1,1,98,117,0.0818445,"\int x^3 (d+i c d x) \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + I*c*d*x)*(a + b*ArcTan[c*x]),x]","\frac{d \left(3 a c^4 x^4 (5+4 i c x)-6 i b \log \left(c^2 x^2+1\right)+3 b \left(4 i c^5 x^5+5 c^4 x^4-5\right) \tan ^{-1}(c x)+b c x \left(-3 i c^3 x^3-5 c^2 x^2+6 i c x+15\right)\right)}{60 c^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{b d \tan ^{-1}(c x)}{4 c^4}+\frac{b d x}{4 c^3}+\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^3}{12 c}-\frac{1}{20} i b d x^4",1,"(d*(3*a*c^4*x^4*(5 + (4*I)*c*x) + b*c*x*(15 + (6*I)*c*x - 5*c^2*x^2 - (3*I)*c^3*x^3) + 3*b*(-5 + 5*c^4*x^4 + (4*I)*c^5*x^5)*ArcTan[c*x] - (6*I)*b*Log[1 + c^2*x^2]))/(60*c^4)","A",1
2,1,88,105,0.0647591,"\int x^2 (d+i c d x) \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + I*c*d*x)*(a + b*ArcTan[c*x]),x]","\frac{d \left(a c^3 x^3 (4+3 i c x)+b c x \left(-i c^2 x^2-2 c x+3 i\right)+2 b \log \left(c^2 x^2+1\right)+b \left(3 i c^4 x^4+4 c^3 x^3-3 i\right) \tan ^{-1}(c x)\right)}{12 c^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{i b d \tan ^{-1}(c x)}{4 c^3}+\frac{i b d x}{4 c^2}+\frac{b d \log \left(c^2 x^2+1\right)}{6 c^3}-\frac{b d x^2}{6 c}-\frac{1}{12} i b d x^3",1,"(d*(a*c^3*x^3*(4 + (3*I)*c*x) + b*c*x*(3*I - 2*c*x - I*c^2*x^2) + b*(-3*I + 4*c^3*x^3 + (3*I)*c^4*x^4)*ArcTan[c*x] + 2*b*Log[1 + c^2*x^2]))/(12*c^3)","A",1
3,1,76,91,0.04944,"\int x (d+i c d x) \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x*(d + I*c*d*x)*(a + b*ArcTan[c*x]),x]","\frac{d \left(c x (a c x (3+2 i c x)+b (-3-i c x))+i b \log \left(c^2 x^2+1\right)+b \left(2 i c^3 x^3+3 c^2 x^2+3\right) \tan ^{-1}(c x)\right)}{6 c^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,"(d*(c*x*(b*(-3 - I*c*x) + a*c*x*(3 + (2*I)*c*x)) + b*(3 + 3*c^2*x^2 + (2*I)*c^3*x^3)*ArcTan[c*x] + I*b*Log[1 + c^2*x^2]))/(6*c^2)","A",1
4,1,84,53,0.0054836,"\int (d+i c d x) \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[(d + I*c*d*x)*(a + b*ArcTan[c*x]),x]","\frac{1}{2} i a c d x^2+a d x-\frac{b d \log \left(c^2 x^2+1\right)}{2 c}+\frac{1}{2} i b c d x^2 \tan ^{-1}(c x)+b d x \tan ^{-1}(c x)+\frac{i b d \tan ^{-1}(c x)}{2 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,"a*d*x - (I/2)*b*d*x + (I/2)*a*c*d*x^2 + ((I/2)*b*d*ArcTan[c*x])/c + b*d*x*ArcTan[c*x] + (I/2)*b*c*d*x^2*ArcTan[c*x] - (b*d*Log[1 + c^2*x^2])/(2*c)","A",1
5,1,76,76,0.0045904,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x,x]","i a c d x+a d \log (x)-\frac{1}{2} i b d \log \left(c^2 x^2+1\right)+\frac{1}{2} i b d \text{Li}_2(-i c x)-\frac{1}{2} i b d \text{Li}_2(i c x)+i b c d x \tan ^{-1}(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)+\frac{1}{2} i b d \text{Li}_2(-i c x)-\frac{1}{2} i b d \text{Li}_2(i c x)+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",1
6,1,75,77,0.0508483,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x^2,x]","\frac{d \left(2 i a c x \log (x)-2 a-b c x \log \left(c^2 x^2+1\right)-b c x \text{Li}_2(-i c x)+b c x \text{Li}_2(i c x)+2 b c x \log (x)-2 b \tan ^{-1}(c x)\right)}{2 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)-\frac{1}{2} b c d \text{Li}_2(-i c x)+\frac{1}{2} b c d \text{Li}_2(i c x)+b c d \log (x)",1,"(d*(-2*a - 2*b*ArcTan[c*x] + (2*I)*a*c*x*Log[x] + 2*b*c*x*Log[x] - b*c*x*Log[1 + c^2*x^2] - b*c*x*PolyLog[2, (-I)*c*x] + b*c*x*PolyLog[2, I*c*x]))/(2*x)","A",1
7,1,88,65,0.0639183,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x^3,x]","-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{i c d \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{b c d \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)}{2 x}+\frac{1}{2} i b c^2 d \left(2 \log (x)-\log \left(c^2 x^2+1\right)\right)","-\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,"-1/2*(d*(a + b*ArcTan[c*x]))/x^2 - (I*c*d*(a + b*ArcTan[c*x]))/x - (b*c*d*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/(2*x) + (I/2)*b*c^2*d*(2*Log[x] - Log[1 + c^2*x^2])","C",1
8,1,94,106,0.0559957,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x^4,x]","-\frac{d \left(3 i a c x+2 a+2 b c^3 x^3 \log (x)+3 i b c^2 x^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)-b c^3 x^3 \log \left(c^2 x^2+1\right)+b c x+b (2+3 i c x) \tan ^{-1}(c x)\right)}{6 x^3}","-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{i c d \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\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{i b c^2 d}{2 x}-\frac{b c d}{6 x^2}",1,"-1/6*(d*(2*a + (3*I)*a*c*x + b*c*x + b*(2 + (3*I)*c*x)*ArcTan[c*x] + (3*I)*b*c^2*x^2*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)] + 2*b*c^3*x^3*Log[x] - b*c^3*x^3*Log[1 + c^2*x^2]))/x^3","C",1
9,1,99,124,0.0555593,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d + I*c*d*x)*(a + b*ArcTan[c*x]))/x^5,x]","-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{i c d \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{b c d \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)}{12 x^3}-\frac{1}{6} i b c^2 d \left(-c^2 \log \left(c^2 x^2+1\right)+2 c^2 \log (x)+\frac{1}{x^2}\right)","-\frac{d \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{i c d \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\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^3 d}{4 x}-\frac{i b c^2 d}{6 x^2}-\frac{b c d}{12 x^3}",1,"-1/4*(d*(a + b*ArcTan[c*x]))/x^4 - ((I/3)*c*d*(a + b*ArcTan[c*x]))/x^3 - (b*c*d*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)])/(12*x^3) - (I/6)*b*c^2*d*(x^(-2) + 2*c^2*Log[x] - c^2*Log[1 + c^2*x^2])","C",1
10,1,124,166,0.1321777,"\int x^3 (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + I*c*d*x)^2*(a + b*ArcTan[c*x]),x]","\frac{d^2 \left(3 a c^4 x^4 \left(-10 c^2 x^2+24 i c x+15\right)-36 i b \log \left(c^2 x^2+1\right)+3 b \left(-10 c^6 x^6+24 i c^5 x^5+15 c^4 x^4-25\right) \tan ^{-1}(c x)+b c x \left(6 c^4 x^4-18 i c^3 x^3-25 c^2 x^2+36 i c x+75\right)\right)}{180 c^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{5 b d^2 \tan ^{-1}(c x)}{12 c^4}+\frac{5 b d^2 x}{12 c^3}+\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{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,"(d^2*(3*a*c^4*x^4*(15 + (24*I)*c*x - 10*c^2*x^2) + b*c*x*(75 + (36*I)*c*x - 25*c^2*x^2 - (18*I)*c^3*x^3 + 6*c^4*x^4) + 3*b*(-25 + 15*c^4*x^4 + (24*I)*c^5*x^5 - 10*c^6*x^6)*ArcTan[c*x] - (36*I)*b*Log[1 + c^2*x^2]))/(180*c^4)","A",1
11,1,116,152,0.1196465,"\int x^2 (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + I*c*d*x)^2*(a + b*ArcTan[c*x]),x]","\frac{d^2 \left(2 a c^3 x^3 \left(-6 c^2 x^2+15 i c x+10\right)+16 b \log \left(c^2 x^2+1\right)+b c x \left(3 c^3 x^3-10 i c^2 x^2-16 c x+30 i\right)+2 b \left(-6 c^5 x^5+15 i c^4 x^4+10 c^3 x^3-15 i\right) \tan ^{-1}(c x)\right)}{60 c^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{i b d^2 \tan ^{-1}(c x)}{2 c^3}+\frac{i b d^2 x}{2 c^2}+\frac{4 b d^2 \log \left(c^2 x^2+1\right)}{15 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,"(d^2*(2*a*c^3*x^3*(10 + (15*I)*c*x - 6*c^2*x^2) + b*c*x*(30*I - 16*c*x - (10*I)*c^2*x^2 + 3*c^3*x^3) + 2*b*(-15*I + 10*c^3*x^3 + (15*I)*c^4*x^4 - 6*c^5*x^5)*ArcTan[c*x] + 16*b*Log[1 + c^2*x^2]))/(60*c^3)","A",1
12,1,101,136,0.0971642,"\int x (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x*(d + I*c*d*x)^2*(a + b*ArcTan[c*x]),x]","\frac{d^2 \left(c x \left(a c x \left(-3 c^2 x^2+8 i c x+6\right)+b \left(c^2 x^2-4 i c x-9\right)\right)+4 i b \log \left(c^2 x^2+1\right)+b \left(-3 c^4 x^4+8 i c^3 x^3+6 c^2 x^2+9\right) \tan ^{-1}(c x)\right)}{12 c^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,"(d^2*(c*x*(a*c*x*(6 + (8*I)*c*x - 3*c^2*x^2) + b*(-9 - (4*I)*c*x + c^2*x^2)) + b*(9 + 6*c^2*x^2 + (8*I)*c^3*x^3 - 3*c^4*x^4)*ArcTan[c*x] + (4*I)*b*Log[1 + c^2*x^2]))/(12*c^2)","A",1
13,1,57,83,0.0417973,"\int (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[(d + I*c*d*x)^2*(a + b*ArcTan[c*x]),x]","\frac{1}{3} d^2 \left(-\frac{(c x-i)^3 \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{1}{2} b x (c x-6 i)-\frac{4 b \log (c x+i)}{c}\right)","-\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,"(d^2*((b*x*(-6*I + c*x))/2 - ((-I + c*x)^3*(a + b*ArcTan[c*x]))/c - (4*b*Log[I + c*x])/c))/3","A",1
14,1,103,129,0.0997368,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x,x]","-\frac{1}{2} d^2 \left(a c^2 x^2-4 i a c x-2 a \log (x)+2 i b \log \left(c^2 x^2+1\right)+b c^2 x^2 \tan ^{-1}(c x)-i b \text{Li}_2(-i c x)+i b \text{Li}_2(i c x)-b c x-4 i b c x \tan ^{-1}(c x)+b \tan ^{-1}(c x)\right)","-\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} i b d^2 \text{Li}_2(-i c x)-\frac{1}{2} i b d^2 \text{Li}_2(i c x)+\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,"-1/2*(d^2*((-4*I)*a*c*x - b*c*x + a*c^2*x^2 + b*ArcTan[c*x] - (4*I)*b*c*x*ArcTan[c*x] + b*c^2*x^2*ArcTan[c*x] - 2*a*Log[x] + (2*I)*b*Log[1 + c^2*x^2] - I*b*PolyLog[2, (-I)*c*x] + I*b*PolyLog[2, I*c*x]))","A",1
15,1,79,89,0.0972651,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^2,x]","-\frac{d^2 \left(a c^2 x^2-2 i a c x \log (x)+a+b c^2 x^2 \tan ^{-1}(c x)+b c x \text{Li}_2(-i c x)-b c x \text{Li}_2(i c x)-b c x \log (c x)+b \tan ^{-1}(c x)\right)}{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 \text{Li}_2(-i c x)+b c d^2 \text{Li}_2(i c x)+b c d^2 \log (x)",1,"-((d^2*(a + a*c^2*x^2 + b*ArcTan[c*x] + b*c^2*x^2*ArcTan[c*x] - (2*I)*a*c*x*Log[x] - b*c*x*Log[c*x] + b*c*x*PolyLog[2, (-I)*c*x] - b*c*x*PolyLog[2, I*c*x]))/x)","A",0
16,1,139,152,0.0780387,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^3,x]","-\frac{d^2 \left(2 a c^2 x^2 \log (x)+4 i a c x+a+b c x \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)+i b c^2 x^2 \text{Li}_2(-i c x)-i b c^2 x^2 \text{Li}_2(i c x)-4 i b c^2 x^2 \log (x)+2 i b c^2 x^2 \log \left(c^2 x^2+1\right)+4 i b c x \tan ^{-1}(c x)+b \tan ^{-1}(c x)\right)}{2 x^2}","-\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)-\frac{1}{2} i b c^2 d^2 \text{Li}_2(-i c x)+\frac{1}{2} i b c^2 d^2 \text{Li}_2(i c 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,"-1/2*(d^2*(a + (4*I)*a*c*x + b*ArcTan[c*x] + (4*I)*b*c*x*ArcTan[c*x] + b*c*x*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)] + 2*a*c^2*x^2*Log[x] - (4*I)*b*c^2*x^2*Log[x] + (2*I)*b*c^2*x^2*Log[1 + c^2*x^2] + I*b*c^2*x^2*PolyLog[2, (-I)*c*x] - I*b*c^2*x^2*PolyLog[2, I*c*x]))/x^2","C",0
17,1,114,87,0.0965466,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^4,x]","-\frac{d^2 \left(-6 a c^2 x^2+6 i a c x+2 a+8 b c^3 x^3 \log (x)+6 i b c^2 x^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)+2 b \left(-3 c^2 x^2+3 i c x+1\right) \tan ^{-1}(c x)-4 b c^3 x^3 \log \left(c^2 x^2+1\right)+b c x\right)}{6 x^3}","-\frac{d^2 (1+i c x)^3 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{4}{3} b c^3 d^2 \log (x)+\frac{4}{3} b c^3 d^2 \log (c x+i)-\frac{i b c^2 d^2}{x}-\frac{b c d^2}{6 x^2}",1,"-1/6*(d^2*(2*a + (6*I)*a*c*x + b*c*x - 6*a*c^2*x^2 + 2*b*(1 + (3*I)*c*x - 3*c^2*x^2)*ArcTan[c*x] + (6*I)*b*c^2*x^2*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)] + 8*b*c^3*x^3*Log[x] - 4*b*c^3*x^3*Log[1 + c^2*x^2]))/x^3","C",1
18,1,152,161,0.0721846,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^5,x]","\frac{d^2 \left(6 a c^2 x^2-8 i a c x-3 a-8 i b c^4 x^4 \log (x)-b c x \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)-4 i b c^2 x^2+6 b c^2 x^2 \tan ^{-1}(c x)+4 i b c^4 x^4 \log \left(c^2 x^2+1\right)+6 b c^3 x^3 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)-8 i b c x \tan ^{-1}(c x)-3 b \tan ^{-1}(c x)\right)}{12 x^4}","\frac{c^2 d^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{2 i c d^2 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\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{3 b c^3 d^2}{4 x}-\frac{i b c^2 d^2}{3 x^2}-\frac{b c d^2}{12 x^3}",1,"(d^2*(-3*a - (8*I)*a*c*x + 6*a*c^2*x^2 - (4*I)*b*c^2*x^2 - 3*b*ArcTan[c*x] - (8*I)*b*c*x*ArcTan[c*x] + 6*b*c^2*x^2*ArcTan[c*x] - b*c*x*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)] + 6*b*c^3*x^3*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)] - (8*I)*b*c^4*x^4*Log[x] + (4*I)*b*c^4*x^4*Log[1 + c^2*x^2]))/(12*x^4)","C",1
19,1,124,171,0.0922811,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + I*c*d*x)^2*(a + b*ArcTan[c*x]))/x^6,x]","\frac{d^2 \left(20 a c^2 x^2-30 i a c x-12 a+32 b c^5 x^5 \log (x)+16 b c^3 x^3-10 i b c^2 x^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)+2 b \left(10 c^2 x^2-15 i c x-6\right) \tan ^{-1}(c x)-16 b c^5 x^5 \log \left(c^2 x^2+1\right)-3 b c x\right)}{60 x^5}","\frac{c^2 d^2 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{i c d^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^4}+\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{i b c^4 d^2}{2 x}+\frac{4 b c^3 d^2}{15 x^2}-\frac{i b c^2 d^2}{6 x^3}-\frac{b c d^2}{20 x^4}",1,"(d^2*(-12*a - (30*I)*a*c*x - 3*b*c*x + 20*a*c^2*x^2 + 16*b*c^3*x^3 + 2*b*(-6 - (15*I)*c*x + 10*c^2*x^2)*ArcTan[c*x] - (10*I)*b*c^2*x^2*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)] + 32*b*c^5*x^5*Log[x] - 16*b*c^5*x^5*Log[1 + c^2*x^2]))/(60*x^5)","C",1
