac{1}{4} a^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) c^2-\frac{1}{4} a \sqrt{a^2 c x^2+c} c^2+\frac{1}{4} a^2 x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3 c-\frac{1}{4} a \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2 c",1,"-(a*c^2*Sqrt[c + a^2*c*x^2])/4 + (a^2*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/4 - (21*a*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/8 - (a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/4 - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x + (7*a^2*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/8 + (a^2*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/4 - (((15*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2] - ((11*I)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((45*I)/8)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((45*I)/8)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((11*I)/2)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (((11*I)/2)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (45*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*Sqrt[c + a^2*c*x^2]) + (45*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*Sqrt[c + a^2*c*x^2]) + (6*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((45*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((45*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",56,15,24,0.6250,1,"{4950, 4944, 4958, 4956, 4183, 2531, 2282, 6589, 4890, 4888, 4181, 6609, 4880, 4886, 4878}"
434,1,1043,0,3.5367164,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3}{x^3} \, dx","Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x^3,x]","-\frac{1}{2} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 a^3+\frac{1}{3} c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3 a^2+2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 a^2+\frac{13 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2 a^2}{\sqrt{a^2 c x^2+c}}+c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) a^2-\frac{5 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}-\frac{6 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) a^2}{\sqrt{a^2 c x^2+c}}-c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right) a^2+\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right) a^2}{2 \sqrt{a^2 c x^2+c}}-\frac{13 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}+\frac{13 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}-\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right) a^2}{2 \sqrt{a^2 c x^2+c}}+\frac{3 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) a^2}{\sqrt{a^2 c x^2+c}}-\frac{3 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) a^2}{\sqrt{a^2 c x^2+c}}-\frac{15 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}+\frac{13 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}-\frac{13 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}+\frac{15 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}-\frac{15 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}-\frac{3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 a}{2 x}-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 x^2}","-\frac{1}{2} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 a^3+\frac{1}{3} c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3 a^2+2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 a^2+\frac{13 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2 a^2}{\sqrt{a^2 c x^2+c}}+c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) a^2-\frac{5 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}-\frac{6 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) a^2}{\sqrt{a^2 c x^2+c}}-c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right) a^2+\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right) a^2}{2 \sqrt{a^2 c x^2+c}}-\frac{13 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}+\frac{13 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}-\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right) a^2}{2 \sqrt{a^2 c x^2+c}}+\frac{3 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) a^2}{\sqrt{a^2 c x^2+c}}-\frac{3 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) a^2}{\sqrt{a^2 c x^2+c}}-\frac{15 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}+\frac{13 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}-\frac{13 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}+\frac{15 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}-\frac{15 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right) a^2}{\sqrt{a^2 c x^2+c}}-\frac{3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 a}{2 x}-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 x^2}",1,"a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - (3*a*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x) - (a^3*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 + ((13*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] + 2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(2*x^2) + (a^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/3 - (5*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - a^2*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]] + (((15*I)/2)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((13*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((13*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((15*I)/2)*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (15*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (13*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (13*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (15*a^2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((15*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((15*I)*a^2*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",87,18,24,0.7500,1,"{4950, 4962, 4944, 4958, 4954, 4956, 4183, 2531, 6609, 2282, 6589, 4930, 4890, 4888, 4181, 4880, 217, 206}"
435,1,1061,0,3.3754532,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3}{x^4} \, dx","Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x^4,x]","\frac{1}{2} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 a^4-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3 a^3}{\sqrt{a^2 c x^2+c}}-\frac{3}{2} c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 a^3-\frac{6 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) a^3}{\sqrt{a^2 c x^2+c}}-\frac{13 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}-c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right) a^3+\frac{13 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right) a^3}{2 \sqrt{a^2 c x^2+c}}-\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right) a^3}{2 \sqrt{a^2 c x^2+c}}-\frac{13 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}+\frac{3 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right) a^3}{\sqrt{a^2 c x^2+c}}-\frac{3 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right) a^3}{\sqrt{a^2 c x^2+c}}-\frac{13 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}-\frac{15 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}+\frac{15 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}+\frac{13 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}-\frac{15 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 a^2}{x}-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) a^2}{x}-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 a}{2 x^2}-\frac{c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3}{3 x^3}","\frac{1}{2} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 a^4-\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3 a^3}{\sqrt{a^2 c x^2+c}}-\frac{3}{2} c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 a^3-\frac{6 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right) a^3}{\sqrt{a^2 c x^2+c}}-\frac{13 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}-c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right) a^3+\frac{13 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right) a^3}{2 \sqrt{a^2 c x^2+c}}-\frac{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right) a^3}{2 \sqrt{a^2 c x^2+c}}-\frac{13 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}+\frac{3 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right) a^3}{\sqrt{a^2 c x^2+c}}-\frac{3 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right) a^3}{\sqrt{a^2 c x^2+c}}-\frac{13 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}-\frac{15 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}+\frac{15 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}+\frac{13 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}-\frac{15 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}+\frac{15 i c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right) a^3}{\sqrt{a^2 c x^2+c}}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 a^2}{x}-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) a^2}{x}-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 a}{2 x^2}-\frac{c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3}{3 x^3}",1,"-((a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) - (3*a^3*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 - (a*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x^2) - (2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x + (a^4*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/2 - (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/(3*x^3) - ((5*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2] - ((6*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (13*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - a^3*c^(5/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + ((13*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((15*I)/2)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((15*I)/2)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((13*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (13*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (15*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (15*a^3*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (13*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((15*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((15*I)*a^3*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",86,18,24,0.7500,1,"{4950, 4944, 4962, 266, 63, 208, 4958, 4956, 4183, 2531, 2282, 6589, 4890, 4888, 4181, 6609, 4880, 4886}"
436,1,408,0,0.7284134,"\int \frac{x^3 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^3*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2],x]","\frac{5 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^4 \sqrt{a^2 c x^2+c}}-\frac{5 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^4 \sqrt{a^2 c x^2+c}}-\frac{5 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^4 \sqrt{a^2 c x^2+c}}+\frac{5 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^4 \sqrt{a^2 c x^2+c}}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{3 a^2 c}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{3 a^4 c}-\frac{5 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^4 \sqrt{a^2 c x^2+c}}-\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^4 c}-\frac{\tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a^4 \sqrt{c}}","\frac{5 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^4 \sqrt{a^2 c x^2+c}}-\frac{5 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^4 \sqrt{a^2 c x^2+c}}-\frac{5 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^4 \sqrt{a^2 c x^2+c}}+\frac{5 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^4 \sqrt{a^2 c x^2+c}}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{3 a^2 c}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{3 a^4 c}-\frac{5 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^4 \sqrt{a^2 c x^2+c}}-\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^4 c}-\frac{\tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a^4 \sqrt{c}}",1,"(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^4*c) - (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a^3*c) - ((5*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^4*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(3*a^2*c) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^4*Sqrt[c]) + ((5*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) - ((5*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) - (5*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) + (5*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2])","A",24,10,24,0.4167,1,"{4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589}"
437,1,625,0,0.4891111,"\int \frac{x^2 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^2*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2],x]","\frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 a^2 c}-\frac{3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a^3 c}-\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a^3 \sqrt{a^2 c x^2+c}}","\frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 a^2 c}-\frac{3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a^3 c}-\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right) \tan ^{-1}(a x)}{a^3 \sqrt{a^2 c x^2+c}}",1,"(-3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a^3*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(2*a^2*c) + (I*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a^3*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - (((3*I)/2)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (((3*I)/2)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2])","A",15,10,24,0.4167,1,"{4952, 4930, 4890, 4886, 4888, 4181, 2531, 6609, 2282, 6589}"
438,1,283,0,0.2335656,"\int \frac{x \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2],x]","-\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{a^2 c}+\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^2 \sqrt{a^2 c x^2+c}}","-\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{a^2 c}+\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^2 \sqrt{a^2 c x^2+c}}",1,"((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(a^2*c) - ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^2*Sqrt[c + a^2*c*x^2])","A",10,7,22,0.3182,1,"{4930, 4890, 4888, 4181, 2531, 2282, 6589}"
439,1,368,0,0.1855349,"\int \frac{\tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^3/Sqrt[c + a^2*c*x^2],x]","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{a \sqrt{a^2 c x^2+c}}","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{a \sqrt{a^2 c x^2+c}}",1,"((-2*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a*Sqrt[c + a^2*c*x^2])","A",11,7,21,0.3333,1,"{4890, 4888, 4181, 2531, 6609, 2282, 6589}"
440,1,327,0,0.280881,"\int \frac{\tan ^{-1}(a x)^3}{x \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^3/(x*Sqrt[c + a^2*c*x^2]),x]","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}",1,"(-2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",11,7,24,0.2917,1,"{4958, 4956, 4183, 2531, 6609, 2282, 6589}"
441,1,260,0,0.3724234,"\int \frac{\tan ^{-1}(a x)^3}{x^2 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^3/(x^2*Sqrt[c + a^2*c*x^2]),x]","\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{c x}-\frac{6 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}","\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{c x}-\frac{6 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}",1,"-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(c*x)) - (6*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",10,7,24,0.2917,1,"{4944, 4958, 4956, 4183, 2531, 2282, 6589}"
442,1,597,0,0.684633,"\int \frac{\tan ^{-1}(a x)^3}{x^3 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^3/(x^3*Sqrt[c + a^2*c*x^2]),x]","\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{2 \sqrt{a^2 c x^2+c}}+\frac{3 i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{2 \sqrt{a^2 c x^2+c}}+\frac{3 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 c x^2}-\frac{3 a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 c x}+\frac{a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}","\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{2 \sqrt{a^2 c x^2+c}}+\frac{3 i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{2 \sqrt{a^2 c x^2+c}}+\frac{3 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 c x^2}-\frac{3 a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 c x}+\frac{a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left(\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}",1,"(-3*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(2*c*x^2) + (a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (((3*I)/2)*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((3*I)/2)*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] + (3*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (3*a^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^2*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",15,10,24,0.4167,1,"{4962, 4944, 4958, 4954, 4956, 4183, 2531, 6609, 2282, 6589}"
443,1,396,0,0.9863348,"\int \frac{\tan ^{-1}(a x)^3}{x^4 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^3/(x^4*Sqrt[c + a^2*c*x^2]),x]","-\frac{5 i a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{5 i a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{5 a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{5 a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{3 c x}-\frac{a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 c x^2}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{3 c x^3}-\frac{a^3 \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{\sqrt{c}}+\frac{5 a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}","-\frac{5 i a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{5 i a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{5 a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{5 a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{3 c x}-\frac{a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 c x^2}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{3 c x^3}-\frac{a^3 \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{\sqrt{c}}+\frac{5 a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}",1,"-((a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c*x)) - (a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*c*x^2) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(3*c*x) + (5*a^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (a^3*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/Sqrt[c] - ((5*I)*a^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((5*I)*a^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (5*a^3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (5*a^3*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]","A",25,11,24,0.4583,1,"{4962, 4944, 266, 63, 208, 4958, 4956, 4183, 2531, 2282, 6589}"
444,1,403,0,0.5159332,"\int \frac{x^3 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^3*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(3/2),x]","-\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^4 c \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{a^4 c^2}+\frac{6 x}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^3}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^4 c \sqrt{a^2 c x^2+c}}-\frac{3 x \tan ^{-1}(a x)^2}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{6 \tan ^{-1}(a x)}{a^4 c \sqrt{a^2 c x^2+c}}","-\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^4 c \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{a^4 c^2}+\frac{6 x}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^3}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^4 c \sqrt{a^2 c x^2+c}}-\frac{3 x \tan ^{-1}(a x)^2}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{6 \tan ^{-1}(a x)}{a^4 c \sqrt{a^2 c x^2+c}}",1,"(6*x)/(a^3*c*Sqrt[c + a^2*c*x^2]) - (6*ArcTan[a*x])/(a^4*c*Sqrt[c + a^2*c*x^2]) - (3*x*ArcTan[a*x]^2)/(a^3*c*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^4*c*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^3/(a^4*c*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(a^4*c^2) - ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^4*c*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^4*c*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^4*c*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^4*c*Sqrt[c + a^2*c*x^2])","A",14,10,24,0.4167,1,"{4964, 4930, 4890, 4888, 4181, 2531, 2282, 6589, 4898, 191}"
445,1,495,0,0.4117547,"\int \frac{x^2 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(3/2),x]","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{6}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)^3}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{3 \tan ^{-1}(a x)^2}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{6 x \tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}}","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{6}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)^3}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{3 \tan ^{-1}(a x)^2}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{6 x \tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}}",1,"6/(a^3*c*Sqrt[c + a^2*c*x^2]) + (6*x*ArcTan[a*x])/(a^2*c*Sqrt[c + a^2*c*x^2]) - (3*ArcTan[a*x]^2)/(a^3*c*Sqrt[c + a^2*c*x^2]) - (x*ArcTan[a*x]^3)/(a^2*c*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a^3*c*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^3*c*Sqrt[c + a^2*c*x^2])","A",14,10,24,0.4167,1,"{4964, 4890, 4888, 4181, 2531, 6609, 2282, 6589, 4898, 4894}"
446,1,107,0,0.1311204,"\int \frac{x \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(3/2),x]","-\frac{6 x}{a c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^3}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{3 x \tan ^{-1}(a x)^2}{a c \sqrt{a^2 c x^2+c}}+\frac{6 \tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}}","-\frac{6 x}{a c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^3}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{3 x \tan ^{-1}(a x)^2}{a c \sqrt{a^2 c x^2+c}}+\frac{6 \tan ^{-1}(a x)}{a^2 c \sqrt{a^2 c x^2+c}}",1,"(-6*x)/(a*c*Sqrt[c + a^2*c*x^2]) + (6*ArcTan[a*x])/(a^2*c*Sqrt[c + a^2*c*x^2]) + (3*x*ArcTan[a*x]^2)/(a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^3/(a^2*c*Sqrt[c + a^2*c*x^2])","A",3,3,22,0.1364,1,"{4930, 4898, 191}"
447,1,100,0,0.0690659,"\int \frac{\tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]^3/(c + a^2*c*x^2)^(3/2),x]","-\frac{6}{a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}+\frac{3 \tan ^{-1}(a x)^2}{a c \sqrt{a^2 c x^2+c}}-\frac{6 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}","-\frac{6}{a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}+\frac{3 \tan ^{-1}(a x)^2}{a c \sqrt{a^2 c x^2+c}}-\frac{6 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}",1,"-6/(a*c*Sqrt[c + a^2*c*x^2]) - (6*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) + (3*ArcTan[a*x]^2)/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^3)/(c*Sqrt[c + a^2*c*x^2])","A",2,2,21,0.09524,1,"{4898, 4894}"
448,1,443,0,0.5578645,"\int \frac{\tan ^{-1}(a x)^3}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]^3/(x*(c + a^2*c*x^2)^(3/2)),x]","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{6 a x}{c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}-\frac{3 a x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}-\frac{6 \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{6 a x}{c \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}-\frac{3 a x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}-\frac{6 \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}",1,"(6*a*x)/(c*Sqrt[c + a^2*c*x^2]) - (6*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (3*a*x*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^3/(c*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2])","A",15,11,24,0.4583,1,"{4966, 4958, 4956, 4183, 2531, 6609, 2282, 6589, 4930, 4898, 191}"
449,1,377,0,0.5843237,"\int \frac{\tan ^{-1}(a x)^3}{x^2 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)^(3/2)),x]","\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{c^2 x}+\frac{6 a}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 x \tan ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}-\frac{3 a \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}+\frac{6 a^2 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}","\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{c^2 x}+\frac{6 a}{c \sqrt{a^2 c x^2+c}}-\frac{a^2 x \tan ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}-\frac{3 a \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}+\frac{6 a^2 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}",1,"(6*a)/(c*Sqrt[c + a^2*c*x^2]) + (6*a^2*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]) - (3*a*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2]) - (a^2*x*ArcTan[a*x]^3)/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(c^2*x) - (6*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + ((6*I)*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - ((6*I)*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) - (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2]) + (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/(c*Sqrt[c + a^2*c*x^2])","A",13,10,24,0.4167,1,"{4966, 4944, 4958, 4956, 4183, 2531, 2282, 6589, 4898, 4894}"
450,1,534,0,1.1109834,"\int \frac{x^5 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^5*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]","-\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{94 x}{9 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 x \tan ^{-1}(a x)^2}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{a^6 c^3}+\frac{5 \tan ^{-1}(a x)^3}{3 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{94 \tan ^{-1}(a x)}{9 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x^3}{27 a^3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^3 \tan ^{-1}(a x)^2}{3 a^3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^3}{3 a^4 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 x^2 \tan ^{-1}(a x)}{9 a^4 c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{94 x}{9 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 x \tan ^{-1}(a x)^2}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{a^6 c^3}+\frac{5 \tan ^{-1}(a x)^3}{3 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{94 \tan ^{-1}(a x)}{9 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x^3}{27 a^3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^3 \tan ^{-1}(a x)^2}{3 a^3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^3}{3 a^4 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 x^2 \tan ^{-1}(a x)}{9 a^4 c \left(a^2 c x^2+c\right)^{3/2}}",1,"(2*x^3)/(27*a^3*c*(c + a^2*c*x^2)^(3/2)) + (94*x)/(9*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (2*x^2*ArcTan[a*x])/(9*a^4*c*(c + a^2*c*x^2)^(3/2)) - (94*ArcTan[a*x])/(9*a^6*c^2*Sqrt[c + a^2*c*x^2]) - (x^3*ArcTan[a*x]^2)/(3*a^3*c*(c + a^2*c*x^2)^(3/2)) - (5*x*ArcTan[a*x]^2)/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^6*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x]^3)/(3*a^4*c*(c + a^2*c*x^2)^(3/2)) + (5*ArcTan[a*x]^3)/(3*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(a^6*c^3) - ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^6*c^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^6*c^2*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^6*c^2*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^6*c^2*Sqrt[c + a^2*c*x^2])","A",22,12,24,0.5000,1,"{4964, 4930, 4890, 4888, 4181, 2531, 2282, 6589, 4898, 191, 4940, 4938}"
451,1,622,0,0.9980009,"\int \frac{x^4 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^4*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{68}{9 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 a^5 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left(a^2 c x^2+c\right)^{3/2}}","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,i e^{i \tan ^{-1}(a x)}\right)}{a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{68}{9 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 a^5 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left(a^2 c x^2+c\right)^{3/2}}",1,"-2/(27*a^5*c*(c + a^2*c*x^2)^(3/2)) + 68/(9*a^5*c^2*Sqrt[c + a^2*c*x^2]) + (2*x^3*ArcTan[a*x])/(9*a^2*c*(c + a^2*c*x^2)^(3/2)) + (22*x*ArcTan[a*x])/(3*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (x^2*ArcTan[a*x]^2)/(3*a^3*c*(c + a^2*c*x^2)^(3/2)) - (11*ArcTan[a*x]^2)/(3*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (x^3*ArcTan[a*x]^3)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (x*ArcTan[a*x]^3)/(a^4*c^2*Sqrt[c + a^2*c*x^2]) - ((2*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^5*c^2*Sqrt[c + a^2*c*x^2])","A",22,15,24,0.6250,1,"{4964, 4890, 4888, 4181, 2531, 6609, 2282, 6589, 4898, 4894, 4944, 4940, 4930, 266, 43}"
452,1,237,0,0.4124887,"\int \frac{x^3 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^3*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]","-\frac{40 x}{9 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^2}{a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)^3}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{40 \tan ^{-1}(a x)}{9 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x^3}{27 a c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^3 \tan ^{-1}(a x)^2}{3 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^3}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 x^2 \tan ^{-1}(a x)}{9 a^2 c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{40 x}{9 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^2}{a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)^3}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{40 \tan ^{-1}(a x)}{9 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x^3}{27 a c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^3 \tan ^{-1}(a x)^2}{3 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^3}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 x^2 \tan ^{-1}(a x)}{9 a^2 c \left(a^2 c x^2+c\right)^{3/2}}",1,"(-2*x^3)/(27*a*c*(c + a^2*c*x^2)^(3/2)) - (40*x)/(9*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (2*x^2*ArcTan[a*x])/(9*a^2*c*(c + a^2*c*x^2)^(3/2)) + (40*ArcTan[a*x])/(9*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (x^3*ArcTan[a*x]^2)/(3*a*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^2)/(a^3*c^2*Sqrt[c + a^2*c*x^2]) - (x^2*ArcTan[a*x]^3)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (2*ArcTan[a*x]^3)/(3*a^4*c^2*Sqrt[c + a^2*c*x^2])","A",7,5,24,0.2083,1,"{4940, 4930, 4898, 191, 4938}"
453,1,199,0,0.4065321,"\int \frac{x^2 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]","-\frac{14}{9 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{4 x \tan ^{-1}(a x)}{3 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \tan ^{-1}(a x)^2}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{27 a^3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^3 \tan ^{-1}(a x)^3}{3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{14}{9 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{4 x \tan ^{-1}(a x)}{3 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \tan ^{-1}(a x)^2}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{27 a^3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^3 \tan ^{-1}(a x)^3}{3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a c \left(a^2 c x^2+c\right)^{3/2}}",1,"2/(27*a^3*c*(c + a^2*c*x^2)^(3/2)) - 14/(9*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (2*x^3*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^(3/2)) - (4*x*ArcTan[a*x])/(3*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (x^2*ArcTan[a*x]^2)/(3*a*c*(c + a^2*c*x^2)^(3/2)) + (2*ArcTan[a*x]^2)/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (x^3*ArcTan[a*x]^3)/(3*c*(c + a^2*c*x^2)^(3/2))","A",7,6,24,0.2500,1,"{4944, 4940, 4930, 4894, 266, 43}"
454,1,199,0,0.1935117,"\int \frac{x \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]","-\frac{40 x}{27 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^2}{3 a c^2 \sqrt{a^2 c x^2+c}}+\frac{4 \tan ^{-1}(a x)}{3 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x}{27 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{\tan ^{-1}(a x)^3}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x \tan ^{-1}(a x)^2}{3 a c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 \tan ^{-1}(a x)}{9 a^2 c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{40 x}{27 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^2}{3 a c^2 \sqrt{a^2 c x^2+c}}+\frac{4 \tan ^{-1}(a x)}{3 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x}{27 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{\tan ^{-1}(a x)^3}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x \tan ^{-1}(a x)^2}{3 a c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 \tan ^{-1}(a x)}{9 a^2 c \left(a^2 c x^2+c\right)^{3/2}}",1,"(-2*x)/(27*a*c*(c + a^2*c*x^2)^(3/2)) - (40*x)/(27*a*c^2*Sqrt[c + a^2*c*x^2]) + (2*ArcTan[a*x])/(9*a^2*c*(c + a^2*c*x^2)^(3/2)) + (4*ArcTan[a*x])/(3*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^2)/(3*a*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^2)/(3*a*c^2*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^3/(3*a^2*c*(c + a^2*c*x^2)^(3/2))","A",6,5,22,0.2273,1,"{4930, 4900, 4898, 191, 192}"
455,1,215,0,0.1785535,"\int \frac{\tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[ArcTan[a*x]^3/(c + a^2*c*x^2)^(5/2),x]","-\frac{40}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \tan ^{-1}(a x)^2}{a c^2 \sqrt{a^2 c x^2+c}}-\frac{40 x \tan ^{-1}(a x)}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 a c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x \tan ^{-1}(a x)^3}{3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{\tan ^{-1}(a x)^2}{3 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 x \tan ^{-1}(a x)}{9 c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{40}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \tan ^{-1}(a x)^2}{a c^2 \sqrt{a^2 c x^2+c}}-\frac{40 x \tan ^{-1}(a x)}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{27 a c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x \tan ^{-1}(a x)^3}{3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{\tan ^{-1}(a x)^2}{3 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 x \tan ^{-1}(a x)}{9 c \left(a^2 c x^2+c\right)^{3/2}}",1,"-2/(27*a*c*(c + a^2*c*x^2)^(3/2)) - 40/(9*a*c^2*Sqrt[c + a^2*c*x^2]) - (2*x*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^(3/2)) - (40*x*ArcTan[a*x])/(9*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^2/(3*a*c*(c + a^2*c*x^2)^(3/2)) + (2*ArcTan[a*x]^2)/(a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^3)/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^3)/(3*c^2*Sqrt[c + a^2*c*x^2])","A",5,4,21,0.1905,1,"{4900, 4898, 4894, 4896}"
456,1,553,0,0.9215975,"\int \frac{\tan ^{-1}(a x)^3}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[ArcTan[a*x]^3/(x*(c + a^2*c*x^2)^(5/2)),x]","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{202 a x}{27 c^2 \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^3}{c^2 \sqrt{a^2 c x^2+c}}-\frac{11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{22 \tan ^{-1}(a x)}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 a x}{27 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{\tan ^{-1}(a x)^3}{3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{a x \tan ^{-1}(a x)^2}{3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 \tan ^{-1}(a x)}{9 c \left(a^2 c x^2+c\right)^{3/2}}","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,-e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{PolyLog}\left(4,e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{202 a x}{27 c^2 \sqrt{a^2 c x^2+c}}+\frac{\tan ^{-1}(a x)^3}{c^2 \sqrt{a^2 c x^2+c}}-\frac{11 a x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{22 \tan ^{-1}(a x)}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 a x}{27 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{\tan ^{-1}(a x)^3}{3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{a x \tan ^{-1}(a x)^2}{3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 \tan ^{-1}(a x)}{9 c \left(a^2 c x^2+c\right)^{3/2}}",1,"(2*a*x)/(27*c*(c + a^2*c*x^2)^(3/2)) + (202*a*x)/(27*c^2*Sqrt[c + a^2*c*x^2]) - (2*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^(3/2)) - (22*ArcTan[a*x])/(3*c^2*Sqrt[c + a^2*c*x^2]) - (a*x*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3/2)) - (11*a*x*ArcTan[a*x]^2)/(3*c^2*Sqrt[c + a^2*c*x^2]) + ArcTan[a*x]^3/(3*c*(c + a^2*c*x^2)^(3/2)) + ArcTan[a*x]^3/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + (6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2])","A",22,13,24,0.5417,1,"{4966, 4958, 4956, 4183, 2531, 6609, 2282, 6589, 4930, 4898, 191, 4900, 192}"
457,1,493,0,0.951968,"\int \frac{\tan ^{-1}(a x)^3}{x^2 \left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)^(5/2)),x]","\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{94 a}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{c^3 x}-\frac{5 a^2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 a \tan ^{-1}(a x)^2}{c^2 \sqrt{a^2 c x^2+c}}+\frac{94 a^2 x \tan ^{-1}(a x)}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 a}{27 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{a \tan ^{-1}(a x)^2}{3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 a^2 x \tan ^{-1}(a x)}{9 c \left(a^2 c x^2+c\right)^{3/2}}","\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,-e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left(2,e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,-e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left(3,e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{94 a}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{c^3 x}-\frac{5 a^2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 a \tan ^{-1}(a x)^2}{c^2 \sqrt{a^2 c x^2+c}}+\frac{94 a^2 x \tan ^{-1}(a x)}{9 c^2 \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 a}{27 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{a^2 x \tan ^{-1}(a x)^3}{3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{a \tan ^{-1}(a x)^2}{3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{2 a^2 x \tan ^{-1}(a x)}{9 c \left(a^2 c x^2+c\right)^{3/2}}",1,"(2*a)/(27*c*(c + a^2*c*x^2)^(3/2)) + (94*a)/(9*c^2*Sqrt[c + a^2*c*x^2]) + (2*a^2*x*ArcTan[a*x])/(9*c*(c + a^2*c*x^2)^(3/2)) + (94*a^2*x*ArcTan[a*x])/(9*c^2*Sqrt[c + a^2*c*x^2]) - (a*ArcTan[a*x]^2)/(3*c*(c + a^2*c*x^2)^(3/2)) - (5*a*ArcTan[a*x]^2)/(c^2*Sqrt[c + a^2*c*x^2]) - (a^2*x*ArcTan[a*x]^3)/(3*c*(c + a^2*c*x^2)^(3/2)) - (5*a^2*x*ArcTan[a*x]^3)/(3*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(c^3*x) - (6*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + ((6*I)*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - ((6*I)*a*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) - (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2]) + (6*a*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/(c^2*Sqrt[c + a^2*c*x^2])","A",19,12,24,0.5000,1,"{4966, 4944, 4958, 4956, 4183, 2531, 2282, 6589, 4898, 4894, 4900, 4896}"
458,0,0,0,0.0544006,"\int x^m \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3 \, dx","Int[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^3,x]","\int x^m \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3 \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)^3,x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
459,0,0,0,0.034215,"\int x^m \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3 \, dx","Int[x^m*(c + a^2*c*x^2)*ArcTan[a*x]^3,x]","\int x^m \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3 \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right) \tan ^{-1}(a x)^3,x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)*ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
460,0,0,0,0.0645185,"\int \frac{x^m \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx","Int[(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2),x]","\int \frac{x^m \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^3}{a^2 c x^2+c},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2), x]","A",0,0,0,0,-1,"{}"
461,0,0,0,0.0618146,"\int \frac{x^m \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2,x]","\int \frac{x^m \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^2} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^3}{\left(a^2 c x^2+c\right)^2},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2, x]","A",0,0,0,0,-1,"{}"
462,0,0,0,0.1093961,"\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3 \, dx","Int[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3,x]","\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3 \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3,x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
463,0,0,0,0.0945365,"\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx","Int[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]","\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx","\text{Int}\left(x^m \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3,x\right)",0,"Defer[Int][x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
464,0,0,0,0.1030921,"\int \frac{x^m \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^m*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2],x]","\int \frac{x^m \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^3}{\sqrt{a^2 c x^2+c}},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2], x]","A",0,0,0,0,-1,"{}"
465,0,0,0,0.1166256,"\int \frac{x^m \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(3/2),x]","\int \frac{x^m \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^3}{\left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(3/2), x]","A",0,0,0,0,-1,"{}"
466,0,0,0,0.0240839,"\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)} \, dx","Int[(x*(c + a^2*c*x^2))/ArcTan[a*x],x]","\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2))/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
467,0,0,0,0.0127628,"\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)/ArcTan[a*x],x]","\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{a^2 c x^2+c}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
468,0,0,0,0.0340749,"\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)/(x*ArcTan[a*x]),x]","\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{a^2 c x^2+c}{x \tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)/(x*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
469,0,0,0,0.0374872,"\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)} \, dx","Int[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x],x]","\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^2)/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
470,0,0,0,0.0240531,"\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)^2/ArcTan[a*x],x]","\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^2/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
471,0,0,0,0.0515064,"\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]),x]","\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{x \tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^2/(x*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
472,0,0,0,0.0381323,"\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)} \, dx","Int[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x],x]","\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^3)/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
473,0,0,0,0.0231705,"\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)^3/ArcTan[a*x],x]","\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^3/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
474,0,0,0,0.050382,"\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]),x]","\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3}{x \tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^3/(x*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
475,0,0,0,0.0660412,"\int \frac{x^2}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","Int[x^2/((c + a^2*c*x^2)*ArcTan[a*x]),x]","\int \frac{x^2}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^2}{\left(a^2 c x^2+c\right) \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^2/((c + a^2*c*x^2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
476,0,0,0,0.0451765,"\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","Int[x/((c + a^2*c*x^2)*ArcTan[a*x]),x]","\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x}{\left(a^2 c x^2+c\right) \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x/((c + a^2*c*x^2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
477,1,12,0,0.0264302,"\int \frac{1}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","Int[1/((c + a^2*c*x^2)*ArcTan[a*x]),x]","\frac{\log \left(\tan ^{-1}(a x)\right)}{a c}","\frac{\log \left(\tan ^{-1}(a x)\right)}{a c}",1,"Log[ArcTan[a*x]]/(a*c)","A",1,1,19,0.05263,1,"{4882}"
478,0,0,0,0.0659304,"\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","Int[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]),x]","\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right) \tan ^{-1}(a x)},x\right)",0,"Defer[Int][1/(x*(c + a^2*c*x^2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
479,0,0,0,0.067958,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","Int[1/(x^2*(c + a^2*c*x^2)*ArcTan[a*x]),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right) \tan ^{-1}(a x)},x\right)",0,"Defer[Int][1/(x^2*(c + a^2*c*x^2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
480,0,0,0,0.068668,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Int[x^4/((c + a^2*c*x^2)^2*ArcTan[a*x]),x]","\int \frac{x^4}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^4}{\left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^4/((c + a^2*c*x^2)^2*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
481,0,0,0,0.069281,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Int[x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]),x]","\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^3}{\left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
482,1,33,0,0.1084642,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Int[x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]),x]","\frac{\log \left(\tan ^{-1}(a x)\right)}{2 a^3 c^2}-\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{2 a^3 c^2}","\frac{\log \left(\tan ^{-1}(a x)\right)}{2 a^3 c^2}-\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{2 a^3 c^2}",1,"-CosIntegral[2*ArcTan[a*x]]/(2*a^3*c^2) + Log[ArcTan[a*x]]/(2*a^3*c^2)","A",4,3,22,0.1364,1,"{4970, 3312, 3302}"
483,1,17,0,0.0709479,"\int \frac{x}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Int[x/((c + a^2*c*x^2)^2*ArcTan[a*x]),x]","\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{2 a^2 c^2}","\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{2 a^2 c^2}",1,"SinIntegral[2*ArcTan[a*x]]/(2*a^2*c^2)","A",4,4,20,0.2000,1,"{4970, 4406, 12, 3299}"
484,1,33,0,0.0664349,"\int \frac{1}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Int[1/((c + a^2*c*x^2)^2*ArcTan[a*x]),x]","\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{2 a c^2}+\frac{\log \left(\tan ^{-1}(a x)\right)}{2 a c^2}","\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{2 a c^2}+\frac{\log \left(\tan ^{-1}(a x)\right)}{2 a c^2}",1,"CosIntegral[2*ArcTan[a*x]]/(2*a*c^2) + Log[ArcTan[a*x]]/(2*a*c^2)","A",4,3,19,0.1579,1,"{4904, 3312, 3302}"
485,0,0,0,0.0612764,"\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Int[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)},x\right)",0,"Defer[Int][1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
486,0,0,0,0.0684653,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)},x\right)",0,"Defer[Int][1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
487,0,0,0,0.0665092,"\int \frac{x^6}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Int[x^6/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]","\int \frac{x^6}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^6}{\left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^6/((c + a^2*c*x^2)^3*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
488,0,0,0,0.0670864,"\int \frac{x^5}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Int[x^5/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]","\int \frac{x^5}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^5}{\left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^5/((c + a^2*c*x^2)^3*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
489,1,50,0,0.1274337,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Int[x^4/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]","-\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{2 a^5 c^3}+\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{8 a^5 c^3}+\frac{3 \log \left(\tan ^{-1}(a x)\right)}{8 a^5 c^3}","-\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{2 a^5 c^3}+\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{8 a^5 c^3}+\frac{3 \log \left(\tan ^{-1}(a x)\right)}{8 a^5 c^3}",1,"-CosIntegral[2*ArcTan[a*x]]/(2*a^5*c^3) + CosIntegral[4*ArcTan[a*x]]/(8*a^5*c^3) + (3*Log[ArcTan[a*x]])/(8*a^5*c^3)","A",5,3,22,0.1364,1,"{4970, 3312, 3302}"
490,1,35,0,0.1133609,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Int[x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]","\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{4 a^4 c^3}-\frac{\text{Si}\left(4 \tan ^{-1}(a x)\right)}{8 a^4 c^3}","\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{4 a^4 c^3}-\frac{\text{Si}\left(4 \tan ^{-1}(a x)\right)}{8 a^4 c^3}",1,"SinIntegral[2*ArcTan[a*x]]/(4*a^4*c^3) - SinIntegral[4*ArcTan[a*x]]/(8*a^4*c^3)","A",5,3,22,0.1364,1,"{4970, 4406, 3299}"
491,1,33,0,0.1100742,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Int[x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]","\frac{\log \left(\tan ^{-1}(a x)\right)}{8 a^3 c^3}-\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{8 a^3 c^3}","\frac{\log \left(\tan ^{-1}(a x)\right)}{8 a^3 c^3}-\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{8 a^3 c^3}",1,"-CosIntegral[4*ArcTan[a*x]]/(8*a^3*c^3) + Log[ArcTan[a*x]]/(8*a^3*c^3)","A",4,3,22,0.1364,1,"{4970, 4406, 3302}"
492,1,35,0,0.0899071,"\int \frac{x}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Int[x/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]","\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{4 a^2 c^3}+\frac{\text{Si}\left(4 \tan ^{-1}(a x)\right)}{8 a^2 c^3}","\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{4 a^2 c^3}+\frac{\text{Si}\left(4 \tan ^{-1}(a x)\right)}{8 a^2 c^3}",1,"SinIntegral[2*ArcTan[a*x]]/(4*a^2*c^3) + SinIntegral[4*ArcTan[a*x]]/(8*a^2*c^3)","A",5,3,20,0.1500,1,"{4970, 4406, 3299}"
493,1,50,0,0.0873405,"\int \frac{1}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Int[1/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]","\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{2 a c^3}+\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{8 a c^3}+\frac{3 \log \left(\tan ^{-1}(a x)\right)}{8 a c^3}","\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{2 a c^3}+\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{8 a c^3}+\frac{3 \log \left(\tan ^{-1}(a x)\right)}{8 a c^3}",1,"CosIntegral[2*ArcTan[a*x]]/(2*a*c^3) + CosIntegral[4*ArcTan[a*x]]/(8*a*c^3) + (3*Log[ArcTan[a*x]])/(8*a*c^3)","A",5,3,19,0.1579,1,"{4904, 3312, 3302}"
494,0,0,0,0.064984,"\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Int[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)},x\right)",0,"Defer[Int][1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
495,0,0,0,0.0779618,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)},x\right)",0,"Defer[Int][1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
496,0,0,0,0.0738188,"\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx","Int[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x],x]","\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x \sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
497,0,0,0,0.038524,"\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx","Int[Sqrt[c + a^2*c*x^2]/ArcTan[a*x],x]","\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][Sqrt[c + a^2*c*x^2]/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
498,0,0,0,0.1046321,"\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)} \, dx","Int[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]),x]","\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{x \tan ^{-1}(a x)},x\right)",0,"Defer[Int][Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
499,0,0,0,0.0850088,"\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)} \, dx","Int[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x],x]","\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
500,0,0,0,0.0439783,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x],x]","\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(3/2)/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
501,0,0,0,0.1285496,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]),x]","\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{x \tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
502,0,0,0,0.0945742,"\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)} \, dx","Int[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x],x]","\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
503,0,0,0,0.0680472,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x],x]","\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(5/2)/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
504,0,0,0,0.13509,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]),x]","\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2}}{x \tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
505,0,0,0,0.0815964,"\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)} \, dx","Int[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]),x]","\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
506,0,0,0,0.04321,"\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)} \, dx","Int[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]),x]","\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)},x\right)",0,"Defer[Int][1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
507,0,0,0,0.1137428,"\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)} \, dx","Int[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]),x]","\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)},x\right)",0,"Defer[Int][1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
508,0,0,0,0.1329558,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","Int[x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]),x]","\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^3}{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
509,0,0,0,0.1311728,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","Int[x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]),x]","\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^2}{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
510,1,39,0,0.1685394,"\int \frac{x}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","Int[x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]),x]","\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{a^2 c \sqrt{a^2 c x^2+c}}","\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{a^2 c \sqrt{a^2 c x^2+c}}",1,"(Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(a^2*c*Sqrt[c + a^2*c*x^2])","A",3,3,22,0.1364,1,"{4971, 4970, 3299}"
511,1,39,0,0.0904266,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","Int[1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]),x]","\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{a c \sqrt{a^2 c x^2+c}}","\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{a c \sqrt{a^2 c x^2+c}}",1,"(Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(a*c*Sqrt[c + a^2*c*x^2])","A",3,3,21,0.1429,1,"{4905, 4904, 3302}"
512,0,0,0,0.1254327,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","Int[1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)},x\right)",0,"Defer[Int][1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
513,0,0,0,0.1245619,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)},x\right)",0,"Defer[Int][1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
514,0,0,0,0.137514,"\int \frac{x^5}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Int[x^5/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]","\int \frac{x^5}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^5}{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^5/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
515,0,0,0,0.1381772,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Int[x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]","\int \frac{x^4}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^4}{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
516,1,87,0,0.2763247,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Int[x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]","\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}","\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}",1,"(3*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(4*a^4*c^2*Sqrt[c + a^2*c*x^2])","A",6,4,24,0.1667,1,"{4971, 4970, 3312, 3299}"
517,1,87,0,0.2730188,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Int[x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]","\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{4 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{4 a^3 c^2 \sqrt{a^2 c x^2+c}}","\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{4 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{4 a^3 c^2 \sqrt{a^2 c x^2+c}}",1,"(Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*a^3*c^2*Sqrt[c + a^2*c*x^2])","A",6,4,24,0.1667,1,"{4971, 4970, 4406, 3302}"
518,1,87,0,0.1988885,"\int \frac{x}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Int[x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]","\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}","\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}",1,"(Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(4*a^2*c^2*Sqrt[c + a^2*c*x^2])","A",6,4,22,0.1818,1,"{4971, 4970, 4406, 3299}"
519,1,87,0,0.1327255,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Int[1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]","\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{4 a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{4 a c^2 \sqrt{a^2 c x^2+c}}","\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{4 a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{4 a c^2 \sqrt{a^2 c x^2+c}}",1,"(3*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*a*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*a*c^2*Sqrt[c + a^2*c*x^2])","A",6,4,21,0.1905,1,"{4905, 4904, 3312, 3302}"
520,0,0,0,0.1247126,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Int[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)},x\right)",0,"Defer[Int][1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
521,0,0,0,0.1279611,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)},x\right)",0,"Defer[Int][1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
522,0,0,0,0.0573032,"\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)} \, dx","Int[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x],x]","\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
523,0,0,0,0.0538758,"\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)} \, dx","Int[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x],x]","\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
524,0,0,0,0.035248,"\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)} \, dx","Int[(x^m*(c + a^2*c*x^2))/ArcTan[a*x],x]","\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2))/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
525,0,0,0,0.0637915,"\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","Int[x^m/((c + a^2*c*x^2)*ArcTan[a*x]),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right) \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
526,0,0,0,0.0626612,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Int[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
527,0,0,0,0.0681296,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Int[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
528,0,0,0,0.1145191,"\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)} \, dx","Int[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x],x]","\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
529,0,0,0,0.1139678,"\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)} \, dx","Int[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x],x]","\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
530,0,0,0,0.0988851,"\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx","Int[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x],x]","\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^m \sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x], x]","A",0,0,0,0,-1,"{}"
531,0,0,0,0.1071552,"\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)} \, dx","Int[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]),x]","\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^m}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
532,0,0,0,0.1211252,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","Int[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
533,0,0,0,0.1235586,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Int[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]","A",0,0,0,0,-1,"{}"
534,0,0,0,0.0234308,"\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^2} \, dx","Int[(x*(c + a^2*c*x^2))/ArcTan[a*x]^2,x]","\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2))/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
535,0,0,0,0.0122972,"\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)^2} \, dx","Int[(c + a^2*c*x^2)/ArcTan[a*x]^2,x]","\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{a^2 c x^2+c}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(c + a^2*c*x^2)/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
536,0,0,0,0.0320409,"\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)^2} \, dx","Int[(c + a^2*c*x^2)/(x*ArcTan[a*x]^2),x]","\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{a^2 c x^2+c}{x \tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(c + a^2*c*x^2)/(x*ArcTan[a*x]^2), x]","A",0,0,0,0,-1,"{}"
537,0,0,0,0.0362287,"\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^2} \, dx","Int[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^2,x]","\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
538,0,0,0,0.0226466,"\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^2} \, dx","Int[(c + a^2*c*x^2)^2/ArcTan[a*x]^2,x]","\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^2/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
539,0,0,0,0.0489749,"\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)^2} \, dx","Int[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^2),x]","\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{x \tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^2), x]","A",0,0,0,0,-1,"{}"
540,0,0,0,0.0358701,"\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^2} \, dx","Int[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^2,x]","\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
541,0,0,0,0.0213235,"\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^2} \, dx","Int[(c + a^2*c*x^2)^3/ArcTan[a*x]^2,x]","\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^3/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
542,0,0,0,0.0505526,"\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)^2} \, dx","Int[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^2),x]","\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3}{x \tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^2), x]","A",0,0,0,0,-1,"{}"
543,0,0,0,0.0824813,"\int \frac{x^3}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Int[x^3/((c + a^2*c*x^2)*ArcTan[a*x]^2),x]","\int \frac{x^3}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","\frac{3 \text{Int}\left(\frac{x^2}{\tan ^{-1}(a x)},x\right)}{a c}-\frac{x^3}{a c \tan ^{-1}(a x)}",0,"-(x^3/(a*c*ArcTan[a*x])) + (3*Defer[Int][x^2/ArcTan[a*x], x])/(a*c)","A",0,0,0,0,-1,"{}"
544,0,0,0,0.0730482,"\int \frac{x^2}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Int[x^2/((c + a^2*c*x^2)*ArcTan[a*x]^2),x]","\int \frac{x^2}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","\frac{2 \text{Int}\left(\frac{x}{\tan ^{-1}(a x)},x\right)}{a c}-\frac{x^2}{a c \tan ^{-1}(a x)}",0,"-(x^2/(a*c*ArcTan[a*x])) + (2*Defer[Int][x/ArcTan[a*x], x])/(a*c)","A",0,0,0,0,-1,"{}"
545,0,0,0,0.047012,"\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Int[x/((c + a^2*c*x^2)*ArcTan[a*x]^2),x]","\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","\frac{\text{Int}\left(\frac{1}{\tan ^{-1}(a x)},x\right)}{a c}-\frac{x}{a c \tan ^{-1}(a x)}",0,"-(x/(a*c*ArcTan[a*x])) + Defer[Int][ArcTan[a*x]^(-1), x]/(a*c)","A",0,0,0,0,-1,"{}"
546,1,14,0,0.0246541,"\int \frac{1}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Int[1/((c + a^2*c*x^2)*ArcTan[a*x]^2),x]","-\frac{1}{a c \tan ^{-1}(a x)}","-\frac{1}{a c \tan ^{-1}(a x)}",1,"-(1/(a*c*ArcTan[a*x]))","A",1,1,19,0.05263,1,"{4884}"
547,0,0,0,0.0778107,"\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Int[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^2),x]","\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)},x\right)}{a c}-\frac{1}{a c x \tan ^{-1}(a x)}",0,"-(1/(a*c*x*ArcTan[a*x])) - Defer[Int][1/(x^2*ArcTan[a*x]), x]/(a*c)","A",0,0,0,0,-1,"{}"
548,0,0,0,0.0818824,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Int[1/(x^2*(c + a^2*c*x^2)*ArcTan[a*x]^2),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^3 \tan ^{-1}(a x)},x\right)}{a c}-\frac{1}{a c x^2 \tan ^{-1}(a x)}",0,"-(1/(a*c*x^2*ArcTan[a*x])) - (2*Defer[Int][1/(x^3*ArcTan[a*x]), x])/(a*c)","A",0,0,0,0,-1,"{}"
549,0,0,0,0.0823133,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Int[1/(x^3*(c + a^2*c*x^2)*ArcTan[a*x]^2),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","-\frac{3 \text{Int}\left(\frac{1}{x^4 \tan ^{-1}(a x)},x\right)}{a c}-\frac{1}{a c x^3 \tan ^{-1}(a x)}",0,"-(1/(a*c*x^3*ArcTan[a*x])) - (3*Defer[Int][1/(x^4*ArcTan[a*x]), x])/(a*c)","A",0,0,0,0,-1,"{}"
550,0,0,0,0.0818898,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Int[1/(x^4*(c + a^2*c*x^2)*ArcTan[a*x]^2),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","-\frac{4 \text{Int}\left(\frac{1}{x^5 \tan ^{-1}(a x)},x\right)}{a c}-\frac{1}{a c x^4 \tan ^{-1}(a x)}",0,"-(1/(a*c*x^4*ArcTan[a*x])) - (4*Defer[Int][1/(x^5*ArcTan[a*x]), x])/(a*c)","A",0,0,0,0,-1,"{}"
551,0,0,0,0.3320467,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Int[x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]","\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","\frac{\text{Int}\left(\frac{1}{\tan ^{-1}(a x)},x\right)}{a^3 c^2}-\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{a^4 c^2}+\frac{x}{a^3 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{x}{a^3 c^2 \tan ^{-1}(a x)}",0,"-(x/(a^3*c^2*ArcTan[a*x])) + x/(a^3*c^2*(1 + a^2*x^2)*ArcTan[a*x]) - CosIntegral[2*ArcTan[a*x]]/(a^4*c^2) + Defer[Int][ArcTan[a*x]^(-1), x]/(a^3*c^2)","A",0,0,0,0,-1,"{}"
552,1,43,0,0.1386059,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Int[x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]","\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{a^3 c^2}-\frac{x^2}{a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}","\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{a^3 c^2}-\frac{x^2}{a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}",1,"-(x^2/(a*c^2*(1 + a^2*x^2)*ArcTan[a*x])) + SinIntegral[2*ArcTan[a*x]]/(a^3*c^2)","A",5,5,22,0.2273,1,"{4942, 4970, 4406, 12, 3299}"
553,1,41,0,0.2104436,"\int \frac{x}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Int[x/((c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]","\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{a^2 c^2}-\frac{x}{a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}","\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{a^2 c^2}-\frac{x}{a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}",1,"-(x/(a*c^2*(1 + a^2*x^2)*ArcTan[a*x])) + CosIntegral[2*ArcTan[a*x]]/(a^2*c^2)","A",9,5,20,0.2500,1,"{4968, 4970, 3312, 3302, 4904}"
554,1,41,0,0.0930027,"\int \frac{1}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Int[1/((c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]","-\frac{1}{a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{a c^2}","-\frac{1}{a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{a c^2}",1,"-(1/(a*c^2*(1 + a^2*x^2)*ArcTan[a*x])) - SinIntegral[2*ArcTan[a*x]]/(a*c^2)","A",5,5,19,0.2632,1,"{4902, 4970, 4406, 12, 3299}"
555,0,0,0,0.3570184,"\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Int[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)},x\right)}{a c^2}+\frac{a x}{c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{c^2}-\frac{1}{a c^2 x \tan ^{-1}(a x)}",0,"-(1/(a*c^2*x*ArcTan[a*x])) + (a*x)/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) - CosIntegral[2*ArcTan[a*x]]/c^2 - Defer[Int][1/(x^2*ArcTan[a*x]), x]/(a*c^2)","A",0,0,0,0,-1,"{}"
556,0,0,0,0.2442669,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^3 \tan ^{-1}(a x)},x\right)}{a c^2}+\frac{a}{c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{a \text{Si}\left(2 \tan ^{-1}(a x)\right)}{c^2}-\frac{1}{a c^2 x^2 \tan ^{-1}(a x)}",0,"-(1/(a*c^2*x^2*ArcTan[a*x])) + a/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) + (a*SinIntegral[2*ArcTan[a*x]])/c^2 - (2*Defer[Int][1/(x^3*ArcTan[a*x]), x])/(a*c^2)","A",0,0,0,0,-1,"{}"
557,0,0,0,0.4992983,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","\frac{a \text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)},x\right)}{c^2}-\frac{3 \text{Int}\left(\frac{1}{x^4 \tan ^{-1}(a x)},x\right)}{a c^2}+\frac{a^2 \text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{c^2}-\frac{a^3 x}{c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{1}{a c^2 x^3 \tan ^{-1}(a x)}+\frac{a}{c^2 x \tan ^{-1}(a x)}",0,"-(1/(a*c^2*x^3*ArcTan[a*x])) + a/(c^2*x*ArcTan[a*x]) - (a^3*x)/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) + (a^2*CosIntegral[2*ArcTan[a*x]])/c^2 - (3*Defer[Int][1/(x^4*ArcTan[a*x]), x])/(a*c^2) + (a*Defer[Int][1/(x^2*ArcTan[a*x]), x])/c^2","A",0,0,0,0,-1,"{}"
558,0,0,0,0.3964492,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","\frac{2 a \text{Int}\left(\frac{1}{x^3 \tan ^{-1}(a x)},x\right)}{c^2}-\frac{4 \text{Int}\left(\frac{1}{x^5 \tan ^{-1}(a x)},x\right)}{a c^2}-\frac{a^3 \text{Si}\left(2 \tan ^{-1}(a x)\right)}{c^2}-\frac{a^3}{c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{a}{c^2 x^2 \tan ^{-1}(a x)}-\frac{1}{a c^2 x^4 \tan ^{-1}(a x)}",0,"-(1/(a*c^2*x^4*ArcTan[a*x])) + a/(c^2*x^2*ArcTan[a*x]) - a^3/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) - (a^3*SinIntegral[2*ArcTan[a*x]])/c^2 - (4*Defer[Int][1/(x^5*ArcTan[a*x]), x])/(a*c^2) + (2*a*Defer[Int][1/(x^3*ArcTan[a*x]), x])/c^2","A",0,0,0,0,-1,"{}"
559,1,86,0,0.5171538,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Int[x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]^2),x]","\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{2 a^4 c^3}-\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{2 a^4 c^3}-\frac{x}{a^3 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{x}{a^3 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}","\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{2 a^4 c^3}-\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{2 a^4 c^3}-\frac{x}{a^3 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{x}{a^3 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}",1,"x/(a^3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - x/(a^3*c^3*(1 + a^2*x^2)*ArcTan[a*x]) + CosIntegral[2*ArcTan[a*x]]/(2*a^4*c^3) - CosIntegral[4*ArcTan[a*x]]/(2*a^4*c^3)","A",20,7,22,0.3182,1,"{4964, 4968, 4970, 3312, 3302, 4904, 4406}"
560,1,67,0,0.2799585,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Int[x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]^2),x]","\frac{\text{Si}\left(4 \tan ^{-1}(a x)\right)}{2 a^3 c^3}-\frac{1}{a^3 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{1}{a^3 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}","\frac{\text{Si}\left(4 \tan ^{-1}(a x)\right)}{2 a^3 c^3}-\frac{1}{a^3 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{1}{a^3 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}",1,"1/(a^3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - 1/(a^3*c^3*(1 + a^2*x^2)*ArcTan[a*x]) + SinIntegral[4*ArcTan[a*x]]/(2*a^3*c^3)","A",12,6,22,0.2727,1,"{4964, 4902, 4970, 4406, 12, 3299}"
561,1,61,0,0.2464196,"\int \frac{x}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Int[x/((c + a^2*c*x^2)^3*ArcTan[a*x]^2),x]","\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{2 a^2 c^3}+\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{2 a^2 c^3}-\frac{x}{a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}","\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{2 a^2 c^3}+\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{2 a^2 c^3}-\frac{x}{a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}",1,"-(x/(a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])) + CosIntegral[2*ArcTan[a*x]]/(2*a^2*c^3) + CosIntegral[4*ArcTan[a*x]]/(2*a^2*c^3)","A",10,6,20,0.3000,1,"{4968, 4970, 4406, 3302, 4904, 3312}"
562,1,58,0,0.1141826,"\int \frac{1}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Int[1/((c + a^2*c*x^2)^3*ArcTan[a*x]^2),x]","-\frac{1}{a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}-\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{a c^3}-\frac{\text{Si}\left(4 \tan ^{-1}(a x)\right)}{2 a c^3}","-\frac{1}{a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}-\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{a c^3}-\frac{\text{Si}\left(4 \tan ^{-1}(a x)\right)}{2 a c^3}",1,"-(1/(a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])) - SinIntegral[2*ArcTan[a*x]]/(a*c^3) - SinIntegral[4*ArcTan[a*x]]/(2*a*c^3)","A",6,4,19,0.2105,1,"{4902, 4970, 4406, 3299}"
563,0,0,0,0.6620137,"\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Int[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]^2),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)},x\right)}{a c^3}+\frac{a x}{c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{a x}{c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}-\frac{3 \text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{2 c^3}-\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{2 c^3}-\frac{1}{a c^3 x \tan ^{-1}(a x)}",0,"-(1/(a*c^3*x*ArcTan[a*x])) + (a*x)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + (a*x)/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (3*CosIntegral[2*ArcTan[a*x]])/(2*c^3) - CosIntegral[4*ArcTan[a*x]]/(2*c^3) - Defer[Int][1/(x^2*ArcTan[a*x]), x]/(a*c^3)","A",0,0,0,0,-1,"{}"
564,0,0,0,0.4257173,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^2),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^3 \tan ^{-1}(a x)},x\right)}{a c^3}+\frac{a}{c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{a}{c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}+\frac{2 a \text{Si}\left(2 \tan ^{-1}(a x)\right)}{c^3}+\frac{a \text{Si}\left(4 \tan ^{-1}(a x)\right)}{2 c^3}-\frac{1}{a c^3 x^2 \tan ^{-1}(a x)}",0,"-(1/(a*c^3*x^2*ArcTan[a*x])) + a/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + a/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) + (2*a*SinIntegral[2*ArcTan[a*x]])/c^3 + (a*SinIntegral[4*ArcTan[a*x]])/(2*c^3) - (2*Defer[Int][1/(x^3*ArcTan[a*x]), x])/(a*c^3)","A",0,0,0,0,-1,"{}"
565,0,0,0,1.2203963,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^2),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","\frac{2 a \text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)},x\right)}{c^3}-\frac{3 \text{Int}\left(\frac{1}{x^4 \tan ^{-1}(a x)},x\right)}{a c^3}+\frac{5 a^2 \text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{2 c^3}+\frac{a^2 \text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{2 c^3}-\frac{2 a^3 x}{c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{a^3 x}{c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}-\frac{1}{a c^3 x^3 \tan ^{-1}(a x)}+\frac{2 a}{c^3 x \tan ^{-1}(a x)}",0,"-(1/(a*c^3*x^3*ArcTan[a*x])) + (2*a)/(c^3*x*ArcTan[a*x]) - (a^3*x)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - (2*a^3*x)/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) + (5*a^2*CosIntegral[2*ArcTan[a*x]])/(2*c^3) + (a^2*CosIntegral[4*ArcTan[a*x]])/(2*c^3) - (3*Defer[Int][1/(x^4*ArcTan[a*x]), x])/(a*c^3) + (2*a*Defer[Int][1/(x^2*ArcTan[a*x]), x])/c^3","A",0,0,0,0,-1,"{}"
566,0,0,0,0.8969239,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^2),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","\frac{4 a \text{Int}\left(\frac{1}{x^3 \tan ^{-1}(a x)},x\right)}{c^3}-\frac{4 \text{Int}\left(\frac{1}{x^5 \tan ^{-1}(a x)},x\right)}{a c^3}-\frac{3 a^3 \text{Si}\left(2 \tan ^{-1}(a x)\right)}{c^3}-\frac{a^3 \text{Si}\left(4 \tan ^{-1}(a x)\right)}{2 c^3}-\frac{2 a^3}{c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{a^3}{c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}+\frac{2 a}{c^3 x^2 \tan ^{-1}(a x)}-\frac{1}{a c^3 x^4 \tan ^{-1}(a x)}",0,"-(1/(a*c^3*x^4*ArcTan[a*x])) + (2*a)/(c^3*x^2*ArcTan[a*x]) - a^3/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - (2*a^3)/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (3*a^3*SinIntegral[2*ArcTan[a*x]])/c^3 - (a^3*SinIntegral[4*ArcTan[a*x]])/(2*c^3) - (4*Defer[Int][1/(x^5*ArcTan[a*x]), x])/(a*c^3) + (4*a*Defer[Int][1/(x^3*ArcTan[a*x]), x])/c^3","A",0,0,0,0,-1,"{}"
567,0,0,0,0.06794,"\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^2} \, dx","Int[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^2,x]","\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x \sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
568,0,0,0,0.0349173,"\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^2} \, dx","Int[Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^2,x]","\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
569,0,0,0,0.0962309,"\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^2} \, dx","Int[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^2),x]","\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{x \tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^2), x]","A",0,0,0,0,-1,"{}"
570,0,0,0,0.076589,"\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^2} \, dx","Int[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^2,x]","\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
571,0,0,0,0.0365105,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^2} \, dx","Int[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^2,x]","\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
572,0,0,0,0.1093232,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)^2} \, dx","Int[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^2),x]","\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{x \tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^2), x]","A",0,0,0,0,-1,"{}"
573,0,0,0,0.077516,"\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^2} \, dx","Int[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^2,x]","\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
574,0,0,0,0.036265,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^2} \, dx","Int[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^2,x]","\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
575,0,0,0,0.1105433,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)^2} \, dx","Int[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^2),x]","\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2}}{x \tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^2), x]","A",0,0,0,0,-1,"{}"
576,0,0,0,0.0758681,"\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx","Int[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2),x]","\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]","A",0,0,0,0,-1,"{}"
577,0,0,0,0.03548,"\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx","Int[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2),x]","\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]","A",0,0,0,0,-1,"{}"
578,0,0,0,0.216659,"\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx","Int[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2),x]","\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)},x\right)}{a}-\frac{\sqrt{a^2 c x^2+c}}{a c x \tan ^{-1}(a x)}",0,"-(Sqrt[c + a^2*c*x^2]/(a*c*x*ArcTan[a*x])) - Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]/a","A",0,0,0,0,-1,"{}"
579,0,0,0,0.3717588,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Int[x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2),x]","\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","\frac{\text{Int}\left(\frac{x}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{a^2 c}-\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{x}{a^3 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}",0,"x/(a^3*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(a^4*c*Sqrt[c + a^2*c*x^2]) + Defer[Int][x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(a^2*c)","A",0,0,0,0,-1,"{}"
580,0,0,0,0.3659399,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Int[x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2),x]","\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","\frac{\text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{a^2 c}+\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{a^3 c \sqrt{a^2 c x^2+c}}+\frac{1}{a^3 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}",0,"1/(a^3*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(a^3*c*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(a^2*c)","A",0,0,0,0,-1,"{}"
581,1,69,0,0.1744922,"\int \frac{x}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Int[x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2),x]","\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{x}{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}","\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{x}{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}",1,"-(x/(a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])) + (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(a^2*c*Sqrt[c + a^2*c*x^2])","A",4,4,22,0.1818,1,"{4942, 4905, 4904, 3302}"