20,1,248,205,0.1151343,"\int x^3 (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + I*c*d*x)^3*(a + b*ArcTan[c*x]),x]","-\frac{1}{7} i a c^3 d^3 x^7-\frac{1}{2} a c^2 d^3 x^6+\frac{3}{5} i a c d^3 x^5+\frac{1}{4} a d^3 x^4-\frac{3 b d^3 \tan ^{-1}(c x)}{4 c^4}-\frac{1}{7} i b c^3 d^3 x^7 \tan ^{-1}(c x)+\frac{3 b d^3 x}{4 c^3}+\frac{1}{42} i b c^2 d^3 x^6-\frac{1}{2} b c^2 d^3 x^6 \tan ^{-1}(c x)+\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{1}{10} b c d^3 x^5+\frac{3}{5} i b c d^3 x^5 \tan ^{-1}(c x)+\frac{1}{4} b d^3 x^4 \tan ^{-1}(c x)-\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{3 b d^3 \tan ^{-1}(c x)}{4 c^4}+\frac{3 b d^3 x}{4 c^3}+\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{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) + (a*d^3*x^4)/4 - ((13*I)/70)*b*d^3*x^4 + ((3*I)/5)*a*c*d^3*x^5 + (b*c*d^3*x^5)/10 - (a*c^2*d^3*x^6)/2 + (I/42)*b*c^2*d^3*x^6 - (I/7)*a*c^3*d^3*x^7 - (3*b*d^3*ArcTan[c*x])/(4*c^4) + (b*d^3*x^4*ArcTan[c*x])/4 + ((3*I)/5)*b*c*d^3*x^5*ArcTan[c*x] - (b*c^2*d^3*x^6*ArcTan[c*x])/2 - (I/7)*b*c^3*d^3*x^7*ArcTan[c*x] - (((13*I)/35)*b*d^3*Log[1 + c^2*x^2])/c^4","A",1
21,1,234,191,0.0939803,"\int x^2 (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + I*c*d*x)^3*(a + b*ArcTan[c*x]),x]","-\frac{1}{6} i a c^3 d^3 x^6-\frac{3}{5} a c^2 d^3 x^5+\frac{3}{4} i a c d^3 x^4+\frac{1}{3} a d^3 x^3-\frac{1}{6} i b c^3 d^3 x^6 \tan ^{-1}(c x)-\frac{11 i b d^3 \tan ^{-1}(c x)}{12 c^3}+\frac{1}{30} i b c^2 d^3 x^5-\frac{3}{5} b c^2 d^3 x^5 \tan ^{-1}(c x)+\frac{11 i b d^3 x}{12 c^2}+\frac{7 b d^3 \log \left(c^2 x^2+1\right)}{15 c^3}+\frac{3}{20} b c d^3 x^4+\frac{3}{4} i b c d^3 x^4 \tan ^{-1}(c x)+\frac{1}{3} b d^3 x^3 \tan ^{-1}(c x)-\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{11 i b d^3 \tan ^{-1}(c x)}{12 c^3}+\frac{1}{30} i b c^2 d^3 x^5+\frac{11 i b d^3 x}{12 c^2}+\frac{7 b d^3 \log \left(c^2 x^2+1\right)}{15 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) + (a*d^3*x^3)/3 - ((11*I)/36)*b*d^3*x^3 + ((3*I)/4)*a*c*d^3*x^4 + (3*b*c*d^3*x^4)/20 - (3*a*c^2*d^3*x^5)/5 + (I/30)*b*c^2*d^3*x^5 - (I/6)*a*c^3*d^3*x^6 - (((11*I)/12)*b*d^3*ArcTan[c*x])/c^3 + (b*d^3*x^3*ArcTan[c*x])/3 + ((3*I)/4)*b*c*d^3*x^4*ArcTan[c*x] - (3*b*c^2*d^3*x^5*ArcTan[c*x])/5 - (I/6)*b*c^3*d^3*x^6*ArcTan[c*x] + (7*b*d^3*Log[1 + c^2*x^2])/(15*c^3)","A",1
22,1,132,157,0.1100302,"\int x (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x*(d + I*c*d*x)^3*(a + b*ArcTan[c*x]),x]","\frac{d^3 \left(c x \left(a c x \left(-4 i c^3 x^3-15 c^2 x^2+20 i c x+10\right)+b \left(i c^3 x^3+5 c^2 x^2-12 i c x-25\right)\right)+12 i b \log \left(c^2 x^2+1\right)+b \left(-4 i c^5 x^5-15 c^4 x^4+20 i c^3 x^3+10 c^2 x^2+25\right) \tan ^{-1}(c x)\right)}{20 c^2}","-\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,"(d^3*(c*x*(b*(-25 - (12*I)*c*x + 5*c^2*x^2 + I*c^3*x^3) + a*c*x*(10 + (20*I)*c*x - 15*c^2*x^2 - (4*I)*c^3*x^3)) + b*(25 + 10*c^2*x^2 + (20*I)*c^3*x^3 - 15*c^4*x^4 - (4*I)*c^5*x^5)*ArcTan[c*x] + (12*I)*b*Log[1 + c^2*x^2]))/(20*c^2)","A",1
23,1,77,100,0.0421884,"\int (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[(d + I*c*d*x)^3*(a + b*ArcTan[c*x]),x]","-\frac{i \left(3 (d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)-b d^4 \left(c^3 x^3-6 i c^2 x^2-21 c x+24 i \log (c x+i)+4 i\right)\right)}{12 c d}","-\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,"((-1/12*I)*(3*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]) - b*d^4*(4*I - 21*c*x - (6*I)*c^2*x^2 + c^3*x^3 + (24*I)*Log[I + c*x])))/(c*d)","A",1
24,1,139,170,0.1404185,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x,x]","-\frac{1}{6} i d^3 \left(2 a c^3 x^3-9 i a c^2 x^2-18 a c x+6 i a \log (x)+2 b c^3 x^3 \tan ^{-1}(c x)-b c^2 x^2+10 b \log \left(c^2 x^2+1\right)-9 i b c^2 x^2 \tan ^{-1}(c x)-3 b \text{Li}_2(-i c x)+3 b \text{Li}_2(i c x)+9 i b c x-18 b c x \tan ^{-1}(c x)-9 i b \tan ^{-1}(c x)\right)","-\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{1}{2} i b d^3 \text{Li}_2(-i c x)-\frac{1}{2} i b d^3 \text{Li}_2(i c x)+\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,"(-1/6*I)*d^3*(-18*a*c*x + (9*I)*b*c*x - (9*I)*a*c^2*x^2 - b*c^2*x^2 + 2*a*c^3*x^3 - (9*I)*b*ArcTan[c*x] - 18*b*c*x*ArcTan[c*x] - (9*I)*b*c^2*x^2*ArcTan[c*x] + 2*b*c^3*x^3*ArcTan[c*x] + (6*I)*a*Log[x] + 10*b*Log[1 + c^2*x^2] - 3*b*PolyLog[2, (-I)*c*x] + 3*b*PolyLog[2, I*c*x])","A",1
25,1,150,162,0.1260339,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^2,x]","\frac{d^3 \left(-i a c^3 x^3-6 a c^2 x^2+6 i a c x \log (x)-2 a-i b c^3 x^3 \tan ^{-1}(c x)+i b c^2 x^2+2 b c x \log \left(c^2 x^2+1\right)-6 b c^2 x^2 \tan ^{-1}(c x)-3 b c x \text{Li}_2(-i c x)+3 b c x \text{Li}_2(i c x)+2 b c x \log (c x)-i b c x \tan ^{-1}(c x)-2 b \tan ^{-1}(c x)\right)}{2 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)-\frac{3}{2} b c d^3 \text{Li}_2(-i c x)+\frac{3}{2} b c d^3 \text{Li}_2(i c x)+b c d^3 \log (x)-\frac{1}{2} i b c d^3 \tan ^{-1}(c x)",1,"(d^3*(-2*a - 6*a*c^2*x^2 + I*b*c^2*x^2 - I*a*c^3*x^3 - 2*b*ArcTan[c*x] - I*b*c*x*ArcTan[c*x] - 6*b*c^2*x^2*ArcTan[c*x] - I*b*c^3*x^3*ArcTan[c*x] + (6*I)*a*c*x*Log[x] + 2*b*c*x*Log[c*x] + 2*b*c*x*Log[1 + c^2*x^2] - 3*b*c*x*PolyLog[2, (-I)*c*x] + 3*b*c*x*PolyLog[2, I*c*x]))/(2*x)","A",0
26,1,164,180,0.1294595,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^3,x]","-\frac{i d^3 \left(2 a c^3 x^3-6 i a c^2 x^2 \log (x)+6 a c x-i a+2 b c^3 x^3 \tan ^{-1}(c x)+3 b c^2 x^2 \text{Li}_2(-i c x)-3 b c^2 x^2 \text{Li}_2(i c x)-6 b c^2 x^2 \log (c x)+2 b c^2 x^2 \log \left(c^2 x^2+1\right)-i b c^2 x^2 \tan ^{-1}(c x)-i b c x+6 b c x \tan ^{-1}(c x)-i b \tan ^{-1}(c x)\right)}{2 x^2}","-\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^3 d^3 x \tan ^{-1}(c x)-\frac{3}{2} i b c^2 d^3 \text{Li}_2(-i c x)+\frac{3}{2} i b c^2 d^3 \text{Li}_2(i c 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)-\frac{b c d^3}{2 x}",1,"((-1/2*I)*d^3*((-I)*a + 6*a*c*x - I*b*c*x + 2*a*c^3*x^3 - I*b*ArcTan[c*x] + 6*b*c*x*ArcTan[c*x] - I*b*c^2*x^2*ArcTan[c*x] + 2*b*c^3*x^3*ArcTan[c*x] - (6*I)*a*c^2*x^2*Log[x] - 6*b*c^2*x^2*Log[c*x] + 2*b*c^2*x^2*Log[1 + c^2*x^2] + 3*b*c^2*x^2*PolyLog[2, (-I)*c*x] - 3*b*c^2*x^2*PolyLog[2, I*c*x]))/x^2","A",0
27,1,170,189,0.1039515,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^4,x]","\frac{d^3 \left(-6 i a c^3 x^3 \log (x)+18 a c^2 x^2-9 i a c x-2 a+3 b c^3 x^3 \text{Li}_2(-i c x)-3 b c^3 x^3 \text{Li}_2(i c x)-20 b c^3 x^3 \log (x)-9 i b c^2 x^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)+18 b c^2 x^2 \tan ^{-1}(c x)+10 b c^3 x^3 \log \left(c^2 x^2+1\right)-b c x-9 i b c x \tan ^{-1}(c x)-2 b \tan ^{-1}(c x)\right)}{6 x^3}","\frac{3 c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}-i a c^3 d^3 \log (x)+\frac{1}{2} b c^3 d^3 \text{Li}_2(-i c x)-\frac{1}{2} b c^3 d^3 \text{Li}_2(i c 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{3 i b c^2 d^3}{2 x}+\frac{5}{3} b c^3 d^3 \log \left(c^2 x^2+1\right)-\frac{b c d^3}{6 x^2}",1,"(d^3*(-2*a - (9*I)*a*c*x - b*c*x + 18*a*c^2*x^2 - 2*b*ArcTan[c*x] - (9*I)*b*c*x*ArcTan[c*x] + 18*b*c^2*x^2*ArcTan[c*x] - (9*I)*b*c^2*x^2*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)] - (6*I)*a*c^3*x^3*Log[x] - 20*b*c^3*x^3*Log[x] + 10*b*c^3*x^3*Log[1 + c^2*x^2] + 3*b*c^3*x^3*PolyLog[2, (-I)*c*x] - 3*b*c^3*x^3*PolyLog[2, I*c*x]))/(6*x^3)","C",0
28,1,165,103,0.1324631,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^5,x]","\frac{d^3 \left(-b c x \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)-3 i \left(-4 a c^3 x^3+6 i a c^2 x^2+4 a c x-i a+8 b c^4 x^4 \log (x)+2 b c^2 x^2-4 b c^4 x^4 \log \left(c^2 x^2+1\right)+6 i b c^3 x^3 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)+b \left(-4 c^3 x^3+6 i c^2 x^2+4 c x-i\right) \tan ^{-1}(c x)\right)\right)}{12 x^4}","-\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-2 i b c^4 d^3 \log (x)+2 i b c^4 d^3 \log (c x+i)+\frac{7 b c^3 d^3}{4 x}-\frac{i b c^2 d^3}{2 x^2}-\frac{b c d^3}{12 x^3}",1,"(d^3*(-(b*c*x*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)]) - (3*I)*((-I)*a + 4*a*c*x + (6*I)*a*c^2*x^2 + 2*b*c^2*x^2 - 4*a*c^3*x^3 + b*(-I + 4*c*x + (6*I)*c^2*x^2 - 4*c^3*x^3)*ArcTan[c*x] + (6*I)*b*c^3*x^3*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)] + 8*b*c^4*x^4*Log[x] - 4*b*c^4*x^4*Log[1 + c^2*x^2])))/(12*x^4)","C",1
29,1,185,150,0.1055768,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^6,x]","\frac{d^3 \left(10 i a c^3 x^3+20 a c^2 x^2-15 i a c x-4 a+24 b c^5 x^5 \log (x)+12 b c^3 x^3+10 i b c^3 x^3 \tan ^{-1}(c x)-5 i b c^2 x^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)+20 b c^2 x^2 \tan ^{-1}(c x)-12 b c^5 x^5 \log \left(c^2 x^2+1\right)+10 i b c^4 x^4 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)-b c x-15 i b c x \tan ^{-1}(c x)-4 b \tan ^{-1}(c x)\right)}{20 x^5}","-\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}+\frac{i c d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)}{20 x^4}+\frac{6}{5} b c^5 d^3 \log (x)-\frac{6}{5} b c^5 d^3 \log (c x+i)+\frac{5 i b c^4 d^3}{4 x}+\frac{3 b c^3 d^3}{5 x^2}-\frac{i b c^2 d^3}{4 x^3}-\frac{b c d^3}{20 x^4}",1,"(d^3*(-4*a - (15*I)*a*c*x - b*c*x + 20*a*c^2*x^2 + (10*I)*a*c^3*x^3 + 12*b*c^3*x^3 - 4*b*ArcTan[c*x] - (15*I)*b*c*x*ArcTan[c*x] + 20*b*c^2*x^2*ArcTan[c*x] + (10*I)*b*c^3*x^3*ArcTan[c*x] - (5*I)*b*c^2*x^2*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)] + (10*I)*b*c^4*x^4*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)] + 24*b*c^5*x^5*Log[x] - 12*b*c^5*x^5*Log[1 + c^2*x^2]))/(20*x^5)","C",1
30,1,188,214,0.1189657,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^7} \, dx","Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x]))/x^7,x]","\frac{d^3 \left(-2 b c x \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-c^2 x^2\right)+i \left(20 a c^3 x^3-45 i a c^2 x^2-36 a c x+10 i a+56 b c^6 x^6 \log (x)+28 b c^4 x^4+20 b c^3 x^3 \tan ^{-1}(c x)-9 b c^2 x^2-45 i b c^2 x^2 \tan ^{-1}(c x)-28 b c^6 x^6 \log \left(c^2 x^2+1\right)-15 i b c^3 x^3 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)-36 b c x \tan ^{-1}(c x)+10 i b \tan ^{-1}(c x)\right)\right)}{60 x^6}","\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{d^3 \left(a+b \tan ^{-1}(c x)\right)}{6 x^6}-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}+\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{11 b c^5 d^3}{12 x}+\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{b c d^3}{30 x^5}",1,"(d^3*(-2*b*c*x*Hypergeometric2F1[-5/2, 1, -3/2, -(c^2*x^2)] + I*((10*I)*a - 36*a*c*x - (45*I)*a*c^2*x^2 - 9*b*c^2*x^2 + 20*a*c^3*x^3 + 28*b*c^4*x^4 + (10*I)*b*ArcTan[c*x] - 36*b*c*x*ArcTan[c*x] - (45*I)*b*c^2*x^2*ArcTan[c*x] + 20*b*c^3*x^3*ArcTan[c*x] - (15*I)*b*c^3*x^3*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)] + 56*b*c^6*x^6*Log[x] - 28*b*c^6*x^6*Log[1 + c^2*x^2])))/(60*x^6)","C",1
31,1,290,238,0.1708124,"\int x^3 (d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]),x]","\frac{1}{8} a c^4 d^4 x^8-\frac{4}{7} i a c^3 d^4 x^7-a c^2 d^4 x^6+\frac{4}{5} i a c d^4 x^5+\frac{1}{4} a d^4 x^4+\frac{1}{8} b c^4 d^4 x^8 \tan ^{-1}(c x)-\frac{11 b d^4 \tan ^{-1}(c x)}{8 c^4}-\frac{1}{56} b c^3 d^4 x^7-\frac{4}{7} i b c^3 d^4 x^7 \tan ^{-1}(c x)+\frac{11 b d^4 x}{8 c^3}+\frac{2}{21} i b c^2 d^4 x^6-b c^2 d^4 x^6 \tan ^{-1}(c x)+\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{9}{40} b c d^4 x^5+\frac{4}{5} i b c d^4 x^5 \tan ^{-1}(c x)+\frac{1}{4} b d^4 x^4 \tan ^{-1}(c x)-\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{11 b d^4 \tan ^{-1}(c x)}{8 c^4}-\frac{1}{56} b c^3 d^4 x^7+\frac{11 b d^4 x}{8 c^3}+\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{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) + (a*d^4*x^4)/4 - ((12*I)/35)*b*d^4*x^4 + ((4*I)/5)*a*c*d^4*x^5 + (9*b*c*d^4*x^5)/40 - a*c^2*d^4*x^6 + ((2*I)/21)*b*c^2*d^4*x^6 - ((4*I)/7)*a*c^3*d^4*x^7 - (b*c^3*d^4*x^7)/56 + (a*c^4*d^4*x^8)/8 - (11*b*d^4*ArcTan[c*x])/(8*c^4) + (b*d^4*x^4*ArcTan[c*x])/4 + ((4*I)/5)*b*c*d^4*x^5*ArcTan[c*x] - b*c^2*d^4*x^6*ArcTan[c*x] - ((4*I)/7)*b*c^3*d^4*x^7*ArcTan[c*x] + (b*c^4*d^4*x^8*ArcTan[c*x])/8 - (((24*I)/35)*b*d^4*Log[1 + c^2*x^2])/c^4","A",1
32,1,276,193,0.1287882,"\int x^2 (d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]),x]","\frac{1}{7} a c^4 d^4 x^7-\frac{2}{3} i a c^3 d^4 x^6-\frac{6}{5} a c^2 d^4 x^5+i a c d^4 x^4+\frac{1}{3} a d^4 x^3+\frac{1}{7} b c^4 d^4 x^7 \tan ^{-1}(c x)-\frac{1}{42} b c^3 d^4 x^6-\frac{2}{3} i b c^3 d^4 x^6 \tan ^{-1}(c x)-\frac{5 i b d^4 \tan ^{-1}(c x)}{3 c^3}+\frac{2}{15} i b c^2 d^4 x^5-\frac{6}{5} b c^2 d^4 x^5 \tan ^{-1}(c x)+\frac{5 i b d^4 x}{3 c^2}+\frac{88 b d^4 \log \left(c^2 x^2+1\right)}{105 c^3}+\frac{47}{140} b c d^4 x^4+i b c d^4 x^4 \tan ^{-1}(c x)+\frac{1}{3} b d^4 x^3 \tan ^{-1}(c x)-\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{176 b d^4 \log (c x+i)}{105 c^3}+\frac{2}{15} i b c^2 d^4 x^5+\frac{5 i b d^4 x}{3 c^2}+\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) + (a*d^4*x^3)/3 - ((5*I)/9)*b*d^4*x^3 + I*a*c*d^4*x^4 + (47*b*c*d^4*x^4)/140 - (6*a*c^2*d^4*x^5)/5 + ((2*I)/15)*b*c^2*d^4*x^5 - ((2*I)/3)*a*c^3*d^4*x^6 - (b*c^3*d^4*x^6)/42 + (a*c^4*d^4*x^7)/7 - (((5*I)/3)*b*d^4*ArcTan[c*x])/c^3 + (b*d^4*x^3*ArcTan[c*x])/3 + I*b*c*d^4*x^4*ArcTan[c*x] - (6*b*c^2*d^4*x^5*ArcTan[c*x])/5 - ((2*I)/3)*b*c^3*d^4*x^6*ArcTan[c*x] + (b*c^4*d^4*x^7*ArcTan[c*x])/7 + (88*b*d^4*Log[1 + c^2*x^2])/(105*c^3)","A",1
33,1,264,178,0.1117,"\int x (d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x*(d + I*c*d*x)^4*(a + b*ArcTan[c*x]),x]","\frac{1}{6} a c^4 d^4 x^6-\frac{4}{5} i a c^3 d^4 x^5-\frac{3}{2} a c^2 d^4 x^4+\frac{4}{3} i a c d^4 x^3+\frac{1}{2} a d^4 x^2+\frac{1}{6} b c^4 d^4 x^6 \tan ^{-1}(c x)-\frac{1}{30} b c^3 d^4 x^5-\frac{4}{5} i b c^3 d^4 x^5 \tan ^{-1}(c x)+\frac{1}{5} i b c^2 d^4 x^4-\frac{3}{2} b c^2 d^4 x^4 \tan ^{-1}(c x)+\frac{16 i b d^4 \log \left(c^2 x^2+1\right)}{15 c^2}+\frac{13 b d^4 \tan ^{-1}(c x)}{6 c^2}+\frac{5}{9} b c d^4 x^3+\frac{4}{3} i b c d^4 x^3 \tan ^{-1}(c x)+\frac{1}{2} b d^4 x^2 \tan ^{-1}(c x)-\frac{13 b d^4 x}{6 c}-\frac{16}{15} i b d^4 x^2","-\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,"(-13*b*d^4*x)/(6*c) + (a*d^4*x^2)/2 - ((16*I)/15)*b*d^4*x^2 + ((4*I)/3)*a*c*d^4*x^3 + (5*b*c*d^4*x^3)/9 - (3*a*c^2*d^4*x^4)/2 + (I/5)*b*c^2*d^4*x^4 - ((4*I)/5)*a*c^3*d^4*x^5 - (b*c^3*d^4*x^5)/30 + (a*c^4*d^4*x^6)/6 + (13*b*d^4*ArcTan[c*x])/(6*c^2) + (b*d^4*x^2*ArcTan[c*x])/2 + ((4*I)/3)*b*c*d^4*x^3*ArcTan[c*x] - (3*b*c^2*d^4*x^4*ArcTan[c*x])/2 - ((4*I)/5)*b*c^3*d^4*x^5*ArcTan[c*x] + (b*c^4*d^4*x^6*ArcTan[c*x])/6 + (((16*I)/15)*b*d^4*Log[1 + c^2*x^2])/c^2","A",1
34,1,77,125,0.0318201,"\int (d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[(d + I*c*d*x)^4*(a + b*ArcTan[c*x]),x]","\frac{d^4 \left(12 (c x-i)^5 \left(a+b \tan ^{-1}(c x)\right)-b \left(3 c^4 x^4-20 i c^3 x^3-66 c^2 x^2+180 i c x+192 \log (c x+i)+35\right)\right)}{60 c}","-\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,"(d^4*(12*(-I + c*x)^5*(a + b*ArcTan[c*x]) - b*(35 + (180*I)*c*x - 66*c^2*x^2 - (20*I)*c^3*x^3 + 3*c^4*x^4 + 192*Log[I + c*x])))/(60*c)","A",1
35,1,174,203,0.1514958,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x,x]","\frac{1}{12} d^4 \left(3 a c^4 x^4-16 i a c^3 x^3-36 a c^2 x^2+48 i a c x+12 a \log (x)+3 b c^4 x^4 \tan ^{-1}(c x)-b c^3 x^3-16 i b c^3 x^3 \tan ^{-1}(c x)+8 i b c^2 x^2-32 i b \log \left(c^2 x^2+1\right)-36 b c^2 x^2 \tan ^{-1}(c x)+6 i b \text{Li}_2(-i c x)-6 i b \text{Li}_2(i c x)+39 b c x+48 i b c x \tan ^{-1}(c x)-39 b \tan ^{-1}(c x)\right)","\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{1}{2} i b d^4 \text{Li}_2(-i c x)-\frac{1}{2} i b d^4 \text{Li}_2(i c x)+\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,"(d^4*((48*I)*a*c*x + 39*b*c*x - 36*a*c^2*x^2 + (8*I)*b*c^2*x^2 - (16*I)*a*c^3*x^3 - b*c^3*x^3 + 3*a*c^4*x^4 - 39*b*ArcTan[c*x] + (48*I)*b*c*x*ArcTan[c*x] - 36*b*c^2*x^2*ArcTan[c*x] - (16*I)*b*c^3*x^3*ArcTan[c*x] + 3*b*c^4*x^4*ArcTan[c*x] + 12*a*Log[x] - (32*I)*b*Log[1 + c^2*x^2] + (6*I)*b*PolyLog[2, (-I)*c*x] - (6*I)*b*PolyLog[2, I*c*x]))/12","A",1
36,1,181,190,0.1545405,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^2,x]","\frac{d^4 \left(2 a c^4 x^4-12 i a c^3 x^3-36 a c^2 x^2+24 i a c x \log (x)-6 a+2 b c^4 x^4 \tan ^{-1}(c x)-b c^3 x^3-12 i b c^3 x^3 \tan ^{-1}(c x)+12 i b c^2 x^2+16 b c x \log \left(c^2 x^2+1\right)-36 b c^2 x^2 \tan ^{-1}(c x)-12 b c x \text{Li}_2(-i c x)+12 b c x \text{Li}_2(i c x)+6 b c x \log (c x)-12 i b c x \tan ^{-1}(c x)-6 b \tan ^{-1}(c x)\right)}{6 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)-2 b c d^4 \text{Li}_2(-i c x)+2 b c d^4 \text{Li}_2(i c x)+b c d^4 \log (x)-2 i b c d^4 \tan ^{-1}(c x)",1,"(d^4*(-6*a - 36*a*c^2*x^2 + (12*I)*b*c^2*x^2 - (12*I)*a*c^3*x^3 - b*c^3*x^3 + 2*a*c^4*x^4 - 6*b*ArcTan[c*x] - (12*I)*b*c*x*ArcTan[c*x] - 36*b*c^2*x^2*ArcTan[c*x] - (12*I)*b*c^3*x^3*ArcTan[c*x] + 2*b*c^4*x^4*ArcTan[c*x] + (24*I)*a*c*x*Log[x] + 6*b*c*x*Log[c*x] + 16*b*c*x*Log[1 + c^2*x^2] - 12*b*c*x*PolyLog[2, (-I)*c*x] + 12*b*c*x*PolyLog[2, I*c*x]))/(6*x)","A",0
37,1,163,173,0.1489046,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^3,x]","\frac{d^4 \left(a c^4 x^4-8 i a c^3 x^3-12 a c^2 x^2 \log (x)-8 i a c x-a+b c^4 x^4 \tan ^{-1}(c x)-b c^3 x^3-8 i b c^3 x^3 \tan ^{-1}(c x)-6 i b c^2 x^2 \text{Li}_2(-i c x)+6 i b c^2 x^2 \text{Li}_2(i c x)+8 i b c^2 x^2 \log (c x)-b c x-8 i b c x \tan ^{-1}(c x)-b \tan ^{-1}(c x)\right)}{2 x^2}","\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^3 d^4 x \tan ^{-1}(c x)-3 i b c^2 d^4 \text{Li}_2(-i c x)+3 i b c^2 d^4 \text{Li}_2(i c x)+4 i b c^2 d^4 \log (x)-\frac{b c d^4}{2 x}",1,"(d^4*(-a - (8*I)*a*c*x - b*c*x - (8*I)*a*c^3*x^3 - b*c^3*x^3 + a*c^4*x^4 - b*ArcTan[c*x] - (8*I)*b*c*x*ArcTan[c*x] - (8*I)*b*c^3*x^3*ArcTan[c*x] + b*c^4*x^4*ArcTan[c*x] - 12*a*c^2*x^2*Log[x] + (8*I)*b*c^2*x^2*Log[c*x] - (6*I)*b*c^2*x^2*PolyLog[2, (-I)*c*x] + (6*I)*b*c^2*x^2*PolyLog[2, I*c*x]))/(2*x^2)","A",0
38,1,193,201,0.158226,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^4,x]","\frac{d^4 \left(6 a c^4 x^4-24 i a c^3 x^3 \log (x)+36 a c^2 x^2-12 i a c x-2 a+6 b c^4 x^4 \tan ^{-1}(c x)+12 b c^3 x^3 \text{Li}_2(-i c x)-12 b c^3 x^3 \text{Li}_2(i c x)-38 b c^3 x^3 \log (c x)-12 i b c^3 x^3 \tan ^{-1}(c x)-12 i b c^2 x^2+36 b c^2 x^2 \tan ^{-1}(c x)+16 b c^3 x^3 \log \left(c^2 x^2+1\right)-b c x-12 i b c x \tan ^{-1}(c x)-2 b \tan ^{-1}(c x)\right)}{6 x^3}","\frac{6 c^2 d^4 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{d^4 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{2 i c d^4 \left(a+b \tan ^{-1}(c x)\right)}{x^2}+a c^4 d^4 x-4 i a c^3 d^4 \log (x)+b c^4 d^4 x \tan ^{-1}(c x)+2 b c^3 d^4 \text{Li}_2(-i c x)-2 b c^3 d^4 \text{Li}_2(i c x)-\frac{19}{3} b c^3 d^4 \log (x)-2 i b c^3 d^4 \tan ^{-1}(c x)-\frac{2 i b c^2 d^4}{x}+\frac{8}{3} b c^3 d^4 \log \left(c^2 x^2+1\right)-\frac{b c d^4}{6 x^2}",1,"(d^4*(-2*a - (12*I)*a*c*x - b*c*x + 36*a*c^2*x^2 - (12*I)*b*c^2*x^2 + 6*a*c^4*x^4 - 2*b*ArcTan[c*x] - (12*I)*b*c*x*ArcTan[c*x] + 36*b*c^2*x^2*ArcTan[c*x] - (12*I)*b*c^3*x^3*ArcTan[c*x] + 6*b*c^4*x^4*ArcTan[c*x] - (24*I)*a*c^3*x^3*Log[x] - 38*b*c^3*x^3*Log[c*x] + 16*b*c^3*x^3*Log[1 + c^2*x^2] + 12*b*c^3*x^3*PolyLog[2, (-I)*c*x] - 12*b*c^3*x^3*PolyLog[2, I*c*x]))/(6*x^3)","A",0
39,1,227,227,0.1250321,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^5,x]","\frac{d^4 \left(12 a c^4 x^4 \log (x)+48 i a c^3 x^3+36 a c^2 x^2-16 i a c x-3 a+6 i b c^4 x^4 \text{Li}_2(-i c x)-6 i b c^4 x^4 \text{Li}_2(i c x)-64 i b c^4 x^4 \log (x)+48 i b c^3 x^3 \tan ^{-1}(c x)-b c x \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)-8 i b c^2 x^2+36 b c^2 x^2 \tan ^{-1}(c x)+32 i b c^4 x^4 \log \left(c^2 x^2+1\right)+36 b c^3 x^3 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)-16 i b c x \tan ^{-1}(c x)-3 b \tan ^{-1}(c x)\right)}{12 x^4}","\frac{4 i c^3 d^4 \left(a+b \tan ^{-1}(c x)\right)}{x}+\frac{3 c^2 d^4 \left(a+b \tan ^{-1}(c x)\right)}{x^2}-\frac{d^4 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{4 i c d^4 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}+a c^4 d^4 \log (x)+\frac{1}{2} i b c^4 d^4 \text{Li}_2(-i c x)-\frac{1}{2} i b c^4 d^4 \text{Li}_2(i c 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{13 b c^3 d^4}{4 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{b c d^4}{12 x^3}",1,"(d^4*(-3*a - (16*I)*a*c*x + 36*a*c^2*x^2 - (8*I)*b*c^2*x^2 + (48*I)*a*c^3*x^3 - 3*b*ArcTan[c*x] - (16*I)*b*c*x*ArcTan[c*x] + 36*b*c^2*x^2*ArcTan[c*x] + (48*I)*b*c^3*x^3*ArcTan[c*x] - b*c*x*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)] + 36*b*c^3*x^3*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)] + 12*a*c^4*x^4*Log[x] - (64*I)*b*c^4*x^4*Log[x] + (32*I)*b*c^4*x^4*Log[1 + c^2*x^2] + (6*I)*b*c^4*x^4*PolyLog[2, (-I)*c*x] - (6*I)*b*c^4*x^4*PolyLog[2, I*c*x]))/(12*x^4)","C",0
40,1,191,117,0.1739925,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^6,x]","-\frac{d^4 \left(3 \left(20 a c^4 x^4-40 i a c^3 x^3-40 a c^2 x^2+20 i a c x+4 a-64 b c^5 x^5 \log (x)-22 b c^3 x^3+32 b c^5 x^5 \log \left(c^2 x^2+1\right)-40 i b c^4 x^4 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)+4 b \left(5 c^4 x^4-10 i c^3 x^3-10 c^2 x^2+5 i c x+1\right) \tan ^{-1}(c x)+b c x\right)+20 i b c^2 x^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)\right)}{60 x^5}","-\frac{d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}+\frac{16}{5} b c^5 d^4 \log (x)-\frac{16}{5} b c^5 d^4 \log (c x+i)+\frac{3 i b c^4 d^4}{x}+\frac{11 b c^3 d^4}{10 x^2}-\frac{i b c^2 d^4}{3 x^3}-\frac{b c d^4}{20 x^4}",1,"-1/60*(d^4*((20*I)*b*c^2*x^2*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)] + 3*(4*a + (20*I)*a*c*x + b*c*x - 40*a*c^2*x^2 - (40*I)*a*c^3*x^3 - 22*b*c^3*x^3 + 20*a*c^4*x^4 + 4*b*(1 + (5*I)*c*x - 10*c^2*x^2 - (10*I)*c^3*x^3 + 5*c^4*x^4)*ArcTan[c*x] - (40*I)*b*c^4*x^4*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)] - 64*b*c^5*x^5*Log[x] + 32*b*c^5*x^5*Log[1 + c^2*x^2])))/x^5","C",1
41,1,235,168,0.130485,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x^7} \, dx","Integrate[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^7,x]","-\frac{d^4 \left(15 a c^4 x^4-40 i a c^3 x^3-45 a c^2 x^2+24 i a c x+5 a-64 i b c^6 x^6 \log (x)-32 i b c^4 x^4+15 b c^4 x^4 \tan ^{-1}(c x)-40 i b c^3 x^3 \tan ^{-1}(c x)+b c x \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-c^2 x^2\right)+6 i b c^2 x^2-45 b c^2 x^2 \tan ^{-1}(c x)+32 i b c^6 x^6 \log \left(c^2 x^2+1\right)+15 b c^5 x^5 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)-15 b c^3 x^3 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)+24 i b c x \tan ^{-1}(c x)+5 b \tan ^{-1}(c x)\right)}{30 x^6}","-\frac{d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{6 x^6}+\frac{i c d^4 (1+i c x)^5 \left(a+b \tan ^{-1}(c x)\right)}{30 x^5}+\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{13 b c^5 d^4}{6 x}+\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{b c d^4}{30 x^5}",1,"-1/30*(d^4*(5*a + (24*I)*a*c*x - 45*a*c^2*x^2 + (6*I)*b*c^2*x^2 - (40*I)*a*c^3*x^3 + 15*a*c^4*x^4 - (32*I)*b*c^4*x^4 + 5*b*ArcTan[c*x] + (24*I)*b*c*x*ArcTan[c*x] - 45*b*c^2*x^2*ArcTan[c*x] - (40*I)*b*c^3*x^3*ArcTan[c*x] + 15*b*c^4*x^4*ArcTan[c*x] + b*c*x*Hypergeometric2F1[-5/2, 1, -3/2, -(c^2*x^2)] - 15*b*c^3*x^3*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)] + 15*b*c^5*x^5*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)] - (64*I)*b*c^6*x^6*Log[x] + (32*I)*b*c^6*x^6*Log[1 + c^2*x^2]))/x^6","C",1
42,1,293,243,0.1067634,"\int \frac{(d+i c d x)^4 \left(a+b \tan ^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^8,x]","-\frac{a c^4 d^4}{3 x^3}+\frac{i a c^3 d^4}{x^4}+\frac{6 a c^2 d^4}{5 x^5}-\frac{2 i a c d^4}{3 x^6}-\frac{a d^4}{7 x^7}-\frac{176}{105} b c^7 d^4 \log (x)-\frac{88 b c^5 d^4}{105 x^2}-\frac{b c^4 d^4 \tan ^{-1}(c x)}{3 x^3}+\frac{47 b c^3 d^4}{140 x^4}+\frac{i b c^3 d^4 \tan ^{-1}(c x)}{x^4}-\frac{2 i b c^2 d^4 \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-c^2 x^2\right)}{15 x^5}+\frac{6 b c^2 d^4 \tan ^{-1}(c x)}{5 x^5}+\frac{88}{105} b c^7 d^4 \log \left(c^2 x^2+1\right)+\frac{i b c^4 d^4 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)}{3 x^3}-\frac{b d^4 \tan ^{-1}(c x)}{7 x^7}-\frac{b c d^4}{42 x^6}-\frac{2 i b c d^4 \tan ^{-1}(c x)}{3 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{d^4 \left(a+b \tan ^{-1}(c x)\right)}{7 x^7}-\frac{2 i c d^4 \left(a+b \tan ^{-1}(c x)\right)}{3 x^6}-\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{5 i b c^6 d^4}{3 x}-\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{b c d^4}{42 x^6}",1,"-1/7*(a*d^4)/x^7 - (((2*I)/3)*a*c*d^4)/x^6 - (b*c*d^4)/(42*x^6) + (6*a*c^2*d^4)/(5*x^5) + (I*a*c^3*d^4)/x^4 + (47*b*c^3*d^4)/(140*x^4) - (a*c^4*d^4)/(3*x^3) - (88*b*c^5*d^4)/(105*x^2) - (b*d^4*ArcTan[c*x])/(7*x^7) - (((2*I)/3)*b*c*d^4*ArcTan[c*x])/x^6 + (6*b*c^2*d^4*ArcTan[c*x])/(5*x^5) + (I*b*c^3*d^4*ArcTan[c*x])/x^4 - (b*c^4*d^4*ArcTan[c*x])/(3*x^3) - (((2*I)/15)*b*c^2*d^4*Hypergeometric2F1[-5/2, 1, -3/2, -(c^2*x^2)])/x^5 + ((I/3)*b*c^4*d^4*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)])/x^3 - (176*b*c^7*d^4*Log[x])/105 + (88*b*c^7*d^4*Log[1 + c^2*x^2])/105","C",1
43,1,166,196,0.4758475,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{d+i c d x} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x]))/(d + I*c*d*x),x]","-\frac{i \left(\tan ^{-1}(c x) \left(6 a+b \left(2 c^3 x^3+3 i c^2 x^2-6 c x+3 i\right)+6 i b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+2 a c^3 x^3+3 i a c^2 x^2-3 i a \log \left(c^2 x^2+1\right)-6 a c x-b c^2 x^2+4 b \log \left(c^2 x^2+1\right)+3 b \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-3 i b c x+6 b \tan ^{-1}(c x)^2-b\right)}{6 c^4 d}","\frac{\log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d}+\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c^2 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 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 c^4 d}+\frac{b \tan ^{-1}(c x)}{2 c^4 d}-\frac{b x}{2 c^3 d}+\frac{i b x \tan ^{-1}(c 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}",1,"((-1/6*I)*(-b - 6*a*c*x - (3*I)*b*c*x + (3*I)*a*c^2*x^2 - b*c^2*x^2 + 2*a*c^3*x^3 + 6*b*ArcTan[c*x]^2 + ArcTan[c*x]*(6*a + b*(3*I - 6*c*x + (3*I)*c^2*x^2 + 2*c^3*x^3) + (6*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - (3*I)*a*Log[1 + c^2*x^2] + 4*b*Log[1 + c^2*x^2] + 3*b*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(c^4*d)","A",0