582,1,69,0,0.2047153,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Int[1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2),x]","-\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{a c \sqrt{a^2 c x^2+c}}-\frac{1}{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}","-\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{a c \sqrt{a^2 c x^2+c}}-\frac{1}{a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}",1,"-(1/(a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])) - (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(a*c*Sqrt[c + a^2*c*x^2])","A",4,4,21,0.1905,1,"{4902, 4971, 4970, 3299}"
583,0,0,0,0.5061614,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Int[1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)},x\right)}{a c}-\frac{\sqrt{a^2 c x^2+c}}{a c^2 x \tan ^{-1}(a x)}-\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{c \sqrt{a^2 c x^2+c}}+\frac{a x}{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}",0,"(a*x)/(c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - Sqrt[c + a^2*c*x^2]/(a*c^2*x*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(c*Sqrt[c + a^2*c*x^2]) - Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]/(a*c)","A",0,0,0,0,-1,"{}"
584,0,0,0,0.4252845,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","\frac{\text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{c}+\frac{a \sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{c \sqrt{a^2 c x^2+c}}+\frac{a}{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}",0,"a/(c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (a*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(c*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c","A",0,0,0,0,-1,"{}"
585,0,0,0,0.7404999,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","\frac{a \text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)},x\right)}{c}+\frac{\text{Int}\left(\frac{1}{x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{c}+\frac{a \sqrt{a^2 c x^2+c}}{c^2 x \tan ^{-1}(a x)}+\frac{a^2 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{c \sqrt{a^2 c x^2+c}}-\frac{a^3 x}{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}",0,"-((a^3*x)/(c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])) + (a*Sqrt[c + a^2*c*x^2])/(c^2*x*ArcTan[a*x]) + (a^2*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(c*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c + (a*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x])/c","A",0,0,0,0,-1,"{}"
586,0,0,0,0.6651913,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","-\frac{a^2 \text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{c}+\frac{\text{Int}\left(\frac{1}{x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{c}-\frac{a^3 \sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{c \sqrt{a^2 c x^2+c}}-\frac{a^3}{c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}",0,"-(a^3/(c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])) - (a^3*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(c*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c - (a^2*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x])/c","A",0,0,0,0,-1,"{}"
587,0,0,0,0.9115948,"\int \frac{x^5}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Int[x^5/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","\int \frac{x^5}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","\frac{\text{Int}\left(\frac{x}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{a^4 c^2}-\frac{7 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{4 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{4 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{x}{a^5 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{x^3}{a^3 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",0,"x^3/(a^3*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + x/(a^5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (7*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*a^6*c^2*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*a^6*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(a^4*c^2)","A",0,0,0,0,-1,"{}"
588,0,0,0,1.0807434,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Int[x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","\int \frac{x^4}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","\frac{\text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{a^4 c^2}+\frac{5 \sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{4 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{4 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{a^5 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{1}{a^5 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",0,"-(1/(a^5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) + 2/(a^5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (5*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(4*a^5*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(a^4*c^2)","A",0,0,0,0,-1,"{}"
589,1,118,0,0.4027284,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Int[x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{x^3}{a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}","\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{x^3}{a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",1,"-(x^3/(a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) + (3*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*a^4*c^2*Sqrt[c + a^2*c*x^2])","A",7,5,24,0.2083,1,"{4942, 4971, 4970, 4406, 3302}"
590,1,142,0,0.5807776,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Int[x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","-\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{4 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{4 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{1}{a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{1}{a^3 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}","-\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{4 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{4 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{1}{a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{1}{a^3 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",1,"1/(a^3*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - 1/(a^3*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(4*a^3*c^2*Sqrt[c + a^2*c*x^2])","A",12,6,24,0.2500,1,"{4964, 4902, 4971, 4970, 3299, 4406}"
591,1,116,0,0.5013456,"\int \frac{x}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Int[x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{x}{a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}","\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{x}{a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",1,"-(x/(a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) + (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*a^2*c^2*Sqrt[c + a^2*c*x^2])","A",13,8,22,0.3636,1,"{4968, 4971, 4970, 4406, 3302, 4905, 4904, 3312}"
592,1,115,0,0.244665,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Int[1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","-\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{4 a c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{4 a c^2 \sqrt{a^2 c x^2+c}}-\frac{1}{a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}","-\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{4 a c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{4 a c^2 \sqrt{a^2 c x^2+c}}-\frac{1}{a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",1,"-(1/(a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) - (3*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*a*c^2*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(4*a*c^2*Sqrt[c + a^2*c*x^2])","A",7,5,21,0.2381,1,"{4902, 4971, 4970, 4406, 3299}"
593,0,0,0,1.1372533,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Int[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)},x\right)}{a c^2}-\frac{5 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{4 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{4 c^2 \sqrt{a^2 c x^2+c}}+\frac{a x}{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{\sqrt{a^2 c x^2+c}}{a c^3 x \tan ^{-1}(a x)}+\frac{a x}{c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",0,"(a*x)/(c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + (a*x)/(c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - Sqrt[c + a^2*c*x^2]/(a*c^3*x*ArcTan[a*x]) - (5*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) - Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]/(a*c^2)","A",0,0,0,0,-1,"{}"
594,0,0,0,0.8040844,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","\frac{\text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{c^2}+\frac{7 a \sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{4 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 a \sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{4 c^2 \sqrt{a^2 c x^2+c}}+\frac{a}{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{a}{c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",0,"a/(c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + a/(c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (7*a*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + (3*a*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c^2","A",0,0,0,0,-1,"{}"
595,0,0,0,2.0482837,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","\frac{2 a \text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)},x\right)}{c^2}+\frac{\text{Int}\left(\frac{1}{x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{c^2}+\frac{9 a^2 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{4 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 a^2 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 a^3 x}{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{2 a \sqrt{a^2 c x^2+c}}{c^3 x \tan ^{-1}(a x)}-\frac{a^3 x}{c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",0,"-((a^3*x)/(c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) - (2*a^3*x)/(c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (2*a*Sqrt[c + a^2*c*x^2])/(c^3*x*ArcTan[a*x]) + (9*a^2*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + (3*a^2*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c^2 + (2*a*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x])/c^2","A",0,0,0,0,-1,"{}"
596,0,0,0,1.6070156,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","-\frac{2 a^2 \text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{c^2}+\frac{\text{Int}\left(\frac{1}{x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{c^2}-\frac{11 a^3 \sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{4 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 a^3 \sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 a^3}{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{a^3}{c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",0,"-(a^3/(c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x])) - (2*a^3)/(c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (11*a^3*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) - (3*a^3*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/c^2 - (2*a^2*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x])/c^2","A",0,0,0,0,-1,"{}"
597,0,0,0,0.1004916,"\int \frac{\sqrt{f x}}{\left(d+c^2 d x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2} \, dx","Int[Sqrt[f*x]/((d + c^2*d*x^2)^2*(a + b*ArcTan[c*x])^2),x]","\int \frac{\sqrt{f x}}{\left(d+c^2 d x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{f x}}{\left(c^2 d x^2+d\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2},x\right)",0,"Defer[Int][Sqrt[f*x]/((d + c^2*d*x^2)^2*(a + b*ArcTan[c*x])^2), x]","A",0,0,0,0,-1,"{}"
598,0,0,0,0.0547625,"\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^2} \, dx","Int[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^2,x]","\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
599,0,0,0,0.0547921,"\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^2} \, dx","Int[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^2,x]","\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
600,0,0,0,0.0339223,"\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^2} \, dx","Int[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^2,x]","\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
601,0,0,0,0.0846464,"\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Int[x^m/((c + a^2*c*x^2)*ArcTan[a*x]^2),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","\frac{m \text{Int}\left(\frac{x^{m-1}}{\tan ^{-1}(a x)},x\right)}{a c}-\frac{x^m}{a c \tan ^{-1}(a x)}",0,"-(x^m/(a*c*ArcTan[a*x])) + (m*Defer[Int][x^(-1 + m)/ArcTan[a*x], x])/(a*c)","A",0,0,0,0,-1,"{}"
602,0,0,0,0.063152,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Int[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^2), x]","A",0,0,0,0,-1,"{}"
603,0,0,0,0.0632904,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Int[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^2),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^2), x]","A",0,0,0,0,-1,"{}"
604,0,0,0,0.1117627,"\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^2} \, dx","Int[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^2,x]","\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
605,0,0,0,0.1140929,"\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^2} \, dx","Int[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^2,x]","\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
606,0,0,0,0.0964304,"\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^2} \, dx","Int[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^2,x]","\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x^m \sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^2, x]","A",0,0,0,0,-1,"{}"
607,0,0,0,0.1045684,"\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx","Int[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2),x]","\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x^m}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]","A",0,0,0,0,-1,"{}"
608,0,0,0,0.1192579,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Int[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x]","A",0,0,0,0,-1,"{}"
609,0,0,0,0.1175044,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Int[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^2},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x]","A",0,0,0,0,-1,"{}"
610,0,0,0,0.0235311,"\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^3} \, dx","Int[(x*(c + a^2*c*x^2))/ArcTan[a*x]^3,x]","\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2))/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
611,0,0,0,0.0126611,"\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)^3} \, dx","Int[(c + a^2*c*x^2)/ArcTan[a*x]^3,x]","\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{a^2 c x^2+c}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(c + a^2*c*x^2)/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
612,0,0,0,0.0320454,"\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)^3} \, dx","Int[(c + a^2*c*x^2)/(x*ArcTan[a*x]^3),x]","\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{a^2 c x^2+c}{x \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(c + a^2*c*x^2)/(x*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
613,0,0,0,0.0354829,"\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^3} \, dx","Int[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^3,x]","\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
614,0,0,0,0.0216738,"\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^3} \, dx","Int[(c + a^2*c*x^2)^2/ArcTan[a*x]^3,x]","\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^2/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
615,0,0,0,0.0483794,"\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)^3} \, dx","Int[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^3),x]","\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{x \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
616,0,0,0,0.0355532,"\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^3} \, dx","Int[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^3,x]","\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
617,0,0,0,0.0218144,"\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^3} \, dx","Int[(c + a^2*c*x^2)^3/ArcTan[a*x]^3,x]","\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^3/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
618,0,0,0,0.0489339,"\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)^3} \, dx","Int[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^3),x]","\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3}{x \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
619,0,0,0,0.0962606,"\int \frac{x^3}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Int[x^3/((c + a^2*c*x^2)*ArcTan[a*x]^3),x]","\int \frac{x^3}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","\frac{3 \text{Int}\left(\frac{x^2}{\tan ^{-1}(a x)^2},x\right)}{2 a c}-\frac{x^3}{2 a c \tan ^{-1}(a x)^2}",0,"-x^3/(2*a*c*ArcTan[a*x]^2) + (3*Defer[Int][x^2/ArcTan[a*x]^2, x])/(2*a*c)","A",0,0,0,0,-1,"{}"
620,0,0,0,0.0750696,"\int \frac{x^2}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Int[x^2/((c + a^2*c*x^2)*ArcTan[a*x]^3),x]","\int \frac{x^2}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","\frac{\text{Int}\left(\frac{x}{\tan ^{-1}(a x)^2},x\right)}{a c}-\frac{x^2}{2 a c \tan ^{-1}(a x)^2}",0,"-x^2/(2*a*c*ArcTan[a*x]^2) + Defer[Int][x/ArcTan[a*x]^2, x]/(a*c)","A",0,0,0,0,-1,"{}"
621,0,0,0,0.0488842,"\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Int[x/((c + a^2*c*x^2)*ArcTan[a*x]^3),x]","\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","\frac{\text{Int}\left(\frac{1}{\tan ^{-1}(a x)^2},x\right)}{2 a c}-\frac{x}{2 a c \tan ^{-1}(a x)^2}",0,"-x/(2*a*c*ArcTan[a*x]^2) + Defer[Int][ArcTan[a*x]^(-2), x]/(2*a*c)","A",0,0,0,0,-1,"{}"
622,1,16,0,0.0245363,"\int \frac{1}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Int[1/((c + a^2*c*x^2)*ArcTan[a*x]^3),x]","-\frac{1}{2 a c \tan ^{-1}(a x)^2}","-\frac{1}{2 a c \tan ^{-1}(a x)^2}",1,"-1/(2*a*c*ArcTan[a*x]^2)","A",1,1,19,0.05263,1,"{4884}"
623,0,0,0,0.0769726,"\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Int[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^3),x]","\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)^2},x\right)}{2 a c}-\frac{1}{2 a c x \tan ^{-1}(a x)^2}",0,"-1/(2*a*c*x*ArcTan[a*x]^2) - Defer[Int][1/(x^2*ArcTan[a*x]^2), x]/(2*a*c)","A",0,0,0,0,-1,"{}"
624,0,0,0,0.081877,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Int[1/(x^2*(c + a^2*c*x^2)*ArcTan[a*x]^3),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{x^3 \tan ^{-1}(a x)^2},x\right)}{a c}-\frac{1}{2 a c x^2 \tan ^{-1}(a x)^2}",0,"-1/(2*a*c*x^2*ArcTan[a*x]^2) - Defer[Int][1/(x^3*ArcTan[a*x]^2), x]/(a*c)","A",0,0,0,0,-1,"{}"
625,0,0,0,0.08074,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Int[1/(x^3*(c + a^2*c*x^2)*ArcTan[a*x]^3),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","-\frac{3 \text{Int}\left(\frac{1}{x^4 \tan ^{-1}(a x)^2},x\right)}{2 a c}-\frac{1}{2 a c x^3 \tan ^{-1}(a x)^2}",0,"-1/(2*a*c*x^3*ArcTan[a*x]^2) - (3*Defer[Int][1/(x^4*ArcTan[a*x]^2), x])/(2*a*c)","A",0,0,0,0,-1,"{}"
626,0,0,0,0.0819838,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Int[1/(x^4*(c + a^2*c*x^2)*ArcTan[a*x]^3),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^5 \tan ^{-1}(a x)^2},x\right)}{a c}-\frac{1}{2 a c x^4 \tan ^{-1}(a x)^2}",0,"-1/(2*a*c*x^4*ArcTan[a*x]^2) - (2*Defer[Int][1/(x^5*ArcTan[a*x]^2), x])/(a*c)","A",0,0,0,0,-1,"{}"
627,0,0,0,0.2438677,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Int[x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^3),x]","\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","\frac{\text{Int}\left(\frac{1}{\tan ^{-1}(a x)^2},x\right)}{2 a^3 c^2}+\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{a^4 c^2}+\frac{x}{2 a^3 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 a^4 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{x}{2 a^3 c^2 \tan ^{-1}(a x)^2}",0,"-x/(2*a^3*c^2*ArcTan[a*x]^2) + x/(2*a^3*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) + (1 - a^2*x^2)/(2*a^4*c^2*(1 + a^2*x^2)*ArcTan[a*x]) + SinIntegral[2*ArcTan[a*x]]/(a^4*c^2) + Defer[Int][ArcTan[a*x]^(-2), x]/(2*a^3*c^2)","A",0,0,0,0,-1,"{}"
628,1,71,0,0.2878457,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Int[x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^3),x]","\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{a^3 c^2}-\frac{x^2}{2 a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}-\frac{x}{a^2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}","\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{a^3 c^2}-\frac{x^2}{2 a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}-\frac{x}{a^2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}",1,"-x^2/(2*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) - x/(a^2*c^2*(1 + a^2*x^2)*ArcTan[a*x]) + CosIntegral[2*ArcTan[a*x]]/(a^3*c^2)","A",10,6,22,0.2727,1,"{4942, 4968, 4970, 3312, 3302, 4904}"
629,1,81,0,0.1192669,"\int \frac{x}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Int[x/((c + a^2*c*x^2)^2*ArcTan[a*x]^3),x]","-\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{a^2 c^2}-\frac{x}{2 a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}-\frac{1-a^2 x^2}{2 a^2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}","-\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{a^2 c^2}-\frac{x}{2 a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}-\frac{1-a^2 x^2}{2 a^2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}",1,"-x/(2*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) - (1 - a^2*x^2)/(2*a^2*c^2*(1 + a^2*x^2)*ArcTan[a*x]) - SinIntegral[2*ArcTan[a*x]]/(a^2*c^2)","A",5,5,20,0.2500,1,"{4932, 4970, 4406, 12, 3299}"
630,1,65,0,0.2459251,"\int \frac{1}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Int[1/((c + a^2*c*x^2)^2*ArcTan[a*x]^3),x]","\frac{x}{c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{1}{2 a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}-\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{a c^2}","\frac{x}{c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{1}{2 a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}-\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{a c^2}",1,"-1/(2*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) + x/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) - CosIntegral[2*ArcTan[a*x]]/(a*c^2)","A",10,6,19,0.3158,1,"{4902, 4968, 4970, 3312, 3302, 4904}"
631,0,0,0,0.2597538,"\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Int[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^3),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)^2},x\right)}{2 a c^2}+\frac{a x}{2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{c^2}-\frac{1}{2 a c^2 x \tan ^{-1}(a x)^2}",0,"-1/(2*a*c^2*x*ArcTan[a*x]^2) + (a*x)/(2*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) + (1 - a^2*x^2)/(2*c^2*(1 + a^2*x^2)*ArcTan[a*x]) + SinIntegral[2*ArcTan[a*x]]/c^2 - Defer[Int][1/(x^2*ArcTan[a*x]^2), x]/(2*a*c^2)","A",0,0,0,0,-1,"{}"
632,0,0,0,0.3902336,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^3),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{x^3 \tan ^{-1}(a x)^2},x\right)}{a c^2}-\frac{a^2 x}{c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{a}{2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}+\frac{a \text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{c^2}-\frac{1}{2 a c^2 x^2 \tan ^{-1}(a x)^2}",0,"-1/(2*a*c^2*x^2*ArcTan[a*x]^2) + a/(2*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) - (a^2*x)/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) + (a*CosIntegral[2*ArcTan[a*x]])/c^2 - Defer[Int][1/(x^3*ArcTan[a*x]^2), x]/(a*c^2)","A",0,0,0,0,-1,"{}"
633,0,0,0,0.4077867,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^3),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","\frac{a \text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)^2},x\right)}{2 c^2}-\frac{3 \text{Int}\left(\frac{1}{x^4 \tan ^{-1}(a x)^2},x\right)}{2 a c^2}-\frac{a^2 \text{Si}\left(2 \tan ^{-1}(a x)\right)}{c^2}-\frac{a^3 x}{2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}-\frac{a^2 \left(1-a^2 x^2\right)}{2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{1}{2 a c^2 x^3 \tan ^{-1}(a x)^2}+\frac{a}{2 c^2 x \tan ^{-1}(a x)^2}",0,"-1/(2*a*c^2*x^3*ArcTan[a*x]^2) + a/(2*c^2*x*ArcTan[a*x]^2) - (a^3*x)/(2*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) - (a^2*(1 - a^2*x^2))/(2*c^2*(1 + a^2*x^2)*ArcTan[a*x]) - (a^2*SinIntegral[2*ArcTan[a*x]])/c^2 - (3*Defer[Int][1/(x^4*ArcTan[a*x]^2), x])/(2*a*c^2) + (a*Defer[Int][1/(x^2*ArcTan[a*x]^2), x])/(2*c^2)","A",0,0,0,0,-1,"{}"
634,0,0,0,0.5467471,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^2*ArcTan[a*x]^3),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","\frac{a \text{Int}\left(\frac{1}{x^3 \tan ^{-1}(a x)^2},x\right)}{c^2}-\frac{2 \text{Int}\left(\frac{1}{x^5 \tan ^{-1}(a x)^2},x\right)}{a c^2}-\frac{a^3 \text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{c^2}+\frac{a^4 x}{c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{a^3}{2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}+\frac{a}{2 c^2 x^2 \tan ^{-1}(a x)^2}-\frac{1}{2 a c^2 x^4 \tan ^{-1}(a x)^2}",0,"-1/(2*a*c^2*x^4*ArcTan[a*x]^2) + a/(2*c^2*x^2*ArcTan[a*x]^2) - a^3/(2*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2) + (a^4*x)/(c^2*(1 + a^2*x^2)*ArcTan[a*x]) - (a^3*CosIntegral[2*ArcTan[a*x]])/c^2 - (2*Defer[Int][1/(x^5*ArcTan[a*x]^2), x])/(a*c^2) + (a*Defer[Int][1/(x^3*ArcTan[a*x]^2), x])/c^2","A",0,0,0,0,-1,"{}"
635,1,177,0,0.642054,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Int[x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]","-\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{2 a^4 c^3}+\frac{\text{Si}\left(4 \tan ^{-1}(a x)\right)}{a^4 c^3}-\frac{x}{2 a^3 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}+\frac{x}{2 a^3 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}-\frac{1-a^2 x^2}{2 a^4 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{3}{2 a^4 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{2}{a^4 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}","-\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{2 a^4 c^3}+\frac{\text{Si}\left(4 \tan ^{-1}(a x)\right)}{a^4 c^3}-\frac{x}{2 a^3 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}+\frac{x}{2 a^3 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}-\frac{1-a^2 x^2}{2 a^4 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{3}{2 a^4 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{2}{a^4 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}",1,"x/(2*a^3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) - x/(2*a^3*c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) + 2/(a^4*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - 3/(2*a^4*c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (1 - a^2*x^2)/(2*a^4*c^3*(1 + a^2*x^2)*ArcTan[a*x]) - SinIntegral[2*ArcTan[a*x]]/(2*a^4*c^3) + SinIntegral[4*ArcTan[a*x]]/(a^4*c^3)","A",25,8,22,0.3636,1,"{4964, 4932, 4970, 4406, 12, 3299, 4968, 4902}"
636,1,120,0,0.5964767,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Int[x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]","\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{a^3 c^3}+\frac{x}{a^2 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{2 x}{a^2 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}-\frac{1}{2 a^3 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}+\frac{1}{2 a^3 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}","\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{a^3 c^3}+\frac{x}{a^2 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{2 x}{a^2 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}-\frac{1}{2 a^3 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}+\frac{1}{2 a^3 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}",1,"1/(2*a^3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) - 1/(2*a^3*c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) - (2*x)/(a^2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + x/(a^2*c^3*(1 + a^2*x^2)*ArcTan[a*x]) + CosIntegral[4*ArcTan[a*x]]/(a^3*c^3)","A",22,8,22,0.3636,1,"{4964, 4902, 4968, 4970, 3312, 3302, 4904, 4406}"
637,1,113,0,0.4493748,"\int \frac{x}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Int[x/((c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]","-\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{2 a^2 c^3}-\frac{\text{Si}\left(4 \tan ^{-1}(a x)\right)}{a^2 c^3}-\frac{x}{2 a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}+\frac{3}{2 a^2 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{2}{a^2 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}","-\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)}{2 a^2 c^3}-\frac{\text{Si}\left(4 \tan ^{-1}(a x)\right)}{a^2 c^3}-\frac{x}{2 a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}+\frac{3}{2 a^2 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{2}{a^2 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}",1,"-x/(2*a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) - 2/(a^2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + 3/(2*a^2*c^3*(1 + a^2*x^2)*ArcTan[a*x]) - SinIntegral[2*ArcTan[a*x]]/(2*a^2*c^3) - SinIntegral[4*ArcTan[a*x]]/(a^2*c^3)","A",19,7,20,0.3500,1,"{4968, 4964, 4902, 4970, 4406, 12, 3299}"
638,1,81,0,0.2659515,"\int \frac{1}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Int[1/((c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]","\frac{2 x}{c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}-\frac{1}{2 a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}-\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{a c^3}-\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{a c^3}","\frac{2 x}{c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}-\frac{1}{2 a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}-\frac{\text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{a c^3}-\frac{\text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{a c^3}",1,"-1/(2*a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) + (2*x)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - CosIntegral[2*ArcTan[a*x]]/(a*c^3) - CosIntegral[4*ArcTan[a*x]]/(a*c^3)","A",11,7,19,0.3684,1,"{4902, 4968, 4970, 4406, 3302, 4904, 3312}"
639,0,0,0,0.7653876,"\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Int[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)^2},x\right)}{2 a c^3}+\frac{a x}{2 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}+\frac{a x}{2 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}+\frac{1-a^2 x^2}{2 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{3}{2 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{2}{c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}+\frac{3 \text{Si}\left(2 \tan ^{-1}(a x)\right)}{2 c^3}+\frac{\text{Si}\left(4 \tan ^{-1}(a x)\right)}{c^3}-\frac{1}{2 a c^3 x \tan ^{-1}(a x)^2}",0,"-1/(2*a*c^3*x*ArcTan[a*x]^2) + (a*x)/(2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) + (a*x)/(2*c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) + 2/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - 3/(2*c^3*(1 + a^2*x^2)*ArcTan[a*x]) + (1 - a^2*x^2)/(2*c^3*(1 + a^2*x^2)*ArcTan[a*x]) + (3*SinIntegral[2*ArcTan[a*x]])/(2*c^3) + SinIntegral[4*ArcTan[a*x]]/c^3 - Defer[Int][1/(x^2*ArcTan[a*x]^2), x]/(2*a*c^3)","A",0,0,0,0,-1,"{}"
640,0,0,0,0.7329278,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{x^3 \tan ^{-1}(a x)^2},x\right)}{a c^3}-\frac{a^2 x}{c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{2 a^2 x}{c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}+\frac{a}{2 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}+\frac{a}{2 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}+\frac{2 a \text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{c^3}+\frac{a \text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{c^3}-\frac{1}{2 a c^3 x^2 \tan ^{-1}(a x)^2}",0,"-1/(2*a*c^3*x^2*ArcTan[a*x]^2) + a/(2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) + a/(2*c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) - (2*a^2*x)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) - (a^2*x)/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) + (2*a*CosIntegral[2*ArcTan[a*x]])/c^3 + (a*CosIntegral[4*ArcTan[a*x]])/c^3 - Defer[Int][1/(x^3*ArcTan[a*x]^2), x]/(a*c^3)","A",0,0,0,0,-1,"{}"
641,0,0,0,1.2810825,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","\frac{a \text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)^2},x\right)}{c^3}-\frac{3 \text{Int}\left(\frac{1}{x^4 \tan ^{-1}(a x)^2},x\right)}{2 a c^3}-\frac{5 a^2 \text{Si}\left(2 \tan ^{-1}(a x)\right)}{2 c^3}-\frac{a^2 \text{Si}\left(4 \tan ^{-1}(a x)\right)}{c^3}-\frac{a^3 x}{c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}-\frac{a^3 x}{2 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}-\frac{a^2 \left(1-a^2 x^2\right)}{c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{3 a^2}{2 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}-\frac{2 a^2}{c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}-\frac{1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}+\frac{a}{c^3 x \tan ^{-1}(a x)^2}",0,"-1/(2*a*c^3*x^3*ArcTan[a*x]^2) + a/(c^3*x*ArcTan[a*x]^2) - (a^3*x)/(2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) - (a^3*x)/(c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) - (2*a^2)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + (3*a^2)/(2*c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (a^2*(1 - a^2*x^2))/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (5*a^2*SinIntegral[2*ArcTan[a*x]])/(2*c^3) - (a^2*SinIntegral[4*ArcTan[a*x]])/c^3 - (3*Defer[Int][1/(x^4*ArcTan[a*x]^2), x])/(2*a*c^3) + (a*Defer[Int][1/(x^2*ArcTan[a*x]^2), x])/c^3","A",0,0,0,0,-1,"{}"
642,0,0,0,1.4031563,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","\frac{2 a \text{Int}\left(\frac{1}{x^3 \tan ^{-1}(a x)^2},x\right)}{c^3}-\frac{2 \text{Int}\left(\frac{1}{x^5 \tan ^{-1}(a x)^2},x\right)}{a c^3}-\frac{3 a^3 \text{CosIntegral}\left(2 \tan ^{-1}(a x)\right)}{c^3}-\frac{a^3 \text{CosIntegral}\left(4 \tan ^{-1}(a x)\right)}{c^3}+\frac{2 a^4 x}{c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}+\frac{2 a^4 x}{c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}-\frac{a^3}{c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}-\frac{a^3}{2 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}+\frac{a}{c^3 x^2 \tan ^{-1}(a x)^2}-\frac{1}{2 a c^3 x^4 \tan ^{-1}(a x)^2}",0,"-1/(2*a*c^3*x^4*ArcTan[a*x]^2) + a/(c^3*x^2*ArcTan[a*x]^2) - a^3/(2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2) - a^3/(c^3*(1 + a^2*x^2)*ArcTan[a*x]^2) + (2*a^4*x)/(c^3*(1 + a^2*x^2)^2*ArcTan[a*x]) + (2*a^4*x)/(c^3*(1 + a^2*x^2)*ArcTan[a*x]) - (3*a^3*CosIntegral[2*ArcTan[a*x]])/c^3 - (a^3*CosIntegral[4*ArcTan[a*x]])/c^3 - (2*Defer[Int][1/(x^5*ArcTan[a*x]^2), x])/(a*c^3) + (2*a*Defer[Int][1/(x^3*ArcTan[a*x]^2), x])/c^3","A",0,0,0,0,-1,"{}"
643,1,16,0,0.090723,"\int \left(\frac{x^3}{\left(1+a^2 x^2\right) \tan ^{-1}(a x)^3}-\frac{3 x^2}{2 a \tan ^{-1}(a x)^2}\right) \, dx","Int[x^3/((1 + a^2*x^2)*ArcTan[a*x]^3) - (3*x^2)/(2*a*ArcTan[a*x]^2),x]","-\frac{x^3}{2 a \tan ^{-1}(a x)^2}","-\frac{x^3}{2 a \tan ^{-1}(a x)^2}",1,"-x^3/(2*a*ArcTan[a*x]^2)","A",2,1,38,0.02632,1,"{4926}"
644,0,0,0,0.0673714,"\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^3} \, dx","Int[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^3,x]","\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x \sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
645,0,0,0,0.0350956,"\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^3} \, dx","Int[Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^3,x]","\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
646,0,0,0,0.115324,"\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^3} \, dx","Int[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^3),x]","\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{x \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
647,0,0,0,0.0800006,"\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^3} \, dx","Int[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^3,x]","\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
648,0,0,0,0.0369999,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^3} \, dx","Int[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^3,x]","\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
649,0,0,0,0.1128139,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)^3} \, dx","Int[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^3),x]","\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{x \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
650,0,0,0,0.0993194,"\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^3} \, dx","Int[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^3,x]","\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
651,0,0,0,0.0472571,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^3} \, dx","Int[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^3,x]","\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
652,0,0,0,0.1260132,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)^3} \, dx","Int[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^3),x]","\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2}}{x \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
653,0,0,0,0.0824185,"\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","Int[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3),x]","\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
654,0,0,0,0.036585,"\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","Int[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3),x]","\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
655,0,0,0,0.2164579,"\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","Int[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3),x]","\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{2 a}-\frac{\sqrt{a^2 c x^2+c}}{2 a c x \tan ^{-1}(a x)^2}",0,"-Sqrt[c + a^2*c*x^2]/(2*a*c*x*ArcTan[a*x]^2) - Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(2*a)","A",0,0,0,0,-1,"{}"
656,0,0,0,0.1082795,"\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","Int[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3),x]","\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
657,0,0,0,0.1105604,"\int \frac{1}{x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","Int[1/(x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3),x]","\int \frac{1}{x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{1}{x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][1/(x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
658,0,0,0,0.5071505,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Int[x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3),x]","\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","\frac{\text{Int}\left(\frac{x}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right)}{a^2 c}+\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{2 a^4 c \sqrt{a^2 c x^2+c}}+\frac{x}{2 a^3 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}+\frac{1}{2 a^4 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}",0,"x/(2*a^3*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) + 1/(2*a^4*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(2*a^4*c*Sqrt[c + a^2*c*x^2]) + Defer[Int][x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/(a^2*c)","A",0,0,0,0,-1,"{}"
659,0,0,0,0.4117948,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Int[x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3),x]","\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","\frac{\text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right)}{a^2 c}+\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{2 a^3 c \sqrt{a^2 c x^2+c}}-\frac{x}{2 a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{1}{2 a^3 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}",0,"1/(2*a^3*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - x/(2*a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(2*a^3*c*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/(a^2*c)","A",0,0,0,0,-1,"{}"
660,1,104,0,0.3220785,"\int \frac{x}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Int[x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3),x]","-\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{2 a^2 c \sqrt{a^2 c x^2+c}}-\frac{x}{2 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}-\frac{1}{2 a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}","-\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{2 a^2 c \sqrt{a^2 c x^2+c}}-\frac{x}{2 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}-\frac{1}{2 a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}",1,"-x/(2*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - 1/(2*a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(2*a^2*c*Sqrt[c + a^2*c*x^2])","A",5,5,22,0.2273,1,"{4942, 4902, 4971, 4970, 3299}"
661,1,101,0,0.2237085,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Int[1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3),x]","-\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{2 a c \sqrt{a^2 c x^2+c}}+\frac{x}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{1}{2 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}","-\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{2 a c \sqrt{a^2 c x^2+c}}+\frac{x}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{1}{2 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}",1,"-1/(2*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) + x/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(2*a*c*Sqrt[c + a^2*c*x^2])","A",5,5,21,0.2381,1,"{4902, 4942, 4905, 4904, 3302}"
662,0,0,0,0.6297893,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Int[1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{2 a c}-\frac{\sqrt{a^2 c x^2+c}}{2 a c^2 x \tan ^{-1}(a x)^2}+\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{2 c \sqrt{a^2 c x^2+c}}+\frac{a x}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}+\frac{1}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}",0,"(a*x)/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - Sqrt[c + a^2*c*x^2]/(2*a*c^2*x*ArcTan[a*x]^2) + 1/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(2*c*Sqrt[c + a^2*c*x^2]) - Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(2*a*c)","A",0,0,0,0,-1,"{}"
663,0,0,0,0.4622287,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","\frac{\text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right)}{c}+\frac{a \sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{2 c \sqrt{a^2 c x^2+c}}-\frac{a^2 x}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{a}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}",0,"a/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - (a^2*x)/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (a*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(2*c*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/c","A",0,0,0,0,-1,"{}"
664,0,0,0,0.8776645,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","\frac{a \text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{2 c}+\frac{\text{Int}\left(\frac{1}{x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right)}{c}+\frac{a \sqrt{a^2 c x^2+c}}{2 c^2 x \tan ^{-1}(a x)^2}-\frac{a^2 \sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{2 c \sqrt{a^2 c x^2+c}}-\frac{a^3 x}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}-\frac{a^2}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}",0,"-(a^3*x)/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) + (a*Sqrt[c + a^2*c*x^2])/(2*c^2*x*ArcTan[a*x]^2) - a^2/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (a^2*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(2*c*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/c + (a*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x])/(2*c)","A",0,0,0,0,-1,"{}"
665,0,0,0,0.683791,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","-\frac{a^2 \text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right)}{c}+\frac{\text{Int}\left(\frac{1}{x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right)}{c}-\frac{a^3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{2 c \sqrt{a^2 c x^2+c}}+\frac{a^4 x}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{a^3}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}",0,"-a^3/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) + (a^4*x)/(2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (a^3*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(2*c*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/c - (a^2*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x])/c","A",0,0,0,0,-1,"{}"
666,0,0,0,1.3582631,"\int \frac{x^5}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Int[x^5/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]","\int \frac{x^5}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","\frac{\text{Int}\left(\frac{x}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right)}{a^4 c^2}+\frac{7 \sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{8 a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{9 \sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{8 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{x}{2 a^5 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}+\frac{2}{a^6 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{x^3}{2 a^3 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}-\frac{3}{2 a^6 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",0,"x^3/(2*a^3*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) + x/(2*a^5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - 3/(2*a^6*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + 2/(a^6*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (7*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(8*a^6*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(8*a^6*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/(a^4*c^2)","A",0,0,0,0,-1,"{}"
667,0,0,0,1.4434187,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Int[x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]","\int \frac{x^4}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","\frac{\text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right)}{a^4 c^2}+\frac{5 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{8 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{9 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{8 a^5 c^2 \sqrt{a^2 c x^2+c}}-\frac{x}{a^4 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{1}{a^5 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}+\frac{3 x}{2 a^4 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}-\frac{1}{2 a^5 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}",0,"-1/(2*a^5*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) + 1/(a^5*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) + (3*x)/(2*a^4*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - x/(a^4*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (5*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(8*a^5*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(8*a^5*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/(a^4*c^2)","A",0,0,0,0,-1,"{}"
668,1,180,0,0.7146576,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Int[x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]","-\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{8 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{9 \sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{8 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{3}{2 a^4 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{x^3}{2 a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}+\frac{3}{2 a^4 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}","-\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{8 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{9 \sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{8 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{3}{2 a^4 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{x^3}{2 a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}+\frac{3}{2 a^4 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",1,"-x^3/(2*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) + 3/(2*a^4*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - 3/(2*a^4*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (3*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(8*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (9*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(8*a^4*c^2*Sqrt[c + a^2*c*x^2])","A",13,7,24,0.2917,1,"{4942, 4964, 4902, 4971, 4970, 3299, 4406}"
669,1,209,0,0.9071023,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Int[x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]","-\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{8 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{9 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{8 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{x}{2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{1}{2 a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}-\frac{3 x}{2 a^2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}+\frac{1}{2 a^3 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}","-\frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{8 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{9 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{8 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{x}{2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{1}{2 a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}-\frac{3 x}{2 a^2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}+\frac{1}{2 a^3 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}",1,"1/(2*a^3*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) - 1/(2*a^3*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - (3*x)/(2*a^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + x/(2*a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(8*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (9*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(8*a^3*c^2*Sqrt[c + a^2*c*x^2])","A",20,11,24,0.4583,1,"{4964, 4902, 4942, 4905, 4904, 3302, 4968, 4971, 4970, 4406, 3312}"
670,1,175,0,0.9311435,"\int \frac{x}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Int[x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]","-\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{8 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{9 \sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{8 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{1}{a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{x}{2 a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}-\frac{3}{2 a^2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}","-\frac{\sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{8 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{9 \sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{8 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{1}{a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{x}{2 a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}-\frac{3}{2 a^2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",1,"-x/(2*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) - 3/(2*a^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) + 1/(a^2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) - (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(8*a^2*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(8*a^2*c^2*Sqrt[c + a^2*c*x^2])","A",20,7,22,0.3182,1,"{4968, 4964, 4902, 4971, 4970, 3299, 4406}"
671,1,145,0,0.5505039,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Int[1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]","-\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{8 a c^2 \sqrt{a^2 c x^2+c}}-\frac{9 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{8 a c^2 \sqrt{a^2 c x^2+c}}+\frac{3 x}{2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}-\frac{1}{2 a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}","-\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{8 a c^2 \sqrt{a^2 c x^2+c}}-\frac{9 \sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{8 a c^2 \sqrt{a^2 c x^2+c}}+\frac{3 x}{2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}-\frac{1}{2 a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}",1,"-1/(2*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) + (3*x)/(2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - (3*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(8*a*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(8*a*c^2*Sqrt[c + a^2*c*x^2])","A",14,9,21,0.4286,1,"{4902, 4968, 4971, 4970, 4406, 3302, 4905, 4904, 3312}"
672,0,0,0,1.7204537,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Int[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right)}{2 a c^2}+\frac{5 \sqrt{a^2 x^2+1} \text{Si}\left(\tan ^{-1}(a x)\right)}{8 c^2 \sqrt{a^2 c x^2+c}}+\frac{9 \sqrt{a^2 x^2+1} \text{Si}\left(3 \tan ^{-1}(a x)\right)}{8 c^2 \sqrt{a^2 c x^2+c}}+\frac{a x}{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}-\frac{1}{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{\sqrt{a^2 c x^2+c}}{2 a c^3 x \tan ^{-1}(a x)^2}+\frac{a x}{2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}+\frac{3}{2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",0,"(a*x)/(2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) + (a*x)/(2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - Sqrt[c + a^2*c*x^2]/(2*a*c^3*x*ArcTan[a*x]^2) + 3/(2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - 1/(2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (5*Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(8*c^2*Sqrt[c + a^2*c*x^2]) + (9*Sqrt[1 + a^2*x^2]*SinIntegral[3*ArcTan[a*x]])/(8*c^2*Sqrt[c + a^2*c*x^2]) - Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(2*a*c^2)","A",0,0,0,0,-1,"{}"
673,0,0,0,1.1517522,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","\frac{\text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right)}{c^2}+\frac{7 a \sqrt{a^2 x^2+1} \text{CosIntegral}\left(\tan ^{-1}(a x)\right)}{8 c^2 \sqrt{a^2 c x^2+c}}+\frac{9 a \sqrt{a^2 x^2+1} \text{CosIntegral}\left(3 \tan ^{-1}(a x)\right)}{8 c^2 \sqrt{a^2 c x^2+c}}-\frac{a^2 x}{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{a}{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}-\frac{3 a^2 x}{2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}+\frac{a}{2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}",0,"a/(2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) + a/(2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) - (3*a^2*x)/(2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]) - (a^2*x)/(2*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (7*a*Sqrt[1 + a^2*x^2]*CosIntegral[ArcTan[a*x]])/(8*c^2*Sqrt[c + a^2*c*x^2]) + (9*a*Sqrt[1 + a^2*x^2]*CosIntegral[3*ArcTan[a*x]])/(8*c^2*Sqrt[c + a^2*c*x^2]) + Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]/c^2","A",0,0,0,0,-1,"{}"
674,0,0,0,0.0564415,"\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^3} \, dx","Int[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^3,x]","\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
675,0,0,0,0.0542615,"\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^3} \, dx","Int[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^3,x]","\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
676,0,0,0,0.0338105,"\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^3} \, dx","Int[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^3,x]","\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
677,0,0,0,0.0816005,"\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Int[x^m/((c + a^2*c*x^2)*ArcTan[a*x]^3),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","\frac{m \text{Int}\left(\frac{x^{m-1}}{\tan ^{-1}(a x)^2},x\right)}{2 a c}-\frac{x^m}{2 a c \tan ^{-1}(a x)^2}",0,"-x^m/(2*a*c*ArcTan[a*x]^2) + (m*Defer[Int][x^(-1 + m)/ArcTan[a*x]^2, x])/(2*a*c)","A",0,0,0,0,-1,"{}"
678,0,0,0,0.0667678,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Int[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^3),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
679,0,0,0,0.0630917,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Int[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
680,0,0,0,0.1136246,"\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^3} \, dx","Int[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^3,x]","\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
681,0,0,0,0.1131649,"\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^3} \, dx","Int[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^3,x]","\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
682,0,0,0,0.1168438,"\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^3} \, dx","Int[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^3,x]","\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x^m \sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^3, x]","A",0,0,0,0,-1,"{}"
683,0,0,0,0.1042403,"\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","Int[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3),x]","\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x^m}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
684,0,0,0,0.1160827,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Int[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
685,0,0,0,0.1172491,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Int[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^3},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x]","A",0,0,0,0,-1,"{}"
686,0,0,0,0.0341483,"\int x^m \left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)} \, dx","Int[x^m*(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]],x]","\int x^m \left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right) \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
687,0,0,0,0.0461141,"\int x \left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)} \, dx","Int[x*(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]],x]","\int x \left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)} \, dx","\frac{c \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}{4 a^2}-\frac{\text{Int}\left(\frac{a^2 c x^2+c}{\sqrt{\tan ^{-1}(a x)}},x\right)}{8 a}",0,"(c*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(4*a^2) - Defer[Int][(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x]/(8*a)","A",0,0,0,0,-1,"{}"
688,0,0,0,0.0131612,"\int \left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]],x]","\int \left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\left(a^2 c x^2+c\right) \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
689,0,0,0,0.039548,"\int \frac{\left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)}}{x} \, dx","Int[((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]])/x,x]","\int \frac{\left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)}}{x} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right) \sqrt{\tan ^{-1}(a x)}}{x},x\right)",0,"Defer[Int][((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]])/x, x]","A",0,0,0,0,-1,"{}"
690,0,0,0,0.0854464,"\int x^m \left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)} \, dx","Int[x^m*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]],x]","\int x^m \left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
691,0,0,0,0.0720387,"\int x \left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)} \, dx","Int[x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]],x]","\int x \left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)} \, dx","\frac{c^2 \left(a^2 x^2+1\right)^3 \sqrt{\tan ^{-1}(a x)}}{6 a^2}-\frac{\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{\sqrt{\tan ^{-1}(a x)}},x\right)}{12 a}",0,"(c^2*(1 + a^2*x^2)^3*Sqrt[ArcTan[a*x]])/(6*a^2) - Defer[Int][(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x]/(12*a)","A",0,0,0,0,-1,"{}"
692,0,0,0,0.0226769,"\int \left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]],x]","\int \left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
693,0,0,0,0.0607245,"\int \frac{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}}{x} \, dx","Int[((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]])/x,x]","\int \frac{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}}{x} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}}{x},x\right)",0,"Defer[Int][((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]])/x, x]","A",0,0,0,0,-1,"{}"
694,0,0,0,0.0583095,"\int x^m \left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)} \, dx","Int[x^m*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]],x]","\int x^m \left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
695,0,0,0,0.0643505,"\int x \left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)} \, dx","Int[x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]],x]","\int x \left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)} \, dx","\frac{c^3 \left(a^2 x^2+1\right)^4 \sqrt{\tan ^{-1}(a x)}}{8 a^2}-\frac{\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3}{\sqrt{\tan ^{-1}(a x)}},x\right)}{16 a}",0,"(c^3*(1 + a^2*x^2)^4*Sqrt[ArcTan[a*x]])/(8*a^2) - Defer[Int][(c + a^2*c*x^2)^3/Sqrt[ArcTan[a*x]], x]/(16*a)","A",0,0,0,0,-1,"{}"
696,0,0,0,0.0240414,"\int \left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]],x]","\int \left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
697,0,0,0,0.052464,"\int \frac{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}}{x} \, dx","Int[((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]])/x,x]","\int \frac{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}}{x} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}}{x},x\right)",0,"Defer[Int][((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]])/x, x]","A",0,0,0,0,-1,"{}"
698,0,0,0,0.0639933,"\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Int[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2),x]","\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","\text{Int}\left(\frac{x^m \sqrt{\tan ^{-1}(a x)}}{a^2 c x^2+c},x\right)",0,"Defer[Int][(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2), x]","A",0,0,0,0,-1,"{}"
699,0,0,0,0.1224341,"\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Int[(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2),x]","\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","\frac{\text{Int}\left(x \sqrt{\tan ^{-1}(a x)},x\right)}{a^2 c}+\frac{2 \text{Int}\left(\tan ^{-1}(a x)^{3/2},x\right)}{3 a^3 c}-\frac{2 x \tan ^{-1}(a x)^{3/2}}{3 a^3 c}",0,"(-2*x*ArcTan[a*x]^(3/2))/(3*a^3*c) + Defer[Int][x*Sqrt[ArcTan[a*x]], x]/(a^2*c) + (2*Defer[Int][ArcTan[a*x]^(3/2), x])/(3*a^3*c)","A",0,0,0,0,-1,"{}"
700,0,0,0,0.0977936,"\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Int[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2),x]","\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","\frac{\text{Int}\left(\sqrt{\tan ^{-1}(a x)},x\right)}{a^2 c}-\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a^3 c}",0,"(-2*ArcTan[a*x]^(3/2))/(3*a^3*c) + Defer[Int][Sqrt[ArcTan[a*x]], x]/(a^2*c)","A",0,0,0,0,-1,"{}"
701,0,0,0,0.0488572,"\int \frac{x \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Int[(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2),x]","\int \frac{x \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","\frac{2 x \tan ^{-1}(a x)^{3/2}}{3 a c}-\frac{2 \text{Int}\left(\tan ^{-1}(a x)^{3/2},x\right)}{3 a c}",0,"(2*x*ArcTan[a*x]^(3/2))/(3*a*c) - (2*Defer[Int][ArcTan[a*x]^(3/2), x])/(3*a*c)","A",0,0,0,0,-1,"{}"
702,1,18,0,0.0242699,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Int[Sqrt[ArcTan[a*x]]/(c + a^2*c*x^2),x]","\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a c}","\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a c}",1,"(2*ArcTan[a*x]^(3/2))/(3*a*c)","A",1,1,21,0.04762,1,"{4884}"
703,0,0,0,0.1072277,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)} \, dx","Int[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)),x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)} \, dx","\frac{i \text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x (a x+i)},x\right)}{c}-\frac{2 i \tan ^{-1}(a x)^{3/2}}{3 c}",0,"(((-2*I)/3)*ArcTan[a*x]^(3/2))/c + (I*Defer[Int][Sqrt[ArcTan[a*x]]/(x*(I + a*x)), x])/c","A",0,0,0,0,-1,"{}"
704,0,0,0,0.1038733,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \left(c+a^2 c x^2\right)} \, dx","Int[Sqrt[ArcTan[a*x]]/(x^2*(c + a^2*c*x^2)),x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \left(c+a^2 c x^2\right)} \, dx","\frac{\text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x^2},x\right)}{c}-\frac{2 a \tan ^{-1}(a x)^{3/2}}{3 c}",0,"(-2*a*ArcTan[a*x]^(3/2))/(3*c) + Defer[Int][Sqrt[ArcTan[a*x]]/x^2, x]/c","A",0,0,0,0,-1,"{}"
705,0,0,0,0.1937675,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^3 \left(c+a^2 c x^2\right)} \, dx","Int[Sqrt[ArcTan[a*x]]/(x^3*(c + a^2*c*x^2)),x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^3 \left(c+a^2 c x^2\right)} \, dx","-\frac{i a^2 \text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x (a x+i)},x\right)}{c}+\frac{\text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x^3},x\right)}{c}+\frac{2 i a^2 \tan ^{-1}(a x)^{3/2}}{3 c}",0,"(((2*I)/3)*a^2*ArcTan[a*x]^(3/2))/c + Defer[Int][Sqrt[ArcTan[a*x]]/x^3, x]/c - (I*a^2*Defer[Int][Sqrt[ArcTan[a*x]]/(x*(I + a*x)), x])/c","A",0,0,0,0,-1,"{}"
706,0,0,0,0.187223,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4 \left(c+a^2 c x^2\right)} \, dx","Int[Sqrt[ArcTan[a*x]]/(x^4*(c + a^2*c*x^2)),x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4 \left(c+a^2 c x^2\right)} \, dx","-\frac{a^2 \text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x^2},x\right)}{c}+\frac{\text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x^4},x\right)}{c}+\frac{2 a^3 \tan ^{-1}(a x)^{3/2}}{3 c}",0,"(2*a^3*ArcTan[a*x]^(3/2))/(3*c) + Defer[Int][Sqrt[ArcTan[a*x]]/x^4, x]/c - (a^2*Defer[Int][Sqrt[ArcTan[a*x]]/x^2, x])/c","A",0,0,0,0,-1,"{}"
707,0,0,0,0.0636876,"\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2,x]","\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","\text{Int}\left(\frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(a^2 c x^2+c\right)^2},x\right)",0,"Defer[Int][(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2, x]","A",0,0,0,0,-1,"{}"
708,0,0,0,0.0637315,"\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2,x]","\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","\text{Int}\left(\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left(a^2 c x^2+c\right)^2},x\right)",0,"Defer[Int][(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2, x]","A",0,0,0,0,-1,"{}"
709,1,80,0,0.1468632,"\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2,x]","\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a^3 c^2}-\frac{x \sqrt{\tan ^{-1}(a x)}}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^{3/2}}{3 a^3 c^2}","\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a^3 c^2}-\frac{x \sqrt{\tan ^{-1}(a x)}}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^{3/2}}{3 a^3 c^2}",1,"-(x*Sqrt[ArcTan[a*x]])/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(3/2)/(3*a^3*c^2) + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(8*a^3*c^2)","A",6,6,24,0.2500,1,"{4936, 4970, 4406, 12, 3305, 3351}"
710,1,79,0,0.1207554,"\int \frac{x \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2,x]","\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a^2 c^2}-\frac{\sqrt{\tan ^{-1}(a x)}}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\sqrt{\tan ^{-1}(a x)}}{4 a^2 c^2}","\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a^2 c^2}-\frac{\sqrt{\tan ^{-1}(a x)}}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\sqrt{\tan ^{-1}(a x)}}{4 a^2 c^2}",1,"Sqrt[ArcTan[a*x]]/(4*a^2*c^2) - Sqrt[ArcTan[a*x]]/(2*a^2*c^2*(1 + a^2*x^2)) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(8*a^2*c^2)","A",6,5,22,0.2273,1,"{4930, 4904, 3312, 3304, 3352}"
711,1,77,0,0.104373,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[Sqrt[ArcTan[a*x]]/(c + a^2*c*x^2)^2,x]","\frac{x \sqrt{\tan ^{-1}(a x)}}{2 c^2 \left(a^2 x^2+1\right)}-\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a c^2}+\frac{\tan ^{-1}(a x)^{3/2}}{3 a c^2}","\frac{x \sqrt{\tan ^{-1}(a x)}}{2 c^2 \left(a^2 x^2+1\right)}-\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a c^2}+\frac{\tan ^{-1}(a x)^{3/2}}{3 a c^2}",1,"(x*Sqrt[ArcTan[a*x]])/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(3/2)/(3*a*c^2) - (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(8*a*c^2)","A",6,6,21,0.2857,1,"{4892, 4970, 4406, 12, 3305, 3351}"
712,0,0,0,0.0595246,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)^2} \, dx","Int[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^2),x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x \left(a^2 c x^2+c\right)^2},x\right)",0,"Defer[Int][Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^2), x]","A",0,0,0,0,-1,"{}"
713,0,0,0,0.0654586,"\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3,x]","\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","\text{Int}\left(\frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(a^2 c x^2+c\right)^3},x\right)",0,"Defer[Int][(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3, x]","A",0,0,0,0,-1,"{}"
714,0,0,0,0.0658467,"\int \frac{x^5 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^5*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3,x]","\int \frac{x^5 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","\text{Int}\left(\frac{x^5 \sqrt{\tan ^{-1}(a x)}}{\left(a^2 c x^2+c\right)^3},x\right)",0,"Defer[Int][(x^5*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3, x]","A",0,0,0,0,-1,"{}"
715,1,139,0,0.1788342,"\int \frac{x^4 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^4*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3,x]","-\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{64 a^5 c^3}+\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a^5 c^3}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a^5 c^3}-\frac{\sqrt{\tan ^{-1}(a x)} \sin \left(2 \tan ^{-1}(a x)\right)}{4 a^5 c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{32 a^5 c^3}","-\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{64 a^5 c^3}+\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a^5 c^3}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a^5 c^3}-\frac{\sqrt{\tan ^{-1}(a x)} \sin \left(2 \tan ^{-1}(a x)\right)}{4 a^5 c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{32 a^5 c^3}",1,"ArcTan[a*x]^(3/2)/(4*a^5*c^3) - (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(64*a^5*c^3) + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(8*a^5*c^3) - (Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(4*a^5*c^3) + (Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(32*a^5*c^3)","A",9,5,24,0.2083,1,"{4970, 3312, 3296, 3305, 3351}"
716,1,118,0,0.2145442,"\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3,x]","-\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{64 a^4 c^3}+\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{16 a^4 c^3}+\frac{x^4 \sqrt{\tan ^{-1}(a x)}}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a^4 c^3}","-\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{64 a^4 c^3}+\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{16 a^4 c^3}+\frac{x^4 \sqrt{\tan ^{-1}(a x)}}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a^4 c^3}",1,"(-3*Sqrt[ArcTan[a*x]])/(32*a^4*c^3) + (x^4*Sqrt[ArcTan[a*x]])/(4*c^3*(1 + a^2*x^2)^2) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(64*a^4*c^3) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(16*a^4*c^3)","A",8,5,24,0.2083,1,"{4944, 4970, 3312, 3304, 3352}"
717,1,83,0,0.1363496,"\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3,x]","\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{64 a^3 c^3}+\frac{\tan ^{-1}(a x)^{3/2}}{12 a^3 c^3}-\frac{\sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{32 a^3 c^3}","\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{64 a^3 c^3}+\frac{\tan ^{-1}(a x)^{3/2}}{12 a^3 c^3}-\frac{\sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{32 a^3 c^3}",1,"ArcTan[a*x]^(3/2)/(12*a^3*c^3) + (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(64*a^3*c^3) - (Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(32*a^3*c^3)","A",6,5,24,0.2083,1,"{4970, 4406, 3296, 3305, 3351}"
718,1,118,0,0.1496301,"\int \frac{x \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3,x]","\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{64 a^2 c^3}+\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{16 a^2 c^3}-\frac{\sqrt{\tan ^{-1}(a x)}}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a^2 c^3}","\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{64 a^2 c^3}+\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{16 a^2 c^3}-\frac{\sqrt{\tan ^{-1}(a x)}}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a^2 c^3}",1,"(3*Sqrt[ArcTan[a*x]])/(32*a^2*c^3) - Sqrt[ArcTan[a*x]]/(4*a^2*c^3*(1 + a^2*x^2)^2) + (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(64*a^2*c^3) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(16*a^2*c^3)","A",8,5,22,0.2273,1,"{4930, 4904, 3312, 3304, 3352}"
719,1,139,0,0.1446505,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[Sqrt[ArcTan[a*x]]/(c + a^2*c*x^2)^3,x]","-\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{64 a c^3}-\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a c^3}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left(2 \tan ^{-1}(a x)\right)}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{32 a c^3}","-\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{64 a c^3}-\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a c^3}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left(2 \tan ^{-1}(a x)\right)}{4 a c^3}+\frac{\sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{32 a c^3}",1,"ArcTan[a*x]^(3/2)/(4*a*c^3) - (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(64*a*c^3) - (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(8*a*c^3) + (Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(4*a*c^3) + (Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(32*a*c^3)","A",9,5,21,0.2381,1,"{4904, 3312, 3296, 3305, 3351}"
720,0,0,0,0.0623277,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)^3} \, dx","Int[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^3),x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)^3} \, dx","\text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x \left(a^2 c x^2+c\right)^3},x\right)",0,"Defer[Int][Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^3), x]","A",0,0,0,0,-1,"{}"
721,0,0,0,0.1006997,"\int x^m \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx","Int[x^m*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]],x]","\int x^m \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(x^m \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^m*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
722,0,0,0,0.1030863,"\int x^2 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx","Int[x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]],x]","\int x^2 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(x^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
723,0,0,0,0.1036734,"\int x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx","Int[x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]],x]","\int x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx","\frac{\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}{3 a^2 c}-\frac{\text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{\tan ^{-1}(a x)}},x\right)}{6 a}",0,"((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(3*a^2*c) - Defer[Int][Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x]/(6*a)","A",0,0,0,0,-1,"{}"
724,0,0,0,0.0320874,"\int \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx","Int[Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]],x]","\int \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
725,0,0,0,0.1089971,"\int x^m \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx","Int[x^m*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]],x]","\int x^m \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
726,0,0,0,0.1238612,"\int x^2 \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx","Int[x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]],x]","\int x^2 \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(x^2 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
727,0,0,0,0.1215301,"\int x \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx","Int[x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]],x]","\int x \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx","\frac{\left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}}{5 a^2 c}-\frac{\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}},x\right)}{10 a}",0,"((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]])/(5*a^2*c) - Defer[Int][(c + a^2*c*x^2)^(3/2)/Sqrt[ArcTan[a*x]], x]/(10*a)","A",0,0,0,0,-1,"{}"
728,0,0,0,0.0357409,"\int \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]],x]","\int \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
729,0,0,0,0.1093197,"\int x^m \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)} \, dx","Int[x^m*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]],x]","\int x^m \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
730,0,0,0,0.1210583,"\int x^2 \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)} \, dx","Int[x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]],x]","\int x^2 \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(x^2 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
731,0,0,0,0.1166301,"\int x \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)} \, dx","Int[x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]],x]","\int x \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)} \, dx","\frac{\left(a^2 c x^2+c\right)^{7/2} \sqrt{\tan ^{-1}(a x)}}{7 a^2 c}-\frac{\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2}}{\sqrt{\tan ^{-1}(a x)}},x\right)}{14 a}",0,"((c + a^2*c*x^2)^(7/2)*Sqrt[ArcTan[a*x]])/(7*a^2*c) - Defer[Int][(c + a^2*c*x^2)^(5/2)/Sqrt[ArcTan[a*x]], x]/(14*a)","A",0,0,0,0,-1,"{}"
732,0,0,0,0.0372892,"\int \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)} \, dx","Int[(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]],x]","\int \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)} \, dx","\text{Int}\left(\left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
733,0,0,0,0.1049493,"\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^m*Sqrt[ArcTan[a*x]])/Sqrt[c + a^2*c*x^2],x]","\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","\text{Int}\left(\frac{x^m \sqrt{\tan ^{-1}(a x)}}{\sqrt{a^2 c x^2+c}},x\right)",0,"Defer[Int][(x^m*Sqrt[ArcTan[a*x]])/Sqrt[c + a^2*c*x^2], x]","A",0,0,0,0,-1,"{}"
734,0,0,0,0.3324006,"\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^3*Sqrt[ArcTan[a*x]])/Sqrt[c + a^2*c*x^2],x]","\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","\frac{\text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)}{3 a^3}-\frac{\text{Int}\left(\frac{x^2}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)}{6 a}+\frac{x^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{3 a^2 c}-\frac{2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{3 a^4 c}",0,"(-2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(3*a^2*c) + Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]/(3*a^3) - Defer[Int][x^2/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]/(6*a)","A",0,0,0,0,-1,"{}"
735,0,0,0,0.2161724,"\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^2*Sqrt[ArcTan[a*x]])/Sqrt[c + a^2*c*x^2],x]","\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","-\frac{\text{Int}\left(\frac{x}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)}{4 a}-\frac{\text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{a^2 c x^2+c}},x\right)}{2 a^2}+\frac{x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{2 a^2 c}",0,"(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(2*a^2*c) - Defer[Int][x/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]/(4*a) - Defer[Int][Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x]/(2*a^2)","A",0,0,0,0,-1,"{}"
736,0,0,0,0.1089744,"\int \frac{x \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x*Sqrt[ArcTan[a*x]])/Sqrt[c + a^2*c*x^2],x]","\int \frac{x \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","\frac{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{a^2 c}-\frac{\text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)}{2 a}",0,"(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(a^2*c) - Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]/(2*a)","A",0,0,0,0,-1,"{}"
737,0,0,0,0.0343619,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Int[Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2],x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","\text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{a^2 c x^2+c}},x\right)",0,"Defer[Int][Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x]","A",0,0,0,0,-1,"{}"
738,0,0,0,0.106914,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \sqrt{c+a^2 c x^2}} \, dx","Int[Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]),x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \sqrt{c+a^2 c x^2}} \, dx","\text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x \sqrt{a^2 c x^2+c}},x\right)",0,"Defer[Int][Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x]","A",0,0,0,0,-1,"{}"
739,0,0,0,0.2142096,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \sqrt{c+a^2 c x^2}} \, dx","Int[Sqrt[ArcTan[a*x]]/(x^2*Sqrt[c + a^2*c*x^2]),x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \sqrt{c+a^2 c x^2}} \, dx","\frac{1}{2} a \text{Int}\left(\frac{1}{x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{c x}",0,"-((Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(c*x)) + (a*Defer[Int][1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/2","A",0,0,0,0,-1,"{}"