44,1,132,156,0.2034299,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{d+i c d x} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x]))/(d + I*c*d*x),x]","-\frac{i \tan ^{-1}(c x) \left(-2 i a+b c^2 x^2+2 i b c x+2 b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+b\right)+i a c^2 x^2-i a \log \left(c^2 x^2+1\right)-2 a c x+b \log \left(c^2 x^2+1\right)+b \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-i b c x+2 b \tan ^{-1}(c x)^2}{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 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 c^3 d}-\frac{i b \tan ^{-1}(c x)}{2 c^3 d}+\frac{i b x}{2 c^2 d}+\frac{b x \tan ^{-1}(c x)}{c^2 d}-\frac{b \log \left(c^2 x^2+1\right)}{2 c^3 d}",1,"-1/2*(-2*a*c*x - I*b*c*x + I*a*c^2*x^2 + 2*b*ArcTan[c*x]^2 + I*ArcTan[c*x]*((-2*I)*a + b + (2*I)*b*c*x + b*c^2*x^2 + 2*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - I*a*Log[1 + c^2*x^2] + b*Log[1 + c^2*x^2] + b*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(c^3*d)","A",0
45,1,108,110,0.1809759,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{d+i c d x} \, dx","Integrate[(x*(a + b*ArcTan[c*x]))/(d + I*c*d*x),x]","\frac{2 i \tan ^{-1}(c x) \left(a-b c x+i b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+a \log \left(c^2 x^2+1\right)-2 i a c x+i b \log \left(c^2 x^2+1\right)+i b \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+2 i b \tan ^{-1}(c x)^2}{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 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 c^2 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,"((-2*I)*a*c*x + (2*I)*b*ArcTan[c*x]^2 + (2*I)*ArcTan[c*x]*(a - b*c*x + I*b*Log[1 + E^((2*I)*ArcTan[c*x])]) + a*Log[1 + c^2*x^2] + I*b*Log[1 + c^2*x^2] + I*b*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(2*c^2*d)","A",0
46,1,60,59,0.0165448,"\int \frac{a+b \tan ^{-1}(c x)}{d+i c d x} \, dx","Integrate[(a + b*ArcTan[c*x])/(d + I*c*d*x),x]","\frac{2 i \log \left(\frac{2 d}{d+i c d x}\right) \left(a+b \tan ^{-1}(c x)\right)-b \text{Li}_2\left(\frac{c x+i}{c x-i}\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{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 c d}",1,"((2*I)*(a + b*ArcTan[c*x])*Log[(2*d)/(d + I*c*d*x)] - b*PolyLog[2, (I + c*x)/(-I + c*x)])/(2*c*d)","A",1
47,1,102,54,0.062192,"\int \frac{a+b \tan ^{-1}(c x)}{x (d+i c d x)} \, dx","Integrate[(a + b*ArcTan[c*x])/(x*(d + I*c*d*x)),x]","\frac{\log \left(\frac{2 i}{-c x+i}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{a \log (x)}{d}+\frac{i b \text{Li}_2(-i c x)}{2 d}-\frac{i b \text{Li}_2(i c x)}{2 d}+\frac{i b \text{Li}_2\left(-\frac{c x+i}{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{Li}_2\left(\frac{2}{i c x+1}-1\right)}{2 d}",1,"(a*Log[x])/d + ((a + b*ArcTan[c*x])*Log[(2*I)/(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, -((I + c*x)/(I - c*x))])/d","A",0
48,1,149,100,0.1010246,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 (d+i c d x)} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^2*(d + I*c*d*x)),x]","-\frac{a+b \tan ^{-1}(c x)}{d x}-\frac{i c \log \left(\frac{2 i}{-c x+i}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{i a c \log (x)}{d}+\frac{b c \left(2 \log (x)-\log \left(c^2 x^2+1\right)\right)}{2 d}+\frac{b c \text{Li}_2(-i c x)}{2 d}-\frac{b c \text{Li}_2(i c x)}{2 d}+\frac{b c \text{Li}_2\left(-\frac{c x+i}{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 \text{Li}_2\left(\frac{2}{i c x+1}-1\right)}{2 d}+\frac{b c \log (x)}{d}",1,"-((a + b*ArcTan[c*x])/(d*x)) - (I*a*c*Log[x])/d - (I*c*(a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)])/d + (b*c*(2*Log[x] - Log[1 + c^2*x^2]))/(2*d) + (b*c*PolyLog[2, (-I)*c*x])/(2*d) - (b*c*PolyLog[2, I*c*x])/(2*d) + (b*c*PolyLog[2, -((I + c*x)/(I - c*x))])/(2*d)","A",0
49,1,178,161,0.186986,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 (d+i c d x)} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^3*(d + I*c*d*x)),x]","-\frac{2 c^2 \log \left(\frac{2 i}{-c x+i}\right) \left(a+b \tan ^{-1}(c x)\right)+\frac{a+b \tan ^{-1}(c x)}{x^2}-\frac{2 i c \left(a+b \tan ^{-1}(c x)\right)}{x}+2 a c^2 \log (x)+\frac{b c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)}{x}+i b c^2 \text{Li}_2(-i c x)-i b c^2 \text{Li}_2(i c x)+i b c^2 \text{Li}_2\left(\frac{c x+i}{c x-i}\right)+i b c^2 \left(2 \log (x)-\log \left(c^2 x^2+1\right)\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 \text{Li}_2\left(\frac{2}{i c x+1}-1\right)}{2 d}+\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,"-1/2*((a + b*ArcTan[c*x])/x^2 - ((2*I)*c*(a + b*ArcTan[c*x]))/x + (b*c*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/x + 2*a*c^2*Log[x] + 2*c^2*(a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)] + I*b*c^2*(2*Log[x] - Log[1 + c^2*x^2]) + I*b*c^2*PolyLog[2, (-I)*c*x] - I*b*c^2*PolyLog[2, I*c*x] + I*b*c^2*PolyLog[2, (I + c*x)/(-I + c*x)])/d","C",0
50,1,254,197,0.1445723,"\int \frac{a+b \tan ^{-1}(c x)}{x^4 (d+i c d x)} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^4*(d + I*c*d*x)),x]","\frac{6 i a c^3 x^3 \log (x)+6 i a c^3 x^3 \log \left(\frac{2 i}{-c x+i}\right)+6 a c^2 x^2+3 i a c x-2 a-3 b c^3 x^3 \text{Li}_2(-i c x)+3 b c^3 x^3 \text{Li}_2(i c x)-3 b c^3 x^3 \text{Li}_2\left(\frac{c x+i}{c x-i}\right)-8 b c^3 x^3 \log (x)+6 i b c^3 x^3 \log \left(\frac{2 i}{-c x+i}\right) \tan ^{-1}(c x)+3 i b c^2 x^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)+6 b c^2 x^2 \tan ^{-1}(c x)+4 b c^3 x^3 \log \left(c^2 x^2+1\right)-b c x+3 i b c x \tan ^{-1}(c x)-2 b \tan ^{-1}(c x)}{6 d x^3}","\frac{i c^3 \log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)}{d x}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)}{2 d x^2}-\frac{b c^3 \text{Li}_2\left(\frac{2}{i c x+1}-1\right)}{2 d}-\frac{4 b c^3 \log (x)}{3 d}+\frac{i b c^3 \tan ^{-1}(c x)}{2 d}+\frac{i b c^2}{2 d x}+\frac{2 b c^3 \log \left(c^2 x^2+1\right)}{3 d}-\frac{b c}{6 d x^2}",1,"(-2*a + (3*I)*a*c*x - b*c*x + 6*a*c^2*x^2 - 2*b*ArcTan[c*x] + (3*I)*b*c*x*ArcTan[c*x] + 6*b*c^2*x^2*ArcTan[c*x] + (3*I)*b*c^2*x^2*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)] + (6*I)*a*c^3*x^3*Log[x] - 8*b*c^3*x^3*Log[x] + (6*I)*a*c^3*x^3*Log[(2*I)/(I - c*x)] + (6*I)*b*c^3*x^3*ArcTan[c*x]*Log[(2*I)/(I - c*x)] + 4*b*c^3*x^3*Log[1 + c^2*x^2] - 3*b*c^3*x^3*PolyLog[2, (-I)*c*x] + 3*b*c^3*x^3*PolyLog[2, I*c*x] - 3*b*c^3*x^3*PolyLog[2, (I + c*x)/(-I + c*x)])/(6*d*x^3)","C",0
51,1,186,203,1.0419974,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{(d+i c d x)^2} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^2,x]","-\frac{2 a c^2 x^2-6 a \log \left(c^2 x^2+1\right)+8 i a c x+\frac{4 i a}{c x-i}-12 i a \tan ^{-1}(c x)+b \left(-4 i \log \left(c^2 x^2+1\right)+2 \tan ^{-1}(c x) \left(c^2 x^2+4 i c x+6 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+i \sin \left(2 \tan ^{-1}(c x)\right)-\cos \left(2 \tan ^{-1}(c x)\right)+1\right)-6 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-2 c x-12 i \tan ^{-1}(c x)^2+\sin \left(2 \tan ^{-1}(c x)\right)+i \cos \left(2 \tan ^{-1}(c x)\right)\right)}{4 c^4 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{x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c^2 d^2}-\frac{2 i a x}{c^3 d^2}-\frac{3 i b \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 c^4 d^2}+\frac{b}{2 c^4 d^2 (-c x+i)}-\frac{b \tan ^{-1}(c x)}{c^4 d^2}+\frac{b x}{2 c^3 d^2}-\frac{2 i b x \tan ^{-1}(c x)}{c^3 d^2}+\frac{i b \log \left(c^2 x^2+1\right)}{c^4 d^2}",1,"-1/4*((8*I)*a*c*x + 2*a*c^2*x^2 + ((4*I)*a)/(-I + c*x) - (12*I)*a*ArcTan[c*x] - 6*a*Log[1 + c^2*x^2] + b*(-2*c*x - (12*I)*ArcTan[c*x]^2 + I*Cos[2*ArcTan[c*x]] - (4*I)*Log[1 + c^2*x^2] - (6*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 2*ArcTan[c*x]*(1 + (4*I)*c*x + c^2*x^2 - Cos[2*ArcTan[c*x]] + 6*Log[1 + E^((2*I)*ArcTan[c*x])] + I*Sin[2*ArcTan[c*x]]) + Sin[2*ArcTan[c*x]]))/(c^4*d^2)","A",0
52,1,153,167,0.8125389,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{(d+i c d x)^2} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^2,x]","-\frac{4 i a \log \left(c^2 x^2+1\right)+4 a c x+\frac{4 a}{c x-i}-8 a \tan ^{-1}(c x)+b \left(-2 \log \left(c^2 x^2+1\right)-4 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-8 \tan ^{-1}(c x)^2-i \sin \left(2 \tan ^{-1}(c x)\right)+\cos \left(2 \tan ^{-1}(c x)\right)+2 \tan ^{-1}(c x) \left(2 c x-4 i \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+\sin \left(2 \tan ^{-1}(c x)\right)+i \cos \left(2 \tan ^{-1}(c x)\right)\right)\right)}{4 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 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c^3 d^2}-\frac{i b}{2 c^3 d^2 (-c x+i)}+\frac{i b \tan ^{-1}(c x)}{2 c^3 d^2}-\frac{b x \tan ^{-1}(c x)}{c^2 d^2}+\frac{b \log \left(c^2 x^2+1\right)}{2 c^3 d^2}",1,"-1/4*(4*a*c*x + (4*a)/(-I + c*x) - 8*a*ArcTan[c*x] + (4*I)*a*Log[1 + c^2*x^2] + b*(-8*ArcTan[c*x]^2 + Cos[2*ArcTan[c*x]] - 2*Log[1 + c^2*x^2] - 4*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - I*Sin[2*ArcTan[c*x]] + 2*ArcTan[c*x]*(2*c*x + I*Cos[2*ArcTan[c*x]] - (4*I)*Log[1 + E^((2*I)*ArcTan[c*x])] + Sin[2*ArcTan[c*x]])))/(c^3*d^2)","A",0
53,1,128,122,0.099258,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{(d+i c d x)^2} \, dx","Integrate[(x*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^2,x]","-\frac{i \left(a+b \tan ^{-1}(c x)\right)}{c^2 d^2 (-c x+i)}+\frac{\log \left(\frac{2 i}{-c x+i}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d^2}+\frac{i b \text{Li}_2\left(-\frac{c x+i}{i-c x}\right)}{2 c^2 d^2}-\frac{b \left(-\frac{\tan ^{-1}(c x)}{c}+\frac{1}{c (-c x+i)}\right)}{2 c 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{i b \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 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,"((-I)*(a + b*ArcTan[c*x]))/(c^2*d^2*(I - c*x)) - (b*(1/(c*(I - c*x)) - ArcTan[c*x]/c))/(2*c*d^2) + ((a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)])/(c^2*d^2) + ((I/2)*b*PolyLog[2, -((I + c*x)/(I - c*x))])/(c^2*d^2)","A",1
54,1,42,69,0.0361733,"\int \frac{a+b \tan ^{-1}(c x)}{(d+i c d x)^2} \, dx","Integrate[(a + b*ArcTan[c*x])/(d + I*c*d*x)^2,x]","\frac{2 a+(b-i b c x) \tan ^{-1}(c x)-i b}{2 c d^2 (c x-i)}","\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,"(2*a - I*b + (b - I*b*c*x)*ArcTan[c*x])/(2*c*d^2*(-I + c*x))","A",1
55,1,128,150,0.1635447,"\int \frac{a+b \tan ^{-1}(c x)}{x (d+i c d x)^2} \, dx","Integrate[(a + b*ArcTan[c*x])/(x*(d + I*c*d*x)^2),x]","\frac{-\frac{2 i \left(a+b \tan ^{-1}(c x)\right)}{c x-i}+2 \log \left(\frac{2 i}{-c x+i}\right) \left(a+b \tan ^{-1}(c x)\right)+2 a \log (x)+i b \text{Li}_2(-i c x)-i b \text{Li}_2(i c x)+i b \text{Li}_2\left(\frac{c x+i}{c x-i}\right)+b \left(-\tan ^{-1}(c x)+\frac{1}{-c x+i}\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{i b \text{Li}_2(-i c x)}{2 d^2}-\frac{i b \text{Li}_2(i c x)}{2 d^2}+\frac{i b \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 d^2}+\frac{b}{2 d^2 (-c x+i)}-\frac{b \tan ^{-1}(c x)}{2 d^2}",1,"(b*((I - c*x)^(-1) - ArcTan[c*x]) - ((2*I)*(a + b*ArcTan[c*x]))/(-I + c*x) + 2*a*Log[x] + 2*(a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)] + I*b*PolyLog[2, (-I)*c*x] - I*b*PolyLog[2, I*c*x] + I*b*PolyLog[2, (I + c*x)/(-I + c*x)])/(2*d^2)","A",1
56,1,165,194,0.3126048,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 (d+i c d x)^2} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^2*(d + I*c*d*x)^2),x]","-\frac{\frac{2 \left(a+b \tan ^{-1}(c x)\right)}{x}+\frac{2 c \left(a+b \tan ^{-1}(c x)\right)}{c x-i}+4 i c \log \left(\frac{2 i}{-c x+i}\right) \left(a+b \tan ^{-1}(c x)\right)+4 i a c \log (x)+b c \left(\log \left(c^2 x^2+1\right)-2 \log (x)\right)-2 b c \text{Li}_2(-i c x)+2 b c \text{Li}_2(i c x)-2 b c \text{Li}_2\left(\frac{c x+i}{c x-i}\right)+i b c \left(-\tan ^{-1}(c x)+\frac{1}{-c x+i}\right)}{2 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{b c \text{Li}_2(-i c x)}{d^2}-\frac{b c \text{Li}_2(i c x)}{d^2}+\frac{b c \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{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,"-1/2*(I*b*c*((I - c*x)^(-1) - ArcTan[c*x]) + (2*(a + b*ArcTan[c*x]))/x + (2*c*(a + b*ArcTan[c*x]))/(-I + c*x) + (4*I)*a*c*Log[x] + (4*I)*c*(a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)] + b*c*(-2*Log[x] + Log[1 + c^2*x^2]) - 2*b*c*PolyLog[2, (-I)*c*x] + 2*b*c*PolyLog[2, I*c*x] - 2*b*c*PolyLog[2, (I + c*x)/(-I + c*x)])/d^2","A",1
57,1,222,244,0.3675553,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 (d+i c d x)^2} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^3*(d + I*c*d*x)^2),x]","-\frac{-\frac{2 i c^2 \left(a+b \tan ^{-1}(c x)\right)}{c x-i}+6 c^2 \log \left(\frac{2 i}{-c x+i}\right) \left(a+b \tan ^{-1}(c x)\right)+\frac{a+b \tan ^{-1}(c x)}{x^2}-\frac{4 i c \left(a+b \tan ^{-1}(c x)\right)}{x}+6 a c^2 \log (x)+\frac{b c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)}{x}+3 i b c^2 \text{Li}_2(-i c x)-3 i b c^2 \text{Li}_2(i c x)+3 i b c^2 \text{Li}_2\left(\frac{c x+i}{c x-i}\right)+2 i b c^2 \left(2 \log (x)-\log \left(c^2 x^2+1\right)\right)-b c^2 \left(\tan ^{-1}(c x)+\frac{1}{c x-i}\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{3 i b c^2 \text{Li}_2(-i c x)}{2 d^2}+\frac{3 i b c^2 \text{Li}_2(i c x)}{2 d^2}-\frac{3 i b c^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 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,"-1/2*(-(b*c^2*((-I + c*x)^(-1) + ArcTan[c*x])) + (a + b*ArcTan[c*x])/x^2 - ((4*I)*c*(a + b*ArcTan[c*x]))/x - ((2*I)*c^2*(a + b*ArcTan[c*x]))/(-I + c*x) + (b*c*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/x + 6*a*c^2*Log[x] + 6*c^2*(a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)] + (2*I)*b*c^2*(2*Log[x] - Log[1 + c^2*x^2]) + (3*I)*b*c^2*PolyLog[2, (-I)*c*x] - (3*I)*b*c^2*PolyLog[2, I*c*x] + (3*I)*b*c^2*PolyLog[2, (I + c*x)/(-I + c*x)])/d^2","C",0
58,1,235,256,1.060632,"\int \frac{x^4 \left(a+b \tan ^{-1}(c x)\right)}{(d+i c d x)^3} \, dx","Integrate[(x^4*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^3,x]","\frac{16 i a c^2 x^2-96 i a \log \left(c^2 x^2+1\right)-96 a c x-\frac{128 a}{c x-i}-\frac{16 i a}{(c x-i)^2}+192 a \tan ^{-1}(c x)+b \left(48 \log \left(c^2 x^2+1\right)+4 i \tan ^{-1}(c x) \left(4 c^2 x^2+24 i c x+48 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+14 i \sin \left(2 \tan ^{-1}(c x)\right)-i \sin \left(4 \tan ^{-1}(c x)\right)-14 \cos \left(2 \tan ^{-1}(c x)\right)+\cos \left(4 \tan ^{-1}(c x)\right)+4\right)+96 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-16 i c x+192 \tan ^{-1}(c x)^2+28 i \sin \left(2 \tan ^{-1}(c x)\right)-i \sin \left(4 \tan ^{-1}(c x)\right)-28 \cos \left(2 \tan ^{-1}(c x)\right)+\cos \left(4 \tan ^{-1}(c x)\right)\right)}{32 c^5 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{i x^2 \left(a+b \tan ^{-1}(c x)\right)}{2 c^3 d^3}-\frac{3 a x}{c^4 d^3}-\frac{3 b \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c^5 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{19 i b \tan ^{-1}(c x)}{8 c^5 d^3}-\frac{i b x}{2 c^4 d^3}-\frac{3 b x \tan ^{-1}(c x)}{c^4 d^3}+\frac{3 b \log \left(c^2 x^2+1\right)}{2 c^5 d^3}",1,"(-96*a*c*x + (16*I)*a*c^2*x^2 - ((16*I)*a)/(-I + c*x)^2 - (128*a)/(-I + c*x) + 192*a*ArcTan[c*x] - (96*I)*a*Log[1 + c^2*x^2] + b*((-16*I)*c*x + 192*ArcTan[c*x]^2 - 28*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] + 48*Log[1 + c^2*x^2] + 96*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (28*I)*Sin[2*ArcTan[c*x]] + (4*I)*ArcTan[c*x]*(4 + (24*I)*c*x + 4*c^2*x^2 - 14*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] + 48*Log[1 + E^((2*I)*ArcTan[c*x])] + (14*I)*Sin[2*ArcTan[c*x]] - I*Sin[4*ArcTan[c*x]]) - I*Sin[4*ArcTan[c*x]]))/(32*c^5*d^3)","A",0
59,1,216,225,0.8621342,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{(d+i c d x)^3} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^3,x]","\frac{-48 a \log \left(c^2 x^2+1\right)+32 i a c x+\frac{96 i a}{c x-i}-\frac{16 a}{(c x-i)^2}-96 i a \tan ^{-1}(c x)+i b \left(-16 \log \left(c^2 x^2+1\right)-48 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-96 \tan ^{-1}(c x)^2-20 i \sin \left(2 \tan ^{-1}(c x)\right)+i \sin \left(4 \tan ^{-1}(c x)\right)+20 \cos \left(2 \tan ^{-1}(c x)\right)-\cos \left(4 \tan ^{-1}(c x)\right)+4 \tan ^{-1}(c x) \left(8 c x-24 i \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+10 \sin \left(2 \tan ^{-1}(c x)\right)-\sin \left(4 \tan ^{-1}(c x)\right)+10 i \cos \left(2 \tan ^{-1}(c x)\right)-i \cos \left(4 \tan ^{-1}(c x)\right)\right)\right)}{32 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{3 i b \text{Li}_2\left(1-\frac{2}{i c x+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{11 b \tan ^{-1}(c x)}{8 c^4 d^3}+\frac{i b x \tan ^{-1}(c x)}{c^3 d^3}-\frac{i b \log \left(c^2 x^2+1\right)}{2 c^4 d^3}",1,"((32*I)*a*c*x - (16*a)/(-I + c*x)^2 + ((96*I)*a)/(-I + c*x) - (96*I)*a*ArcTan[c*x] - 48*a*Log[1 + c^2*x^2] + I*b*(-96*ArcTan[c*x]^2 + 20*Cos[2*ArcTan[c*x]] - Cos[4*ArcTan[c*x]] - 16*Log[1 + c^2*x^2] - 48*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (20*I)*Sin[2*ArcTan[c*x]] + 4*ArcTan[c*x]*(8*c*x + (10*I)*Cos[2*ArcTan[c*x]] - I*Cos[4*ArcTan[c*x]] - (24*I)*Log[1 + E^((2*I)*ArcTan[c*x])] + 10*Sin[2*ArcTan[c*x]] - Sin[4*ArcTan[c*x]]) + I*Sin[4*ArcTan[c*x]]))/(32*c^4*d^3)","A",0
60,1,187,176,0.1705357,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{(d+i c d x)^3} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^3,x]","-\frac{i \left(8 a c^2 x^2 \log \left(\frac{2 i}{-c x+i}\right)+16 i a c x-16 i a c x \log \left(\frac{2 i}{-c x+i}\right)-8 a \log \left(\frac{2 i}{-c x+i}\right)+12 a+b \left(7 c^2 x^2+2 i c x+8 (c x-i)^2 \log \left(\frac{2 i}{-c x+i}\right)+5\right) \tan ^{-1}(c x)+4 i b (c x-i)^2 \text{Li}_2\left(\frac{c x+i}{c x-i}\right)+7 b c x-6 i b\right)}{8 c^3 d^3 (c x-i)^2}","-\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{b \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 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,"((-1/8*I)*(12*a - (6*I)*b + (16*I)*a*c*x + 7*b*c*x - 8*a*Log[(2*I)/(I - c*x)] - (16*I)*a*c*x*Log[(2*I)/(I - c*x)] + 8*a*c^2*x^2*Log[(2*I)/(I - c*x)] + b*ArcTan[c*x]*(5 + (2*I)*c*x + 7*c^2*x^2 + 8*(-I + c*x)^2*Log[(2*I)/(I - c*x)]) + (4*I)*b*(-I + c*x)^2*PolyLog[2, (I + c*x)/(-I + c*x)]))/(c^3*d^3*(-I + c*x)^2)","A",1
61,1,63,88,0.088326,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{(d+i c d x)^3} \, dx","Integrate[(x*(a + b*ArcTan[c*x]))/(d + I*c*d*x)^3,x]","\frac{a (-4-8 i c x)-b \left(3 c^2 x^2+2 i c x+1\right) \tan ^{-1}(c x)+b (-3 c x+2 i)}{8 c^2 d^3 (c x-i)^2}","\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,"(b*(2*I - 3*c*x) + a*(-4 - (8*I)*c*x) - b*(1 + (2*I)*c*x + 3*c^2*x^2)*ArcTan[c*x])/(8*c^2*d^3*(-I + c*x)^2)","A",1
62,1,55,92,0.0434489,"\int \frac{a+b \tan ^{-1}(c x)}{(d+i c d x)^3} \, dx","Integrate[(a + b*ArcTan[c*x])/(d + I*c*d*x)^3,x]","-\frac{i \left(4 a+b \left(c^2 x^2-2 i c x+3\right) \tan ^{-1}(c x)+b (c x-2 i)\right)}{8 c d^3 (c x-i)^2}","\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,"((-1/8*I)*(4*a + b*(-2*I + c*x) + b*(3 - (2*I)*c*x + c^2*x^2)*ArcTan[c*x]))/(c*d^3*(-I + c*x)^2)","A",1
63,1,162,195,0.2351529,"\int \frac{a+b \tan ^{-1}(c x)}{x (d+i c d x)^3} \, dx","Integrate[(a + b*ArcTan[c*x])/(x*(d + I*c*d*x)^3),x]","\frac{-\frac{8 i \left(a+b \tan ^{-1}(c x)\right)}{c x-i}-\frac{4 \left(a+b \tan ^{-1}(c x)\right)}{(c x-i)^2}+8 \log \left(\frac{2 i}{-c x+i}\right) \left(a+b \tan ^{-1}(c x)\right)+8 a \log (x)+4 i b \text{Li}_2(-i c x)-4 i b \text{Li}_2(i c x)+4 i b \text{Li}_2\left(\frac{c x+i}{c x-i}\right)+\frac{5 b}{-c x+i}+\frac{i b}{(c x-i)^2}-5 b \tan ^{-1}(c x)}{8 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{i b \text{Li}_2(-i c x)}{2 d^3}-\frac{i b \text{Li}_2(i c x)}{2 d^3}+\frac{i b \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 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,"((5*b)/(I - c*x) + (I*b)/(-I + c*x)^2 - 5*b*ArcTan[c*x] - (4*(a + b*ArcTan[c*x]))/(-I + c*x)^2 - ((8*I)*(a + b*ArcTan[c*x]))/(-I + c*x) + 8*a*Log[x] + 8*(a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)] + (4*I)*b*PolyLog[2, (-I)*c*x] - (4*I)*b*PolyLog[2, I*c*x] + (4*I)*b*PolyLog[2, (I + c*x)/(-I + c*x)])/(8*d^3)","A",1
64,1,227,250,0.3324953,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 (d+i c d x)^3} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^2*(d + I*c*d*x)^3),x]","\frac{-\frac{8 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{16 c \left(a+b \tan ^{-1}(c x)\right)}{c x-i}+\frac{4 i c \left(a+b \tan ^{-1}(c x)\right)}{(c x-i)^2}-24 i c \log \left(\frac{2 i}{-c x+i}\right) \left(a+b \tan ^{-1}(c x)\right)-24 i a c \log (x)+4 b c \left(2 \log (x)-\log \left(c^2 x^2+1\right)\right)+12 b c \text{Li}_2(-i c x)-12 b c \text{Li}_2(i c x)+12 b c \text{Li}_2\left(\frac{c x+i}{c x-i}\right)-8 i b c \left(-\tan ^{-1}(c x)+\frac{1}{-c x+i}\right)+\frac{b c \left(i c x+i (c x-i)^2 \tan ^{-1}(c x)+2\right)}{(c x-i)^2}}{8 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{3 b c \text{Li}_2(-i c x)}{2 d^3}-\frac{3 b c \text{Li}_2(i c x)}{2 d^3}+\frac{3 b c \text{Li}_2\left(1-\frac{2}{i c x+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,"((-8*I)*b*c*((I - c*x)^(-1) - ArcTan[c*x]) - (8*(a + b*ArcTan[c*x]))/x + ((4*I)*c*(a + b*ArcTan[c*x]))/(-I + c*x)^2 - (16*c*(a + b*ArcTan[c*x]))/(-I + c*x) + (b*c*(2 + I*c*x + I*(-I + c*x)^2*ArcTan[c*x]))/(-I + c*x)^2 - (24*I)*a*c*Log[x] - (24*I)*c*(a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)] + 4*b*c*(2*Log[x] - Log[1 + c^2*x^2]) + 12*b*c*PolyLog[2, (-I)*c*x] - 12*b*c*PolyLog[2, I*c*x] + 12*b*c*PolyLog[2, (I + c*x)/(-I + c*x)])/(8*d^3)","A",1
65,1,285,306,0.5861114,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 (d+i c d x)^3} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^3*(d + I*c*d*x)^3),x]","-\frac{-\frac{24 i c^2 \left(a+b \tan ^{-1}(c x)\right)}{c x-i}-\frac{4 c^2 \left(a+b \tan ^{-1}(c x)\right)}{(c x-i)^2}+48 c^2 \log \left(\frac{2 i}{-c x+i}\right) \left(a+b \tan ^{-1}(c x)\right)+\frac{4 \left(a+b \tan ^{-1}(c x)\right)}{x^2}-\frac{24 i c \left(a+b \tan ^{-1}(c x)\right)}{x}+48 a c^2 \log (x)+\frac{4 b c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)}{x}+24 i b c^2 \text{Li}_2(-i c x)-24 i b c^2 \text{Li}_2(i c x)+24 i b c^2 \text{Li}_2\left(\frac{c x+i}{c x-i}\right)+12 i b c^2 \left(2 \log (x)-\log \left(c^2 x^2+1\right)\right)+12 b c^2 \left(-\tan ^{-1}(c x)+\frac{1}{-c x+i}\right)-\frac{b c^2 \left(c x+(c x-i)^2 \tan ^{-1}(c x)-2 i\right)}{(c x-i)^2}}{8 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 \text{Li}_2(-i c x)}{d^3}+\frac{3 i b c^2 \text{Li}_2(i c x)}{d^3}-\frac{3 i b c^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{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,"-1/8*(12*b*c^2*((I - c*x)^(-1) - ArcTan[c*x]) + (4*(a + b*ArcTan[c*x]))/x^2 - ((24*I)*c*(a + b*ArcTan[c*x]))/x - (4*c^2*(a + b*ArcTan[c*x]))/(-I + c*x)^2 - ((24*I)*c^2*(a + b*ArcTan[c*x]))/(-I + c*x) - (b*c^2*(-2*I + c*x + (-I + c*x)^2*ArcTan[c*x]))/(-I + c*x)^2 + (4*b*c*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/x + 48*a*c^2*Log[x] + 48*c^2*(a + b*ArcTan[c*x])*Log[(2*I)/(I - c*x)] + (12*I)*b*c^2*(2*Log[x] - Log[1 + c^2*x^2]) + (24*I)*b*c^2*PolyLog[2, (-I)*c*x] - (24*I)*b*c^2*PolyLog[2, I*c*x] + (24*I)*b*c^2*PolyLog[2, (I + c*x)/(-I + c*x)])/d^3","C",0
66,1,73,100,0.048633,"\int \frac{a+b \tan ^{-1}(c x)}{(1+i c x)^4} \, dx","Integrate[(a + b*ArcTan[c*x])/(1 + I*c*x)^4,x]","\frac{-24 a+b \left(-3 i c^2 x^2-9 c x+10 i\right)+3 b \left(-i c^3 x^3-3 c^2 x^2+3 i c x-7\right) \tan ^{-1}(c x)}{72 c (c x-i)^3}","\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,"(-24*a + b*(10*I - 9*c*x - (3*I)*c^2*x^2) + 3*b*(-7 + (3*I)*c*x - 3*c^2*x^2 - I*c^3*x^3)*ArcTan[c*x])/(72*c*(-I + c*x)^3)","A",1
67,1,88,49,0.0270737,"\int \frac{\tan ^{-1}(a x)}{c x+i a c x^2} \, dx","Integrate[ArcTan[a*x]/(c*x + I*a*c*x^2),x]","\frac{i \text{Li}_2(-i a x)}{2 c}-\frac{i \text{Li}_2(i a x)}{2 c}+\frac{i \text{Li}_2\left(-\frac{a x+i}{i-a x}\right)}{2 c}+\frac{\log \left(\frac{2 i}{-a x+i}\right) \tan ^{-1}(a x)}{c}","\frac{i \text{Li}_2\left(\frac{2}{i a x+1}-1\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*I)/(I - a*x)])/c + ((I/2)*PolyLog[2, (-I)*a*x])/c - ((I/2)*PolyLog[2, I*a*x])/c + ((I/2)*PolyLog[2, -((I + a*x)/(I - a*x))])/c","A",0
68,1,285,287,0.8594042,"\int x^3 (d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d + I*c*d*x)*(a + b*ArcTan[c*x])^2,x]","\frac{d \left(12 i a^2 c^5 x^5+15 a^2 c^4 x^4-6 i a b c^4 x^4-10 a b c^3 x^3+12 i a b c^2 x^2-12 i a b \log \left(c^2 x^2+1\right)+2 b \tan ^{-1}(c x) \left(3 a \left(4 i c^5 x^5+5 c^4 x^4-5\right)+b \left(-3 i c^4 x^4-5 c^3 x^3+6 i c^2 x^2+15 c x+9 i\right)+12 i b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+30 a b c x+18 i a b+2 i b^2 c^3 x^3+5 b^2 c^2 x^2-20 b^2 \log \left(c^2 x^2+1\right)+3 b^2 \left(4 i c^5 x^5+5 c^4 x^4-1\right) \tan ^{-1}(c x)^2+12 b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-18 i b^2 c x+5 b^2\right)}{60 c^4}","-\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{a b d x}{2 c^3}+\frac{i b d x^2 \left(a+b \tan ^{-1}(c x)\right)}{5 c^2}+\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 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{5 c^4}+\frac{3 i b^2 d \tan ^{-1}(c x)}{10 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{b^2 d x^2}{12 c^2}-\frac{b^2 d \log \left(c^2 x^2+1\right)}{3 c^4}+\frac{i b^2 d x^3}{30 c}",1,"(d*((18*I)*a*b + 5*b^2 + 30*a*b*c*x - (18*I)*b^2*c*x + (12*I)*a*b*c^2*x^2 + 5*b^2*c^2*x^2 - 10*a*b*c^3*x^3 + (2*I)*b^2*c^3*x^3 + 15*a^2*c^4*x^4 - (6*I)*a*b*c^4*x^4 + (12*I)*a^2*c^5*x^5 + 3*b^2*(-1 + 5*c^4*x^4 + (4*I)*c^5*x^5)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b*(9*I + 15*c*x + (6*I)*c^2*x^2 - 5*c^3*x^3 - (3*I)*c^4*x^4) + 3*a*(-5 + 5*c^4*x^4 + (4*I)*c^5*x^5) + (12*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - (12*I)*a*b*Log[1 + c^2*x^2] - 20*b^2*Log[1 + c^2*x^2] + 12*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(60*c^4)","A",0