740,0,0,0,0.3292389,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^3 \sqrt{c+a^2 c x^2}} \, dx","Int[Sqrt[ArcTan[a*x]]/(x^3*Sqrt[c + a^2*c*x^2]),x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^3 \sqrt{c+a^2 c x^2}} \, dx","-\frac{1}{2} a^2 \text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x \sqrt{a^2 c x^2+c}},x\right)+\frac{1}{4} a \text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{2 c x^2}",0,"-(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(2*c*x^2) + (a*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/4 - (a^2*Defer[Int][Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x])/2","A",0,0,0,0,-1,"{}"
741,0,0,0,0.4241381,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4 \sqrt{c+a^2 c x^2}} \, dx","Int[Sqrt[ArcTan[a*x]]/(x^4*Sqrt[c + a^2*c*x^2]),x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4 \sqrt{c+a^2 c x^2}} \, dx","-\frac{1}{3} a^3 \text{Int}\left(\frac{1}{x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{1}{6} a \text{Int}\left(\frac{1}{x^3 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{2 a^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{3 c x}-\frac{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{3 c x^3}",0,"-(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(3*c*x) + (a*Defer[Int][1/(x^3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/6 - (a^3*Defer[Int][1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/3","A",0,0,0,0,-1,"{}"
742,0,0,0,0.1151555,"\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2),x]","\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2), x]","A",0,0,0,0,-1,"{}"
743,0,0,0,0.1237839,"\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2),x]","\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2), x]","A",0,0,0,0,-1,"{}"
744,0,0,0,0.1165692,"\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2),x]","\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^2 \sqrt{\tan ^{-1}(a x)}}{\left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2), x]","A",0,0,0,0,-1,"{}"
745,1,93,0,0.1824257,"\int \frac{x \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2),x]","\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\tan ^{-1}(a x)}}{a^2 c \sqrt{a^2 c x^2+c}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\tan ^{-1}(a x)}}{a^2 c \sqrt{a^2 c x^2+c}}",1,"-(Sqrt[ArcTan[a*x]]/(a^2*c*Sqrt[c + a^2*c*x^2])) + (Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c*Sqrt[c + a^2*c*x^2])","A",5,5,24,0.2083,1,"{4930, 4905, 4904, 3304, 3352}"
746,1,91,0,0.1111987,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[Sqrt[ArcTan[a*x]]/(c + a^2*c*x^2)^(3/2),x]","\frac{x \sqrt{\tan ^{-1}(a x)}}{c \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c \sqrt{a^2 c x^2+c}}","\frac{x \sqrt{\tan ^{-1}(a x)}}{c \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c \sqrt{a^2 c x^2+c}}",1,"(x*Sqrt[ArcTan[a*x]])/(c*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a*c*Sqrt[c + a^2*c*x^2])","A",5,5,23,0.2174,1,"{4905, 4904, 3296, 3305, 3351}"
747,0,0,0,0.1162413,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^(3/2)),x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x \left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^(3/2)), x]","A",0,0,0,0,-1,"{}"
748,0,0,0,0.1181018,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[Sqrt[ArcTan[a*x]]/(x^2*(c + a^2*c*x^2)^(3/2)),x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][Sqrt[ArcTan[a*x]]/(x^2*(c + a^2*c*x^2)^(3/2)), x]","A",0,0,0,0,-1,"{}"
749,0,0,0,0.1142425,"\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2),x]","\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(a^2 c x^2+c\right)^{5/2}},x\right)",0,"Defer[Int][(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2), x]","A",0,0,0,0,-1,"{}"
750,0,0,0,0.1222436,"\int \frac{x^4 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^4*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2),x]","\int \frac{x^4 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^4 \sqrt{\tan ^{-1}(a x)}}{\left(a^2 c x^2+c\right)^{5/2}},x\right)",0,"Defer[Int][(x^4*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2), x]","A",0,0,0,0,-1,"{}"
751,1,215,0,0.3473231,"\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2),x]","\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{12 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{\tan ^{-1}(a x)}}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \cos \left(3 \tan ^{-1}(a x)\right)}{12 a^4 c^2 \sqrt{a^2 c x^2+c}}","\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{12 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{\tan ^{-1}(a x)}}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \cos \left(3 \tan ^{-1}(a x)\right)}{12 a^4 c^2 \sqrt{a^2 c x^2+c}}",1,"(-3*Sqrt[ArcTan[a*x]])/(4*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]]*Cos[3*ArcTan[a*x]])/(12*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(12*a^4*c^2*Sqrt[c + a^2*c*x^2])","A",10,6,26,0.2308,1,"{4971, 4970, 3312, 3296, 3304, 3352}"
752,1,163,0,0.4248998,"\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2),x]","-\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{12 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{3 c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{12 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{3 c \left(a^2 c x^2+c\right)^{3/2}}",1,"(x^3*Sqrt[ArcTan[a*x]])/(3*c*(c + a^2*c*x^2)^(3/2)) - (Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(12*a^3*c^2*Sqrt[c + a^2*c*x^2])","A",9,6,26,0.2308,1,"{4944, 4971, 4970, 3312, 3305, 3351}"
753,1,163,0,0.2504062,"\int \frac{x \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{12 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\tan ^{-1}(a x)}}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{12 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\tan ^{-1}(a x)}}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}",1,"-Sqrt[ArcTan[a*x]]/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) + (Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(12*a^2*c^2*Sqrt[c + a^2*c*x^2])","A",9,6,24,0.2500,1,"{4930, 4905, 4904, 3312, 3304, 3352}"
754,1,213,0,0.1878946,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[Sqrt[ArcTan[a*x]]/(c + a^2*c*x^2)^(5/2),x]","-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4 a c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{12 a c^2 \sqrt{a^2 c x^2+c}}+\frac{3 x \sqrt{\tan ^{-1}(a x)}}{4 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \sin \left(3 \tan ^{-1}(a x)\right)}{12 a c^2 \sqrt{a^2 c x^2+c}}","-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4 a c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{12 a c^2 \sqrt{a^2 c x^2+c}}+\frac{3 x \sqrt{\tan ^{-1}(a x)}}{4 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \sin \left(3 \tan ^{-1}(a x)\right)}{12 a c^2 \sqrt{a^2 c x^2+c}}",1,"(3*x*Sqrt[ArcTan[a*x]])/(4*c^2*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*a*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(12*a*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]]*Sin[3*ArcTan[a*x]])/(12*a*c^2*Sqrt[c + a^2*c*x^2])","A",10,6,23,0.2609,1,"{4905, 4904, 3312, 3296, 3305, 3351}"
755,0,0,0,0.1214996,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^(5/2)),x]","\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x \left(a^2 c x^2+c\right)^{5/2}},x\right)",0,"Defer[Int][Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^(5/2)), x]","A",0,0,0,0,-1,"{}"
756,0,0,0,0.0351817,"\int x^m \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2} \, dx","Int[x^m*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2),x]","\int x^m \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right) \tan ^{-1}(a x)^{3/2},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
757,0,0,0,0.0361001,"\int x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2} \, dx","Int[x^2*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2),x]","\int x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(x^2 \left(a^2 c x^2+c\right) \tan ^{-1}(a x)^{3/2},x\right)",0,"Defer[Int][x^2*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
758,0,0,0,0.0382859,"\int x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2} \, dx","Int[x*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2),x]","\int x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2} \, dx","\frac{c \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}{4 a^2}-\frac{3 \text{Int}\left(\left(a^2 c x^2+c\right) \sqrt{\tan ^{-1}(a x)},x\right)}{8 a}",0,"(c*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(4*a^2) - (3*Defer[Int][(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x])/(8*a)","A",0,0,0,0,-1,"{}"
759,0,0,0,0.0235658,"\int \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2} \, dx","Int[(c + a^2*c*x^2)*ArcTan[a*x]^(3/2),x]","\int \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2} \, dx","\frac{1}{8} c \text{Int}\left(\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right)+\frac{2}{3} c \text{Int}\left(\tan ^{-1}(a x)^{3/2},x\right)+\frac{1}{3} c x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}-\frac{c \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}{4 a}",0,"-(c*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(4*a) + (c*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/3 + (c*Defer[Int][1/Sqrt[ArcTan[a*x]], x])/8 + (2*c*Defer[Int][ArcTan[a*x]^(3/2), x])/3","A",0,0,0,0,-1,"{}"
760,0,0,0,0.0320078,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}}{x} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x]^(3/2))/x,x]","\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}}{x} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right) \tan ^{-1}(a x)^{3/2}}{x},x\right)",0,"Defer[Int][((c + a^2*c*x^2)*ArcTan[a*x]^(3/2))/x, x]","A",0,0,0,0,-1,"{}"
761,0,0,0,0.0350468,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}}{x^2} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x]^(3/2))/x^2,x]","\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}}{x^2} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right) \tan ^{-1}(a x)^{3/2}}{x^2},x\right)",0,"Defer[Int][((c + a^2*c*x^2)*ArcTan[a*x]^(3/2))/x^2, x]","A",0,0,0,0,-1,"{}"
762,0,0,0,0.0541324,"\int x^m \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2} \, dx","Int[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2),x]","\int x^m \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)^{3/2},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
763,0,0,0,0.0560478,"\int x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2} \, dx","Int[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2),x]","\int x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(x^2 \left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)^{3/2},x\right)",0,"Defer[Int][x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
764,0,0,0,0.0617982,"\int x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2} \, dx","Int[x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2),x]","\int x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2} \, dx","\frac{c^2 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^{3/2}}{6 a^2}-\frac{\text{Int}\left(\left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)},x\right)}{4 a}",0,"(c^2*(1 + a^2*x^2)^3*ArcTan[a*x]^(3/2))/(6*a^2) - Defer[Int][(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]], x]/(4*a)","A",0,0,0,0,-1,"{}"
765,0,0,0,0.0663589,"\int \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2} \, dx","Int[(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2),x]","\int \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2} \, dx","\frac{3}{80} c \text{Int}\left(\frac{a^2 c x^2+c}{\sqrt{\tan ^{-1}(a x)}},x\right)+\frac{1}{10} c^2 \text{Int}\left(\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right)+\frac{8}{15} c^2 \text{Int}\left(\tan ^{-1}(a x)^{3/2},x\right)+\frac{1}{5} c^2 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}+\frac{4}{15} c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}-\frac{3 c^2 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}{40 a}-\frac{c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}{5 a}",0,"-(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(5*a) - (3*c^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(40*a) + (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/5 + (c^2*Defer[Int][1/Sqrt[ArcTan[a*x]], x])/10 + (3*c*Defer[Int][(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/80 + (8*c^2*Defer[Int][ArcTan[a*x]^(3/2), x])/15","A",0,0,0,0,-1,"{}"
766,0,0,0,0.0503347,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}}{x} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2))/x,x]","\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}}{x} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)^{3/2}}{x},x\right)",0,"Defer[Int][((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2))/x, x]","A",0,0,0,0,-1,"{}"
767,0,0,0,0.0560523,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}}{x^2} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2))/x^2,x]","\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}}{x^2} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)^{3/2}}{x^2},x\right)",0,"Defer[Int][((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2))/x^2, x]","A",0,0,0,0,-1,"{}"
768,0,0,0,0.0573169,"\int x^m \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2} \, dx","Int[x^m*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2),x]","\int x^m \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)^{3/2},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
769,0,0,0,0.0566498,"\int x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2} \, dx","Int[x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2),x]","\int x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(x^2 \left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)^{3/2},x\right)",0,"Defer[Int][x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
770,0,0,0,0.0620576,"\int x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2} \, dx","Int[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2),x]","\int x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2} \, dx","\frac{c^3 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x)^{3/2}}{8 a^2}-\frac{3 \text{Int}\left(\left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)},x\right)}{16 a}",0,"(c^3*(1 + a^2*x^2)^4*ArcTan[a*x]^(3/2))/(8*a^2) - (3*Defer[Int][(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]], x])/(16*a)","A",0,0,0,0,-1,"{}"
771,0,0,0,0.1243989,"\int \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2} \, dx","Int[(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2),x]","\int \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2} \, dx","\frac{9}{280} c^2 \text{Int}\left(\frac{a^2 c x^2+c}{\sqrt{\tan ^{-1}(a x)}},x\right)+\frac{1}{56} c \text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{\sqrt{\tan ^{-1}(a x)}},x\right)+\frac{3}{35} c^3 \text{Int}\left(\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right)+\frac{16}{35} c^3 \text{Int}\left(\tan ^{-1}(a x)^{3/2},x\right)+\frac{1}{7} c^3 x \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^{3/2}+\frac{6}{35} c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}+\frac{8}{35} c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}-\frac{c^3 \left(a^2 x^2+1\right)^3 \sqrt{\tan ^{-1}(a x)}}{28 a}-\frac{9 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}{140 a}-\frac{6 c^3 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}{35 a}",0,"(-6*c^3*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(35*a) - (9*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(140*a) - (c^3*(1 + a^2*x^2)^3*Sqrt[ArcTan[a*x]])/(28*a) + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^(3/2))/7 + (3*c^3*Defer[Int][1/Sqrt[ArcTan[a*x]], x])/35 + (9*c^2*Defer[Int][(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/280 + (c*Defer[Int][(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x])/56 + (16*c^3*Defer[Int][ArcTan[a*x]^(3/2), x])/35","A",0,0,0,0,-1,"{}"
772,0,0,0,0.052133,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}}{x} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2))/x,x]","\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}}{x} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)^{3/2}}{x},x\right)",0,"Defer[Int][((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2))/x, x]","A",0,0,0,0,-1,"{}"
773,0,0,0,0.0571167,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}}{x^2} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2))/x^2,x]","\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}}{x^2} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)^{3/2}}{x^2},x\right)",0,"Defer[Int][((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2))/x^2, x]","A",0,0,0,0,-1,"{}"
774,0,0,0,0.0627471,"\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx","Int[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2),x]","\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^{3/2}}{a^2 c x^2+c},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2), x]","A",0,0,0,0,-1,"{}"
775,0,0,0,0.1228814,"\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx","Int[(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2),x]","\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx","\frac{\text{Int}\left(x \tan ^{-1}(a x)^{3/2},x\right)}{a^2 c}+\frac{2 \text{Int}\left(\tan ^{-1}(a x)^{5/2},x\right)}{5 a^3 c}-\frac{2 x \tan ^{-1}(a x)^{5/2}}{5 a^3 c}",0,"(-2*x*ArcTan[a*x]^(5/2))/(5*a^3*c) + Defer[Int][x*ArcTan[a*x]^(3/2), x]/(a^2*c) + (2*Defer[Int][ArcTan[a*x]^(5/2), x])/(5*a^3*c)","A",0,0,0,0,-1,"{}"
776,0,0,0,0.0964175,"\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx","Int[(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2),x]","\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx","\frac{\text{Int}\left(\tan ^{-1}(a x)^{3/2},x\right)}{a^2 c}-\frac{2 \tan ^{-1}(a x)^{5/2}}{5 a^3 c}",0,"(-2*ArcTan[a*x]^(5/2))/(5*a^3*c) + Defer[Int][ArcTan[a*x]^(3/2), x]/(a^2*c)","A",0,0,0,0,-1,"{}"
777,0,0,0,0.047974,"\int \frac{x \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx","Int[(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2),x]","\int \frac{x \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx","\frac{2 x \tan ^{-1}(a x)^{5/2}}{5 a c}-\frac{2 \text{Int}\left(\tan ^{-1}(a x)^{5/2},x\right)}{5 a c}",0,"(2*x*ArcTan[a*x]^(5/2))/(5*a*c) - (2*Defer[Int][ArcTan[a*x]^(5/2), x])/(5*a*c)","A",0,0,0,0,-1,"{}"
778,1,18,0,0.0251957,"\int \frac{\tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx","Int[ArcTan[a*x]^(3/2)/(c + a^2*c*x^2),x]","\frac{2 \tan ^{-1}(a x)^{5/2}}{5 a c}","\frac{2 \tan ^{-1}(a x)^{5/2}}{5 a c}",1,"(2*ArcTan[a*x]^(5/2))/(5*a*c)","A",1,1,21,0.04762,1,"{4884}"
779,0,0,0,0.1083104,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)),x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)} \, dx","\frac{i \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x (a x+i)},x\right)}{c}-\frac{2 i \tan ^{-1}(a x)^{5/2}}{5 c}",0,"(((-2*I)/5)*ArcTan[a*x]^(5/2))/c + (I*Defer[Int][ArcTan[a*x]^(3/2)/(x*(I + a*x)), x])/c","A",0,0,0,0,-1,"{}"
780,0,0,0,0.1024089,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^(3/2)/(x^2*(c + a^2*c*x^2)),x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \left(c+a^2 c x^2\right)} \, dx","\frac{\text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x^2},x\right)}{c}-\frac{2 a \tan ^{-1}(a x)^{5/2}}{5 c}",0,"(-2*a*ArcTan[a*x]^(5/2))/(5*c) + Defer[Int][ArcTan[a*x]^(3/2)/x^2, x]/c","A",0,0,0,0,-1,"{}"
781,0,0,0,0.1923334,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^(3/2)/(x^3*(c + a^2*c*x^2)),x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \left(c+a^2 c x^2\right)} \, dx","-\frac{i a^2 \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x (a x+i)},x\right)}{c}+\frac{\text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x^3},x\right)}{c}+\frac{2 i a^2 \tan ^{-1}(a x)^{5/2}}{5 c}",0,"(((2*I)/5)*a^2*ArcTan[a*x]^(5/2))/c + Defer[Int][ArcTan[a*x]^(3/2)/x^3, x]/c - (I*a^2*Defer[Int][ArcTan[a*x]^(3/2)/(x*(I + a*x)), x])/c","A",0,0,0,0,-1,"{}"
782,0,0,0,0.1862473,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^4 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^(3/2)/(x^4*(c + a^2*c*x^2)),x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x^4 \left(c+a^2 c x^2\right)} \, dx","-\frac{a^2 \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x^2},x\right)}{c}+\frac{\text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x^4},x\right)}{c}+\frac{2 a^3 \tan ^{-1}(a x)^{5/2}}{5 c}",0,"(2*a^3*ArcTan[a*x]^(5/2))/(5*c) + Defer[Int][ArcTan[a*x]^(3/2)/x^4, x]/c - (a^2*Defer[Int][ArcTan[a*x]^(3/2)/x^2, x])/c","A",0,0,0,0,-1,"{}"
783,0,0,0,0.0640575,"\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2,x]","\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^2} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(a^2 c x^2+c\right)^2},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2, x]","A",0,0,0,0,-1,"{}"
784,0,0,0,0.0643157,"\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2,x]","\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^2} \, dx","\text{Int}\left(\frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left(a^2 c x^2+c\right)^2},x\right)",0,"Defer[Int][(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2, x]","A",0,0,0,0,-1,"{}"
785,1,127,0,0.1884363,"\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2,x]","\frac{3 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a^3 c^2}-\frac{x \tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left(a^2 x^2+1\right)}-\frac{3 \sqrt{\tan ^{-1}(a x)}}{8 a^3 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^{5/2}}{5 a^3 c^2}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{16 a^3 c^2}","\frac{3 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a^3 c^2}-\frac{x \tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left(a^2 x^2+1\right)}-\frac{3 \sqrt{\tan ^{-1}(a x)}}{8 a^3 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^{5/2}}{5 a^3 c^2}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{16 a^3 c^2}",1,"(3*Sqrt[ArcTan[a*x]])/(16*a^3*c^2) - (3*Sqrt[ArcTan[a*x]])/(8*a^3*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x]^(3/2))/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(5/2)/(5*a^3*c^2) + (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a^3*c^2)","A",7,6,24,0.2500,1,"{4936, 4930, 4904, 3312, 3304, 3352}"
786,1,109,0,0.1520579,"\int \frac{x \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2,x]","-\frac{3 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a^2 c^2}-\frac{\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{3 x \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2}","-\frac{3 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a^2 c^2}-\frac{\tan ^{-1}(a x)^{3/2}}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{3 x \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^2}",1,"(3*x*Sqrt[ArcTan[a*x]])/(8*a*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(3/2)/(4*a^2*c^2) - ArcTan[a*x]^(3/2)/(2*a^2*c^2*(1 + a^2*x^2)) - (3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a^2*c^2)","A",7,7,22,0.3182,1,"{4930, 4892, 4970, 4406, 12, 3305, 3351}"
787,1,124,0,0.1476864,"\int \frac{\tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]^(3/2)/(c + a^2*c*x^2)^2,x]","\frac{x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left(a^2 x^2+1\right)}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left(a^2 x^2+1\right)}-\frac{3 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a c^2}+\frac{\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac{3 \sqrt{\tan ^{-1}(a x)}}{16 a c^2}","\frac{x \tan ^{-1}(a x)^{3/2}}{2 c^2 \left(a^2 x^2+1\right)}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \left(a^2 x^2+1\right)}-\frac{3 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a c^2}+\frac{\tan ^{-1}(a x)^{5/2}}{5 a c^2}-\frac{3 \sqrt{\tan ^{-1}(a x)}}{16 a c^2}",1,"(-3*Sqrt[ArcTan[a*x]])/(16*a*c^2) + (3*Sqrt[ArcTan[a*x]])/(8*a*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x]^(3/2))/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(5/2)/(5*a*c^2) - (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a*c^2)","A",7,6,21,0.2857,1,"{4892, 4930, 4904, 3312, 3304, 3352}"
788,0,0,0,0.0594933,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^2),x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)^2} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x \left(a^2 c x^2+c\right)^2},x\right)",0,"Defer[Int][ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^2), x]","A",0,0,0,0,-1,"{}"
789,0,0,0,0.0629787,"\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3,x]","\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(a^2 c x^2+c\right)^3},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3, x]","A",0,0,0,0,-1,"{}"
790,0,0,0,0.064012,"\int \frac{x^5 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^5*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3,x]","\int \frac{x^5 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","\text{Int}\left(\frac{x^5 \tan ^{-1}(a x)^{3/2}}{\left(a^2 c x^2+c\right)^3},x\right)",0,"Defer[Int][(x^5*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3, x]","A",0,0,0,0,-1,"{}"
791,1,230,0,0.4111296,"\int \frac{x^4 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^4*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3,x]","-\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{512 a^5 c^3}+\frac{3 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a^5 c^3}+\frac{3 x^4 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left(a^2 x^2+1\right)^2}-\frac{x^3 \tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}-\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 a^4 c^3 \left(a^2 x^2+1\right)}-\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a^5 c^3 \left(a^2 x^2+1\right)}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a^5 c^3}+\frac{27 \sqrt{\tan ^{-1}(a x)}}{256 a^5 c^3}","-\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{512 a^5 c^3}+\frac{3 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a^5 c^3}+\frac{3 x^4 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left(a^2 x^2+1\right)^2}-\frac{x^3 \tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}-\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 a^4 c^3 \left(a^2 x^2+1\right)}-\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a^5 c^3 \left(a^2 x^2+1\right)}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a^5 c^3}+\frac{27 \sqrt{\tan ^{-1}(a x)}}{256 a^5 c^3}",1,"(27*Sqrt[ArcTan[a*x]])/(256*a^5*c^3) + (3*x^4*Sqrt[ArcTan[a*x]])/(32*a*c^3*(1 + a^2*x^2)^2) - (9*Sqrt[ArcTan[a*x]])/(32*a^5*c^3*(1 + a^2*x^2)) - (x^3*ArcTan[a*x]^(3/2))/(4*a^2*c^3*(1 + a^2*x^2)^2) - (3*x*ArcTan[a*x]^(3/2))/(8*a^4*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(5/2))/(20*a^5*c^3) - (3*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*a^5*c^3) + (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a^5*c^3)","A",15,8,24,0.3333,1,"{4940, 4936, 4930, 4904, 3312, 3304, 3352, 4970}"
792,1,168,0,0.2471148,"\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3,x]","\frac{3 \sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{512 a^4 c^3}-\frac{3 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{64 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^{3/2}}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{3 \tan ^{-1}(a x)^{3/2}}{32 a^4 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left(2 \tan ^{-1}(a x)\right)}{32 a^4 c^3}-\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{256 a^4 c^3}","\frac{3 \sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{512 a^4 c^3}-\frac{3 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{64 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^{3/2}}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{3 \tan ^{-1}(a x)^{3/2}}{32 a^4 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left(2 \tan ^{-1}(a x)\right)}{32 a^4 c^3}-\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{256 a^4 c^3}",1,"(-3*ArcTan[a*x]^(3/2))/(32*a^4*c^3) + (x^4*ArcTan[a*x]^(3/2))/(4*c^3*(1 + a^2*x^2)^2) + (3*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*a^4*c^3) - (3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(64*a^4*c^3) + (3*Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(32*a^4*c^3) - (3*Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(256*a^4*c^3)","A",10,6,24,0.2500,1,"{4944, 4970, 3312, 3296, 3305, 3351}"
793,1,108,0,0.1626137,"\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3,x]","\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{512 a^3 c^3}+\frac{\tan ^{-1}(a x)^{5/2}}{20 a^3 c^3}-\frac{\tan ^{-1}(a x)^{3/2} \sin \left(4 \tan ^{-1}(a x)\right)}{32 a^3 c^3}-\frac{3 \sqrt{\tan ^{-1}(a x)} \cos \left(4 \tan ^{-1}(a x)\right)}{256 a^3 c^3}","\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{512 a^3 c^3}+\frac{\tan ^{-1}(a x)^{5/2}}{20 a^3 c^3}-\frac{\tan ^{-1}(a x)^{3/2} \sin \left(4 \tan ^{-1}(a x)\right)}{32 a^3 c^3}-\frac{3 \sqrt{\tan ^{-1}(a x)} \cos \left(4 \tan ^{-1}(a x)\right)}{256 a^3 c^3}",1,"ArcTan[a*x]^(5/2)/(20*a^3*c^3) - (3*Sqrt[ArcTan[a*x]]*Cos[4*ArcTan[a*x]])/(256*a^3*c^3) + (3*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*a^3*c^3) - (ArcTan[a*x]^(3/2)*Sin[4*ArcTan[a*x]])/(32*a^3*c^3)","A",7,5,24,0.2083,1,"{4970, 4406, 3296, 3304, 3352}"
794,1,168,0,0.1857522,"\int \frac{x \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3,x]","-\frac{3 \sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{512 a^2 c^3}-\frac{3 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{64 a^2 c^3}-\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)^{3/2}}{32 a^2 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left(2 \tan ^{-1}(a x)\right)}{32 a^2 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{256 a^2 c^3}","-\frac{3 \sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{512 a^2 c^3}-\frac{3 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{64 a^2 c^3}-\frac{\tan ^{-1}(a x)^{3/2}}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)^{3/2}}{32 a^2 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left(2 \tan ^{-1}(a x)\right)}{32 a^2 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{256 a^2 c^3}",1,"(3*ArcTan[a*x]^(3/2))/(32*a^2*c^3) - ArcTan[a*x]^(3/2)/(4*a^2*c^3*(1 + a^2*x^2)^2) - (3*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*a^2*c^3) - (3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(64*a^2*c^3) + (3*Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(32*a^2*c^3) + (3*Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(256*a^2*c^3)","A",10,6,22,0.2727,1,"{4930, 4904, 3312, 3296, 3305, 3351}"
795,1,219,0,0.2909587,"\int \frac{\tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]^(3/2)/(c + a^2*c*x^2)^3,x]","\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left(a^2 x^2+1\right)}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left(a^2 x^2+1\right)^2}-\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{512 a c^3}-\frac{3 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a c^3}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a c^3}","\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 c^3 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)^{3/2}}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{9 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left(a^2 x^2+1\right)}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a c^3 \left(a^2 x^2+1\right)^2}-\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{512 a c^3}-\frac{3 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a c^3}+\frac{3 \tan ^{-1}(a x)^{5/2}}{20 a c^3}-\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a c^3}",1,"(-45*Sqrt[ArcTan[a*x]])/(256*a*c^3) + (3*Sqrt[ArcTan[a*x]])/(32*a*c^3*(1 + a^2*x^2)^2) + (9*Sqrt[ArcTan[a*x]])/(32*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x]^(3/2))/(4*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x]^(3/2))/(8*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(5/2))/(20*a*c^3) - (3*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(512*a*c^3) - (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a*c^3)","A",15,7,21,0.3333,1,"{4900, 4892, 4930, 4904, 3312, 3304, 3352}"
796,0,0,0,0.0606894,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^3),x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)^3} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x \left(a^2 c x^2+c\right)^3},x\right)",0,"Defer[Int][ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^3), x]","A",0,0,0,0,-1,"{}"
797,0,0,0,0.0980957,"\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx","Int[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2),x]","\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(x^m \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2},x\right)",0,"Defer[Int][x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
798,0,0,0,0.1010847,"\int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx","Int[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2),x]","\int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2},x\right)",0,"Defer[Int][x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
799,0,0,0,0.1056234,"\int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx","Int[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2),x]","\int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx","\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}{3 a^2 c}-\frac{\text{Int}\left(\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)},x\right)}{2 a}",0,"((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(3*a^2*c) - Defer[Int][Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]/(2*a)","A",0,0,0,0,-1,"{}"
800,0,0,0,0.1050269,"\int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx","Int[Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2),x]","\int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx","\frac{3}{8} c \text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{1}{2} c \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right)+\frac{1}{2} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}-\frac{3 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{4 a}",0,"(-3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(4*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/2 + (3*c*Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/8 + (c*Defer[Int][ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/2","A",0,0,0,0,-1,"{}"
801,0,0,0,0.0966777,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{x} \, dx","Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/x,x]","\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{x} \, dx","\text{Int}\left(\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{x},x\right)",0,"Defer[Int][(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/x, x]","A",0,0,0,0,-1,"{}"
802,0,0,0,0.109922,"\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2} \, dx","Int[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2),x]","\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
803,0,0,0,0.1151813,"\int x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2} \, dx","Int[x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2),x]","\int x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(x^2 \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2},x\right)",0,"Defer[Int][x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
804,0,0,0,0.117776,"\int x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2} \, dx","Int[x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2),x]","\int x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2} \, dx","\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^{3/2}}{5 a^2 c}-\frac{3 \text{Int}\left(\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)},x\right)}{10 a}",0,"((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/(5*a^2*c) - (3*Defer[Int][(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x])/(10*a)","A",0,0,0,0,-1,"{}"
805,0,0,0,0.1784878,"\int \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2} \, dx","Int[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2),x]","\int \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2} \, dx","\frac{9}{32} c^2 \text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{3}{8} c^2 \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right)+\frac{1}{16} c \text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{\tan ^{-1}(a x)}},x\right)+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}-\frac{9 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{16 a}+\frac{1}{4} x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}-\frac{\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}{8 a}",0,"(-9*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(16*a) - ((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(8*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/4 + (9*c^2*Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/32 + (c*Defer[Int][Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x])/16 + (3*c^2*Defer[Int][ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/8","A",0,0,0,0,-1,"{}"
806,0,0,0,0.1132962,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}}{x} \, dx","Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/x,x]","\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}}{x} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}{x},x\right)",0,"Defer[Int][((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/x, x]","A",0,0,0,0,-1,"{}"
807,0,0,0,0.1134657,"\int x^m \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2} \, dx","Int[x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2),x]","\int x^m \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^{3/2},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
808,0,0,0,0.1141536,"\int x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2} \, dx","Int[x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2),x]","\int x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2} \, dx","\text{Int}\left(x^2 \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^{3/2},x\right)",0,"Defer[Int][x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
809,0,0,0,0.1164191,"\int x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2} \, dx","Int[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2),x]","\int x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2} \, dx","\frac{\left(a^2 c x^2+c\right)^{7/2} \tan ^{-1}(a x)^{3/2}}{7 a^2 c}-\frac{3 \text{Int}\left(\left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)},x\right)}{14 a}",0,"((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^(3/2))/(7*a^2*c) - (3*Defer[Int][(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x])/(14*a)","A",0,0,0,0,-1,"{}"
810,0,0,0,0.2738118,"\int \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2} \, dx","Int[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2),x]","\int \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2} \, dx","\frac{15}{64} c^3 \text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{5}{16} c^3 \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right)+\frac{5}{96} c^2 \text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{\tan ^{-1}(a x)}},x\right)+\frac{1}{40} c \text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}},x\right)+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}-\frac{15 c^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{32 a}+\frac{5}{24} c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}-\frac{5 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}{48 a}+\frac{1}{6} x \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^{3/2}-\frac{\left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}}{20 a}",0,"(-15*c^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(32*a) - (5*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(48*a) - ((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]])/(20*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/6 + (15*c^3*Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/64 + (5*c^2*Defer[Int][Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x])/96 + (c*Defer[Int][(c + a^2*c*x^2)^(3/2)/Sqrt[ArcTan[a*x]], x])/40 + (5*c^3*Defer[Int][ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/16","A",0,0,0,0,-1,"{}"
811,0,0,0,0.1118375,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}}{x} \, dx","Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/x,x]","\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}}{x} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^{3/2}}{x},x\right)",0,"Defer[Int][((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/x, x]","A",0,0,0,0,-1,"{}"
812,0,0,0,0.1015751,"\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^m*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^2],x]","\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^2], x]","A",0,0,0,0,-1,"{}"
813,0,0,0,0.450052,"\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^3*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^2],x]","\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","\frac{\text{Int}\left(\frac{x}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)}{8 a^2}+\frac{5 \text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{a^2 c x^2+c}},x\right)}{4 a^3}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{3 a^2 c}-\frac{x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{4 a^3 c}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{3 a^4 c}",0,"-(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(4*a^3*c) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(3*a^2*c) + Defer[Int][x/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]/(8*a^2) + (5*Defer[Int][Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/(4*a^3)","A",0,0,0,0,-1,"{}"
814,0,0,0,0.2586601,"\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^2*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^2],x]","\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","\frac{3 \text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)}{8 a^2}-\frac{\text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right)}{2 a^2}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{2 a^2 c}-\frac{3 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{4 a^3 c}",0,"(-3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(4*a^3*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(2*a^2*c) + (3*Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/(8*a^2) - Defer[Int][ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x]/(2*a^2)","A",0,0,0,0,-1,"{}"
815,0,0,0,0.1078759,"\int \frac{x \tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^2],x]","\int \frac{x \tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{a^2 c}-\frac{3 \text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{a^2 c x^2+c}},x\right)}{2 a}",0,"(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(a^2*c) - (3*Defer[Int][Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/(2*a)","A",0,0,0,0,-1,"{}"
816,0,0,0,0.0353936,"\int \frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2],x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right)",0,"Defer[Int][ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x]","A",0,0,0,0,-1,"{}"
817,0,0,0,0.1053117,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^(3/2)/(x*Sqrt[c + a^2*c*x^2]),x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{c+a^2 c x^2}} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{a^2 c x^2+c}},x\right)",0,"Defer[Int][ArcTan[a*x]^(3/2)/(x*Sqrt[c + a^2*c*x^2]), x]","A",0,0,0,0,-1,"{}"
818,0,0,0,0.2134365,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^(3/2)/(x^2*Sqrt[c + a^2*c*x^2]),x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \sqrt{c+a^2 c x^2}} \, dx","\frac{3}{2} a \text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x \sqrt{a^2 c x^2+c}},x\right)-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{c x}",0,"-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(c*x)) + (3*a*Defer[Int][Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x])/2","A",0,0,0,0,-1,"{}"
819,0,0,0,0.4228466,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^(3/2)/(x^3*Sqrt[c + a^2*c*x^2]),x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx","\frac{3}{8} a^2 \text{Int}\left(\frac{1}{x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{1}{2} a^2 \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{a^2 c x^2+c}},x\right)-\frac{3 a \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{4 c x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{2 c x^2}",0,"(-3*a*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(4*c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(2*c*x^2) + (3*a^2*Defer[Int][1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/8 - (a^2*Defer[Int][ArcTan[a*x]^(3/2)/(x*Sqrt[c + a^2*c*x^2]), x])/2","A",0,0,0,0,-1,"{}"
820,0,0,0,0.6558273,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^4 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^(3/2)/(x^4*Sqrt[c + a^2*c*x^2]),x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x^4 \sqrt{c+a^2 c x^2}} \, dx","-\frac{5}{4} a^3 \text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x \sqrt{a^2 c x^2+c}},x\right)+\frac{1}{8} a^2 \text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{2 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{3 c x}-\frac{a \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{4 c x^2}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{3 c x^3}",0,"-(a*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(4*c*x^2) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(3*c*x) + (a^2*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/8 - (5*a^3*Defer[Int][Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x])/4","A",0,0,0,0,-1,"{}"
821,0,0,0,0.115381,"\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2),x]","\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2), x]","A",0,0,0,0,-1,"{}"
822,0,0,0,0.1352348,"\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2),x]","\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2), x]","A",0,0,0,0,-1,"{}"
823,0,0,0,0.151754,"\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2),x]","\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^2 \tan ^{-1}(a x)^{3/2}}{\left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2), x]","A",0,0,0,0,-1,"{}"
824,1,129,0,0.2113569,"\int \frac{x \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2),x]","-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^2 c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^{3/2}}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{3 x \sqrt{\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}}","-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^2 c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^{3/2}}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{3 x \sqrt{\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}}",1,"(3*x*Sqrt[ArcTan[a*x]])/(2*a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^(3/2)/(a^2*c*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(2*a^2*c*Sqrt[c + a^2*c*x^2])","A",6,6,24,0.2500,1,"{4930, 4905, 4904, 3296, 3305, 3351}"
825,1,125,0,0.1422504,"\int \frac{\tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]^(3/2)/(c + a^2*c*x^2)^(3/2),x]","-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^{3/2}}{c \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}}","-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^{3/2}}{c \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{2 a c \sqrt{a^2 c x^2+c}}",1,"(3*Sqrt[ArcTan[a*x]])/(2*a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^(3/2))/(c*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(2*a*c*Sqrt[c + a^2*c*x^2])","A",5,5,23,0.2174,1,"{4898, 4905, 4904, 3304, 3352}"
826,0,0,0,0.1172062,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^(3/2)),x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x \left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^(3/2)), x]","A",0,0,0,0,-1,"{}"
827,0,0,0,0.1178608,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]^(3/2)/(x^2*(c + a^2*c*x^2)^(3/2)),x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x^2 \left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][ArcTan[a*x]^(3/2)/(x^2*(c + a^2*c*x^2)^(3/2)), x]","A",0,0,0,0,-1,"{}"
828,0,0,0,0.1179569,"\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2),x]","\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(a^2 c x^2+c\right)^{5/2}},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x]","A",0,0,0,0,-1,"{}"
829,0,0,0,0.1239344,"\int \frac{x^5 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^5*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2),x]","\int \frac{x^5 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^5 \tan ^{-1}(a x)^{3/2}}{\left(a^2 c x^2+c\right)^{5/2}},x\right)",0,"Defer[Int][(x^5*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x]","A",0,0,0,0,-1,"{}"
830,0,0,0,0.1229932,"\int \frac{x^4 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^4*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2),x]","\int \frac{x^4 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^4 \tan ^{-1}(a x)^{3/2}}{\left(a^2 c x^2+c\right)^{5/2}},x\right)",0,"Defer[Int][(x^4*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x]","A",0,0,0,0,-1,"{}"
831,1,263,0,0.6414703,"\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2),x]","-\frac{9 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{24 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{\tan ^{-1}(a x)}}{a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{6 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^{3/2}}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{9 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{24 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{\tan ^{-1}(a x)}}{a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{6 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^{3/2}}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}",1,"(x^3*Sqrt[ArcTan[a*x]])/(6*a*c*(c + a^2*c*x^2)^(3/2)) + (x*Sqrt[ArcTan[a*x]])/(a^3*c^2*Sqrt[c + a^2*c*x^2]) - (x^2*ArcTan[a*x]^(3/2))/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (2*ArcTan[a*x]^(3/2))/(3*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(24*a^4*c^2*Sqrt[c + a^2*c*x^2])","A",15,10,26,0.3846,1,"{4940, 4930, 4905, 4904, 3296, 3305, 3351, 4971, 4970, 3312}"
832,1,247,0,0.4762091,"\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2),x]","-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{24 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{8 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \cos \left(3 \tan ^{-1}(a x)\right)}{24 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^3 \tan ^{-1}(a x)^{3/2}}{3 c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{24 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{8 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \cos \left(3 \tan ^{-1}(a x)\right)}{24 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^3 \tan ^{-1}(a x)^{3/2}}{3 c \left(a^2 c x^2+c\right)^{3/2}}",1,"(3*Sqrt[ArcTan[a*x]])/(8*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (x^3*ArcTan[a*x]^(3/2))/(3*c*(c + a^2*c*x^2)^(3/2)) - (Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]]*Cos[3*ArcTan[a*x]])/(24*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(24*a^3*c^2*Sqrt[c + a^2*c*x^2])","A",11,7,26,0.2692,1,"{4944, 4971, 4970, 3312, 3296, 3304, 3352}"
833,1,248,0,0.2814744,"\int \frac{x \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2),x]","-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{24 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 x \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \sin \left(3 \tan ^{-1}(a x)\right)}{24 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^{3/2}}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{24 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 x \sqrt{\tan ^{-1}(a x)}}{8 a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \sin \left(3 \tan ^{-1}(a x)\right)}{24 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^{3/2}}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}",1,"(3*x*Sqrt[ArcTan[a*x]])/(8*a*c^2*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^(3/2)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a^2*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(24*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]]*Sin[3*ArcTan[a*x]])/(24*a^2*c^2*Sqrt[c + a^2*c*x^2])","A",11,7,24,0.2917,1,"{4930, 4905, 4904, 3312, 3296, 3305, 3351}"
834,1,252,0,0.3490528,"\int \frac{\tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[ArcTan[a*x]^(3/2)/(c + a^2*c*x^2)^(5/2),x]","-\frac{9 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{24 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^{3/2}}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^{3/2}}{3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{\sqrt{\tan ^{-1}(a x)}}{6 a c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{9 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{24 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^{3/2}}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^{3/2}}{3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{\sqrt{\tan ^{-1}(a x)}}{6 a c \left(a^2 c x^2+c\right)^{3/2}}",1,"Sqrt[ArcTan[a*x]]/(6*a*c*(c + a^2*c*x^2)^(3/2)) + Sqrt[ArcTan[a*x]]/(a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^(3/2))/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^(3/2))/(3*c^2*Sqrt[c + a^2*c*x^2]) - (9*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(24*a*c^2*Sqrt[c + a^2*c*x^2])","A",14,7,23,0.3043,1,"{4900, 4898, 4905, 4904, 3304, 3352, 3312}"
835,0,0,0,0.1155609,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^(5/2)),x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x \left(a^2 c x^2+c\right)^{5/2}},x\right)",0,"Defer[Int][ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^(5/2)), x]","A",0,0,0,0,-1,"{}"
836,0,0,0,0.1166897,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[ArcTan[a*x]^(3/2)/(x^2*(c + a^2*c*x^2)^(5/2)),x]","\int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \left(c+a^2 c x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x^2 \left(a^2 c x^2+c\right)^{5/2}},x\right)",0,"Defer[Int][ArcTan[a*x]^(3/2)/(x^2*(c + a^2*c*x^2)^(5/2)), x]","A",0,0,0,0,-1,"{}"
837,0,0,0,0.0344447,"\int x^m \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2} \, dx","Int[x^m*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2),x]","\int x^m \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right) \tan ^{-1}(a x)^{5/2},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
838,0,0,0,0.0349865,"\int x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2} \, dx","Int[x^2*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2),x]","\int x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2} \, dx","\text{Int}\left(x^2 \left(a^2 c x^2+c\right) \tan ^{-1}(a x)^{5/2},x\right)",0,"Defer[Int][x^2*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
839,0,0,0,0.0551784,"\int x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2} \, dx","Int[x*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2),x]","\int x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2} \, dx","-\frac{5 c \text{Int}\left(\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right)}{64 a}-\frac{5 c \text{Int}\left(\tan ^{-1}(a x)^{3/2},x\right)}{12 a}+\frac{c \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{5/2}}{4 a^2}-\frac{5 c x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}{24 a}+\frac{5 c \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}{32 a^2}",0,"(5*c*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(32*a^2) - (5*c*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(24*a) + (c*(1 + a^2*x^2)^2*ArcTan[a*x]^(5/2))/(4*a^2) - (5*c*Defer[Int][1/Sqrt[ArcTan[a*x]], x])/(64*a) - (5*c*Defer[Int][ArcTan[a*x]^(3/2), x])/(12*a)","A",0,0,0,0,-1,"{}"
840,0,0,0,0.0249849,"\int \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2} \, dx","Int[(c + a^2*c*x^2)*ArcTan[a*x]^(5/2),x]","\int \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2} \, dx","\frac{5}{8} c \text{Int}\left(\sqrt{\tan ^{-1}(a x)},x\right)+\frac{2}{3} c \text{Int}\left(\tan ^{-1}(a x)^{5/2},x\right)+\frac{1}{3} c x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{5/2}-\frac{5 c \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}{12 a}",0,"(-5*c*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(12*a) + (c*x*(1 + a^2*x^2)*ArcTan[a*x]^(5/2))/3 + (5*c*Defer[Int][Sqrt[ArcTan[a*x]], x])/8 + (2*c*Defer[Int][ArcTan[a*x]^(5/2), x])/3","A",0,0,0,0,-1,"{}"
841,0,0,0,0.0323027,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}}{x} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x]^(5/2))/x,x]","\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}}{x} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right) \tan ^{-1}(a x)^{5/2}}{x},x\right)",0,"Defer[Int][((c + a^2*c*x^2)*ArcTan[a*x]^(5/2))/x, x]","A",0,0,0,0,-1,"{}"
842,0,0,0,0.0332529,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}}{x^2} \, dx","Int[((c + a^2*c*x^2)*ArcTan[a*x]^(5/2))/x^2,x]","\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}}{x^2} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right) \tan ^{-1}(a x)^{5/2}}{x^2},x\right)",0,"Defer[Int][((c + a^2*c*x^2)*ArcTan[a*x]^(5/2))/x^2, x]","A",0,0,0,0,-1,"{}"
843,0,0,0,0.0540261,"\int x^m \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2} \, dx","Int[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2),x]","\int x^m \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)^{5/2},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
844,0,0,0,0.0556613,"\int x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2} \, dx","Int[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2),x]","\int x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2} \, dx","\text{Int}\left(x^2 \left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)^{5/2},x\right)",0,"Defer[Int][x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
845,0,0,0,0.1139812,"\int x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2} \, dx","Int[x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2),x]","\int x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2} \, dx","-\frac{c \text{Int}\left(\frac{a^2 c x^2+c}{\sqrt{\tan ^{-1}(a x)}},x\right)}{64 a}-\frac{c^2 \text{Int}\left(\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right)}{24 a}-\frac{2 c^2 \text{Int}\left(\tan ^{-1}(a x)^{3/2},x\right)}{9 a}+\frac{c^2 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^{5/2}}{6 a^2}-\frac{c^2 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}{12 a}+\frac{c^2 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}{32 a^2}-\frac{c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}{9 a}+\frac{c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}{12 a^2}",0,"(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(12*a^2) + (c^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(32*a^2) - (c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(9*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(12*a) + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x]^(5/2))/(6*a^2) - (c^2*Defer[Int][1/Sqrt[ArcTan[a*x]], x])/(24*a) - (c*Defer[Int][(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/(64*a) - (2*c^2*Defer[Int][ArcTan[a*x]^(3/2), x])/(9*a)","A",0,0,0,0,-1,"{}"
846,0,0,0,0.0698693,"\int \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2} \, dx","Int[(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2),x]","\int \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2} \, dx","\frac{3}{16} c \text{Int}\left(\left(a^2 c x^2+c\right) \sqrt{\tan ^{-1}(a x)},x\right)+\frac{1}{2} c^2 \text{Int}\left(\sqrt{\tan ^{-1}(a x)},x\right)+\frac{8}{15} c^2 \text{Int}\left(\tan ^{-1}(a x)^{5/2},x\right)+\frac{1}{5} c^2 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{5/2}+\frac{4}{15} c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{5/2}-\frac{c^2 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}{8 a}-\frac{c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}{3 a}",0,"-(c^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(3*a) - (c^2*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(8*a) + (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^(5/2))/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(5/2))/5 + (c^2*Defer[Int][Sqrt[ArcTan[a*x]], x])/2 + (3*c*Defer[Int][(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x])/16 + (8*c^2*Defer[Int][ArcTan[a*x]^(5/2), x])/15","A",0,0,0,0,-1,"{}"
847,0,0,0,0.0482354,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}}{x} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2))/x,x]","\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}}{x} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)^{5/2}}{x},x\right)",0,"Defer[Int][((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2))/x, x]","A",0,0,0,0,-1,"{}"
848,0,0,0,0.0542138,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}}{x^2} \, dx","Int[((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2))/x^2,x]","\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}}{x^2} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)^{5/2}}{x^2},x\right)",0,"Defer[Int][((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2))/x^2, x]","A",0,0,0,0,-1,"{}"
849,0,0,0,0.0546779,"\int x^m \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2} \, dx","Int[x^m*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2),x]","\int x^m \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)^{5/2},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
850,0,0,0,0.054122,"\int x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2} \, dx","Int[x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2),x]","\int x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2} \, dx","\text{Int}\left(x^2 \left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)^{5/2},x\right)",0,"Defer[Int][x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
851,0,0,0,0.1874919,"\int x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2} \, dx","Int[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2),x]","\int x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2} \, dx","-\frac{9 c^2 \text{Int}\left(\frac{a^2 c x^2+c}{\sqrt{\tan ^{-1}(a x)}},x\right)}{896 a}-\frac{5 c \text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{\sqrt{\tan ^{-1}(a x)}},x\right)}{896 a}-\frac{3 c^3 \text{Int}\left(\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right)}{112 a}-\frac{c^3 \text{Int}\left(\tan ^{-1}(a x)^{3/2},x\right)}{7 a}+\frac{c^3 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x)^{5/2}}{8 a^2}-\frac{5 c^3 x \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^{3/2}}{112 a}+\frac{5 c^3 \left(a^2 x^2+1\right)^3 \sqrt{\tan ^{-1}(a x)}}{448 a^2}-\frac{3 c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}{56 a}+\frac{9 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}{448 a^2}-\frac{c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}{14 a}+\frac{3 c^3 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}{56 a^2}",0,"(3*c^3*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])/(56*a^2) + (9*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])/(448*a^2) + (5*c^3*(1 + a^2*x^2)^3*Sqrt[ArcTan[a*x]])/(448*a^2) - (c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(14*a) - (3*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(56*a) - (5*c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^(3/2))/(112*a) + (c^3*(1 + a^2*x^2)^4*ArcTan[a*x]^(5/2))/(8*a^2) - (3*c^3*Defer[Int][1/Sqrt[ArcTan[a*x]], x])/(112*a) - (9*c^2*Defer[Int][(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x])/(896*a) - (5*c*Defer[Int][(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x])/(896*a) - (c^3*Defer[Int][ArcTan[a*x]^(3/2), x])/(7*a)","A",0,0,0,0,-1,"{}"
852,0,0,0,0.126401,"\int \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2} \, dx","Int[(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2),x]","\int \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2} \, dx","\frac{9}{56} c^2 \text{Int}\left(\left(a^2 c x^2+c\right) \sqrt{\tan ^{-1}(a x)},x\right)+\frac{5}{56} c \text{Int}\left(\left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)},x\right)+\frac{3}{7} c^3 \text{Int}\left(\sqrt{\tan ^{-1}(a x)},x\right)+\frac{16}{35} c^3 \text{Int}\left(\tan ^{-1}(a x)^{5/2},x\right)+\frac{1}{7} c^3 x \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^{5/2}+\frac{6}{35} c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{5/2}+\frac{8}{35} c^3 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{5/2}-\frac{5 c^3 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^{3/2}}{84 a}-\frac{3 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}{28 a}-\frac{2 c^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}{7 a}",0,"(-2*c^3*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))/(7*a) - (3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))/(28*a) - (5*c^3*(1 + a^2*x^2)^3*ArcTan[a*x]^(3/2))/(84*a) + (8*c^3*x*(1 + a^2*x^2)*ArcTan[a*x]^(5/2))/35 + (6*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(5/2))/35 + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^(5/2))/7 + (3*c^3*Defer[Int][Sqrt[ArcTan[a*x]], x])/7 + (9*c^2*Defer[Int][(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x])/56 + (5*c*Defer[Int][(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]], x])/56 + (16*c^3*Defer[Int][ArcTan[a*x]^(5/2), x])/35","A",0,0,0,0,-1,"{}"
853,0,0,0,0.0499419,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}}{x} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2))/x,x]","\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}}{x} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)^{5/2}}{x},x\right)",0,"Defer[Int][((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2))/x, x]","A",0,0,0,0,-1,"{}"
854,0,0,0,0.0538417,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}}{x^2} \, dx","Int[((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2))/x^2,x]","\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}}{x^2} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)^{5/2}}{x^2},x\right)",0,"Defer[Int][((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2))/x^2, x]","A",0,0,0,0,-1,"{}"
855,0,0,0,0.0629983,"\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx","Int[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2),x]","\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^{5/2}}{a^2 c x^2+c},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2), x]","A",0,0,0,0,-1,"{}"
856,0,0,0,0.1197537,"\int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx","Int[(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2),x]","\int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx","\frac{\text{Int}\left(x \tan ^{-1}(a x)^{5/2},x\right)}{a^2 c}+\frac{2 \text{Int}\left(\tan ^{-1}(a x)^{7/2},x\right)}{7 a^3 c}-\frac{2 x \tan ^{-1}(a x)^{7/2}}{7 a^3 c}",0,"(-2*x*ArcTan[a*x]^(7/2))/(7*a^3*c) + Defer[Int][x*ArcTan[a*x]^(5/2), x]/(a^2*c) + (2*Defer[Int][ArcTan[a*x]^(7/2), x])/(7*a^3*c)","A",0,0,0,0,-1,"{}"
857,0,0,0,0.0911277,"\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx","Int[(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2),x]","\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx","\frac{\text{Int}\left(\tan ^{-1}(a x)^{5/2},x\right)}{a^2 c}-\frac{2 \tan ^{-1}(a x)^{7/2}}{7 a^3 c}",0,"(-2*ArcTan[a*x]^(7/2))/(7*a^3*c) + Defer[Int][ArcTan[a*x]^(5/2), x]/(a^2*c)","A",0,0,0,0,-1,"{}"
858,0,0,0,0.0538538,"\int \frac{x \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx","Int[(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2),x]","\int \frac{x \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx","\frac{2 x \tan ^{-1}(a x)^{7/2}}{7 a c}-\frac{2 \text{Int}\left(\tan ^{-1}(a x)^{7/2},x\right)}{7 a c}",0,"(2*x*ArcTan[a*x]^(7/2))/(7*a*c) - (2*Defer[Int][ArcTan[a*x]^(7/2), x])/(7*a*c)","A",0,0,0,0,-1,"{}"
859,1,18,0,0.0252277,"\int \frac{\tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx","Int[ArcTan[a*x]^(5/2)/(c + a^2*c*x^2),x]","\frac{2 \tan ^{-1}(a x)^{7/2}}{7 a c}","\frac{2 \tan ^{-1}(a x)^{7/2}}{7 a c}",1,"(2*ArcTan[a*x]^(7/2))/(7*a*c)","A",1,1,21,0.04762,1,"{4884}"
860,0,0,0,0.1149663,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)),x]","\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)} \, dx","\frac{i \text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{x (a x+i)},x\right)}{c}-\frac{2 i \tan ^{-1}(a x)^{7/2}}{7 c}",0,"(((-2*I)/7)*ArcTan[a*x]^(7/2))/c + (I*Defer[Int][ArcTan[a*x]^(5/2)/(x*(I + a*x)), x])/c","A",0,0,0,0,-1,"{}"
861,0,0,0,0.1037631,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x^2 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^(5/2)/(x^2*(c + a^2*c*x^2)),x]","\int \frac{\tan ^{-1}(a x)^{5/2}}{x^2 \left(c+a^2 c x^2\right)} \, dx","\frac{\text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{x^2},x\right)}{c}-\frac{2 a \tan ^{-1}(a x)^{7/2}}{7 c}",0,"(-2*a*ArcTan[a*x]^(7/2))/(7*c) + Defer[Int][ArcTan[a*x]^(5/2)/x^2, x]/c","A",0,0,0,0,-1,"{}"
862,0,0,0,0.1866412,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x^3 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^(5/2)/(x^3*(c + a^2*c*x^2)),x]","\int \frac{\tan ^{-1}(a x)^{5/2}}{x^3 \left(c+a^2 c x^2\right)} \, dx","-\frac{i a^2 \text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{x (a x+i)},x\right)}{c}+\frac{\text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{x^3},x\right)}{c}+\frac{2 i a^2 \tan ^{-1}(a x)^{7/2}}{7 c}",0,"(((2*I)/7)*a^2*ArcTan[a*x]^(7/2))/c + Defer[Int][ArcTan[a*x]^(5/2)/x^3, x]/c - (I*a^2*Defer[Int][ArcTan[a*x]^(5/2)/(x*(I + a*x)), x])/c","A",0,0,0,0,-1,"{}"
863,0,0,0,0.1790403,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x^4 \left(c+a^2 c x^2\right)} \, dx","Int[ArcTan[a*x]^(5/2)/(x^4*(c + a^2*c*x^2)),x]","\int \frac{\tan ^{-1}(a x)^{5/2}}{x^4 \left(c+a^2 c x^2\right)} \, dx","-\frac{a^2 \text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{x^2},x\right)}{c}+\frac{\text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{x^4},x\right)}{c}+\frac{2 a^3 \tan ^{-1}(a x)^{7/2}}{7 c}",0,"(2*a^3*ArcTan[a*x]^(7/2))/(7*c) + Defer[Int][ArcTan[a*x]^(5/2)/x^4, x]/c - (a^2*Defer[Int][ArcTan[a*x]^(5/2)/x^2, x])/c","A",0,0,0,0,-1,"{}"
864,0,0,0,0.0638905,"\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2,x]","\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^2} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(a^2 c x^2+c\right)^2},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2, x]","A",0,0,0,0,-1,"{}"
865,0,0,0,0.06566,"\int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2,x]","\int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^2} \, dx","\text{Int}\left(\frac{x^3 \tan ^{-1}(a x)^{5/2}}{\left(a^2 c x^2+c\right)^2},x\right)",0,"Defer[Int][(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2, x]","A",0,0,0,0,-1,"{}"
866,1,157,0,0.2244778,"\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2,x]","-\frac{15 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{128 a^3 c^2}-\frac{x \tan ^{-1}(a x)^{5/2}}{2 a^2 c^2 \left(a^2 x^2+1\right)}-\frac{5 \tan ^{-1}(a x)^{3/2}}{8 a^3 c^2 \left(a^2 x^2+1\right)}+\frac{15 x \sqrt{\tan ^{-1}(a x)}}{32 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^{7/2}}{7 a^3 c^2}+\frac{5 \tan ^{-1}(a x)^{3/2}}{16 a^3 c^2}","-\frac{15 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{128 a^3 c^2}-\frac{x \tan ^{-1}(a x)^{5/2}}{2 a^2 c^2 \left(a^2 x^2+1\right)}-\frac{5 \tan ^{-1}(a x)^{3/2}}{8 a^3 c^2 \left(a^2 x^2+1\right)}+\frac{15 x \sqrt{\tan ^{-1}(a x)}}{32 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^{7/2}}{7 a^3 c^2}+\frac{5 \tan ^{-1}(a x)^{3/2}}{16 a^3 c^2}",1,"(15*x*Sqrt[ArcTan[a*x]])/(32*a^2*c^2*(1 + a^2*x^2)) + (5*ArcTan[a*x]^(3/2))/(16*a^3*c^2) - (5*ArcTan[a*x]^(3/2))/(8*a^3*c^2*(1 + a^2*x^2)) - (x*ArcTan[a*x]^(5/2))/(2*a^2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(7/2)/(7*a^3*c^2) - (15*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(128*a^3*c^2)","A",8,8,24,0.3333,1,"{4936, 4930, 4892, 4970, 4406, 12, 3305, 3351}"
867,1,156,0,0.1993455,"\int \frac{x \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2,x]","-\frac{15 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{128 a^2 c^2}-\frac{\tan ^{-1}(a x)^{5/2}}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{8 a c^2 \left(a^2 x^2+1\right)}+\frac{15 \sqrt{\tan ^{-1}(a x)}}{32 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^2}-\frac{15 \sqrt{\tan ^{-1}(a x)}}{64 a^2 c^2}","-\frac{15 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{128 a^2 c^2}-\frac{\tan ^{-1}(a x)^{5/2}}{2 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{8 a c^2 \left(a^2 x^2+1\right)}+\frac{15 \sqrt{\tan ^{-1}(a x)}}{32 a^2 c^2 \left(a^2 x^2+1\right)}+\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^2}-\frac{15 \sqrt{\tan ^{-1}(a x)}}{64 a^2 c^2}",1,"(-15*Sqrt[ArcTan[a*x]])/(64*a^2*c^2) + (15*Sqrt[ArcTan[a*x]])/(32*a^2*c^2*(1 + a^2*x^2)) + (5*x*ArcTan[a*x]^(3/2))/(8*a*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(5/2)/(4*a^2*c^2) - ArcTan[a*x]^(5/2)/(2*a^2*c^2*(1 + a^2*x^2)) - (15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(128*a^2*c^2)","A",8,6,22,0.2727,1,"{4930, 4892, 4904, 3312, 3304, 3352}"
868,1,151,0,0.1839277,"\int \frac{\tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^2,x]","\frac{x \tan ^{-1}(a x)^{5/2}}{2 c^2 \left(a^2 x^2+1\right)}+\frac{5 \tan ^{-1}(a x)^{3/2}}{8 a c^2 \left(a^2 x^2+1\right)}-\frac{15 x \sqrt{\tan ^{-1}(a x)}}{32 c^2 \left(a^2 x^2+1\right)}+\frac{15 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{128 a c^2}+\frac{\tan ^{-1}(a x)^{7/2}}{7 a c^2}-\frac{5 \tan ^{-1}(a x)^{3/2}}{16 a c^2}","\frac{x \tan ^{-1}(a x)^{5/2}}{2 c^2 \left(a^2 x^2+1\right)}+\frac{5 \tan ^{-1}(a x)^{3/2}}{8 a c^2 \left(a^2 x^2+1\right)}-\frac{15 x \sqrt{\tan ^{-1}(a x)}}{32 c^2 \left(a^2 x^2+1\right)}+\frac{15 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{128 a c^2}+\frac{\tan ^{-1}(a x)^{7/2}}{7 a c^2}-\frac{5 \tan ^{-1}(a x)^{3/2}}{16 a c^2}",1,"(-15*x*Sqrt[ArcTan[a*x]])/(32*c^2*(1 + a^2*x^2)) - (5*ArcTan[a*x]^(3/2))/(16*a*c^2) + (5*ArcTan[a*x]^(3/2))/(8*a*c^2*(1 + a^2*x^2)) + (x*ArcTan[a*x]^(5/2))/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^(7/2)/(7*a*c^2) + (15*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(128*a*c^2)","A",8,7,21,0.3333,1,"{4892, 4930, 4970, 4406, 12, 3305, 3351}"
869,0,0,0,0.0590948,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)^2} \, dx","Int[ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^2),x]","\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)^2} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{x \left(a^2 c x^2+c\right)^2},x\right)",0,"Defer[Int][ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^2), x]","A",0,0,0,0,-1,"{}"
870,0,0,0,0.0631842,"\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3,x]","\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(a^2 c x^2+c\right)^3},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3, x]","A",0,0,0,0,-1,"{}"
871,0,0,0,0.0615847,"\int \frac{x^5 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^5*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3,x]","\int \frac{x^5 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","\text{Int}\left(\frac{x^5 \tan ^{-1}(a x)^{5/2}}{\left(a^2 c x^2+c\right)^3},x\right)",0,"Defer[Int][(x^5*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3, x]","A",0,0,0,0,-1,"{}"
872,1,310,0,0.4863932,"\int \frac{x^4 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^4*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3,x]","\frac{15 \sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4096 a^5 c^3}-\frac{15 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{128 a^5 c^3}+\frac{5 x^4 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left(a^2 x^2+1\right)^2}-\frac{x^3 \tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}-\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 a^4 c^3 \left(a^2 x^2+1\right)}+\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 a^4 c^3 \left(a^2 x^2+1\right)}-\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a^5 c^3 \left(a^2 x^2+1\right)}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a^5 c^3}+\frac{45 \tan ^{-1}(a x)^{3/2}}{256 a^5 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left(2 \tan ^{-1}(a x)\right)}{256 a^5 c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{2048 a^5 c^3}","\frac{15 \sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4096 a^5 c^3}-\frac{15 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{128 a^5 c^3}+\frac{5 x^4 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left(a^2 x^2+1\right)^2}-\frac{x^3 \tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}-\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 a^4 c^3 \left(a^2 x^2+1\right)}+\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 a^4 c^3 \left(a^2 x^2+1\right)}-\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a^5 c^3 \left(a^2 x^2+1\right)}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a^5 c^3}+\frac{45 \tan ^{-1}(a x)^{3/2}}{256 a^5 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left(2 \tan ^{-1}(a x)\right)}{256 a^5 c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{2048 a^5 c^3}",1,"(45*x*Sqrt[ArcTan[a*x]])/(128*a^4*c^3*(1 + a^2*x^2)) + (45*ArcTan[a*x]^(3/2))/(256*a^5*c^3) + (5*x^4*ArcTan[a*x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)^2) - (15*ArcTan[a*x]^(3/2))/(32*a^5*c^3*(1 + a^2*x^2)) - (x^3*ArcTan[a*x]^(5/2))/(4*a^2*c^3*(1 + a^2*x^2)^2) - (3*x*ArcTan[a*x]^(5/2))/(8*a^4*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(7/2))/(28*a^5*c^3) + (15*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a^5*c^3) - (15*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(128*a^5*c^3) + (15*Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(256*a^5*c^3) - (15*Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(2048*a^5*c^3)","A",18,11,24,0.4583,1,"{4940, 4936, 4930, 4892, 4970, 4406, 12, 3305, 3351, 3312, 3296}"
873,1,256,0,0.4916248,"\int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3,x]","\frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4096 a^4 c^3}-\frac{15 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{256 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{15 x^4 \sqrt{\tan ^{-1}(a x)}}{256 c^3 \left(a^2 x^2+1\right)^2}+\frac{5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left(a^2 x^2+1\right)^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left(a^2 x^2+1\right)}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^4 c^3 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}-\frac{135 \sqrt{\tan ^{-1}(a x)}}{2048 a^4 c^3}","\frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4096 a^4 c^3}-\frac{15 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{256 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)^{5/2}}{4 c^3 \left(a^2 x^2+1\right)^2}-\frac{15 x^4 \sqrt{\tan ^{-1}(a x)}}{256 c^3 \left(a^2 x^2+1\right)^2}+\frac{5 x^3 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left(a^2 x^2+1\right)^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left(a^2 x^2+1\right)}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^4 c^3 \left(a^2 x^2+1\right)}-\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^4 c^3}-\frac{135 \sqrt{\tan ^{-1}(a x)}}{2048 a^4 c^3}",1,"(-135*Sqrt[ArcTan[a*x]])/(2048*a^4*c^3) - (15*x^4*Sqrt[ArcTan[a*x]])/(256*c^3*(1 + a^2*x^2)^2) + (45*Sqrt[ArcTan[a*x]])/(256*a^4*c^3*(1 + a^2*x^2)) + (5*x^3*ArcTan[a*x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)^2) + (15*x*ArcTan[a*x]^(3/2))/(64*a^3*c^3*(1 + a^2*x^2)) - (3*ArcTan[a*x]^(5/2))/(32*a^4*c^3) + (x^4*ArcTan[a*x]^(5/2))/(4*c^3*(1 + a^2*x^2)^2) + (15*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a^4*c^3) - (15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(256*a^4*c^3)","A",16,9,24,0.3750,1,"{4944, 4940, 4936, 4930, 4904, 3312, 3304, 3352, 4970}"
874,1,133,0,0.1750994,"\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3,x]","-\frac{15 \sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4096 a^3 c^3}+\frac{\tan ^{-1}(a x)^{7/2}}{28 a^3 c^3}-\frac{\tan ^{-1}(a x)^{5/2} \sin \left(4 \tan ^{-1}(a x)\right)}{32 a^3 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{2048 a^3 c^3}-\frac{5 \tan ^{-1}(a x)^{3/2} \cos \left(4 \tan ^{-1}(a x)\right)}{256 a^3 c^3}","-\frac{15 \sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4096 a^3 c^3}+\frac{\tan ^{-1}(a x)^{7/2}}{28 a^3 c^3}-\frac{\tan ^{-1}(a x)^{5/2} \sin \left(4 \tan ^{-1}(a x)\right)}{32 a^3 c^3}+\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{2048 a^3 c^3}-\frac{5 \tan ^{-1}(a x)^{3/2} \cos \left(4 \tan ^{-1}(a x)\right)}{256 a^3 c^3}",1,"ArcTan[a*x]^(7/2)/(28*a^3*c^3) - (5*ArcTan[a*x]^(3/2)*Cos[4*ArcTan[a*x]])/(256*a^3*c^3) - (15*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a^3*c^3) + (15*Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(2048*a^3*c^3) - (ArcTan[a*x]^(5/2)*Sin[4*ArcTan[a*x]])/(32*a^3*c^3)","A",8,5,24,0.2083,1,"{4970, 4406, 3296, 3305, 3351}"
875,1,254,0,0.3376979,"\int \frac{x \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3,x]","-\frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4096 a^2 c^3}-\frac{15 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{256 a^2 c^3}-\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a c^3 \left(a^2 x^2+1\right)}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left(a^2 x^2+1\right)^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left(a^2 x^2+1\right)}+\frac{15 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^2 c^3}-\frac{225 \sqrt{\tan ^{-1}(a x)}}{2048 a^2 c^3}","-\frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4096 a^2 c^3}-\frac{15 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{256 a^2 c^3}-\frac{\tan ^{-1}(a x)^{5/2}}{4 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a c^3 \left(a^2 x^2+1\right)}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left(a^2 x^2+1\right)^2}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left(a^2 x^2+1\right)}+\frac{15 \sqrt{\tan ^{-1}(a x)}}{256 a^2 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}(a x)^{5/2}}{32 a^2 c^3}-\frac{225 \sqrt{\tan ^{-1}(a x)}}{2048 a^2 c^3}",1,"(-225*Sqrt[ArcTan[a*x]])/(2048*a^2*c^3) + (15*Sqrt[ArcTan[a*x]])/(256*a^2*c^3*(1 + a^2*x^2)^2) + (45*Sqrt[ArcTan[a*x]])/(256*a^2*c^3*(1 + a^2*x^2)) + (5*x*ArcTan[a*x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)^2) + (15*x*ArcTan[a*x]^(3/2))/(64*a*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(5/2))/(32*a^2*c^3) - ArcTan[a*x]^(5/2)/(4*a^2*c^3*(1 + a^2*x^2)^2) - (15*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a^2*c^3) - (15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(256*a^2*c^3)","A",16,7,22,0.3182,1,"{4930, 4900, 4892, 4904, 3312, 3304, 3352}"