69,1,241,255,0.6367547,"\int x^2 (d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d + I*c*d*x)*(a + b*ArcTan[c*x])^2,x]","\frac{i d \left(3 a^2 c^4 x^4-4 i a^2 c^3 x^3-2 a b c^3 x^3+4 i a b c^2 x^2-4 i a b \log \left(c^2 x^2+1\right)+2 b \tan ^{-1}(c x) \left(a \left(3 c^4 x^4-4 i c^3 x^3-3\right)+b \left(-c^3 x^3+2 i c^2 x^2+3 c x+2 i\right)+4 i b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+6 a b c x+b^2 c^2 x^2-4 b^2 \log \left(c^2 x^2+1\right)+b^2 \left(3 c^4 x^4-4 i c^3 x^3+1\right) \tan ^{-1}(c x)^2+4 b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-4 i b^2 c x+b^2\right)}{12 c^3}","-\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{i a b d x}{2 c^2}+\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 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{3 c^3}-\frac{b^2 d \tan ^{-1}(c x)}{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{i b^2 d \log \left(c^2 x^2+1\right)}{3 c^3}+\frac{i b^2 d x^2}{12 c}",1,"((I/12)*d*(b^2 + 6*a*b*c*x - (4*I)*b^2*c*x + (4*I)*a*b*c^2*x^2 + b^2*c^2*x^2 - (4*I)*a^2*c^3*x^3 - 2*a*b*c^3*x^3 + 3*a^2*c^4*x^4 + b^2*(1 - (4*I)*c^3*x^3 + 3*c^4*x^4)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b*(2*I + 3*c*x + (2*I)*c^2*x^2 - c^3*x^3) + a*(-3 - (4*I)*c^3*x^3 + 3*c^4*x^4) + (4*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - (4*I)*a*b*Log[1 + c^2*x^2] - 4*b^2*Log[1 + c^2*x^2] + 4*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/c^3","A",0
70,1,208,211,0.5388332,"\int x (d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d + I*c*d*x)*(a + b*ArcTan[c*x])^2,x]","\frac{d \left(2 i a^2 c^3 x^3+3 a^2 c^2 x^2-2 i a b c^2 x^2+2 i a b \log \left(c^2 x^2+1\right)+2 b \tan ^{-1}(c x) \left(a \left(2 i c^3 x^3+3 c^2 x^2+3\right)-i b \left(c^2 x^2-3 i c x+1\right)-2 i b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)-6 a b c x+3 b^2 \log \left(c^2 x^2+1\right)+b^2 \left(2 i c^3 x^3+3 c^2 x^2+1\right) \tan ^{-1}(c x)^2-2 b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+2 i b^2 c x\right)}{6 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 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{3 c^2}+\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,"(d*(-6*a*b*c*x + (2*I)*b^2*c*x + 3*a^2*c^2*x^2 - (2*I)*a*b*c^2*x^2 + (2*I)*a^2*c^3*x^3 + b^2*(1 + 3*c^2*x^2 + (2*I)*c^3*x^3)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*((-I)*b*(1 - (3*I)*c*x + c^2*x^2) + a*(3 + 3*c^2*x^2 + (2*I)*c^3*x^3) - (2*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) + (2*I)*a*b*Log[1 + c^2*x^2] + 3*b^2*Log[1 + c^2*x^2] - 2*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(6*c^2)","A",0
71,1,151,130,0.2691049,"\int (d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[(d + I*c*d*x)*(a + b*ArcTan[c*x])^2,x]","\frac{i d \left(a^2 c^2 x^2-2 i a^2 c x+2 i a b \log \left(c^2 x^2+1\right)+2 b \tan ^{-1}(c x) \left(a c^2 x^2-2 i a c x+a-b c x-2 i b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)-2 a b c x+b^2 \log \left(c^2 x^2+1\right)-2 b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+b^2 (c x-i)^2 \tan ^{-1}(c x)^2\right)}{2 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}-\frac{i b^2 d \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{c}-i b^2 d x \tan ^{-1}(c x)",1,"((I/2)*d*((-2*I)*a^2*c*x - 2*a*b*c*x + a^2*c^2*x^2 + b^2*(-I + c*x)^2*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(a - (2*I)*a*c*x - b*c*x + a*c^2*x^2 - (2*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) + (2*I)*a*b*Log[1 + c^2*x^2] + b^2*Log[1 + c^2*x^2] - 2*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/c","A",0
72,1,272,216,0.4848503,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d + I*c*d*x)*(a + b*ArcTan[c*x])^2)/x,x]","d \left(i a^2 c x+a^2 \log (c x)+i a b \left(2 c x \tan ^{-1}(c x)-\log \left(c^2 x^2+1\right)\right)+i a b (\text{Li}_2(-i c x)-\text{Li}_2(i c x))+b^2 \left(\text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+\tan ^{-1}(c x) \left((1+i c x) \tan ^{-1}(c x)+2 i \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)\right)+b^2 \left(i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+i \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+\frac{2}{3} i \tan ^{-1}(c x)^3+\tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-\tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-\frac{i \pi ^3}{24}\right)\right)","-i b d \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(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+b^2 (-d) \text{Li}_2\left(1-\frac{2}{i c x+1}\right)-\frac{1}{2} b^2 d \text{Li}_3\left(1-\frac{2}{i c x+1}\right)+\frac{1}{2} b^2 d \text{Li}_3\left(\frac{2}{i c x+1}-1\right)",1,"d*(I*a^2*c*x + a^2*Log[c*x] + I*a*b*(2*c*x*ArcTan[c*x] - Log[1 + c^2*x^2]) + b^2*(ArcTan[c*x]*((1 + I*c*x)*ArcTan[c*x] + (2*I)*Log[1 + E^((2*I)*ArcTan[c*x])]) + PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + I*a*b*(PolyLog[2, (-I)*c*x] - PolyLog[2, I*c*x]) + b^2*((-1/24*I)*Pi^3 + ((2*I)/3)*ArcTan[c*x]^3 + ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + PolyLog[3, E^((-2*I)*ArcTan[c*x])]/2 - PolyLog[3, -E^((2*I)*ArcTan[c*x])]/2))","A",0
73,1,289,228,0.4384782,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d + I*c*d*x)*(a + b*ArcTan[c*x])^2)/x^2,x]","\frac{i d \left(a^2 c x \log (x)+i a^2+i a b \left(c x \left(\log \left(c^2 x^2+1\right)-2 \log (c x)\right)+2 \tan ^{-1}(c x)\right)+i a b c x (\text{Li}_2(-i c x)-\text{Li}_2(i c x))+i b^2 \left(i c x \left(\tan ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)\right)+\tan ^{-1}(c x)^2-2 c x \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)\right)+\frac{1}{24} b^2 c x \left(24 i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+24 i \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+12 \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-12 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+16 i \tan ^{-1}(c x)^3+24 \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-24 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-i \pi ^3\right)\right)}{x}","b c d \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)-b c d \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(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-i b^2 c d \text{Li}_2\left(\frac{2}{1-i c x}-1\right)-\frac{1}{2} i b^2 c d \text{Li}_3\left(1-\frac{2}{i c x+1}\right)+\frac{1}{2} i b^2 c d \text{Li}_3\left(\frac{2}{i c x+1}-1\right)",1,"(I*d*(I*a^2 + a^2*c*x*Log[x] + I*a*b*(2*ArcTan[c*x] + c*x*(-2*Log[c*x] + Log[1 + c^2*x^2])) + I*b^2*(ArcTan[c*x]^2 - 2*c*x*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + I*c*x*(ArcTan[c*x]^2 + PolyLog[2, E^((2*I)*ArcTan[c*x])])) + I*a*b*c*x*(PolyLog[2, (-I)*c*x] - PolyLog[2, I*c*x]) + (b^2*c*x*((-I)*Pi^3 + (16*I)*ArcTan[c*x]^3 + 24*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 24*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + (24*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - 12*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/24))/x","A",0
74,1,190,159,0.2994864,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d + I*c*d*x)*(a + b*ArcTan[c*x])^2)/x^3,x]","-\frac{d \left(2 i a^2 c x+a^2-4 i a b c^2 x^2 \log (c x)+2 i a b c^2 x^2 \log \left(c^2 x^2+1\right)+2 b \tan ^{-1}(c x) \left(a c^2 x^2+2 i a c x+a-2 i b c^2 x^2 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+b c x\right)+2 a b c x-2 b^2 c^2 x^2 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)-2 b^2 c^2 x^2 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)-b^2 (c x-i)^2 \tan ^{-1}(c x)^2\right)}{2 x^2}","\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}+b^2 c^2 d \text{Li}_2\left(\frac{2}{1-i c x}-1\right)-\frac{1}{2} b^2 c^2 d \log \left(c^2 x^2+1\right)+b^2 c^2 d \log (x)",1,"-1/2*(d*(a^2 + (2*I)*a^2*c*x + 2*a*b*c*x - b^2*(-I + c*x)^2*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(a + (2*I)*a*c*x + b*c*x + a*c^2*x^2 - (2*I)*b*c^2*x^2*Log[1 - E^((2*I)*ArcTan[c*x])]) - (4*I)*a*b*c^2*x^2*Log[c*x] - 2*b^2*c^2*x^2*Log[(c*x)/Sqrt[1 + c^2*x^2]] + (2*I)*a*b*c^2*x^2*Log[1 + c^2*x^2] - 2*b^2*c^2*x^2*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/x^2","A",0
75,1,240,224,0.5514653,"\int \frac{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d + I*c*d*x)*(a + b*ArcTan[c*x])^2)/x^4,x]","\frac{d \left(-3 i a^2 c x-2 a^2-4 a b c^3 x^3 \log (c x)-6 i a b c^2 x^2+2 a b c^3 x^3 \log \left(c^2 x^2+1\right)-2 b \tan ^{-1}(c x) \left(a \left(3 i c^3 x^3+3 i c x+2\right)+2 b c^3 x^3 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+b c x \left(c^2 x^2+3 i c x+1\right)\right)-2 a b c x+2 i b^2 c^3 x^3 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)-i b^2 \left(c^3 x^3+3 c x-2 i\right) \tan ^{-1}(c x)^2-2 b^2 c^2 x^2+6 i b^2 c^3 x^3 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)\right)}{6 x^3}","-\frac{1}{6} i c^3 d \left(a+b \tan ^{-1}(c x)\right)^2-\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 b c^2 d \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{3 x^3}-\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{1}{3} i b^2 c^3 d \text{Li}_2\left(\frac{2}{1-i c x}-1\right)+i b^2 c^3 d \log (x)-\frac{1}{3} b^2 c^3 d \tan ^{-1}(c x)-\frac{b^2 c^2 d}{3 x}-\frac{1}{2} i b^2 c^3 d \log \left(c^2 x^2+1\right)",1,"(d*(-2*a^2 - (3*I)*a^2*c*x - 2*a*b*c*x - (6*I)*a*b*c^2*x^2 - 2*b^2*c^2*x^2 - I*b^2*(-2*I + 3*c*x + c^3*x^3)*ArcTan[c*x]^2 - 2*b*ArcTan[c*x]*(b*c*x*(1 + (3*I)*c*x + c^2*x^2) + a*(2 + (3*I)*c*x + (3*I)*c^3*x^3) + 2*b*c^3*x^3*Log[1 - E^((2*I)*ArcTan[c*x])]) - 4*a*b*c^3*x^3*Log[c*x] + (6*I)*b^2*c^3*x^3*Log[(c*x)/Sqrt[1 + c^2*x^2]] + 2*a*b*c^3*x^3*Log[1 + c^2*x^2] + (2*I)*b^2*c^3*x^3*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/(6*x^3)","A",0
76,1,342,373,1.2752789,"\int x^3 (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2,x]","\frac{d^2 \left(-30 a^2 c^6 x^6+72 i a^2 c^5 x^5+45 a^2 c^4 x^4+12 a b c^5 x^5-36 i a b c^4 x^4-50 a b c^3 x^3+72 i a b c^2 x^2-72 i a b \log \left(c^2 x^2+1\right)+2 b \tan ^{-1}(c x) \left(a \left(-30 c^6 x^6+72 i c^5 x^5+45 c^4 x^4-75\right)+b \left(6 c^5 x^5-18 i c^4 x^4-25 c^3 x^3+36 i c^2 x^2+75 c x+54 i\right)+72 i b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+150 a b c x+108 i a b-3 b^2 c^4 x^4+12 i b^2 c^3 x^3+31 b^2 c^2 x^2-106 b^2 \log \left(c^2 x^2+1\right)-3 b^2 \left(10 c^6 x^6-24 i c^5 x^5-15 c^4 x^4+1\right) \tan ^{-1}(c x)^2+72 b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-108 i b^2 c x+34 b^2\right)}{180 c^4}","-\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{5 a b d^2 x}{6 c^3}-\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{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{2 b^2 d^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{5 c^4}+\frac{3 i b^2 d^2 \tan ^{-1}(c x)}{5 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{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{i b^2 d^2 x^3}{15 c}-\frac{1}{60} b^2 d^2 x^4",1,"(d^2*((108*I)*a*b + 34*b^2 + 150*a*b*c*x - (108*I)*b^2*c*x + (72*I)*a*b*c^2*x^2 + 31*b^2*c^2*x^2 - 50*a*b*c^3*x^3 + (12*I)*b^2*c^3*x^3 + 45*a^2*c^4*x^4 - (36*I)*a*b*c^4*x^4 - 3*b^2*c^4*x^4 + (72*I)*a^2*c^5*x^5 + 12*a*b*c^5*x^5 - 30*a^2*c^6*x^6 - 3*b^2*(1 - 15*c^4*x^4 - (24*I)*c^5*x^5 + 10*c^6*x^6)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b*(54*I + 75*c*x + (36*I)*c^2*x^2 - 25*c^3*x^3 - (18*I)*c^4*x^4 + 6*c^5*x^5) + a*(-75 + 45*c^4*x^4 + (72*I)*c^5*x^5 - 30*c^6*x^6) + (72*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - (72*I)*a*b*Log[1 + c^2*x^2] - 106*b^2*Log[1 + c^2*x^2] + 72*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(180*c^4)","A",0
77,1,306,333,1.2515113,"\int x^2 (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2,x]","-\frac{d^2 \left(6 a^2 c^5 x^5-15 i a^2 c^4 x^4-10 a^2 c^3 x^3-3 a b c^4 x^4+10 i a b c^3 x^3+16 a b c^2 x^2-16 a b \log \left(c^2 x^2+1\right)+b \tan ^{-1}(c x) \left(2 a \left(6 c^5 x^5-15 i c^4 x^4-10 c^3 x^3+15 i\right)+b \left(-3 c^4 x^4+10 i c^3 x^3+16 c^2 x^2-30 i c x+19\right)+32 b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)-30 i a b c x+9 a b+b^2 c^3 x^3-5 i b^2 c^2 x^2+20 i b^2 \log \left(c^2 x^2+1\right)+b^2 (c x-i)^3 \left(6 c^2 x^2+3 i c x-1\right) \tan ^{-1}(c x)^2-16 i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-19 b^2 c x-5 i b^2\right)}{30 c^3}","-\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}{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{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{8 i b^2 d^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{15 c^3}-\frac{19 b^2 d^2 \tan ^{-1}(c x)}{30 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{2 i b^2 d^2 \log \left(c^2 x^2+1\right)}{3 c^3}+\frac{i b^2 d^2 x^2}{6 c}-\frac{1}{30} b^2 d^2 x^3",1,"-1/30*(d^2*(9*a*b - (5*I)*b^2 - (30*I)*a*b*c*x - 19*b^2*c*x + 16*a*b*c^2*x^2 - (5*I)*b^2*c^2*x^2 - 10*a^2*c^3*x^3 + (10*I)*a*b*c^3*x^3 + b^2*c^3*x^3 - (15*I)*a^2*c^4*x^4 - 3*a*b*c^4*x^4 + 6*a^2*c^5*x^5 + b^2*(-I + c*x)^3*(-1 + (3*I)*c*x + 6*c^2*x^2)*ArcTan[c*x]^2 + b*ArcTan[c*x]*(b*(19 - (30*I)*c*x + 16*c^2*x^2 + (10*I)*c^3*x^3 - 3*c^4*x^4) + 2*a*(15*I - 10*c^3*x^3 - (15*I)*c^4*x^4 + 6*c^5*x^5) + 32*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - 16*a*b*Log[1 + c^2*x^2] + (20*I)*b^2*Log[1 + c^2*x^2] - (16*I)*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/c^3","A",0
78,1,257,293,0.8220764,"\int x (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2,x]","-\frac{d^2 \left(3 a^2 c^4 x^4-8 i a^2 c^3 x^3-6 a^2 c^2 x^2-2 a b c^3 x^3+8 i a b c^2 x^2-8 i a b \log \left(c^2 x^2+1\right)+2 b \tan ^{-1}(c x) \left(a \left(3 c^4 x^4-8 i c^3 x^3-6 c^2 x^2-9\right)+b \left(-c^3 x^3+4 i c^2 x^2+9 c x+4 i\right)+8 i b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+18 a b c x+b^2 c^2 x^2-10 b^2 \log \left(c^2 x^2+1\right)+8 b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-8 i b^2 c x+b^2 (c x-i)^3 (3 c x+i) \tan ^{-1}(c x)^2+b^2\right)}{12 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{2 b^2 d^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{3 c^2}+\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,"-1/12*(d^2*(b^2 + 18*a*b*c*x - (8*I)*b^2*c*x - 6*a^2*c^2*x^2 + (8*I)*a*b*c^2*x^2 + b^2*c^2*x^2 - (8*I)*a^2*c^3*x^3 - 2*a*b*c^3*x^3 + 3*a^2*c^4*x^4 + b^2*(-I + c*x)^3*(I + 3*c*x)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b*(4*I + 9*c*x + (4*I)*c^2*x^2 - c^3*x^3) + a*(-9 - 6*c^2*x^2 - (8*I)*c^3*x^3 + 3*c^4*x^4) + (8*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - (8*I)*a*b*Log[1 + c^2*x^2] - 10*b^2*Log[1 + c^2*x^2] + 8*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/c^2","A",0
79,1,205,192,0.6688871,"\int (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2,x]","-\frac{d^2 \left(a^2 c^3 x^3-3 i a^2 c^2 x^2-3 a^2 c x-a b c^2 x^2+4 a b \log \left(c^2 x^2+1\right)-b \tan ^{-1}(c x) \left(a \left(-2 c^3 x^3+6 i c^2 x^2+6 c x+6 i\right)+b \left(c^2 x^2-6 i c x+1\right)+8 b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+6 i a b c x-3 i b^2 \log \left(c^2 x^2+1\right)+4 i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+b^2 c x+b^2 (c x-i)^3 \tan ^{-1}(c x)^2\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{4 i b^2 d^2 \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{3 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,"-1/3*(d^2*(-3*a^2*c*x + (6*I)*a*b*c*x + b^2*c*x - (3*I)*a^2*c^2*x^2 - a*b*c^2*x^2 + a^2*c^3*x^3 + b^2*(-I + c*x)^3*ArcTan[c*x]^2 - b*ArcTan[c*x]*(b*(1 - (6*I)*c*x + c^2*x^2) + a*(6*I + 6*c*x + (6*I)*c^2*x^2 - 2*c^3*x^3) + 8*b*Log[1 + E^((2*I)*ArcTan[c*x])]) + 4*a*b*Log[1 + c^2*x^2] - (3*I)*b^2*Log[1 + c^2*x^2] + (4*I)*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/c","A",0
80,1,360,300,0.6626976,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2)/x,x]","\frac{1}{2} d^2 \left(a^2 \left(-c^2\right) x^2+4 i a^2 c x+2 a^2 \log (c x)-2 a b \left(\left(c^2 x^2+1\right) \tan ^{-1}(c x)-c x\right)+4 i a b \left(2 c x \tan ^{-1}(c x)-\log \left(c^2 x^2+1\right)\right)+2 i a b (\text{Li}_2(-i c x)-\text{Li}_2(i c x))-b^2 \log \left(c^2 x^2+1\right)-b^2 \left(c^2 x^2+1\right) \tan ^{-1}(c x)^2+4 b^2 \left(\text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+\tan ^{-1}(c x) \left((1+i c x) \tan ^{-1}(c x)+2 i \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)\right)+2 b^2 \left(i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+i \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+\frac{2}{3} i \tan ^{-1}(c x)^3+\tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-\tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-\frac{i \pi ^3}{24}\right)+2 b^2 c x \tan ^{-1}(c x)\right)","-\frac{1}{2} c^2 d^2 x^2 \left(a+b \tan ^{-1}(c x)\right)^2-i b d^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d^2 \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)+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)-2 b^2 d^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)-\frac{1}{2} b^2 d^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)+\frac{1}{2} b^2 d^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)+b^2 c d^2 x \tan ^{-1}(c x)",1,"(d^2*((4*I)*a^2*c*x - a^2*c^2*x^2 + 2*b^2*c*x*ArcTan[c*x] - b^2*(1 + c^2*x^2)*ArcTan[c*x]^2 - 2*a*b*(-(c*x) + (1 + c^2*x^2)*ArcTan[c*x]) + 2*a^2*Log[c*x] + (4*I)*a*b*(2*c*x*ArcTan[c*x] - Log[1 + c^2*x^2]) - b^2*Log[1 + c^2*x^2] + 4*b^2*(ArcTan[c*x]*((1 + I*c*x)*ArcTan[c*x] + (2*I)*Log[1 + E^((2*I)*ArcTan[c*x])]) + PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + (2*I)*a*b*(PolyLog[2, (-I)*c*x] - PolyLog[2, I*c*x]) + 2*b^2*((-1/24*I)*Pi^3 + ((2*I)/3)*ArcTan[c*x]^3 + ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + PolyLog[3, E^((-2*I)*ArcTan[c*x])]/2 - PolyLog[3, -E^((2*I)*ArcTan[c*x])]/2)))/2","A",0
81,1,378,317,0.676097,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2)/x^2,x]","-\frac{d^2 \left(12 a^2 c^2 x^2-24 i a^2 c x \log (c x)+12 a^2+24 a b c^2 x^2 \tan ^{-1}(c x)+24 a b c x \text{Li}_2(-i c x)-24 a b c x \text{Li}_2(i c x)-24 a b c x \log (c x)+24 a b \tan ^{-1}(c x)+12 b^2 c^2 x^2 \tan ^{-1}(c x)^2+24 b^2 c x \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+12 b^2 c x \left(2 \tan ^{-1}(c x)-i\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+12 i b^2 c x \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)-12 i b^2 c x \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)+12 i b^2 c x \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)-\pi ^3 b^2 c x+16 b^2 c x \tan ^{-1}(c x)^3+12 b^2 \tan ^{-1}(c x)^2-24 i b^2 c x \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)+24 i b^2 c x \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-24 b^2 c x \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+24 b^2 c x \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{12 x}","-c^2 d^2 x \left(a+b \tan ^{-1}(c x)\right)^2+2 b c d^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)-2 b c d^2 \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)-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-i b^2 c d^2 \text{Li}_2\left(\frac{2}{1-i c x}-1\right)-i b^2 c d^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)-i b^2 c d^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)+i b^2 c d^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)",1,"-1/12*(d^2*(12*a^2 - b^2*c*Pi^3*x + 12*a^2*c^2*x^2 + 24*a*b*ArcTan[c*x] + 24*a*b*c^2*x^2*ArcTan[c*x] + 12*b^2*ArcTan[c*x]^2 + 12*b^2*c^2*x^2*ArcTan[c*x]^2 + 16*b^2*c*x*ArcTan[c*x]^3 - (24*I)*b^2*c*x*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 24*b^2*c*x*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + 24*b^2*c*x*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*b^2*c*x*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - (24*I)*a^2*c*x*Log[c*x] - 24*a*b*c*x*Log[c*x] + 24*b^2*c*x*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + 12*b^2*c*x*(-I + 2*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (12*I)*b^2*c*x*PolyLog[2, E^((2*I)*ArcTan[c*x])] + 24*a*b*c*x*PolyLog[2, (-I)*c*x] - 24*a*b*c*x*PolyLog[2, I*c*x] - (12*I)*b^2*c*x*PolyLog[3, E^((-2*I)*ArcTan[c*x])] + (12*I)*b^2*c*x*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/x","A",0
82,1,388,337,0.9033716,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2)/x^3,x]","-\frac{d^2 \left(2 a^2 c^2 x^2 \log (x)+4 i a^2 c x+a^2+2 i a b c^2 x^2 (\text{Li}_2(-i c x)-\text{Li}_2(i c x))+4 i a b c x \left(c x \left(\log \left(c^2 x^2+1\right)-2 \log (c x)\right)+2 \tan ^{-1}(c x)\right)+2 a b \left(\tan ^{-1}(c x)+c x \left(c x \tan ^{-1}(c x)+1\right)\right)+\frac{1}{12} b^2 c^2 x^2 \left(24 i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+24 i \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+12 \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-12 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+16 i \tan ^{-1}(c x)^3+24 \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-24 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-i \pi ^3\right)+b^2 \left(-2 c^2 x^2 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+\left(c^2 x^2+1\right) \tan ^{-1}(c x)^2+2 c x \tan ^{-1}(c x)\right)+4 i b^2 c x \left(i c x \left(\tan ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)\right)+\tan ^{-1}(c x)^2-2 c x \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)\right)\right)}{2 x^2}","i b c^2 d^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)-i b c^2 d^2 \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(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}+2 b^2 c^2 d^2 \text{Li}_2\left(\frac{2}{1-i c x}-1\right)+\frac{1}{2} b^2 c^2 d^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)-\frac{1}{2} b^2 c^2 d^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)-\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,"-1/2*(d^2*(a^2 + (4*I)*a^2*c*x + 2*a*b*(ArcTan[c*x] + c*x*(1 + c*x*ArcTan[c*x])) + 2*a^2*c^2*x^2*Log[x] + b^2*(2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 - 2*c^2*x^2*Log[(c*x)/Sqrt[1 + c^2*x^2]]) + (4*I)*a*b*c*x*(2*ArcTan[c*x] + c*x*(-2*Log[c*x] + Log[1 + c^2*x^2])) + (4*I)*b^2*c*x*(ArcTan[c*x]^2 - 2*c*x*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + I*c*x*(ArcTan[c*x]^2 + PolyLog[2, E^((2*I)*ArcTan[c*x])])) + (2*I)*a*b*c^2*x^2*(PolyLog[2, (-I)*c*x] - PolyLog[2, I*c*x]) + (b^2*c^2*x^2*((-I)*Pi^3 + (16*I)*ArcTan[c*x]^3 + 24*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 24*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + (24*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - 12*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/12))/x^2","A",0
83,1,253,267,0.7028902,"\int \frac{(d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d + I*c*d*x)^2*(a + b*ArcTan[c*x])^2)/x^4,x]","\frac{d^2 \left(3 a^2 c^2 x^2-3 i a^2 c x-a^2-8 a b c^3 x^3 \log (c x)-6 i a b c^2 x^2+4 a b c^3 x^3 \log \left(c^2 x^2+1\right)-b \tan ^{-1}(c x) \left(a \left(6 i c^3 x^3-6 c^2 x^2+6 i c x+2\right)+8 b c^3 x^3 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+b c x \left(c^2 x^2+6 i c x+1\right)\right)-a b c x+4 i b^2 c^3 x^3 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)-b^2 c^2 x^2+6 i b^2 c^3 x^3 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+b^2 (-1-i c x)^3 \tan ^{-1}(c x)^2\right)}{3 x^3}","-\frac{8}{3} a b c^3 d^2 \log (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{2 i b c^2 d^2 \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{d^2 (1+i c x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 x^3}-\frac{b c d^2 \left(a+b \tan ^{-1}(c x)\right)}{3 x^2}-\frac{4}{3} i b^2 c^3 d^2 \text{Li}_2(-i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{Li}_2(i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{Li}_2\left(1-\frac{2}{1-i c x}\right)+2 i b^2 c^3 d^2 \log (x)-\frac{1}{3} b^2 c^3 d^2 \tan ^{-1}(c x)-\frac{b^2 c^2 d^2}{3 x}-i b^2 c^3 d^2 \log \left(c^2 x^2+1\right)",1,"(d^2*(-a^2 - (3*I)*a^2*c*x - a*b*c*x + 3*a^2*c^2*x^2 - (6*I)*a*b*c^2*x^2 - b^2*c^2*x^2 + b^2*(-1 - I*c*x)^3*ArcTan[c*x]^2 - b*ArcTan[c*x]*(b*c*x*(1 + (6*I)*c*x + c^2*x^2) + a*(2 + (6*I)*c*x - 6*c^2*x^2 + (6*I)*c^3*x^3) + 8*b*c^3*x^3*Log[1 - E^((2*I)*ArcTan[c*x])]) - 8*a*b*c^3*x^3*Log[c*x] + (6*I)*b^2*c^3*x^3*Log[(c*x)/Sqrt[1 + c^2*x^2]] + 4*a*b*c^3*x^3*Log[1 + c^2*x^2] + (4*I)*b^2*c^3*x^3*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/(3*x^3)","A",0
84,1,408,438,1.9700795,"\int x^3 (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2,x]","\frac{d^3 \left(-180 i a^2 c^7 x^7-630 a^2 c^6 x^6+756 i a^2 c^5 x^5+315 a^2 c^4 x^4+60 i a b c^6 x^6+252 a b c^5 x^5-468 i a b c^4 x^4-630 a b c^3 x^3+936 i a b c^2 x^2-936 i a b \log \left(c^2 x^2+1\right)+6 b \tan ^{-1}(c x) \left(3 a \left(-20 i c^7 x^7-70 c^6 x^6+84 i c^5 x^5+35 c^4 x^4-105\right)+b \left(10 i c^6 x^6+42 c^5 x^5-78 i c^4 x^4-105 c^3 x^3+156 i c^2 x^2+315 c x+244 i\right)+312 i b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+1890 a b c x+1464 i a b-12 i b^2 c^5 x^5-63 b^2 c^4 x^4+176 i b^2 c^3 x^3+441 b^2 c^2 x^2-1386 b^2 \log \left(c^2 x^2+1\right)+9 b^2 (c x-i)^4 \left(-20 i c^3 x^3+10 c^2 x^2+4 i c x-1\right) \tan ^{-1}(c x)^2+936 b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-1464 i b^2 c x+504 b^2\right)}{1260 c^4}","-\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{1}{7} i c^3 d^3 x^7 \left(a+b \tan ^{-1}(c x)\right)^2+\frac{3 a b d^3 x}{2 c^3}-\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}{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{26 b^2 d^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{35 c^4}+\frac{122 i b^2 d^3 \tan ^{-1}(c x)}{105 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{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{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,"(d^3*((1464*I)*a*b + 504*b^2 + 1890*a*b*c*x - (1464*I)*b^2*c*x + (936*I)*a*b*c^2*x^2 + 441*b^2*c^2*x^2 - 630*a*b*c^3*x^3 + (176*I)*b^2*c^3*x^3 + 315*a^2*c^4*x^4 - (468*I)*a*b*c^4*x^4 - 63*b^2*c^4*x^4 + (756*I)*a^2*c^5*x^5 + 252*a*b*c^5*x^5 - (12*I)*b^2*c^5*x^5 - 630*a^2*c^6*x^6 + (60*I)*a*b*c^6*x^6 - (180*I)*a^2*c^7*x^7 + 9*b^2*(-I + c*x)^4*(-1 + (4*I)*c*x + 10*c^2*x^2 - (20*I)*c^3*x^3)*ArcTan[c*x]^2 + 6*b*ArcTan[c*x]*(b*(244*I + 315*c*x + (156*I)*c^2*x^2 - 105*c^3*x^3 - (78*I)*c^4*x^4 + 42*c^5*x^5 + (10*I)*c^6*x^6) + 3*a*(-105 + 35*c^4*x^4 + (84*I)*c^5*x^5 - 70*c^6*x^6 - (20*I)*c^7*x^7) + (312*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - (936*I)*a*b*Log[1 + c^2*x^2] - 1386*b^2*Log[1 + c^2*x^2] + 936*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(1260*c^4)","A",0