876,1,296,0,0.3688571,"\int \frac{\tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^3,x]","\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 c^3 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)^{5/2}}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left(a^2 x^2+1\right)}+\frac{5 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left(a^2 x^2+1\right)^2}-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 c^3 \left(a^2 x^2+1\right)}+\frac{15 \sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4096 a c^3}+\frac{15 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{128 a c^3}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a c^3}-\frac{75 \tan ^{-1}(a x)^{3/2}}{256 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left(2 \tan ^{-1}(a x)\right)}{256 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{2048 a c^3}","\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 c^3 \left(a^2 x^2+1\right)}+\frac{x \tan ^{-1}(a x)^{5/2}}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left(a^2 x^2+1\right)}+\frac{5 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left(a^2 x^2+1\right)^2}-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 c^3 \left(a^2 x^2+1\right)}+\frac{15 \sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4096 a c^3}+\frac{15 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{128 a c^3}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a c^3}-\frac{75 \tan ^{-1}(a x)^{3/2}}{256 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left(2 \tan ^{-1}(a x)\right)}{256 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left(4 \tan ^{-1}(a x)\right)}{2048 a c^3}",1,"(-45*x*Sqrt[ArcTan[a*x]])/(128*c^3*(1 + a^2*x^2)) - (75*ArcTan[a*x]^(3/2))/(256*a*c^3) + (5*ArcTan[a*x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)^2) + (15*ArcTan[a*x]^(3/2))/(32*a*c^3*(1 + a^2*x^2)) + (x*ArcTan[a*x]^(5/2))/(4*c^3*(1 + a^2*x^2)^2) + (3*x*ArcTan[a*x]^(5/2))/(8*c^3*(1 + a^2*x^2)) + (3*ArcTan[a*x]^(7/2))/(28*a*c^3) + (15*Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4096*a*c^3) + (15*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(128*a*c^3) - (15*Sqrt[ArcTan[a*x]]*Sin[2*ArcTan[a*x]])/(256*a*c^3) - (15*Sqrt[ArcTan[a*x]]*Sin[4*ArcTan[a*x]])/(2048*a*c^3)","A",18,11,21,0.5238,1,"{4900, 4892, 4930, 4970, 4406, 12, 3305, 3351, 4904, 3312, 3296}"
877,0,0,0,0.0571301,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)^3} \, dx","Int[ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^3),x]","\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)^3} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{x \left(a^2 c x^2+c\right)^3},x\right)",0,"Defer[Int][ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^3), x]","A",0,0,0,0,-1,"{}"
878,0,0,0,0.0951104,"\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx","Int[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2),x]","\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx","\text{Int}\left(x^m \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2},x\right)",0,"Defer[Int][x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
879,0,0,0,0.1086933,"\int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx","Int[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2),x]","\int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx","\text{Int}\left(x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2},x\right)",0,"Defer[Int][x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
880,0,0,0,0.1787213,"\int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx","Int[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2),x]","\int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx","-\frac{5 c \text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)}{16 a}-\frac{5 c \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right)}{12 a}+\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{5/2}}{3 a^2 c}-\frac{5 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{12 a}+\frac{5 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{8 a^2}",0,"(5*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(8*a^2) - (5*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(12*a) + ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/(3*a^2*c) - (5*c*Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/(16*a) - (5*c*Defer[Int][ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/(12*a)","A",0,0,0,0,-1,"{}"
881,0,0,0,0.1032237,"\int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx","Int[Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2),x]","\int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx","\frac{15}{8} c \text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{a^2 c x^2+c}},x\right)+\frac{1}{2} c \text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{a^2 c x^2+c}},x\right)+\frac{1}{2} x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}-\frac{5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{4 a}",0,"(-5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(4*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/2 + (15*c*Defer[Int][Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/8 + (c*Defer[Int][ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x])/2","A",0,0,0,0,-1,"{}"
882,0,0,0,0.0940205,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{x} \, dx","Int[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/x,x]","\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{x} \, dx","\text{Int}\left(\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{x},x\right)",0,"Defer[Int][(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/x, x]","A",0,0,0,0,-1,"{}"
883,0,0,0,0.1069282,"\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2} \, dx","Int[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2),x]","\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{5/2},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
884,0,0,0,0.1162185,"\int x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2} \, dx","Int[x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2),x]","\int x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2} \, dx","\text{Int}\left(x^2 \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{5/2},x\right)",0,"Defer[Int][x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
885,0,0,0,0.2724724,"\int x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2} \, dx","Int[x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2),x]","\int x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2} \, dx","-\frac{9 c^2 \text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)}{64 a}-\frac{3 c^2 \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right)}{16 a}-\frac{c \text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{\tan ^{-1}(a x)}},x\right)}{32 a}+\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^{5/2}}{5 a^2 c}-\frac{x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}{8 a}-\frac{3 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{16 a}+\frac{\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}{16 a^2}+\frac{9 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{32 a^2}",0,"(9*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(32*a^2) + ((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(16*a^2) - (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(16*a) - (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(8*a) + ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/(5*a^2*c) - (9*c^2*Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/(64*a) - (c*Defer[Int][Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x])/(32*a) - (3*c^2*Defer[Int][ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/(16*a)","A",0,0,0,0,-1,"{}"
886,0,0,0,0.1793463,"\int \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2} \, dx","Int[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2),x]","\int \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2} \, dx","\frac{45}{32} c^2 \text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{a^2 c x^2+c}},x\right)+\frac{3}{8} c^2 \text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{a^2 c x^2+c}},x\right)+\frac{5}{16} c \text{Int}\left(\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)},x\right)+\frac{1}{4} x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{5/2}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}-\frac{5 \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}{24 a}-\frac{15 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{16 a}",0,"(-15*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(16*a) - (5*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(24*a) + (3*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/8 + (x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/4 + (45*c^2*Defer[Int][Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/32 + (5*c*Defer[Int][Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x])/16 + (3*c^2*Defer[Int][ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x])/8","A",0,0,0,0,-1,"{}"
887,0,0,0,0.1079888,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}}{x} \, dx","Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/x,x]","\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}}{x} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{5/2}}{x},x\right)",0,"Defer[Int][((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/x, x]","A",0,0,0,0,-1,"{}"
888,0,0,0,0.1082168,"\int x^m \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2} \, dx","Int[x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2),x]","\int x^m \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2} \, dx","\text{Int}\left(x^m \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^{5/2},x\right)",0,"Defer[Int][x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
889,0,0,0,0.1139453,"\int x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2} \, dx","Int[x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2),x]","\int x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2} \, dx","\text{Int}\left(x^2 \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^{5/2},x\right)",0,"Defer[Int][x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
890,0,0,0,0.3635451,"\int x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2} \, dx","Int[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2),x]","\int x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2} \, dx","-\frac{75 c^3 \text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)}{896 a}-\frac{25 c^2 \text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{\tan ^{-1}(a x)}},x\right)}{1344 a}-\frac{25 c^3 \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right)}{224 a}-\frac{c \text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}},x\right)}{112 a}-\frac{25 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{224 a}+\frac{75 c^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{448 a^2}+\frac{\left(a^2 c x^2+c\right)^{7/2} \tan ^{-1}(a x)^{5/2}}{7 a^2 c}-\frac{5 x \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^{3/2}}{84 a}+\frac{\left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}}{56 a^2}-\frac{25 c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}{336 a}+\frac{25 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}{672 a^2}",0,"(75*c^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(448*a^2) + (25*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])/(672*a^2) + ((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]])/(56*a^2) - (25*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(224*a) - (25*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(336*a) - (5*x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/(84*a) + ((c + a^2*c*x^2)^(7/2)*ArcTan[a*x]^(5/2))/(7*a^2*c) - (75*c^3*Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/(896*a) - (25*c^2*Defer[Int][Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x])/(1344*a) - (c*Defer[Int][(c + a^2*c*x^2)^(3/2)/Sqrt[ArcTan[a*x]], x])/(112*a) - (25*c^3*Defer[Int][ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/(224*a)","A",0,0,0,0,-1,"{}"
891,0,0,0,0.2665225,"\int \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2} \, dx","Int[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2),x]","\int \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2} \, dx","\frac{75}{64} c^3 \text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{a^2 c x^2+c}},x\right)+\frac{5}{16} c^3 \text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{a^2 c x^2+c}},x\right)+\frac{25}{96} c^2 \text{Int}\left(\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)},x\right)+\frac{1}{8} c \text{Int}\left(\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)},x\right)+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}-\frac{25 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{32 a}+\frac{5}{24} c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{5/2}-\frac{25 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}{144 a}+\frac{1}{6} x \left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^{5/2}-\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^{3/2}}{12 a}",0,"(-25*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(32*a) - (25*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/(144*a) - ((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/(12*a) + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/16 + (5*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/24 + (x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/6 + (75*c^3*Defer[Int][Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/64 + (25*c^2*Defer[Int][Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x])/96 + (c*Defer[Int][(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x])/8 + (5*c^3*Defer[Int][ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x])/16","A",0,0,0,0,-1,"{}"
892,0,0,0,0.1089534,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}}{x} \, dx","Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/x,x]","\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}}{x} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^{5/2}}{x},x\right)",0,"Defer[Int][((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/x, x]","A",0,0,0,0,-1,"{}"
893,0,0,0,0.1022218,"\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^m*ArcTan[a*x]^(5/2))/Sqrt[c + a^2*c*x^2],x]","\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^{5/2}}{\sqrt{a^2 c x^2+c}},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^(5/2))/Sqrt[c + a^2*c*x^2], x]","A",0,0,0,0,-1,"{}"
894,0,0,0,0.4720489,"\int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^3*ArcTan[a*x]^(5/2))/Sqrt[c + a^2*c*x^2],x]","\int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","-\frac{5 \text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)}{16 a^3}+\frac{25 \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right)}{12 a^3}+\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{3 a^2 c}-\frac{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{3 a^4 c}-\frac{5 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{12 a^3 c}+\frac{5 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{8 a^4 c}",0,"(5*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(8*a^4*c) - (5*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(12*a^3*c) - (2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*a^4*c) + (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*a^2*c) - (5*Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/(16*a^3) + (25*Defer[Int][ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/(12*a^3)","A",0,0,0,0,-1,"{}"
895,0,0,0,0.2537952,"\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x^2*ArcTan[a*x]^(5/2))/Sqrt[c + a^2*c*x^2],x]","\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","\frac{15 \text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{a^2 c x^2+c}},x\right)}{8 a^2}-\frac{\text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{a^2 c x^2+c}},x\right)}{2 a^2}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{2 a^2 c}-\frac{5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{4 a^3 c}",0,"(-5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(4*a^3*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(2*a^2*c) + (15*Defer[Int][Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x])/(8*a^2) - Defer[Int][ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x]/(2*a^2)","A",0,0,0,0,-1,"{}"
896,0,0,0,0.1092675,"\int \frac{x \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","Int[(x*ArcTan[a*x]^(5/2))/Sqrt[c + a^2*c*x^2],x]","\int \frac{x \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{a^2 c}-\frac{5 \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right)}{2 a}",0,"(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(a^2*c) - (5*Defer[Int][ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x])/(2*a)","A",0,0,0,0,-1,"{}"
897,0,0,0,0.0349281,"\int \frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2],x]","\int \frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{a^2 c x^2+c}},x\right)",0,"Defer[Int][ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x]","A",0,0,0,0,-1,"{}"
898,0,0,0,0.1031262,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^(5/2)/(x*Sqrt[c + a^2*c*x^2]),x]","\int \frac{\tan ^{-1}(a x)^{5/2}}{x \sqrt{c+a^2 c x^2}} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{x \sqrt{a^2 c x^2+c}},x\right)",0,"Defer[Int][ArcTan[a*x]^(5/2)/(x*Sqrt[c + a^2*c*x^2]), x]","A",0,0,0,0,-1,"{}"
899,0,0,0,0.2101975,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x^2 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^(5/2)/(x^2*Sqrt[c + a^2*c*x^2]),x]","\int \frac{\tan ^{-1}(a x)^{5/2}}{x^2 \sqrt{c+a^2 c x^2}} \, dx","\frac{5}{2} a \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{a^2 c x^2+c}},x\right)-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{c x}",0,"-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(c*x)) + (5*a*Defer[Int][ArcTan[a*x]^(3/2)/(x*Sqrt[c + a^2*c*x^2]), x])/2","A",0,0,0,0,-1,"{}"
900,0,0,0,0.4232588,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^(5/2)/(x^3*Sqrt[c + a^2*c*x^2]),x]","\int \frac{\tan ^{-1}(a x)^{5/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx","\frac{15}{8} a^2 \text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{x \sqrt{a^2 c x^2+c}},x\right)-\frac{1}{2} a^2 \text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{x \sqrt{a^2 c x^2+c}},x\right)-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{2 c x^2}-\frac{5 a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{4 c x}",0,"(-5*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(4*c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(2*c*x^2) + (15*a^2*Defer[Int][Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x])/8 - (a^2*Defer[Int][ArcTan[a*x]^(5/2)/(x*Sqrt[c + a^2*c*x^2]), x])/2","A",0,0,0,0,-1,"{}"
901,0,0,0,0.7436083,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x^4 \sqrt{c+a^2 c x^2}} \, dx","Int[ArcTan[a*x]^(5/2)/(x^4*Sqrt[c + a^2*c*x^2]),x]","\int \frac{\tan ^{-1}(a x)^{5/2}}{x^4 \sqrt{c+a^2 c x^2}} \, dx","\frac{5}{16} a^3 \text{Int}\left(\frac{1}{x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{25}{12} a^3 \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{a^2 c x^2+c}},x\right)+\frac{2 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{3 c x}-\frac{5 a^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{8 c x}-\frac{5 a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{12 c x^2}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}}{3 c x^3}",0,"(-5*a^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])/(8*c*x) - (5*a*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/(12*c*x^2) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*c*x^3) + (2*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/(3*c*x) + (5*a^3*Defer[Int][1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/16 - (25*a^3*Defer[Int][ArcTan[a*x]^(3/2)/(x*Sqrt[c + a^2*c*x^2]), x])/12","A",0,0,0,0,-1,"{}"
902,0,0,0,0.1136731,"\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(3/2),x]","\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(3/2), x]","A",0,0,0,0,-1,"{}"
903,0,0,0,0.1215479,"\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(3/2),x]","\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^2 \tan ^{-1}(a x)^{5/2}}{\left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(3/2), x]","A",0,0,0,0,-1,"{}"
904,1,161,0,0.22684,"\int \frac{x \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(3/2),x]","-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4 a^2 c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^{5/2}}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt{a^2 c x^2+c}}+\frac{15 \sqrt{\tan ^{-1}(a x)}}{4 a^2 c \sqrt{a^2 c x^2+c}}","-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4 a^2 c \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^{5/2}}{a^2 c \sqrt{a^2 c x^2+c}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt{a^2 c x^2+c}}+\frac{15 \sqrt{\tan ^{-1}(a x)}}{4 a^2 c \sqrt{a^2 c x^2+c}}",1,"(15*Sqrt[ArcTan[a*x]])/(4*a^2*c*Sqrt[c + a^2*c*x^2]) + (5*x*ArcTan[a*x]^(3/2))/(2*a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^(5/2)/(a^2*c*Sqrt[c + a^2*c*x^2]) - (15*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*a^2*c*Sqrt[c + a^2*c*x^2])","A",6,6,24,0.2500,1,"{4930, 4898, 4905, 4904, 3304, 3352}"
905,1,155,0,0.1583849,"\int \frac{\tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^(3/2),x]","\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4 a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^{5/2}}{c \sqrt{a^2 c x^2+c}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt{a^2 c x^2+c}}-\frac{15 x \sqrt{\tan ^{-1}(a x)}}{4 c \sqrt{a^2 c x^2+c}}","\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4 a c \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^{5/2}}{c \sqrt{a^2 c x^2+c}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{2 a c \sqrt{a^2 c x^2+c}}-\frac{15 x \sqrt{\tan ^{-1}(a x)}}{4 c \sqrt{a^2 c x^2+c}}",1,"(-15*x*Sqrt[ArcTan[a*x]])/(4*c*Sqrt[c + a^2*c*x^2]) + (5*ArcTan[a*x]^(3/2))/(2*a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^(5/2))/(c*Sqrt[c + a^2*c*x^2]) + (15*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*a*c*Sqrt[c + a^2*c*x^2])","A",6,6,23,0.2609,1,"{4898, 4905, 4904, 3296, 3305, 3351}"
906,0,0,0,0.119166,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","Int[ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^(3/2)),x]","\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{x \left(a^2 c x^2+c\right)^{3/2}},x\right)",0,"Defer[Int][ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^(3/2)), x]","A",0,0,0,0,-1,"{}"
907,0,0,0,0.1166722,"\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2),x]","\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(a^2 c x^2+c\right)^{5/2}},x\right)",0,"Defer[Int][(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2), x]","A",0,0,0,0,-1,"{}"
908,0,0,0,0.1185268,"\int \frac{x^4 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^4*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2),x]","\int \frac{x^4 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^4 \tan ^{-1}(a x)^{5/2}}{\left(a^2 c x^2+c\right)^{5/2}},x\right)",0,"Defer[Int][(x^4*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2), x]","A",0,0,0,0,-1,"{}"
909,1,350,0,0.7080645,"\int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2),x]","-\frac{45 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{16 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{144 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)^{5/2}}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{16 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 \sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \cos \left(3 \tan ^{-1}(a x)\right)}{144 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 x^3 \tan ^{-1}(a x)^{3/2}}{18 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^{5/2}}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{45 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{16 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{144 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)^{5/2}}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{45 \sqrt{\tan ^{-1}(a x)}}{16 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 \sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \cos \left(3 \tan ^{-1}(a x)\right)}{144 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 x^3 \tan ^{-1}(a x)^{3/2}}{18 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^{5/2}}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}",1,"(45*Sqrt[ArcTan[a*x]])/(16*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (5*x^3*ArcTan[a*x]^(3/2))/(18*a*c*(c + a^2*c*x^2)^(3/2)) + (5*x*ArcTan[a*x]^(3/2))/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (x^2*ArcTan[a*x]^(5/2))/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (2*ArcTan[a*x]^(5/2))/(3*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (5*Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]]*Cos[3*ArcTan[a*x]])/(144*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (45*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(16*a^4*c^2*Sqrt[c + a^2*c*x^2]) + (5*Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(144*a^4*c^2*Sqrt[c + a^2*c*x^2])","A",17,11,26,0.4231,1,"{4940, 4930, 4898, 4905, 4904, 3304, 3352, 4971, 4970, 3312, 3296}"
910,1,295,0,0.7753871,"\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2),x]","\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{16 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 \sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{144 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 x \sqrt{\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{9 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^3 \tan ^{-1}(a x)^{5/2}}{3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{5 x^3 \sqrt{\tan ^{-1}(a x)}}{36 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{5 x^2 \tan ^{-1}(a x)^{3/2}}{18 a c \left(a^2 c x^2+c\right)^{3/2}}","\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{16 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 \sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{144 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 x \sqrt{\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{9 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^3 \tan ^{-1}(a x)^{5/2}}{3 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{5 x^3 \sqrt{\tan ^{-1}(a x)}}{36 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{5 x^2 \tan ^{-1}(a x)^{3/2}}{18 a c \left(a^2 c x^2+c\right)^{3/2}}",1,"(-5*x^3*Sqrt[ArcTan[a*x]])/(36*c*(c + a^2*c*x^2)^(3/2)) - (5*x*Sqrt[ArcTan[a*x]])/(6*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (5*x^2*ArcTan[a*x]^(3/2))/(18*a*c*(c + a^2*c*x^2)^(3/2)) + (5*ArcTan[a*x]^(3/2))/(9*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (x^3*ArcTan[a*x]^(5/2))/(3*c*(c + a^2*c*x^2)^(3/2)) + (15*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(16*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (5*Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(144*a^3*c^2*Sqrt[c + a^2*c*x^2])","A",16,11,26,0.4231,1,"{4944, 4940, 4930, 4905, 4904, 3296, 3305, 3351, 4971, 4970, 3312}"
911,1,293,0,0.4509787,"\int \frac{x \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2),x]","-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{16 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 \sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{144 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \sqrt{\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^{5/2}}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{18 a c \left(a^2 c x^2+c\right)^{3/2}}+\frac{5 \sqrt{\tan ^{-1}(a x)}}{36 a^2 c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{16 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 \sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{144 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{9 a c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \sqrt{\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{\tan ^{-1}(a x)^{5/2}}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{5 x \tan ^{-1}(a x)^{3/2}}{18 a c \left(a^2 c x^2+c\right)^{3/2}}+\frac{5 \sqrt{\tan ^{-1}(a x)}}{36 a^2 c \left(a^2 c x^2+c\right)^{3/2}}",1,"(5*Sqrt[ArcTan[a*x]])/(36*a^2*c*(c + a^2*c*x^2)^(3/2)) + (5*Sqrt[ArcTan[a*x]])/(6*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (5*x*ArcTan[a*x]^(3/2))/(18*a*c*(c + a^2*c*x^2)^(3/2)) + (5*x*ArcTan[a*x]^(3/2))/(9*a*c^2*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^(5/2)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)) - (15*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(16*a^2*c^2*Sqrt[c + a^2*c*x^2]) - (5*Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(144*a^2*c^2*Sqrt[c + a^2*c*x^2])","A",15,8,24,0.3333,1,"{4930, 4900, 4898, 4905, 4904, 3304, 3352, 3312}"
912,1,337,0,0.4140088,"\int \frac{\tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^(5/2),x]","\frac{45 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{16 a c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{144 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^{5/2}}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{3 a c^2 \sqrt{a^2 c x^2+c}}-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{16 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 \sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \sin \left(3 \tan ^{-1}(a x)\right)}{144 a c^2 \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^{5/2}}{3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{18 a c \left(a^2 c x^2+c\right)^{3/2}}","\frac{45 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{16 a c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{144 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \tan ^{-1}(a x)^{5/2}}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{3 a c^2 \sqrt{a^2 c x^2+c}}-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{16 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 \sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)} \sin \left(3 \tan ^{-1}(a x)\right)}{144 a c^2 \sqrt{a^2 c x^2+c}}+\frac{x \tan ^{-1}(a x)^{5/2}}{3 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{5 \tan ^{-1}(a x)^{3/2}}{18 a c \left(a^2 c x^2+c\right)^{3/2}}",1,"(-45*x*Sqrt[ArcTan[a*x]])/(16*c^2*Sqrt[c + a^2*c*x^2]) + (5*ArcTan[a*x]^(3/2))/(18*a*c*(c + a^2*c*x^2)^(3/2)) + (5*ArcTan[a*x]^(3/2))/(3*a*c^2*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^(5/2))/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x*ArcTan[a*x]^(5/2))/(3*c^2*Sqrt[c + a^2*c*x^2]) + (45*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(16*a*c^2*Sqrt[c + a^2*c*x^2]) + (5*Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(144*a*c^2*Sqrt[c + a^2*c*x^2]) - (5*Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]]*Sin[3*ArcTan[a*x]])/(144*a*c^2*Sqrt[c + a^2*c*x^2])","A",17,8,23,0.3478,1,"{4900, 4898, 4905, 4904, 3296, 3305, 3351, 3312}"
913,0,0,0,0.1185602,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","Int[ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^(5/2)),x]","\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{\tan ^{-1}(a x)^{5/2}}{x \left(a^2 c x^2+c\right)^{5/2}},x\right)",0,"Defer[Int][ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^(5/2)), x]","A",0,0,0,0,-1,"{}"
914,0,0,0,0.034926,"\int \frac{x^m \left(c+a^2 c x^2\right)}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(x^m*(c + a^2*c*x^2))/Sqrt[ArcTan[a*x]],x]","\int \frac{x^m \left(c+a^2 c x^2\right)}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2))/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
915,0,0,0,0.0230053,"\int \frac{x \left(c+a^2 c x^2\right)}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(x*(c + a^2*c*x^2))/Sqrt[ArcTan[a*x]],x]","\int \frac{x \left(c+a^2 c x^2\right)}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2))/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
916,0,0,0,0.0122409,"\int \frac{c+a^2 c x^2}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]],x]","\int \frac{c+a^2 c x^2}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{a^2 c x^2+c}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
917,0,0,0,0.0322063,"\int \frac{c+a^2 c x^2}{x \sqrt{\tan ^{-1}(a x)}} \, dx","Int[(c + a^2*c*x^2)/(x*Sqrt[ArcTan[a*x]]),x]","\int \frac{c+a^2 c x^2}{x \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{a^2 c x^2+c}{x \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)/(x*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
918,0,0,0,0.0553061,"\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(x^m*(c + a^2*c*x^2)^2)/Sqrt[ArcTan[a*x]],x]","\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^2}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^2)/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
919,0,0,0,0.0387091,"\int \frac{x \left(c+a^2 c x^2\right)^2}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(x*(c + a^2*c*x^2)^2)/Sqrt[ArcTan[a*x]],x]","\int \frac{x \left(c+a^2 c x^2\right)^2}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^2}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^2)/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
920,0,0,0,0.0219361,"\int \frac{\left(c+a^2 c x^2\right)^2}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]],x]","\int \frac{\left(c+a^2 c x^2\right)^2}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
921,0,0,0,0.0501793,"\int \frac{\left(c+a^2 c x^2\right)^2}{x \sqrt{\tan ^{-1}(a x)}} \, dx","Int[(c + a^2*c*x^2)^2/(x*Sqrt[ArcTan[a*x]]),x]","\int \frac{\left(c+a^2 c x^2\right)^2}{x \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{x \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^2/(x*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
922,0,0,0,0.0557871,"\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(x^m*(c + a^2*c*x^2)^3)/Sqrt[ArcTan[a*x]],x]","\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^3}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^3)/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
923,0,0,0,0.0365296,"\int \frac{x \left(c+a^2 c x^2\right)^3}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(x*(c + a^2*c*x^2)^3)/Sqrt[ArcTan[a*x]],x]","\int \frac{x \left(c+a^2 c x^2\right)^3}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^3}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^3)/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
924,0,0,0,0.0216785,"\int \frac{\left(c+a^2 c x^2\right)^3}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(c + a^2*c*x^2)^3/Sqrt[ArcTan[a*x]],x]","\int \frac{\left(c+a^2 c x^2\right)^3}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^3/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
925,0,0,0,0.0515009,"\int \frac{\left(c+a^2 c x^2\right)^3}{x \sqrt{\tan ^{-1}(a x)}} \, dx","Int[(c + a^2*c*x^2)^3/(x*Sqrt[ArcTan[a*x]]),x]","\int \frac{\left(c+a^2 c x^2\right)^3}{x \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3}{x \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^3/(x*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
926,0,0,0,0.0677465,"\int \frac{x^m}{\left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^m/((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right) \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
927,0,0,0,0.0490175,"\int \frac{x}{\left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x/((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]),x]","\int \frac{x}{\left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)}} \, dx","\frac{2 x \sqrt{\tan ^{-1}(a x)}}{a c}-\frac{2 \text{Int}\left(\sqrt{\tan ^{-1}(a x)},x\right)}{a c}",0,"(2*x*Sqrt[ArcTan[a*x]])/(a*c) - (2*Defer[Int][Sqrt[ArcTan[a*x]], x])/(a*c)","A",0,0,0,0,-1,"{}"
928,1,16,0,0.0243045,"\int \frac{1}{\left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)}} \, dx","Int[1/((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]),x]","\frac{2 \sqrt{\tan ^{-1}(a x)}}{a c}","\frac{2 \sqrt{\tan ^{-1}(a x)}}{a c}",1,"(2*Sqrt[ArcTan[a*x]])/(a*c)","A",1,1,21,0.04762,1,"{4884}"
929,0,0,0,0.0645273,"\int \frac{1}{x \left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)}} \, dx","Int[1/(x*(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]),x]","\int \frac{1}{x \left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right) \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][1/(x*(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
930,0,0,0,0.0671878,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^m/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
931,0,0,0,0.0656968,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^3/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]),x]","\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^3}{\left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][x^3/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
932,1,47,0,0.1067109,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^2/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{\tan ^{-1}(a x)}}{a^3 c^2}-\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a^3 c^2}","\frac{\sqrt{\tan ^{-1}(a x)}}{a^3 c^2}-\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a^3 c^2}",1,"Sqrt[ArcTan[a*x]]/(a^3*c^2) - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(2*a^3*c^2)","A",5,4,24,0.1667,1,"{4970, 3312, 3304, 3352}"
933,1,31,0,0.0744631,"\int \frac{x}{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a^2 c^2}","\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a^2 c^2}",1,"(Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(2*a^2*c^2)","A",5,5,22,0.2273,1,"{4970, 4406, 12, 3305, 3351}"
934,1,47,0,0.0695104,"\int \frac{1}{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","Int[1/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a c^2}+\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2}","\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a c^2}+\frac{\sqrt{\tan ^{-1}(a x)}}{a c^2}",1,"Sqrt[ArcTan[a*x]]/(a*c^2) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(2*a*c^2)","A",5,4,21,0.1905,1,"{4904, 3312, 3304, 3352}"
935,0,0,0,0.0595416,"\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","Int[1/(x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][1/(x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
936,0,0,0,0.0635359,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^m/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
937,0,0,0,0.0633327,"\int \frac{x^5}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^5/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]","\int \frac{x^5}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^5}{\left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][x^5/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
938,1,89,0,0.1373624,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^4/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^5 c^3}-\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a^5 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{4 a^5 c^3}","\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^5 c^3}-\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a^5 c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{4 a^5 c^3}",1,"(3*Sqrt[ArcTan[a*x]])/(4*a^5*c^3) + (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a^5*c^3) - (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(2*a^5*c^3)","A",7,4,24,0.1667,1,"{4970, 3312, 3304, 3352}"
939,1,71,0,0.1281904,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^3/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{4 a^4 c^3}-\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^4 c^3}","\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{4 a^4 c^3}-\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^4 c^3}",1,"-(Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a^4*c^3) + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(4*a^4*c^3)","A",7,4,24,0.1667,1,"{4970, 4406, 3305, 3351}"
940,1,58,0,0.1189236,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^2/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{\tan ^{-1}(a x)}}{4 a^3 c^3}-\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^3 c^3}","\frac{\sqrt{\tan ^{-1}(a x)}}{4 a^3 c^3}-\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^3 c^3}",1,"Sqrt[ArcTan[a*x]]/(4*a^3*c^3) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a^3*c^3)","A",5,4,24,0.1667,1,"{4970, 4406, 3304, 3352}"
941,1,71,0,0.1061917,"\int \frac{x}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^2 c^3}+\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{4 a^2 c^3}","\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^2 c^3}+\frac{\sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{4 a^2 c^3}",1,"(Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a^2*c^3) + (Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(4*a^2*c^3)","A",7,4,22,0.1818,1,"{4970, 4406, 3305, 3351}"
942,1,89,0,0.0955638,"\int \frac{1}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Int[1/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a c^3}+\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{4 a c^3}","\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a c^3}+\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a c^3}+\frac{3 \sqrt{\tan ^{-1}(a x)}}{4 a c^3}",1,"(3*Sqrt[ArcTan[a*x]])/(4*a*c^3) + (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a*c^3) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(2*a*c^3)","A",7,4,21,0.1905,1,"{4904, 3312, 3304, 3352}"
943,0,0,0,0.0594507,"\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Int[1/(x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][1/(x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
944,0,0,0,0.094579,"\int \frac{x^m \sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(x^m*Sqrt[c + a^2*c*x^2])/Sqrt[ArcTan[a*x]],x]","\int \frac{x^m \sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^m \sqrt{a^2 c x^2+c}}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(x^m*Sqrt[c + a^2*c*x^2])/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
945,0,0,0,0.0667061,"\int \frac{x \sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(x*Sqrt[c + a^2*c*x^2])/Sqrt[ArcTan[a*x]],x]","\int \frac{x \sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x \sqrt{a^2 c x^2+c}}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(x*Sqrt[c + a^2*c*x^2])/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
946,0,0,0,0.0338757,"\int \frac{\sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]],x]","\int \frac{\sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
947,0,0,0,0.0972284,"\int \frac{\sqrt{c+a^2 c x^2}}{x \sqrt{\tan ^{-1}(a x)}} \, dx","Int[Sqrt[c + a^2*c*x^2]/(x*Sqrt[ArcTan[a*x]]),x]","\int \frac{\sqrt{c+a^2 c x^2}}{x \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{x \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][Sqrt[c + a^2*c*x^2]/(x*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
948,0,0,0,0.1085271,"\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(x^m*(c + a^2*c*x^2)^(3/2))/Sqrt[ArcTan[a*x]],x]","\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^(3/2))/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
949,0,0,0,0.08012,"\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(x*(c + a^2*c*x^2)^(3/2))/Sqrt[ArcTan[a*x]],x]","\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^(3/2))/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
950,0,0,0,0.0363264,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(c + a^2*c*x^2)^(3/2)/Sqrt[ArcTan[a*x]],x]","\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(3/2)/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
951,0,0,0,0.1098241,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \sqrt{\tan ^{-1}(a x)}} \, dx","Int[(c + a^2*c*x^2)^(3/2)/(x*Sqrt[ArcTan[a*x]]),x]","\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{x \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(3/2)/(x*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
952,0,0,0,0.1111949,"\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(x^m*(c + a^2*c*x^2)^(5/2))/Sqrt[ArcTan[a*x]],x]","\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^{5/2}}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^(5/2))/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
953,0,0,0,0.0778781,"\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(x*(c + a^2*c*x^2)^(5/2))/Sqrt[ArcTan[a*x]],x]","\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^{5/2}}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^(5/2))/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
954,0,0,0,0.0372537,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Int[(c + a^2*c*x^2)^(5/2)/Sqrt[ArcTan[a*x]],x]","\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2}}{\sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(5/2)/Sqrt[ArcTan[a*x]], x]","A",0,0,0,0,-1,"{}"
955,0,0,0,0.1200957,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \sqrt{\tan ^{-1}(a x)}} \, dx","Int[(c + a^2*c*x^2)^(5/2)/(x*Sqrt[ArcTan[a*x]]),x]","\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2}}{x \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(5/2)/(x*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
956,0,0,0,0.1056529,"\int \frac{x^m}{\sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^m/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]),x]","\int \frac{x^m}{\sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^m}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][x^m/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
957,0,0,0,0.070693,"\int \frac{x}{\sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]),x]","\int \frac{x}{\sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][x/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
958,0,0,0,0.0349825,"\int \frac{1}{\sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]),x]","\int \frac{1}{\sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
959,0,0,0,0.1067161,"\int \frac{1}{x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]),x]","\int \frac{1}{x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
960,0,0,0,0.1156961,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^m/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
961,0,0,0,0.1183221,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^2/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]),x]","\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^2}{\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][x^2/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
962,1,60,0,0.1648097,"\int \frac{x}{\left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c \sqrt{a^2 c x^2+c}}","\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c \sqrt{a^2 c x^2+c}}",1,"(Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c*Sqrt[c + a^2*c*x^2])","A",4,4,24,0.1667,1,"{4971, 4970, 3305, 3351}"
963,1,60,0,0.0931624,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[1/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c \sqrt{a^2 c x^2+c}}","\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c \sqrt{a^2 c x^2+c}}",1,"(Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a*c*Sqrt[c + a^2*c*x^2])","A",4,4,23,0.1739,1,"{4905, 4904, 3304, 3352}"
964,0,0,0,0.1168049,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[1/(x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][1/(x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
965,0,0,0,0.1179162,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^m/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
966,0,0,0,0.1204154,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^4/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]),x]","\int \frac{x^4}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{x^4}{\left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][x^4/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
967,1,131,0,0.2914518,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^3/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]),x]","\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^4 c^2 \sqrt{a^2 c x^2+c}}","\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^4 c^2 \sqrt{a^2 c x^2+c}}",1,"(3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(2*a^4*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(2*a^4*c^2*Sqrt[c + a^2*c*x^2])","A",8,5,26,0.1923,1,"{4971, 4970, 3312, 3305, 3351}"
968,1,131,0,0.2962791,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x^2/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^3 c^2 \sqrt{a^2 c x^2+c}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^3 c^2 \sqrt{a^2 c x^2+c}}",1,"(Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(2*a^3*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(2*a^3*c^2*Sqrt[c + a^2*c*x^2])","A",8,5,26,0.1923,1,"{4971, 4970, 4406, 3304, 3352}"
969,1,131,0,0.2141888,"\int \frac{x}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[x/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^2 c^2 \sqrt{a^2 c x^2+c}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^2 c^2 \sqrt{a^2 c x^2+c}}",1,"(Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(2*a^2*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(2*a^2*c^2*Sqrt[c + a^2*c*x^2])","A",8,5,24,0.2083,1,"{4971, 4970, 4406, 3305, 3351}"
970,1,131,0,0.146135,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[1/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]),x]","\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a c^2 \sqrt{a^2 c x^2+c}}","\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a c^2 \sqrt{a^2 c x^2+c}}",1,"(3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(2*a*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(2*a*c^2*Sqrt[c + a^2*c*x^2])","A",8,5,23,0.2174,1,"{4905, 4904, 3312, 3304, 3352}"
971,0,0,0,0.1179574,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Int[1/(x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)",0,"Defer[Int][1/(x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
972,0,0,0,0.0351689,"\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^(3/2),x]","\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
973,0,0,0,0.0229703,"\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(x*(c + a^2*c*x^2))/ArcTan[a*x]^(3/2),x]","\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2))/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
974,0,0,0,0.0120097,"\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(c + a^2*c*x^2)/ArcTan[a*x]^(3/2),x]","\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{a^2 c x^2+c}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
975,0,0,0,0.0318866,"\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)^{3/2}} \, dx","Int[(c + a^2*c*x^2)/(x*ArcTan[a*x]^(3/2)),x]","\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{a^2 c x^2+c}{x \tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)/(x*ArcTan[a*x]^(3/2)), x]","A",0,0,0,0,-1,"{}"
976,0,0,0,0.0540973,"\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(3/2),x]","\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
977,0,0,0,0.0360699,"\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(3/2),x]","\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
978,0,0,0,0.0218943,"\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(c + a^2*c*x^2)^2/ArcTan[a*x]^(3/2),x]","\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^2/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
979,0,0,0,0.048979,"\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)^{3/2}} \, dx","Int[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^(3/2)),x]","\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{x \tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^(3/2)), x]","A",0,0,0,0,-1,"{}"
980,0,0,0,0.055548,"\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(3/2),x]","\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
981,0,0,0,0.0366894,"\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(3/2),x]","\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
982,0,0,0,0.02225,"\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(c + a^2*c*x^2)^3/ArcTan[a*x]^(3/2),x]","\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^3/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
983,0,0,0,0.0497706,"\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)^{3/2}} \, dx","Int[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^(3/2)),x]","\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3}{x \tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^(3/2)), x]","A",0,0,0,0,-1,"{}"
984,0,0,0,0.0805591,"\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^m/((c + a^2*c*x^2)*ArcTan[a*x]^(3/2)),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}} \, dx","\frac{2 m \text{Int}\left(\frac{x^{m-1}}{\sqrt{\tan ^{-1}(a x)}},x\right)}{a c}-\frac{2 x^m}{a c \sqrt{\tan ^{-1}(a x)}}",0,"(-2*x^m)/(a*c*Sqrt[ArcTan[a*x]]) + (2*m*Defer[Int][x^(-1 + m)/Sqrt[ArcTan[a*x]], x])/(a*c)","A",0,0,0,0,-1,"{}"
985,0,0,0,0.0480899,"\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}} \, dx","Int[x/((c + a^2*c*x^2)*ArcTan[a*x]^(3/2)),x]","\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}} \, dx","\frac{2 \text{Int}\left(\frac{1}{\sqrt{\tan ^{-1}(a x)}},x\right)}{a c}-\frac{2 x}{a c \sqrt{\tan ^{-1}(a x)}}",0,"(-2*x)/(a*c*Sqrt[ArcTan[a*x]]) + (2*Defer[Int][1/Sqrt[ArcTan[a*x]], x])/(a*c)","A",0,0,0,0,-1,"{}"
986,1,16,0,0.0249069,"\int \frac{1}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/((c + a^2*c*x^2)*ArcTan[a*x]^(3/2)),x]","-\frac{2}{a c \sqrt{\tan ^{-1}(a x)}}","-\frac{2}{a c \sqrt{\tan ^{-1}(a x)}}",1,"-2/(a*c*Sqrt[ArcTan[a*x]])","A",1,1,21,0.04762,1,"{4884}"
987,0,0,0,0.0764422,"\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^2 \sqrt{\tan ^{-1}(a x)}},x\right)}{a c}-\frac{2}{a c x \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c*x*Sqrt[ArcTan[a*x]]) - (2*Defer[Int][1/(x^2*Sqrt[ArcTan[a*x]]), x])/(a*c)","A",0,0,0,0,-1,"{}"
988,0,0,0,0.0623191,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x]","A",0,0,0,0,-1,"{}"
989,0,0,0,0.1988461,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^4/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)),x]","\int \frac{x^4}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","\frac{8 \text{Int}\left(\frac{x^3}{\left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)}{a}+4 a \text{Int}\left(\frac{x^5}{\left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{2 x^4}{a c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}",0,"(-2*x^4)/(a*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (8*Defer[Int][x^3/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a + 4*a*Defer[Int][x^5/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
990,0,0,0,0.2463229,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)),x]","\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","2 a \text{Int}\left(\frac{x^4}{\left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{3 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^4 c^2}-\frac{2 x^3}{a c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}+\frac{6 \sqrt{\tan ^{-1}(a x)}}{a^4 c^2}",0,"(-2*x^3)/(a*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (6*Sqrt[ArcTan[a*x]])/(a^4*c^2) - (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a^4*c^2) + 2*a*Defer[Int][x^4/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
991,1,60,0,0.1449058,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)),x]","\frac{2 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^3 c^2}-\frac{2 x^2}{a c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}","\frac{2 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^3 c^2}-\frac{2 x^2}{a c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x^2)/(a*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (2*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a^3*c^2)","A",6,6,24,0.2500,1,"{4942, 4970, 4406, 12, 3305, 3351}"
992,1,138,0,0.1695855,"\int \frac{x}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Int[x/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)),x]","\frac{2 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^2 c^2}-\frac{2 x}{a c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}+\frac{4 \left(1-a^2 x^2\right) \sqrt{\tan ^{-1}(a x)}}{a^2 c^2 \left(a^2 x^2+1\right)}-\frac{8 \sqrt{\tan ^{-1}(a x)}}{a^2 c^2 \left(a^2 x^2+1\right)}+\frac{4 \sqrt{\tan ^{-1}(a x)}}{a^2 c^2}","\frac{2 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^2 c^2}-\frac{2 x}{a c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}+\frac{4 \left(1-a^2 x^2\right) \sqrt{\tan ^{-1}(a x)}}{a^2 c^2 \left(a^2 x^2+1\right)}-\frac{8 \sqrt{\tan ^{-1}(a x)}}{a^2 c^2 \left(a^2 x^2+1\right)}+\frac{4 \sqrt{\tan ^{-1}(a x)}}{a^2 c^2}",1,"(-2*x)/(a*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (4*Sqrt[ArcTan[a*x]])/(a^2*c^2) - (8*Sqrt[ArcTan[a*x]])/(a^2*c^2*(1 + a^2*x^2)) + (4*(1 - a^2*x^2)*Sqrt[ArcTan[a*x]])/(a^2*c^2*(1 + a^2*x^2)) + (2*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a^2*c^2)","A",7,6,22,0.2727,1,"{4932, 4930, 4904, 3312, 3304, 3352}"
993,1,57,0,0.1035879,"\int \frac{1}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)),x]","-\frac{2}{a c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}-\frac{2 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a c^2}","-\frac{2}{a c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}-\frac{2 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a c^2}",1,"-2/(a*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (2*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a*c^2)","A",6,6,21,0.2857,1,"{4902, 4970, 4406, 12, 3305, 3351}"
994,0,0,0,0.1991163,"\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-\frac{2}{a c^2 x \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}-\frac{3 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{c^2}-\frac{6 \sqrt{\tan ^{-1}(a x)}}{c^2}",0,"-2/(a*c^2*x*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (6*Sqrt[ArcTan[a*x]])/c^2 - (3*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^2 - (2*Defer[Int][1/(x^2*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a","A",0,0,0,0,-1,"{}"
995,0,0,0,0.1928707,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{4 \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-8 a \text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{2}{a c^2 x^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c^2*x^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (4*Defer[Int][1/(x^3*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a - 8*a*Defer[Int][1/(x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
996,0,0,0,0.1987583,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{6 \text{Int}\left(\frac{1}{x^4 \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-10 a \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{2}{a c^2 x^3 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c^2*x^3*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (6*Defer[Int][1/(x^4*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a - 10*a*Defer[Int][1/(x^2*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
997,0,0,0,0.2014753,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{8 \text{Int}\left(\frac{1}{x^5 \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-12 a \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{2}{a c^2 x^4 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c^2*x^4*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (8*Defer[Int][1/(x^5*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a - 12*a*Defer[Int][1/(x^3*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
998,0,0,0,0.0633806,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x]","A",0,0,0,0,-1,"{}"
999,1,96,0,0.3340846,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]","-\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^4 c^3}+\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^4 c^3}-\frac{2 x^3}{a c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}","-\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^4 c^3}+\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^4 c^3}-\frac{2 x^3}{a c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x^3)/(a*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^4*c^3) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a^4*c^3)","A",13,6,24,0.2500,1,"{4968, 4970, 3312, 3304, 3352, 4406}"
1000,1,67,0,0.313015,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]","\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3 c^3}-\frac{2 x^2}{a c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}","\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3 c^3}-\frac{2 x^2}{a c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x^2)/(a*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^3*c^3)","A",15,5,24,0.2083,1,"{4968, 4970, 4406, 3305, 3351}"
1001,1,93,0,0.2712726,"\int \frac{x}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Int[x/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]","\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c^3}+\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^2 c^3}-\frac{2 x}{a c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}","\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c^3}+\frac{\sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^2 c^3}-\frac{2 x}{a c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x)/(a*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c^3) + (Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a^2*c^3)","A",13,7,22,0.3182,1,"{4968, 4970, 4406, 3304, 3352, 4904, 3312}"
1002,1,94,0,0.1376984,"\int \frac{1}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]","-\frac{2}{a c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c^3}-\frac{2 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a c^3}","-\frac{2}{a c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{\sqrt{\frac{\pi }{2}} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c^3}-\frac{2 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a c^3}",1,"-2/(a*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (Sqrt[Pi/2]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a*c^3) - (2*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a*c^3)","A",8,5,21,0.2381,1,"{4902, 4970, 4406, 3305, 3351}"
1003,0,0,0,0.2327383,"\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-\frac{2}{a c^3 x \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{5 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4 c^3}-\frac{5 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)}}{2 c^3}",0,"-2/(a*c^3*x*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (15*Sqrt[ArcTan[a*x]])/(2*c^3) - (5*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(4*c^3) - (5*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^3 - (2*Defer[Int][1/(x^2*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a","A",0,0,0,0,-1,"{}"
1004,0,0,0,0.1951143,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{4 \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-12 a \text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{2}{a c^3 x^2 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c^3*x^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (4*Defer[Int][1/(x^3*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a - 12*a*Defer[Int][1/(x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1005,0,0,0,0.2172304,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{6 \text{Int}\left(\frac{1}{x^4 \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-14 a \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{2}{a c^3 x^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c^3*x^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (6*Defer[Int][1/(x^4*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a - 14*a*Defer[Int][1/(x^2*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1006,0,0,0,0.2036167,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{8 \text{Int}\left(\frac{1}{x^5 \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-16 a \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{2}{a c^3 x^4 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c^3*x^4*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (8*Defer[Int][1/(x^5*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a - 16*a*Defer[Int][1/(x^3*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1007,0,0,0,0.0983858,"\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(3/2),x]","\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
1008,0,0,0,0.0696731,"\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(3/2),x]","\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x \sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
1009,0,0,0,0.0362968,"\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^(3/2),x]","\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
1010,0,0,0,0.1003496,"\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^{3/2}} \, dx","Int[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^(3/2)),x]","\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{x \tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^(3/2)), x]","A",0,0,0,0,-1,"{}"
1011,0,0,0,0.1122263,"\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(3/2),x]","\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
1012,0,0,0,0.0773068,"\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(3/2),x]","\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
1013,0,0,0,0.035829,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^(3/2),x]","\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
1014,0,0,0,0.111514,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)^{3/2}} \, dx","Int[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^(3/2)),x]","\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{x \tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^(3/2)), x]","A",0,0,0,0,-1,"{}"
1015,0,0,0,0.1115333,"\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(3/2),x]","\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
1016,0,0,0,0.0785105,"\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(3/2),x]","\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
1017,0,0,0,0.0377011,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Int[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^(3/2),x]","\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^(3/2), x]","A",0,0,0,0,-1,"{}"
1018,0,0,0,0.1132425,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)^{3/2}} \, dx","Int[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^(3/2)),x]","\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2}}{x \tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^(3/2)), x]","A",0,0,0,0,-1,"{}"
1019,0,0,0,0.1076099,"\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)),x]","\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x^m}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x]","A",0,0,0,0,-1,"{}"
1020,0,0,0,0.0722918,"\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)),x]","\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x]","A",0,0,0,0,-1,"{}"
1021,0,0,0,0.0350177,"\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x]","A",0,0,0,0,-1,"{}"
1022,0,0,0,0.2109742,"\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-\frac{2 \sqrt{a^2 c x^2+c}}{a c x \sqrt{\tan ^{-1}(a x)}}",0,"(-2*Sqrt[c + a^2*c*x^2])/(a*c*x*Sqrt[ArcTan[a*x]]) - (2*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x])/a","A",0,0,0,0,-1,"{}"
1023,0,0,0,0.1053295,"\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x]","A",0,0,0,0,-1,"{}"
1024,0,0,0,0.1165322,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x]","A",0,0,0,0,-1,"{}"
1025,0,0,0,0.3732114,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)),x]","\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","\frac{6 \text{Int}\left(\frac{x^2}{\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{a}+4 a \text{Int}\left(\frac{x^4}{\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{2 x^3}{a c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",0,"(-2*x^3)/(a*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (6*Defer[Int][x^2/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a + 4*a*Defer[Int][x^4/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1026,0,0,0,0.4313616,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)),x]","\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","2 a \text{Int}\left(\frac{x^3}{\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{4 \sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3 c \sqrt{a^2 c x^2+c}}-\frac{2 x^2}{a c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",0,"(-2*x^2)/(a*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (4*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^3*c*Sqrt[c + a^2*c*x^2]) + 2*a*Defer[Int][x^3/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1027,1,93,0,0.1781843,"\int \frac{x}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)),x]","\frac{2 \sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{2 x}{a c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}","\frac{2 \sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c \sqrt{a^2 c x^2+c}}-\frac{2 x}{a c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x)/(a*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (2*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c*Sqrt[c + a^2*c*x^2])","A",5,5,24,0.2083,1,"{4942, 4905, 4904, 3304, 3352}"
1028,1,92,0,0.2127642,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)),x]","-\frac{2 \sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c \sqrt{a^2 c x^2+c}}-\frac{2}{a c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}","-\frac{2 \sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c \sqrt{a^2 c x^2+c}}-\frac{2}{a c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",1,"-2/(a*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (2*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a*c*Sqrt[c + a^2*c*x^2])","A",5,5,23,0.2174,1,"{4902, 4971, 4970, 3305, 3351}"
1029,0,0,0,0.3323131,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-\frac{4 \sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{2}{a c x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c*x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (4*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(c*Sqrt[c + a^2*c*x^2]) - (2*Defer[Int][1/(x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a","A",0,0,0,0,-1,"{}"
1030,0,0,0,0.3599097,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{4 \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-6 a \text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{2}{a c x^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c*x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (4*Defer[Int][1/(x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a - 6*a*Defer[Int][1/(x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1031,0,0,0,0.3652512,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{6 \text{Int}\left(\frac{1}{x^4 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-8 a \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{2}{a c x^3 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c*x^3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (6*Defer[Int][1/(x^4*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a - 8*a*Defer[Int][1/(x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1032,0,0,0,0.3630496,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{8 \text{Int}\left(\frac{1}{x^5 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-10 a \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{2}{a c x^4 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c*x^4*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (8*Defer[Int][1/(x^5*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a - 10*a*Defer[Int][1/(x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1033,0,0,0,0.1148765,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^{3/2}},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x]","A",0,0,0,0,-1,"{}"
1034,1,160,0,0.4335013,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]","\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{3 \pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x^3}{a c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}","\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{3 \pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x^3}{a c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x^3)/(a*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^4*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[(3*Pi)/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^4*c^2*Sqrt[c + a^2*c*x^2])","A",9,6,26,0.2308,1,"{4942, 4971, 4970, 4406, 3304, 3352}"
1035,1,281,0,0.6664943,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]","\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{2 \pi }{3}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x^2}{a c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}","\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{2 \pi }{3}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x^2}{a c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x^2)/(a*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^3*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^3*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^3*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[(2*Pi)/3]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^3*c^2*Sqrt[c + a^2*c*x^2])","A",17,7,26,0.2692,1,"{4968, 4971, 4970, 3312, 3305, 3351, 4406}"
1036,1,280,0,0.5464273,"\int \frac{x}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]","-\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{2 \pi }{3}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x}{a c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}","-\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{2 \pi }{3}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x}{a c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x)/(a*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[Pi/6]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[(2*Pi)/3]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c^2*Sqrt[c + a^2*c*x^2])","A",17,9,24,0.3750,1,"{4968, 4971, 4970, 4406, 3304, 3352, 4905, 4904, 3312}"
1037,1,157,0,0.2652721,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]","-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{3 \pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{a c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}","-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{\frac{3 \pi }{2}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{a c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",1,"-2/(a*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (3*Sqrt[Pi/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[(3*Pi)/2]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a*c^2*Sqrt[c + a^2*c*x^2])","A",9,6,23,0.2609,1,"{4902, 4971, 4970, 4406, 3305, 3351}"
1038,0,0,0,0.3968914,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-\frac{6 \sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{\frac{2 \pi }{3}} \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2}{a c x \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c*x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (6*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[(2*Pi)/3]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Defer[Int][1/(x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/a","A",0,0,0,0,-1,"{}"
1039,0,0,0,0.3692582,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{4 \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-10 a \text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{2}{a c x^2 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c*x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (4*Defer[Int][1/(x^3*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/a - 10*a*Defer[Int][1/(x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1040,0,0,0,0.364653,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{6 \text{Int}\left(\frac{1}{x^4 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-12 a \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{2}{a c x^3 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c*x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (6*Defer[Int][1/(x^4*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/a - 12*a*Defer[Int][1/(x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1041,0,0,0,0.3687551,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","-\frac{8 \text{Int}\left(\frac{1}{x^5 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-14 a \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)-\frac{2}{a c x^4 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",0,"-2/(a*c*x^4*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (8*Defer[Int][1/(x^5*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/a - 14*a*Defer[Int][1/(x^3*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1042,0,0,0,0.0360636,"\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^(5/2),x]","\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1043,0,0,0,0.0227916,"\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(x*(c + a^2*c*x^2))/ArcTan[a*x]^(5/2),x]","\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2))/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1044,0,0,0,0.0127875,"\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(c + a^2*c*x^2)/ArcTan[a*x]^(5/2),x]","\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{a^2 c x^2+c}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1045,0,0,0,0.0319057,"\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)^{5/2}} \, dx","Int[(c + a^2*c*x^2)/(x*ArcTan[a*x]^(5/2)),x]","\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{a^2 c x^2+c}{x \tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)/(x*ArcTan[a*x]^(5/2)), x]","A",0,0,0,0,-1,"{}"
1046,0,0,0,0.0556037,"\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(5/2),x]","\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1047,0,0,0,0.0372311,"\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(5/2),x]","\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1048,0,0,0,0.0220499,"\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(c + a^2*c*x^2)^2/ArcTan[a*x]^(5/2),x]","\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^2/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1049,0,0,0,0.0504415,"\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)^{5/2}} \, dx","Int[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^(5/2)),x]","\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^2}{x \tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^(5/2)), x]","A",0,0,0,0,-1,"{}"
1050,0,0,0,0.0564165,"\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(5/2),x]","\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1051,0,0,0,0.0366252,"\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(5/2),x]","\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1052,0,0,0,0.0217256,"\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(c + a^2*c*x^2)^3/ArcTan[a*x]^(5/2),x]","\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^3/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1053,0,0,0,0.0511725,"\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)^{5/2}} \, dx","Int[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^(5/2)),x]","\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^3}{x \tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^(5/2)), x]","A",0,0,0,0,-1,"{}"
1054,0,0,0,0.0817706,"\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}} \, dx","Int[x^m/((c + a^2*c*x^2)*ArcTan[a*x]^(5/2)),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}} \, dx","\frac{2 m \text{Int}\left(\frac{x^{m-1}}{\tan ^{-1}(a x)^{3/2}},x\right)}{3 a c}-\frac{2 x^m}{3 a c \tan ^{-1}(a x)^{3/2}}",0,"(-2*x^m)/(3*a*c*ArcTan[a*x]^(3/2)) + (2*m*Defer[Int][x^(-1 + m)/ArcTan[a*x]^(3/2), x])/(3*a*c)","A",0,0,0,0,-1,"{}"
1055,0,0,0,0.0488155,"\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}} \, dx","Int[x/((c + a^2*c*x^2)*ArcTan[a*x]^(5/2)),x]","\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}} \, dx","\frac{2 \text{Int}\left(\frac{1}{\tan ^{-1}(a x)^{3/2}},x\right)}{3 a c}-\frac{2 x}{3 a c \tan ^{-1}(a x)^{3/2}}",0,"(-2*x)/(3*a*c*ArcTan[a*x]^(3/2)) + (2*Defer[Int][ArcTan[a*x]^(-3/2), x])/(3*a*c)","A",0,0,0,0,-1,"{}"
1056,1,18,0,0.0243907,"\int \frac{1}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/((c + a^2*c*x^2)*ArcTan[a*x]^(5/2)),x]","-\frac{2}{3 a c \tan ^{-1}(a x)^{3/2}}","-\frac{2}{3 a c \tan ^{-1}(a x)^{3/2}}",1,"-2/(3*a*c*ArcTan[a*x]^(3/2))","A",1,1,21,0.04762,1,"{4884}"
1057,0,0,0,0.0772083,"\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)^{3/2}},x\right)}{3 a c}-\frac{2}{3 a c x \tan ^{-1}(a x)^{3/2}}",0,"-2/(3*a*c*x*ArcTan[a*x]^(3/2)) - (2*Defer[Int][1/(x^2*ArcTan[a*x]^(3/2)), x])/(3*a*c)","A",0,0,0,0,-1,"{}"
1058,0,0,0,0.063027,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Int[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^2 \tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x]","A",0,0,0,0,-1,"{}"
1059,0,0,0,0.4153365,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Int[x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]","\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","\frac{16}{3} \text{Int}\left(\frac{x^3}{\left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{8}{3} a^2 \text{Int}\left(\frac{x^5}{\left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{4 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^4 c^2}-\frac{4 x^4}{3 c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}-\frac{2 x^3}{3 a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}",0,"(-2*x^3)/(3*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) - (4*x^2)/(a^2*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (4*x^4)/(3*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a^4*c^2) + (16*Defer[Int][x^3/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/3 + (8*a^2*Defer[Int][x^5/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/3","A",0,0,0,0,-1,"{}"
1060,1,180,0,0.2631001,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Int[x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]","\frac{8 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a^3 c^2}-\frac{2 x^2}{3 a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}+\frac{16 \left(1-a^2 x^2\right) \sqrt{\tan ^{-1}(a x)}}{3 a^3 c^2 \left(a^2 x^2+1\right)}-\frac{32 \sqrt{\tan ^{-1}(a x)}}{3 a^3 c^2 \left(a^2 x^2+1\right)}+\frac{16 \sqrt{\tan ^{-1}(a x)}}{3 a^3 c^2}","\frac{8 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a^3 c^2}-\frac{2 x^2}{3 a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}+\frac{16 \left(1-a^2 x^2\right) \sqrt{\tan ^{-1}(a x)}}{3 a^3 c^2 \left(a^2 x^2+1\right)}-\frac{32 \sqrt{\tan ^{-1}(a x)}}{3 a^3 c^2 \left(a^2 x^2+1\right)}+\frac{16 \sqrt{\tan ^{-1}(a x)}}{3 a^3 c^2}",1,"(-2*x^2)/(3*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) - (8*x)/(3*a^2*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (16*Sqrt[ArcTan[a*x]])/(3*a^3*c^2) - (32*Sqrt[ArcTan[a*x]])/(3*a^3*c^2*(1 + a^2*x^2)) + (16*(1 - a^2*x^2)*Sqrt[ArcTan[a*x]])/(3*a^3*c^2*(1 + a^2*x^2)) + (8*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*a^3*c^2)","A",8,7,24,0.2917,1,"{4942, 4932, 4930, 4904, 3312, 3304, 3352}"
1061,1,101,0,0.1215473,"\int \frac{x}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Int[x/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]","-\frac{8 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a^2 c^2}-\frac{2 x}{3 a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}-\frac{4 \left(1-a^2 x^2\right)}{3 a^2 c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}","-\frac{8 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a^2 c^2}-\frac{2 x}{3 a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}-\frac{4 \left(1-a^2 x^2\right)}{3 a^2 c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x)/(3*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) - (4*(1 - a^2*x^2))/(3*a^2*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (8*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*a^2*c^2)","A",6,6,22,0.2727,1,"{4932, 4970, 4406, 12, 3305, 3351}"
1062,1,174,0,0.2060009,"\int \frac{1}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]","\frac{8 x}{3 c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}-\frac{16 \left(1-a^2 x^2\right) \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left(a^2 x^2+1\right)}+\frac{32 \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left(a^2 x^2+1\right)}-\frac{2}{3 a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a c^2}-\frac{16 \sqrt{\tan ^{-1}(a x)}}{3 a c^2}","\frac{8 x}{3 c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}-\frac{16 \left(1-a^2 x^2\right) \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left(a^2 x^2+1\right)}+\frac{32 \sqrt{\tan ^{-1}(a x)}}{3 a c^2 \left(a^2 x^2+1\right)}-\frac{2}{3 a c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}-\frac{8 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a c^2}-\frac{16 \sqrt{\tan ^{-1}(a x)}}{3 a c^2}",1,"-2/(3*a*c^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) + (8*x)/(3*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - (16*Sqrt[ArcTan[a*x]])/(3*a*c^2) + (32*Sqrt[ArcTan[a*x]])/(3*a*c^2*(1 + a^2*x^2)) - (16*(1 - a^2*x^2)*Sqrt[ArcTan[a*x]])/(3*a*c^2*(1 + a^2*x^2)) - (8*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*a*c^2)","A",8,7,21,0.3333,1,"{4902, 4932, 4930, 4904, 3312, 3304, 3352}"
1063,0,0,0,0.3604798,"\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","\frac{8 \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)}{3 a^2}+\frac{16}{3} \text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{4}{c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}+\frac{4}{3 a^2 c^2 x^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c^2 x \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}+\frac{4 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{c^2}",0,"-2/(3*a*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) + 4/(c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + 4/(3*a^2*c^2*x^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^2 + (8*Defer[Int][1/(x^3*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + (16*Defer[Int][1/(x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/3","A",0,0,0,0,-1,"{}"
1064,0,0,0,0.483022,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","\frac{56}{3} \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{8 \text{Int}\left(\frac{1}{x^4 \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)}{a^2}+\frac{16}{3 c^2 x \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c^2 x^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c^2 x^3 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}+\frac{8 \sqrt{\pi } a \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{c^2}+\frac{16 a \sqrt{\tan ^{-1}(a x)}}{c^2}",0,"-2/(3*a*c^2*x^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) + 8/(3*a^2*c^2*x^3*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + 16/(3*c^2*x*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (16*a*Sqrt[ArcTan[a*x]])/c^2 + (8*a*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^2 + (8*Defer[Int][1/(x^4*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a^2 + (56*Defer[Int][1/(x^2*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/3","A",0,0,0,0,-1,"{}"
1065,0,0,0,0.4639373,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","\frac{80}{3} a^2 \text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{112}{3} \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{16 \text{Int}\left(\frac{1}{x^5 \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)}{a^2}+\frac{20}{3 c^2 x^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c^2 x^3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}+\frac{4}{a^2 c^2 x^4 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}",0,"-2/(3*a*c^2*x^3*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) + 4/(a^2*c^2*x^4*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + 20/(3*c^2*x^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (16*Defer[Int][1/(x^5*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/a^2 + (112*Defer[Int][1/(x^3*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/3 + (80*a^2*Defer[Int][1/(x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/3","A",0,0,0,0,-1,"{}"
1066,0,0,0,0.4714329,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","40 a^2 \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{184}{3} \text{Int}\left(\frac{1}{x^4 \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{80 \text{Int}\left(\frac{1}{x^6 \left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)}{3 a^2}+\frac{8}{c^2 x^3 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c^2 x^4 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}+\frac{16}{3 a^2 c^2 x^5 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}",0,"-2/(3*a*c^2*x^4*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)) + 16/(3*a^2*c^2*x^5*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + 8/(c^2*x^3*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (80*Defer[Int][1/(x^6*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + (184*Defer[Int][1/(x^4*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x])/3 + 40*a^2*Defer[Int][1/(x^2*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1067,0,0,0,0.0647661,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Int[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^3 \tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x]","A",0,0,0,0,-1,"{}"
1068,1,160,0,0.5901589,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Int[x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)),x]","\frac{4 \sqrt{2 \pi } S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^4 c^3}-\frac{4 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a^4 c^3}+\frac{4 x^4}{3 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2 x^3}{3 a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}","\frac{4 \sqrt{2 \pi } S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^4 c^3}-\frac{4 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a^4 c^3}+\frac{4 x^4}{3 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2 x^3}{3 a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x^3)/(3*a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) - (4*x^2)/(a^2*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (4*x^4)/(3*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (4*Sqrt[2*Pi]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a^4*c^3) - (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*a^4*c^3)","A",24,6,24,0.2500,1,"{4968, 4942, 4970, 4406, 3305, 3351}"
1069,1,129,0,0.6884088,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Int[x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)),x]","\frac{4 \sqrt{2 \pi } \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^3 c^3}+\frac{8 x^3}{3 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2 x^2}{3 a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}","\frac{4 \sqrt{2 \pi } \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^3 c^3}+\frac{8 x^3}{3 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2 x^2}{3 a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x^2)/(3*a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) - (8*x)/(3*a^2*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (8*x^3)/(3*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (4*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a^3*c^3)","A",27,7,24,0.2917,1,"{4968, 4970, 3312, 3304, 3352, 4406, 4904}"
1070,1,155,0,0.4998613,"\int \frac{x}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Int[x/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)),x]","-\frac{4 \sqrt{2 \pi } S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^2 c^3}-\frac{4 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a^2 c^3}+\frac{4 x^2}{c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2 x}{3 a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}","-\frac{4 \sqrt{2 \pi } S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^2 c^3}-\frac{4 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a^2 c^3}+\frac{4 x^2}{c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2 x}{3 a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x)/(3*a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) - 4/(3*a^2*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (4*x^2)/(c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (4*Sqrt[2*Pi]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a^2*c^3) - (4*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*a^2*c^3)","A",24,6,22,0.2727,1,"{4968, 4970, 4406, 3305, 3351, 4902}"
1071,1,125,0,0.2968871,"\int \frac{1}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)),x]","\frac{16 x}{3 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}-\frac{4 \sqrt{2 \pi } \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a c^3}-\frac{8 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a c^3}","\frac{16 x}{3 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}-\frac{4 \sqrt{2 \pi } \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a c^3}-\frac{8 \sqrt{\pi } \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a c^3}",1,"-2/(3*a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) + (16*x)/(3*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) - (4*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a*c^3) - (8*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*a*c^3)","A",14,8,21,0.3810,1,"{4902, 4968, 4970, 4406, 3304, 3352, 4904, 3312}"
1072,0,0,0,0.4116244,"\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","\frac{8 \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)}{3 a^2}+8 \text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{4}{3 a^2 c^3 x^2 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}+\frac{20}{3 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c^3 x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}+\frac{5 \sqrt{2 \pi } S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 c^3}+\frac{20 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 c^3}",0,"-2/(3*a*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) + 20/(3*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + 4/(3*a^2*c^3*x^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (5*Sqrt[2*Pi]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*c^3) + (20*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*c^3) + (8*Defer[Int][1/(x^3*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + 8*Defer[Int][1/(x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1073,0,0,0,0.5080722,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","\frac{80}{3} \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{8 \text{Int}\left(\frac{1}{x^4 \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)}{a^2}+\frac{8}{c^3 x \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c^3 x^2 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c^3 x^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}+\frac{5 \sqrt{\frac{\pi }{2}} a \text{FresnelC}\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{c^3}+\frac{20 \sqrt{\pi } a \text{FresnelC}\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{c^3}+\frac{30 a \sqrt{\tan ^{-1}(a x)}}{c^3}",0,"-2/(3*a*c^3*x^2*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) + 8/(3*a^2*c^3*x^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + 8/(c^3*x*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (30*a*Sqrt[ArcTan[a*x]])/c^3 + (5*a*Sqrt[Pi/2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/c^3 + (20*a*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/c^3 + (8*Defer[Int][1/(x^4*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a^2 + (80*Defer[Int][1/(x^2*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/3","A",0,0,0,0,-1,"{}"
1074,0,0,0,0.461449,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","56 a^2 \text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{152}{3} \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{16 \text{Int}\left(\frac{1}{x^5 \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)}{a^2}+\frac{28}{3 c^3 x^2 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c^3 x^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}+\frac{4}{a^2 c^3 x^4 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",0,"-2/(3*a*c^3*x^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) + 4/(a^2*c^3*x^4*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + 28/(3*c^3*x^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (16*Defer[Int][1/(x^5*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/a^2 + (152*Defer[Int][1/(x^3*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/3 + 56*a^2*Defer[Int][1/(x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1075,0,0,0,0.4691818,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","\frac{224}{3} a^2 \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)+80 \text{Int}\left(\frac{1}{x^4 \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{80 \text{Int}\left(\frac{1}{x^6 \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)}{3 a^2}+\frac{32}{3 c^3 x^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c^3 x^4 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}+\frac{16}{3 a^2 c^3 x^5 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",0,"-2/(3*a*c^3*x^4*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) + 16/(3*a^2*c^3*x^5*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + 32/(3*c^3*x^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (80*Defer[Int][1/(x^6*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + 80*Defer[Int][1/(x^4*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x] + (224*a^2*Defer[Int][1/(x^2*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x])/3","A",0,0,0,0,-1,"{}"
1076,0,0,0,0.0981857,"\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(5/2),x]","\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x^m \sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1077,0,0,0,0.0665015,"\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(5/2),x]","\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x \sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1078,0,0,0,0.0336495,"\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^(5/2),x]","\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1079,0,0,0,0.0987067,"\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^{5/2}} \, dx","Int[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^(5/2)),x]","\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{\sqrt{a^2 c x^2+c}}{x \tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^(5/2)), x]","A",0,0,0,0,-1,"{}"
1080,0,0,0,0.1088277,"\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(5/2),x]","\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1081,0,0,0,0.0764499,"\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(5/2),x]","\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1082,0,0,0,0.0364513,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^(5/2),x]","\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1083,0,0,0,0.1112824,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)^{5/2}} \, dx","Int[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^(5/2)),x]","\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{3/2}}{x \tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^(5/2)), x]","A",0,0,0,0,-1,"{}"
1084,0,0,0,0.1100002,"\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(5/2),x]","\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x^m \left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1085,0,0,0,0.0791245,"\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(5/2),x]","\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x \left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1086,0,0,0,0.0425756,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Int[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^(5/2),x]","\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2}}{\tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^(5/2), x]","A",0,0,0,0,-1,"{}"
1087,0,0,0,0.1138908,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)^{5/2}} \, dx","Int[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^(5/2)),x]","\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{\left(a^2 c x^2+c\right)^{5/2}}{x \tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^(5/2)), x]","A",0,0,0,0,-1,"{}"
1088,0,0,0,0.1143479,"\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)),x]","\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x^m}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]","A",0,0,0,0,-1,"{}"
1089,0,0,0,0.0752147,"\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)),x]","\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]","A",0,0,0,0,-1,"{}"
1090,0,0,0,0.0361965,"\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]","A",0,0,0,0,-1,"{}"
1091,0,0,0,0.216067,"\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}},x\right)}{3 a}-\frac{2 \sqrt{a^2 c x^2+c}}{3 a c x \tan ^{-1}(a x)^{3/2}}",0,"(-2*Sqrt[c + a^2*c*x^2])/(3*a*c*x*ArcTan[a*x]^(3/2)) - (2*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x])/(3*a)","A",0,0,0,0,-1,"{}"
1092,0,0,0,0.1086751,"\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]","A",0,0,0,0,-1,"{}"
1093,0,0,0,0.1182381,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x]","A",0,0,0,0,-1,"{}"
1094,0,0,0,0.9276429,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]","\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","\frac{44}{3} \text{Int}\left(\frac{x^3}{\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)+8 a^2 \text{Int}\left(\frac{x^5}{\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{8 \sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^4 c \sqrt{a^2 c x^2+c}}-\frac{8 x^4}{3 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}-\frac{2 x^3}{3 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",0,"(-2*x^3)/(3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) - (4*x^2)/(a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (8*x^4)/(3*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (8*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^4*c*Sqrt[c + a^2*c*x^2]) + (44*Defer[Int][x^3/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/3 + 8*a^2*Defer[Int][x^5/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1095,0,0,0,0.6803972,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]","\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","4 \text{Int}\left(\frac{x^2}{\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{8}{3} a^2 \text{Int}\left(\frac{x^4}{\left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{8 \sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^3 c \sqrt{a^2 c x^2+c}}-\frac{4 x^3}{3 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}-\frac{2 x^2}{3 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",0,"(-2*x^2)/(3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) - (8*x)/(3*a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (4*x^3)/(3*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (8*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a^3*c*Sqrt[c + a^2*c*x^2]) + 4*Defer[Int][x^2/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x] + (8*a^2*Defer[Int][x^4/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/3","A",0,0,0,0,-1,"{}"
1096,1,129,0,0.3033094,"\int \frac{x}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]","-\frac{4 \sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^2 c \sqrt{a^2 c x^2+c}}-\frac{2 x}{3 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}","-\frac{4 \sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^2 c \sqrt{a^2 c x^2+c}}-\frac{2 x}{3 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x)/(3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) - 4/(3*a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (4*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a^2*c*Sqrt[c + a^2*c*x^2])","A",6,6,24,0.2500,1,"{4942, 4902, 4971, 4970, 3305, 3351}"
1097,1,126,0,0.2284619,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]","-\frac{4 \sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a c \sqrt{a^2 c x^2+c}}+\frac{4 x}{3 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}","-\frac{4 \sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a c \sqrt{a^2 c x^2+c}}+\frac{4 x}{3 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}",1,"-2/(3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) + (4*x)/(3*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) - (4*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a*c*Sqrt[c + a^2*c*x^2])","A",6,6,23,0.2609,1,"{4902, 4942, 4905, 4904, 3304, 3352}"
1098,0,0,0,0.6974711,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","\frac{8 \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{3 a^2}+4 \text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{8 \sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 c \sqrt{a^2 c x^2+c}}+\frac{8}{3 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}+\frac{4}{3 a^2 c x^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}",0,"-2/(3*a*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) + 8/(3*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + 4/(3*a^2*c*x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (8*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*c*Sqrt[c + a^2*c*x^2]) + (8*Defer[Int][1/(x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + 4*Defer[Int][1/(x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1099,0,0,0,0.8332881,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","\frac{8 \text{Int}\left(\frac{1}{x^4 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{a^2}+\frac{44}{3} \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{8 \sqrt{2 \pi } a \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{4}{c x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}+\frac{8}{3 a^2 c x^3 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}",0,"-2/(3*a*c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) + 8/(3*a^2*c*x^3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + 4/(c*x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (8*a*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(c*Sqrt[c + a^2*c*x^2]) + (8*Defer[Int][1/(x^4*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a^2 + (44*Defer[Int][1/(x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/3","A",0,0,0,0,-1,"{}"
1100,0,0,0,0.8489398,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","16 a^2 \text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{92}{3} \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{16 \text{Int}\left(\frac{1}{x^5 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{a^2}+\frac{16}{3 c x^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}+\frac{4}{a^2 c x^4 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",0,"-2/(3*a*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) + 4/(a^2*c*x^4*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + 16/(3*c*x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (16*Defer[Int][1/(x^5*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/a^2 + (92*Defer[Int][1/(x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/3 + 16*a^2*Defer[Int][1/(x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1101,0,0,0,0.8520861,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","\frac{80}{3} a^2 \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)+52 \text{Int}\left(\frac{1}{x^4 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{80 \text{Int}\left(\frac{1}{x^6 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{3 a^2}+\frac{20}{3 c x^3 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}+\frac{16}{3 a^2 c x^5 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",0,"-2/(3*a*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)) + 16/(3*a^2*c*x^5*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + 20/(3*c*x^3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (80*Defer[Int][1/(x^6*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + 52*Defer[Int][1/(x^4*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x] + (80*a^2*Defer[Int][1/(x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x])/3","A",0,0,0,0,-1,"{}"
1102,0,0,0,0.1186947,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]","\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","\text{Int}\left(\frac{x^m}{\left(a^2 c x^2+c\right)^{5/2} \tan ^{-1}(a x)^{5/2}},x\right)",0,"Defer[Int][x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x]","A",0,0,0,0,-1,"{}"
1103,1,190,0,0.8081293,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]","-\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{6 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x^3}{3 a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}","-\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{6 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 x^3}{3 a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{4 x^2}{a^2 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x^3)/(3*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)) - (4*x^2)/(a^2*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^4*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[6*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^4*c^2*Sqrt[c + a^2*c*x^2])","A",18,8,26,0.3077,1,"{4942, 4968, 4971, 4970, 3312, 3305, 3351, 4406}"
1104,1,224,0,1.1395629,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]","-\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{6 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 x^3}{3 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{2 x^2}{3 a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}","-\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{6 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 x^3}{3 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{2 x^2}{3 a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{8 x}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x^2)/(3*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)) - (8*x)/(3*a^2*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (4*x^3)/(3*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a^3*c^2*Sqrt[c + a^2*c*x^2]) + (Sqrt[6*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^3*c^2*Sqrt[c + a^2*c*x^2])","A",27,10,26,0.3846,1,"{4968, 4942, 4971, 4970, 4406, 3304, 3352, 4905, 4904, 3312}"
1105,1,222,0,1.0573084,"\int \frac{x}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]","-\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{6 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{8 x^2}{3 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{2 x}{3 a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}","-\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{6 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{8 x^2}{3 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{2 x}{3 a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}-\frac{4}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",1,"(-2*x)/(3*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)) - 4/(3*a^2*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (8*x^2)/(3*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*a^2*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[6*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c^2*Sqrt[c + a^2*c*x^2])","A",27,8,24,0.3333,1,"{4968, 4971, 4970, 3312, 3305, 3351, 4406, 4902}"
1106,1,183,0,0.6168143,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]","-\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{6 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{4 x}{c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}","-\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{6 \pi } \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{4 x}{c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}",1,"-2/(3*a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)) + (4*x)/(c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a*c^2*Sqrt[c + a^2*c*x^2]) - (Sqrt[6*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(a*c^2*Sqrt[c + a^2*c*x^2])","A",18,10,23,0.4348,1,"{4902, 4968, 4971, 4970, 4406, 3304, 3352, 4905, 4904, 3312}"
1107,0,0,0,0.767769,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","\frac{8 \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{3 a^2}+\frac{20}{3} \text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{4 \sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{4 \sqrt{\frac{2 \pi }{3}} \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{16}{3 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}+\frac{4}{3 a^2 c x^2 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c x \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}",0,"-2/(3*a*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)) + 16/(3*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + 4/(3*a^2*c*x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (4*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(c^2*Sqrt[c + a^2*c*x^2]) + (4*Sqrt[(2*Pi)/3]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(c^2*Sqrt[c + a^2*c*x^2]) + (8*Defer[Int][1/(x^3*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + (20*Defer[Int][1/(x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/3","A",0,0,0,0,-1,"{}"
1108,0,0,0,0.9317952,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","\frac{8 \text{Int}\left(\frac{1}{x^4 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{a^2}+\frac{68}{3} \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{20 \sqrt{2 \pi } a \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{c^2 \sqrt{a^2 c x^2+c}}+\frac{20 \sqrt{\frac{2 \pi }{3}} a \sqrt{a^2 x^2+1} \text{FresnelC}\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{20}{3 c x \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}+\frac{8}{3 a^2 c x^3 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c x^2 \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}",0,"-2/(3*a*c*x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)) + 8/(3*a^2*c*x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + 20/(3*c*x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (20*a*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(c^2*Sqrt[c + a^2*c*x^2]) + (20*a*Sqrt[(2*Pi)/3]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(3*c^2*Sqrt[c + a^2*c*x^2]) + (8*Defer[Int][1/(x^4*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/a^2 + (68*Defer[Int][1/(x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/3","A",0,0,0,0,-1,"{}"
1109,0,0,0,0.8992048,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","40 a^2 \text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)+44 \text{Int}\left(\frac{1}{x^3 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{16 \text{Int}\left(\frac{1}{x^5 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{a^2}+\frac{8}{c x^2 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c x^3 \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}+\frac{4}{a^2 c x^4 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",0,"-2/(3*a*c*x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)) + 4/(a^2*c*x^4*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + 8/(c*x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (16*Defer[Int][1/(x^5*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/a^2 + 44*Defer[Int][1/(x^3*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x] + 40*a^2*Defer[Int][1/(x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1110,0,0,0,0.9177273,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Int[1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]","\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","56 a^2 \text{Int}\left(\frac{1}{x^2 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{212}{3} \text{Int}\left(\frac{1}{x^4 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)+\frac{80 \text{Int}\left(\frac{1}{x^6 \left(a^2 c x^2+c\right)^{5/2} \sqrt{\tan ^{-1}(a x)}},x\right)}{3 a^2}+\frac{28}{3 c x^3 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}-\frac{2}{3 a c x^4 \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}+\frac{16}{3 a^2 c x^5 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",0,"-2/(3*a*c*x^4*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)) + 16/(3*a^2*c*x^5*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + 28/(3*c*x^3*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) + (80*Defer[Int][1/(x^6*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/(3*a^2) + (212*Defer[Int][1/(x^4*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/3 + 56*a^2*Defer[Int][1/(x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]","A",0,0,0,0,-1,"{}"
1111,0,0,0,0.084999,"\int \frac{x \tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx","Int[(x*ArcTan[a*x]^n)/(c + a^2*c*x^2),x]","\int \frac{x \tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx","\frac{x \tan ^{-1}(a x)^{n+1}}{a c (n+1)}-\frac{\text{Int}\left(\tan ^{-1}(a x)^{n+1},x\right)}{a c (n+1)}",0,"(x*ArcTan[a*x]^(1 + n))/(a*c*(1 + n)) - Defer[Int][ArcTan[a*x]^(1 + n), x]/(a*c*(1 + n))","A",0,0,0,0,-1,"{}"
1112,1,20,0,0.0402534,"\int \frac{\tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx","Int[ArcTan[a*x]^n/(c + a^2*c*x^2),x]","\frac{\tan ^{-1}(a x)^{n+1}}{a c (n+1)}","\frac{\tan ^{-1}(a x)^{n+1}}{a c (n+1)}",1,"ArcTan[a*x]^(1 + n)/(a*c*(1 + n))","A",1,1,19,0.05263,1,"{4884}"
1113,0,0,0,0.0986459,"\int (f x)^m \left(d+c^2 d x^2\right)^q \left(a+b \tan ^{-1}(c x)\right)^p \, dx","Int[(f*x)^m*(d + c^2*d*x^2)^q*(a + b*ArcTan[c*x])^p,x]","\int (f x)^m \left(d+c^2 d x^2\right)^q \left(a+b \tan ^{-1}(c x)\right)^p \, dx","\text{Int}\left((f x)^m \left(c^2 d x^2+d\right)^q \left(a+b \tan ^{-1}(c x)\right)^p,x\right)",0,"Defer[Int][(f*x)^m*(d + c^2*d*x^2)^q*(a + b*ArcTan[c*x])^p, x]","A",0,0,0,0,-1,"{}"
1114,1,107,0,0.1102978,"\int x^3 \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^3*(d + e*x^2)*(a + b*ArcTan[c*x]),x]","\frac{1}{4} d x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{6} e x^6 \left(a+b \tan ^{-1}(c x)\right)-\frac{b x^3 \left(3 c^2 d-2 e\right)}{36 c^3}+\frac{b x \left(3 c^2 d-2 e\right)}{12 c^5}-\frac{b \left(3 c^2 d-2 e\right) \tan ^{-1}(c x)}{12 c^6}-\frac{b e x^5}{30 c}","\frac{1}{4} d x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{6} e x^6 \left(a+b \tan ^{-1}(c x)\right)-\frac{b x^3 \left(3 c^2 d-2 e\right)}{36 c^3}+\frac{b x \left(3 c^2 d-2 e\right)}{12 c^5}-\frac{b \left(3 c^2 d-2 e\right) \tan ^{-1}(c x)}{12 c^6}-\frac{b e x^5}{30 c}",1,"(b*(3*c^2*d - 2*e)*x)/(12*c^5) - (b*(3*c^2*d - 2*e)*x^3)/(36*c^3) - (b*e*x^5)/(30*c) - (b*(3*c^2*d - 2*e)*ArcTan[c*x])/(12*c^6) + (d*x^4*(a + b*ArcTan[c*x]))/4 + (e*x^6*(a + b*ArcTan[c*x]))/6","A",5,5,19,0.2632,1,"{14, 4976, 459, 302, 203}"
1115,1,94,0,0.1394068,"\int x^2 \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^2*(d + e*x^2)*(a + b*ArcTan[c*x]),x]","\frac{1}{3} d x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{5} e x^5 \left(a+b \tan ^{-1}(c x)\right)-\frac{b x^2 \left(5 c^2 d-3 e\right)}{30 c^3}+\frac{b \left(5 c^2 d-3 e\right) \log \left(c^2 x^2+1\right)}{30 c^5}-\frac{b e x^4}{20 c}","\frac{1}{3} d x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{5} e x^5 \left(a+b \tan ^{-1}(c x)\right)-\frac{b x^2 \left(5 c^2 d-3 e\right)}{30 c^3}+\frac{b \left(5 c^2 d-3 e\right) \log \left(c^2 x^2+1\right)}{30 c^5}-\frac{b e x^4}{20 c}",1,"-(b*(5*c^2*d - 3*e)*x^2)/(30*c^3) - (b*e*x^4)/(20*c) + (d*x^3*(a + b*ArcTan[c*x]))/3 + (e*x^5*(a + b*ArcTan[c*x]))/5 + (b*(5*c^2*d - 3*e)*Log[1 + c^2*x^2])/(30*c^5)","A",4,4,19,0.2105,1,"{14, 4976, 446, 77}"
1116,1,82,0,0.0698475,"\int x \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x*(d + e*x^2)*(a + b*ArcTan[c*x]),x]","\frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{4 e}-\frac{b x \left(2 c^2 d-e\right)}{4 c^3}-\frac{b \left(c^2 d-e\right)^2 \tan ^{-1}(c x)}{4 c^4 e}-\frac{b e x^3}{12 c}","\frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{4 e}-\frac{b x \left(2 c^2 d-e\right)}{4 c^3}-\frac{b \left(c^2 d-e\right)^2 \tan ^{-1}(c x)}{4 c^4 e}-\frac{b e x^3}{12 c}",1,"-(b*(2*c^2*d - e)*x)/(4*c^3) - (b*e*x^3)/(12*c) - (b*(c^2*d - e)^2*ArcTan[c*x])/(4*c^4*e) + ((d + e*x^2)^2*(a + b*ArcTan[c*x]))/(4*e)","A",4,3,17,0.1765,1,"{4974, 390, 203}"
1117,1,68,0,0.0718078,"\int \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[(d + e*x^2)*(a + b*ArcTan[c*x]),x]","d x \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} e x^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{b \left(3 c^2 d-e\right) \log \left(c^2 x^2+1\right)}{6 c^3}-\frac{b e x^2}{6 c}","d x \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} e x^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{b \left(3 c^2 d-e\right) \log \left(c^2 x^2+1\right)}{6 c^3}-\frac{b e x^2}{6 c}",1,"-(b*e*x^2)/(6*c) + d*x*(a + b*ArcTan[c*x]) + (e*x^3*(a + b*ArcTan[c*x]))/3 - (b*(3*c^2*d - e)*Log[1 + c^2*x^2])/(6*c^3)","A",5,4,16,0.2500,1,"{4912, 1593, 444, 43}"
1118,1,77,0,0.0928198,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Int[((d + e*x^2)*(a + b*ArcTan[c*x]))/x,x]","\frac{1}{2} i b d \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d \text{PolyLog}(2,i c x)+\frac{1}{2} e x^2 \left(a+b \tan ^{-1}(c x)\right)+a d \log (x)+\frac{b e \tan ^{-1}(c x)}{2 c^2}-\frac{b e x}{2 c}","\frac{1}{2} i b d \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d \text{PolyLog}(2,i c x)+\frac{1}{2} e x^2 \left(a+b \tan ^{-1}(c x)\right)+a d \log (x)+\frac{b e \tan ^{-1}(c x)}{2 c^2}-\frac{b e x}{2 c}",1,"-(b*e*x)/(2*c) + (b*e*ArcTan[c*x])/(2*c^2) + (e*x^2*(a + b*ArcTan[c*x]))/2 + a*d*Log[x] + (I/2)*b*d*PolyLog[2, (-I)*c*x] - (I/2)*b*d*PolyLog[2, I*c*x]","A",8,6,19,0.3158,1,"{4980, 4848, 2391, 4852, 321, 203}"
1119,1,57,0,0.0765387,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Int[((d + e*x^2)*(a + b*ArcTan[c*x]))/x^2,x]","-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{x}+e x \left(a+b \tan ^{-1}(c x)\right)-\frac{b \left(c^2 d+e\right) \log \left(c^2 x^2+1\right)}{2 c}+b c d \log (x)","-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{x}+e x \left(a+b \tan ^{-1}(c x)\right)-\frac{b \left(c^2 d+e\right) \log \left(c^2 x^2+1\right)}{2 c}+b c d \log (x)",1,"-((d*(a + b*ArcTan[c*x]))/x) + e*x*(a + b*ArcTan[c*x]) + b*c*d*Log[x] - (b*(c^2*d + e)*Log[1 + c^2*x^2])/(2*c)","A",4,4,19,0.2105,1,"{14, 4976, 446, 72}"
1120,1,77,0,0.0991435,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Int[((d + e*x^2)*(a + b*ArcTan[c*x]))/x^3,x]","\frac{1}{2} i b e \text{PolyLog}(2,-i c x)-\frac{1}{2} i b e \text{PolyLog}(2,i c x)-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}+a e \log (x)-\frac{1}{2} b c^2 d \tan ^{-1}(c x)-\frac{b c d}{2 x}","\frac{1}{2} i b e \text{PolyLog}(2,-i c x)-\frac{1}{2} i b e \text{PolyLog}(2,i c x)-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}+a e \log (x)-\frac{1}{2} b c^2 d \tan ^{-1}(c x)-\frac{b c d}{2 x}",1,"-(b*c*d)/(2*x) - (b*c^2*d*ArcTan[c*x])/2 - (d*(a + b*ArcTan[c*x]))/(2*x^2) + a*e*Log[x] + (I/2)*b*e*PolyLog[2, (-I)*c*x] - (I/2)*b*e*PolyLog[2, I*c*x]","A",8,6,19,0.3158,1,"{4980, 4852, 325, 203, 4848, 2391}"
1121,1,83,0,0.119147,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Int[((d + e*x^2)*(a + b*ArcTan[c*x]))/x^4,x]","-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{x}+\frac{1}{6} b c \left(c^2 d-3 e\right) \log \left(c^2 x^2+1\right)-\frac{1}{3} b c \log (x) \left(c^2 d-3 e\right)-\frac{b c d}{6 x^2}","-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{x}+\frac{1}{6} b c \left(c^2 d-3 e\right) \log \left(c^2 x^2+1\right)-\frac{1}{3} b c \log (x) \left(c^2 d-3 e\right)-\frac{b c d}{6 x^2}",1,"-(b*c*d)/(6*x^2) - (d*(a + b*ArcTan[c*x]))/(3*x^3) - (e*(a + b*ArcTan[c*x]))/x - (b*c*(c^2*d - 3*e)*Log[x])/3 + (b*c*(c^2*d - 3*e)*Log[1 + c^2*x^2])/6","A",5,5,19,0.2632,1,"{14, 4976, 12, 446, 77}"
1122,1,82,0,0.0904979,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Int[((d + e*x^2)*(a + b*ArcTan[c*x]))/x^5,x]","-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}+\frac{b c \left(c^2 d-2 e\right)}{4 x}+\frac{1}{4} b c^2 \left(c^2 d-2 e\right) \tan ^{-1}(c x)-\frac{b c d}{12 x^3}","-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}+\frac{b c \left(c^2 d-2 e\right)}{4 x}+\frac{1}{4} b c^2 \left(c^2 d-2 e\right) \tan ^{-1}(c x)-\frac{b c d}{12 x^3}",1,"-(b*c*d)/(12*x^3) + (b*c*(c^2*d - 2*e))/(4*x) + (b*c^2*(c^2*d - 2*e)*ArcTan[c*x])/4 - (d*(a + b*ArcTan[c*x]))/(4*x^4) - (e*(a + b*ArcTan[c*x]))/(2*x^2)","A",5,6,19,0.3158,1,"{14, 4976, 12, 453, 325, 203}"
1123,1,110,0,0.1288246,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Int[((d + e*x^2)*(a + b*ArcTan[c*x]))/x^6,x]","-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}+\frac{b c \left(3 c^2 d-5 e\right)}{30 x^2}-\frac{1}{30} b c^3 \left(3 c^2 d-5 e\right) \log \left(c^2 x^2+1\right)+\frac{1}{15} b c^3 \log (x) \left(3 c^2 d-5 e\right)-\frac{b c d}{20 x^4}","-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}+\frac{b c \left(3 c^2 d-5 e\right)}{30 x^2}-\frac{1}{30} b c^3 \left(3 c^2 d-5 e\right) \log \left(c^2 x^2+1\right)+\frac{1}{15} b c^3 \log (x) \left(3 c^2 d-5 e\right)-\frac{b c d}{20 x^4}",1,"-(b*c*d)/(20*x^4) + (b*c*(3*c^2*d - 5*e))/(30*x^2) - (d*(a + b*ArcTan[c*x]))/(5*x^5) - (e*(a + b*ArcTan[c*x]))/(3*x^3) + (b*c^3*(3*c^2*d - 5*e)*Log[x])/15 - (b*c^3*(3*c^2*d - 5*e)*Log[1 + c^2*x^2])/30","A",5,5,19,0.2632,1,"{14, 4976, 12, 446, 77}"
1124,1,105,0,0.1090909,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{x^7} \, dx","Int[((d + e*x^2)*(a + b*ArcTan[c*x]))/x^7,x]","-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{6 x^6}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}+\frac{b c \left(2 c^2 d-3 e\right)}{36 x^3}-\frac{b c^3 \left(2 c^2 d-3 e\right)}{12 x}-\frac{1}{12} b c^4 \left(2 c^2 d-3 e\right) \tan ^{-1}(c x)-\frac{b c d}{30 x^5}","-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{6 x^6}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}+\frac{b c \left(2 c^2 d-3 e\right)}{36 x^3}-\frac{b c^3 \left(2 c^2 d-3 e\right)}{12 x}-\frac{1}{12} b c^4 \left(2 c^2 d-3 e\right) \tan ^{-1}(c x)-\frac{b c d}{30 x^5}",1,"-(b*c*d)/(30*x^5) + (b*c*(2*c^2*d - 3*e))/(36*x^3) - (b*c^3*(2*c^2*d - 3*e))/(12*x) - (b*c^4*(2*c^2*d - 3*e)*ArcTan[c*x])/12 - (d*(a + b*ArcTan[c*x]))/(6*x^6) - (e*(a + b*ArcTan[c*x]))/(4*x^4)","A",6,6,19,0.3158,1,"{14, 4976, 12, 453, 325, 203}"
1125,1,185,0,0.1931972,"\int x^3 \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^3*(d + e*x^2)^2*(a + b*ArcTan[c*x]),x]","\frac{1}{4} d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} d e x^6 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{8} e^2 x^8 \left(a+b \tan ^{-1}(c x)\right)-\frac{b x^3 \left(6 c^4 d^2-8 c^2 d e+3 e^2\right)}{72 c^5}+\frac{b x \left(6 c^4 d^2-8 c^2 d e+3 e^2\right)}{24 c^7}-\frac{b \left(6 c^4 d^2-8 c^2 d e+3 e^2\right) \tan ^{-1}(c x)}{24 c^8}-\frac{b e x^5 \left(8 c^2 d-3 e\right)}{120 c^3}-\frac{b e^2 x^7}{56 c}","\frac{1}{4} d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} d e x^6 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{8} e^2 x^8 \left(a+b \tan ^{-1}(c x)\right)-\frac{b x^3 \left(6 c^4 d^2-8 c^2 d e+3 e^2\right)}{72 c^5}+\frac{b x \left(6 c^4 d^2-8 c^2 d e+3 e^2\right)}{24 c^7}-\frac{b \left(6 c^4 d^2-8 c^2 d e+3 e^2\right) \tan ^{-1}(c x)}{24 c^8}-\frac{b e x^5 \left(8 c^2 d-3 e\right)}{120 c^3}-\frac{b e^2 x^7}{56 c}",1,"(b*(6*c^4*d^2 - 8*c^2*d*e + 3*e^2)*x)/(24*c^7) - (b*(6*c^4*d^2 - 8*c^2*d*e + 3*e^2)*x^3)/(72*c^5) - (b*(8*c^2*d - 3*e)*e*x^5)/(120*c^3) - (b*e^2*x^7)/(56*c) - (b*(6*c^4*d^2 - 8*c^2*d*e + 3*e^2)*ArcTan[c*x])/(24*c^8) + (d^2*x^4*(a + b*ArcTan[c*x]))/4 + (d*e*x^6*(a + b*ArcTan[c*x]))/3 + (e^2*x^8*(a + b*ArcTan[c*x]))/8","A",4,5,21,0.2381,1,"{266, 43, 4976, 1261, 203}"
1126,1,161,0,0.2458055,"\int x^2 \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^2*(d + e*x^2)^2*(a + b*ArcTan[c*x]),x]","\frac{1}{3} d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{2}{5} d e x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{7} e^2 x^7 \left(a+b \tan ^{-1}(c x)\right)-\frac{b x^2 \left(35 c^4 d^2-42 c^2 d e+15 e^2\right)}{210 c^5}+\frac{b \left(35 c^4 d^2-42 c^2 d e+15 e^2\right) \log \left(c^2 x^2+1\right)}{210 c^7}-\frac{b e x^4 \left(14 c^2 d-5 e\right)}{140 c^3}-\frac{b e^2 x^6}{42 c}","\frac{1}{3} d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{2}{5} d e x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{7} e^2 x^7 \left(a+b \tan ^{-1}(c x)\right)-\frac{b x^2 \left(35 c^4 d^2-42 c^2 d e+15 e^2\right)}{210 c^5}+\frac{b \left(35 c^4 d^2-42 c^2 d e+15 e^2\right) \log \left(c^2 x^2+1\right)}{210 c^7}-\frac{b e x^4 \left(14 c^2 d-5 e\right)}{140 c^3}-\frac{b e^2 x^6}{42 c}",1,"-(b*(35*c^4*d^2 - 42*c^2*d*e + 15*e^2)*x^2)/(210*c^5) - (b*(14*c^2*d - 5*e)*e*x^4)/(140*c^3) - (b*e^2*x^6)/(42*c) + (d^2*x^3*(a + b*ArcTan[c*x]))/3 + (2*d*e*x^5*(a + b*ArcTan[c*x]))/5 + (e^2*x^7*(a + b*ArcTan[c*x]))/7 + (b*(35*c^4*d^2 - 42*c^2*d*e + 15*e^2)*Log[1 + c^2*x^2])/(210*c^7)","A",5,5,21,0.2381,1,"{270, 4976, 12, 1251, 771}"
1127,1,115,0,0.1138208,"\int x \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x*(d + e*x^2)^2*(a + b*ArcTan[c*x]),x]","\frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{6 e}-\frac{b x \left(3 c^4 d^2-3 c^2 d e+e^2\right)}{6 c^5}-\frac{b e x^3 \left(3 c^2 d-e\right)}{18 c^3}-\frac{b \left(c^2 d-e\right)^3 \tan ^{-1}(c x)}{6 c^6 e}-\frac{b e^2 x^5}{30 c}","\frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{6 e}-\frac{b x \left(3 c^4 d^2-3 c^2 d e+e^2\right)}{6 c^5}-\frac{b e x^3 \left(3 c^2 d-e\right)}{18 c^3}-\frac{b \left(c^2 d-e\right)^3 \tan ^{-1}(c x)}{6 c^6 e}-\frac{b e^2 x^5}{30 c}",1,"-(b*(3*c^4*d^2 - 3*c^2*d*e + e^2)*x)/(6*c^5) - (b*(3*c^2*d - e)*e*x^3)/(18*c^3) - (b*e^2*x^5)/(30*c) - (b*(c^2*d - e)^3*ArcTan[c*x])/(6*c^6*e) + ((d + e*x^2)^3*(a + b*ArcTan[c*x]))/(6*e)","A",4,3,19,0.1579,1,"{4974, 390, 203}"
1128,1,124,0,0.159716,"\int \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[(d + e*x^2)^2*(a + b*ArcTan[c*x]),x]","d^2 x \left(a+b \tan ^{-1}(c x)\right)+\frac{2}{3} d e x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{5} e^2 x^5 \left(a+b \tan ^{-1}(c x)\right)-\frac{b \left(15 c^4 d^2-10 c^2 d e+3 e^2\right) \log \left(c^2 x^2+1\right)}{30 c^5}-\frac{b e x^2 \left(10 c^2 d-3 e\right)}{30 c^3}-\frac{b e^2 x^4}{20 c}","d^2 x \left(a+b \tan ^{-1}(c x)\right)+\frac{2}{3} d e x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{5} e^2 x^5 \left(a+b \tan ^{-1}(c x)\right)-\frac{b \left(15 c^4 d^2-10 c^2 d e+3 e^2\right) \log \left(c^2 x^2+1\right)}{30 c^5}-\frac{b e x^2 \left(10 c^2 d-3 e\right)}{30 c^3}-\frac{b e^2 x^4}{20 c}",1,"-(b*(10*c^2*d - 3*e)*e*x^2)/(30*c^3) - (b*e^2*x^4)/(20*c) + d^2*x*(a + b*ArcTan[c*x]) + (2*d*e*x^3*(a + b*ArcTan[c*x]))/3 + (e^2*x^5*(a + b*ArcTan[c*x]))/5 - (b*(15*c^4*d^2 - 10*c^2*d*e + 3*e^2)*Log[1 + c^2*x^2])/(30*c^5)","A",5,5,18,0.2778,1,"{194, 4912, 1594, 1247, 698}"
1129,1,137,0,0.1795166,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Int[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x,x]","\frac{1}{2} i b d^2 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^2 \text{PolyLog}(2,i c x)+d e x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} e^2 x^4 \left(a+b \tan ^{-1}(c x)\right)+a d^2 \log (x)+\frac{b d e \tan ^{-1}(c x)}{c^2}+\frac{b e^2 x}{4 c^3}-\frac{b e^2 \tan ^{-1}(c x)}{4 c^4}-\frac{b d e x}{c}-\frac{b e^2 x^3}{12 c}","\frac{1}{2} i b d^2 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^2 \text{PolyLog}(2,i c x)+d e x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} e^2 x^4 \left(a+b \tan ^{-1}(c x)\right)+a d^2 \log (x)+\frac{b d e \tan ^{-1}(c x)}{c^2}+\frac{b e^2 x}{4 c^3}-\frac{b e^2 \tan ^{-1}(c x)}{4 c^4}-\frac{b d e x}{c}-\frac{b e^2 x^3}{12 c}",1,"-((b*d*e*x)/c) + (b*e^2*x)/(4*c^3) - (b*e^2*x^3)/(12*c) + (b*d*e*ArcTan[c*x])/c^2 - (b*e^2*ArcTan[c*x])/(4*c^4) + d*e*x^2*(a + b*ArcTan[c*x]) + (e^2*x^4*(a + b*ArcTan[c*x]))/4 + a*d^2*Log[x] + (I/2)*b*d^2*PolyLog[2, (-I)*c*x] - (I/2)*b*d^2*PolyLog[2, I*c*x]","A",12,7,21,0.3333,1,"{4980, 4848, 2391, 4852, 321, 203, 302}"
1130,1,109,0,0.1556355,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Int[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^2,x]","-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{x}+2 d e x \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} e^2 x^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{b \left(3 c^4 d^2+6 c^2 d e-e^2\right) \log \left(c^2 x^2+1\right)}{6 c^3}+b c d^2 \log (x)-\frac{b e^2 x^2}{6 c}","-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{x}+2 d e x \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} e^2 x^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{b \left(3 c^4 d^2+6 c^2 d e-e^2\right) \log \left(c^2 x^2+1\right)}{6 c^3}+b c d^2 \log (x)-\frac{b e^2 x^2}{6 c}",1,"-(b*e^2*x^2)/(6*c) - (d^2*(a + b*ArcTan[c*x]))/x + 2*d*e*x*(a + b*ArcTan[c*x]) + (e^2*x^3*(a + b*ArcTan[c*x]))/3 + b*c*d^2*Log[x] - (b*(3*c^4*d^2 + 6*c^2*d*e - e^2)*Log[1 + c^2*x^2])/(6*c^3)","A",4,4,21,0.1905,1,"{270, 4976, 1251, 893}"
1131,1,128,0,0.1632657,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Int[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^3,x]","i b d e \text{PolyLog}(2,-i c x)-i b d e \text{PolyLog}(2,i c x)-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}+\frac{1}{2} e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)+2 a d e \log (x)-\frac{1}{2} b c^2 d^2 \tan ^{-1}(c x)+\frac{b e^2 \tan ^{-1}(c x)}{2 c^2}-\frac{b c d^2}{2 x}-\frac{b e^2 x}{2 c}","i b d e \text{PolyLog}(2,-i c x)-i b d e \text{PolyLog}(2,i c x)-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}+\frac{1}{2} e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)+2 a d e \log (x)-\frac{1}{2} b c^2 d^2 \tan ^{-1}(c x)+\frac{b e^2 \tan ^{-1}(c x)}{2 c^2}-\frac{b c d^2}{2 x}-\frac{b e^2 x}{2 c}",1,"-(b*c*d^2)/(2*x) - (b*e^2*x)/(2*c) - (b*c^2*d^2*ArcTan[c*x])/2 + (b*e^2*ArcTan[c*x])/(2*c^2) - (d^2*(a + b*ArcTan[c*x]))/(2*x^2) + (e^2*x^2*(a + b*ArcTan[c*x]))/2 + 2*a*d*e*Log[x] + I*b*d*e*PolyLog[2, (-I)*c*x] - I*b*d*e*PolyLog[2, I*c*x]","A",11,7,21,0.3333,1,"{4980, 4852, 325, 203, 4848, 2391, 321}"
1132,1,115,0,0.1682813,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Int[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^4,x]","-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{2 d e \left(a+b \tan ^{-1}(c x)\right)}{x}+e^2 x \left(a+b \tan ^{-1}(c x)\right)+\frac{b \left(c^4 d^2-6 c^2 d e-3 e^2\right) \log \left(c^2 x^2+1\right)}{6 c}-\frac{1}{3} b c d \log (x) \left(c^2 d-6 e\right)-\frac{b c d^2}{6 x^2}","-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{2 d e \left(a+b \tan ^{-1}(c x)\right)}{x}+e^2 x \left(a+b \tan ^{-1}(c x)\right)+\frac{b \left(c^4 d^2-6 c^2 d e-3 e^2\right) \log \left(c^2 x^2+1\right)}{6 c}-\frac{1}{3} b c d \log (x) \left(c^2 d-6 e\right)-\frac{b c d^2}{6 x^2}",1,"-(b*c*d^2)/(6*x^2) - (d^2*(a + b*ArcTan[c*x]))/(3*x^3) - (2*d*e*(a + b*ArcTan[c*x]))/x + e^2*x*(a + b*ArcTan[c*x]) - (b*c*d*(c^2*d - 6*e)*Log[x])/3 + (b*(c^4*d^2 - 6*c^2*d*e - 3*e^2)*Log[1 + c^2*x^2])/(6*c)","A",5,5,21,0.2381,1,"{270, 4976, 12, 1251, 893}"
1133,1,139,0,0.1682794,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Int[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^5,x]","\frac{1}{2} i b e^2 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b e^2 \text{PolyLog}(2,i c x)-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{d e \left(a+b \tan ^{-1}(c x)\right)}{x^2}+a e^2 \log (x)+\frac{b c^3 d^2}{4 x}+\frac{1}{4} b c^4 d^2 \tan ^{-1}(c x)-b c^2 d e \tan ^{-1}(c x)-\frac{b c d^2}{12 x^3}-\frac{b c d e}{x}","\frac{1}{2} i b e^2 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b e^2 \text{PolyLog}(2,i c x)-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{d e \left(a+b \tan ^{-1}(c x)\right)}{x^2}+a e^2 \log (x)+\frac{b c^3 d^2}{4 x}+\frac{1}{4} b c^4 d^2 \tan ^{-1}(c x)-b c^2 d e \tan ^{-1}(c x)-\frac{b c d^2}{12 x^3}-\frac{b c d e}{x}",1,"-(b*c*d^2)/(12*x^3) + (b*c^3*d^2)/(4*x) - (b*c*d*e)/x + (b*c^4*d^2*ArcTan[c*x])/4 - b*c^2*d*e*ArcTan[c*x] - (d^2*(a + b*ArcTan[c*x]))/(4*x^4) - (d*e*(a + b*ArcTan[c*x]))/x^2 + a*e^2*Log[x] + (I/2)*b*e^2*PolyLog[2, (-I)*c*x] - (I/2)*b*e^2*PolyLog[2, I*c*x]","A",12,6,21,0.2857,1,"{4980, 4852, 325, 203, 4848, 2391}"
1134,1,150,0,0.1850436,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Int[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^6,x]","-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{2 d e \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{1}{30} b c \left(3 c^4 d^2-10 c^2 d e+15 e^2\right) \log \left(c^2 x^2+1\right)+\frac{1}{15} b c \log (x) \left(3 c^4 d^2-10 c^2 d e+15 e^2\right)+\frac{b c d \left(3 c^2 d-10 e\right)}{30 x^2}-\frac{b c d^2}{20 x^4}","-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{2 d e \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{1}{30} b c \left(3 c^4 d^2-10 c^2 d e+15 e^2\right) \log \left(c^2 x^2+1\right)+\frac{1}{15} b c \log (x) \left(3 c^4 d^2-10 c^2 d e+15 e^2\right)+\frac{b c d \left(3 c^2 d-10 e\right)}{30 x^2}-\frac{b c d^2}{20 x^4}",1,"-(b*c*d^2)/(20*x^4) + (b*c*d*(3*c^2*d - 10*e))/(30*x^2) - (d^2*(a + b*ArcTan[c*x]))/(5*x^5) - (2*d*e*(a + b*ArcTan[c*x]))/(3*x^3) - (e^2*(a + b*ArcTan[c*x]))/x + (b*c*(3*c^4*d^2 - 10*c^2*d*e + 15*e^2)*Log[x])/15 - (b*c*(3*c^4*d^2 - 10*c^2*d*e + 15*e^2)*Log[1 + c^2*x^2])/30","A",5,5,21,0.2381,1,"{270, 4976, 12, 1251, 893}"
1135,1,111,0,0.1478152,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^7} \, dx","Int[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^7,x]","-\frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{6 d x^6}-\frac{b c \left(c^4 d^2-3 c^2 d e+3 e^2\right)}{6 x}+\frac{b c d \left(c^2 d-3 e\right)}{18 x^3}-\frac{b \left(c^2 d-e\right)^3 \tan ^{-1}(c x)}{6 d}-\frac{b c d^2}{30 x^5}","-\frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{6 d x^6}-\frac{b c \left(c^4 d^2-3 c^2 d e+3 e^2\right)}{6 x}+\frac{b c d \left(c^2 d-3 e\right)}{18 x^3}-\frac{b \left(c^2 d-e\right)^3 \tan ^{-1}(c x)}{6 d}-\frac{b c d^2}{30 x^5}",1,"-(b*c*d^2)/(30*x^5) + (b*c*d*(c^2*d - 3*e))/(18*x^3) - (b*c*(c^4*d^2 - 3*c^2*d*e + 3*e^2))/(6*x) - (b*(c^2*d - e)^3*ArcTan[c*x])/(6*d) - ((d + e*x^2)^3*(a + b*ArcTan[c*x]))/(6*d*x^6)","A",5,5,21,0.2381,1,"{264, 4976, 12, 461, 203}"
1136,1,186,0,0.2323649,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^8} \, dx","Int[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^8,x]","-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{7 x^7}-\frac{2 d e \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{b c \left(15 c^4 d^2-42 c^2 d e+35 e^2\right)}{210 x^2}+\frac{1}{210} b c^3 \left(15 c^4 d^2-42 c^2 d e+35 e^2\right) \log \left(c^2 x^2+1\right)-\frac{1}{105} b c^3 \log (x) \left(15 c^4 d^2-42 c^2 d e+35 e^2\right)+\frac{b c d \left(5 c^2 d-14 e\right)}{140 x^4}-\frac{b c d^2}{42 x^6}","-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{7 x^7}-\frac{2 d e \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{b c \left(15 c^4 d^2-42 c^2 d e+35 e^2\right)}{210 x^2}+\frac{1}{210} b c^3 \left(15 c^4 d^2-42 c^2 d e+35 e^2\right) \log \left(c^2 x^2+1\right)-\frac{1}{105} b c^3 \log (x) \left(15 c^4 d^2-42 c^2 d e+35 e^2\right)+\frac{b c d \left(5 c^2 d-14 e\right)}{140 x^4}-\frac{b c d^2}{42 x^6}",1,"-(b*c*d^2)/(42*x^6) + (b*c*d*(5*c^2*d - 14*e))/(140*x^4) - (b*c*(15*c^4*d^2 - 42*c^2*d*e + 35*e^2))/(210*x^2) - (d^2*(a + b*ArcTan[c*x]))/(7*x^7) - (2*d*e*(a + b*ArcTan[c*x]))/(5*x^5) - (e^2*(a + b*ArcTan[c*x]))/(3*x^3) - (b*c^3*(15*c^4*d^2 - 42*c^2*d*e + 35*e^2)*Log[x])/105 + (b*c^3*(15*c^4*d^2 - 42*c^2*d*e + 35*e^2)*Log[1 + c^2*x^2])/210","A",5,5,21,0.2381,1,"{270, 4976, 12, 1251, 893}"
1137,1,285,0,0.4588011,"\int x^3 \left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^3*(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]","\frac{\left(d+e x^2\right)^5 \left(a+b \tan ^{-1}(c x)\right)}{10 e^2}-\frac{d \left(d+e x^2\right)^4 \left(a+b \tan ^{-1}(c x)\right)}{8 e^2}-\frac{b x \left(25 c^4 d^2-135 c^2 d e+84 e^2\right) \left(d+e x^2\right)^2}{4200 c^5 e}+\frac{b x \left(750 c^4 d^2 e+5 c^6 d^3-1071 c^2 d e^2+420 e^3\right) \left(d+e x^2\right)}{12600 c^7 e}+\frac{b x \left(-4977 c^4 d^2 e^2+1815 c^6 d^3 e+325 c^8 d^4+4305 c^2 d e^3-1260 e^4\right)}{12600 c^9 e}+\frac{b \left(c^2 d-e\right)^4 \left(c^2 d+4 e\right) \tan ^{-1}(c x)}{40 c^{10} e^2}-\frac{b x \left(23 c^2 d-36 e\right) \left(d+e x^2\right)^3}{2520 c^3 e}-\frac{b x \left(d+e x^2\right)^4}{90 c e}","\frac{\left(d+e x^2\right)^5 \left(a+b \tan ^{-1}(c x)\right)}{10 e^2}-\frac{d \left(d+e x^2\right)^4 \left(a+b \tan ^{-1}(c x)\right)}{8 e^2}-\frac{b e x^5 \left(20 c^4 d^2-15 c^2 d e+4 e^2\right)}{200 c^5}-\frac{b x^3 \left(-20 c^4 d^2 e+10 c^6 d^3+15 c^2 d e^2-4 e^3\right)}{120 c^7}+\frac{b x \left(-20 c^4 d^2 e+10 c^6 d^3+15 c^2 d e^2-4 e^3\right)}{40 c^9}-\frac{b e^2 x^7 \left(15 c^2 d-4 e\right)}{280 c^3}+\frac{b \left(c^2 d-e\right)^4 \left(c^2 d+4 e\right) \tan ^{-1}(c x)}{40 c^{10} e^2}-\frac{b e^3 x^9}{90 c}",1,"(b*(325*c^8*d^4 + 1815*c^6*d^3*e - 4977*c^4*d^2*e^2 + 4305*c^2*d*e^3 - 1260*e^4)*x)/(12600*c^9*e) + (b*(5*c^6*d^3 + 750*c^4*d^2*e - 1071*c^2*d*e^2 + 420*e^3)*x*(d + e*x^2))/(12600*c^7*e) - (b*(25*c^4*d^2 - 135*c^2*d*e + 84*e^2)*x*(d + e*x^2)^2)/(4200*c^5*e) - (b*(23*c^2*d - 36*e)*x*(d + e*x^2)^3)/(2520*c^3*e) - (b*x*(d + e*x^2)^4)/(90*c*e) + (b*(c^2*d - e)^4*(c^2*d + 4*e)*ArcTan[c*x])/(40*c^10*e^2) - (d*(d + e*x^2)^4*(a + b*ArcTan[c*x]))/(8*e^2) + ((d + e*x^2)^5*(a + b*ArcTan[c*x]))/(10*e^2)","A",8,7,21,0.3333,1,"{266, 43, 4976, 12, 528, 388, 203}"
1138,1,239,0,0.384412,"\int x^2 \left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^2*(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]","\frac{3}{5} d^2 e x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{3}{7} d e^2 x^7 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{9} e^3 x^9 \left(a+b \tan ^{-1}(c x)\right)-\frac{b e x^4 \left(189 c^4 d^2-135 c^2 d e+35 e^2\right)}{1260 c^5}-\frac{b x^2 \left(-189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2-35 e^3\right)}{630 c^7}+\frac{b \left(-189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2-35 e^3\right) \log \left(c^2 x^2+1\right)}{630 c^9}-\frac{b e^2 x^6 \left(27 c^2 d-7 e\right)}{378 c^3}-\frac{b e^3 x^8}{72 c}","\frac{3}{5} d^2 e x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{3}{7} d e^2 x^7 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{9} e^3 x^9 \left(a+b \tan ^{-1}(c x)\right)-\frac{b e x^4 \left(189 c^4 d^2-135 c^2 d e+35 e^2\right)}{1260 c^5}-\frac{b x^2 \left(-189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2-35 e^3\right)}{630 c^7}+\frac{b \left(-189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2-35 e^3\right) \log \left(c^2 x^2+1\right)}{630 c^9}-\frac{b e^2 x^6 \left(27 c^2 d-7 e\right)}{378 c^3}-\frac{b e^3 x^8}{72 c}",1,"-(b*(105*c^6*d^3 - 189*c^4*d^2*e + 135*c^2*d*e^2 - 35*e^3)*x^2)/(630*c^7) - (b*e*(189*c^4*d^2 - 135*c^2*d*e + 35*e^2)*x^4)/(1260*c^5) - (b*(27*c^2*d - 7*e)*e^2*x^6)/(378*c^3) - (b*e^3*x^8)/(72*c) + (d^3*x^3*(a + b*ArcTan[c*x]))/3 + (3*d^2*e*x^5*(a + b*ArcTan[c*x]))/5 + (3*d*e^2*x^7*(a + b*ArcTan[c*x]))/7 + (e^3*x^9*(a + b*ArcTan[c*x]))/9 + (b*(105*c^6*d^3 - 189*c^4*d^2*e + 135*c^2*d*e^2 - 35*e^3)*Log[1 + c^2*x^2])/(630*c^9)","A",5,5,21,0.2381,1,"{270, 4976, 12, 1799, 1620}"
1139,1,158,0,0.1466625,"\int x \left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x*(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]","\frac{\left(d+e x^2\right)^4 \left(a+b \tan ^{-1}(c x)\right)}{8 e}-\frac{b e x^3 \left(6 c^4 d^2-4 c^2 d e+e^2\right)}{24 c^5}-\frac{b x \left(2 c^2 d-e\right) \left(2 c^4 d^2-2 c^2 d e+e^2\right)}{8 c^7}-\frac{b e^2 x^5 \left(4 c^2 d-e\right)}{40 c^3}-\frac{b \left(c^2 d-e\right)^4 \tan ^{-1}(c x)}{8 c^8 e}-\frac{b e^3 x^7}{56 c}","\frac{\left(d+e x^2\right)^4 \left(a+b \tan ^{-1}(c x)\right)}{8 e}-\frac{b e x^3 \left(6 c^4 d^2-4 c^2 d e+e^2\right)}{24 c^5}-\frac{b x \left(2 c^2 d-e\right) \left(2 c^4 d^2-2 c^2 d e+e^2\right)}{8 c^7}-\frac{b e^2 x^5 \left(4 c^2 d-e\right)}{40 c^3}-\frac{b \left(c^2 d-e\right)^4 \tan ^{-1}(c x)}{8 c^8 e}-\frac{b e^3 x^7}{56 c}",1,"-(b*(2*c^2*d - e)*(2*c^4*d^2 - 2*c^2*d*e + e^2)*x)/(8*c^7) - (b*e*(6*c^4*d^2 - 4*c^2*d*e + e^2)*x^3)/(24*c^5) - (b*(4*c^2*d - e)*e^2*x^5)/(40*c^3) - (b*e^3*x^7)/(56*c) - (b*(c^2*d - e)^4*ArcTan[c*x])/(8*c^8*e) + ((d + e*x^2)^4*(a + b*ArcTan[c*x]))/(8*e)","A",4,3,19,0.1579,1,"{4974, 390, 203}"
1140,1,188,0,0.1505366,"\int \left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]","d^2 e x^3 \left(a+b \tan ^{-1}(c x)\right)+d^3 x \left(a+b \tan ^{-1}(c x)\right)+\frac{3}{5} d e^2 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{7} e^3 x^7 \left(a+b \tan ^{-1}(c x)\right)-\frac{b e x^2 \left(35 c^4 d^2-21 c^2 d e+5 e^2\right)}{70 c^5}-\frac{b \left(-35 c^4 d^2 e+35 c^6 d^3+21 c^2 d e^2-5 e^3\right) \log \left(c^2 x^2+1\right)}{70 c^7}-\frac{b e^2 x^4 \left(21 c^2 d-5 e\right)}{140 c^3}-\frac{b e^3 x^6}{42 c}","d^2 e x^3 \left(a+b \tan ^{-1}(c x)\right)+d^3 x \left(a+b \tan ^{-1}(c x)\right)+\frac{3}{5} d e^2 x^5 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{7} e^3 x^7 \left(a+b \tan ^{-1}(c x)\right)-\frac{b e x^2 \left(35 c^4 d^2-21 c^2 d e+5 e^2\right)}{70 c^5}-\frac{b \left(-35 c^4 d^2 e+35 c^6 d^3+21 c^2 d e^2-5 e^3\right) \log \left(c^2 x^2+1\right)}{70 c^7}-\frac{b e^2 x^4 \left(21 c^2 d-5 e\right)}{140 c^3}-\frac{b e^3 x^6}{42 c}",1,"-(b*e*(35*c^4*d^2 - 21*c^2*d*e + 5*e^2)*x^2)/(70*c^5) - (b*(21*c^2*d - 5*e)*e^2*x^4)/(140*c^3) - (b*e^3*x^6)/(42*c) + d^3*x*(a + b*ArcTan[c*x]) + d^2*e*x^3*(a + b*ArcTan[c*x]) + (3*d*e^2*x^5*(a + b*ArcTan[c*x]))/5 + (e^3*x^7*(a + b*ArcTan[c*x]))/7 - (b*(35*c^6*d^3 - 35*c^4*d^2*e + 21*c^2*d*e^2 - 5*e^3)*Log[1 + c^2*x^2])/(70*c^7)","A",4,4,18,0.2222,1,"{194, 4912, 1810, 260}"
1141,1,228,0,0.2212789,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Int[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x,x]","\frac{1}{2} i b d^3 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^3 \text{PolyLog}(2,i c x)+\frac{3}{2} d^2 e x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{3}{4} d e^2 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{6} e^3 x^6 \left(a+b \tan ^{-1}(c x)\right)+a d^3 \log (x)+\frac{3 b d^2 e \tan ^{-1}(c x)}{2 c^2}+\frac{3 b d e^2 x}{4 c^3}-\frac{3 b d e^2 \tan ^{-1}(c x)}{4 c^4}+\frac{b e^3 x^3}{18 c^3}-\frac{b e^3 x}{6 c^5}+\frac{b e^3 \tan ^{-1}(c x)}{6 c^6}-\frac{3 b d^2 e x}{2 c}-\frac{b d e^2 x^3}{4 c}-\frac{b e^3 x^5}{30 c}","\frac{1}{2} i b d^3 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d^3 \text{PolyLog}(2,i c x)+\frac{3}{2} d^2 e x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{3}{4} d e^2 x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{6} e^3 x^6 \left(a+b \tan ^{-1}(c x)\right)+a d^3 \log (x)+\frac{3 b d^2 e \tan ^{-1}(c x)}{2 c^2}+\frac{3 b d e^2 x}{4 c^3}-\frac{3 b d e^2 \tan ^{-1}(c x)}{4 c^4}+\frac{b e^3 x^3}{18 c^3}-\frac{b e^3 x}{6 c^5}+\frac{b e^3 \tan ^{-1}(c x)}{6 c^6}-\frac{3 b d^2 e x}{2 c}-\frac{b d e^2 x^3}{4 c}-\frac{b e^3 x^5}{30 c}",1,"(-3*b*d^2*e*x)/(2*c) + (3*b*d*e^2*x)/(4*c^3) - (b*e^3*x)/(6*c^5) - (b*d*e^2*x^3)/(4*c) + (b*e^3*x^3)/(18*c^3) - (b*e^3*x^5)/(30*c) + (3*b*d^2*e*ArcTan[c*x])/(2*c^2) - (3*b*d*e^2*ArcTan[c*x])/(4*c^4) + (b*e^3*ArcTan[c*x])/(6*c^6) + (3*d^2*e*x^2*(a + b*ArcTan[c*x]))/2 + (3*d*e^2*x^4*(a + b*ArcTan[c*x]))/4 + (e^3*x^6*(a + b*ArcTan[c*x]))/6 + a*d^3*Log[x] + (I/2)*b*d^3*PolyLog[2, (-I)*c*x] - (I/2)*b*d^3*PolyLog[2, I*c*x]","A",16,7,21,0.3333,1,"{4980, 4848, 2391, 4852, 321, 203, 302}"
1142,1,160,0,0.2581388,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Int[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^2,x]","3 d^2 e x \left(a+b \tan ^{-1}(c x)\right)-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{x}+d e^2 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{5} e^3 x^5 \left(a+b \tan ^{-1}(c x)\right)-\frac{b \left(15 c^4 d^2 e+5 c^6 d^3-5 c^2 d e^2+e^3\right) \log \left(c^2 x^2+1\right)}{10 c^5}-\frac{b e^2 x^2 \left(5 c^2 d-e\right)}{10 c^3}+b c d^3 \log (x)-\frac{b e^3 x^4}{20 c}","3 d^2 e x \left(a+b \tan ^{-1}(c x)\right)-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{x}+d e^2 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{5} e^3 x^5 \left(a+b \tan ^{-1}(c x)\right)-\frac{b \left(15 c^4 d^2 e+5 c^6 d^3-5 c^2 d e^2+e^3\right) \log \left(c^2 x^2+1\right)}{10 c^5}-\frac{b e^2 x^2 \left(5 c^2 d-e\right)}{10 c^3}+b c d^3 \log (x)-\frac{b e^3 x^4}{20 c}",1,"-(b*(5*c^2*d - e)*e^2*x^2)/(10*c^3) - (b*e^3*x^4)/(20*c) - (d^3*(a + b*ArcTan[c*x]))/x + 3*d^2*e*x*(a + b*ArcTan[c*x]) + d*e^2*x^3*(a + b*ArcTan[c*x]) + (e^3*x^5*(a + b*ArcTan[c*x]))/5 + b*c*d^3*Log[x] - (b*(5*c^6*d^3 + 15*c^4*d^2*e - 5*c^2*d*e^2 + e^3)*Log[1 + c^2*x^2])/(10*c^5)","A",4,4,21,0.1905,1,"{270, 4976, 1799, 1620}"
1143,1,200,0,0.2105373,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Int[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^3,x]","\frac{3}{2} i b d^2 e \text{PolyLog}(2,-i c x)-\frac{3}{2} i b d^2 e \text{PolyLog}(2,i c x)-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}+\frac{3}{2} d e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} e^3 x^4 \left(a+b \tan ^{-1}(c x)\right)+3 a d^2 e \log (x)-\frac{1}{2} b c^2 d^3 \tan ^{-1}(c x)+\frac{3 b d e^2 \tan ^{-1}(c x)}{2 c^2}+\frac{b e^3 x}{4 c^3}-\frac{b e^3 \tan ^{-1}(c x)}{4 c^4}-\frac{b c d^3}{2 x}-\frac{3 b d e^2 x}{2 c}-\frac{b e^3 x^3}{12 c}","\frac{3}{2} i b d^2 e \text{PolyLog}(2,-i c x)-\frac{3}{2} i b d^2 e \text{PolyLog}(2,i c x)-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}+\frac{3}{2} d e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{4} e^3 x^4 \left(a+b \tan ^{-1}(c x)\right)+3 a d^2 e \log (x)-\frac{1}{2} b c^2 d^3 \tan ^{-1}(c x)+\frac{3 b d e^2 \tan ^{-1}(c x)}{2 c^2}+\frac{b e^3 x}{4 c^3}-\frac{b e^3 \tan ^{-1}(c x)}{4 c^4}-\frac{b c d^3}{2 x}-\frac{3 b d e^2 x}{2 c}-\frac{b e^3 x^3}{12 c}",1,"-(b*c*d^3)/(2*x) - (3*b*d*e^2*x)/(2*c) + (b*e^3*x)/(4*c^3) - (b*e^3*x^3)/(12*c) - (b*c^2*d^3*ArcTan[c*x])/2 + (3*b*d*e^2*ArcTan[c*x])/(2*c^2) - (b*e^3*ArcTan[c*x])/(4*c^4) - (d^3*(a + b*ArcTan[c*x]))/(2*x^2) + (3*d*e^2*x^2*(a + b*ArcTan[c*x]))/2 + (e^3*x^4*(a + b*ArcTan[c*x]))/4 + 3*a*d^2*e*Log[x] + ((3*I)/2)*b*d^2*e*PolyLog[2, (-I)*c*x] - ((3*I)/2)*b*d^2*e*PolyLog[2, I*c*x]","A",15,8,21,0.3810,1,"{4980, 4852, 325, 203, 4848, 2391, 321, 302}"
1144,1,158,0,0.2646646,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Int[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^4,x]","-\frac{3 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}+3 d e^2 x \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} e^3 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{b \left(c^2 d+e\right) \left(c^4 d^2-10 c^2 d e+e^2\right) \log \left(c^2 x^2+1\right)}{6 c^3}-\frac{1}{3} b c d^2 \log (x) \left(c^2 d-9 e\right)-\frac{b c d^3}{6 x^2}-\frac{b e^3 x^2}{6 c}","-\frac{3 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}+3 d e^2 x \left(a+b \tan ^{-1}(c x)\right)+\frac{1}{3} e^3 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{b \left(c^2 d+e\right) \left(c^4 d^2-10 c^2 d e+e^2\right) \log \left(c^2 x^2+1\right)}{6 c^3}-\frac{1}{3} b c d^2 \log (x) \left(c^2 d-9 e\right)-\frac{b c d^3}{6 x^2}-\frac{b e^3 x^2}{6 c}",1,"-(b*c*d^3)/(6*x^2) - (b*e^3*x^2)/(6*c) - (d^3*(a + b*ArcTan[c*x]))/(3*x^3) - (3*d^2*e*(a + b*ArcTan[c*x]))/x + 3*d*e^2*x*(a + b*ArcTan[c*x]) + (e^3*x^3*(a + b*ArcTan[c*x]))/3 - (b*c*d^2*(c^2*d - 9*e)*Log[x])/3 + (b*(c^2*d + e)*(c^4*d^2 - 10*c^2*d*e + e^2)*Log[1 + c^2*x^2])/(6*c^3)","A",5,5,21,0.2381,1,"{270, 4976, 12, 1799, 1620}"
1145,1,200,0,0.2070304,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Int[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^5,x]","\frac{3}{2} i b d e^2 \text{PolyLog}(2,-i c x)-\frac{3}{2} i b d e^2 \text{PolyLog}(2,i c x)-\frac{3 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}+\frac{1}{2} e^3 x^2 \left(a+b \tan ^{-1}(c x)\right)+3 a d e^2 \log (x)-\frac{3}{2} b c^2 d^2 e \tan ^{-1}(c x)+\frac{b c^3 d^3}{4 x}+\frac{1}{4} b c^4 d^3 \tan ^{-1}(c x)+\frac{b e^3 \tan ^{-1}(c x)}{2 c^2}-\frac{3 b c d^2 e}{2 x}-\frac{b c d^3}{12 x^3}-\frac{b e^3 x}{2 c}","\frac{3}{2} i b d e^2 \text{PolyLog}(2,-i c x)-\frac{3}{2} i b d e^2 \text{PolyLog}(2,i c x)-\frac{3 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}+\frac{1}{2} e^3 x^2 \left(a+b \tan ^{-1}(c x)\right)+3 a d e^2 \log (x)-\frac{3}{2} b c^2 d^2 e \tan ^{-1}(c x)+\frac{b c^3 d^3}{4 x}+\frac{1}{4} b c^4 d^3 \tan ^{-1}(c x)+\frac{b e^3 \tan ^{-1}(c x)}{2 c^2}-\frac{3 b c d^2 e}{2 x}-\frac{b c d^3}{12 x^3}-\frac{b e^3 x}{2 c}",1,"-(b*c*d^3)/(12*x^3) + (b*c^3*d^3)/(4*x) - (3*b*c*d^2*e)/(2*x) - (b*e^3*x)/(2*c) + (b*c^4*d^3*ArcTan[c*x])/4 - (3*b*c^2*d^2*e*ArcTan[c*x])/2 + (b*e^3*ArcTan[c*x])/(2*c^2) - (d^3*(a + b*ArcTan[c*x]))/(4*x^4) - (3*d^2*e*(a + b*ArcTan[c*x]))/(2*x^2) + (e^3*x^2*(a + b*ArcTan[c*x]))/2 + 3*a*d*e^2*Log[x] + ((3*I)/2)*b*d*e^2*PolyLog[2, (-I)*c*x] - ((3*I)/2)*b*d*e^2*PolyLog[2, I*c*x]","A",15,7,21,0.3333,1,"{4980, 4852, 325, 203, 4848, 2391, 321}"
1146,1,177,0,0.285452,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Int[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^6,x]","-\frac{d^2 e \left(a+b \tan ^{-1}(c x)\right)}{x^3}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{3 d e^2 \left(a+b \tan ^{-1}(c x)\right)}{x}+e^3 x \left(a+b \tan ^{-1}(c x)\right)-\frac{b \left(-5 c^4 d^2 e+c^6 d^3+15 c^2 d e^2+5 e^3\right) \log \left(c^2 x^2+1\right)}{10 c}+\frac{1}{5} b c d \log (x) \left(c^4 d^2-5 c^2 d e+15 e^2\right)+\frac{b c d^2 \left(c^2 d-5 e\right)}{10 x^2}-\frac{b c d^3}{20 x^4}","-\frac{d^2 e \left(a+b \tan ^{-1}(c x)\right)}{x^3}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{3 d e^2 \left(a+b \tan ^{-1}(c x)\right)}{x}+e^3 x \left(a+b \tan ^{-1}(c x)\right)-\frac{b \left(-5 c^4 d^2 e+c^6 d^3+15 c^2 d e^2+5 e^3\right) \log \left(c^2 x^2+1\right)}{10 c}+\frac{1}{5} b c d \log (x) \left(c^4 d^2-5 c^2 d e+15 e^2\right)+\frac{b c d^2 \left(c^2 d-5 e\right)}{10 x^2}-\frac{b c d^3}{20 x^4}",1,"-(b*c*d^3)/(20*x^4) + (b*c*d^2*(c^2*d - 5*e))/(10*x^2) - (d^3*(a + b*ArcTan[c*x]))/(5*x^5) - (d^2*e*(a + b*ArcTan[c*x]))/x^3 - (3*d*e^2*(a + b*ArcTan[c*x]))/x + e^3*x*(a + b*ArcTan[c*x]) + (b*c*d*(c^4*d^2 - 5*c^2*d*e + 15*e^2)*Log[x])/5 - (b*(c^6*d^3 - 5*c^4*d^2*e + 15*c^2*d*e^2 + 5*e^3)*Log[1 + c^2*x^2])/(10*c)","A",5,5,21,0.2381,1,"{270, 4976, 12, 1799, 1620}"
1147,1,228,0,0.2308826,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^7} \, dx","Int[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^7,x]","\frac{1}{2} i b e^3 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b e^3 \text{PolyLog}(2,i c x)-\frac{3 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{6 x^6}-\frac{3 d e^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}+a e^3 \log (x)+\frac{3 b c^3 d^2 e}{4 x}+\frac{3}{4} b c^4 d^2 e \tan ^{-1}(c x)+\frac{b c^3 d^3}{18 x^3}-\frac{b c^5 d^3}{6 x}-\frac{1}{6} b c^6 d^3 \tan ^{-1}(c x)-\frac{3}{2} b c^2 d e^2 \tan ^{-1}(c x)-\frac{b c d^2 e}{4 x^3}-\frac{b c d^3}{30 x^5}-\frac{3 b c d e^2}{2 x}","\frac{1}{2} i b e^3 \text{PolyLog}(2,-i c x)-\frac{1}{2} i b e^3 \text{PolyLog}(2,i c x)-\frac{3 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{6 x^6}-\frac{3 d e^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}+a e^3 \log (x)+\frac{3 b c^3 d^2 e}{4 x}+\frac{3}{4} b c^4 d^2 e \tan ^{-1}(c x)+\frac{b c^3 d^3}{18 x^3}-\frac{b c^5 d^3}{6 x}-\frac{1}{6} b c^6 d^3 \tan ^{-1}(c x)-\frac{3}{2} b c^2 d e^2 \tan ^{-1}(c x)-\frac{b c d^2 e}{4 x^3}-\frac{b c d^3}{30 x^5}-\frac{3 b c d e^2}{2 x}",1,"-(b*c*d^3)/(30*x^5) + (b*c^3*d^3)/(18*x^3) - (b*c*d^2*e)/(4*x^3) - (b*c^5*d^3)/(6*x) + (3*b*c^3*d^2*e)/(4*x) - (3*b*c*d*e^2)/(2*x) - (b*c^6*d^3*ArcTan[c*x])/6 + (3*b*c^4*d^2*e*ArcTan[c*x])/4 - (3*b*c^2*d*e^2*ArcTan[c*x])/2 - (d^3*(a + b*ArcTan[c*x]))/(6*x^6) - (3*d^2*e*(a + b*ArcTan[c*x]))/(4*x^4) - (3*d*e^2*(a + b*ArcTan[c*x]))/(2*x^2) + a*e^3*Log[x] + (I/2)*b*e^3*PolyLog[2, (-I)*c*x] - (I/2)*b*e^3*PolyLog[2, I*c*x]","A",17,6,21,0.2857,1,"{4980, 4852, 325, 203, 4848, 2391}"
1148,1,224,0,0.3270385,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^8} \, dx","Int[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^8,x]","-\frac{3 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{7 x^7}-\frac{d e^2 \left(a+b \tan ^{-1}(c x)\right)}{x^3}-\frac{e^3 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{b c d \left(5 c^4 d^2-21 c^2 d e+35 e^2\right)}{70 x^2}+\frac{1}{70} b c \left(-21 c^4 d^2 e+5 c^6 d^3+35 c^2 d e^2-35 e^3\right) \log \left(c^2 x^2+1\right)-\frac{1}{35} b c \log (x) \left(-21 c^4 d^2 e+5 c^6 d^3+35 c^2 d e^2-35 e^3\right)+\frac{b c d^2 \left(5 c^2 d-21 e\right)}{140 x^4}-\frac{b c d^3}{42 x^6}","-\frac{3 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{7 x^7}-\frac{d e^2 \left(a+b \tan ^{-1}(c x)\right)}{x^3}-\frac{e^3 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{b c d \left(5 c^4 d^2-21 c^2 d e+35 e^2\right)}{70 x^2}+\frac{1}{70} b c \left(-21 c^4 d^2 e+5 c^6 d^3+35 c^2 d e^2-35 e^3\right) \log \left(c^2 x^2+1\right)-\frac{1}{35} b c \log (x) \left(-21 c^4 d^2 e+5 c^6 d^3+35 c^2 d e^2-35 e^3\right)+\frac{b c d^2 \left(5 c^2 d-21 e\right)}{140 x^4}-\frac{b c d^3}{42 x^6}",1,"-(b*c*d^3)/(42*x^6) + (b*c*d^2*(5*c^2*d - 21*e))/(140*x^4) - (b*c*d*(5*c^4*d^2 - 21*c^2*d*e + 35*e^2))/(70*x^2) - (d^3*(a + b*ArcTan[c*x]))/(7*x^7) - (3*d^2*e*(a + b*ArcTan[c*x]))/(5*x^5) - (d*e^2*(a + b*ArcTan[c*x]))/x^3 - (e^3*(a + b*ArcTan[c*x]))/x - (b*c*(5*c^6*d^3 - 21*c^4*d^2*e + 35*c^2*d*e^2 - 35*e^3)*Log[x])/35 + (b*c*(5*c^6*d^3 - 21*c^4*d^2*e + 35*c^2*d*e^2 - 35*e^3)*Log[1 + c^2*x^2])/70","A",5,5,21,0.2381,1,"{270, 4976, 12, 1799, 1620}"
1149,1,152,0,0.195143,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^9} \, dx","Int[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^9,x]","-\frac{\left(d+e x^2\right)^4 \left(a+b \tan ^{-1}(c x)\right)}{8 d x^8}-\frac{b c d \left(c^4 d^2-4 c^2 d e+6 e^2\right)}{24 x^3}+\frac{b c \left(c^2 d-2 e\right) \left(c^4 d^2-2 c^2 d e+2 e^2\right)}{8 x}+\frac{b c d^2 \left(c^2 d-4 e\right)}{40 x^5}+\frac{b \left(c^2 d-e\right)^4 \tan ^{-1}(c x)}{8 d}-\frac{b c d^3}{56 x^7}","-\frac{\left(d+e x^2\right)^4 \left(a+b \tan ^{-1}(c x)\right)}{8 d x^8}-\frac{b c d \left(c^4 d^2-4 c^2 d e+6 e^2\right)}{24 x^3}+\frac{b c \left(c^2 d-2 e\right) \left(c^4 d^2-2 c^2 d e+2 e^2\right)}{8 x}+\frac{b c d^2 \left(c^2 d-4 e\right)}{40 x^5}+\frac{b \left(c^2 d-e\right)^4 \tan ^{-1}(c x)}{8 d}-\frac{b c d^3}{56 x^7}",1,"-(b*c*d^3)/(56*x^7) + (b*c*d^2*(c^2*d - 4*e))/(40*x^5) - (b*c*d*(c^4*d^2 - 4*c^2*d*e + 6*e^2))/(24*x^3) + (b*c*(c^2*d - 2*e)*(c^4*d^2 - 2*c^2*d*e + 2*e^2))/(8*x) + (b*(c^2*d - e)^4*ArcTan[c*x])/(8*d) - ((d + e*x^2)^4*(a + b*ArcTan[c*x]))/(8*d*x^8)","A",5,5,21,0.2381,1,"{264, 4976, 12, 461, 203}"
1150,1,244,0,0.176337,"\int \left(c+d x^2\right)^4 \tan ^{-1}(a x) \, dx","Int[(c + d*x^2)^4*ArcTan[a*x],x]","-\frac{d^2 x^4 \left(378 a^4 c^2-180 a^2 c d+35 d^2\right)}{1260 a^5}-\frac{d x^2 \left(-378 a^4 c^2 d+420 a^6 c^3+180 a^2 c d^2-35 d^3\right)}{630 a^7}-\frac{\left(378 a^4 c^2 d^2-420 a^6 c^3 d+315 a^8 c^4-180 a^2 c d^3+35 d^4\right) \log \left(a^2 x^2+1\right)}{630 a^9}-\frac{d^3 x^6 \left(36 a^2 c-7 d\right)}{378 a^3}+\frac{6}{5} c^2 d^2 x^5 \tan ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \tan ^{-1}(a x)+c^4 x \tan ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \tan ^{-1}(a x)-\frac{d^4 x^8}{72 a}+\frac{1}{9} d^4 x^9 \tan ^{-1}(a x)","-\frac{d^2 x^4 \left(378 a^4 c^2-180 a^2 c d+35 d^2\right)}{1260 a^5}-\frac{d x^2 \left(-378 a^4 c^2 d+420 a^6 c^3+180 a^2 c d^2-35 d^3\right)}{630 a^7}-\frac{\left(378 a^4 c^2 d^2-420 a^6 c^3 d+315 a^8 c^4-180 a^2 c d^3+35 d^4\right) \log \left(a^2 x^2+1\right)}{630 a^9}-\frac{d^3 x^6 \left(36 a^2 c-7 d\right)}{378 a^3}+\frac{6}{5} c^2 d^2 x^5 \tan ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \tan ^{-1}(a x)+c^4 x \tan ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \tan ^{-1}(a x)-\frac{d^4 x^8}{72 a}+\frac{1}{9} d^4 x^9 \tan ^{-1}(a x)",1,"-(d*(420*a^6*c^3 - 378*a^4*c^2*d + 180*a^2*c*d^2 - 35*d^3)*x^2)/(630*a^7) - (d^2*(378*a^4*c^2 - 180*a^2*c*d + 35*d^2)*x^4)/(1260*a^5) - ((36*a^2*c - 7*d)*d^3*x^6)/(378*a^3) - (d^4*x^8)/(72*a) + c^4*x*ArcTan[a*x] + (4*c^3*d*x^3*ArcTan[a*x])/3 + (6*c^2*d^2*x^5*ArcTan[a*x])/5 + (4*c*d^3*x^7*ArcTan[a*x])/7 + (d^4*x^9*ArcTan[a*x])/9 - ((315*a^8*c^4 - 420*a^6*c^3*d + 378*a^4*c^2*d^2 - 180*a^2*c*d^3 + 35*d^4)*Log[1 + a^2*x^2])/(630*a^9)","A",4,4,14,0.2857,1,"{194, 4912, 1810, 260}"
1151,1,361,0,0.3696315,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{d+e x^2} \, dx","Int[(x^3*(a + b*ArcTan[c*x]))/(d + e*x^2),x]","-\frac{i b d \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e^2}+\frac{i b d \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 e^2}+\frac{i b d \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 e^2}+\frac{d \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e^2}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 e}+\frac{b \tan ^{-1}(c x)}{2 c^2 e}-\frac{b x}{2 c e}","-\frac{i b d \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e^2}+\frac{i b d \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 e^2}+\frac{i b d \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 e^2}+\frac{d \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e^2}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 e}+\frac{b \tan ^{-1}(c x)}{2 c^2 e}-\frac{b x}{2 c e}",1,"-(b*x)/(2*c*e) + (b*ArcTan[c*x])/(2*c^2*e) + (x^2*(a + b*ArcTan[c*x]))/(2*e) + (d*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e^2 - (d*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) - (d*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) - ((I/2)*b*d*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^2 + ((I/4)*b*d*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/e^2 + ((I/4)*b*d*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/e^2","A",14,9,21,0.4286,1,"{4916, 4852, 321, 203, 4980, 4856, 2402, 2315, 2447}"
1152,1,311,0,0.2433349,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{d+e x^2} \, dx","Int[(x*(a + b*ArcTan[c*x]))/(d + e*x^2),x]","-\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 e}-\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 e}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e}+\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e}+\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e}","-\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 e}-\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 e}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e}+\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e}+\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e}",1,"-(((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e) + ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e) + ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e) + ((I/2)*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/e - ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/e - ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/e","A",10,5,19,0.2632,1,"{4980, 4856, 2402, 2315, 2447}"
1153,1,353,0,0.3855247,"\int \frac{a+b \tan ^{-1}(c x)}{x \left(d+e x^2\right)} \, dx","Int[(a + b*ArcTan[c*x])/(x*(d + e*x^2)),x]","\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 d}+\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 d}+\frac{i b \text{PolyLog}(2,-i c x)}{2 d}-\frac{i b \text{PolyLog}(2,i c x)}{2 d}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d}-\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d}-\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d}+\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{a \log (x)}{d}","\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 d}+\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 d}+\frac{i b \text{PolyLog}(2,-i c x)}{2 d}-\frac{i b \text{PolyLog}(2,i c x)}{2 d}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d}-\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d}-\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d}+\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{a \log (x)}{d}",1,"(a*Log[x])/d + ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d) - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d) + ((I/2)*b*PolyLog[2, (-I)*c*x])/d - ((I/2)*b*PolyLog[2, I*c*x])/d - ((I/2)*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/d + ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d + ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d","A",15,8,21,0.3810,1,"{4928, 4848, 2391, 4980, 4856, 2402, 2315, 2447}"
1154,1,409,0,0.4823757,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 \left(d+e x^2\right)} \, dx","Int[(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)),x]","-\frac{i b e \text{PolyLog}(2,-i c x)}{2 d^2}+\frac{i b e \text{PolyLog}(2,i c x)}{2 d^2}+\frac{i b e \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d^2}-\frac{i b e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 d^2}-\frac{i b e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 d^2}-\frac{e \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d^2}+\frac{e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d^2}-\frac{a+b \tan ^{-1}(c x)}{2 d x^2}-\frac{a e \log (x)}{d^2}-\frac{b c^2 \tan ^{-1}(c x)}{2 d}-\frac{b c}{2 d x}","-\frac{i b e \text{PolyLog}(2,-i c x)}{2 d^2}+\frac{i b e \text{PolyLog}(2,i c x)}{2 d^2}+\frac{i b e \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d^2}-\frac{i b e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 d^2}-\frac{i b e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 d^2}-\frac{e \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d^2}+\frac{e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d^2}-\frac{a+b \tan ^{-1}(c x)}{2 d x^2}-\frac{a e \log (x)}{d^2}-\frac{b c^2 \tan ^{-1}(c x)}{2 d}-\frac{b c}{2 d x}",1,"-(b*c)/(2*d*x) - (b*c^2*ArcTan[c*x])/(2*d) - (a + b*ArcTan[c*x])/(2*d*x^2) - (a*e*Log[x])/d^2 - (e*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d^2 + (e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) + (e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) - ((I/2)*b*e*PolyLog[2, (-I)*c*x])/d^2 + ((I/2)*b*e*PolyLog[2, I*c*x])/d^2 + ((I/2)*b*e*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^2 - ((I/4)*b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d^2 - ((I/4)*b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d^2","A",19,12,21,0.5714,1,"{4918, 4852, 325, 203, 4928, 4848, 2391, 4980, 4856, 2402, 2315, 2447}"
1155,1,555,0,0.6308969,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{d+e x^2} \, dx","Int[(x^2*(a + b*ArcTan[c*x]))/(d + e*x^2),x]","\frac{i b \sqrt{-d} \text{PolyLog}\left(2,\frac{\sqrt{e} (-c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 e^{3/2}}-\frac{i b \sqrt{-d} \text{PolyLog}\left(2,\frac{\sqrt{e} (1-i c x)}{\sqrt{e}+i c \sqrt{-d}}\right)}{4 e^{3/2}}-\frac{i b \sqrt{-d} \text{PolyLog}\left(2,\frac{\sqrt{e} (1+i c x)}{\sqrt{e}+i c \sqrt{-d}}\right)}{4 e^{3/2}}+\frac{i b \sqrt{-d} \text{PolyLog}\left(2,\frac{\sqrt{e} (c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 e^{3/2}}-\frac{a \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{3/2}}+\frac{a x}{e}-\frac{b \log \left(c^2 x^2+1\right)}{2 c e}-\frac{i b \sqrt{-d} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 e^{3/2}}+\frac{i b \sqrt{-d} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 e^{3/2}}-\frac{i b \sqrt{-d} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 e^{3/2}}+\frac{i b \sqrt{-d} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 e^{3/2}}+\frac{b x \tan ^{-1}(c x)}{e}","\frac{i b \sqrt{-d} \text{PolyLog}\left(2,\frac{\sqrt{e} (-c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 e^{3/2}}-\frac{i b \sqrt{-d} \text{PolyLog}\left(2,\frac{\sqrt{e} (1-i c x)}{\sqrt{e}+i c \sqrt{-d}}\right)}{4 e^{3/2}}-\frac{i b \sqrt{-d} \text{PolyLog}\left(2,\frac{\sqrt{e} (1+i c x)}{\sqrt{e}+i c \sqrt{-d}}\right)}{4 e^{3/2}}+\frac{i b \sqrt{-d} \text{PolyLog}\left(2,\frac{\sqrt{e} (c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 e^{3/2}}-\frac{a \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{3/2}}+\frac{a x}{e}-\frac{b \log \left(c^2 x^2+1\right)}{2 c e}-\frac{i b \sqrt{-d} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 e^{3/2}}+\frac{i b \sqrt{-d} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 e^{3/2}}-\frac{i b \sqrt{-d} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 e^{3/2}}+\frac{i b \sqrt{-d} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 e^{3/2}}+\frac{b x \tan ^{-1}(c x)}{e}",1,"(a*x)/e + (b*x*ArcTan[c*x])/e - (a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(3/2) - ((I/4)*b*Sqrt[-d]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/e^(3/2) + ((I/4)*b*Sqrt[-d]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/e^(3/2) - ((I/4)*b*Sqrt[-d]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/e^(3/2) + ((I/4)*b*Sqrt[-d]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/e^(3/2) - (b*Log[1 + c^2*x^2])/(2*c*e) + ((I/4)*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/e^(3/2) - ((I/4)*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/e^(3/2) - ((I/4)*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/e^(3/2) + ((I/4)*b*Sqrt[-d]*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/e^(3/2)","A",23,10,21,0.4762,1,"{4916, 4846, 260, 4910, 205, 4908, 2409, 2394, 2393, 2391}"
1156,1,517,0,0.4059129,"\int \frac{a+b \tan ^{-1}(c x)}{d+e x^2} \, dx","Int[(a + b*ArcTan[c*x])/(d + e*x^2),x]","\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (-c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (1-i c x)}{\sqrt{e}+i c \sqrt{-d}}\right)}{4 \sqrt{-d} \sqrt{e}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (1+i c x)}{\sqrt{e}+i c \sqrt{-d}}\right)}{4 \sqrt{-d} \sqrt{e}}+\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} \sqrt{e}}-\frac{i b \log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}+\frac{i b \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}-\frac{i b \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}+\frac{i b \log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}","\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (-c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (1-i c x)}{\sqrt{e}+i c \sqrt{-d}}\right)}{4 \sqrt{-d} \sqrt{e}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (1+i c x)}{\sqrt{e}+i c \sqrt{-d}}\right)}{4 \sqrt{-d} \sqrt{e}}+\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}+\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} \sqrt{e}}-\frac{i b \log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}+\frac{i b \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}-\frac{i b \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}+\frac{i b \log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}",1,"(a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*Sqrt[e]) - ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) + ((I/4)*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) - ((I/4)*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) + ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) + ((I/4)*b*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) - ((I/4)*b*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) - ((I/4)*b*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(Sqrt[-d]*Sqrt[e]) + ((I/4)*b*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*Sqrt[e])","A",19,7,18,0.3889,1,"{4910, 205, 4908, 2409, 2394, 2393, 2391}"
1157,1,561,0,0.5284722,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 \left(d+e x^2\right)} \, dx","Int[(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)),x]","\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (-c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (1-i c x)}{\sqrt{e}+i c \sqrt{-d}}\right)}{4 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (1+i c x)}{\sqrt{e}+i c \sqrt{-d}}\right)}{4 (-d)^{3/2}}+\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 (-d)^{3/2}}-\frac{a+b \tan ^{-1}(c x)}{d x}-\frac{a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{3/2}}-\frac{b c \log \left(c^2 x^2+1\right)}{2 d}-\frac{i b \sqrt{e} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 (-d)^{3/2}}+\frac{i b \sqrt{e} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 (-d)^{3/2}}-\frac{i b \sqrt{e} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 (-d)^{3/2}}+\frac{i b \sqrt{e} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 (-d)^{3/2}}+\frac{b c \log (x)}{d}","\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (-c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (1-i c x)}{\sqrt{e}+i c \sqrt{-d}}\right)}{4 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (1+i c x)}{\sqrt{e}+i c \sqrt{-d}}\right)}{4 (-d)^{3/2}}+\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (c x+i)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 (-d)^{3/2}}-\frac{a+b \tan ^{-1}(c x)}{d x}-\frac{a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{3/2}}-\frac{b c \log \left(c^2 x^2+1\right)}{2 d}-\frac{i b \sqrt{e} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 (-d)^{3/2}}+\frac{i b \sqrt{e} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 (-d)^{3/2}}-\frac{i b \sqrt{e} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 (-d)^{3/2}}+\frac{i b \sqrt{e} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)}{4 (-d)^{3/2}}+\frac{b c \log (x)}{d}",1,"-((a + b*ArcTan[c*x])/(d*x)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + (b*c*Log[x])/d - ((I/4)*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(-d)^(3/2) + ((I/4)*b*Sqrt[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(3/2) - ((I/4)*b*Sqrt[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(-d)^(3/2) + ((I/4)*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(3/2) - (b*c*Log[1 + c^2*x^2])/(2*d) + ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(3/2) - ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(-d)^(3/2) - ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(-d)^(3/2) + ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(3/2)","A",25,13,21,0.6190,1,"{4918, 4852, 266, 36, 29, 31, 4910, 205, 4908, 2409, 2394, 2393, 2391}"
1158,1,403,0,0.448857,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Int[(x^3*(a + b*ArcTan[c*x]))/(d + e*x^2)^2,x]","-\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 e^2}-\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 e^2}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e^2}+\frac{d \left(a+b \tan ^{-1}(c x)\right)}{2 e^2 \left(d+e x^2\right)}+\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e^2}+\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e^2}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^2}-\frac{b c^2 d \tan ^{-1}(c x)}{2 e^2 \left(c^2 d-e\right)}+\frac{b c \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 e^{3/2} \left(c^2 d-e\right)}","-\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 e^2}-\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 e^2}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e^2}+\frac{d \left(a+b \tan ^{-1}(c x)\right)}{2 e^2 \left(d+e x^2\right)}+\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e^2}+\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e^2}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^2}-\frac{b c^2 d \tan ^{-1}(c x)}{2 e^2 \left(c^2 d-e\right)}+\frac{b c \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 e^{3/2} \left(c^2 d-e\right)}",1,"-(b*c^2*d*ArcTan[c*x])/(2*(c^2*d - e)*e^2) + (d*(a + b*ArcTan[c*x]))/(2*e^2*(d + e*x^2)) + (b*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*(c^2*d - e)*e^(3/2)) - ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e^2 + ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) + ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) + ((I/2)*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^2 - ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/e^2 - ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/e^2","A",16,9,21,0.4286,1,"{4980, 4974, 391, 203, 205, 4856, 2402, 2315, 2447}"
1159,1,91,0,0.0662689,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Int[(x*(a + b*ArcTan[c*x]))/(d + e*x^2)^2,x]","-\frac{a+b \tan ^{-1}(c x)}{2 e \left(d+e x^2\right)}+\frac{b c^2 \tan ^{-1}(c x)}{2 e \left(c^2 d-e\right)}-\frac{b c \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 \sqrt{d} \sqrt{e} \left(c^2 d-e\right)}","-\frac{a+b \tan ^{-1}(c x)}{2 e \left(d+e x^2\right)}+\frac{b c^2 \tan ^{-1}(c x)}{2 e \left(c^2 d-e\right)}-\frac{b c \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 \sqrt{d} \sqrt{e} \left(c^2 d-e\right)}",1,"(b*c^2*ArcTan[c*x])/(2*(c^2*d - e)*e) - (a + b*ArcTan[c*x])/(2*e*(d + e*x^2)) - (b*c*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*Sqrt[d]*(c^2*d - e)*Sqrt[e])","A",4,4,19,0.2105,1,"{4974, 391, 203, 205}"
1160,1,443,0,0.4894097,"\int \frac{a+b \tan ^{-1}(c x)}{x \left(d+e x^2\right)^2} \, dx","Int[(a + b*ArcTan[c*x])/(x*(d + e*x^2)^2),x]","\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 d^2}+\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 d^2}+\frac{i b \text{PolyLog}(2,-i c x)}{2 d^2}-\frac{i b \text{PolyLog}(2,i c x)}{2 d^2}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d^2}+\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{a+b \tan ^{-1}(c x)}{2 d \left(d+e x^2\right)}+\frac{a \log (x)}{d^2}+\frac{b c \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{3/2} \left(c^2 d-e\right)}-\frac{b c^2 \tan ^{-1}(c x)}{2 d \left(c^2 d-e\right)}","\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 d^2}+\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 d^2}+\frac{i b \text{PolyLog}(2,-i c x)}{2 d^2}-\frac{i b \text{PolyLog}(2,i c x)}{2 d^2}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d^2}+\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{a+b \tan ^{-1}(c x)}{2 d \left(d+e x^2\right)}+\frac{a \log (x)}{d^2}+\frac{b c \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{3/2} \left(c^2 d-e\right)}-\frac{b c^2 \tan ^{-1}(c x)}{2 d \left(c^2 d-e\right)}",1,"-(b*c^2*ArcTan[c*x])/(2*d*(c^2*d - e)) + (a + b*ArcTan[c*x])/(2*d*(d + e*x^2)) + (b*c*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*(c^2*d - e)) + (a*Log[x])/d^2 + ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d^2 - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) + ((I/2)*b*PolyLog[2, (-I)*c*x])/d^2 - ((I/2)*b*PolyLog[2, I*c*x])/d^2 - ((I/2)*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^2 + ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d^2 + ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d^2","A",19,11,21,0.5238,1,"{4980, 4848, 2391, 4974, 391, 203, 205, 4856, 2402, 2315, 2447}"
1161,1,489,0,0.5127792,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 \left(d+e x^2\right)^2} \, dx","Int[(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)^2),x]","-\frac{i b e \text{PolyLog}(2,-i c x)}{d^3}+\frac{i b e \text{PolyLog}(2,i c x)}{d^3}+\frac{i b e \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{d^3}-\frac{i b e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d^3}-\frac{i b e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d^3}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{2 d^2 \left(d+e x^2\right)}-\frac{2 e \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}+\frac{e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{d^3}+\frac{e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{d^3}-\frac{a+b \tan ^{-1}(c x)}{2 d^2 x^2}-\frac{2 a e \log (x)}{d^3}-\frac{b c e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{5/2} \left(c^2 d-e\right)}+\frac{b c^2 e \tan ^{-1}(c x)}{2 d^2 \left(c^2 d-e\right)}-\frac{b c^2 \tan ^{-1}(c x)}{2 d^2}-\frac{b c}{2 d^2 x}","-\frac{i b e \text{PolyLog}(2,-i c x)}{d^3}+\frac{i b e \text{PolyLog}(2,i c x)}{d^3}+\frac{i b e \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{d^3}-\frac{i b e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d^3}-\frac{i b e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d^3}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{2 d^2 \left(d+e x^2\right)}-\frac{2 e \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}+\frac{e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{d^3}+\frac{e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{d^3}-\frac{a+b \tan ^{-1}(c x)}{2 d^2 x^2}-\frac{2 a e \log (x)}{d^3}-\frac{b c e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{5/2} \left(c^2 d-e\right)}+\frac{b c^2 e \tan ^{-1}(c x)}{2 d^2 \left(c^2 d-e\right)}-\frac{b c^2 \tan ^{-1}(c x)}{2 d^2}-\frac{b c}{2 d^2 x}",1,"-(b*c)/(2*d^2*x) - (b*c^2*ArcTan[c*x])/(2*d^2) + (b*c^2*e*ArcTan[c*x])/(2*d^2*(c^2*d - e)) - (a + b*ArcTan[c*x])/(2*d^2*x^2) - (e*(a + b*ArcTan[c*x]))/(2*d^2*(d + e*x^2)) - (b*c*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(5/2)*(c^2*d - e)) - (2*a*e*Log[x])/d^3 - (2*e*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d^3 + (e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d^3 + (e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d^3 - (I*b*e*PolyLog[2, (-I)*c*x])/d^3 + (I*b*e*PolyLog[2, I*c*x])/d^3 + (I*b*e*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^3 - ((I/2)*b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d^3 - ((I/2)*b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d^3","A",22,13,21,0.6190,1,"{4980, 4852, 325, 203, 4848, 2391, 4974, 391, 205, 4856, 2402, 2315, 2447}"
1162,1,1335,0,1.964143,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Int[(x^2*(a + b*ArcTan[c*x]))/(d + e*x^2)^2,x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 \sqrt{d} e^{3/2}}-\frac{x \left(a+b \tan ^{-1}(c x)\right)}{2 e \left(e x^2+d\right)}+\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} e^{3/2}}-\frac{i b \log (i c x+1) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b \log (1-i c x) \log \left(\frac{c \left(\sqrt{e} x+\sqrt{-d}\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (i c x+1) \log \left(\frac{c \left(\sqrt{e} x+\sqrt{-d}\right)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b c \log \left(\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{i b c \log \left(-\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{i b c \log \left(-\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}-\frac{i b c \log \left(\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{b c \log \left(c^2 x^2+1\right)}{4 \left(c^2 d-e\right) e}-\frac{b c \log \left(e x^2+d\right)}{4 \left(c^2 d-e\right) e}+\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}-\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 \sqrt{d} e^{3/2}}-\frac{x \left(a+b \tan ^{-1}(c x)\right)}{2 e \left(e x^2+d\right)}+\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} e^{3/2}}-\frac{i b \log (i c x+1) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b \log (1-i c x) \log \left(\frac{c \left(\sqrt{e} x+\sqrt{-d}\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{i b \log (i c x+1) \log \left(\frac{c \left(\sqrt{e} x+\sqrt{-d}\right)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b c \log \left(\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{i b c \log \left(-\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{i b c \log \left(-\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}-\frac{i b c \log \left(\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{b c \log \left(c^2 x^2+1\right)}{4 \left(c^2 d-e\right) e}-\frac{b c \log \left(e x^2+d\right)}{4 \left(c^2 d-e\right) e}+\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}-\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}+\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right)}{8 \sqrt{-c^2} \sqrt{d} e^{3/2}}",1,"-(x*(a + b*ArcTan[c*x]))/(2*e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*e^(3/2)) - ((a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*Sqrt[d]*e^(3/2)) - ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/4)*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/8)*b*c*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + ((I/8)*b*c*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + ((I/8)*b*c*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) - ((I/8)*b*c*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + (b*c*Log[1 + c^2*x^2])/(4*(c^2*d - e)*e) - (b*c*Log[d + e*x^2])/(4*(c^2*d - e)*e) + ((I/4)*b*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/4)*b*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/4)*b*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(Sqrt[-d]*e^(3/2)) - ((I/8)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + ((I/8)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) - ((I/8)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2)) + ((I/8)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*Sqrt[d]*e^(3/2))","A",45,14,21,0.6667,1,"{4980, 199, 205, 4912, 6725, 444, 36, 31, 4908, 2409, 2394, 2393, 2391, 4910}"
1163,1,819,0,0.8928975,"\int \frac{a+b \tan ^{-1}(c x)}{\left(d+e x^2\right)^2} \, dx","Int[(a + b*ArcTan[c*x])/(d + e*x^2)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 d^{3/2} \sqrt{e}}+\frac{x \left(a+b \tan ^{-1}(c x)\right)}{2 d \left(e x^2+d\right)}+\frac{i b c \log \left(\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \log \left(-\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \log \left(-\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}+\frac{i b c \log \left(\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{b c \log \left(c^2 x^2+1\right)}{4 d \left(c^2 d-e\right)}+\frac{b c \log \left(e x^2+d\right)}{4 d \left(c^2 d-e\right)}+\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}+\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}","\frac{\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 d^{3/2} \sqrt{e}}+\frac{x \left(a+b \tan ^{-1}(c x)\right)}{2 d \left(e x^2+d\right)}+\frac{i b c \log \left(\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \log \left(-\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \log \left(-\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}+\frac{i b c \log \left(\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{b c \log \left(c^2 x^2+1\right)}{4 d \left(c^2 d-e\right)}+\frac{b c \log \left(e x^2+d\right)}{4 d \left(c^2 d-e\right)}+\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}+\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}-\frac{i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{3/2} \sqrt{e}}",1,"(x*(a + b*ArcTan[c*x]))/(2*d*(d + e*x^2)) + ((a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*Sqrt[e]) + ((I/8)*b*c*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*Sqrt[e]) - ((I/8)*b*c*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*Sqrt[e]) - ((I/8)*b*c*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*Sqrt[e]) + ((I/8)*b*c*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*Sqrt[e]) - (b*c*Log[1 + c^2*x^2])/(4*d*(c^2*d - e)) + (b*c*Log[d + e*x^2])/(4*d*(c^2*d - e)) + ((I/8)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*Sqrt[e]) - ((I/8)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*Sqrt[e]) + ((I/8)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*Sqrt[e]) - ((I/8)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*Sqrt[e])","A",24,12,18,0.6667,1,"{199, 205, 4912, 6725, 444, 36, 31, 4908, 2409, 2394, 2393, 2391}"
1164,1,1382,0,1.5803684,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 \left(d+e x^2\right)^2} \, dx","Int[(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^2),x]","-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 d^{5/2}}-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{e x \left(a+b \tan ^{-1}(c x)\right)}{2 d^2 \left(e x^2+d\right)}-\frac{a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{5/2}}+\frac{b c \log (x)}{d^2}+\frac{i b \sqrt{e} \log (i c x+1) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \log (1-i c x) \log \left(\frac{c \left(\sqrt{e} x+\sqrt{-d}\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (i c x+1) \log \left(\frac{c \left(\sqrt{e} x+\sqrt{-d}\right)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 (-d)^{5/2}}-\frac{i b c \sqrt{e} \log \left(\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \log \left(-\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \log \left(-\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right)}{8 \sqrt{-c^2} d^{5/2}}-\frac{i b c \sqrt{e} \log \left(\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right)}{8 \sqrt{-c^2} d^{5/2}}+\frac{b c e \log \left(c^2 x^2+1\right)}{4 d^2 \left(c^2 d-e\right)}-\frac{b c \log \left(c^2 x^2+1\right)}{2 d^2}-\frac{b c e \log \left(e x^2+d\right)}{4 d^2 \left(c^2 d-e\right)}-\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 (-d)^{5/2}}-\frac{i b c \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{5/2}}-\frac{i b c \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{5/2}}","-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 d^{5/2}}-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{e x \left(a+b \tan ^{-1}(c x)\right)}{2 d^2 \left(e x^2+d\right)}-\frac{a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{5/2}}+\frac{b c \log (x)}{d^2}+\frac{i b \sqrt{e} \log (i c x+1) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \log (1-i c x) \log \left(\frac{c \left(\sqrt{e} x+\sqrt{-d}\right)}{c \sqrt{-d}-i \sqrt{e}}\right)}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \log (i c x+1) \log \left(\frac{c \left(\sqrt{e} x+\sqrt{-d}\right)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 (-d)^{5/2}}-\frac{i b c \sqrt{e} \log \left(\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \log \left(-\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \log \left(-\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right)}{8 \sqrt{-c^2} d^{5/2}}-\frac{i b c \sqrt{e} \log \left(\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right)}{8 \sqrt{-c^2} d^{5/2}}+\frac{b c e \log \left(c^2 x^2+1\right)}{4 d^2 \left(c^2 d-e\right)}-\frac{b c \log \left(c^2 x^2+1\right)}{2 d^2}-\frac{b c e \log \left(e x^2+d\right)}{4 d^2 \left(c^2 d-e\right)}-\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 (-d)^{5/2}}-\frac{i b c \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{5/2}}-\frac{i b c \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{5/2}}+\frac{i b c \sqrt{e} \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right)}{8 \sqrt{-c^2} d^{5/2}}",1,"-((a + b*ArcTan[c*x])/(d^2*x)) - (e*x*(a + b*ArcTan[c*x]))/(2*d^2*(d + e*x^2)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(5/2) - (Sqrt[e]*(a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(5/2)) + (b*c*Log[x])/d^2 + ((I/4)*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(-d)^(5/2) - ((I/4)*b*Sqrt[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(5/2) + ((I/4)*b*Sqrt[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(-d)^(5/2) - ((I/4)*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(5/2) - ((I/8)*b*c*Sqrt[e]*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)) + ((I/8)*b*c*Sqrt[e]*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)) + ((I/8)*b*c*Sqrt[e]*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)) - ((I/8)*b*c*Sqrt[e]*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)) - (b*c*Log[1 + c^2*x^2])/(2*d^2) + (b*c*e*Log[1 + c^2*x^2])/(4*d^2*(c^2*d - e)) - (b*c*e*Log[d + e*x^2])/(4*d^2*(c^2*d - e)) - ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(5/2) + ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(-d)^(5/2) + ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(-d)^(5/2) - ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(5/2) - ((I/8)*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)) + ((I/8)*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)) - ((I/8)*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)) + ((I/8)*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2))","A",50,17,21,0.8095,1,"{4980, 4852, 266, 36, 29, 31, 199, 205, 4912, 6725, 444, 4908, 2409, 2394, 2393, 2391, 4910}"
1165,1,532,0,0.6501022,"\int \frac{x^5 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Int[(x^5*(a + b*ArcTan[c*x]))/(d + e*x^2)^3,x]","-\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 e^3}-\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 e^3}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e^3}-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{4 e^3 \left(d+e x^2\right)^2}+\frac{d \left(a+b \tan ^{-1}(c x)\right)}{e^3 \left(d+e x^2\right)}+\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e^3}+\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e^3}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^3}+\frac{b c^4 d^2 \tan ^{-1}(c x)}{4 e^3 \left(c^2 d-e\right)^2}-\frac{b c d x}{8 e^2 \left(c^2 d-e\right) \left(d+e x^2\right)}-\frac{b c^2 d \tan ^{-1}(c x)}{e^3 \left(c^2 d-e\right)}-\frac{b c \sqrt{d} \left(3 c^2 d-e\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 e^{5/2} \left(c^2 d-e\right)^2}+\frac{b c \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{5/2} \left(c^2 d-e\right)}","-\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 e^3}-\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 e^3}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 e^3}-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{4 e^3 \left(d+e x^2\right)^2}+\frac{d \left(a+b \tan ^{-1}(c x)\right)}{e^3 \left(d+e x^2\right)}+\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e^3}+\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e^3}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^3}+\frac{b c^4 d^2 \tan ^{-1}(c x)}{4 e^3 \left(c^2 d-e\right)^2}-\frac{b c d x}{8 e^2 \left(c^2 d-e\right) \left(d+e x^2\right)}-\frac{b c^2 d \tan ^{-1}(c x)}{e^3 \left(c^2 d-e\right)}-\frac{b c \sqrt{d} \left(3 c^2 d-e\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 e^{5/2} \left(c^2 d-e\right)^2}+\frac{b c \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{5/2} \left(c^2 d-e\right)}",1,"-(b*c*d*x)/(8*(c^2*d - e)*e^2*(d + e*x^2)) + (b*c^4*d^2*ArcTan[c*x])/(4*(c^2*d - e)^2*e^3) - (b*c^2*d*ArcTan[c*x])/((c^2*d - e)*e^3) - (d^2*(a + b*ArcTan[c*x]))/(4*e^3*(d + e*x^2)^2) + (d*(a + b*ArcTan[c*x]))/(e^3*(d + e*x^2)) + (b*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/((c^2*d - e)*e^(5/2)) - (b*c*Sqrt[d]*(3*c^2*d - e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*(c^2*d - e)^2*e^(5/2)) - ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/e^3 + ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^3) + ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^3) + ((I/2)*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^3 - ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/e^3 - ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/e^3","A",21,11,21,0.5238,1,"{4980, 4974, 414, 522, 203, 205, 391, 4856, 2402, 2315, 2447}"
1166,1,130,0,0.1859468,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Int[(x^3*(a + b*ArcTan[c*x]))/(d + e*x^2)^3,x]","\frac{x^4 \left(a+b \tan ^{-1}(c x)\right)}{4 d \left(d+e x^2\right)^2}-\frac{b c \left(c^2 d-3 e\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{d} e^{3/2} \left(c^2 d-e\right)^2}+\frac{b c x}{8 e \left(c^2 d-e\right) \left(d+e x^2\right)}-\frac{b \tan ^{-1}(c x)}{4 d \left(c^2 d-e\right)^2}","\frac{x^4 \left(a+b \tan ^{-1}(c x)\right)}{4 d \left(d+e x^2\right)^2}-\frac{b c \left(c^2 d-3 e\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{d} e^{3/2} \left(c^2 d-e\right)^2}+\frac{b c x}{8 e \left(c^2 d-e\right) \left(d+e x^2\right)}-\frac{b \tan ^{-1}(c x)}{4 d \left(c^2 d-e\right)^2}",1,"(b*c*x)/(8*(c^2*d - e)*e*(d + e*x^2)) - (b*ArcTan[c*x])/(4*d*(c^2*d - e)^2) + (x^4*(a + b*ArcTan[c*x]))/(4*d*(d + e*x^2)^2) - (b*c*(c^2*d - 3*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[d]*(c^2*d - e)^2*e^(3/2))","A",6,6,21,0.2857,1,"{264, 4976, 12, 470, 522, 205}"
1167,1,131,0,0.1134006,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Int[(x*(a + b*ArcTan[c*x]))/(d + e*x^2)^3,x]","-\frac{a+b \tan ^{-1}(c x)}{4 e \left(d+e x^2\right)^2}-\frac{b c \left(3 c^2 d-e\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{3/2} \sqrt{e} \left(c^2 d-e\right)^2}-\frac{b c x}{8 d \left(c^2 d-e\right) \left(d+e x^2\right)}+\frac{b c^4 \tan ^{-1}(c x)}{4 e \left(c^2 d-e\right)^2}","-\frac{a+b \tan ^{-1}(c x)}{4 e \left(d+e x^2\right)^2}-\frac{b c \left(3 c^2 d-e\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{3/2} \sqrt{e} \left(c^2 d-e\right)^2}-\frac{b c x}{8 d \left(c^2 d-e\right) \left(d+e x^2\right)}+\frac{b c^4 \tan ^{-1}(c x)}{4 e \left(c^2 d-e\right)^2}",1,"-(b*c*x)/(8*d*(c^2*d - e)*(d + e*x^2)) + (b*c^4*ArcTan[c*x])/(4*(c^2*d - e)^2*e) - (a + b*ArcTan[c*x])/(4*e*(d + e*x^2)^2) - (b*c*(3*c^2*d - e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*(c^2*d - e)^2*Sqrt[e])","A",5,5,19,0.2632,1,"{4974, 414, 522, 203, 205}"
1168,1,574,0,0.6311106,"\int \frac{a+b \tan ^{-1}(c x)}{x \left(d+e x^2\right)^3} \, dx","Int[(a + b*ArcTan[c*x])/(x*(d + e*x^2)^3),x]","\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 d^3}+\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 d^3}+\frac{i b \text{PolyLog}(2,-i c x)}{2 d^3}-\frac{i b \text{PolyLog}(2,i c x)}{2 d^3}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d^3}+\frac{a+b \tan ^{-1}(c x)}{2 d^2 \left(d+e x^2\right)}-\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d^3}-\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d^3}+\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}+\frac{a+b \tan ^{-1}(c x)}{4 d \left(d+e x^2\right)^2}+\frac{a \log (x)}{d^3}+\frac{b c e x}{8 d^2 \left(c^2 d-e\right) \left(d+e x^2\right)}-\frac{b c^2 \tan ^{-1}(c x)}{2 d^2 \left(c^2 d-e\right)}+\frac{b c \sqrt{e} \left(3 c^2 d-e\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{5/2} \left(c^2 d-e\right)^2}+\frac{b c \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{5/2} \left(c^2 d-e\right)}-\frac{b c^4 \tan ^{-1}(c x)}{4 d \left(c^2 d-e\right)^2}","\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 d^3}+\frac{i b \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 d^3}+\frac{i b \text{PolyLog}(2,-i c x)}{2 d^3}-\frac{i b \text{PolyLog}(2,i c x)}{2 d^3}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d^3}+\frac{a+b \tan ^{-1}(c x)}{2 d^2 \left(d+e x^2\right)}-\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d^3}-\frac{\left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d^3}+\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}+\frac{a+b \tan ^{-1}(c x)}{4 d \left(d+e x^2\right)^2}+\frac{a \log (x)}{d^3}+\frac{b c e x}{8 d^2 \left(c^2 d-e\right) \left(d+e x^2\right)}-\frac{b c^2 \tan ^{-1}(c x)}{2 d^2 \left(c^2 d-e\right)}+\frac{b c \sqrt{e} \left(3 c^2 d-e\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{5/2} \left(c^2 d-e\right)^2}+\frac{b c \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{5/2} \left(c^2 d-e\right)}-\frac{b c^4 \tan ^{-1}(c x)}{4 d \left(c^2 d-e\right)^2}",1,"(b*c*e*x)/(8*d^2*(c^2*d - e)*(d + e*x^2)) - (b*c^4*ArcTan[c*x])/(4*d*(c^2*d - e)^2) - (b*c^2*ArcTan[c*x])/(2*d^2*(c^2*d - e)) + (a + b*ArcTan[c*x])/(4*d*(d + e*x^2)^2) + (a + b*ArcTan[c*x])/(2*d^2*(d + e*x^2)) + (b*c*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(5/2)*(c^2*d - e)) + (b*c*(3*c^2*d - e)*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*(c^2*d - e)^2) + (a*Log[x])/d^3 + ((a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d^3 - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^3) - ((a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^3) + ((I/2)*b*PolyLog[2, (-I)*c*x])/d^3 - ((I/2)*b*PolyLog[2, I*c*x])/d^3 - ((I/2)*b*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^3 + ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d^3 + ((I/4)*b*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d^3","A",24,13,21,0.6190,1,"{4980, 4848, 2391, 4974, 414, 522, 203, 205, 391, 4856, 2402, 2315, 2447}"
1169,1,629,0,0.6801349,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 \left(d+e x^2\right)^3} \, dx","Int[(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)^3),x]","-\frac{3 i b e \text{PolyLog}(2,-i c x)}{2 d^4}+\frac{3 i b e \text{PolyLog}(2,i c x)}{2 d^4}+\frac{3 i b e \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d^4}-\frac{3 i b e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 d^4}-\frac{3 i b e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 d^4}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{d^3 \left(d+e x^2\right)}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{4 d^2 \left(d+e x^2\right)^2}-\frac{3 e \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^4}+\frac{3 e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d^4}+\frac{3 e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d^4}-\frac{a+b \tan ^{-1}(c x)}{2 d^3 x^2}-\frac{3 a e \log (x)}{d^4}-\frac{b c e^2 x}{8 d^3 \left(c^2 d-e\right) \left(d+e x^2\right)}-\frac{b c e^{3/2} \left(3 c^2 d-e\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{7/2} \left(c^2 d-e\right)^2}-\frac{b c e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{7/2} \left(c^2 d-e\right)}+\frac{b c^4 e \tan ^{-1}(c x)}{4 d^2 \left(c^2 d-e\right)^2}+\frac{b c^2 e \tan ^{-1}(c x)}{d^3 \left(c^2 d-e\right)}-\frac{b c^2 \tan ^{-1}(c x)}{2 d^3}-\frac{b c}{2 d^3 x}","-\frac{3 i b e \text{PolyLog}(2,-i c x)}{2 d^4}+\frac{3 i b e \text{PolyLog}(2,i c x)}{2 d^4}+\frac{3 i b e \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 d^4}-\frac{3 i b e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 d^4}-\frac{3 i b e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 d^4}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{d^3 \left(d+e x^2\right)}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{4 d^2 \left(d+e x^2\right)^2}-\frac{3 e \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^4}+\frac{3 e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d^4}+\frac{3 e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d^4}-\frac{a+b \tan ^{-1}(c x)}{2 d^3 x^2}-\frac{3 a e \log (x)}{d^4}-\frac{b c e^2 x}{8 d^3 \left(c^2 d-e\right) \left(d+e x^2\right)}-\frac{b c e^{3/2} \left(3 c^2 d-e\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{7/2} \left(c^2 d-e\right)^2}-\frac{b c e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{7/2} \left(c^2 d-e\right)}+\frac{b c^4 e \tan ^{-1}(c x)}{4 d^2 \left(c^2 d-e\right)^2}+\frac{b c^2 e \tan ^{-1}(c x)}{d^3 \left(c^2 d-e\right)}-\frac{b c^2 \tan ^{-1}(c x)}{2 d^3}-\frac{b c}{2 d^3 x}",1,"-(b*c)/(2*d^3*x) - (b*c*e^2*x)/(8*d^3*(c^2*d - e)*(d + e*x^2)) - (b*c^2*ArcTan[c*x])/(2*d^3) + (b*c^4*e*ArcTan[c*x])/(4*d^2*(c^2*d - e)^2) + (b*c^2*e*ArcTan[c*x])/(d^3*(c^2*d - e)) - (a + b*ArcTan[c*x])/(2*d^3*x^2) - (e*(a + b*ArcTan[c*x]))/(4*d^2*(d + e*x^2)^2) - (e*(a + b*ArcTan[c*x]))/(d^3*(d + e*x^2)) - (b*c*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(7/2)*(c^2*d - e)) - (b*c*(3*c^2*d - e)*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(7/2)*(c^2*d - e)^2) - (3*a*e*Log[x])/d^4 - (3*e*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/d^4 + (3*e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^4) + (3*e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^4) - (((3*I)/2)*b*e*PolyLog[2, (-I)*c*x])/d^4 + (((3*I)/2)*b*e*PolyLog[2, I*c*x])/d^4 + (((3*I)/2)*b*e*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^4 - (((3*I)/4)*b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d^4 - (((3*I)/4)*b*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d^4","A",27,15,21,0.7143,1,"{4980, 4852, 325, 203, 4848, 2391, 4974, 414, 522, 205, 391, 4856, 2402, 2315, 2447}"
1170,1,966,0,2.2601574,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Int[(x^2*(a + b*ArcTan[c*x]))/(d + e*x^2)^3,x]","\frac{i b \log \left(\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}-\frac{i b \log \left(-\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}-\frac{i b \log \left(-\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}+\frac{i b \log \left(\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}-\frac{b \log \left(c^2 x^2+1\right) c}{4 d \left(c^2 d-e\right) e}+\frac{b \left(5 c^2 d-3 e\right) \log \left(c^2 x^2+1\right) c}{16 d \left(c^2 d-e\right)^2 e}+\frac{b \log \left(e x^2+d\right) c}{4 d \left(c^2 d-e\right) e}-\frac{b \left(5 c^2 d-3 e\right) \log \left(e x^2+d\right) c}{16 d \left(c^2 d-e\right)^2 e}+\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}+\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}+\frac{b c}{8 \left(c^2 d-e\right) e \left(e x^2+d\right)}+\frac{x \left(a+b \tan ^{-1}(c x)\right)}{8 d e \left(e x^2+d\right)}-\frac{x \left(a+b \tan ^{-1}(c x)\right)}{4 e \left(e x^2+d\right)^2}+\frac{\left(a+b \tan ^{-1}(c x)\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{3/2} e^{3/2}}","\frac{i b \log \left(\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}-\frac{i b \log \left(-\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}-\frac{i b \log \left(-\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}+\frac{i b \log \left(\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}-\frac{b \log \left(c^2 x^2+1\right) c}{4 d \left(c^2 d-e\right) e}+\frac{b \left(5 c^2 d-3 e\right) \log \left(c^2 x^2+1\right) c}{16 d \left(c^2 d-e\right)^2 e}+\frac{b \log \left(e x^2+d\right) c}{4 d \left(c^2 d-e\right) e}-\frac{b \left(5 c^2 d-3 e\right) \log \left(e x^2+d\right) c}{16 d \left(c^2 d-e\right)^2 e}+\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}+\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{3/2} e^{3/2}}+\frac{b c}{8 \left(c^2 d-e\right) e \left(e x^2+d\right)}+\frac{x \left(a+b \tan ^{-1}(c x)\right)}{8 d e \left(e x^2+d\right)}-\frac{x \left(a+b \tan ^{-1}(c x)\right)}{4 e \left(e x^2+d\right)^2}+\frac{\left(a+b \tan ^{-1}(c x)\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{3/2} e^{3/2}}",1,"(b*c)/(8*(c^2*d - e)*e*(d + e*x^2)) - (x*(a + b*ArcTan[c*x]))/(4*e*(d + e*x^2)^2) + (x*(a + b*ArcTan[c*x]))/(8*d*e*(d + e*x^2)) + ((a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) + ((I/32)*b*c*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) - ((I/32)*b*c*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) - ((I/32)*b*c*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) + ((I/32)*b*c*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) + (b*c*(5*c^2*d - 3*e)*Log[1 + c^2*x^2])/(16*d*(c^2*d - e)^2*e) - (b*c*Log[1 + c^2*x^2])/(4*d*(c^2*d - e)*e) - (b*c*(5*c^2*d - 3*e)*Log[d + e*x^2])/(16*d*(c^2*d - e)^2*e) + (b*c*Log[d + e*x^2])/(4*d*(c^2*d - e)*e) + ((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) - ((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) + ((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) - ((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*e^(3/2))","A",49,15,21,0.7143,1,"{4980, 199, 205, 4912, 6725, 571, 77, 4908, 2409, 2394, 2393, 2391, 444, 36, 31}"
1171,1,893,0,0.9485145,"\int \frac{a+b \tan ^{-1}(c x)}{\left(d+e x^2\right)^3} \, dx","Int[(a + b*ArcTan[c*x])/(d + e*x^2)^3,x]","\frac{3 i b \log \left(\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \log \left(-\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \log \left(-\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}+\frac{3 i b \log \left(\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{b \left(5 c^2 d-3 e\right) \log \left(c^2 x^2+1\right) c}{16 d^2 \left(c^2 d-e\right)^2}+\frac{b \left(5 c^2 d-3 e\right) \log \left(e x^2+d\right) c}{16 d^2 \left(c^2 d-e\right)^2}+\frac{3 i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}+\frac{3 i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{b c}{8 d \left(c^2 d-e\right) \left(e x^2+d\right)}+\frac{3 x \left(a+b \tan ^{-1}(c x)\right)}{8 d^2 \left(e x^2+d\right)}+\frac{x \left(a+b \tan ^{-1}(c x)\right)}{4 d \left(e x^2+d\right)^2}+\frac{3 \left(a+b \tan ^{-1}(c x)\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{5/2} \sqrt{e}}","\frac{3 i b \log \left(\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \log \left(-\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \log \left(-\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}+\frac{3 i b \log \left(\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{b \left(5 c^2 d-3 e\right) \log \left(c^2 x^2+1\right) c}{16 d^2 \left(c^2 d-e\right)^2}+\frac{b \left(5 c^2 d-3 e\right) \log \left(e x^2+d\right) c}{16 d^2 \left(c^2 d-e\right)^2}+\frac{3 i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}+\frac{3 i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{3 i b \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right) c}{32 \sqrt{-c^2} d^{5/2} \sqrt{e}}-\frac{b c}{8 d \left(c^2 d-e\right) \left(e x^2+d\right)}+\frac{3 x \left(a+b \tan ^{-1}(c x)\right)}{8 d^2 \left(e x^2+d\right)}+\frac{x \left(a+b \tan ^{-1}(c x)\right)}{4 d \left(e x^2+d\right)^2}+\frac{3 \left(a+b \tan ^{-1}(c x)\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{5/2} \sqrt{e}}",1,"-(b*c)/(8*d*(c^2*d - e)*(d + e*x^2)) + (x*(a + b*ArcTan[c*x]))/(4*d*(d + e*x^2)^2) + (3*x*(a + b*ArcTan[c*x]))/(8*d^2*(d + e*x^2)) + (3*(a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]) + (((3*I)/32)*b*c*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (((3*I)/32)*b*c*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (((3*I)/32)*b*c*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) + (((3*I)/32)*b*c*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (b*c*(5*c^2*d - 3*e)*Log[1 + c^2*x^2])/(16*d^2*(c^2*d - e)^2) + (b*c*(5*c^2*d - 3*e)*Log[d + e*x^2])/(16*d^2*(c^2*d - e)^2) + (((3*I)/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (((3*I)/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) + (((3*I)/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e]) - (((3*I)/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)*Sqrt[e])","A",23,11,18,0.6111,1,"{199, 205, 4912, 6725, 571, 77, 4908, 2409, 2394, 2393, 2391}"
1172,1,1518,0,2.6359645,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 \left(d+e x^2\right)^3} \, dx","Int[(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^3),x]","-\frac{7 x \left(a+b \tan ^{-1}(c x)\right) e}{8 d^3 \left(e x^2+d\right)}-\frac{x \left(a+b \tan ^{-1}(c x)\right) e}{4 d^2 \left(e x^2+d\right)^2}+\frac{b c \log \left(c^2 x^2+1\right) e}{4 d^3 \left(c^2 d-e\right)}+\frac{b c \left(5 c^2 d-3 e\right) \log \left(c^2 x^2+1\right) e}{16 d^3 \left(c^2 d-e\right)^2}-\frac{b c \log \left(e x^2+d\right) e}{4 d^3 \left(c^2 d-e\right)}-\frac{b c \left(5 c^2 d-3 e\right) \log \left(e x^2+d\right) e}{16 d^3 \left(c^2 d-e\right)^2}+\frac{b c e}{8 d^2 \left(c^2 d-e\right) \left(e x^2+d\right)}-\frac{7 \left(a+b \tan ^{-1}(c x)\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \sqrt{e}}{8 d^{7/2}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \sqrt{e}}{d^{7/2}}-\frac{i b \log (i c x+1) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}+\frac{i b \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{\sqrt{-d} c+i \sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}-\frac{i b \log (1-i c x) \log \left(\frac{c \left(\sqrt{e} x+\sqrt{-d}\right)}{c \sqrt{-d}-i \sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}+\frac{i b \log (i c x+1) \log \left(\frac{c \left(\sqrt{e} x+\sqrt{-d}\right)}{\sqrt{-d} c+i \sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}-\frac{7 i b c \log \left(\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}+\frac{7 i b c \log \left(-\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}+\frac{7 i b c \log \left(-\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}-\frac{7 i b c \log \left(\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}+\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}+\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+i \sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}-\frac{7 i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}+\frac{7 i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}-\frac{7 i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}+\frac{7 i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}-\frac{a+b \tan ^{-1}(c x)}{d^3 x}+\frac{b c \log (x)}{d^3}-\frac{b c \log \left(c^2 x^2+1\right)}{2 d^3}","-\frac{7 x \left(a+b \tan ^{-1}(c x)\right) e}{8 d^3 \left(e x^2+d\right)}-\frac{x \left(a+b \tan ^{-1}(c x)\right) e}{4 d^2 \left(e x^2+d\right)^2}+\frac{b c \log \left(c^2 x^2+1\right) e}{4 d^3 \left(c^2 d-e\right)}+\frac{b c \left(5 c^2 d-3 e\right) \log \left(c^2 x^2+1\right) e}{16 d^3 \left(c^2 d-e\right)^2}-\frac{b c \log \left(e x^2+d\right) e}{4 d^3 \left(c^2 d-e\right)}-\frac{b c \left(5 c^2 d-3 e\right) \log \left(e x^2+d\right) e}{16 d^3 \left(c^2 d-e\right)^2}+\frac{b c e}{8 d^2 \left(c^2 d-e\right) \left(e x^2+d\right)}-\frac{7 \left(a+b \tan ^{-1}(c x)\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \sqrt{e}}{8 d^{7/2}}-\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) \sqrt{e}}{d^{7/2}}-\frac{i b \log (i c x+1) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}+\frac{i b \log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{\sqrt{-d} c+i \sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}-\frac{i b \log (1-i c x) \log \left(\frac{c \left(\sqrt{e} x+\sqrt{-d}\right)}{c \sqrt{-d}-i \sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}+\frac{i b \log (i c x+1) \log \left(\frac{c \left(\sqrt{e} x+\sqrt{-d}\right)}{\sqrt{-d} c+i \sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}-\frac{7 i b c \log \left(\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}+\frac{7 i b c \log \left(-\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(1-\frac{i \sqrt{e} x}{\sqrt{d}}\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}+\frac{7 i b c \log \left(-\frac{\sqrt{e} \left(1-\sqrt{-c^2} x\right)}{i \sqrt{-c^2} \sqrt{d}-\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}-\frac{7 i b c \log \left(\frac{\sqrt{e} \left(\sqrt{-c^2} x+1\right)}{i \sqrt{-c^2} \sqrt{d}+\sqrt{e}}\right) \log \left(\frac{i \sqrt{e} x}{\sqrt{d}}+1\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}+\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}-\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}+\frac{i b \text{PolyLog}\left(2,\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+i \sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}-\frac{7 i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}+\frac{7 i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}-\frac{7 i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}-i \sqrt{e}}\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}+\frac{7 i b c \text{PolyLog}\left(2,\frac{\sqrt{-c^2} \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{-c^2} \sqrt{d}+i \sqrt{e}}\right) \sqrt{e}}{32 \sqrt{-c^2} d^{7/2}}-\frac{a+b \tan ^{-1}(c x)}{d^3 x}+\frac{b c \log (x)}{d^3}-\frac{b c \log \left(c^2 x^2+1\right)}{2 d^3}",1,"(b*c*e)/(8*d^2*(c^2*d - e)*(d + e*x^2)) - (a + b*ArcTan[c*x])/(d^3*x) - (e*x*(a + b*ArcTan[c*x]))/(4*d^2*(d + e*x^2)^2) - (7*e*x*(a + b*ArcTan[c*x]))/(8*d^3*(d + e*x^2)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(7/2) - (7*Sqrt[e]*(a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(7/2)) + (b*c*Log[x])/d^3 - ((I/4)*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(-d)^(7/2) + ((I/4)*b*Sqrt[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(7/2) - ((I/4)*b*Sqrt[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(-d)^(7/2) + ((I/4)*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(7/2) - (((7*I)/32)*b*c*Sqrt[e]*Log[(Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(7/2)) + (((7*I)/32)*b*c*Sqrt[e]*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(7/2)) + (((7*I)/32)*b*c*Sqrt[e]*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(7/2)) - (((7*I)/32)*b*c*Sqrt[e]*Log[(Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(7/2)) - (b*c*Log[1 + c^2*x^2])/(2*d^3) + (b*c*(5*c^2*d - 3*e)*e*Log[1 + c^2*x^2])/(16*d^3*(c^2*d - e)^2) + (b*c*e*Log[1 + c^2*x^2])/(4*d^3*(c^2*d - e)) - (b*c*(5*c^2*d - 3*e)*e*Log[d + e*x^2])/(16*d^3*(c^2*d - e)^2) - (b*c*e*Log[d + e*x^2])/(4*d^3*(c^2*d - e)) + ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(7/2) - ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(-d)^(7/2) - ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(-d)^(7/2) + ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(7/2) - (((7*I)/32)*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(7/2)) + (((7*I)/32)*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(7/2)) - (((7*I)/32)*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(7/2)) + (((7*I)/32)*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(7/2))","A",73,19,21,0.9048,1,"{4980, 4852, 266, 36, 29, 31, 199, 205, 4912, 6725, 571, 77, 4908, 2409, 2394, 2393, 2391, 444, 4910}"
1173,1,223,0,0.3676281,"\int x^3 \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^3*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]),x]","\frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{5 e^2}-\frac{d \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{3 e^2}+\frac{b \left(15 c^4 d^2+20 c^2 d e-24 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{120 c^5 e^{3/2}}+\frac{b \left(c^2 d-e\right)^{3/2} \left(2 c^2 d+3 e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{15 c^5 e^2}-\frac{b x \left(c^2 d-12 e\right) \sqrt{d+e x^2}}{120 c^3 e}-\frac{b x \left(d+e x^2\right)^{3/2}}{20 c e}","\frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{5 e^2}-\frac{d \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{3 e^2}+\frac{b \left(15 c^4 d^2+20 c^2 d e-24 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{120 c^5 e^{3/2}}+\frac{b \left(c^2 d-e\right)^{3/2} \left(2 c^2 d+3 e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{15 c^5 e^2}-\frac{b x \left(c^2 d-12 e\right) \sqrt{d+e x^2}}{120 c^3 e}-\frac{b x \left(d+e x^2\right)^{3/2}}{20 c e}",1,"-(b*(c^2*d - 12*e)*x*Sqrt[d + e*x^2])/(120*c^3*e) - (b*x*(d + e*x^2)^(3/2))/(20*c*e) - (d*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(3*e^2) + ((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/(5*e^2) + (b*(c^2*d - e)^(3/2)*(2*c^2*d + 3*e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(15*c^5*e^2) + (b*(15*c^4*d^2 + 20*c^2*d*e - 24*e^2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(120*c^5*e^(3/2))","A",9,10,23,0.4348,1,"{266, 43, 4976, 12, 528, 523, 217, 206, 377, 203}"
1174,0,0,0,0.1581551,"\int x^2 \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^2*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]),x]","\int x^2 \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right) \, dx","b \text{Int}\left(x^2 \tan ^{-1}(c x) \sqrt{d+e x^2},x\right)-\frac{a d^2 \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{8 e^{3/2}}+\frac{1}{4} a x^3 \sqrt{d+e x^2}+\frac{a d x \sqrt{d+e x^2}}{8 e}",0,"(a*d*x*Sqrt[d + e*x^2])/(8*e) + (a*x^3*Sqrt[d + e*x^2])/4 - (a*d^2*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(8*e^(3/2)) + b*Defer[Int][x^2*Sqrt[d + e*x^2]*ArcTan[c*x], x]","A",0,0,0,0,-1,"{}"
1175,1,140,0,0.1422838,"\int x \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]),x]","\frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{3 e}-\frac{b \left(c^2 d-e\right)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{3 c^3 e}-\frac{b \left(3 c^2 d-2 e\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{6 c^3 \sqrt{e}}-\frac{b x \sqrt{d+e x^2}}{6 c}","\frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{3 e}-\frac{b \left(c^2 d-e\right)^{3/2} \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{3 c^3 e}-\frac{b \left(3 c^2 d-2 e\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{6 c^3 \sqrt{e}}-\frac{b x \sqrt{d+e x^2}}{6 c}",1,"-(b*x*Sqrt[d + e*x^2])/(6*c) + ((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(3*e) - (b*(c^2*d - e)^(3/2)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(3*c^3*e) - (b*(3*c^2*d - 2*e)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(6*c^3*Sqrt[e])","A",7,7,21,0.3333,1,"{4974, 416, 523, 217, 206, 377, 203}"
1176,0,0,0,0.0238289,"\int \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]),x]","\int \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right) \, dx","\text{Int}\left(\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right),x\right)",0,"Defer[Int][Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]), x]","A",0,0,0,0,-1,"{}"
1177,0,0,0,0.1630504,"\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Int[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x) \sqrt{d+e x^2}}{x},x\right)+a \sqrt{d+e x^2}+a \left(-\sqrt{d}\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)",0,"a*Sqrt[d + e*x^2] - a*Sqrt[d]*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + b*Defer[Int][(Sqrt[d + e*x^2]*ArcTan[c*x])/x, x]","A",0,0,0,0,-1,"{}"
1178,0,0,0,0.146594,"\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Int[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^2,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x) \sqrt{d+e x^2}}{x^2},x\right)-\frac{a \sqrt{d+e x^2}}{x}+a \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)",0,"-((a*Sqrt[d + e*x^2])/x) + a*Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + b*Defer[Int][(Sqrt[d + e*x^2]*ArcTan[c*x])/x^2, x]","A",0,0,0,0,-1,"{}"
1179,0,0,0,0.1652424,"\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Int[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^3,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x) \sqrt{d+e x^2}}{x^3},x\right)-\frac{a \sqrt{d+e x^2}}{2 x^2}-\frac{a e \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{2 \sqrt{d}}",0,"-(a*Sqrt[d + e*x^2])/(2*x^2) - (a*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*Sqrt[d]) + b*Defer[Int][(Sqrt[d + e*x^2]*ArcTan[c*x])/x^3, x]","A",0,0,0,0,-1,"{}"
1180,1,137,0,0.2790432,"\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Int[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^4,x]","-\frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{3 d x^3}-\frac{b \left(c^2 d-e\right)^{3/2} \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{3 d}+\frac{b c \left(2 c^2 d-3 e\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{6 \sqrt{d}}-\frac{b c \sqrt{d+e x^2}}{6 x^2}","-\frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{3 d x^3}-\frac{b \left(c^2 d-e\right)^{3/2} \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{3 d}+\frac{b c \left(2 c^2 d-3 e\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{6 \sqrt{d}}-\frac{b c \sqrt{d+e x^2}}{6 x^2}",1,"-(b*c*Sqrt[d + e*x^2])/(6*x^2) - ((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(3*d*x^3) + (b*c*(2*c^2*d - 3*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(6*Sqrt[d]) - (b*(c^2*d - e)^(3/2)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d)","A",9,8,23,0.3478,1,"{264, 4976, 12, 446, 98, 156, 63, 208}"
1181,0,0,0,0.1758134,"\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Int[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^5,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x) \sqrt{d+e x^2}}{x^5},x\right)+\frac{a e^2 \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{8 d^{3/2}}-\frac{a e \sqrt{d+e x^2}}{8 d x^2}-\frac{a \sqrt{d+e x^2}}{4 x^4}",0,"-(a*Sqrt[d + e*x^2])/(4*x^4) - (a*e*Sqrt[d + e*x^2])/(8*d*x^2) + (a*e^2*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(8*d^(3/2)) + b*Defer[Int][(Sqrt[d + e*x^2]*ArcTan[c*x])/x^5, x]","A",0,0,0,0,-1,"{}"
1182,1,224,0,0.3522879,"\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Int[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^6,x]","\frac{2 e \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{15 d^2 x^3}-\frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{5 d x^5}-\frac{b c \left(24 c^4 d^2-20 c^2 d e-15 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{120 d^{3/2}}+\frac{b \left(3 c^2 d+2 e\right) \left(c^2 d-e\right)^{3/2} \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{15 d^2}+\frac{b c \left(12 c^2 d-e\right) \sqrt{d+e x^2}}{120 d x^2}-\frac{b c \left(d+e x^2\right)^{3/2}}{20 d x^4}","\frac{2 e \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{15 d^2 x^3}-\frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{5 d x^5}-\frac{b c \left(24 c^4 d^2-20 c^2 d e-15 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{120 d^{3/2}}+\frac{b \left(3 c^2 d+2 e\right) \left(c^2 d-e\right)^{3/2} \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{15 d^2}+\frac{b c \left(12 c^2 d-e\right) \sqrt{d+e x^2}}{120 d x^2}-\frac{b c \left(d+e x^2\right)^{3/2}}{20 d x^4}",1,"(b*c*(12*c^2*d - e)*Sqrt[d + e*x^2])/(120*d*x^2) - (b*c*(d + e*x^2)^(3/2))/(20*d*x^4) - ((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(5*d*x^5) + (2*e*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(15*d^2*x^3) - (b*c*(24*c^4*d^2 - 20*c^2*d*e - 15*e^2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(120*d^(3/2)) + (b*(c^2*d - e)^(3/2)*(3*c^2*d + 2*e)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(15*d^2)","A",10,9,23,0.3913,1,"{271, 264, 4976, 12, 573, 149, 156, 63, 208}"
1183,1,279,0,0.4610884,"\int x^3 \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^3*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]),x]","\frac{\left(d+e x^2\right)^{7/2} \left(a+b \tan ^{-1}(c x)\right)}{7 e^2}-\frac{d \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{5 e^2}+\frac{b x \left(3 c^4 d^2+54 c^2 d e-40 e^2\right) \sqrt{d+e x^2}}{560 c^5 e}+\frac{b \left(70 c^4 d^2 e+35 c^6 d^3-168 c^2 d e^2+80 e^3\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{560 c^7 e^{3/2}}+\frac{b \left(c^2 d-e\right)^{5/2} \left(2 c^2 d+5 e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{35 c^7 e^2}-\frac{b x \left(13 c^2 d-30 e\right) \left(d+e x^2\right)^{3/2}}{840 c^3 e}-\frac{b x \left(d+e x^2\right)^{5/2}}{42 c e}","\frac{\left(d+e x^2\right)^{7/2} \left(a+b \tan ^{-1}(c x)\right)}{7 e^2}-\frac{d \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{5 e^2}+\frac{b x \left(3 c^4 d^2+54 c^2 d e-40 e^2\right) \sqrt{d+e x^2}}{560 c^5 e}+\frac{b \left(70 c^4 d^2 e+35 c^6 d^3-168 c^2 d e^2+80 e^3\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{560 c^7 e^{3/2}}+\frac{b \left(c^2 d-e\right)^{5/2} \left(2 c^2 d+5 e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{35 c^7 e^2}-\frac{b x \left(13 c^2 d-30 e\right) \left(d+e x^2\right)^{3/2}}{840 c^3 e}-\frac{b x \left(d+e x^2\right)^{5/2}}{42 c e}",1,"(b*(3*c^4*d^2 + 54*c^2*d*e - 40*e^2)*x*Sqrt[d + e*x^2])/(560*c^5*e) - (b*(13*c^2*d - 30*e)*x*(d + e*x^2)^(3/2))/(840*c^3*e) - (b*x*(d + e*x^2)^(5/2))/(42*c*e) - (d*(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/(5*e^2) + ((d + e*x^2)^(7/2)*(a + b*ArcTan[c*x]))/(7*e^2) + (b*(c^2*d - e)^(5/2)*(2*c^2*d + 5*e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(35*c^7*e^2) + (b*(35*c^6*d^3 + 70*c^4*d^2*e - 168*c^2*d*e^2 + 80*e^3)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(560*c^7*e^(3/2))","A",10,10,23,0.4348,1,"{266, 43, 4976, 12, 528, 523, 217, 206, 377, 203}"
1184,0,0,0,0.1948604,"\int x^2 \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^2*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]),x]","\int x^2 \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","b \text{Int}\left(x^2 \tan ^{-1}(c x) \left(d+e x^2\right)^{3/2},x\right)-\frac{a d^3 \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{16 e^{3/2}}+\frac{a d^2 x \sqrt{d+e x^2}}{16 e}+\frac{1}{8} a d x^3 \sqrt{d+e x^2}+\frac{1}{6} a x^3 \left(d+e x^2\right)^{3/2}",0,"(a*d^2*x*Sqrt[d + e*x^2])/(16*e) + (a*d*x^3*Sqrt[d + e*x^2])/8 + (a*x^3*(d + e*x^2)^(3/2))/6 - (a*d^3*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(16*e^(3/2)) + b*Defer[Int][x^2*(d + e*x^2)^(3/2)*ArcTan[c*x], x]","A",0,0,0,0,-1,"{}"
1185,1,181,0,0.233115,"\int x \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]),x]","\frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{5 e}-\frac{b \left(15 c^4 d^2-20 c^2 d e+8 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{40 c^5 \sqrt{e}}-\frac{b x \left(7 c^2 d-4 e\right) \sqrt{d+e x^2}}{40 c^3}-\frac{b \left(c^2 d-e\right)^{5/2} \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{5 c^5 e}-\frac{b x \left(d+e x^2\right)^{3/2}}{20 c}","\frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{5 e}-\frac{b \left(15 c^4 d^2-20 c^2 d e+8 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{40 c^5 \sqrt{e}}-\frac{b x \left(7 c^2 d-4 e\right) \sqrt{d+e x^2}}{40 c^3}-\frac{b \left(c^2 d-e\right)^{5/2} \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{5 c^5 e}-\frac{b x \left(d+e x^2\right)^{3/2}}{20 c}",1,"-(b*(7*c^2*d - 4*e)*x*Sqrt[d + e*x^2])/(40*c^3) - (b*x*(d + e*x^2)^(3/2))/(20*c) + ((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/(5*e) - (b*(c^2*d - e)^(5/2)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(5*c^5*e) - (b*(15*c^4*d^2 - 20*c^2*d*e + 8*e^2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(40*c^5*Sqrt[e])","A",8,8,21,0.3810,1,"{4974, 416, 528, 523, 217, 206, 377, 203}"
1186,0,0,0,0.0283884,"\int \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]),x]","\int \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","\text{Int}\left(\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right),x\right)",0,"Defer[Int][(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]), x]","A",0,0,0,0,-1,"{}"
1187,0,0,0,0.1927809,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Int[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x) \left(d+e x^2\right)^{3/2}}{x},x\right)-a d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)+a d \sqrt{d+e x^2}+\frac{1}{3} a \left(d+e x^2\right)^{3/2}",0,"a*d*Sqrt[d + e*x^2] + (a*(d + e*x^2)^(3/2))/3 - a*d^(3/2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + b*Defer[Int][((d + e*x^2)^(3/2)*ArcTan[c*x])/x, x]","A",0,0,0,0,-1,"{}"
1188,0,0,0,0.1740187,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Int[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^2,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x) \left(d+e x^2\right)^{3/2}}{x^2},x\right)-\frac{a \left(d+e x^2\right)^{3/2}}{x}+\frac{3}{2} a e x \sqrt{d+e x^2}+\frac{3}{2} a d \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)",0,"(3*a*e*x*Sqrt[d + e*x^2])/2 - (a*(d + e*x^2)^(3/2))/x + (3*a*d*Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/2 + b*Defer[Int][((d + e*x^2)^(3/2)*ArcTan[c*x])/x^2, x]","A",0,0,0,0,-1,"{}"
1189,0,0,0,0.2040548,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Int[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^3,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x) \left(d+e x^2\right)^{3/2}}{x^3},x\right)-\frac{a \left(d+e x^2\right)^{3/2}}{2 x^2}+\frac{3}{2} a e \sqrt{d+e x^2}-\frac{3}{2} a \sqrt{d} e \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)",0,"(3*a*e*Sqrt[d + e*x^2])/2 - (a*(d + e*x^2)^(3/2))/(2*x^2) - (3*a*Sqrt[d]*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/2 + b*Defer[Int][((d + e*x^2)^(3/2)*ArcTan[c*x])/x^3, x]","A",0,0,0,0,-1,"{}"
1190,0,0,0,0.1748584,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Int[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^4,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x) \left(d+e x^2\right)^{3/2}}{x^4},x\right)+a e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)-\frac{a e \sqrt{d+e x^2}}{x}-\frac{a \left(d+e x^2\right)^{3/2}}{3 x^3}",0,"-((a*e*Sqrt[d + e*x^2])/x) - (a*(d + e*x^2)^(3/2))/(3*x^3) + a*e^(3/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]] + b*Defer[Int][((d + e*x^2)^(3/2)*ArcTan[c*x])/x^4, x]","A",0,0,0,0,-1,"{}"
1191,0,0,0,0.1961051,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Int[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^5,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x) \left(d+e x^2\right)^{3/2}}{x^5},x\right)-\frac{3 a e^2 \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{8 \sqrt{d}}-\frac{3 a e \sqrt{d+e x^2}}{8 x^2}-\frac{a \left(d+e x^2\right)^{3/2}}{4 x^4}",0,"(-3*a*e*Sqrt[d + e*x^2])/(8*x^2) - (a*(d + e*x^2)^(3/2))/(4*x^4) - (3*a*e^2*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(8*Sqrt[d]) + b*Defer[Int][((d + e*x^2)^(3/2)*ArcTan[c*x])/x^5, x]","A",0,0,0,0,-1,"{}"
1192,1,178,0,0.3245997,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Int[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^6,x]","-\frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{5 d x^5}-\frac{b c \left(8 c^4 d^2-20 c^2 d e+15 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{40 \sqrt{d}}+\frac{b c \left(4 c^2 d-7 e\right) \sqrt{d+e x^2}}{40 x^2}+\frac{b \left(c^2 d-e\right)^{5/2} \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{5 d}-\frac{b c \left(d+e x^2\right)^{3/2}}{20 x^4}","-\frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{5 d x^5}-\frac{b c \left(8 c^4 d^2-20 c^2 d e+15 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{40 \sqrt{d}}+\frac{b c \left(4 c^2 d-7 e\right) \sqrt{d+e x^2}}{40 x^2}+\frac{b \left(c^2 d-e\right)^{5/2} \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{5 d}-\frac{b c \left(d+e x^2\right)^{3/2}}{20 x^4}",1,"(b*c*(4*c^2*d - 7*e)*Sqrt[d + e*x^2])/(40*x^2) - (b*c*(d + e*x^2)^(3/2))/(20*x^4) - ((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/(5*d*x^5) - (b*c*(8*c^4*d^2 - 20*c^2*d*e + 15*e^2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(40*Sqrt[d]) + (b*(c^2*d - e)^(5/2)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(5*d)","A",10,9,23,0.3913,1,"{264, 4976, 12, 446, 98, 149, 156, 63, 208}"
1193,1,345,0,0.5833096,"\int x^3 \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^3*(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]),x]","\frac{\left(d+e x^2\right)^{9/2} \left(a+b \tan ^{-1}(c x)\right)}{9 e^2}-\frac{d \left(d+e x^2\right)^{7/2} \left(a+b \tan ^{-1}(c x)\right)}{7 e^2}-\frac{b x \left(69 c^4 d^2-520 c^2 d e+336 e^2\right) \left(d+e x^2\right)^{3/2}}{12096 c^5 e}+\frac{b x \left(712 c^4 d^2 e+59 c^6 d^3-1104 c^2 d e^2+448 e^3\right) \sqrt{d+e x^2}}{8064 c^7 e}+\frac{b \left(-3024 c^4 d^2 e^2+840 c^6 d^3 e+315 c^8 d^4+2880 c^2 d e^3-896 e^4\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{8064 c^9 e^{3/2}}+\frac{b \left(c^2 d-e\right)^{7/2} \left(2 c^2 d+7 e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{63 c^9 e^2}-\frac{b x \left(33 c^2 d-56 e\right) \left(d+e x^2\right)^{5/2}}{3024 c^3 e}-\frac{b x \left(d+e x^2\right)^{7/2}}{72 c e}","\frac{\left(d+e x^2\right)^{9/2} \left(a+b \tan ^{-1}(c x)\right)}{9 e^2}-\frac{d \left(d+e x^2\right)^{7/2} \left(a+b \tan ^{-1}(c x)\right)}{7 e^2}-\frac{b x \left(69 c^4 d^2-520 c^2 d e+336 e^2\right) \left(d+e x^2\right)^{3/2}}{12096 c^5 e}+\frac{b x \left(712 c^4 d^2 e+59 c^6 d^3-1104 c^2 d e^2+448 e^3\right) \sqrt{d+e x^2}}{8064 c^7 e}+\frac{b \left(-3024 c^4 d^2 e^2+840 c^6 d^3 e+315 c^8 d^4+2880 c^2 d e^3-896 e^4\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{8064 c^9 e^{3/2}}+\frac{b \left(c^2 d-e\right)^{7/2} \left(2 c^2 d+7 e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{63 c^9 e^2}-\frac{b x \left(33 c^2 d-56 e\right) \left(d+e x^2\right)^{5/2}}{3024 c^3 e}-\frac{b x \left(d+e x^2\right)^{7/2}}{72 c e}",1,"(b*(59*c^6*d^3 + 712*c^4*d^2*e - 1104*c^2*d*e^2 + 448*e^3)*x*Sqrt[d + e*x^2])/(8064*c^7*e) - (b*(69*c^4*d^2 - 520*c^2*d*e + 336*e^2)*x*(d + e*x^2)^(3/2))/(12096*c^5*e) - (b*(33*c^2*d - 56*e)*x*(d + e*x^2)^(5/2))/(3024*c^3*e) - (b*x*(d + e*x^2)^(7/2))/(72*c*e) - (d*(d + e*x^2)^(7/2)*(a + b*ArcTan[c*x]))/(7*e^2) + ((d + e*x^2)^(9/2)*(a + b*ArcTan[c*x]))/(9*e^2) + (b*(c^2*d - e)^(7/2)*(2*c^2*d + 7*e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(63*c^9*e^2) + (b*(315*c^8*d^4 + 840*c^6*d^3*e - 3024*c^4*d^2*e^2 + 2880*c^2*d*e^3 - 896*e^4)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(8064*c^9*e^(3/2))","A",11,10,23,0.4348,1,"{266, 43, 4976, 12, 528, 523, 217, 206, 377, 203}"
1194,0,0,0,0.2272191,"\int x^2 \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^2*(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]),x]","\int x^2 \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","b \text{Int}\left(x^2 \tan ^{-1}(c x) \left(d+e x^2\right)^{5/2},x\right)-\frac{5 a d^4 \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{128 e^{3/2}}+\frac{5 a d^3 x \sqrt{d+e x^2}}{128 e}+\frac{5}{64} a d^2 x^3 \sqrt{d+e x^2}+\frac{5}{48} a d x^3 \left(d+e x^2\right)^{3/2}+\frac{1}{8} a x^3 \left(d+e x^2\right)^{5/2}",0,"(5*a*d^3*x*Sqrt[d + e*x^2])/(128*e) + (5*a*d^2*x^3*Sqrt[d + e*x^2])/64 + (5*a*d*x^3*(d + e*x^2)^(3/2))/48 + (a*x^3*(d + e*x^2)^(5/2))/8 - (5*a*d^4*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(128*e^(3/2)) + b*Defer[Int][x^2*(d + e*x^2)^(5/2)*ArcTan[c*x], x]","A",0,0,0,0,-1,"{}"
1195,1,233,0,0.3306215,"\int x \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x*(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]),x]","\frac{\left(d+e x^2\right)^{7/2} \left(a+b \tan ^{-1}(c x)\right)}{7 e}-\frac{b x \left(19 c^4 d^2-22 c^2 d e+8 e^2\right) \sqrt{d+e x^2}}{112 c^5}-\frac{b \left(-70 c^4 d^2 e+35 c^6 d^3+56 c^2 d e^2-16 e^3\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{112 c^7 \sqrt{e}}-\frac{b x \left(11 c^2 d-6 e\right) \left(d+e x^2\right)^{3/2}}{168 c^3}-\frac{b \left(c^2 d-e\right)^{7/2} \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{7 c^7 e}-\frac{b x \left(d+e x^2\right)^{5/2}}{42 c}","\frac{\left(d+e x^2\right)^{7/2} \left(a+b \tan ^{-1}(c x)\right)}{7 e}-\frac{b x \left(19 c^4 d^2-22 c^2 d e+8 e^2\right) \sqrt{d+e x^2}}{112 c^5}-\frac{b \left(-70 c^4 d^2 e+35 c^6 d^3+56 c^2 d e^2-16 e^3\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{112 c^7 \sqrt{e}}-\frac{b x \left(11 c^2 d-6 e\right) \left(d+e x^2\right)^{3/2}}{168 c^3}-\frac{b \left(c^2 d-e\right)^{7/2} \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{7 c^7 e}-\frac{b x \left(d+e x^2\right)^{5/2}}{42 c}",1,"-(b*(19*c^4*d^2 - 22*c^2*d*e + 8*e^2)*x*Sqrt[d + e*x^2])/(112*c^5) - (b*(11*c^2*d - 6*e)*x*(d + e*x^2)^(3/2))/(168*c^3) - (b*x*(d + e*x^2)^(5/2))/(42*c) + ((d + e*x^2)^(7/2)*(a + b*ArcTan[c*x]))/(7*e) - (b*(c^2*d - e)^(7/2)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(7*c^7*e) - (b*(35*c^6*d^3 - 70*c^4*d^2*e + 56*c^2*d*e^2 - 16*e^3)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(112*c^7*Sqrt[e])","A",9,8,21,0.3810,1,"{4974, 416, 528, 523, 217, 206, 377, 203}"
1196,0,0,0,0.0278654,"\int \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]),x]","\int \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","\text{Int}\left(\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right),x\right)",0,"Defer[Int][(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]), x]","A",0,0,0,0,-1,"{}"
1197,0,0,0,0.2025903,"\int \frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Int[((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/x,x]","\int \frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x) \left(d+e x^2\right)^{5/2}}{x},x\right)+a d^2 \sqrt{d+e x^2}-a d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)+\frac{1}{3} a d \left(d+e x^2\right)^{3/2}+\frac{1}{5} a \left(d+e x^2\right)^{5/2}",0,"a*d^2*Sqrt[d + e*x^2] + (a*d*(d + e*x^2)^(3/2))/3 + (a*(d + e*x^2)^(5/2))/5 - a*d^(5/2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + b*Defer[Int][((d + e*x^2)^(5/2)*ArcTan[c*x])/x, x]","A",0,0,0,0,-1,"{}"
1198,0,0,0,0.1869941,"\int \frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Int[((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/x^2,x]","\int \frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x) \left(d+e x^2\right)^{5/2}}{x^2},x\right)+\frac{15}{8} a d^2 \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)-\frac{a \left(d+e x^2\right)^{5/2}}{x}+\frac{5}{4} a e x \left(d+e x^2\right)^{3/2}+\frac{15}{8} a d e x \sqrt{d+e x^2}",0,"(15*a*d*e*x*Sqrt[d + e*x^2])/8 + (5*a*e*x*(d + e*x^2)^(3/2))/4 - (a*(d + e*x^2)^(5/2))/x + (15*a*d^2*Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/8 + b*Defer[Int][((d + e*x^2)^(5/2)*ArcTan[c*x])/x^2, x]","A",0,0,0,0,-1,"{}"
1199,0,0,0,0.2080682,"\int \frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Int[((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/x^3,x]","\int \frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x) \left(d+e x^2\right)^{5/2}}{x^3},x\right)-\frac{5}{2} a d^{3/2} e \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)-\frac{a \left(d+e x^2\right)^{5/2}}{2 x^2}+\frac{5}{6} a e \left(d+e x^2\right)^{3/2}+\frac{5}{2} a d e \sqrt{d+e x^2}",0,"(5*a*d*e*Sqrt[d + e*x^2])/2 + (5*a*e*(d + e*x^2)^(3/2))/6 - (a*(d + e*x^2)^(5/2))/(2*x^2) - (5*a*d^(3/2)*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/2 + b*Defer[Int][((d + e*x^2)^(5/2)*ArcTan[c*x])/x^3, x]","A",0,0,0,0,-1,"{}"
1200,0,0,0,0.187136,"\int \frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Int[((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/x^4,x]","\int \frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x) \left(d+e x^2\right)^{5/2}}{x^4},x\right)+\frac{5}{2} a e^2 x \sqrt{d+e x^2}+\frac{5}{2} a d e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)-\frac{a \left(d+e x^2\right)^{5/2}}{3 x^3}-\frac{5 a e \left(d+e x^2\right)^{3/2}}{3 x}",0,"(5*a*e^2*x*Sqrt[d + e*x^2])/2 - (5*a*e*(d + e*x^2)^(3/2))/(3*x) - (a*(d + e*x^2)^(5/2))/(3*x^3) + (5*a*d*e^(3/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/2 + b*Defer[Int][((d + e*x^2)^(5/2)*ArcTan[c*x])/x^4, x]","A",0,0,0,0,-1,"{}"
1201,1,176,0,0.2475935,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Int[(x^3*(a + b*ArcTan[c*x]))/Sqrt[d + e*x^2],x]","\frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{3 e^2}-\frac{d \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{e^2}+\frac{b \sqrt{c^2 d-e} \left(2 c^2 d+e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{3 c^3 e^2}+\frac{b \left(3 c^2 d+2 e\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{6 c^3 e^{3/2}}-\frac{b x \sqrt{d+e x^2}}{6 c e}","\frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{3 e^2}-\frac{d \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{e^2}+\frac{b \sqrt{c^2 d-e} \left(2 c^2 d+e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{3 c^3 e^2}+\frac{b \left(3 c^2 d+2 e\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{6 c^3 e^{3/2}}-\frac{b x \sqrt{d+e x^2}}{6 c e}",1,"-(b*x*Sqrt[d + e*x^2])/(6*c*e) - (d*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/e^2 + ((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/(3*e^2) + (b*Sqrt[c^2*d - e]*(2*c^2*d + e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(3*c^3*e^2) + (b*(3*c^2*d + 2*e)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(6*c^3*e^(3/2))","A",8,10,23,0.4348,1,"{266, 43, 4976, 12, 528, 523, 217, 206, 377, 203}"
1202,0,0,0,0.1555805,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Int[(x^2*(a + b*ArcTan[c*x]))/Sqrt[d + e*x^2],x]","\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","b \text{Int}\left(\frac{x^2 \tan ^{-1}(c x)}{\sqrt{d+e x^2}},x\right)-\frac{a d \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{2 e^{3/2}}+\frac{a x \sqrt{d+e x^2}}{2 e}",0,"(a*x*Sqrt[d + e*x^2])/(2*e) - (a*d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*e^(3/2)) + b*Defer[Int][(x^2*ArcTan[c*x])/Sqrt[d + e*x^2], x]","A",0,0,0,0,-1,"{}"
1203,1,103,0,0.0986641,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Int[(x*(a + b*ArcTan[c*x]))/Sqrt[d + e*x^2],x]","\frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{e}-\frac{b \sqrt{c^2 d-e} \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{c e}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{c \sqrt{e}}","\frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{e}-\frac{b \sqrt{c^2 d-e} \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{c e}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{c \sqrt{e}}",1,"(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/e - (b*Sqrt[c^2*d - e]*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(c*e) - (b*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(c*Sqrt[e])","A",6,6,21,0.2857,1,"{4974, 402, 217, 206, 377, 203}"
1204,0,0,0,0.0249126,"\int \frac{a+b \tan ^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","Int[(a + b*ArcTan[c*x])/Sqrt[d + e*x^2],x]","\int \frac{a+b \tan ^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \tan ^{-1}(c x)}{\sqrt{d+e x^2}},x\right)",0,"Defer[Int][(a + b*ArcTan[c*x])/Sqrt[d + e*x^2], x]","A",0,0,0,0,-1,"{}"
1205,0,0,0,0.1630192,"\int \frac{a+b \tan ^{-1}(c x)}{x \sqrt{d+e x^2}} \, dx","Int[(a + b*ArcTan[c*x])/(x*Sqrt[d + e*x^2]),x]","\int \frac{a+b \tan ^{-1}(c x)}{x \sqrt{d+e x^2}} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x)}{x \sqrt{d+e x^2}},x\right)-\frac{a \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{\sqrt{d}}",0,"-((a*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/Sqrt[d]) + b*Defer[Int][ArcTan[c*x]/(x*Sqrt[d + e*x^2]), x]","A",0,0,0,0,-1,"{}"
1206,1,100,0,0.1768066,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 \sqrt{d+e x^2}} \, dx","Int[(a + b*ArcTan[c*x])/(x^2*Sqrt[d + e*x^2]),x]","-\frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{d x}+\frac{b \sqrt{c^2 d-e} \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{d}-\frac{b c \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{\sqrt{d}}","-\frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{d x}+\frac{b \sqrt{c^2 d-e} \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{d}-\frac{b c \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{\sqrt{d}}",1,"-((Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/(d*x)) - (b*c*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/Sqrt[d] + (b*Sqrt[c^2*d - e]*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/d","A",7,6,23,0.2609,1,"{264, 4976, 446, 83, 63, 208}"
1207,0,0,0,0.1721509,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 \sqrt{d+e x^2}} \, dx","Int[(a + b*ArcTan[c*x])/(x^3*Sqrt[d + e*x^2]),x]","\int \frac{a+b \tan ^{-1}(c x)}{x^3 \sqrt{d+e x^2}} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x)}{x^3 \sqrt{d+e x^2}},x\right)+\frac{a e \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{2 d^{3/2}}-\frac{a \sqrt{d+e x^2}}{2 d x^2}",0,"-(a*Sqrt[d + e*x^2])/(2*d*x^2) + (a*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*d^(3/2)) + b*Defer[Int][ArcTan[c*x]/(x^3*Sqrt[d + e*x^2]), x]","A",0,0,0,0,-1,"{}"
1208,1,179,0,0.2660889,"\int \frac{a+b \tan ^{-1}(c x)}{x^4 \sqrt{d+e x^2}} \, dx","Int[(a + b*ArcTan[c*x])/(x^4*Sqrt[d + e*x^2]),x]","\frac{2 e \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{3 d^2 x}-\frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{3 d x^3}+\frac{b c \left(2 c^2 d+3 e\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{6 d^{3/2}}-\frac{b \sqrt{c^2 d-e} \left(c^2 d+2 e\right) \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{3 d^2}-\frac{b c \sqrt{d+e x^2}}{6 d x^2}","\frac{2 e \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{3 d^2 x}-\frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{3 d x^3}+\frac{b c \left(2 c^2 d+3 e\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{6 d^{3/2}}-\frac{b \sqrt{c^2 d-e} \left(c^2 d+2 e\right) \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{3 d^2}-\frac{b c \sqrt{d+e x^2}}{6 d x^2}",1,"-(b*c*Sqrt[d + e*x^2])/(6*d*x^2) - (Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/(3*d*x^3) + (2*e*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/(3*d^2*x) + (b*c*(2*c^2*d + 3*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(6*d^(3/2)) - (b*Sqrt[c^2*d - e]*(c^2*d + 2*e)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d^2)","A",9,9,23,0.3913,1,"{271, 264, 4976, 12, 573, 149, 156, 63, 208}"
1209,1,137,0,0.1843639,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Int[(x^3*(a + b*ArcTan[c*x]))/(d + e*x^2)^(3/2),x]","\frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{e^2}+\frac{d \left(a+b \tan ^{-1}(c x)\right)}{e^2 \sqrt{d+e x^2}}-\frac{b \left(2 c^2 d-e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{c e^2 \sqrt{c^2 d-e}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{c e^{3/2}}","\frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{e^2}+\frac{d \left(a+b \tan ^{-1}(c x)\right)}{e^2 \sqrt{d+e x^2}}-\frac{b \left(2 c^2 d-e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{c e^2 \sqrt{c^2 d-e}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{c e^{3/2}}",1,"(d*(a + b*ArcTan[c*x]))/(e^2*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/e^2 - (b*(2*c^2*d - e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(c*Sqrt[c^2*d - e]*e^2) - (b*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(c*e^(3/2))","A",7,9,23,0.3913,1,"{266, 43, 4976, 12, 523, 217, 206, 377, 203}"
1210,0,0,0,0.1675327,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Int[(x^2*(a + b*ArcTan[c*x]))/(d + e*x^2)^(3/2),x]","\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","b \text{Int}\left(\frac{x^2 \tan ^{-1}(c x)}{\left(d+e x^2\right)^{3/2}},x\right)+\frac{a \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{e^{3/2}}-\frac{a x}{e \sqrt{d+e x^2}}",0,"-((a*x)/(e*Sqrt[d + e*x^2])) + (a*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/e^(3/2) + b*Defer[Int][(x^2*ArcTan[c*x])/(d + e*x^2)^(3/2), x]","A",0,0,0,0,-1,"{}"
1211,1,71,0,0.0739139,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Int[(x*(a + b*ArcTan[c*x]))/(d + e*x^2)^(3/2),x]","\frac{b c \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{e \sqrt{c^2 d-e}}-\frac{a+b \tan ^{-1}(c x)}{e \sqrt{d+e x^2}}","\frac{b c \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{e \sqrt{c^2 d-e}}-\frac{a+b \tan ^{-1}(c x)}{e \sqrt{d+e x^2}}",1,"-((a + b*ArcTan[c*x])/(e*Sqrt[d + e*x^2])) + (b*c*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(Sqrt[c^2*d - e]*e)","A",3,3,21,0.1429,1,"{4974, 377, 203}"
1212,1,70,0,0.0769833,"\int \frac{a+b \tan ^{-1}(c x)}{\left(d+e x^2\right)^{3/2}} \, dx","Int[(a + b*ArcTan[c*x])/(d + e*x^2)^(3/2),x]","\frac{x \left(a+b \tan ^{-1}(c x)\right)}{d \sqrt{d+e x^2}}+\frac{b \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{d \sqrt{c^2 d-e}}","\frac{x \left(a+b \tan ^{-1}(c x)\right)}{d \sqrt{d+e x^2}}+\frac{b \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{d \sqrt{c^2 d-e}}",1,"(x*(a + b*ArcTan[c*x]))/(d*Sqrt[d + e*x^2]) + (b*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(d*Sqrt[c^2*d - e])","A",5,6,20,0.3000,1,"{191, 4912, 12, 444, 63, 208}"
1213,0,0,0,0.176169,"\int \frac{a+b \tan ^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}} \, dx","Int[(a + b*ArcTan[c*x])/(x*(d + e*x^2)^(3/2)),x]","\int \frac{a+b \tan ^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}},x\right)-\frac{a \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{d^{3/2}}+\frac{a}{d \sqrt{d+e x^2}}",0,"a/(d*Sqrt[d + e*x^2]) - (a*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/d^(3/2) + b*Defer[Int][ArcTan[c*x]/(x*(d + e*x^2)^(3/2)), x]","A",0,0,0,0,-1,"{}"
1214,1,135,0,0.2299666,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 \left(d+e x^2\right)^{3/2}} \, dx","Int[(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^(3/2)),x]","-\frac{2 e x \left(a+b \tan ^{-1}(c x)\right)}{d^2 \sqrt{d+e x^2}}-\frac{a+b \tan ^{-1}(c x)}{d x \sqrt{d+e x^2}}+\frac{b \left(c^2 d-2 e\right) \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{d^2 \sqrt{c^2 d-e}}-\frac{b c \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{d^{3/2}}","-\frac{2 e x \left(a+b \tan ^{-1}(c x)\right)}{d^2 \sqrt{d+e x^2}}-\frac{a+b \tan ^{-1}(c x)}{d x \sqrt{d+e x^2}}+\frac{b \left(c^2 d-2 e\right) \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{d^2 \sqrt{c^2 d-e}}-\frac{b c \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{d^{3/2}}",1,"-((a + b*ArcTan[c*x])/(d*x*Sqrt[d + e*x^2])) - (2*e*x*(a + b*ArcTan[c*x]))/(d^2*Sqrt[d + e*x^2]) - (b*c*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/d^(3/2) + (b*(c^2*d - 2*e)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(d^2*Sqrt[c^2*d - e])","A",8,8,23,0.3478,1,"{271, 191, 4976, 12, 573, 156, 63, 208}"
1215,0,0,0,0.1964315,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}} \, dx","Int[(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)^(3/2)),x]","\int \frac{a+b \tan ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}},x\right)-\frac{3 a e}{2 d^2 \sqrt{d+e x^2}}+\frac{3 a e \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{2 d^{5/2}}-\frac{a}{2 d x^2 \sqrt{d+e x^2}}",0,"a/(d*x^2*Sqrt[d + e*x^2]) - (3*a*Sqrt[d + e*x^2])/(2*d^2*x^2) + (3*a*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*d^(5/2)) + b*Defer[Int][ArcTan[c*x]/(x^3*(d + e*x^2)^(3/2)), x]","A",0,0,0,0,-1,"{}"
1216,1,249,0,0.8736212,"\int \frac{a+b \tan ^{-1}(c x)}{x^4 \left(d+e x^2\right)^{3/2}} \, dx","Int[(a + b*ArcTan[c*x])/(x^4*(d + e*x^2)^(3/2)),x]","\frac{8 e^2 x \left(a+b \tan ^{-1}(c x)\right)}{3 d^3 \sqrt{d+e x^2}}+\frac{4 e \left(a+b \tan ^{-1}(c x)\right)}{3 d^2 x \sqrt{d+e x^2}}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}-\frac{b \left(c^4 d^2+4 c^2 d e-8 e^2\right) \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{3 d^3 \sqrt{c^2 d-e}}+\frac{b c \left(c^2 d+4 e\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{3 d^{5/2}}-\frac{b c \sqrt{d+e x^2}}{6 d^2 x^2}+\frac{b c e \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{6 d^{5/2}}","\frac{8 e^2 x \left(a+b \tan ^{-1}(c x)\right)}{3 d^3 \sqrt{d+e x^2}}+\frac{4 e \left(a+b \tan ^{-1}(c x)\right)}{3 d^2 x \sqrt{d+e x^2}}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}-\frac{b \left(c^4 d^2+4 c^2 d e-8 e^2\right) \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{3 d^3 \sqrt{c^2 d-e}}+\frac{b c \left(c^2 d+4 e\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{3 d^{5/2}}-\frac{b c \sqrt{d+e x^2}}{6 d^2 x^2}+\frac{b c e \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{6 d^{5/2}}",1,"-(b*c*Sqrt[d + e*x^2])/(6*d^2*x^2) - (a + b*ArcTan[c*x])/(3*d*x^3*Sqrt[d + e*x^2]) + (4*e*(a + b*ArcTan[c*x]))/(3*d^2*x*Sqrt[d + e*x^2]) + (8*e^2*x*(a + b*ArcTan[c*x]))/(3*d^3*Sqrt[d + e*x^2]) + (b*c*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(6*d^(5/2)) + (b*c*(c^2*d + 4*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(3*d^(5/2)) - (b*(c^4*d^2 + 4*c^2*d*e - 8*e^2)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d^3*Sqrt[c^2*d - e])","A",14,10,23,0.4348,1,"{271, 191, 4976, 12, 6725, 266, 51, 63, 208, 444}"
1217,0,0,0,0.1795456,"\int \frac{x^4 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Int[(x^4*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2),x]","\int \frac{x^4 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","b \text{Int}\left(\frac{x^4 \tan ^{-1}(c x)}{\left(d+e x^2\right)^{5/2}},x\right)-\frac{a x}{e^2 \sqrt{d+e x^2}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{e^{5/2}}-\frac{a x^3}{3 e \left(d+e x^2\right)^{3/2}}",0,"-(a*x^3)/(3*e*(d + e*x^2)^(3/2)) - (a*x)/(e^2*Sqrt[d + e*x^2]) + (a*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/e^(5/2) + b*Defer[Int][(x^4*ArcTan[c*x])/(d + e*x^2)^(5/2), x]","A",0,0,0,0,-1,"{}"
1218,1,143,0,0.2087241,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Int[(x^3*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2),x]","-\frac{a+b \tan ^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{d \left(a+b \tan ^{-1}(c x)\right)}{3 e^2 \left(d+e x^2\right)^{3/2}}+\frac{b c \left(2 c^2 d-3 e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{3 e^2 \left(c^2 d-e\right)^{3/2}}+\frac{b c x}{3 e \left(c^2 d-e\right) \sqrt{d+e x^2}}","-\frac{a+b \tan ^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{d \left(a+b \tan ^{-1}(c x)\right)}{3 e^2 \left(d+e x^2\right)^{3/2}}+\frac{b c \left(2 c^2 d-3 e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{3 e^2 \left(c^2 d-e\right)^{3/2}}+\frac{b c x}{3 e \left(c^2 d-e\right) \sqrt{d+e x^2}}",1,"(b*c*x)/(3*(c^2*d - e)*e*Sqrt[d + e*x^2]) + (d*(a + b*ArcTan[c*x]))/(3*e^2*(d + e*x^2)^(3/2)) - (a + b*ArcTan[c*x])/(e^2*Sqrt[d + e*x^2]) + (b*c*(2*c^2*d - 3*e)*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(3*(c^2*d - e)^(3/2)*e^2)","A",6,7,23,0.3043,1,"{266, 43, 4976, 12, 527, 377, 203}"
1219,1,109,0,0.1993038,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Int[(x^2*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2),x]","\frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}+\frac{b c}{3 e \left(c^2 d-e\right) \sqrt{d+e x^2}}-\frac{b \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{3 d \left(c^2 d-e\right)^{3/2}}","\frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}+\frac{b c}{3 e \left(c^2 d-e\right) \sqrt{d+e x^2}}-\frac{b \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{3 d \left(c^2 d-e\right)^{3/2}}",1,"(b*c)/(3*(c^2*d - e)*e*Sqrt[d + e*x^2]) + (x^3*(a + b*ArcTan[c*x]))/(3*d*(d + e*x^2)^(3/2)) - (b*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d*(c^2*d - e)^(3/2))","A",5,6,23,0.2609,1,"{264, 4976, 446, 78, 63, 208}"
1220,1,110,0,0.093573,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Int[(x*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2),x]","-\frac{a+b \tan ^{-1}(c x)}{3 e \left(d+e x^2\right)^{3/2}}-\frac{b c x}{3 d \left(c^2 d-e\right) \sqrt{d+e x^2}}+\frac{b c^3 \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{3 e \left(c^2 d-e\right)^{3/2}}","-\frac{a+b \tan ^{-1}(c x)}{3 e \left(d+e x^2\right)^{3/2}}-\frac{b c x}{3 d \left(c^2 d-e\right) \sqrt{d+e x^2}}+\frac{b c^3 \tan ^{-1}\left(\frac{x \sqrt{c^2 d-e}}{\sqrt{d+e x^2}}\right)}{3 e \left(c^2 d-e\right)^{3/2}}",1,"-(b*c*x)/(3*d*(c^2*d - e)*Sqrt[d + e*x^2]) - (a + b*ArcTan[c*x])/(3*e*(d + e*x^2)^(3/2)) + (b*c^3*ArcTan[(Sqrt[c^2*d - e]*x)/Sqrt[d + e*x^2]])/(3*(c^2*d - e)^(3/2)*e)","A",4,4,21,0.1905,1,"{4974, 382, 377, 203}"
1221,1,144,0,0.3143356,"\int \frac{a+b \tan ^{-1}(c x)}{\left(d+e x^2\right)^{5/2}} \, dx","Int[(a + b*ArcTan[c*x])/(d + e*x^2)^(5/2),x]","\frac{2 x \left(a+b \tan ^{-1}(c x)\right)}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left(a+b \tan ^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}+\frac{b \left(3 c^2 d-2 e\right) \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{3 d^2 \left(c^2 d-e\right)^{3/2}}-\frac{b c}{3 d \left(c^2 d-e\right) \sqrt{d+e x^2}}","\frac{2 x \left(a+b \tan ^{-1}(c x)\right)}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left(a+b \tan ^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}+\frac{b \left(3 c^2 d-2 e\right) \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{3 d^2 \left(c^2 d-e\right)^{3/2}}-\frac{b c}{3 d \left(c^2 d-e\right) \sqrt{d+e x^2}}",1,"-(b*c)/(3*d*(c^2*d - e)*Sqrt[d + e*x^2]) + (x*(a + b*ArcTan[c*x]))/(3*d*(d + e*x^2)^(3/2)) + (2*x*(a + b*ArcTan[c*x]))/(3*d^2*Sqrt[d + e*x^2]) + (b*(3*c^2*d - 2*e)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d^2*(c^2*d - e)^(3/2))","A",7,9,20,0.4500,1,"{192, 191, 4912, 6688, 12, 571, 78, 63, 208}"
1222,0,0,0,0.1932077,"\int \frac{a+b \tan ^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}} \, dx","Int[(a + b*ArcTan[c*x])/(x*(d + e*x^2)^(5/2)),x]","\int \frac{a+b \tan ^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}},x\right)+\frac{a}{d^2 \sqrt{d+e x^2}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{d^{5/2}}+\frac{a}{3 d \left(d+e x^2\right)^{3/2}}",0,"a/(3*d*(d + e*x^2)^(3/2)) + a/(d^2*Sqrt[d + e*x^2]) - (a*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/d^(5/2) + b*Defer[Int][ArcTan[c*x]/(x*(d + e*x^2)^(5/2)), x]","A",0,0,0,0,-1,"{}"
1223,1,274,0,0.9322202,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 \left(d+e x^2\right)^{5/2}} \, dx","Int[(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^(5/2)),x]","-\frac{8 e x \left(a+b \tan ^{-1}(c x)\right)}{3 d^3 \sqrt{d+e x^2}}-\frac{4 e x \left(a+b \tan ^{-1}(c x)\right)}{3 d^2 \left(d+e x^2\right)^{3/2}}-\frac{a+b \tan ^{-1}(c x)}{d x \left(d+e x^2\right)^{3/2}}-\frac{b \left(3 c^4 d^2-12 c^2 d e+8 e^2\right)}{3 c d^3 \left(c^2 d-e\right) \sqrt{d+e x^2}}+\frac{b \left(3 c^4 d^2-12 c^2 d e+8 e^2\right) \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{3 d^3 \left(c^2 d-e\right)^{3/2}}+\frac{b c}{d^2 \sqrt{d+e x^2}}-\frac{8 b e}{3 c d^3 \sqrt{d+e x^2}}-\frac{b c \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{d^{5/2}}","-\frac{8 e x \left(a+b \tan ^{-1}(c x)\right)}{3 d^3 \sqrt{d+e x^2}}-\frac{4 e x \left(a+b \tan ^{-1}(c x)\right)}{3 d^2 \left(d+e x^2\right)^{3/2}}-\frac{a+b \tan ^{-1}(c x)}{d x \left(d+e x^2\right)^{3/2}}-\frac{b \left(3 c^4 d^2-12 c^2 d e+8 e^2\right)}{3 c d^3 \left(c^2 d-e\right) \sqrt{d+e x^2}}+\frac{b \left(3 c^4 d^2-12 c^2 d e+8 e^2\right) \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{3 d^3 \left(c^2 d-e\right)^{3/2}}+\frac{b c}{d^2 \sqrt{d+e x^2}}-\frac{8 b e}{3 c d^3 \sqrt{d+e x^2}}-\frac{b c \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{d^{5/2}}",1,"(b*c)/(d^2*Sqrt[d + e*x^2]) - (8*b*e)/(3*c*d^3*Sqrt[d + e*x^2]) - (b*(3*c^4*d^2 - 12*c^2*d*e + 8*e^2))/(3*c*d^3*(c^2*d - e)*Sqrt[d + e*x^2]) - (a + b*ArcTan[c*x])/(d*x*(d + e*x^2)^(3/2)) - (4*e*x*(a + b*ArcTan[c*x]))/(3*d^2*(d + e*x^2)^(3/2)) - (8*e*x*(a + b*ArcTan[c*x]))/(3*d^3*Sqrt[d + e*x^2]) - (b*c*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/d^(5/2) + (b*(3*c^4*d^2 - 12*c^2*d*e + 8*e^2)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d^3*(c^2*d - e)^(3/2))","A",13,12,23,0.5217,1,"{271, 192, 191, 4976, 12, 6725, 266, 51, 63, 208, 261, 444}"
1224,0,0,0,0.2086842,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}} \, dx","Int[(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)^(5/2)),x]","\int \frac{a+b \tan ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}} \, dx","b \text{Int}\left(\frac{\tan ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}},x\right)-\frac{5 a e}{2 d^3 \sqrt{d+e x^2}}-\frac{5 a e}{6 d^2 \left(d+e x^2\right)^{3/2}}+\frac{5 a e \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{2 d^{7/2}}-\frac{a}{2 d x^2 \left(d+e x^2\right)^{3/2}}",0,"a/(3*d*x^2*(d + e*x^2)^(3/2)) + (5*a)/(3*d^2*x^2*Sqrt[d + e*x^2]) - (5*a*Sqrt[d + e*x^2])/(2*d^3*x^2) + (5*a*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*d^(7/2)) + b*Defer[Int][ArcTan[c*x]/(x^3*(d + e*x^2)^(5/2)), x]","A",0,0,0,0,-1,"{}"
1225,1,425,0,1.0954335,"\int \frac{a+b \tan ^{-1}(c x)}{x^4 \left(d+e x^2\right)^{5/2}} \, dx","Int[(a + b*ArcTan[c*x])/(x^4*(d + e*x^2)^(5/2)),x]","\frac{16 e^2 x \left(a+b \tan ^{-1}(c x)\right)}{3 d^4 \sqrt{d+e x^2}}+\frac{8 e^2 x \left(a+b \tan ^{-1}(c x)\right)}{3 d^3 \left(d+e x^2\right)^{3/2}}+\frac{2 e \left(a+b \tan ^{-1}(c x)\right)}{d^2 x \left(d+e x^2\right)^{3/2}}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \left(d+e x^2\right)^{3/2}}+\frac{b \left(c^2 d-2 e\right) \left(c^4 d^2+8 c^2 d e-8 e^2\right)}{3 c d^4 \left(c^2 d-e\right) \sqrt{d+e x^2}}-\frac{b \left(c^2 d-2 e\right) \left(c^4 d^2+8 c^2 d e-8 e^2\right) \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{3 d^4 \left(c^2 d-e\right)^{3/2}}-\frac{b c \left(c^2 d+6 e\right)}{3 d^3 \sqrt{d+e x^2}}+\frac{b c \left(c^2 d+6 e\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{3 d^{7/2}}+\frac{16 b e^2}{3 c d^4 \sqrt{d+e x^2}}-\frac{b c \sqrt{d+e x^2}}{2 d^3 x^2}+\frac{b c}{3 d^2 x^2 \sqrt{d+e x^2}}+\frac{b c e \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{2 d^{7/2}}","\frac{16 e^2 x \left(a+b \tan ^{-1}(c x)\right)}{3 d^4 \sqrt{d+e x^2}}+\frac{8 e^2 x \left(a+b \tan ^{-1}(c x)\right)}{3 d^3 \left(d+e x^2\right)^{3/2}}+\frac{2 e \left(a+b \tan ^{-1}(c x)\right)}{d^2 x \left(d+e x^2\right)^{3/2}}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3 \left(d+e x^2\right)^{3/2}}+\frac{b \left(c^2 d-2 e\right) \left(c^4 d^2+8 c^2 d e-8 e^2\right)}{3 c d^4 \left(c^2 d-e\right) \sqrt{d+e x^2}}-\frac{b \left(c^2 d-2 e\right) \left(c^4 d^2+8 c^2 d e-8 e^2\right) \tanh ^{-1}\left(\frac{c \sqrt{d+e x^2}}{\sqrt{c^2 d-e}}\right)}{3 d^4 \left(c^2 d-e\right)^{3/2}}-\frac{b c \left(c^2 d+6 e\right)}{3 d^3 \sqrt{d+e x^2}}+\frac{b c \left(c^2 d+6 e\right) \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{3 d^{7/2}}+\frac{16 b e^2}{3 c d^4 \sqrt{d+e x^2}}-\frac{b c e}{2 d^3 \sqrt{d+e x^2}}-\frac{b c}{6 d^2 x^2 \sqrt{d+e x^2}}+\frac{b c e \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{2 d^{7/2}}",1,"(16*b*e^2)/(3*c*d^4*Sqrt[d + e*x^2]) - (b*c*(c^2*d + 6*e))/(3*d^3*Sqrt[d + e*x^2]) + (b*(c^2*d - 2*e)*(c^4*d^2 + 8*c^2*d*e - 8*e^2))/(3*c*d^4*(c^2*d - e)*Sqrt[d + e*x^2]) + (b*c)/(3*d^2*x^2*Sqrt[d + e*x^2]) - (b*c*Sqrt[d + e*x^2])/(2*d^3*x^2) - (a + b*ArcTan[c*x])/(3*d*x^3*(d + e*x^2)^(3/2)) + (2*e*(a + b*ArcTan[c*x]))/(d^2*x*(d + e*x^2)^(3/2)) + (8*e^2*x*(a + b*ArcTan[c*x]))/(3*d^3*(d + e*x^2)^(3/2)) + (16*e^2*x*(a + b*ArcTan[c*x]))/(3*d^4*Sqrt[d + e*x^2]) + (b*c*e*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(2*d^(7/2)) + (b*c*(c^2*d + 6*e)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/(3*d^(7/2)) - (b*(c^2*d - 2*e)*(c^4*d^2 + 8*c^2*d*e - 8*e^2)*ArcTanh[(c*Sqrt[d + e*x^2])/Sqrt[c^2*d - e]])/(3*d^4*(c^2*d - e)^(3/2))","A",18,12,23,0.5217,1,"{271, 192, 191, 4976, 12, 6725, 266, 51, 63, 208, 261, 444}"
1226,1,208,0,0.9958606,"\int \frac{\tan ^{-1}(a x)}{\left(c+d x^2\right)^{7/2}} \, dx","Int[ArcTan[a*x]/(c + d*x^2)^(7/2),x]","\frac{\left(15 a^4 c^2-20 a^2 c d+8 d^2\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{15 c^3 \left(a^2 c-d\right)^{5/2}}-\frac{a \left(7 a^2 c-4 d\right)}{15 c^2 \left(a^2 c-d\right)^2 \sqrt{c+d x^2}}-\frac{a}{15 c \left(a^2 c-d\right) \left(c+d x^2\right)^{3/2}}+\frac{8 x \tan ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}+\frac{4 x \tan ^{-1}(a x)}{15 c^2 \left(c+d x^2\right)^{3/2}}+\frac{x \tan ^{-1}(a x)}{5 c \left(c+d x^2\right)^{5/2}}","\frac{\left(15 a^4 c^2-20 a^2 c d+8 d^2\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{15 c^3 \left(a^2 c-d\right)^{5/2}}-\frac{a \left(7 a^2 c-4 d\right)}{15 c^2 \left(a^2 c-d\right)^2 \sqrt{c+d x^2}}-\frac{a}{15 c \left(a^2 c-d\right) \left(c+d x^2\right)^{3/2}}+\frac{8 x \tan ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}+\frac{4 x \tan ^{-1}(a x)}{15 c^2 \left(c+d x^2\right)^{3/2}}+\frac{x \tan ^{-1}(a x)}{5 c \left(c+d x^2\right)^{5/2}}",1,"-a/(15*c*(a^2*c - d)*(c + d*x^2)^(3/2)) - (a*(7*a^2*c - 4*d))/(15*c^2*(a^2*c - d)^2*Sqrt[c + d*x^2]) + (x*ArcTan[a*x])/(5*c*(c + d*x^2)^(5/2)) + (4*x*ArcTan[a*x])/(15*c^2*(c + d*x^2)^(3/2)) + (8*x*ArcTan[a*x])/(15*c^3*Sqrt[c + d*x^2]) + ((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]])/(15*c^3*(a^2*c - d)^(5/2))","A",8,9,16,0.5625,1,"{192, 191, 4912, 6688, 12, 6715, 897, 1261, 208}"
1227,1,293,0,1.2309911,"\int \frac{\tan ^{-1}(a x)}{\left(c+d x^2\right)^{9/2}} \, dx","Int[ArcTan[a*x]/(c + d*x^2)^(9/2),x]","-\frac{a \left(19 a^4 c^2-22 a^2 c d+8 d^2\right)}{35 c^3 \left(a^2 c-d\right)^3 \sqrt{c+d x^2}}+\frac{\left(-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{35 c^4 \left(a^2 c-d\right)^{7/2}}-\frac{a \left(11 a^2 c-6 d\right)}{105 c^2 \left(a^2 c-d\right)^2 \left(c+d x^2\right)^{3/2}}-\frac{a}{35 c \left(a^2 c-d\right) \left(c+d x^2\right)^{5/2}}+\frac{16 x \tan ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \tan ^{-1}(a x)}{35 c^3 \left(c+d x^2\right)^{3/2}}+\frac{6 x \tan ^{-1}(a x)}{35 c^2 \left(c+d x^2\right)^{5/2}}+\frac{x \tan ^{-1}(a x)}{7 c \left(c+d x^2\right)^{7/2}}","-\frac{a \left(19 a^4 c^2-22 a^2 c d+8 d^2\right)}{35 c^3 \left(a^2 c-d\right)^3 \sqrt{c+d x^2}}+\frac{\left(-70 a^4 c^2 d+35 a^6 c^3+56 a^2 c d^2-16 d^3\right) \tanh ^{-1}\left(\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c-d}}\right)}{35 c^4 \left(a^2 c-d\right)^{7/2}}-\frac{a \left(11 a^2 c-6 d\right)}{105 c^2 \left(a^2 c-d\right)^2 \left(c+d x^2\right)^{3/2}}-\frac{a}{35 c \left(a^2 c-d\right) \left(c+d x^2\right)^{5/2}}+\frac{16 x \tan ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \tan ^{-1}(a x)}{35 c^3 \left(c+d x^2\right)^{3/2}}+\frac{6 x \tan ^{-1}(a x)}{35 c^2 \left(c+d x^2\right)^{5/2}}+\frac{x \tan ^{-1}(a x)}{7 c \left(c+d x^2\right)^{7/2}}",1,"-a/(35*c*(a^2*c - d)*(c + d*x^2)^(5/2)) - (a*(11*a^2*c - 6*d))/(105*c^2*(a^2*c - d)^2*(c + d*x^2)^(3/2)) - (a*(19*a^4*c^2 - 22*a^2*c*d + 8*d^2))/(35*c^3*(a^2*c - d)^3*Sqrt[c + d*x^2]) + (x*ArcTan[a*x])/(7*c*(c + d*x^2)^(7/2)) + (6*x*ArcTan[a*x])/(35*c^2*(c + d*x^2)^(5/2)) + (8*x*ArcTan[a*x])/(35*c^3*(c + d*x^2)^(3/2)) + (16*x*ArcTan[a*x])/(35*c^4*Sqrt[c + d*x^2]) + ((35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c - d]])/(35*c^4*(a^2*c - d)^(7/2))","A",8,9,16,0.5625,1,"{192, 191, 4912, 6688, 12, 6715, 1619, 63, 208}"
1228,1,374,0,1.9832197,"\int x^m \left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^m*(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]","\frac{3 d^2 e x^{m+3} \left(a+b \tan ^{-1}(c x)\right)}{m+3}+\frac{d^3 x^{m+1} \left(a+b \tan ^{-1}(c x)\right)}{m+1}+\frac{3 d e^2 x^{m+5} \left(a+b \tan ^{-1}(c x)\right)}{m+5}+\frac{e^3 x^{m+7} \left(a+b \tan ^{-1}(c x)\right)}{m+7}+\frac{b x^{m+2} \left(3 c^4 d^2 e \left(m^3+13 m^2+47 m+35\right)+c^6 \left(-d^3\right) \left(m^3+15 m^2+71 m+105\right)-3 c^2 d e^2 \left(m^3+11 m^2+31 m+21\right)+e^3 \left(m^3+9 m^2+23 m+15\right)\right) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{c^5 (m+1) (m+2) (m+3) (m+5) (m+7)}-\frac{b e x^{m+2} \left(3 c^4 d^2 \left(m^2+12 m+35\right)-3 c^2 d e \left(m^2+10 m+21\right)+e^2 \left(m^2+8 m+15\right)\right)}{c^5 (m+2) (m+3) (m+5) (m+7)}-\frac{b e^2 x^{m+4} \left(\frac{3 c^2 d}{m+5}-\frac{e}{m+7}\right)}{c^3 (m+4)}-\frac{b e^3 x^{m+6}}{c (m+6) (m+7)}","\frac{3 d^2 e x^{m+3} \left(a+b \tan ^{-1}(c x)\right)}{m+3}+\frac{d^3 x^{m+1} \left(a+b \tan ^{-1}(c x)\right)}{m+1}+\frac{3 d e^2 x^{m+5} \left(a+b \tan ^{-1}(c x)\right)}{m+5}+\frac{e^3 x^{m+7} \left(a+b \tan ^{-1}(c x)\right)}{m+7}+\frac{b x^{m+2} \left(3 c^4 d^2 e \left(m^3+13 m^2+47 m+35\right)+c^6 \left(-d^3\right) \left(m^3+15 m^2+71 m+105\right)-3 c^2 d e^2 \left(m^3+11 m^2+31 m+21\right)+e^3 \left(m^3+9 m^2+23 m+15\right)\right) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{c^5 (m+1) (m+2) (m+3) (m+5) (m+7)}-\frac{b e x^{m+2} \left(3 c^4 d^2 \left(m^2+12 m+35\right)-3 c^2 d e \left(m^2+10 m+21\right)+e^2 \left(m^2+8 m+15\right)\right)}{c^5 (m+2) (m+3) (m+5) (m+7)}+\frac{b e^2 x^{m+4} \left(e (m+5)-3 c^2 d (m+7)\right)}{c^3 (m+4) (m+5) (m+7)}-\frac{b e^3 x^{m+6}}{c (m+6) (m+7)}",1,"-((b*e*(e^2*(15 + 8*m + m^2) - 3*c^2*d*e*(21 + 10*m + m^2) + 3*c^4*d^2*(35 + 12*m + m^2))*x^(2 + m))/(c^5*(2 + m)*(3 + m)*(5 + m)*(7 + m))) - (b*e^2*((3*c^2*d)/(5 + m) - e/(7 + m))*x^(4 + m))/(c^3*(4 + m)) - (b*e^3*x^(6 + m))/(c*(6 + m)*(7 + m)) + (d^3*x^(1 + m)*(a + b*ArcTan[c*x]))/(1 + m) + (3*d^2*e*x^(3 + m)*(a + b*ArcTan[c*x]))/(3 + m) + (3*d*e^2*x^(5 + m)*(a + b*ArcTan[c*x]))/(5 + m) + (e^3*x^(7 + m)*(a + b*ArcTan[c*x]))/(7 + m) + (b*(e^3*(15 + 23*m + 9*m^2 + m^3) - 3*c^2*d*e^2*(21 + 31*m + 11*m^2 + m^3) + 3*c^4*d^2*e*(35 + 47*m + 13*m^2 + m^3) - c^6*d^3*(105 + 71*m + 15*m^2 + m^3))*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/(c^5*(1 + m)*(2 + m)*(3 + m)*(5 + m)*(7 + m))","A",4,4,21,0.1905,1,"{270, 4976, 1802, 364}"
1229,1,226,0,0.2936793,"\int x^m \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^m*(d + e*x^2)^2*(a + b*ArcTan[c*x]),x]","\frac{d^2 x^{m+1} \left(a+b \tan ^{-1}(c x)\right)}{m+1}+\frac{2 d e x^{m+3} \left(a+b \tan ^{-1}(c x)\right)}{m+3}+\frac{e^2 x^{m+5} \left(a+b \tan ^{-1}(c x)\right)}{m+5}-\frac{b x^{m+2} \left(c^4 d^2 \left(m^2+8 m+15\right)-2 c^2 d e \left(m^2+6 m+5\right)+e^2 \left(m^2+4 m+3\right)\right) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{c^3 (m+1) (m+2) (m+3) (m+5)}-\frac{b e x^{m+2} \left(\frac{2 c^2 d}{m+3}-\frac{e}{m+5}\right)}{c^3 (m+2)}-\frac{b e^2 x^{m+4}}{c (m+4) (m+5)}","\frac{d^2 x^{m+1} \left(a+b \tan ^{-1}(c x)\right)}{m+1}+\frac{2 d e x^{m+3} \left(a+b \tan ^{-1}(c x)\right)}{m+3}+\frac{e^2 x^{m+5} \left(a+b \tan ^{-1}(c x)\right)}{m+5}-\frac{b x^{m+2} \left(c^4 d^2 \left(m^2+8 m+15\right)-2 c^2 d e \left(m^2+6 m+5\right)+e^2 \left(m^2+4 m+3\right)\right) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{c^3 (m+1) (m+2) (m+3) (m+5)}+\frac{b e x^{m+2} \left(e (m+3)-2 c^2 d (m+5)\right)}{c^3 (m+2) (m+3) (m+5)}-\frac{b e^2 x^{m+4}}{c (m+4) (m+5)}",1,"-((b*e*((2*c^2*d)/(3 + m) - e/(5 + m))*x^(2 + m))/(c^3*(2 + m))) - (b*e^2*x^(4 + m))/(c*(4 + m)*(5 + m)) + (d^2*x^(1 + m)*(a + b*ArcTan[c*x]))/(1 + m) + (2*d*e*x^(3 + m)*(a + b*ArcTan[c*x]))/(3 + m) + (e^2*x^(5 + m)*(a + b*ArcTan[c*x]))/(5 + m) - (b*(e^2*(3 + 4*m + m^2) - 2*c^2*d*e*(5 + 6*m + m^2) + c^4*d^2*(15 + 8*m + m^2))*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/(c^3*(1 + m)*(2 + m)*(3 + m)*(5 + m))","A",4,4,21,0.1905,1,"{270, 4976, 1261, 364}"
1230,1,122,0,0.1269678,"\int x^m \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^m*(d + e*x^2)*(a + b*ArcTan[c*x]),x]","\frac{d x^{m+1} \left(a+b \tan ^{-1}(c x)\right)}{m+1}+\frac{e x^{m+3} \left(a+b \tan ^{-1}(c x)\right)}{m+3}-\frac{b x^{m+2} \left(\frac{c^2 d}{m+1}-\frac{e}{m+3}\right) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{c (m+2)}-\frac{b e x^{m+2}}{c \left(m^2+5 m+6\right)}","\frac{d x^{m+1} \left(a+b \tan ^{-1}(c x)\right)}{m+1}+\frac{e x^{m+3} \left(a+b \tan ^{-1}(c x)\right)}{m+3}-\frac{b x^{m+2} \left(\frac{c^2 d}{m+1}-\frac{e}{m+3}\right) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{c (m+2)}-\frac{b e x^{m+2}}{c \left(m^2+5 m+6\right)}",1,"-((b*e*x^(2 + m))/(c*(6 + 5*m + m^2))) + (d*x^(1 + m)*(a + b*ArcTan[c*x]))/(1 + m) + (e*x^(3 + m)*(a + b*ArcTan[c*x]))/(3 + m) - (b*((c^2*d)/(1 + m) - e/(3 + m))*x^(2 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/(c*(2 + m))","A",3,4,19,0.2105,1,"{14, 4976, 459, 364}"
1231,0,0,0,0.1209626,"\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{d+e x^2} \, dx","Int[(x^m*(a + b*ArcTan[c*x]))/(d + e*x^2),x]","\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{d+e x^2} \, dx","b \text{Int}\left(\frac{x^m \tan ^{-1}(c x)}{d+e x^2},x\right)+\frac{a x^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\frac{e x^2}{d}\right)}{d (m+1)}",0,"(a*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/(d*(1 + m)) + b*Defer[Int][(x^m*ArcTan[c*x])/(d + e*x^2), x]","A",0,0,0,0,-1,"{}"
1232,0,0,0,0.1168446,"\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Int[(x^m*(a + b*ArcTan[c*x]))/(d + e*x^2)^2,x]","\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","b \text{Int}\left(\frac{x^m \tan ^{-1}(c x)}{\left(d+e x^2\right)^2},x\right)+\frac{a x^{m+1} \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};-\frac{e x^2}{d}\right)}{d^2 (m+1)}",0,"(a*x^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/(d^2*(1 + m)) + b*Defer[Int][(x^m*ArcTan[c*x])/(d + e*x^2)^2, x]","A",0,0,0,0,-1,"{}"
1233,0,0,0,0.1760792,"\int x^m \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^m*(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]),x]","\int x^m \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","b \text{Int}\left(x^m \tan ^{-1}(c x) \left(d+e x^2\right)^{5/2},x\right)+\frac{a x^{m+1} \left(d+e x^2\right)^{7/2} \, _2F_1\left(1,\frac{m+8}{2};\frac{m+3}{2};-\frac{e x^2}{d}\right)}{d (m+1)}",0,"(a*d^2*x^(1 + m)*Sqrt[d + e*x^2]*Hypergeometric2F1[-5/2, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/((1 + m)*Sqrt[1 + (e*x^2)/d]) + b*Defer[Int][x^m*(d + e*x^2)^(5/2)*ArcTan[c*x], x]","A",0,0,0,0,-1,"{}"
1234,0,0,0,0.1715752,"\int x^m \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^m*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]),x]","\int x^m \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","b \text{Int}\left(x^m \tan ^{-1}(c x) \left(d+e x^2\right)^{3/2},x\right)+\frac{a x^{m+1} \left(d+e x^2\right)^{5/2} \, _2F_1\left(1,\frac{m+6}{2};\frac{m+3}{2};-\frac{e x^2}{d}\right)}{d (m+1)}",0,"(a*d*x^(1 + m)*Sqrt[d + e*x^2]*Hypergeometric2F1[-3/2, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/((1 + m)*Sqrt[1 + (e*x^2)/d]) + b*Defer[Int][x^m*(d + e*x^2)^(3/2)*ArcTan[c*x], x]","A",0,0,0,0,-1,"{}"
1235,0,0,0,0.1535881,"\int x^m \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^m*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]),x]","\int x^m \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right) \, dx","b \text{Int}\left(x^m \tan ^{-1}(c x) \sqrt{d+e x^2},x\right)+\frac{a x^{m+1} \left(d+e x^2\right)^{3/2} \, _2F_1\left(1,\frac{m+4}{2};\frac{m+3}{2};-\frac{e x^2}{d}\right)}{d (m+1)}",0,"(a*x^(1 + m)*Sqrt[d + e*x^2]*Hypergeometric2F1[-1/2, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/((1 + m)*Sqrt[1 + (e*x^2)/d]) + b*Defer[Int][x^m*Sqrt[d + e*x^2]*ArcTan[c*x], x]","A",0,0,0,0,-1,"{}"
1236,0,0,0,0.1564054,"\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Int[(x^m*(a + b*ArcTan[c*x]))/Sqrt[d + e*x^2],x]","\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","b \text{Int}\left(\frac{x^m \tan ^{-1}(c x)}{\sqrt{d+e x^2}},x\right)+\frac{a x^{m+1} \sqrt{d+e x^2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+3}{2};-\frac{e x^2}{d}\right)}{d (m+1)}",0,"(a*x^(1 + m)*Sqrt[1 + (e*x^2)/d]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/((1 + m)*Sqrt[d + e*x^2]) + b*Defer[Int][(x^m*ArcTan[c*x])/Sqrt[d + e*x^2], x]","A",0,0,0,0,-1,"{}"
1237,0,0,0,0.1736916,"\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Int[(x^m*(a + b*ArcTan[c*x]))/(d + e*x^2)^(3/2),x]","\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","b \text{Int}\left(\frac{x^m \tan ^{-1}(c x)}{\left(d+e x^2\right)^{3/2}},x\right)+\frac{a x^{m+1} \, _2F_1\left(1,\frac{m}{2};\frac{m+3}{2};-\frac{e x^2}{d}\right)}{d (m+1) \sqrt{d+e x^2}}",0,"(a*x^(1 + m)*Sqrt[1 + (e*x^2)/d]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/(d*(1 + m)*Sqrt[d + e*x^2]) + b*Defer[Int][(x^m*ArcTan[c*x])/(d + e*x^2)^(3/2), x]","A",0,0,0,0,-1,"{}"
1238,0,0,0,0.1764066,"\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Int[(x^m*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2),x]","\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","b \text{Int}\left(\frac{x^m \tan ^{-1}(c x)}{\left(d+e x^2\right)^{5/2}},x\right)+\frac{a x^{m+1} \, _2F_1\left(1,\frac{m-2}{2};\frac{m+3}{2};-\frac{e x^2}{d}\right)}{d (m+1) \left(d+e x^2\right)^{3/2}}",0,"(a*x^(1 + m)*Sqrt[1 + (e*x^2)/d]*Hypergeometric2F1[5/2, (1 + m)/2, (3 + m)/2, -((e*x^2)/d)])/(d^2*(1 + m)*Sqrt[d + e*x^2]) + b*Defer[Int][(x^m*ArcTan[c*x])/(d + e*x^2)^(5/2), x]","A",0,0,0,0,-1,"{}"
1239,0,0,0,0.1184398,"\int x^m \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^m*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]","\int x^m \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","b \text{Int}\left(x^m \tan ^{-1}(c x) \left(d+e x^2\right)^p,x\right)+\frac{a x^{m+1} \left(d+e x^2\right)^{p+1} \, _2F_1\left(1,\frac{1}{2} (m+2 p+3);\frac{m+3}{2};-\frac{e x^2}{d}\right)}{d (m+1)}",0,"(a*x^(1 + m)*(d + e*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((e*x^2)/d)])/((1 + m)*(1 + (e*x^2)/d)^p) + b*Defer[Int][x^m*(d + e*x^2)^p*ArcTan[c*x], x]","A",0,0,0,0,-1,"{}"
1240,0,0,0,0.1423803,"\int x^{-2-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^(-2 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]","\int x^{-2-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","b \text{Int}\left(x^{-2 p-2} \tan ^{-1}(c x) \left(d+e x^2\right)^p,x\right)-\frac{a x^{-2 p-1} \left(d+e x^2\right)^{p+1} \, _2F_1\left(\frac{1}{2},1;\frac{1}{2} (1-2 p);-\frac{e x^2}{d}\right)}{d (2 p+1)}",0,"-((a*x^(-1 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(-1 - 2*p)/2, -p, (1 - 2*p)/2, -((e*x^2)/d)])/((1 + 2*p)*(1 + (e*x^2)/d)^p)) + b*Defer[Int][x^(-2 - 2*p)*(d + e*x^2)^p*ArcTan[c*x], x]","A",0,0,0,0,-1,"{}"
1241,1,129,0,0.1664921,"\int x^{-3-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^(-3 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]","-\frac{x^{-2 (p+1)} \left(d+e x^2\right)^{p+1} \left(a+b \tan ^{-1}(c x)\right)}{2 d (p+1)}-\frac{b c x^{-2 p-1} \left(d+e x^2\right)^p \left(\frac{e x^2}{d}+1\right)^{-p} F_1\left(\frac{1}{2} (-2 p-1);1,-p-1;\frac{1}{2} (1-2 p);-c^2 x^2,-\frac{e x^2}{d}\right)}{2 \left(2 p^2+3 p+1\right)}","-\frac{x^{-2 (p+1)} \left(d+e x^2\right)^{p+1} \left(a+b \tan ^{-1}(c x)\right)}{2 d (p+1)}-\frac{b c x^{-2 p-1} \left(d+e x^2\right)^p \left(\frac{e x^2}{d}+1\right)^{-p} F_1\left(\frac{1}{2} (-2 p-1);1,-p-1;\frac{1}{2} (1-2 p);-c^2 x^2,-\frac{e x^2}{d}\right)}{2 \left(2 p^2+3 p+1\right)}",1,"-(b*c*x^(-1 - 2*p)*(d + e*x^2)^p*AppellF1[(-1 - 2*p)/2, 1, -1 - p, (1 - 2*p)/2, -(c^2*x^2), -((e*x^2)/d)])/(2*(1 + 3*p + 2*p^2)*(1 + (e*x^2)/d)^p) - ((d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(2*d*(1 + p)*x^(2*(1 + p)))","A",4,5,25,0.2000,1,"{264, 4976, 12, 511, 510}"
1242,0,0,0,0.1400935,"\int x^{-4-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^(-4 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]","\int x^{-4-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","b \text{Int}\left(x^{-2 p-4} \tan ^{-1}(c x) \left(d+e x^2\right)^p,x\right)-\frac{a x^{-2 p-3} \left(d+e x^2\right)^{p+1} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2} (-2 p-1);-\frac{e x^2}{d}\right)}{d (2 p+3)}",0,"-((a*x^(-3 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(-3 - 2*p)/2, -p, (-1 - 2*p)/2, -((e*x^2)/d)])/((3 + 2*p)*(1 + (e*x^2)/d)^p)) + b*Defer[Int][x^(-4 - 2*p)*(d + e*x^2)^p*ArcTan[c*x], x]","A",0,0,0,0,-1,"{}"
1243,1,285,0,0.3747954,"\int x^{-5-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^(-5 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]","\frac{e x^{-2 (p+1)} \left(d+e x^2\right)^{p+1} \left(a+b \tan ^{-1}(c x)\right)}{2 d^2 (p+1) (p+2)}-\frac{x^{-2 (p+2)} \left(d+e x^2\right)^{p+1} \left(a+b \tan ^{-1}(c x)\right)}{2 d (p+2)}-\frac{b x^{-2 p-3} \left(c^2 d (p+1)+e\right) \left(d+e x^2\right)^p \left(\frac{e x^2}{d}+1\right)^{-p} F_1\left(\frac{1}{2} (-2 p-3);1,-p-1;\frac{1}{2} (-2 p-1);-c^2 x^2,-\frac{e x^2}{d}\right)}{2 c d (p+1) (p+2) (2 p+3)}+\frac{b e x^{-2 p-3} \left(d+e x^2\right)^p \left(\frac{e x^2}{d}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} (-2 p-3),-p-1;\frac{1}{2} (-2 p-1);-\frac{e x^2}{d}\right)}{2 c d \left(2 p^3+9 p^2+13 p+6\right)}","\frac{e x^{-2 (p+1)} \left(d+e x^2\right)^{p+1} \left(a+b \tan ^{-1}(c x)\right)}{2 d^2 (p+1) (p+2)}-\frac{x^{-2 (p+2)} \left(d+e x^2\right)^{p+1} \left(a+b \tan ^{-1}(c x)\right)}{2 d (p+2)}-\frac{b x^{-2 p-3} \left(c^2 d (p+1)+e\right) \left(d+e x^2\right)^p \left(\frac{e x^2}{d}+1\right)^{-p} F_1\left(\frac{1}{2} (-2 p-3);1,-p-1;\frac{1}{2} (-2 p-1);-c^2 x^2,-\frac{e x^2}{d}\right)}{2 c d (p+1) (p+2) (2 p+3)}+\frac{b e x^{-2 p-3} \left(d+e x^2\right)^p \left(\frac{e x^2}{d}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} (-2 p-3),-p-1;\frac{1}{2} (-2 p-1);-\frac{e x^2}{d}\right)}{2 c d \left(2 p^3+9 p^2+13 p+6\right)}",1,"-(b*(e + c^2*d*(1 + p))*x^(-3 - 2*p)*(d + e*x^2)^p*AppellF1[(-3 - 2*p)/2, 1, -1 - p, (-1 - 2*p)/2, -(c^2*x^2), -((e*x^2)/d)])/(2*c*d*(1 + p)*(2 + p)*(3 + 2*p)*(1 + (e*x^2)/d)^p) + (e*(d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(2*d^2*(1 + p)*(2 + p)*x^(2*(1 + p))) - ((d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(2*d*(2 + p)*x^(2*(2 + p))) + (b*e*x^(-3 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(-3 - 2*p)/2, -1 - p, (-1 - 2*p)/2, -((e*x^2)/d)])/(2*c*d*(6 + 13*p + 9*p^2 + 2*p^3)*(1 + (e*x^2)/d)^p)","A",8,9,25,0.3600,1,"{271, 264, 4976, 12, 584, 365, 364, 511, 510}"
1244,0,0,0,0.1437779,"\int x^{-6-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^(-6 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]","\int x^{-6-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","b \text{Int}\left(x^{-2 p-6} \tan ^{-1}(c x) \left(d+e x^2\right)^p,x\right)-\frac{a x^{-2 p-5} \left(d+e x^2\right)^{p+1} \, _2F_1\left(-\frac{3}{2},1;\frac{1}{2} (-2 p-3);-\frac{e x^2}{d}\right)}{d (2 p+5)}",0,"-((a*x^(-5 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(-5 - 2*p)/2, -p, (-3 - 2*p)/2, -((e*x^2)/d)])/((5 + 2*p)*(1 + (e*x^2)/d)^p)) + b*Defer[Int][x^(-6 - 2*p)*(d + e*x^2)^p*ArcTan[c*x], x]","A",0,0,0,0,-1,"{}"
1245,1,466,0,1.4257786,"\int x^{-7-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^(-7 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]","-\frac{e^2 x^{-2 (p+1)} \left(d+e x^2\right)^{p+1} \left(a+b \tan ^{-1}(c x)\right)}{d^3 (p+1) (p+2) (p+3)}+\frac{e x^{-2 (p+2)} \left(d+e x^2\right)^{p+1} \left(a+b \tan ^{-1}(c x)\right)}{d^2 (p+2) (p+3)}-\frac{x^{-2 (p+3)} \left(d+e x^2\right)^{p+1} \left(a+b \tan ^{-1}(c x)\right)}{2 d (p+3)}-\frac{b x^{-2 p-5} \left(c^4 d^2 \left(p^2+3 p+2\right)+2 c^2 d e (p+1)+2 e^2\right) \left(d+e x^2\right)^p \left(\frac{e x^2}{d}+1\right)^{-p} F_1\left(\frac{1}{2} (-2 p-5);1,-p-1;\frac{1}{2} (-2 p-3);-c^2 x^2,-\frac{e x^2}{d}\right)}{2 c^3 d^2 (p+1) (p+2) (p+3) (2 p+5)}+\frac{b e x^{-2 p-5} \left(c^2 d (p+1)+e\right) \left(d+e x^2\right)^p \left(\frac{e x^2}{d}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} (-2 p-5),-p-1;\frac{1}{2} (-2 p-3);-\frac{e x^2}{d}\right)}{c^3 d^2 (p+1) (p+2) (p+3) (2 p+5)}-\frac{b e^2 x^{-2 p-3} \left(d+e x^2\right)^p \left(\frac{e x^2}{d}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} (-2 p-3),-p-1;\frac{1}{2} (-2 p-1);-\frac{e x^2}{d}\right)}{c d^2 (p+1) (p+2) (p+3) (2 p+3)}","-\frac{e^2 x^{-2 (p+1)} \left(d+e x^2\right)^{p+1} \left(a+b \tan ^{-1}(c x)\right)}{d^3 (p+1) (p+2) (p+3)}+\frac{e x^{-2 (p+2)} \left(d+e x^2\right)^{p+1} \left(a+b \tan ^{-1}(c x)\right)}{d^2 (p+2) (p+3)}-\frac{x^{-2 (p+3)} \left(d+e x^2\right)^{p+1} \left(a+b \tan ^{-1}(c x)\right)}{2 d (p+3)}-\frac{b x^{-2 p-5} \left(c^4 d^2 \left(p^2+3 p+2\right)+2 c^2 d e (p+1)+2 e^2\right) \left(d+e x^2\right)^p \left(\frac{e x^2}{d}+1\right)^{-p} F_1\left(\frac{1}{2} (-2 p-5);1,-p-1;\frac{1}{2} (-2 p-3);-c^2 x^2,-\frac{e x^2}{d}\right)}{2 c^3 d^2 (p+1) (p+2) (p+3) (2 p+5)}+\frac{b e x^{-2 p-5} \left(c^2 d (p+1)+e\right) \left(d+e x^2\right)^p \left(\frac{e x^2}{d}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} (-2 p-5),-p-1;\frac{1}{2} (-2 p-3);-\frac{e x^2}{d}\right)}{c^3 d^2 (p+1) (p+2) (p+3) (2 p+5)}-\frac{b e^2 x^{-2 p-3} \left(d+e x^2\right)^p \left(\frac{e x^2}{d}+1\right)^{-p} \, _2F_1\left(\frac{1}{2} (-2 p-3),-p-1;\frac{1}{2} (-2 p-1);-\frac{e x^2}{d}\right)}{c d^2 (p+1) (p+2) (p+3) (2 p+3)}",1,"-(b*(2*e^2 + 2*c^2*d*e*(1 + p) + c^4*d^2*(2 + 3*p + p^2))*x^(-5 - 2*p)*(d + e*x^2)^p*AppellF1[(-5 - 2*p)/2, 1, -1 - p, (-3 - 2*p)/2, -(c^2*x^2), -((e*x^2)/d)])/(2*c^3*d^2*(1 + p)*(2 + p)*(3 + p)*(5 + 2*p)*(1 + (e*x^2)/d)^p) - (e^2*(d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(d^3*(1 + p)*(2 + p)*(3 + p)*x^(2*(1 + p))) + (e*(d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(d^2*(2 + p)*(3 + p)*x^(2*(2 + p))) - ((d + e*x^2)^(1 + p)*(a + b*ArcTan[c*x]))/(2*d*(3 + p)*x^(2*(3 + p))) + (b*e*(e + c^2*d*(1 + p))*x^(-5 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(-5 - 2*p)/2, -1 - p, (-3 - 2*p)/2, -((e*x^2)/d)])/(c^3*d^2*(1 + p)*(2 + p)*(3 + p)*(5 + 2*p)*(1 + (e*x^2)/d)^p) - (b*e^2*x^(-3 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(-3 - 2*p)/2, -1 - p, (-1 - 2*p)/2, -((e*x^2)/d)])/(c*d^2*(1 + p)*(2 + p)*(3 + p)*(3 + 2*p)*(1 + (e*x^2)/d)^p)","A",10,9,25,0.3600,1,"{271, 264, 4976, 12, 6725, 365, 364, 511, 510}"
1246,0,0,0,0.1409338,"\int x^{-8-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Int[x^(-8 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]","\int x^{-8-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","b \text{Int}\left(x^{-2 p-8} \tan ^{-1}(c x) \left(d+e x^2\right)^p,x\right)-\frac{a x^{-2 p-7} \left(d+e x^2\right)^{p+1} \, _2F_1\left(-\frac{5}{2},1;\frac{1}{2} (-2 p-5);-\frac{e x^2}{d}\right)}{d (2 p+7)}",0,"-((a*x^(-7 - 2*p)*(d + e*x^2)^p*Hypergeometric2F1[(-7 - 2*p)/2, -p, (-5 - 2*p)/2, -((e*x^2)/d)])/((7 + 2*p)*(1 + (e*x^2)/d)^p)) + b*Defer[Int][x^(-8 - 2*p)*(d + e*x^2)^p*ArcTan[c*x], x]","A",0,0,0,0,-1,"{}"
1247,1,271,0,0.651159,"\int x^3 \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x^3*(d + e*x^2)*(a + b*ArcTan[c*x])^2,x]","\frac{a b d x}{2 c^3}-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}+\frac{b e x^3 \left(a+b \tan ^{-1}(c x)\right)}{9 c^3}-\frac{a b e x}{3 c^5}+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{6 c^6}+\frac{1}{4} d x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b d x^3 \left(a+b \tan ^{-1}(c x)\right)}{6 c}+\frac{1}{6} e x^6 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e x^5 \left(a+b \tan ^{-1}(c x)\right)}{15 c}+\frac{b^2 d x^2}{12 c^2}-\frac{b^2 d \log \left(c^2 x^2+1\right)}{3 c^4}+\frac{b^2 d x \tan ^{-1}(c x)}{2 c^3}+\frac{b^2 e x^4}{60 c^2}-\frac{4 b^2 e x^2}{45 c^4}+\frac{23 b^2 e \log \left(c^2 x^2+1\right)}{90 c^6}-\frac{b^2 e x \tan ^{-1}(c x)}{3 c^5}","\frac{a b d x}{2 c^3}-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}+\frac{b e x^3 \left(a+b \tan ^{-1}(c x)\right)}{9 c^3}-\frac{a b e x}{3 c^5}+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{6 c^6}+\frac{1}{4} d x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b d x^3 \left(a+b \tan ^{-1}(c x)\right)}{6 c}+\frac{1}{6} e x^6 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e x^5 \left(a+b \tan ^{-1}(c x)\right)}{15 c}+\frac{b^2 d x^2}{12 c^2}-\frac{b^2 d \log \left(c^2 x^2+1\right)}{3 c^4}+\frac{b^2 d x \tan ^{-1}(c x)}{2 c^3}+\frac{b^2 e x^4}{60 c^2}-\frac{4 b^2 e x^2}{45 c^4}+\frac{23 b^2 e \log \left(c^2 x^2+1\right)}{90 c^6}-\frac{b^2 e x \tan ^{-1}(c x)}{3 c^5}",1,"(a*b*d*x)/(2*c^3) - (a*b*e*x)/(3*c^5) + (b^2*d*x^2)/(12*c^2) - (4*b^2*e*x^2)/(45*c^4) + (b^2*e*x^4)/(60*c^2) + (b^2*d*x*ArcTan[c*x])/(2*c^3) - (b^2*e*x*ArcTan[c*x])/(3*c^5) - (b*d*x^3*(a + b*ArcTan[c*x]))/(6*c) + (b*e*x^3*(a + b*ArcTan[c*x]))/(9*c^3) - (b*e*x^5*(a + b*ArcTan[c*x]))/(15*c) - (d*(a + b*ArcTan[c*x])^2)/(4*c^4) + (e*(a + b*ArcTan[c*x])^2)/(6*c^6) + (d*x^4*(a + b*ArcTan[c*x])^2)/4 + (e*x^6*(a + b*ArcTan[c*x])^2)/6 - (b^2*d*Log[1 + c^2*x^2])/(3*c^4) + (23*b^2*e*Log[1 + c^2*x^2])/(90*c^6)","A",29,8,21,0.3810,1,"{4980, 4852, 4916, 266, 43, 4846, 260, 4884}"
1248,1,323,0,0.5900954,"\int x^2 \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x^2*(d + e*x^2)*(a + b*ArcTan[c*x])^2,x]","-\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^3}+\frac{i b^2 e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{5 c^5}-\frac{i d \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^3}-\frac{2 b d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}+\frac{b e x^2 \left(a+b \tan ^{-1}(c x)\right)}{5 c^3}+\frac{i e \left(a+b \tan ^{-1}(c x)\right)^2}{5 c^5}+\frac{2 b e \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{5 c^5}+\frac{1}{3} d x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b d x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c}+\frac{1}{5} e x^5 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e x^4 \left(a+b \tan ^{-1}(c x)\right)}{10 c}+\frac{b^2 d x}{3 c^2}-\frac{b^2 d \tan ^{-1}(c x)}{3 c^3}+\frac{b^2 e x^3}{30 c^2}-\frac{3 b^2 e x}{10 c^4}+\frac{3 b^2 e \tan ^{-1}(c x)}{10 c^5}","-\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^3}+\frac{i b^2 e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{5 c^5}-\frac{i d \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^3}-\frac{2 b d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}+\frac{b e x^2 \left(a+b \tan ^{-1}(c x)\right)}{5 c^3}+\frac{i e \left(a+b \tan ^{-1}(c x)\right)^2}{5 c^5}+\frac{2 b e \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{5 c^5}+\frac{1}{3} d x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b d x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c}+\frac{1}{5} e x^5 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e x^4 \left(a+b \tan ^{-1}(c x)\right)}{10 c}+\frac{b^2 d x}{3 c^2}-\frac{b^2 d \tan ^{-1}(c x)}{3 c^3}+\frac{b^2 e x^3}{30 c^2}-\frac{3 b^2 e x}{10 c^4}+\frac{3 b^2 e \tan ^{-1}(c x)}{10 c^5}",1,"(b^2*d*x)/(3*c^2) - (3*b^2*e*x)/(10*c^4) + (b^2*e*x^3)/(30*c^2) - (b^2*d*ArcTan[c*x])/(3*c^3) + (3*b^2*e*ArcTan[c*x])/(10*c^5) - (b*d*x^2*(a + b*ArcTan[c*x]))/(3*c) + (b*e*x^2*(a + b*ArcTan[c*x]))/(5*c^3) - (b*e*x^4*(a + b*ArcTan[c*x]))/(10*c) - ((I/3)*d*(a + b*ArcTan[c*x])^2)/c^3 + ((I/5)*e*(a + b*ArcTan[c*x])^2)/c^5 + (d*x^3*(a + b*ArcTan[c*x])^2)/3 + (e*x^5*(a + b*ArcTan[c*x])^2)/5 - (2*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) + (2*b*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(5*c^5) - ((I/3)*b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3 + ((I/5)*b^2*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^5","A",25,10,21,0.4762,1,"{4980, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 302}"
1249,1,199,0,0.3983268,"\int x \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x*(d + e*x^2)*(a + b*ArcTan[c*x])^2,x]","\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2}+\frac{a b e x}{2 c^3}-\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}+\frac{1}{2} d x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{a b d x}{c}+\frac{1}{4} e x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e x^3 \left(a+b \tan ^{-1}(c x)\right)}{6 c}+\frac{b^2 d \log \left(c^2 x^2+1\right)}{2 c^2}+\frac{b^2 e x^2}{12 c^2}-\frac{b^2 e \log \left(c^2 x^2+1\right)}{3 c^4}+\frac{b^2 e x \tan ^{-1}(c x)}{2 c^3}-\frac{b^2 d x \tan ^{-1}(c x)}{c}","\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2}+\frac{a b e x}{2 c^3}-\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}+\frac{1}{2} d x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{a b d x}{c}+\frac{1}{4} e x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e x^3 \left(a+b \tan ^{-1}(c x)\right)}{6 c}+\frac{b^2 d \log \left(c^2 x^2+1\right)}{2 c^2}+\frac{b^2 e x^2}{12 c^2}-\frac{b^2 e \log \left(c^2 x^2+1\right)}{3 c^4}+\frac{b^2 e x \tan ^{-1}(c x)}{2 c^3}-\frac{b^2 d x \tan ^{-1}(c x)}{c}",1,"-((a*b*d*x)/c) + (a*b*e*x)/(2*c^3) + (b^2*e*x^2)/(12*c^2) - (b^2*d*x*ArcTan[c*x])/c + (b^2*e*x*ArcTan[c*x])/(2*c^3) - (b*e*x^3*(a + b*ArcTan[c*x]))/(6*c) + (d*(a + b*ArcTan[c*x])^2)/(2*c^2) - (e*(a + b*ArcTan[c*x])^2)/(4*c^4) + (d*x^2*(a + b*ArcTan[c*x])^2)/2 + (e*x^4*(a + b*ArcTan[c*x])^2)/4 + (b^2*d*Log[1 + c^2*x^2])/(2*c^2) - (b^2*e*Log[1 + c^2*x^2])/(3*c^4)","A",19,8,19,0.4211,1,"{4980, 4852, 4916, 4846, 260, 4884, 266, 43}"
1250,1,231,0,0.3575402,"\int \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[(d + e*x^2)*(a + b*ArcTan[c*x])^2,x]","-\frac{i b^2 e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^3}+\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c}-\frac{i e \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^3}-\frac{2 b e \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}+d x \left(a+b \tan ^{-1}(c x)\right)^2+\frac{i d \left(a+b \tan ^{-1}(c x)\right)^2}{c}+\frac{2 b d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{1}{3} e x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c}+\frac{b^2 e x}{3 c^2}-\frac{b^2 e \tan ^{-1}(c x)}{3 c^3}","-\frac{i b^2 e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^3}+\frac{i b^2 d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c}-\frac{i e \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^3}-\frac{2 b e \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}+d x \left(a+b \tan ^{-1}(c x)\right)^2+\frac{i d \left(a+b \tan ^{-1}(c x)\right)^2}{c}+\frac{2 b d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{1}{3} e x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c}+\frac{b^2 e x}{3 c^2}-\frac{b^2 e \tan ^{-1}(c x)}{3 c^3}",1,"(b^2*e*x)/(3*c^2) - (b^2*e*ArcTan[c*x])/(3*c^3) - (b*e*x^2*(a + b*ArcTan[c*x]))/(3*c) + (I*d*(a + b*ArcTan[c*x])^2)/c - ((I/3)*e*(a + b*ArcTan[c*x])^2)/c^3 + d*x*(a + b*ArcTan[c*x])^2 + (e*x^3*(a + b*ArcTan[c*x])^2)/3 + (2*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c - (2*b*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) + (I*b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/c - ((I/3)*b^2*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3","A",16,10,18,0.5556,1,"{4914, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 321, 203}"
1251,1,217,0,0.4423589,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)^2}{x} \, dx","Int[((d + e*x^2)*(a + b*ArcTan[c*x])^2)/x,x]","-i b d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{1}{2} b^2 d \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{1}{2} b^2 d \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2}+2 d \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{2} e x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{a b e x}{c}+\frac{b^2 e \log \left(c^2 x^2+1\right)}{2 c^2}-\frac{b^2 e x \tan ^{-1}(c x)}{c}","-i b d \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{1}{2} b^2 d \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{1}{2} b^2 d \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2}+2 d \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{2} e x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{a b e x}{c}+\frac{b^2 e \log \left(c^2 x^2+1\right)}{2 c^2}-\frac{b^2 e x \tan ^{-1}(c x)}{c}",1,"-((a*b*e*x)/c) - (b^2*e*x*ArcTan[c*x])/c + (e*(a + b*ArcTan[c*x])^2)/(2*c^2) + (e*x^2*(a + b*ArcTan[c*x])^2)/2 + 2*d*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + (b^2*e*Log[1 + c^2*x^2])/(2*c^2) - I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (b^2*d*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*d*PolyLog[3, -1 + 2/(1 + I*c*x)])/2","A",14,10,21,0.4762,1,"{4980, 4850, 4988, 4884, 4994, 6610, 4852, 4916, 4846, 260}"
1252,1,172,0,0.326766,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)^2}{x^2} \, dx","Int[((d + e*x^2)*(a + b*ArcTan[c*x])^2)/x^2,x]","-i b^2 c d \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)+\frac{i b^2 e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c}-i c d \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{x}+2 b c d \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+\frac{i e \left(a+b \tan ^{-1}(c x)\right)^2}{c}+e x \left(a+b \tan ^{-1}(c x)\right)^2+\frac{2 b e \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}","-i b^2 c d \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)+\frac{i b^2 e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c}-i c d \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{x}+2 b c d \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+\frac{i e \left(a+b \tan ^{-1}(c x)\right)^2}{c}+e x \left(a+b \tan ^{-1}(c x)\right)^2+\frac{2 b e \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}",1,"(-I)*c*d*(a + b*ArcTan[c*x])^2 + (I*e*(a + b*ArcTan[c*x])^2)/c - (d*(a + b*ArcTan[c*x])^2)/x + e*x*(a + b*ArcTan[c*x])^2 + (2*b*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c + 2*b*c*d*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - I*b^2*c*d*PolyLog[2, -1 + 2/(1 - I*c*x)] + (I*b^2*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/c","A",11,10,21,0.4762,1,"{4980, 4846, 4920, 4854, 2402, 2315, 4852, 4924, 4868, 2447}"
1253,1,220,0,0.4607683,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)^2}{x^3} \, dx","Int[((d + e*x^2)*(a + b*ArcTan[c*x])^2)/x^3,x]","-i b e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+i b e \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{1}{2} b^2 e \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{1}{2} b^2 e \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-\frac{1}{2} c^2 d \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{b c d \left(a+b \tan ^{-1}(c x)\right)}{x}+2 e \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{2} b^2 c^2 d \log \left(c^2 x^2+1\right)+b^2 c^2 d \log (x)","-i b e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+i b e \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{1}{2} b^2 e \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{1}{2} b^2 e \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-\frac{1}{2} c^2 d \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{b c d \left(a+b \tan ^{-1}(c x)\right)}{x}+2 e \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-\frac{1}{2} b^2 c^2 d \log \left(c^2 x^2+1\right)+b^2 c^2 d \log (x)",1,"-((b*c*d*(a + b*ArcTan[c*x]))/x) - (c^2*d*(a + b*ArcTan[c*x])^2)/2 - (d*(a + b*ArcTan[c*x])^2)/(2*x^2) + 2*e*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + b^2*c^2*d*Log[x] - (b^2*c^2*d*Log[1 + c^2*x^2])/2 - I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (b^2*e*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*e*PolyLog[3, -1 + 2/(1 + I*c*x)])/2","A",16,12,21,0.5714,1,"{4980, 4852, 4918, 266, 36, 29, 31, 4884, 4850, 4988, 4994, 6610}"
1254,1,502,0,1.1401066,"\int x^3 \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x^3*(d + e*x^2)^2*(a + b*ArcTan[c*x])^2,x]","\frac{a b d^2 x}{2 c^3}-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}+\frac{2 b d e x^3 \left(a+b \tan ^{-1}(c x)\right)}{9 c^3}-\frac{2 a b d e x}{3 c^5}+\frac{d e \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^6}+\frac{b e^2 x^5 \left(a+b \tan ^{-1}(c x)\right)}{20 c^3}-\frac{b e^2 x^3 \left(a+b \tan ^{-1}(c x)\right)}{12 c^5}+\frac{a b e^2 x}{4 c^7}-\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{8 c^8}+\frac{1}{4} d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)}{6 c}+\frac{1}{3} d e x^6 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{2 b d e x^5 \left(a+b \tan ^{-1}(c x)\right)}{15 c}+\frac{1}{8} e^2 x^8 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e^2 x^7 \left(a+b \tan ^{-1}(c x)\right)}{28 c}+\frac{b^2 d^2 x^2}{12 c^2}-\frac{b^2 d^2 \log \left(c^2 x^2+1\right)}{3 c^4}+\frac{b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}+\frac{b^2 d e x^4}{30 c^2}-\frac{8 b^2 d e x^2}{45 c^4}+\frac{23 b^2 d e \log \left(c^2 x^2+1\right)}{45 c^6}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{3 c^5}+\frac{b^2 e^2 x^6}{168 c^2}-\frac{3 b^2 e^2 x^4}{140 c^4}+\frac{71 b^2 e^2 x^2}{840 c^6}-\frac{22 b^2 e^2 \log \left(c^2 x^2+1\right)}{105 c^8}+\frac{b^2 e^2 x \tan ^{-1}(c x)}{4 c^7}","\frac{a b d^2 x}{2 c^3}-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}+\frac{2 b d e x^3 \left(a+b \tan ^{-1}(c x)\right)}{9 c^3}-\frac{2 a b d e x}{3 c^5}+\frac{d e \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^6}+\frac{b e^2 x^5 \left(a+b \tan ^{-1}(c x)\right)}{20 c^3}-\frac{b e^2 x^3 \left(a+b \tan ^{-1}(c x)\right)}{12 c^5}+\frac{a b e^2 x}{4 c^7}-\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{8 c^8}+\frac{1}{4} d^2 x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)}{6 c}+\frac{1}{3} d e x^6 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{2 b d e x^5 \left(a+b \tan ^{-1}(c x)\right)}{15 c}+\frac{1}{8} e^2 x^8 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e^2 x^7 \left(a+b \tan ^{-1}(c x)\right)}{28 c}+\frac{b^2 d^2 x^2}{12 c^2}-\frac{b^2 d^2 \log \left(c^2 x^2+1\right)}{3 c^4}+\frac{b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}+\frac{b^2 d e x^4}{30 c^2}-\frac{8 b^2 d e x^2}{45 c^4}+\frac{23 b^2 d e \log \left(c^2 x^2+1\right)}{45 c^6}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{3 c^5}+\frac{b^2 e^2 x^6}{168 c^2}-\frac{3 b^2 e^2 x^4}{140 c^4}+\frac{71 b^2 e^2 x^2}{840 c^6}-\frac{22 b^2 e^2 \log \left(c^2 x^2+1\right)}{105 c^8}+\frac{b^2 e^2 x \tan ^{-1}(c x)}{4 c^7}",1,"(a*b*d^2*x)/(2*c^3) - (2*a*b*d*e*x)/(3*c^5) + (a*b*e^2*x)/(4*c^7) + (b^2*d^2*x^2)/(12*c^2) - (8*b^2*d*e*x^2)/(45*c^4) + (71*b^2*e^2*x^2)/(840*c^6) + (b^2*d*e*x^4)/(30*c^2) - (3*b^2*e^2*x^4)/(140*c^4) + (b^2*e^2*x^6)/(168*c^2) + (b^2*d^2*x*ArcTan[c*x])/(2*c^3) - (2*b^2*d*e*x*ArcTan[c*x])/(3*c^5) + (b^2*e^2*x*ArcTan[c*x])/(4*c^7) - (b*d^2*x^3*(a + b*ArcTan[c*x]))/(6*c) + (2*b*d*e*x^3*(a + b*ArcTan[c*x]))/(9*c^3) - (b*e^2*x^3*(a + b*ArcTan[c*x]))/(12*c^5) - (2*b*d*e*x^5*(a + b*ArcTan[c*x]))/(15*c) + (b*e^2*x^5*(a + b*ArcTan[c*x]))/(20*c^3) - (b*e^2*x^7*(a + b*ArcTan[c*x]))/(28*c) - (d^2*(a + b*ArcTan[c*x])^2)/(4*c^4) + (d*e*(a + b*ArcTan[c*x])^2)/(3*c^6) - (e^2*(a + b*ArcTan[c*x])^2)/(8*c^8) + (d^2*x^4*(a + b*ArcTan[c*x])^2)/4 + (d*e*x^6*(a + b*ArcTan[c*x])^2)/3 + (e^2*x^8*(a + b*ArcTan[c*x])^2)/8 - (b^2*d^2*Log[1 + c^2*x^2])/(3*c^4) + (23*b^2*d*e*Log[1 + c^2*x^2])/(45*c^6) - (22*b^2*e^2*Log[1 + c^2*x^2])/(105*c^8)","A",50,8,23,0.3478,1,"{4980, 4852, 4916, 266, 43, 4846, 260, 4884}"
1255,1,580,0,1.068153,"\int x^2 \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x^2*(d + e*x^2)^2*(a + b*ArcTan[c*x])^2,x]","-\frac{i b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^3}+\frac{2 i b^2 d e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{5 c^5}-\frac{i b^2 e^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{7 c^7}-\frac{i d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^3}-\frac{2 b d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}+\frac{2 b d e x^2 \left(a+b \tan ^{-1}(c x)\right)}{5 c^3}+\frac{2 i d e \left(a+b \tan ^{-1}(c x)\right)^2}{5 c^5}+\frac{4 b d e \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{5 c^5}+\frac{b e^2 x^4 \left(a+b \tan ^{-1}(c x)\right)}{14 c^3}-\frac{b e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{7 c^5}-\frac{i e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{7 c^7}-\frac{2 b e^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{7 c^7}+\frac{1}{3} d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c}+\frac{2}{5} d e x^5 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b d e x^4 \left(a+b \tan ^{-1}(c x)\right)}{5 c}+\frac{1}{7} e^2 x^7 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e^2 x^6 \left(a+b \tan ^{-1}(c x)\right)}{21 c}+\frac{b^2 d^2 x}{3 c^2}-\frac{b^2 d^2 \tan ^{-1}(c x)}{3 c^3}+\frac{b^2 d e x^3}{15 c^2}-\frac{3 b^2 d e x}{5 c^4}+\frac{3 b^2 d e \tan ^{-1}(c x)}{5 c^5}+\frac{b^2 e^2 x^5}{105 c^2}-\frac{5 b^2 e^2 x^3}{126 c^4}+\frac{11 b^2 e^2 x}{42 c^6}-\frac{11 b^2 e^2 \tan ^{-1}(c x)}{42 c^7}","-\frac{i b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^3}+\frac{2 i b^2 d e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{5 c^5}-\frac{i b^2 e^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{7 c^7}-\frac{i d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^3}-\frac{2 b d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}+\frac{2 b d e x^2 \left(a+b \tan ^{-1}(c x)\right)}{5 c^3}+\frac{2 i d e \left(a+b \tan ^{-1}(c x)\right)^2}{5 c^5}+\frac{4 b d e \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{5 c^5}+\frac{b e^2 x^4 \left(a+b \tan ^{-1}(c x)\right)}{14 c^3}-\frac{b e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{7 c^5}-\frac{i e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{7 c^7}-\frac{2 b e^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{7 c^7}+\frac{1}{3} d^2 x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c}+\frac{2}{5} d e x^5 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b d e x^4 \left(a+b \tan ^{-1}(c x)\right)}{5 c}+\frac{1}{7} e^2 x^7 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e^2 x^6 \left(a+b \tan ^{-1}(c x)\right)}{21 c}+\frac{b^2 d^2 x}{3 c^2}-\frac{b^2 d^2 \tan ^{-1}(c x)}{3 c^3}+\frac{b^2 d e x^3}{15 c^2}-\frac{3 b^2 d e x}{5 c^4}+\frac{3 b^2 d e \tan ^{-1}(c x)}{5 c^5}+\frac{b^2 e^2 x^5}{105 c^2}-\frac{5 b^2 e^2 x^3}{126 c^4}+\frac{11 b^2 e^2 x}{42 c^6}-\frac{11 b^2 e^2 \tan ^{-1}(c x)}{42 c^7}",1,"(b^2*d^2*x)/(3*c^2) - (3*b^2*d*e*x)/(5*c^4) + (11*b^2*e^2*x)/(42*c^6) + (b^2*d*e*x^3)/(15*c^2) - (5*b^2*e^2*x^3)/(126*c^4) + (b^2*e^2*x^5)/(105*c^2) - (b^2*d^2*ArcTan[c*x])/(3*c^3) + (3*b^2*d*e*ArcTan[c*x])/(5*c^5) - (11*b^2*e^2*ArcTan[c*x])/(42*c^7) - (b*d^2*x^2*(a + b*ArcTan[c*x]))/(3*c) + (2*b*d*e*x^2*(a + b*ArcTan[c*x]))/(5*c^3) - (b*e^2*x^2*(a + b*ArcTan[c*x]))/(7*c^5) - (b*d*e*x^4*(a + b*ArcTan[c*x]))/(5*c) + (b*e^2*x^4*(a + b*ArcTan[c*x]))/(14*c^3) - (b*e^2*x^6*(a + b*ArcTan[c*x]))/(21*c) - ((I/3)*d^2*(a + b*ArcTan[c*x])^2)/c^3 + (((2*I)/5)*d*e*(a + b*ArcTan[c*x])^2)/c^5 - ((I/7)*e^2*(a + b*ArcTan[c*x])^2)/c^7 + (d^2*x^3*(a + b*ArcTan[c*x])^2)/3 + (2*d*e*x^5*(a + b*ArcTan[c*x])^2)/5 + (e^2*x^7*(a + b*ArcTan[c*x])^2)/7 - (2*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) + (4*b*d*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(5*c^5) - (2*b*e^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(7*c^7) - ((I/3)*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3 + (((2*I)/5)*b^2*d*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^5 - ((I/7)*b^2*e^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^7","A",44,10,23,0.4348,1,"{4980, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 302}"
1256,1,380,0,0.7536302,"\int x \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[x*(d + e*x^2)^2*(a + b*ArcTan[c*x])^2,x]","\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2}+\frac{a b d e x}{c^3}-\frac{d e \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^4}+\frac{b e^2 x^3 \left(a+b \tan ^{-1}(c x)\right)}{9 c^3}-\frac{a b e^2 x}{3 c^5}+\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{6 c^6}+\frac{1}{2} d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{a b d^2 x}{c}+\frac{1}{2} d e x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b d e x^3 \left(a+b \tan ^{-1}(c x)\right)}{3 c}+\frac{1}{6} e^2 x^6 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e^2 x^5 \left(a+b \tan ^{-1}(c x)\right)}{15 c}+\frac{b^2 d^2 \log \left(c^2 x^2+1\right)}{2 c^2}+\frac{b^2 d e x^2}{6 c^2}-\frac{2 b^2 d e \log \left(c^2 x^2+1\right)}{3 c^4}+\frac{b^2 d e x \tan ^{-1}(c x)}{c^3}+\frac{b^2 e^2 x^4}{60 c^2}-\frac{4 b^2 e^2 x^2}{45 c^4}+\frac{23 b^2 e^2 \log \left(c^2 x^2+1\right)}{90 c^6}-\frac{b^2 e^2 x \tan ^{-1}(c x)}{3 c^5}-\frac{b^2 d^2 x \tan ^{-1}(c x)}{c}","\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2}+\frac{a b d e x}{c^3}-\frac{d e \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^4}+\frac{b e^2 x^3 \left(a+b \tan ^{-1}(c x)\right)}{9 c^3}-\frac{a b e^2 x}{3 c^5}+\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{6 c^6}+\frac{1}{2} d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{a b d^2 x}{c}+\frac{1}{2} d e x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b d e x^3 \left(a+b \tan ^{-1}(c x)\right)}{3 c}+\frac{1}{6} e^2 x^6 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e^2 x^5 \left(a+b \tan ^{-1}(c x)\right)}{15 c}+\frac{b^2 d^2 \log \left(c^2 x^2+1\right)}{2 c^2}+\frac{b^2 d e x^2}{6 c^2}-\frac{2 b^2 d e \log \left(c^2 x^2+1\right)}{3 c^4}+\frac{b^2 d e x \tan ^{-1}(c x)}{c^3}+\frac{b^2 e^2 x^4}{60 c^2}-\frac{4 b^2 e^2 x^2}{45 c^4}+\frac{23 b^2 e^2 \log \left(c^2 x^2+1\right)}{90 c^6}-\frac{b^2 e^2 x \tan ^{-1}(c x)}{3 c^5}-\frac{b^2 d^2 x \tan ^{-1}(c x)}{c}",1,"-((a*b*d^2*x)/c) + (a*b*d*e*x)/c^3 - (a*b*e^2*x)/(3*c^5) + (b^2*d*e*x^2)/(6*c^2) - (4*b^2*e^2*x^2)/(45*c^4) + (b^2*e^2*x^4)/(60*c^2) - (b^2*d^2*x*ArcTan[c*x])/c + (b^2*d*e*x*ArcTan[c*x])/c^3 - (b^2*e^2*x*ArcTan[c*x])/(3*c^5) - (b*d*e*x^3*(a + b*ArcTan[c*x]))/(3*c) + (b*e^2*x^3*(a + b*ArcTan[c*x]))/(9*c^3) - (b*e^2*x^5*(a + b*ArcTan[c*x]))/(15*c) + (d^2*(a + b*ArcTan[c*x])^2)/(2*c^2) - (d*e*(a + b*ArcTan[c*x])^2)/(2*c^4) + (e^2*(a + b*ArcTan[c*x])^2)/(6*c^6) + (d^2*x^2*(a + b*ArcTan[c*x])^2)/2 + (d*e*x^4*(a + b*ArcTan[c*x])^2)/2 + (e^2*x^6*(a + b*ArcTan[c*x])^2)/6 + (b^2*d^2*Log[1 + c^2*x^2])/(2*c^2) - (2*b^2*d*e*Log[1 + c^2*x^2])/(3*c^4) + (23*b^2*e^2*Log[1 + c^2*x^2])/(90*c^6)","A",35,8,21,0.3810,1,"{4980, 4852, 4916, 4846, 260, 4884, 266, 43}"
1257,1,442,0,0.6894191,"\int \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Int[(d + e*x^2)^2*(a + b*ArcTan[c*x])^2,x]","-\frac{2 i b^2 d e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^3}+\frac{i b^2 e^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{5 c^5}+\frac{i b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c}-\frac{2 i d e \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^3}-\frac{4 b d e \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}+\frac{b e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{5 c^3}+\frac{i e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{5 c^5}+\frac{2 b e^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{5 c^5}+d^2 x \left(a+b \tan ^{-1}(c x)\right)^2+\frac{i d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{c}+\frac{2 b d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{2}{3} d e x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{2 b d e x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c}+\frac{1}{5} e^2 x^5 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e^2 x^4 \left(a+b \tan ^{-1}(c x)\right)}{10 c}+\frac{2 b^2 d e x}{3 c^2}-\frac{2 b^2 d e \tan ^{-1}(c x)}{3 c^3}+\frac{b^2 e^2 x^3}{30 c^2}-\frac{3 b^2 e^2 x}{10 c^4}+\frac{3 b^2 e^2 \tan ^{-1}(c x)}{10 c^5}","-\frac{2 i b^2 d e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^3}+\frac{i b^2 e^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{5 c^5}+\frac{i b^2 d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c}-\frac{2 i d e \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^3}-\frac{4 b d e \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}+\frac{b e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{5 c^3}+\frac{i e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{5 c^5}+\frac{2 b e^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{5 c^5}+d^2 x \left(a+b \tan ^{-1}(c x)\right)^2+\frac{i d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{c}+\frac{2 b d^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{2}{3} d e x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{2 b d e x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c}+\frac{1}{5} e^2 x^5 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e^2 x^4 \left(a+b \tan ^{-1}(c x)\right)}{10 c}+\frac{2 b^2 d e x}{3 c^2}-\frac{2 b^2 d e \tan ^{-1}(c x)}{3 c^3}+\frac{b^2 e^2 x^3}{30 c^2}-\frac{3 b^2 e^2 x}{10 c^4}+\frac{3 b^2 e^2 \tan ^{-1}(c x)}{10 c^5}",1,"(2*b^2*d*e*x)/(3*c^2) - (3*b^2*e^2*x)/(10*c^4) + (b^2*e^2*x^3)/(30*c^2) - (2*b^2*d*e*ArcTan[c*x])/(3*c^3) + (3*b^2*e^2*ArcTan[c*x])/(10*c^5) - (2*b*d*e*x^2*(a + b*ArcTan[c*x]))/(3*c) + (b*e^2*x^2*(a + b*ArcTan[c*x]))/(5*c^3) - (b*e^2*x^4*(a + b*ArcTan[c*x]))/(10*c) + (I*d^2*(a + b*ArcTan[c*x])^2)/c - (((2*I)/3)*d*e*(a + b*ArcTan[c*x])^2)/c^3 + ((I/5)*e^2*(a + b*ArcTan[c*x])^2)/c^5 + d^2*x*(a + b*ArcTan[c*x])^2 + (2*d*e*x^3*(a + b*ArcTan[c*x])^2)/3 + (e^2*x^5*(a + b*ArcTan[c*x])^2)/5 + (2*b*d^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c - (4*b*d*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) + (2*b*e^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(5*c^5) + (I*b^2*d^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c - (((2*I)/3)*b^2*d*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3 + ((I/5)*b^2*e^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^5","A",30,11,20,0.5500,1,"{4914, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 321, 203, 302}"
1258,1,355,0,0.6899846,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x} \, dx","Int[((d + e*x^2)^2*(a + b*ArcTan[c*x])^2)/x,x]","-i b d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{1}{2} b^2 d^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{1}{2} b^2 d^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)+\frac{d e \left(a+b \tan ^{-1}(c x)\right)^2}{c^2}+\frac{a b e^2 x}{2 c^3}-\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}+2 d^2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2+d e x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{2 a b d e x}{c}+\frac{1}{4} e^2 x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e^2 x^3 \left(a+b \tan ^{-1}(c x)\right)}{6 c}+\frac{b^2 d e \log \left(c^2 x^2+1\right)}{c^2}+\frac{b^2 e^2 x^2}{12 c^2}-\frac{b^2 e^2 \log \left(c^2 x^2+1\right)}{3 c^4}+\frac{b^2 e^2 x \tan ^{-1}(c x)}{2 c^3}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{c}","-i b d^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d^2 \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-\frac{1}{2} b^2 d^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+\frac{1}{2} b^2 d^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)+\frac{d e \left(a+b \tan ^{-1}(c x)\right)^2}{c^2}+\frac{a b e^2 x}{2 c^3}-\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}+2 d^2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2+d e x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{2 a b d e x}{c}+\frac{1}{4} e^2 x^4 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e^2 x^3 \left(a+b \tan ^{-1}(c x)\right)}{6 c}+\frac{b^2 d e \log \left(c^2 x^2+1\right)}{c^2}+\frac{b^2 e^2 x^2}{12 c^2}-\frac{b^2 e^2 \log \left(c^2 x^2+1\right)}{3 c^4}+\frac{b^2 e^2 x \tan ^{-1}(c x)}{2 c^3}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{c}",1,"(-2*a*b*d*e*x)/c + (a*b*e^2*x)/(2*c^3) + (b^2*e^2*x^2)/(12*c^2) - (2*b^2*d*e*x*ArcTan[c*x])/c + (b^2*e^2*x*ArcTan[c*x])/(2*c^3) - (b*e^2*x^3*(a + b*ArcTan[c*x]))/(6*c) + (d*e*(a + b*ArcTan[c*x])^2)/c^2 - (e^2*(a + b*ArcTan[c*x])^2)/(4*c^4) + d*e*x^2*(a + b*ArcTan[c*x])^2 + (e^2*x^4*(a + b*ArcTan[c*x])^2)/4 + 2*d^2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + (b^2*d*e*Log[1 + c^2*x^2])/c^2 - (b^2*e^2*Log[1 + c^2*x^2])/(3*c^4) - I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + I*b*d^2*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - (b^2*d^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/2 + (b^2*d^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/2","A",25,12,23,0.5217,1,"{4980, 4850, 4988, 4884, 4994, 6610, 4852, 4916, 4846, 260, 266, 43}"
1259,1,343,0,0.5757158,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x^2} \, dx","Int[((d + e*x^2)^2*(a + b*ArcTan[c*x])^2)/x^2,x]","-\frac{i b^2 e^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^3}-i b^2 c d^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)+\frac{2 i b^2 d e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c}-\frac{i e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^3}-\frac{2 b e^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}-i c d^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x}+2 b c d^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 d e x \left(a+b \tan ^{-1}(c x)\right)^2+\frac{2 i d e \left(a+b \tan ^{-1}(c x)\right)^2}{c}+\frac{4 b d e \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{1}{3} e^2 x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c}+\frac{b^2 e^2 x}{3 c^2}-\frac{b^2 e^2 \tan ^{-1}(c x)}{3 c^3}","-\frac{i b^2 e^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{3 c^3}-i b^2 c d^2 \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)+\frac{2 i b^2 d e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c}-\frac{i e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^3}-\frac{2 b e^2 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}-i c d^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x}+2 b c d^2 \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 d e x \left(a+b \tan ^{-1}(c x)\right)^2+\frac{2 i d e \left(a+b \tan ^{-1}(c x)\right)^2}{c}+\frac{4 b d e \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{1}{3} e^2 x^3 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{b e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c}+\frac{b^2 e^2 x}{3 c^2}-\frac{b^2 e^2 \tan ^{-1}(c x)}{3 c^3}",1,"(b^2*e^2*x)/(3*c^2) - (b^2*e^2*ArcTan[c*x])/(3*c^3) - (b*e^2*x^2*(a + b*ArcTan[c*x]))/(3*c) - I*c*d^2*(a + b*ArcTan[c*x])^2 + ((2*I)*d*e*(a + b*ArcTan[c*x])^2)/c - ((I/3)*e^2*(a + b*ArcTan[c*x])^2)/c^3 - (d^2*(a + b*ArcTan[c*x])^2)/x + 2*d*e*x*(a + b*ArcTan[c*x])^2 + (e^2*x^3*(a + b*ArcTan[c*x])^2)/3 + (4*b*d*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c - (2*b*e^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) + 2*b*c*d^2*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)] - I*b^2*c*d^2*PolyLog[2, -1 + 2/(1 - I*c*x)] + ((2*I)*b^2*d*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/c - ((I/3)*b^2*e^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3","A",20,13,23,0.5652,1,"{4980, 4846, 4920, 4854, 2402, 2315, 4852, 4924, 4868, 2447, 4916, 321, 203}"
1260,1,320,0,0.6086364,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x^3} \, dx","Int[((d + e*x^2)^2*(a + b*ArcTan[c*x])^2)/x^3,x]","-2 i b d e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 i b d e \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-b^2 d e \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+b^2 d e \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-\frac{1}{2} c^2 d^2 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2}-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{b c d^2 \left(a+b \tan ^{-1}(c x)\right)}{x}+4 d e \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{2} e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{a b e^2 x}{c}-\frac{1}{2} b^2 c^2 d^2 \log \left(c^2 x^2+1\right)+b^2 c^2 d^2 \log (x)+\frac{b^2 e^2 \log \left(c^2 x^2+1\right)}{2 c^2}-\frac{b^2 e^2 x \tan ^{-1}(c x)}{c}","-2 i b d e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 i b d e \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)-b^2 d e \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)+b^2 d e \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)-\frac{1}{2} c^2 d^2 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2}-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{b c d^2 \left(a+b \tan ^{-1}(c x)\right)}{x}+4 d e \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2+\frac{1}{2} e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)^2-\frac{a b e^2 x}{c}-\frac{1}{2} b^2 c^2 d^2 \log \left(c^2 x^2+1\right)+b^2 c^2 d^2 \log (x)+\frac{b^2 e^2 \log \left(c^2 x^2+1\right)}{2 c^2}-\frac{b^2 e^2 x \tan ^{-1}(c x)}{c}",1,"-((a*b*e^2*x)/c) - (b^2*e^2*x*ArcTan[c*x])/c - (b*c*d^2*(a + b*ArcTan[c*x]))/x - (c^2*d^2*(a + b*ArcTan[c*x])^2)/2 + (e^2*(a + b*ArcTan[c*x])^2)/(2*c^2) - (d^2*(a + b*ArcTan[c*x])^2)/(2*x^2) + (e^2*x^2*(a + b*ArcTan[c*x])^2)/2 + 4*d*e*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)] + b^2*c^2*d^2*Log[x] - (b^2*c^2*d^2*Log[1 + c^2*x^2])/2 + (b^2*e^2*Log[1 + c^2*x^2])/(2*c^2) - (2*I)*b*d*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)] + (2*I)*b*d*e*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)] - b^2*d*e*PolyLog[3, 1 - 2/(1 + I*c*x)] + b^2*d*e*PolyLog[3, -1 + 2/(1 + I*c*x)]","A",22,15,23,0.6522,1,"{4980, 4852, 4918, 266, 36, 29, 31, 4884, 4850, 4988, 4994, 6610, 4916, 4846, 260}"
1261,1,590,0,0.4988802,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{d+e x^2} \, dx","Int[(x^3*(a + b*ArcTan[c*x])^2)/(d + e*x^2),x]","-\frac{i b d \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^2}+\frac{i b d \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e^2}+\frac{i b d \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e^2}+\frac{b^2 d \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 e^2}-\frac{b^2 d \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 e^2}-\frac{b^2 d \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 e^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 e}+\frac{d \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{e^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e^2}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 e}-\frac{a b x}{c e}+\frac{b^2 \log \left(c^2 x^2+1\right)}{2 c^2 e}-\frac{b^2 x \tan ^{-1}(c x)}{c e}","-\frac{i b d \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e^2}+\frac{i b d \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e^2}+\frac{i b d \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e^2}+\frac{b^2 d \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 e^2}-\frac{b^2 d \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 e^2}-\frac{b^2 d \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 e^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 e}+\frac{d \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{e^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e^2}-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e^2}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 e}-\frac{a b x}{c e}+\frac{b^2 \log \left(c^2 x^2+1\right)}{2 c^2 e}-\frac{b^2 x \tan ^{-1}(c x)}{c e}",1,"-((a*b*x)/(c*e)) - (b^2*x*ArcTan[c*x])/(c*e) + (a + b*ArcTan[c*x])^2/(2*c^2*e) + (x^2*(a + b*ArcTan[c*x])^2)/(2*e) + (d*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e^2 - (d*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) - (d*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) + (b^2*Log[1 + c^2*x^2])/(2*c^2*e) - (I*b*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^2 + ((I/2)*b*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/e^2 + ((I/2)*b*d*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/e^2 + (b^2*d*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e^2) - (b^2*d*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*e^2) - (b^2*d*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*e^2)","A",11,7,23,0.3043,1,"{4916, 4852, 4846, 260, 4884, 4980, 4858}"
1262,1,554,0,0.4799092,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{d+e x^2} \, dx","Int[(x^2*(a + b*ArcTan[c*x])^2)/(d + e*x^2),x]","-\frac{i b \sqrt{-d} \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e^{3/2}}+\frac{i b \sqrt{-d} \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e^{3/2}}+\frac{b^2 \sqrt{-d} \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 e^{3/2}}-\frac{b^2 \sqrt{-d} \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 e^{3/2}}+\frac{i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c e}+\frac{\sqrt{-d} \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e^{3/2}}-\frac{\sqrt{-d} \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e^{3/2}}+\frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{e}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{c e}+\frac{2 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c e}","-\frac{i b \sqrt{-d} \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e^{3/2}}+\frac{i b \sqrt{-d} \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e^{3/2}}+\frac{b^2 \sqrt{-d} \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 e^{3/2}}-\frac{b^2 \sqrt{-d} \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 e^{3/2}}+\frac{i b^2 \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right)}{c e}+\frac{\sqrt{-d} \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e^{3/2}}-\frac{\sqrt{-d} \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e^{3/2}}+\frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{e}+\frac{i \left(a+b \tan ^{-1}(c x)\right)^2}{c e}+\frac{2 b \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c e}",1,"(I*(a + b*ArcTan[c*x])^2)/(c*e) + (x*(a + b*ArcTan[c*x])^2)/e + (2*b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(c*e) + (Sqrt[-d]*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^(3/2)) - (Sqrt[-d]*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^(3/2)) + (I*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/(c*e) - ((I/2)*b*Sqrt[-d]*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/e^(3/2) + ((I/2)*b*Sqrt[-d]*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/e^(3/2) + (b^2*Sqrt[-d]*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*e^(3/2)) - (b^2*Sqrt[-d]*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*e^(3/2))","A",10,8,23,0.3478,1,"{4916, 4846, 4920, 4854, 2402, 2315, 4914, 4858}"
1263,1,492,0,0.2495054,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{d+e x^2} \, dx","Int[(x*(a + b*ArcTan[c*x])^2)/(d + e*x^2),x]","-\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e}-\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 e}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 e}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 e}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{e}","-\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e}-\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e}+\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 e}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 e}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 e}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 e}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 e}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{e}",1,"-(((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e - ((I/2)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/e - ((I/2)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/e - (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*e) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*e)","A",4,2,21,0.09524,1,"{4980, 4858}"
1264,1,460,0,0.2465684,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d+e x^2} \, dx","Int[(a + b*ArcTan[c*x])^2/(d + e*x^2),x]","-\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 \sqrt{-d} \sqrt{e}}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 \sqrt{-d} \sqrt{e}}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 \sqrt{-d} \sqrt{e}}","-\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 \sqrt{-d} \sqrt{e}}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 \sqrt{-d} \sqrt{e}}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 \sqrt{-d} \sqrt{e}}",1,"((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*Sqrt[-d]*Sqrt[e]) - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*Sqrt[-d]*Sqrt[e]) - ((I/2)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(Sqrt[-d]*Sqrt[e]) + ((I/2)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(Sqrt[-d]*Sqrt[e]) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*Sqrt[e]) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*Sqrt[e])","A",4,2,20,0.1000,1,"{4914, 4858}"
1265,1,637,0,0.66702,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x \left(d+e x^2\right)} \, dx","Int[(a + b*ArcTan[c*x])^2/(x*(d + e*x^2)),x]","\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d}+\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{i b \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 d}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 d}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 d}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 d}+\frac{b^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d}+\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}","\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d}+\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{i b \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{i b \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 d}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 d}+\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 d}-\frac{b^2 \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 d}+\frac{b^2 \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d}+\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}+\frac{2 \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}",1,"(2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d + ((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d) - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d) - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d + ((I/2)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d + ((I/2)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d + (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*d) - (b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*d) + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*d) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*d)","A",12,7,23,0.3043,1,"{4980, 4850, 4988, 4884, 4994, 6610, 4858}"
1266,1,553,0,0.5160373,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^2 \left(d+e x^2\right)} \, dx","Int[(a + b*ArcTan[c*x])^2/(x^2*(d + e*x^2)),x]","-\frac{i b \sqrt{e} \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 (-d)^{3/2}}+\frac{i b \sqrt{e} \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 (-d)^{3/2}}+\frac{b^2 \sqrt{e} \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 (-d)^{3/2}}-\frac{b^2 \sqrt{e} \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 (-d)^{3/2}}-\frac{i b^2 c \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d}+\frac{\sqrt{e} \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 (-d)^{3/2}}-\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d x}+\frac{2 b c \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}","-\frac{i b \sqrt{e} \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 (-d)^{3/2}}+\frac{i b \sqrt{e} \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 (-d)^{3/2}}+\frac{b^2 \sqrt{e} \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 (-d)^{3/2}}-\frac{b^2 \sqrt{e} \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 (-d)^{3/2}}-\frac{i b^2 c \text{PolyLog}\left(2,-1+\frac{2}{1-i c x}\right)}{d}+\frac{\sqrt{e} \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 (-d)^{3/2}}-\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{d}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d x}+\frac{2 b c \log \left(2-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}",1,"((-I)*c*(a + b*ArcTan[c*x])^2)/d - (a + b*ArcTan[c*x])^2/(d*x) + (Sqrt[e]*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(3/2)) - (Sqrt[e]*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(3/2)) + (2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d - (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)])/d - ((I/2)*b*Sqrt[e]*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(-d)^(3/2) + ((I/2)*b*Sqrt[e]*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(-d)^(3/2) + (b^2*Sqrt[e]*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(3/2)) - (b^2*Sqrt[e]*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(3/2))","A",9,7,23,0.3043,1,"{4918, 4852, 4924, 4868, 2447, 4914, 4858}"
1267,1,745,0,0.9232723,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^3 \left(d+e x^2\right)} \, dx","Int[(a + b*ArcTan[c*x])^2/(x^3*(d + e*x^2)),x]","\frac{i b e \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{i b e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{i b e \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{i b e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d^2}-\frac{i b e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d^2}-\frac{b^2 e \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 d^2}+\frac{b^2 e \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 d^2}-\frac{b^2 e \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d^2}+\frac{b^2 e \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 d^2}+\frac{b^2 e \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 d^2}-\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}-\frac{e \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d^2}+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d^2}-\frac{2 e \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 d x^2}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{d x}-\frac{b^2 c^2 \log \left(c^2 x^2+1\right)}{2 d}+\frac{b^2 c^2 \log (x)}{d}","\frac{i b e \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{i b e \text{PolyLog}\left(2,1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{i b e \text{PolyLog}\left(2,-1+\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{i b e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d^2}-\frac{i b e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d^2}-\frac{b^2 e \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right)}{2 d^2}+\frac{b^2 e \text{PolyLog}\left(3,1-\frac{2}{1+i c x}\right)}{2 d^2}-\frac{b^2 e \text{PolyLog}\left(3,-1+\frac{2}{1+i c x}\right)}{2 d^2}+\frac{b^2 e \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 d^2}+\frac{b^2 e \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 d^2}-\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}-\frac{e \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 d^2}+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 d^2}-\frac{2 e \tanh ^{-1}\left(1-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 d x^2}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{d x}-\frac{b^2 c^2 \log \left(c^2 x^2+1\right)}{2 d}+\frac{b^2 c^2 \log (x)}{d}",1,"-((b*c*(a + b*ArcTan[c*x]))/(d*x)) - (c^2*(a + b*ArcTan[c*x])^2)/(2*d) - (a + b*ArcTan[c*x])^2/(2*d*x^2) - (2*e*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^2 + (b^2*c^2*Log[x])/d - (e*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d^2 + (e*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) + (e*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) - (b^2*c^2*Log[1 + c^2*x^2])/(2*d) + (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^2 + (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^2 - (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^2 - ((I/2)*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d^2 - ((I/2)*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d^2 - (b^2*e*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*d^2) + (b^2*e*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*d^2) - (b^2*e*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^2) + (b^2*e*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*d^2) + (b^2*e*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*d^2)","A",21,13,23,0.5652,1,"{4918, 4852, 266, 36, 29, 31, 4884, 4980, 4850, 4988, 4994, 6610, 4858}"
1268,1,943,0,1.7589625,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^2} \, dx","Int[(x^3*(a + b*ArcTan[c*x])^2)/(d + e*x^2)^2,x]","\frac{i c \sqrt{-d} \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 \left(c^2 d-e\right) e^{3/2}}-\frac{i c \sqrt{-d} \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 \left(c^2 d-e\right) e^{3/2}}-\frac{\text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right) b^2}{2 e^2}+\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 e^2}+\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 e^2}-\frac{c \sqrt{-d} \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 \left(c^2 d-e\right) e^{3/2}}+\frac{c \sqrt{-d} \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 \left(c^2 d-e\right) e^{3/2}}+\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) b}{e^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 e^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 e^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 e^2 \left(1-\frac{\sqrt{e} x}{\sqrt{-d}}\right)}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 e^2 \left(\frac{\sqrt{e} x}{\sqrt{-d}}+1\right)}-\frac{c^2 d \left(a+b \tan ^{-1}(c x)\right)^2}{2 \left(c^2 d-e\right) e^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2}{1-i c x}\right)}{e^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{2 e^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{2 e^2}","\frac{i c \sqrt{-d} \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 \left(c^2 d-e\right) e^{3/2}}-\frac{i c \sqrt{-d} \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 \left(c^2 d-e\right) e^{3/2}}-\frac{\text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right) b^2}{2 e^2}+\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 e^2}+\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 e^2}-\frac{c \sqrt{-d} \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 \left(c^2 d-e\right) e^{3/2}}+\frac{c \sqrt{-d} \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 \left(c^2 d-e\right) e^{3/2}}+\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) b}{e^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 e^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 e^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 e^2 \left(1-\frac{\sqrt{e} x}{\sqrt{-d}}\right)}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 e^2 \left(\frac{\sqrt{e} x}{\sqrt{-d}}+1\right)}-\frac{c^2 d \left(a+b \tan ^{-1}(c x)\right)^2}{2 \left(c^2 d-e\right) e^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2}{1-i c x}\right)}{e^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{2 e^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{2 e^2}",1,"-(c^2*d*(a + b*ArcTan[c*x])^2)/(2*(c^2*d - e)*e^2) + (a + b*ArcTan[c*x])^2/(4*e^2*(1 - (Sqrt[e]*x)/Sqrt[-d])) + (a + b*ArcTan[c*x])^2/(4*e^2*(1 + (Sqrt[e]*x)/Sqrt[-d])) - ((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e^2 - (b*c*Sqrt[-d]*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*(c^2*d - e)*e^(3/2)) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) + (b*c*Sqrt[-d]*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*(c^2*d - e)*e^(3/2)) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^2 + ((I/4)*b^2*c*Sqrt[-d]*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/((c^2*d - e)*e^(3/2)) - ((I/2)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/e^2 - ((I/4)*b^2*c*Sqrt[-d]*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/((c^2*d - e)*e^(3/2)) - ((I/2)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/e^2 - (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e^2) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*e^2) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*e^2)","A",33,12,23,0.5217,1,"{4980, 4978, 4864, 4856, 2402, 2315, 2447, 4984, 4884, 4920, 4854, 4858}"
1269,1,1033,0,1.9476996,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^2} \, dx","Int[(x^2*(a + b*ArcTan[c*x])^2)/(d + e*x^2)^2,x]","-\frac{i c \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) b^2}{2 \left(c^2 d-e\right) e}-\frac{i c \text{PolyLog}\left(2,1-\frac{2}{i c x+1}\right) b^2}{2 \left(c^2 d-e\right) e}+\frac{i c \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 \left(c^2 d-e\right) e}+\frac{i c \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 \left(c^2 d-e\right) e}+\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{8 \sqrt{-d} e^{3/2}}-\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{8 \sqrt{-d} e^{3/2}}+\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2}{1-i c x}\right) b}{\left(c^2 d-e\right) e}-\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2}{i c x+1}\right) b}{\left(c^2 d-e\right) e}-\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 \left(c^2 d-e\right) e}-\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 \left(c^2 d-e\right) e}-\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{4 \sqrt{-d} e^{3/2}}+\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{4 \sqrt{-d} e^{3/2}}-\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{2 \left(c^2 d-e\right) e}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 e^{3/2} \left(\sqrt{-d}-\sqrt{e} x\right)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 e^{3/2} \left(\sqrt{e} x+\sqrt{-d}\right)}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{4 \sqrt{-d} e^{3/2}}","-\frac{i c \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) b^2}{2 \left(c^2 d-e\right) e}-\frac{i c \text{PolyLog}\left(2,1-\frac{2}{i c x+1}\right) b^2}{2 \left(c^2 d-e\right) e}+\frac{i c \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 \left(c^2 d-e\right) e}+\frac{i c \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 \left(c^2 d-e\right) e}+\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{8 \sqrt{-d} e^{3/2}}-\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{8 \sqrt{-d} e^{3/2}}+\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2}{1-i c x}\right) b}{\left(c^2 d-e\right) e}-\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2}{i c x+1}\right) b}{\left(c^2 d-e\right) e}-\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 \left(c^2 d-e\right) e}-\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 \left(c^2 d-e\right) e}-\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{4 \sqrt{-d} e^{3/2}}+\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{4 \sqrt{-d} e^{3/2}}-\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{2 \left(c^2 d-e\right) e}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 e^{3/2} \left(\sqrt{-d}-\sqrt{e} x\right)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 e^{3/2} \left(\sqrt{e} x+\sqrt{-d}\right)}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{4 \sqrt{-d} e^{3/2}}",1,"((-I/2)*c*(a + b*ArcTan[c*x])^2)/((c^2*d - e)*e) + (a + b*ArcTan[c*x])^2/(4*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) - (a + b*ArcTan[c*x])^2/(4*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/((c^2*d - e)*e) - (b*c*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/((c^2*d - e)*e) - (b*c*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*(c^2*d - e)*e) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*e^(3/2)) - (b*c*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*(c^2*d - e)*e) - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*Sqrt[-d]*e^(3/2)) - ((I/2)*b^2*c*PolyLog[2, 1 - 2/(1 - I*c*x)])/((c^2*d - e)*e) - ((I/2)*b^2*c*PolyLog[2, 1 - 2/(1 + I*c*x)])/((c^2*d - e)*e) + ((I/4)*b^2*c*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/((c^2*d - e)*e) - ((I/4)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b^2*c*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/((c^2*d - e)*e) + ((I/4)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(Sqrt[-d]*e^(3/2)) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(8*Sqrt[-d]*e^(3/2)) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(8*Sqrt[-d]*e^(3/2))","A",38,12,23,0.5217,1,"{4980, 4914, 4864, 4856, 2402, 2315, 2447, 4984, 4884, 4920, 4854, 4858}"
1270,1,457,0,1.0876787,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^2} \, dx","Int[(x*(a + b*ArcTan[c*x])^2)/(d + e*x^2)^2,x]","\frac{i b^2 c \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 \sqrt{-d} \sqrt{e} \left(c^2 d-e\right)}-\frac{i b^2 c \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 \sqrt{-d} \sqrt{e} \left(c^2 d-e\right)}+\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 e \left(c^2 d-e\right)}-\frac{b c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 \sqrt{-d} \sqrt{e} \left(c^2 d-e\right)}+\frac{b c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 \sqrt{-d} \sqrt{e} \left(c^2 d-e\right)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 d e \left(1-\frac{\sqrt{e} x}{\sqrt{-d}}\right)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 d e \left(\frac{\sqrt{e} x}{\sqrt{-d}}+1\right)}","\frac{i b^2 c \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{4 \sqrt{-d} \sqrt{e} \left(c^2 d-e\right)}-\frac{i b^2 c \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{4 \sqrt{-d} \sqrt{e} \left(c^2 d-e\right)}+\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 e \left(c^2 d-e\right)}-\frac{b c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}-i \sqrt{e}\right)}\right)}{2 \sqrt{-d} \sqrt{e} \left(c^2 d-e\right)}+\frac{b c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}+\sqrt{e} x\right)}{(1-i c x) \left(c \sqrt{-d}+i \sqrt{e}\right)}\right)}{2 \sqrt{-d} \sqrt{e} \left(c^2 d-e\right)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 d e \left(1-\frac{\sqrt{e} x}{\sqrt{-d}}\right)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 d e \left(\frac{\sqrt{e} x}{\sqrt{-d}}+1\right)}",1,"(c^2*(a + b*ArcTan[c*x])^2)/(2*(c^2*d - e)*e) - (a + b*ArcTan[c*x])^2/(4*d*e*(1 - (Sqrt[e]*x)/Sqrt[-d])) - (a + b*ArcTan[c*x])^2/(4*d*e*(1 + (Sqrt[e]*x)/Sqrt[-d])) - (b*c*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*Sqrt[-d]*(c^2*d - e)*Sqrt[e]) + (b*c*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*Sqrt[-d]*(c^2*d - e)*Sqrt[e]) + ((I/4)*b^2*c*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(Sqrt[-d]*(c^2*d - e)*Sqrt[e]) - ((I/4)*b^2*c*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(Sqrt[-d]*(c^2*d - e)*Sqrt[e])","A",27,10,21,0.4762,1,"{4978, 4864, 4856, 2402, 2315, 2447, 4984, 4884, 4920, 4854}"
1271,1,1039,0,1.3288696,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^2} \, dx","Int[(a + b*ArcTan[c*x])^2/(d + e*x^2)^2,x]","\frac{i c \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) b^2}{2 d \left(c^2 d-e\right)}+\frac{i c \text{PolyLog}\left(2,1-\frac{2}{i c x+1}\right) b^2}{2 d \left(c^2 d-e\right)}-\frac{i c \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 d \left(c^2 d-e\right)}-\frac{i c \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 d \left(c^2 d-e\right)}-\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{8 (-d)^{3/2} \sqrt{e}}+\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{8 (-d)^{3/2} \sqrt{e}}-\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2}{1-i c x}\right) b}{d \left(c^2 d-e\right)}+\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2}{i c x+1}\right) b}{d \left(c^2 d-e\right)}+\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 d \left(c^2 d-e\right)}+\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 d \left(c^2 d-e\right)}+\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{4 (-d)^{3/2} \sqrt{e}}-\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{4 (-d)^{3/2} \sqrt{e}}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{2 d \left(c^2 d-e\right)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 d \sqrt{e} \left(\sqrt{-d}-\sqrt{e} x\right)}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 d \sqrt{e} \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{4 (-d)^{3/2} \sqrt{e}}","\frac{i c \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) b^2}{2 d \left(c^2 d-e\right)}+\frac{i c \text{PolyLog}\left(2,1-\frac{2}{i c x+1}\right) b^2}{2 d \left(c^2 d-e\right)}-\frac{i c \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 d \left(c^2 d-e\right)}-\frac{i c \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 d \left(c^2 d-e\right)}-\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{8 (-d)^{3/2} \sqrt{e}}+\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{8 (-d)^{3/2} \sqrt{e}}-\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2}{1-i c x}\right) b}{d \left(c^2 d-e\right)}+\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2}{i c x+1}\right) b}{d \left(c^2 d-e\right)}+\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 d \left(c^2 d-e\right)}+\frac{c \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 d \left(c^2 d-e\right)}+\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{4 (-d)^{3/2} \sqrt{e}}-\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{4 (-d)^{3/2} \sqrt{e}}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{2 d \left(c^2 d-e\right)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 d \sqrt{e} \left(\sqrt{-d}-\sqrt{e} x\right)}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 d \sqrt{e} \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{4 (-d)^{3/2} \sqrt{e}}",1,"((I/2)*c*(a + b*ArcTan[c*x])^2)/(d*(c^2*d - e)) - (a + b*ArcTan[c*x])^2/(4*d*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcTan[c*x])^2/(4*d*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) - (b*c*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/(d*(c^2*d - e)) + (b*c*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(d*(c^2*d - e)) + (b*c*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d*(c^2*d - e)) - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(3/2)*Sqrt[e]) + (b*c*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d*(c^2*d - e)) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(3/2)*Sqrt[e]) + ((I/2)*b^2*c*PolyLog[2, 1 - 2/(1 - I*c*x)])/(d*(c^2*d - e)) + ((I/2)*b^2*c*PolyLog[2, 1 - 2/(1 + I*c*x)])/(d*(c^2*d - e)) - ((I/4)*b^2*c*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(d*(c^2*d - e)) + ((I/4)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/((-d)^(3/2)*Sqrt[e]) - ((I/4)*b^2*c*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(d*(c^2*d - e)) - ((I/4)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/((-d)^(3/2)*Sqrt[e]) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(8*(-d)^(3/2)*Sqrt[e]) + (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(8*(-d)^(3/2)*Sqrt[e])","A",32,11,20,0.5500,1,"{4914, 4864, 4856, 2402, 2315, 2447, 4984, 4884, 4920, 4854, 4858}"
1272,1,1087,0,1.8609698,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x \left(d+e x^2\right)^2} \, dx","Int[(a + b*ArcTan[c*x])^2/(x*(d + e*x^2)^2),x]","\frac{i c \sqrt{e} \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 (-d)^{3/2} \left(c^2 d-e\right)}-\frac{i c \sqrt{e} \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 (-d)^{3/2} \left(c^2 d-e\right)}+\frac{\text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right) b^2}{2 d^2}-\frac{\text{PolyLog}\left(3,1-\frac{2}{i c x+1}\right) b^2}{2 d^2}+\frac{\text{PolyLog}\left(3,\frac{2}{i c x+1}-1\right) b^2}{2 d^2}-\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 d^2}-\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 d^2}-\frac{c \sqrt{e} \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 (-d)^{3/2} \left(c^2 d-e\right)}+\frac{c \sqrt{e} \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 (-d)^{3/2} \left(c^2 d-e\right)}-\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) b}{d^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2}{i c x+1}\right) b}{d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,\frac{2}{i c x+1}-1\right) b}{d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 d^2}-\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d \left(c^2 d-e\right)}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 d^2 \left(1-\frac{\sqrt{e} x}{\sqrt{-d}}\right)}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 d^2 \left(\frac{\sqrt{e} x}{\sqrt{-d}}+1\right)}+\frac{2 \left(a+b \tan ^{-1}(c x)\right)^2 \tanh ^{-1}\left(1-\frac{2}{i c x+1}\right)}{d^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2}{1-i c x}\right)}{d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{2 d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{2 d^2}","\frac{i c \sqrt{e} \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 (-d)^{3/2} \left(c^2 d-e\right)}-\frac{i c \sqrt{e} \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 (-d)^{3/2} \left(c^2 d-e\right)}+\frac{\text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right) b^2}{2 d^2}-\frac{\text{PolyLog}\left(3,1-\frac{2}{i c x+1}\right) b^2}{2 d^2}+\frac{\text{PolyLog}\left(3,\frac{2}{i c x+1}-1\right) b^2}{2 d^2}-\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 d^2}-\frac{\text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 d^2}-\frac{c \sqrt{e} \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 (-d)^{3/2} \left(c^2 d-e\right)}+\frac{c \sqrt{e} \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 (-d)^{3/2} \left(c^2 d-e\right)}-\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) b}{d^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2}{i c x+1}\right) b}{d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,\frac{2}{i c x+1}-1\right) b}{d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 d^2}-\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d \left(c^2 d-e\right)}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 d^2 \left(1-\frac{\sqrt{e} x}{\sqrt{-d}}\right)}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{4 d^2 \left(\frac{\sqrt{e} x}{\sqrt{-d}}+1\right)}+\frac{2 \left(a+b \tan ^{-1}(c x)\right)^2 \tanh ^{-1}\left(1-\frac{2}{i c x+1}\right)}{d^2}+\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2}{1-i c x}\right)}{d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{2 d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{2 d^2}",1,"-(c^2*(a + b*ArcTan[c*x])^2)/(2*d*(c^2*d - e)) + (a + b*ArcTan[c*x])^2/(4*d^2*(1 - (Sqrt[e]*x)/Sqrt[-d])) + (a + b*ArcTan[c*x])^2/(4*d^2*(1 + (Sqrt[e]*x)/Sqrt[-d])) + (2*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^2 + ((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d^2 - (b*c*Sqrt[e]*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(3/2)*(c^2*d - e)) - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) + (b*c*Sqrt[e]*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(3/2)*(c^2*d - e)) - ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^2) - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^2 - (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^2 + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^2 + ((I/4)*b^2*c*Sqrt[e]*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/((-d)^(3/2)*(c^2*d - e)) + ((I/2)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d^2 - ((I/4)*b^2*c*Sqrt[e]*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/((-d)^(3/2)*(c^2*d - e)) + ((I/2)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d^2 + (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*d^2) - (b^2*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*d^2) + (b^2*PolyLog[3, -1 + 2/(1 + I*c*x)])/(2*d^2) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*d^2) - (b^2*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*d^2)","A",39,16,23,0.6957,1,"{4980, 4850, 4988, 4884, 4994, 6610, 4978, 4864, 4856, 2402, 2315, 2447, 4984, 4920, 4854, 4858}"
1273,1,1141,0,2.0471042,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^2 \left(d+e x^2\right)^2} \, dx","Int[(a + b*ArcTan[c*x])^2/(x^2*(d + e*x^2)^2),x]","-\frac{i c e \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) b^2}{2 d^2 \left(c^2 d-e\right)}-\frac{i c \text{PolyLog}\left(2,\frac{2}{1-i c x}-1\right) b^2}{d^2}-\frac{i c e \text{PolyLog}\left(2,1-\frac{2}{i c x+1}\right) b^2}{2 d^2 \left(c^2 d-e\right)}+\frac{i c e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 d^2 \left(c^2 d-e\right)}+\frac{i c e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 d^2 \left(c^2 d-e\right)}-\frac{3 \sqrt{e} \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{8 (-d)^{5/2}}+\frac{3 \sqrt{e} \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{8 (-d)^{5/2}}+\frac{c e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2}{1-i c x}\right) b}{d^2 \left(c^2 d-e\right)}-\frac{c e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2}{i c x+1}\right) b}{d^2 \left(c^2 d-e\right)}-\frac{c e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 d^2 \left(c^2 d-e\right)}-\frac{c e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 d^2 \left(c^2 d-e\right)}+\frac{2 c \left(a+b \tan ^{-1}(c x)\right) \log \left(2-\frac{2}{1-i c x}\right) b}{d^2}+\frac{3 i \sqrt{e} \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{4 (-d)^{5/2}}-\frac{3 i \sqrt{e} \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{4 (-d)^{5/2}}-\frac{i c e \left(a+b \tan ^{-1}(c x)\right)^2}{2 d^2 \left(c^2 d-e\right)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d^2 x}+\frac{\sqrt{e} \left(a+b \tan ^{-1}(c x)\right)^2}{4 d^2 \left(\sqrt{-d}-\sqrt{e} x\right)}-\frac{\sqrt{e} \left(a+b \tan ^{-1}(c x)\right)^2}{4 d^2 \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{3 \sqrt{e} \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{4 (-d)^{5/2}}","-\frac{i c e \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) b^2}{2 d^2 \left(c^2 d-e\right)}-\frac{i c \text{PolyLog}\left(2,\frac{2}{1-i c x}-1\right) b^2}{d^2}-\frac{i c e \text{PolyLog}\left(2,1-\frac{2}{i c x+1}\right) b^2}{2 d^2 \left(c^2 d-e\right)}+\frac{i c e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 d^2 \left(c^2 d-e\right)}+\frac{i c e \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 d^2 \left(c^2 d-e\right)}-\frac{3 \sqrt{e} \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{8 (-d)^{5/2}}+\frac{3 \sqrt{e} \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{8 (-d)^{5/2}}+\frac{c e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2}{1-i c x}\right) b}{d^2 \left(c^2 d-e\right)}-\frac{c e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2}{i c x+1}\right) b}{d^2 \left(c^2 d-e\right)}-\frac{c e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 d^2 \left(c^2 d-e\right)}-\frac{c e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 d^2 \left(c^2 d-e\right)}+\frac{2 c \left(a+b \tan ^{-1}(c x)\right) \log \left(2-\frac{2}{1-i c x}\right) b}{d^2}+\frac{3 i \sqrt{e} \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{4 (-d)^{5/2}}-\frac{3 i \sqrt{e} \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{4 (-d)^{5/2}}-\frac{i c e \left(a+b \tan ^{-1}(c x)\right)^2}{2 d^2 \left(c^2 d-e\right)}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d^2 x}+\frac{\sqrt{e} \left(a+b \tan ^{-1}(c x)\right)^2}{4 d^2 \left(\sqrt{-d}-\sqrt{e} x\right)}-\frac{\sqrt{e} \left(a+b \tan ^{-1}(c x)\right)^2}{4 d^2 \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{d^2}-\frac{3 \sqrt{e} \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{4 (-d)^{5/2}}",1,"((-I)*c*(a + b*ArcTan[c*x])^2)/d^2 - ((I/2)*c*e*(a + b*ArcTan[c*x])^2)/(d^2*(c^2*d - e)) - (a + b*ArcTan[c*x])^2/(d^2*x) + (Sqrt[e]*(a + b*ArcTan[c*x])^2)/(4*d^2*(Sqrt[-d] - Sqrt[e]*x)) - (Sqrt[e]*(a + b*ArcTan[c*x])^2)/(4*d^2*(Sqrt[-d] + Sqrt[e]*x)) + (b*c*e*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/(d^2*(c^2*d - e)) - (b*c*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(d^2*(c^2*d - e)) - (b*c*e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^2*(c^2*d - e)) - (3*Sqrt[e]*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(5/2)) - (b*c*e*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^2*(c^2*d - e)) + (3*Sqrt[e]*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*(-d)^(5/2)) + (2*b*c*(a + b*ArcTan[c*x])*Log[2 - 2/(1 - I*c*x)])/d^2 - ((I/2)*b^2*c*e*PolyLog[2, 1 - 2/(1 - I*c*x)])/(d^2*(c^2*d - e)) - (I*b^2*c*PolyLog[2, -1 + 2/(1 - I*c*x)])/d^2 - ((I/2)*b^2*c*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/(d^2*(c^2*d - e)) + ((I/4)*b^2*c*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(d^2*(c^2*d - e)) + (((3*I)/4)*b*Sqrt[e]*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(-d)^(5/2) + ((I/4)*b^2*c*e*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(d^2*(c^2*d - e)) - (((3*I)/4)*b*Sqrt[e]*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(-d)^(5/2) - (3*b^2*Sqrt[e]*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(8*(-d)^(5/2)) + (3*b^2*Sqrt[e]*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(8*(-d)^(5/2))","A",42,15,23,0.6522,1,"{4980, 4852, 4924, 4868, 2447, 4914, 4864, 4856, 2402, 2315, 4984, 4884, 4920, 4854, 4858}"
1274,1,1181,0,2.0170013,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^3 \left(d+e x^2\right)^2} \, dx","Int[(a + b*ArcTan[c*x])^2/(x^3*(d + e*x^2)^2),x]","\frac{c^2 \log (x) b^2}{d^2}-\frac{c^2 \log \left(c^2 x^2+1\right) b^2}{2 d^2}+\frac{i c e^{3/2} \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 (-d)^{5/2} \left(c^2 d-e\right)}-\frac{i c e^{3/2} \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 (-d)^{5/2} \left(c^2 d-e\right)}-\frac{e \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right) b^2}{d^3}+\frac{e \text{PolyLog}\left(3,1-\frac{2}{i c x+1}\right) b^2}{d^3}-\frac{e \text{PolyLog}\left(3,\frac{2}{i c x+1}-1\right) b^2}{d^3}+\frac{e \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{2 d^3}+\frac{e \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{2 d^3}-\frac{c \left(a+b \tan ^{-1}(c x)\right) b}{d^2 x}-\frac{c e^{3/2} \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 (-d)^{5/2} \left(c^2 d-e\right)}+\frac{c e^{3/2} \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 (-d)^{5/2} \left(c^2 d-e\right)}+\frac{2 i e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) b}{d^3}+\frac{2 i e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2}{i c x+1}\right) b}{d^3}-\frac{2 i e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,\frac{2}{i c x+1}-1\right) b}{d^3}-\frac{i e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{d^3}-\frac{i e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{d^3}+\frac{c^2 e \left(a+b \tan ^{-1}(c x)\right)^2}{2 d^2 \left(c^2 d-e\right)}-\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{4 d^3 \left(1-\frac{\sqrt{e} x}{\sqrt{-d}}\right)}-\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{4 d^3 \left(\frac{\sqrt{e} x}{\sqrt{-d}}+1\right)}-\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 d^2 x^2}-\frac{4 e \left(a+b \tan ^{-1}(c x)\right)^2 \tanh ^{-1}\left(1-\frac{2}{i c x+1}\right)}{d^3}-\frac{2 e \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2}{1-i c x}\right)}{d^3}+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{d^3}+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{d^3}","\frac{c^2 \log (x) b^2}{d^2}-\frac{c^2 \log \left(c^2 x^2+1\right) b^2}{2 d^2}+\frac{i c e^{3/2} \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 (-d)^{5/2} \left(c^2 d-e\right)}-\frac{i c e^{3/2} \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{4 (-d)^{5/2} \left(c^2 d-e\right)}-\frac{e \text{PolyLog}\left(3,1-\frac{2}{1-i c x}\right) b^2}{d^3}+\frac{e \text{PolyLog}\left(3,1-\frac{2}{i c x+1}\right) b^2}{d^3}-\frac{e \text{PolyLog}\left(3,\frac{2}{i c x+1}-1\right) b^2}{d^3}+\frac{e \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b^2}{2 d^3}+\frac{e \text{PolyLog}\left(3,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b^2}{2 d^3}-\frac{c \left(a+b \tan ^{-1}(c x)\right) b}{d^2 x}-\frac{c e^{3/2} \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{2 (-d)^{5/2} \left(c^2 d-e\right)}+\frac{c e^{3/2} \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{2 (-d)^{5/2} \left(c^2 d-e\right)}+\frac{2 i e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right) b}{d^3}+\frac{2 i e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2}{i c x+1}\right) b}{d^3}-\frac{2 i e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,\frac{2}{i c x+1}-1\right) b}{d^3}-\frac{i e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right) b}{d^3}-\frac{i e \left(a+b \tan ^{-1}(c x)\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right) b}{d^3}+\frac{c^2 e \left(a+b \tan ^{-1}(c x)\right)^2}{2 d^2 \left(c^2 d-e\right)}-\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{4 d^3 \left(1-\frac{\sqrt{e} x}{\sqrt{-d}}\right)}-\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{4 d^3 \left(\frac{\sqrt{e} x}{\sqrt{-d}}+1\right)}-\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 d^2 x^2}-\frac{4 e \left(a+b \tan ^{-1}(c x)\right)^2 \tanh ^{-1}\left(1-\frac{2}{i c x+1}\right)}{d^3}-\frac{2 e \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2}{1-i c x}\right)}{d^3}+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{d^3}+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{d^3}",1,"-((b*c*(a + b*ArcTan[c*x]))/(d^2*x)) - (c^2*(a + b*ArcTan[c*x])^2)/(2*d^2) + (c^2*e*(a + b*ArcTan[c*x])^2)/(2*d^2*(c^2*d - e)) - (a + b*ArcTan[c*x])^2/(2*d^2*x^2) - (e*(a + b*ArcTan[c*x])^2)/(4*d^3*(1 - (Sqrt[e]*x)/Sqrt[-d])) - (e*(a + b*ArcTan[c*x])^2)/(4*d^3*(1 + (Sqrt[e]*x)/Sqrt[-d])) - (4*e*(a + b*ArcTan[c*x])^2*ArcTanh[1 - 2/(1 + I*c*x)])/d^3 + (b^2*c^2*Log[x])/d^2 - (2*e*(a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/d^3 - (b*c*e^(3/2)*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(5/2)*(c^2*d - e)) + (e*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d^3 + (b*c*e^(3/2)*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*(-d)^(5/2)*(c^2*d - e)) + (e*(a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d^3 - (b^2*c^2*Log[1 + c^2*x^2])/(2*d^2) + ((2*I)*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/d^3 + ((2*I)*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/d^3 - ((2*I)*b*e*(a + b*ArcTan[c*x])*PolyLog[2, -1 + 2/(1 + I*c*x)])/d^3 + ((I/4)*b^2*c*e^(3/2)*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/((-d)^(5/2)*(c^2*d - e)) - (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/d^3 - ((I/4)*b^2*c*e^(3/2)*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/((-d)^(5/2)*(c^2*d - e)) - (I*b*e*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/d^3 - (b^2*e*PolyLog[3, 1 - 2/(1 - I*c*x)])/d^3 + (b^2*e*PolyLog[3, 1 - 2/(1 + I*c*x)])/d^3 - (b^2*e*PolyLog[3, -1 + 2/(1 + I*c*x)])/d^3 + (b^2*e*PolyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(2*d^3) + (b^2*e*PolyLog[3, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*d^3)","A",47,22,23,0.9565,1,"{4980, 4852, 4918, 266, 36, 29, 31, 4884, 4850, 4988, 4994, 6610, 4978, 4864, 4856, 2402, 2315, 2447, 4984, 4920, 4854, 4858}"
1275,1,111,0,0.4440912,"\int x^4 \tan ^{-1}(x) \log \left(1+x^2\right) \, dx","Int[x^4*ArcTan[x]*Log[1 + x^2],x]","\frac{9 x^4}{200}-\frac{77 x^2}{300}-\frac{1}{20} \log ^2\left(x^2+1\right)-\frac{1}{20} x^4 \log \left(x^2+1\right)+\frac{1}{10} x^2 \log \left(x^2+1\right)+\frac{137}{300} \log \left(x^2+1\right)-\frac{2}{25} x^5 \tan ^{-1}(x)+\frac{2}{15} x^3 \tan ^{-1}(x)+\frac{1}{5} x^5 \log \left(x^2+1\right) \tan ^{-1}(x)-\frac{2}{5} x \tan ^{-1}(x)+\frac{1}{5} \tan ^{-1}(x)^2","\frac{9 x^4}{200}-\frac{77 x^2}{300}-\frac{1}{20} \log ^2\left(x^2+1\right)-\frac{1}{20} x^4 \log \left(x^2+1\right)+\frac{1}{10} x^2 \log \left(x^2+1\right)+\frac{137}{300} \log \left(x^2+1\right)-\frac{2}{25} x^5 \tan ^{-1}(x)+\frac{2}{15} x^3 \tan ^{-1}(x)+\frac{1}{5} x^5 \log \left(x^2+1\right) \tan ^{-1}(x)-\frac{2}{5} x \tan ^{-1}(x)+\frac{1}{5} \tan ^{-1}(x)^2",1,"(-77*x^2)/300 + (9*x^4)/200 - (2*x*ArcTan[x])/5 + (2*x^3*ArcTan[x])/15 - (2*x^5*ArcTan[x])/25 + ArcTan[x]^2/5 + (137*Log[1 + x^2])/300 + (x^2*Log[1 + x^2])/10 - (x^4*Log[1 + x^2])/20 + (x^5*ArcTan[x]*Log[1 + x^2])/5 - Log[1 + x^2]^2/20","A",24,14,12,1.167,1,"{4852, 266, 43, 5021, 6725, 446, 77, 4916, 4846, 260, 4884, 2475, 2390, 2301}"
1276,1,88,0,0.117512,"\int x^3 \tan ^{-1}(x) \log \left(1+x^2\right) \, dx","Int[x^3*ArcTan[x]*Log[1 + x^2],x]","\frac{7 x^3}{72}-\frac{1}{12} x^3 \log \left(x^2+1\right)+\frac{1}{4} x \log \left(x^2+1\right)-\frac{1}{8} x^4 \tan ^{-1}(x)+\frac{1}{4} x^2 \tan ^{-1}(x)+\frac{1}{4} x^4 \log \left(x^2+1\right) \tan ^{-1}(x)-\frac{1}{4} \log \left(x^2+1\right) \tan ^{-1}(x)-\frac{25 x}{24}+\frac{25}{24} \tan ^{-1}(x)","\frac{7 x^3}{72}-\frac{1}{12} x^3 \log \left(x^2+1\right)+\frac{1}{4} x \log \left(x^2+1\right)-\frac{1}{8} x^4 \tan ^{-1}(x)+\frac{1}{4} x^2 \tan ^{-1}(x)+\frac{1}{4} x^4 \log \left(x^2+1\right) \tan ^{-1}(x)-\frac{1}{4} \log \left(x^2+1\right) \tan ^{-1}(x)-\frac{25 x}{24}+\frac{25}{24} \tan ^{-1}(x)",1,"(-25*x)/24 + (7*x^3)/72 + (25*ArcTan[x])/24 + (x^2*ArcTan[x])/4 - (x^4*ArcTan[x])/8 + (x*Log[1 + x^2])/4 - (x^3*Log[1 + x^2])/12 - (ArcTan[x]*Log[1 + x^2])/4 + (x^4*ArcTan[x]*Log[1 + x^2])/4","A",14,12,12,1.000,1,"{4852, 302, 203, 2454, 2395, 43, 5019, 459, 321, 2471, 2448, 2455}"
1277,1,82,0,0.3251578,"\int x^2 \tan ^{-1}(x) \log \left(1+x^2\right) \, dx","Int[x^2*ArcTan[x]*Log[1 + x^2],x]","\frac{5 x^2}{18}+\frac{1}{12} \log ^2\left(x^2+1\right)-\frac{1}{6} x^2 \log \left(x^2+1\right)-\frac{11}{18} \log \left(x^2+1\right)-\frac{2}{9} x^3 \tan ^{-1}(x)+\frac{1}{3} x^3 \log \left(x^2+1\right) \tan ^{-1}(x)+\frac{2}{3} x \tan ^{-1}(x)-\frac{1}{3} \tan ^{-1}(x)^2","\frac{5 x^2}{18}+\frac{1}{12} \log ^2\left(x^2+1\right)-\frac{1}{6} x^2 \log \left(x^2+1\right)-\frac{11}{18} \log \left(x^2+1\right)-\frac{2}{9} x^3 \tan ^{-1}(x)+\frac{1}{3} x^3 \log \left(x^2+1\right) \tan ^{-1}(x)+\frac{2}{3} x \tan ^{-1}(x)-\frac{1}{3} \tan ^{-1}(x)^2",1,"(5*x^2)/18 + (2*x*ArcTan[x])/3 - (2*x^3*ArcTan[x])/9 - ArcTan[x]^2/3 - (11*Log[1 + x^2])/18 - (x^2*Log[1 + x^2])/6 + (x^3*ArcTan[x]*Log[1 + x^2])/3 + Log[1 + x^2]^2/12","A",19,12,12,1.000,1,"{4852, 266, 43, 5021, 6725, 4916, 4846, 260, 4884, 2475, 2390, 2301}"
1278,1,49,0,0.0499575,"\int x \tan ^{-1}(x) \log \left(1+x^2\right) \, dx","Int[x*ArcTan[x]*Log[1 + x^2],x]","-\frac{1}{2} x \log \left(x^2+1\right)-\frac{1}{2} x^2 \tan ^{-1}(x)+\frac{1}{2} \left(x^2+1\right) \log \left(x^2+1\right) \tan ^{-1}(x)+\frac{3 x}{2}-\frac{3}{2} \tan ^{-1}(x)","-\frac{1}{2} x \log \left(x^2+1\right)-\frac{1}{2} x^2 \tan ^{-1}(x)+\frac{1}{2} \left(x^2+1\right) \log \left(x^2+1\right) \tan ^{-1}(x)+\frac{3 x}{2}-\frac{3}{2} \tan ^{-1}(x)",1,"(3*x)/2 - (3*ArcTan[x])/2 - (x^2*ArcTan[x])/2 - (x*Log[1 + x^2])/2 + ((1 + x^2)*ArcTan[x]*Log[1 + x^2])/2","A",7,8,10,0.8000,1,"{4852, 321, 203, 2454, 2389, 2295, 5019, 2448}"
1279,1,38,0,0.1060844,"\int \tan ^{-1}(x) \log \left(1+x^2\right) \, dx","Int[ArcTan[x]*Log[1 + x^2],x]","-\frac{1}{4} \log ^2\left(x^2+1\right)+\log \left(x^2+1\right)+x \log \left(x^2+1\right) \tan ^{-1}(x)+\tan ^{-1}(x)^2-2 x \tan ^{-1}(x)","-\frac{1}{4} \log ^2\left(x^2+1\right)+\log \left(x^2+1\right)+x \log \left(x^2+1\right) \tan ^{-1}(x)+\tan ^{-1}(x)^2-2 x \tan ^{-1}(x)",1,"-2*x*ArcTan[x] + ArcTan[x]^2 + Log[1 + x^2] + x*ArcTan[x]*Log[1 + x^2] - Log[1 + x^2]^2/4","A",8,8,9,0.8889,1,"{4846, 260, 5009, 2475, 2390, 2301, 4916, 4884}"
1280,1,189,0,0.1810399,"\int \frac{\tan ^{-1}(x) \log \left(1+x^2\right)}{x} \, dx","Int[(ArcTan[x]*Log[1 + x^2])/x,x]","-\frac{1}{2} i \left(-\log \left(x^2+1\right)+\log (1-i x)+\log (1+i x)\right) \text{PolyLog}(2,-i x)+\frac{1}{2} i \left(-\log \left(x^2+1\right)+\log (1-i x)+\log (1+i x)\right) \text{PolyLog}(2,i x)-i \text{PolyLog}(3,1-i x)+i \text{PolyLog}(3,1+i x)+i \log (1-i x) \text{PolyLog}(2,1-i x)-i \log (1+i x) \text{PolyLog}(2,1+i x)+\frac{1}{2} i \log (i x) \log ^2(1-i x)-\frac{1}{2} i \log ^2(1+i x) \log (-i x)","-\frac{1}{2} i \left(-\log \left(x^2+1\right)+\log (1-i x)+\log (1+i x)\right) \text{PolyLog}(2,-i x)+\frac{1}{2} i \left(-\log \left(x^2+1\right)+\log (1-i x)+\log (1+i x)\right) \text{PolyLog}(2,i x)-i \text{PolyLog}(3,1-i x)+i \text{PolyLog}(3,1+i x)+i \log (1-i x) \text{PolyLog}(2,1-i x)-i \log (1+i x) \text{PolyLog}(2,1+i x)+\frac{1}{2} i \log (i x) \log ^2(1-i x)-\frac{1}{2} i \log ^2(1+i x) \log (-i x)",1,"(-I/2)*Log[1 + I*x]^2*Log[(-I)*x] + (I/2)*Log[1 - I*x]^2*Log[I*x] + I*Log[1 - I*x]*PolyLog[2, 1 - I*x] - I*Log[1 + I*x]*PolyLog[2, 1 + I*x] - (I/2)*(Log[1 - I*x] + Log[1 + I*x] - Log[1 + x^2])*PolyLog[2, (-I)*x] + (I/2)*(Log[1 - I*x] + Log[1 + I*x] - Log[1 + x^2])*PolyLog[2, I*x] - I*PolyLog[3, 1 - I*x] + I*PolyLog[3, 1 + I*x]","A",12,7,12,0.5833,1,"{4848, 2391, 5011, 2396, 2433, 2374, 6589}"
1281,1,41,0,0.1251738,"\int \frac{\tan ^{-1}(x) \log \left(1+x^2\right)}{x^2} \, dx","Int[(ArcTan[x]*Log[1 + x^2])/x^2,x]","-\frac{1}{2} \text{PolyLog}\left(2,-x^2\right)-\frac{1}{4} \log ^2\left(x^2+1\right)-\frac{\log \left(x^2+1\right) \tan ^{-1}(x)}{x}+\tan ^{-1}(x)^2","-\frac{1}{2} \text{PolyLog}\left(2,-x^2\right)-\frac{1}{4} \log ^2\left(x^2+1\right)-\frac{\log \left(x^2+1\right) \tan ^{-1}(x)}{x}+\tan ^{-1}(x)^2",1,"ArcTan[x]^2 - (ArcTan[x]*Log[1 + x^2])/x - Log[1 + x^2]^2/4 - PolyLog[2, -x^2]/2","A",8,12,12,1.000,1,"{4852, 266, 36, 29, 31, 5017, 2475, 2410, 2390, 2301, 2391, 4884}"
1282,1,69,0,0.0752657,"\int \frac{\tan ^{-1}(x) \log \left(1+x^2\right)}{x^3} \, dx","Int[(ArcTan[x]*Log[1 + x^2])/x^3,x]","\frac{1}{2} i \text{PolyLog}(2,-i x)-\frac{1}{2} i \text{PolyLog}(2,i x)-\frac{\log \left(x^2+1\right)}{2 x}-\frac{\log \left(x^2+1\right) \tan ^{-1}(x)}{2 x^2}-\frac{1}{2} \log \left(x^2+1\right) \tan ^{-1}(x)+\tan ^{-1}(x)","\frac{1}{2} i \text{PolyLog}(2,-i x)-\frac{1}{2} i \text{PolyLog}(2,i x)-\frac{\log \left(x^2+1\right)}{2 x}-\frac{\log \left(x^2+1\right) \tan ^{-1}(x)}{2 x^2}-\frac{1}{2} \log \left(x^2+1\right) \tan ^{-1}(x)+\tan ^{-1}(x)",1,"ArcTan[x] - Log[1 + x^2]/(2*x) - (ArcTan[x]*Log[1 + x^2])/2 - (ArcTan[x]*Log[1 + x^2])/(2*x^2) + (I/2)*PolyLog[2, (-I)*x] - (I/2)*PolyLog[2, I*x]","A",6,6,12,0.5000,1,"{4852, 325, 203, 5021, 4848, 2391}"
1283,1,81,0,0.2096357,"\int \frac{\tan ^{-1}(x) \log \left(1+x^2\right)}{x^4} \, dx","Int[(ArcTan[x]*Log[1 + x^2])/x^4,x]","\frac{1}{6} \text{PolyLog}\left(2,-x^2\right)+\frac{1}{12} \log ^2\left(x^2+1\right)-\frac{\log \left(x^2+1\right)}{6 x^2}-\frac{1}{2} \log \left(x^2+1\right)-\frac{\log \left(x^2+1\right) \tan ^{-1}(x)}{3 x^3}+\log (x)-\frac{1}{3} \tan ^{-1}(x)^2-\frac{2 \tan ^{-1}(x)}{3 x}","\frac{1}{6} \text{PolyLog}\left(2,-x^2\right)+\frac{1}{12} \log ^2\left(x^2+1\right)-\frac{\log \left(x^2+1\right)}{6 x^2}-\frac{1}{2} \log \left(x^2+1\right)-\frac{\log \left(x^2+1\right) \tan ^{-1}(x)}{3 x^3}+\log (x)-\frac{1}{3} \tan ^{-1}(x)^2-\frac{2 \tan ^{-1}(x)}{3 x}",1,"(-2*ArcTan[x])/(3*x) - ArcTan[x]^2/3 + Log[x] - Log[1 + x^2]/2 - Log[1 + x^2]/(6*x^2) - (ArcTan[x]*Log[1 + x^2])/(3*x^3) + Log[1 + x^2]^2/12 + PolyLog[2, -x^2]/6","A",18,15,12,1.250,1,"{4852, 266, 44, 5017, 2475, 2410, 2395, 36, 29, 31, 2391, 2390, 2301, 4918, 4884}"
1284,1,102,0,0.1292377,"\int \frac{\tan ^{-1}(x) \log \left(1+x^2\right)}{x^5} \, dx","Int[(ArcTan[x]*Log[1 + x^2])/x^5,x]","-\frac{1}{4} i \text{PolyLog}(2,-i x)+\frac{1}{4} i \text{PolyLog}(2,i x)+\frac{\log \left(x^2+1\right)}{4 x}-\frac{\log \left(x^2+1\right)}{12 x^3}-\frac{\tan ^{-1}(x)}{4 x^2}-\frac{\log \left(x^2+1\right) \tan ^{-1}(x)}{4 x^4}+\frac{1}{4} \log \left(x^2+1\right) \tan ^{-1}(x)-\frac{5}{12 x}-\frac{11}{12} \tan ^{-1}(x)","-\frac{1}{4} i \text{PolyLog}(2,-i x)+\frac{1}{4} i \text{PolyLog}(2,i x)+\frac{\log \left(x^2+1\right)}{4 x}-\frac{\log \left(x^2+1\right)}{12 x^3}-\frac{\tan ^{-1}(x)}{4 x^2}-\frac{\log \left(x^2+1\right) \tan ^{-1}(x)}{4 x^4}+\frac{1}{4} \log \left(x^2+1\right) \tan ^{-1}(x)-\frac{5}{12 x}-\frac{11}{12} \tan ^{-1}(x)",1,"-5/(12*x) - (11*ArcTan[x])/12 - ArcTan[x]/(4*x^2) - Log[1 + x^2]/(12*x^3) + Log[1 + x^2]/(4*x) + (ArcTan[x]*Log[1 + x^2])/4 - (ArcTan[x]*Log[1 + x^2])/(4*x^4) - (I/4)*PolyLog[2, (-I)*x] + (I/4)*PolyLog[2, I*x]","A",12,8,12,0.6667,1,"{4852, 325, 203, 5021, 453, 4980, 4848, 2391}"
1285,1,114,0,0.279356,"\int \frac{\tan ^{-1}(x) \log \left(1+x^2\right)}{x^6} \, dx","Int[(ArcTan[x]*Log[1 + x^2])/x^6,x]","-\frac{1}{10} \text{PolyLog}\left(2,-x^2\right)-\frac{7}{60 x^2}-\frac{1}{20} \log ^2\left(x^2+1\right)+\frac{\log \left(x^2+1\right)}{10 x^2}-\frac{\log \left(x^2+1\right)}{20 x^4}+\frac{5}{12} \log \left(x^2+1\right)-\frac{2 \tan ^{-1}(x)}{15 x^3}-\frac{\log \left(x^2+1\right) \tan ^{-1}(x)}{5 x^5}-\frac{5 \log (x)}{6}+\frac{1}{5} \tan ^{-1}(x)^2+\frac{2 \tan ^{-1}(x)}{5 x}","-\frac{1}{10} \text{PolyLog}\left(2,-x^2\right)-\frac{7}{60 x^2}-\frac{1}{20} \log ^2\left(x^2+1\right)+\frac{\log \left(x^2+1\right)}{10 x^2}-\frac{\log \left(x^2+1\right)}{20 x^4}+\frac{5}{12} \log \left(x^2+1\right)-\frac{2 \tan ^{-1}(x)}{15 x^3}-\frac{\log \left(x^2+1\right) \tan ^{-1}(x)}{5 x^5}-\frac{5 \log (x)}{6}+\frac{1}{5} \tan ^{-1}(x)^2+\frac{2 \tan ^{-1}(x)}{5 x}",1,"-7/(60*x^2) - (2*ArcTan[x])/(15*x^3) + (2*ArcTan[x])/(5*x) + ArcTan[x]^2/5 - (5*Log[x])/6 + (5*Log[1 + x^2])/12 - Log[1 + x^2]/(20*x^4) + Log[1 + x^2]/(10*x^2) - (ArcTan[x]*Log[1 + x^2])/(5*x^5) - Log[1 + x^2]^2/20 - PolyLog[2, -x^2]/10","A",26,15,12,1.250,1,"{4852, 266, 44, 5017, 2475, 2410, 2390, 2301, 2395, 36, 29, 31, 2391, 4918, 4884}"
1286,1,278,0,0.6928847,"\int x^4 \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right) \, dx","Int[x^4*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]),x]","\frac{1}{5} x^5 \left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)+\frac{2 a e x^3}{15 c^2}-\frac{2 a e x}{5 c^4}+\frac{2 a e \tan ^{-1}(c x)}{5 c^5}-\frac{2}{25} a e x^5-\frac{b x^4 \left(e \log \left(c^2 x^2+1\right)+d\right)}{20 c}+\frac{b x^2 \left(e \log \left(c^2 x^2+1\right)+d\right)}{10 c^3}-\frac{b \log \left(c^2 x^2+1\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{10 c^5}-\frac{77 b e x^2}{300 c^3}+\frac{b e \log ^2\left(c^2 x^2+1\right)}{20 c^5}+\frac{137 b e \log \left(c^2 x^2+1\right)}{300 c^5}+\frac{2 b e x^3 \tan ^{-1}(c x)}{15 c^2}-\frac{2 b e x \tan ^{-1}(c x)}{5 c^4}+\frac{b e \tan ^{-1}(c x)^2}{5 c^5}+\frac{9 b e x^4}{200 c}-\frac{2}{25} b e x^5 \tan ^{-1}(c x)","\frac{1}{5} x^5 \left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)+\frac{2 a e x^3}{15 c^2}-\frac{2 a e x}{5 c^4}+\frac{2 a e \tan ^{-1}(c x)}{5 c^5}-\frac{2}{25} a e x^5-\frac{b x^4 \left(e \log \left(c^2 x^2+1\right)+d\right)}{20 c}+\frac{b x^2 \left(e \log \left(c^2 x^2+1\right)+d\right)}{10 c^3}-\frac{b \log \left(c^2 x^2+1\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{10 c^5}-\frac{77 b e x^2}{300 c^3}+\frac{b e \log ^2\left(c^2 x^2+1\right)}{20 c^5}+\frac{137 b e \log \left(c^2 x^2+1\right)}{300 c^5}+\frac{2 b e x^3 \tan ^{-1}(c x)}{15 c^2}-\frac{2 b e x \tan ^{-1}(c x)}{5 c^4}+\frac{b e \tan ^{-1}(c x)^2}{5 c^5}+\frac{9 b e x^4}{200 c}-\frac{2}{25} b e x^5 \tan ^{-1}(c x)",1,"(-2*a*e*x)/(5*c^4) - (77*b*e*x^2)/(300*c^3) + (2*a*e*x^3)/(15*c^2) + (9*b*e*x^4)/(200*c) - (2*a*e*x^5)/25 + (2*a*e*ArcTan[c*x])/(5*c^5) - (2*b*e*x*ArcTan[c*x])/(5*c^4) + (2*b*e*x^3*ArcTan[c*x])/(15*c^2) - (2*b*e*x^5*ArcTan[c*x])/25 + (b*e*ArcTan[c*x]^2)/(5*c^5) + (137*b*e*Log[1 + c^2*x^2])/(300*c^5) + (b*e*Log[1 + c^2*x^2]^2)/(20*c^5) + (b*x^2*(d + e*Log[1 + c^2*x^2]))/(10*c^3) - (b*x^4*(d + e*Log[1 + c^2*x^2]))/(20*c) + (x^5*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/5 - (b*Log[1 + c^2*x^2]*(d + e*Log[1 + c^2*x^2]))/(10*c^5)","A",26,15,26,0.5769,1,"{4852, 266, 43, 5021, 6725, 1802, 635, 203, 260, 4916, 4846, 4884, 2475, 2390, 2301}"
1287,1,221,0,0.2439927,"\int x^3 \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right) \, dx","Int[x^3*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]),x]","\frac{1}{4} x^4 \left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)+\frac{e x^2 \left(a+b \tan ^{-1}(c x)\right)}{4 c^2}-\frac{e \log \left(c^2 x^2+1\right) \left(a+b \tan ^{-1}(c x)\right)}{4 c^4}-\frac{1}{8} e x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{b x (2 d-3 e)}{8 c^3}-\frac{b (2 d-3 e) \tan ^{-1}(c x)}{8 c^4}-\frac{b e x^3 \log \left(c^2 x^2+1\right)}{12 c}+\frac{b e x \log \left(c^2 x^2+1\right)}{4 c^3}-\frac{2 b e x}{3 c^3}+\frac{2 b e \tan ^{-1}(c x)}{3 c^4}-\frac{b x^3 (2 d-e)}{24 c}+\frac{b e x^3}{18 c}","\frac{1}{4} x^4 \left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)+\frac{e x^2 \left(a+b \tan ^{-1}(c x)\right)}{4 c^2}-\frac{e \log \left(c^2 x^2+1\right) \left(a+b \tan ^{-1}(c x)\right)}{4 c^4}-\frac{1}{8} e x^4 \left(a+b \tan ^{-1}(c x)\right)+\frac{b x (2 d-3 e)}{8 c^3}-\frac{b (2 d-3 e) \tan ^{-1}(c x)}{8 c^4}-\frac{b e x^3 \log \left(c^2 x^2+1\right)}{12 c}+\frac{b e x \log \left(c^2 x^2+1\right)}{4 c^3}-\frac{2 b e x}{3 c^3}+\frac{2 b e \tan ^{-1}(c x)}{3 c^4}-\frac{b x^3 (2 d-e)}{24 c}+\frac{b e x^3}{18 c}",1,"(b*(2*d - 3*e)*x)/(8*c^3) - (2*b*e*x)/(3*c^3) - (b*(2*d - e)*x^3)/(24*c) + (b*e*x^3)/(18*c) - (b*(2*d - 3*e)*ArcTan[c*x])/(8*c^4) + (2*b*e*ArcTan[c*x])/(3*c^4) + (e*x^2*(a + b*ArcTan[c*x]))/(4*c^2) - (e*x^4*(a + b*ArcTan[c*x]))/8 + (b*e*x*Log[1 + c^2*x^2])/(4*c^3) - (b*e*x^3*Log[1 + c^2*x^2])/(12*c) - (e*(a + b*ArcTan[c*x])*Log[1 + c^2*x^2])/(4*c^4) + (x^4*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/4","A",14,11,26,0.4231,1,"{2454, 2395, 43, 5019, 459, 321, 203, 2471, 2448, 2455, 302}"
1288,1,213,0,0.5692884,"\int x^2 \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right) \, dx","Int[x^2*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]),x]","\frac{1}{3} x^3 \left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)+\frac{2 a e x}{3 c^2}-\frac{2 a e \tan ^{-1}(c x)}{3 c^3}-\frac{2}{9} a e x^3-\frac{b x^2 \left(e \log \left(c^2 x^2+1\right)+d\right)}{6 c}+\frac{b \log \left(c^2 x^2+1\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{6 c^3}-\frac{b e \log ^2\left(c^2 x^2+1\right)}{12 c^3}-\frac{11 b e \log \left(c^2 x^2+1\right)}{18 c^3}+\frac{2 b e x \tan ^{-1}(c x)}{3 c^2}-\frac{b e \tan ^{-1}(c x)^2}{3 c^3}+\frac{5 b e x^2}{18 c}-\frac{2}{9} b e x^3 \tan ^{-1}(c x)","\frac{1}{3} x^3 \left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)+\frac{2 a e x}{3 c^2}-\frac{2 a e \tan ^{-1}(c x)}{3 c^3}-\frac{2}{9} a e x^3-\frac{b x^2 \left(e \log \left(c^2 x^2+1\right)+d\right)}{6 c}+\frac{b \log \left(c^2 x^2+1\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{6 c^3}-\frac{b e \log ^2\left(c^2 x^2+1\right)}{12 c^3}-\frac{11 b e \log \left(c^2 x^2+1\right)}{18 c^3}+\frac{2 b e x \tan ^{-1}(c x)}{3 c^2}-\frac{b e \tan ^{-1}(c x)^2}{3 c^3}+\frac{5 b e x^2}{18 c}-\frac{2}{9} b e x^3 \tan ^{-1}(c x)",1,"(2*a*e*x)/(3*c^2) + (5*b*e*x^2)/(18*c) - (2*a*e*x^3)/9 - (2*a*e*ArcTan[c*x])/(3*c^3) + (2*b*e*x*ArcTan[c*x])/(3*c^2) - (2*b*e*x^3*ArcTan[c*x])/9 - (b*e*ArcTan[c*x]^2)/(3*c^3) - (11*b*e*Log[1 + c^2*x^2])/(18*c^3) - (b*e*Log[1 + c^2*x^2]^2)/(12*c^3) - (b*x^2*(d + e*Log[1 + c^2*x^2]))/(6*c) + (x^3*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/3 + (b*Log[1 + c^2*x^2]*(d + e*Log[1 + c^2*x^2]))/(6*c^3)","A",21,15,26,0.5769,1,"{4852, 266, 43, 5021, 6725, 801, 635, 203, 260, 4916, 4846, 4884, 2475, 2390, 2301}"
1289,1,137,0,0.1107924,"\int x \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right) \, dx","Int[x*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]),x]","\frac{e \left(c^2 x^2+1\right) \log \left(c^2 x^2+1\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c^2}+\frac{1}{2} d x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{1}{2} e x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{b (d-e) \tan ^{-1}(c x)}{2 c^2}-\frac{b e x \log \left(c^2 x^2+1\right)}{2 c}-\frac{b e \tan ^{-1}(c x)}{c^2}-\frac{b x (d-e)}{2 c}+\frac{b e x}{c}","\frac{e \left(c^2 x^2+1\right) \log \left(c^2 x^2+1\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c^2}+\frac{1}{2} d x^2 \left(a+b \tan ^{-1}(c x)\right)-\frac{1}{2} e x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{b (d-e) \tan ^{-1}(c x)}{2 c^2}-\frac{b e x \log \left(c^2 x^2+1\right)}{2 c}-\frac{b e \tan ^{-1}(c x)}{c^2}-\frac{b x (d-e)}{2 c}+\frac{b e x}{c}",1,"-(b*(d - e)*x)/(2*c) + (b*e*x)/c + (b*(d - e)*ArcTan[c*x])/(2*c^2) - (b*e*ArcTan[c*x])/c^2 + (d*x^2*(a + b*ArcTan[c*x]))/2 - (e*x^2*(a + b*ArcTan[c*x]))/2 - (b*e*x*Log[1 + c^2*x^2])/(2*c) + (e*(1 + c^2*x^2)*(a + b*ArcTan[c*x])*Log[1 + c^2*x^2])/(2*c^2)","A",7,7,24,0.2917,1,"{2454, 2389, 2295, 5019, 321, 203, 2448}"
1290,1,100,0,0.1890075,"\int \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right) \, dx","Int[(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]),x]","x \left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{b c}-2 a e x-\frac{b \left(e \log \left(c^2 x^2+1\right)+d\right)^2}{4 c e}+\frac{b e \log \left(c^2 x^2+1\right)}{c}-2 b e x \tan ^{-1}(c x)","x \left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)+\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{b c}-2 a e x-\frac{b \left(e \log \left(c^2 x^2+1\right)+d\right)^2}{4 c e}+\frac{b e \log \left(c^2 x^2+1\right)}{c}-2 b e x \tan ^{-1}(c x)",1,"-2*a*e*x - 2*b*e*x*ArcTan[c*x] + (e*(a + b*ArcTan[c*x])^2)/(b*c) + (b*e*Log[1 + c^2*x^2])/c + x*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]) - (b*(d + e*Log[1 + c^2*x^2])^2)/(4*c*e)","A",9,8,23,0.3478,1,"{5009, 2475, 2390, 2301, 4916, 4846, 260, 4884}"
1291,1,282,0,0.3418669,"\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right)}{x} \, dx","Int[((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x,x]","-\frac{1}{2} a e \text{PolyLog}\left(2,-c^2 x^2\right)-\frac{1}{2} i b e \left(-\log \left(c^2 x^2+1\right)+\log (1-i c x)+\log (1+i c x)\right) \text{PolyLog}(2,-i c x)+\frac{1}{2} i b e \left(-\log \left(c^2 x^2+1\right)+\log (1-i c x)+\log (1+i c x)\right) \text{PolyLog}(2,i c x)+\frac{1}{2} i b d \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d \text{PolyLog}(2,i c x)-i b e \text{PolyLog}(3,1-i c x)+i b e \text{PolyLog}(3,1+i c x)+i b e \log (1-i c x) \text{PolyLog}(2,1-i c x)-i b e \log (1+i c x) \text{PolyLog}(2,1+i c x)+a d \log (x)+\frac{1}{2} i b e \log (i c x) \log ^2(1-i c x)-\frac{1}{2} i b e \log (-i c x) \log ^2(1+i c x)","-\frac{1}{2} a e \text{PolyLog}\left(2,-c^2 x^2\right)-\frac{1}{2} i b e \left(-\log \left(c^2 x^2+1\right)+\log (1-i c x)+\log (1+i c x)\right) \text{PolyLog}(2,-i c x)+\frac{1}{2} i b e \left(-\log \left(c^2 x^2+1\right)+\log (1-i c x)+\log (1+i c x)\right) \text{PolyLog}(2,i c x)+\frac{1}{2} i b d \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d \text{PolyLog}(2,i c x)-i b e \text{PolyLog}(3,1-i c x)+i b e \text{PolyLog}(3,1+i c x)+i b e \log (1-i c x) \text{PolyLog}(2,1-i c x)-i b e \log (1+i c x) \text{PolyLog}(2,1+i c x)+a d \log (x)+\frac{1}{2} i b e \log (i c x) \log ^2(1-i c x)-\frac{1}{2} i b e \log (-i c x) \log ^2(1+i c x)",1,"a*d*Log[x] + (I/2)*b*e*Log[I*c*x]*Log[1 - I*c*x]^2 - (I/2)*b*e*Log[(-I)*c*x]*Log[1 + I*c*x]^2 + (I/2)*b*d*PolyLog[2, (-I)*c*x] - (I/2)*b*e*(Log[1 - I*c*x] + Log[1 + I*c*x] - Log[1 + c^2*x^2])*PolyLog[2, (-I)*c*x] - (I/2)*b*d*PolyLog[2, I*c*x] + (I/2)*b*e*(Log[1 - I*c*x] + Log[1 + I*c*x] - Log[1 + c^2*x^2])*PolyLog[2, I*c*x] - (a*e*PolyLog[2, -(c^2*x^2)])/2 + I*b*e*Log[1 - I*c*x]*PolyLog[2, 1 - I*c*x] - I*b*e*Log[1 + I*c*x]*PolyLog[2, 1 + I*c*x] - I*b*e*PolyLog[3, 1 - I*c*x] + I*b*e*PolyLog[3, 1 + I*c*x]","A",18,9,26,0.3462,1,"{5015, 4848, 2391, 5013, 5011, 2396, 2433, 2374, 6589}"
1292,1,92,0,0.2499104,"\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right)}{x^2} \, dx","Int[((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x^2,x]","-\frac{1}{2} b c e \text{PolyLog}\left(2,-c^2 x^2\right)-\frac{\left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{x}+\frac{c e \left(a+b \tan ^{-1}(c x)\right)^2}{b}-\frac{b c \left(e \log \left(c^2 x^2+1\right)+d\right)^2}{4 e}+b c d \log (x)","-\frac{1}{2} b c e \text{PolyLog}\left(2,\frac{1}{c^2 x^2+1}\right)-\frac{\left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{x}+\frac{c e \left(a+b \tan ^{-1}(c x)\right)^2}{b}+\frac{1}{2} b c \log \left(1-\frac{1}{c^2 x^2+1}\right) \left(e \log \left(c^2 x^2+1\right)+d\right)",1,"(c*e*(a + b*ArcTan[c*x])^2)/b + b*c*d*Log[x] - ((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x - (b*c*(d + e*Log[1 + c^2*x^2])^2)/(4*e) - (b*c*e*PolyLog[2, -(c^2*x^2)])/2","A",8,8,26,0.3077,1,"{5017, 2475, 2411, 2344, 2301, 2316, 2315, 4884}"
1293,1,154,0,0.1401451,"\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right)}{x^3} \, dx","Int[((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x^3,x]","\frac{1}{2} i b c^2 e \text{PolyLog}(2,-i c x)-\frac{1}{2} i b c^2 e \text{PolyLog}(2,i c x)-\frac{\left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{2 x^2}-\frac{1}{2} a c^2 e \log \left(c^2 x^2+1\right)+a c^2 e \log (x)-\frac{b c \left(e \log \left(c^2 x^2+1\right)+d\right)}{2 x}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left(e \log \left(c^2 x^2+1\right)+d\right)+b c^2 e \tan ^{-1}(c x)","\frac{1}{2} i b c^2 e \text{PolyLog}(2,-i c x)-\frac{1}{2} i b c^2 e \text{PolyLog}(2,i c x)-\frac{\left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{2 x^2}-\frac{1}{2} a c^2 e \log \left(c^2 x^2+1\right)+a c^2 e \log (x)-\frac{b c \left(e \log \left(c^2 x^2+1\right)+d\right)}{2 x}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left(e \log \left(c^2 x^2+1\right)+d\right)+b c^2 e \tan ^{-1}(c x)",1,"b*c^2*e*ArcTan[c*x] + a*c^2*e*Log[x] - (a*c^2*e*Log[1 + c^2*x^2])/2 - (b*c*(d + e*Log[1 + c^2*x^2]))/(2*x) - (b*c^2*ArcTan[c*x]*(d + e*Log[1 + c^2*x^2]))/2 - ((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/(2*x^2) + (I/2)*b*c^2*e*PolyLog[2, (-I)*c*x] - (I/2)*b*c^2*e*PolyLog[2, I*c*x]","A",10,9,26,0.3462,1,"{4852, 325, 203, 5021, 801, 635, 260, 4848, 2391}"
1294,1,186,0,0.4284422,"\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right)}{x^4} \, dx","Int[((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x^4,x]","\frac{1}{6} b c^3 e \text{PolyLog}\left(2,-c^2 x^2\right)-\frac{\left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{3 x^3}-\frac{c^3 e \left(a+b \tan ^{-1}(c x)\right)^2}{3 b}-\frac{2 c^2 e \left(a+b \tan ^{-1}(c x)\right)}{3 x}+\frac{b c^3 \left(e \log \left(c^2 x^2+1\right)+d\right)^2}{12 e}-\frac{b c \left(c^2 x^2+1\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{6 x^2}-\frac{1}{3} b c^3 d \log (x)-\frac{1}{3} b c^3 e \log \left(c^2 x^2+1\right)+b c^3 e \log (x)","\frac{1}{6} b c^3 e \text{PolyLog}\left(2,\frac{1}{c^2 x^2+1}\right)-\frac{\left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{3 x^3}-\frac{c^3 e \left(a+b \tan ^{-1}(c x)\right)^2}{3 b}-\frac{2 c^2 e \left(a+b \tan ^{-1}(c x)\right)}{3 x}-\frac{1}{6} b c^3 \log \left(1-\frac{1}{c^2 x^2+1}\right) \left(e \log \left(c^2 x^2+1\right)+d\right)-\frac{b c \left(c^2 x^2+1\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{6 x^2}-\frac{1}{3} b c^3 e \log \left(c^2 x^2+1\right)+b c^3 e \log (x)",1,"(-2*c^2*e*(a + b*ArcTan[c*x]))/(3*x) - (c^3*e*(a + b*ArcTan[c*x])^2)/(3*b) - (b*c^3*d*Log[x])/3 + b*c^3*e*Log[x] - (b*c^3*e*Log[1 + c^2*x^2])/3 - (b*c*(1 + c^2*x^2)*(d + e*Log[1 + c^2*x^2]))/(6*x^2) - ((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/(3*x^3) + (b*c^3*(d + e*Log[1 + c^2*x^2])^2)/(12*e) + (b*c^3*e*PolyLog[2, -(c^2*x^2)])/6","A",17,16,26,0.6154,1,"{5017, 2475, 2411, 2347, 2344, 2301, 2316, 2315, 2314, 31, 4918, 4852, 266, 36, 29, 4884}"
1295,1,225,0,0.2557089,"\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right)}{x^5} \, dx","Int[((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x^5,x]","-\frac{1}{4} i b c^4 e \text{PolyLog}(2,-i c x)+\frac{1}{4} i b c^4 e \text{PolyLog}(2,i c x)-\frac{\left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{4 x^4}-\frac{a c^2 e}{4 x^2}+\frac{1}{4} a c^4 e \log \left(c^2 x^2+1\right)-\frac{1}{2} a c^4 e \log (x)+\frac{b c^3 \left(e \log \left(c^2 x^2+1\right)+d\right)}{4 x}-\frac{b c \left(e \log \left(c^2 x^2+1\right)+d\right)}{12 x^3}+\frac{1}{4} b c^4 \tan ^{-1}(c x) \left(e \log \left(c^2 x^2+1\right)+d\right)-\frac{b c^2 e \tan ^{-1}(c x)}{4 x^2}-\frac{5 b c^3 e}{12 x}-\frac{11}{12} b c^4 e \tan ^{-1}(c x)","-\frac{1}{4} i b c^4 e \text{PolyLog}(2,-i c x)+\frac{1}{4} i b c^4 e \text{PolyLog}(2,i c x)-\frac{\left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{4 x^4}-\frac{a c^2 e}{4 x^2}+\frac{1}{4} a c^4 e \log \left(c^2 x^2+1\right)-\frac{1}{2} a c^4 e \log (x)+\frac{b c^3 \left(e \log \left(c^2 x^2+1\right)+d\right)}{4 x}-\frac{b c \left(e \log \left(c^2 x^2+1\right)+d\right)}{12 x^3}+\frac{1}{4} b c^4 \tan ^{-1}(c x) \left(e \log \left(c^2 x^2+1\right)+d\right)-\frac{b c^2 e \tan ^{-1}(c x)}{4 x^2}-\frac{5 b c^3 e}{12 x}-\frac{11}{12} b c^4 e \tan ^{-1}(c x)",1,"-(a*c^2*e)/(4*x^2) - (5*b*c^3*e)/(12*x) - (11*b*c^4*e*ArcTan[c*x])/12 - (b*c^2*e*ArcTan[c*x])/(4*x^2) - (a*c^4*e*Log[x])/2 + (a*c^4*e*Log[1 + c^2*x^2])/4 - (b*c*(d + e*Log[1 + c^2*x^2]))/(12*x^3) + (b*c^3*(d + e*Log[1 + c^2*x^2]))/(4*x) + (b*c^4*ArcTan[c*x]*(d + e*Log[1 + c^2*x^2]))/4 - ((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/(4*x^4) - (I/4)*b*c^4*e*PolyLog[2, (-I)*c*x] + (I/4)*b*c^4*e*PolyLog[2, I*c*x]","A",15,10,26,0.3846,1,"{4852, 325, 203, 5021, 1802, 635, 260, 4980, 4848, 2391}"
1296,1,245,0,0.6251434,"\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right)}{x^6} \, dx","Int[((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x^6,x]","-\frac{1}{10} b c^5 e \text{PolyLog}\left(2,-c^2 x^2\right)-\frac{\left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{5 x^5}-\frac{2 c^2 e \left(a+b \tan ^{-1}(c x)\right)}{15 x^3}+\frac{c^5 e \left(a+b \tan ^{-1}(c x)\right)^2}{5 b}+\frac{2 c^4 e \left(a+b \tan ^{-1}(c x)\right)}{5 x}-\frac{b c^5 \left(e \log \left(c^2 x^2+1\right)+d\right)^2}{20 e}+\frac{b c^3 \left(c^2 x^2+1\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{10 x^2}-\frac{b c \left(e \log \left(c^2 x^2+1\right)+d\right)}{20 x^4}+\frac{1}{5} b c^5 d \log (x)-\frac{7 b c^3 e}{60 x^2}+\frac{19}{60} b c^5 e \log \left(c^2 x^2+1\right)-\frac{5}{6} b c^5 e \log (x)","-\frac{1}{10} b c^5 e \text{PolyLog}\left(2,\frac{1}{c^2 x^2+1}\right)-\frac{\left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{5 x^5}-\frac{2 c^2 e \left(a+b \tan ^{-1}(c x)\right)}{15 x^3}+\frac{c^5 e \left(a+b \tan ^{-1}(c x)\right)^2}{5 b}+\frac{2 c^4 e \left(a+b \tan ^{-1}(c x)\right)}{5 x}+\frac{1}{10} b c^5 \log \left(1-\frac{1}{c^2 x^2+1}\right) \left(e \log \left(c^2 x^2+1\right)+d\right)+\frac{b c^3 \left(c^2 x^2+1\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{10 x^2}-\frac{b c \left(e \log \left(c^2 x^2+1\right)+d\right)}{20 x^4}-\frac{7 b c^3 e}{60 x^2}+\frac{19}{60} b c^5 e \log \left(c^2 x^2+1\right)-\frac{5}{6} b c^5 e \log (x)",1,"(-7*b*c^3*e)/(60*x^2) - (2*c^2*e*(a + b*ArcTan[c*x]))/(15*x^3) + (2*c^4*e*(a + b*ArcTan[c*x]))/(5*x) + (c^5*e*(a + b*ArcTan[c*x])^2)/(5*b) + (b*c^5*d*Log[x])/5 - (5*b*c^5*e*Log[x])/6 + (19*b*c^5*e*Log[1 + c^2*x^2])/60 - (b*c*(d + e*Log[1 + c^2*x^2]))/(20*x^4) + (b*c^3*(1 + c^2*x^2)*(d + e*Log[1 + c^2*x^2]))/(10*x^2) - ((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/(5*x^5) - (b*c^5*(d + e*Log[1 + c^2*x^2])^2)/(20*e) - (b*c^5*e*PolyLog[2, -(c^2*x^2)])/10","A",26,18,26,0.6923,1,"{5017, 2475, 2411, 2347, 2344, 2301, 2316, 2315, 2314, 31, 2319, 44, 4918, 4852, 266, 36, 29, 4884}"
1297,1,562,0,0.7119487,"\int x \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right) \, dx","Int[x*(a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]),x]","\frac{i b e \left(c^2 f-g\right) \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 c^2 g}-\frac{i b e \left(c^2 f-g\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}-i \sqrt{g}\right)}\right)}{4 c^2 g}-\frac{i b e \left(c^2 f-g\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-f}+\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}+i \sqrt{g}\right)}\right)}{4 c^2 g}+\frac{1}{2} d x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{e \left(f+g x^2\right) \log \left(f+g x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{2 g}-\frac{1}{2} e x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{b (d-e) \tan ^{-1}(c x)}{2 c^2}-\frac{b e \left(c^2 f-g\right) \tan ^{-1}(c x) \log \left(f+g x^2\right)}{2 c^2 g}-\frac{b e \left(c^2 f-g\right) \log \left(\frac{2}{1-i c x}\right) \tan ^{-1}(c x)}{c^2 g}+\frac{b e \left(c^2 f-g\right) \tan ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}-i \sqrt{g}\right)}\right)}{2 c^2 g}+\frac{b e \left(c^2 f-g\right) \tan ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}+\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}+i \sqrt{g}\right)}\right)}{2 c^2 g}-\frac{b x (d-e)}{2 c}-\frac{b e x \log \left(f+g x^2\right)}{2 c}-\frac{b e \sqrt{f} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{c \sqrt{g}}+\frac{b e x}{c}","\frac{i b e \left(c^2 f-g\right) \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 c^2 g}-\frac{i b e \left(c^2 f-g\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}-i \sqrt{g}\right)}\right)}{4 c^2 g}-\frac{i b e \left(c^2 f-g\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-f}+\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}+i \sqrt{g}\right)}\right)}{4 c^2 g}+\frac{1}{2} d x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{e \left(f+g x^2\right) \log \left(f+g x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{2 g}-\frac{1}{2} e x^2 \left(a+b \tan ^{-1}(c x)\right)+\frac{b (d-e) \tan ^{-1}(c x)}{2 c^2}-\frac{b e \left(c^2 f-g\right) \tan ^{-1}(c x) \log \left(f+g x^2\right)}{2 c^2 g}-\frac{b e \left(c^2 f-g\right) \log \left(\frac{2}{1-i c x}\right) \tan ^{-1}(c x)}{c^2 g}+\frac{b e \left(c^2 f-g\right) \tan ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}-i \sqrt{g}\right)}\right)}{2 c^2 g}+\frac{b e \left(c^2 f-g\right) \tan ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}+\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}+i \sqrt{g}\right)}\right)}{2 c^2 g}-\frac{b x (d-e)}{2 c}-\frac{b e x \log \left(f+g x^2\right)}{2 c}-\frac{b e \sqrt{f} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{c \sqrt{g}}+\frac{b e x}{c}",1,"-(b*(d - e)*x)/(2*c) + (b*e*x)/c + (b*(d - e)*ArcTan[c*x])/(2*c^2) + (d*x^2*(a + b*ArcTan[c*x]))/2 - (e*x^2*(a + b*ArcTan[c*x]))/2 - (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/(c*Sqrt[g]) - (b*e*(c^2*f - g)*ArcTan[c*x]*Log[2/(1 - I*c*x)])/(c^2*g) + (b*e*(c^2*f - g)*ArcTan[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - I*Sqrt[g])*(1 - I*c*x))])/(2*c^2*g) + (b*e*(c^2*f - g)*ArcTan[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + I*Sqrt[g])*(1 - I*c*x))])/(2*c^2*g) - (b*e*x*Log[f + g*x^2])/(2*c) - (b*e*(c^2*f - g)*ArcTan[c*x]*Log[f + g*x^2])/(2*c^2*g) + (e*(f + g*x^2)*(a + b*ArcTan[c*x])*Log[f + g*x^2])/(2*g) + ((I/2)*b*e*(c^2*f - g)*PolyLog[2, 1 - 2/(1 - I*c*x)])/(c^2*g) - ((I/4)*b*e*(c^2*f - g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - I*Sqrt[g])*(1 - I*c*x))])/(c^2*g) - ((I/4)*b*e*(c^2*f - g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + I*Sqrt[g])*(1 - I*c*x))])/(c^2*g)","A",21,16,22,0.7273,1,"{2454, 2389, 2295, 5019, 321, 203, 2528, 2448, 205, 2470, 12, 4928, 4856, 2402, 2315, 2447}"
1298,1,656,0,0.830916,"\int \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right) \, dx","Int[(a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]),x]","-\frac{b e \text{PolyLog}\left(2,\frac{c^2 \left(f+g x^2\right)}{c^2 f-g}\right)}{2 c}-\frac{i b e \sqrt{-f} \text{PolyLog}\left(2,\frac{\sqrt{g} (-c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{g}}+\frac{i b e \sqrt{-f} \text{PolyLog}\left(2,\frac{\sqrt{g} (1-i c x)}{\sqrt{g}+i c \sqrt{-f}}\right)}{2 \sqrt{g}}+\frac{i b e \sqrt{-f} \text{PolyLog}\left(2,\frac{\sqrt{g} (1+i c x)}{\sqrt{g}+i c \sqrt{-f}}\right)}{2 \sqrt{g}}-\frac{i b e \sqrt{-f} \text{PolyLog}\left(2,\frac{\sqrt{g} (c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{g}}+x \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)+\frac{2 a e \sqrt{f} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{g}}-2 a e x-\frac{b \log \left(-\frac{g \left(c^2 x^2+1\right)}{c^2 f-g}\right) \left(d+e \log \left(f+g x^2\right)\right)}{2 c}+\frac{b e \log \left(c^2 x^2+1\right)}{c}+\frac{i b e \sqrt{-f} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-f}-\sqrt{g} x\right)}{c \sqrt{-f}-i \sqrt{g}}\right)}{2 \sqrt{g}}-\frac{i b e \sqrt{-f} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-f}-\sqrt{g} x\right)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{g}}+\frac{i b e \sqrt{-f} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-f}+\sqrt{g} x\right)}{c \sqrt{-f}-i \sqrt{g}}\right)}{2 \sqrt{g}}-\frac{i b e \sqrt{-f} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-f}+\sqrt{g} x\right)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{g}}-2 b e x \tan ^{-1}(c x)","-\frac{b e \text{PolyLog}\left(2,\frac{c^2 \left(f+g x^2\right)}{c^2 f-g}\right)}{2 c}-\frac{i b e \sqrt{-f} \text{PolyLog}\left(2,\frac{\sqrt{g} (-c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{g}}+\frac{i b e \sqrt{-f} \text{PolyLog}\left(2,\frac{\sqrt{g} (1-i c x)}{\sqrt{g}+i c \sqrt{-f}}\right)}{2 \sqrt{g}}+\frac{i b e \sqrt{-f} \text{PolyLog}\left(2,\frac{\sqrt{g} (1+i c x)}{\sqrt{g}+i c \sqrt{-f}}\right)}{2 \sqrt{g}}-\frac{i b e \sqrt{-f} \text{PolyLog}\left(2,\frac{\sqrt{g} (c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{g}}+x \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)+\frac{2 a e \sqrt{f} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{g}}-2 a e x-\frac{b \log \left(-\frac{g \left(c^2 x^2+1\right)}{c^2 f-g}\right) \left(d+e \log \left(f+g x^2\right)\right)}{2 c}+\frac{b e \log \left(c^2 x^2+1\right)}{c}+\frac{i b e \sqrt{-f} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-f}-\sqrt{g} x\right)}{c \sqrt{-f}-i \sqrt{g}}\right)}{2 \sqrt{g}}-\frac{i b e \sqrt{-f} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-f}-\sqrt{g} x\right)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{g}}+\frac{i b e \sqrt{-f} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-f}+\sqrt{g} x\right)}{c \sqrt{-f}-i \sqrt{g}}\right)}{2 \sqrt{g}}-\frac{i b e \sqrt{-f} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-f}+\sqrt{g} x\right)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{g}}-2 b e x \tan ^{-1}(c x)",1,"-2*a*e*x - 2*b*e*x*ArcTan[c*x] + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] + ((I/2)*b*e*Sqrt[-f]*Log[1 + I*c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])])/Sqrt[g] - ((I/2)*b*e*Sqrt[-f]*Log[1 - I*c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[g] + ((I/2)*b*e*Sqrt[-f]*Log[1 - I*c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])])/Sqrt[g] - ((I/2)*b*e*Sqrt[-f]*Log[1 + I*c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[g] + (b*e*Log[1 + c^2*x^2])/c + x*(a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]) - (b*Log[-((g*(1 + c^2*x^2))/(c^2*f - g))]*(d + e*Log[f + g*x^2]))/(2*c) - ((I/2)*b*e*Sqrt[-f]*PolyLog[2, (Sqrt[g]*(I - c*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[g] + ((I/2)*b*e*Sqrt[-f]*PolyLog[2, (Sqrt[g]*(1 - I*c*x))/(I*c*Sqrt[-f] + Sqrt[g])])/Sqrt[g] + ((I/2)*b*e*Sqrt[-f]*PolyLog[2, (Sqrt[g]*(1 + I*c*x))/(I*c*Sqrt[-f] + Sqrt[g])])/Sqrt[g] - ((I/2)*b*e*Sqrt[-f]*PolyLog[2, (Sqrt[g]*(I + c*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[g] - (b*e*PolyLog[2, (c^2*(f + g*x^2))/(c^2*f - g)])/(2*c)","A",28,12,21,0.5714,1,"{5009, 2475, 2394, 2393, 2391, 4916, 4846, 260, 4910, 205, 4908, 2409}"
1299,0,0,0,0.2807501,"\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x} \, dx","Int[((a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]))/x,x]","\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x} \, dx","\frac{1}{2} a e \text{PolyLog}\left(2,\frac{g x^2}{f}+1\right)+\frac{1}{2} i b d \text{PolyLog}(2,-i c x)-\frac{1}{2} i b d \text{PolyLog}(2,i c x)+b e \text{Int}\left(\frac{\tan ^{-1}(c x) \log \left(f+g x^2\right)}{x},x\right)+a d \log (x)+\frac{1}{2} a e \log \left(-\frac{g x^2}{f}\right) \log \left(f+g x^2\right)",0,"a*d*Log[x] + (a*e*Log[-((g*x^2)/f)]*Log[f + g*x^2])/2 + (I/2)*b*d*PolyLog[2, (-I)*c*x] - (I/2)*b*d*PolyLog[2, I*c*x] + (a*e*PolyLog[2, 1 + (g*x^2)/f])/2 + b*e*Defer[Int][(ArcTan[c*x]*Log[f + g*x^2])/x, x]","A",0,0,0,0,-1,"{}"
1300,1,672,0,0.7567694,"\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x^2} \, dx","Int[((a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]))/x^2,x]","-\frac{1}{2} b c e \text{PolyLog}\left(2,\frac{c^2 \left(f+g x^2\right)}{c^2 f-g}\right)+\frac{1}{2} b c e \text{PolyLog}\left(2,\frac{g x^2}{f}+1\right)+\frac{i b e \sqrt{g} \text{PolyLog}\left(2,\frac{\sqrt{g} (-c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \text{PolyLog}\left(2,\frac{\sqrt{g} (1-i c x)}{\sqrt{g}+i c \sqrt{-f}}\right)}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \text{PolyLog}\left(2,\frac{\sqrt{g} (1+i c x)}{\sqrt{g}+i c \sqrt{-f}}\right)}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \text{PolyLog}\left(2,\frac{\sqrt{g} (c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{-f}}-\frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x}+\frac{2 a e \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{f}}-\frac{1}{2} b c \log \left(-\frac{g \left(c^2 x^2+1\right)}{c^2 f-g}\right) \left(d+e \log \left(f+g x^2\right)\right)+\frac{1}{2} b c \log \left(-\frac{g x^2}{f}\right) \left(d+e \log \left(f+g x^2\right)\right)-\frac{i b e \sqrt{g} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-f}-\sqrt{g} x\right)}{c \sqrt{-f}-i \sqrt{g}}\right)}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-f}-\sqrt{g} x\right)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-f}+\sqrt{g} x\right)}{c \sqrt{-f}-i \sqrt{g}}\right)}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-f}+\sqrt{g} x\right)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{-f}}","-\frac{1}{2} b c e \text{PolyLog}\left(2,\frac{c^2 \left(f+g x^2\right)}{c^2 f-g}\right)+\frac{1}{2} b c e \text{PolyLog}\left(2,\frac{g x^2}{f}+1\right)+\frac{i b e \sqrt{g} \text{PolyLog}\left(2,\frac{\sqrt{g} (-c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \text{PolyLog}\left(2,\frac{\sqrt{g} (1-i c x)}{\sqrt{g}+i c \sqrt{-f}}\right)}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \text{PolyLog}\left(2,\frac{\sqrt{g} (1+i c x)}{\sqrt{g}+i c \sqrt{-f}}\right)}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \text{PolyLog}\left(2,\frac{\sqrt{g} (c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{-f}}-\frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x}+\frac{2 a e \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{f}}-\frac{1}{2} b c \log \left(-\frac{g \left(c^2 x^2+1\right)}{c^2 f-g}\right) \left(d+e \log \left(f+g x^2\right)\right)+\frac{1}{2} b c \log \left(-\frac{g x^2}{f}\right) \left(d+e \log \left(f+g x^2\right)\right)-\frac{i b e \sqrt{g} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-f}-\sqrt{g} x\right)}{c \sqrt{-f}-i \sqrt{g}}\right)}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-f}-\sqrt{g} x\right)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \log (1-i c x) \log \left(\frac{c \left(\sqrt{-f}+\sqrt{g} x\right)}{c \sqrt{-f}-i \sqrt{g}}\right)}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \log (1+i c x) \log \left(\frac{c \left(\sqrt{-f}+\sqrt{g} x\right)}{c \sqrt{-f}+i \sqrt{g}}\right)}{2 \sqrt{-f}}",1,"(2*a*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] - ((I/2)*b*e*Sqrt[g]*Log[1 + I*c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])])/Sqrt[-f] + ((I/2)*b*e*Sqrt[g]*Log[1 - I*c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[-f] - ((I/2)*b*e*Sqrt[g]*Log[1 - I*c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])])/Sqrt[-f] + ((I/2)*b*e*Sqrt[g]*Log[1 + I*c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[-f] - ((a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]))/x + (b*c*Log[-((g*x^2)/f)]*(d + e*Log[f + g*x^2]))/2 - (b*c*Log[-((g*(1 + c^2*x^2))/(c^2*f - g))]*(d + e*Log[f + g*x^2]))/2 + ((I/2)*b*e*Sqrt[g]*PolyLog[2, (Sqrt[g]*(I - c*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[-f] - ((I/2)*b*e*Sqrt[g]*PolyLog[2, (Sqrt[g]*(1 - I*c*x))/(I*c*Sqrt[-f] + Sqrt[g])])/Sqrt[-f] - ((I/2)*b*e*Sqrt[g]*PolyLog[2, (Sqrt[g]*(1 + I*c*x))/(I*c*Sqrt[-f] + Sqrt[g])])/Sqrt[-f] + ((I/2)*b*e*Sqrt[g]*PolyLog[2, (Sqrt[g]*(I + c*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[-f] - (b*c*e*PolyLog[2, (c^2*(f + g*x^2))/(c^2*f - g)])/2 + (b*c*e*PolyLog[2, 1 + (g*x^2)/f])/2","A",28,14,24,0.5833,1,"{5017, 2475, 36, 29, 31, 2416, 2394, 2315, 2393, 2391, 4910, 205, 4908, 2409}"
1301,1,528,0,0.769275,"\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x^3} \, dx","Int[((a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]))/x^3,x]","\frac{i b e \left(c^2 f-g\right) \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 f}-\frac{i b e \left(c^2 f-g\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}-i \sqrt{g}\right)}\right)}{4 f}-\frac{i b e \left(c^2 f-g\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-f}+\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}+i \sqrt{g}\right)}\right)}{4 f}+\frac{i b e g \text{PolyLog}(2,-i c x)}{2 f}-\frac{i b e g \text{PolyLog}(2,i c x)}{2 f}-\frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{2 x^2}-\frac{a e g \log \left(f+g x^2\right)}{2 f}+\frac{a e g \log (x)}{f}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left(d+e \log \left(f+g x^2\right)\right)-\frac{b e \left(c^2 f-g\right) \log \left(\frac{2}{1-i c x}\right) \tan ^{-1}(c x)}{f}+\frac{b e \left(c^2 f-g\right) \tan ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}-i \sqrt{g}\right)}\right)}{2 f}+\frac{b e \left(c^2 f-g\right) \tan ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}+\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}+i \sqrt{g}\right)}\right)}{2 f}-\frac{b c \left(d+e \log \left(f+g x^2\right)\right)}{2 x}+\frac{b c e \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{f}}","\frac{i b e \left(c^2 f-g\right) \text{PolyLog}\left(2,1-\frac{2}{1-i c x}\right)}{2 f}-\frac{i b e \left(c^2 f-g\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}-i \sqrt{g}\right)}\right)}{4 f}-\frac{i b e \left(c^2 f-g\right) \text{PolyLog}\left(2,1-\frac{2 c \left(\sqrt{-f}+\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}+i \sqrt{g}\right)}\right)}{4 f}+\frac{i b e g \text{PolyLog}(2,-i c x)}{2 f}-\frac{i b e g \text{PolyLog}(2,i c x)}{2 f}-\frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{2 x^2}-\frac{a e g \log \left(f+g x^2\right)}{2 f}+\frac{a e g \log (x)}{f}-\frac{1}{2} b c^2 \tan ^{-1}(c x) \left(d+e \log \left(f+g x^2\right)\right)-\frac{b e \left(c^2 f-g\right) \log \left(\frac{2}{1-i c x}\right) \tan ^{-1}(c x)}{f}+\frac{b e \left(c^2 f-g\right) \tan ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}-i \sqrt{g}\right)}\right)}{2 f}+\frac{b e \left(c^2 f-g\right) \tan ^{-1}(c x) \log \left(\frac{2 c \left(\sqrt{-f}+\sqrt{g} x\right)}{(1-i c x) \left(c \sqrt{-f}+i \sqrt{g}\right)}\right)}{2 f}-\frac{b c \left(d+e \log \left(f+g x^2\right)\right)}{2 x}+\frac{b c e \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{f}}",1,"(b*c*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] + (a*e*g*Log[x])/f - (b*e*(c^2*f - g)*ArcTan[c*x]*Log[2/(1 - I*c*x)])/f + (b*e*(c^2*f - g)*ArcTan[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - I*Sqrt[g])*(1 - I*c*x))])/(2*f) + (b*e*(c^2*f - g)*ArcTan[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + I*Sqrt[g])*(1 - I*c*x))])/(2*f) - (a*e*g*Log[f + g*x^2])/(2*f) - (b*c*(d + e*Log[f + g*x^2]))/(2*x) - (b*c^2*ArcTan[c*x]*(d + e*Log[f + g*x^2]))/2 - ((a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]))/(2*x^2) + ((I/2)*b*e*g*PolyLog[2, (-I)*c*x])/f - ((I/2)*b*e*g*PolyLog[2, I*c*x])/f + ((I/2)*b*e*(c^2*f - g)*PolyLog[2, 1 - 2/(1 - I*c*x)])/f - ((I/4)*b*e*(c^2*f - g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - I*Sqrt[g])*(1 - I*c*x))])/f - ((I/4)*b*e*(c^2*f - g)*PolyLog[2, 1 - (2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + I*Sqrt[g])*(1 - I*c*x))])/f","A",22,18,24,0.7500,1,"{4852, 325, 203, 5021, 801, 635, 205, 260, 446, 72, 6725, 4848, 2391, 4928, 4856, 2402, 2315, 2447}"