85,1,369,402,1.510463,"\int x^2 (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2,x]","\frac{d^3 \left(-30 i a^2 c^6 x^6-108 a^2 c^5 x^5+135 i a^2 c^4 x^4+60 a^2 c^3 x^3+12 i a b c^5 x^5+54 a b c^4 x^4-110 i a b c^3 x^3-168 a b c^2 x^2+168 a b \log \left(c^2 x^2+1\right)+2 b \tan ^{-1}(c x) \left(3 a \left(-10 i c^6 x^6-36 c^5 x^5+45 i c^4 x^4+20 c^3 x^3-55 i\right)+b \left(6 i c^5 x^5+27 c^4 x^4-55 i c^3 x^3-84 c^2 x^2+165 i c x-111\right)-168 b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+330 i a b c x-162 a b-3 i b^2 c^4 x^4-18 b^2 c^3 x^3+61 i b^2 c^2 x^2-226 i b^2 \log \left(c^2 x^2+1\right)+3 b^2 (c x-i)^4 \left(-10 i c^2 x^2+4 c x+i\right) \tan ^{-1}(c x)^2+168 i b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+222 b^2 c x+64 i b^2\right)}{180 c^3}","-\frac{1}{6} i c^3 d^3 x^6 \left(a+b \tan ^{-1}(c x)\right)^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}{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{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{14 i b^2 d^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{15 c^3}-\frac{37 b^2 d^3 \tan ^{-1}(c x)}{30 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{113 i b^2 d^3 \log \left(c^2 x^2+1\right)}{90 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,"(d^3*(-162*a*b + (64*I)*b^2 + (330*I)*a*b*c*x + 222*b^2*c*x - 168*a*b*c^2*x^2 + (61*I)*b^2*c^2*x^2 + 60*a^2*c^3*x^3 - (110*I)*a*b*c^3*x^3 - 18*b^2*c^3*x^3 + (135*I)*a^2*c^4*x^4 + 54*a*b*c^4*x^4 - (3*I)*b^2*c^4*x^4 - 108*a^2*c^5*x^5 + (12*I)*a*b*c^5*x^5 - (30*I)*a^2*c^6*x^6 + 3*b^2*(-I + c*x)^4*(I + 4*c*x - (10*I)*c^2*x^2)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b*(-111 + (165*I)*c*x - 84*c^2*x^2 - (55*I)*c^3*x^3 + 27*c^4*x^4 + (6*I)*c^5*x^5) + 3*a*(-55*I + 20*c^3*x^3 + (45*I)*c^4*x^4 - 36*c^5*x^5 - (10*I)*c^6*x^6) - 168*b*Log[1 + E^((2*I)*ArcTan[c*x])]) + 168*a*b*Log[1 + c^2*x^2] - (226*I)*b^2*Log[1 + c^2*x^2] + (168*I)*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(180*c^3)","A",0
86,1,325,307,1.4017755,"\int x (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2,x]","\frac{d^3 \left(-12 i a^2 c^5 x^5-45 a^2 c^4 x^4+60 i a^2 c^3 x^3+30 a^2 c^2 x^2+6 i a b c^4 x^4+30 a b c^3 x^3-72 i a b c^2 x^2+72 i a b \log \left(c^2 x^2+1\right)+6 b \tan ^{-1}(c x) \left(a \left(-4 i c^5 x^5-15 c^4 x^4+20 i c^3 x^3+10 c^2 x^2+25\right)+b \left(i c^4 x^4+5 c^3 x^3-12 i c^2 x^2-25 c x-13 i\right)-24 i b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)-150 a b c x-18 i a b-2 i b^2 c^3 x^3-15 b^2 c^2 x^2+90 b^2 \log \left(c^2 x^2+1\right)-72 b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+78 i b^2 c x+3 b^2 (1-4 i c x) (c x-i)^4 \tan ^{-1}(c x)^2-15 b^2\right)}{60 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{6 b^2 d^3 \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{5 c^2}+\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,"(d^3*((-18*I)*a*b - 15*b^2 - 150*a*b*c*x + (78*I)*b^2*c*x + 30*a^2*c^2*x^2 - (72*I)*a*b*c^2*x^2 - 15*b^2*c^2*x^2 + (60*I)*a^2*c^3*x^3 + 30*a*b*c^3*x^3 - (2*I)*b^2*c^3*x^3 - 45*a^2*c^4*x^4 + (6*I)*a*b*c^4*x^4 - (12*I)*a^2*c^5*x^5 + 3*b^2*(1 - (4*I)*c*x)*(-I + c*x)^4*ArcTan[c*x]^2 + 6*b*ArcTan[c*x]*(b*(-13*I - 25*c*x - (12*I)*c^2*x^2 + 5*c^3*x^3 + I*c^4*x^4) + a*(25 + 10*c^2*x^2 + (20*I)*c^3*x^3 - 15*c^4*x^4 - (4*I)*c^5*x^5) - (24*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) + (72*I)*a*b*Log[1 + c^2*x^2] + 90*b^2*Log[1 + c^2*x^2] - 72*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(60*c^2)","A",0
87,1,267,226,0.9542543,"\int (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2,x]","-\frac{i d^3 \left(3 a^2 c^4 x^4-12 i a^2 c^3 x^3-18 a^2 c^2 x^2+12 i a^2 c x-2 a b c^3 x^3+12 i a b c^2 x^2-24 i a b \log \left(c^2 x^2+1\right)+2 b \tan ^{-1}(c x) \left(3 a \left(c^4 x^4-4 i c^3 x^3-6 c^2 x^2+4 i c x-7\right)+b \left(-c^3 x^3+6 i c^2 x^2+21 c x+6 i\right)+24 i b \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+42 a b c x+b^2 c^2 x^2-22 b^2 \log \left(c^2 x^2+1\right)+24 b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-12 i b^2 c x+3 b^2 (c x-i)^4 \tan ^{-1}(c x)^2+b^2\right)}{12 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{2 i b^2 d^3 \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{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,"((-1/12*I)*d^3*(b^2 + (12*I)*a^2*c*x + 42*a*b*c*x - (12*I)*b^2*c*x - 18*a^2*c^2*x^2 + (12*I)*a*b*c^2*x^2 + b^2*c^2*x^2 - (12*I)*a^2*c^3*x^3 - 2*a*b*c^3*x^3 + 3*a^2*c^4*x^4 + 3*b^2*(-I + c*x)^4*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b*(6*I + 21*c*x + (6*I)*c^2*x^2 - c^3*x^3) + 3*a*(-7 + (4*I)*c*x - 6*c^2*x^2 - (4*I)*c^3*x^3 + c^4*x^4) + (24*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - (24*I)*a*b*Log[1 + c^2*x^2] - 22*b^2*Log[1 + c^2*x^2] + 24*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/c","A",0
88,1,465,385,0.9654139,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x,x]","-\frac{1}{24} i d^3 \left(8 a^2 c^3 x^3-36 i a^2 c^2 x^2-72 a^2 c x+24 i a^2 \log (c x)+16 a b c^3 x^3 \tan ^{-1}(c x)-8 a b c^2 x^2+80 a b \log \left(c^2 x^2+1\right)-72 i a b c^2 x^2 \tan ^{-1}(c x)-24 a b \text{Li}_2(-i c x)+24 a b \text{Li}_2(i c x)+72 i a b c x-144 a b c x \tan ^{-1}(c x)-72 i a b \tan ^{-1}(c x)+8 b^2 c^3 x^3 \tan ^{-1}(c x)^2-36 i b^2 \log \left(c^2 x^2+1\right)-36 i b^2 c^2 x^2 \tan ^{-1}(c x)^2-8 b^2 c^2 x^2 \tan ^{-1}(c x)-24 b^2 \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)-8 b^2 \left(3 \tan ^{-1}(c x)-10 i\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+12 i b^2 \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-12 i b^2 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+8 b^2 c x-72 b^2 c x \tan ^{-1}(c x)^2+72 i b^2 c x \tan ^{-1}(c x)-16 b^2 \tan ^{-1}(c x)^3+44 i b^2 \tan ^{-1}(c x)^2-8 b^2 \tan ^{-1}(c x)+24 i b^2 \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-24 i b^2 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-160 b^2 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+\pi ^3 b^2\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)-i b d^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d^3 \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \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{10}{3} b^2 d^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)-\frac{1}{2} b^2 d^3 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)+\frac{1}{2} b^2 d^3 \text{Li}_3\left(\frac{2}{i c x+1}-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,"(-1/24*I)*d^3*(b^2*Pi^3 - 72*a^2*c*x + (72*I)*a*b*c*x + 8*b^2*c*x - (36*I)*a^2*c^2*x^2 - 8*a*b*c^2*x^2 + 8*a^2*c^3*x^3 - (72*I)*a*b*ArcTan[c*x] - 8*b^2*ArcTan[c*x] - 144*a*b*c*x*ArcTan[c*x] + (72*I)*b^2*c*x*ArcTan[c*x] - (72*I)*a*b*c^2*x^2*ArcTan[c*x] - 8*b^2*c^2*x^2*ArcTan[c*x] + 16*a*b*c^3*x^3*ArcTan[c*x] + (44*I)*b^2*ArcTan[c*x]^2 - 72*b^2*c*x*ArcTan[c*x]^2 - (36*I)*b^2*c^2*x^2*ArcTan[c*x]^2 + 8*b^2*c^3*x^3*ArcTan[c*x]^2 - 16*b^2*ArcTan[c*x]^3 + (24*I)*b^2*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 160*b^2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - (24*I)*b^2*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*a^2*Log[c*x] + 80*a*b*Log[1 + c^2*x^2] - (36*I)*b^2*Log[1 + c^2*x^2] - 24*b^2*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - 8*b^2*(-10*I + 3*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - 24*a*b*PolyLog[2, (-I)*c*x] + 24*a*b*PolyLog[2, I*c*x] + (12*I)*b^2*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - (12*I)*b^2*PolyLog[3, -E^((2*I)*ArcTan[c*x])])","A",0
89,1,512,402,0.6306683,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^2,x]","\frac{d^3 \left(-4 i a^2 c^3 x^3-24 a^2 c^2 x^2+24 i a^2 c x \log (x)-8 a^2-8 i a b c^3 x^3 \tan ^{-1}(c x)+8 i a b c^2 x^2+16 a b c x \log \left(c^2 x^2+1\right)-48 a b c^2 x^2 \tan ^{-1}(c x)-24 a b c x \text{Li}_2(-i c x)+24 a b c x \text{Li}_2(i c x)+16 a b c x \log (c x)-8 i a b c x \tan ^{-1}(c x)-16 a b \tan ^{-1}(c x)-4 i b^2 c^3 x^3 \tan ^{-1}(c x)^2-4 i b^2 c x \log \left(c^2 x^2+1\right)-24 b^2 c^2 x^2 \tan ^{-1}(c x)^2+8 i b^2 c^2 x^2 \tan ^{-1}(c x)-24 b^2 c x \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)-24 b^2 c x \left(\tan ^{-1}(c x)-i\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-8 i b^2 c x \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+12 i b^2 c x \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-12 i b^2 c x \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+\pi ^3 b^2 c x-16 b^2 c x \tan ^{-1}(c x)^3+12 i b^2 c x \tan ^{-1}(c x)^2-8 b^2 \tan ^{-1}(c x)^2+24 i b^2 c x \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)+16 b^2 c x \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)-24 i b^2 c x \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-48 b^2 c x \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{8 x}","-\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+3 b c d^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)-3 b c d^3 \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)-\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)-i b^2 c d^3 \text{Li}_2\left(\frac{2}{1-i c x}-1\right)-3 i b^2 c d^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)-\frac{3}{2} i b^2 c d^3 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)+\frac{3}{2} i b^2 c d^3 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)",1,"(d^3*(-8*a^2 + b^2*c*Pi^3*x - 24*a^2*c^2*x^2 + (8*I)*a*b*c^2*x^2 - (4*I)*a^2*c^3*x^3 - 16*a*b*ArcTan[c*x] - (8*I)*a*b*c*x*ArcTan[c*x] - 48*a*b*c^2*x^2*ArcTan[c*x] + (8*I)*b^2*c^2*x^2*ArcTan[c*x] - (8*I)*a*b*c^3*x^3*ArcTan[c*x] - 8*b^2*ArcTan[c*x]^2 + (12*I)*b^2*c*x*ArcTan[c*x]^2 - 24*b^2*c^2*x^2*ArcTan[c*x]^2 - (4*I)*b^2*c^3*x^3*ArcTan[c*x]^2 - 16*b^2*c*x*ArcTan[c*x]^3 + (24*I)*b^2*c*x*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + 16*b^2*c*x*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - 48*b^2*c*x*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - (24*I)*b^2*c*x*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*a^2*c*x*Log[x] + 16*a*b*c*x*Log[c*x] + 16*a*b*c*x*Log[1 + c^2*x^2] - (4*I)*b^2*c*x*Log[1 + c^2*x^2] - 24*b^2*c*x*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - 24*b^2*c*x*(-I + ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (8*I)*b^2*c*x*PolyLog[2, E^((2*I)*ArcTan[c*x])] - 24*a*b*c*x*PolyLog[2, (-I)*c*x] + 24*a*b*c*x*PolyLog[2, I*c*x] + (12*I)*b^2*c*x*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - (12*I)*b^2*c*x*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/(8*x)","A",0
90,1,500,416,1.1480829,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^3,x]","\frac{1}{2} d^3 \left(-2 i a^2 c^3 x-6 a^2 c^2 \log (x)-\frac{6 i a^2 c}{x}-\frac{a^2}{x^2}-6 i a b c^2 (\text{Li}_2(-i c x)-\text{Li}_2(i c x))-2 i a b c^2 \left(2 c x \tan ^{-1}(c x)-\log \left(c^2 x^2+1\right)\right)-\frac{6 i a b c \left(c x \left(\log \left(c^2 x^2+1\right)-2 \log (c x)\right)+2 \tan ^{-1}(c x)\right)}{x}-\frac{2 a b \left(\tan ^{-1}(c x)+c x \left(c x \tan ^{-1}(c x)+1\right)\right)}{x^2}-2 i b^2 c^2 \left(\tan ^{-1}(c x) \left((c x-i) \tan ^{-1}(c x)+2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)-i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)\right)+6 b^2 c^2 \left(-i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)-i \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)-\frac{2}{3} i \tan ^{-1}(c x)^3-\tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)+\tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+\frac{i \pi ^3}{24}\right)-\frac{b^2 \left(-2 c^2 x^2 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+\left(c^2 x^2+1\right) \tan ^{-1}(c x)^2+2 c x \tan ^{-1}(c x)\right)}{x^2}+\frac{6 b^2 c \left(c x \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+\tan ^{-1}(c x) \left((c x-i) \tan ^{-1}(c x)+2 i c x \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)\right)\right)}{x}\right)","-i c^3 d^3 x \left(a+b \tan ^{-1}(c x)\right)^2+3 i b c^2 d^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)-3 i b c^2 d^3 \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)+\frac{7}{2} c^2 d^3 \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}+3 b^2 c^2 d^3 \text{Li}_2\left(\frac{2}{1-i c x}-1\right)+b^2 c^2 d^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)+\frac{3}{2} b^2 c^2 d^3 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)-\frac{3}{2} b^2 c^2 d^3 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)-\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,"(d^3*(-(a^2/x^2) - ((6*I)*a^2*c)/x - (2*I)*a^2*c^3*x - (2*a*b*(ArcTan[c*x] + c*x*(1 + c*x*ArcTan[c*x])))/x^2 - 6*a^2*c^2*Log[x] - (b^2*(2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 - 2*c^2*x^2*Log[(c*x)/Sqrt[1 + c^2*x^2]]))/x^2 - (2*I)*a*b*c^2*(2*c*x*ArcTan[c*x] - Log[1 + c^2*x^2]) - ((6*I)*a*b*c*(2*ArcTan[c*x] + c*x*(-2*Log[c*x] + Log[1 + c^2*x^2])))/x - (2*I)*b^2*c^2*(ArcTan[c*x]*((-I + c*x)*ArcTan[c*x] + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + (6*b^2*c*(ArcTan[c*x]*((-I + c*x)*ArcTan[c*x] + (2*I)*c*x*Log[1 - E^((2*I)*ArcTan[c*x])]) + c*x*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/x - (6*I)*a*b*c^2*(PolyLog[2, (-I)*c*x] - PolyLog[2, I*c*x]) + 6*b^2*c^2*((I/24)*Pi^3 - ((2*I)/3)*ArcTan[c*x]^3 - ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - I*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - PolyLog[3, E^((-2*I)*ArcTan[c*x])]/2 + PolyLog[3, -E^((2*I)*ArcTan[c*x])]/2)))/2","A",0
91,1,595,429,0.7966449,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^4,x]","\frac{d^3 \left(-24 i a^2 c^3 x^3 \log (x)+72 a^2 c^2 x^2-36 i a^2 c x-8 a^2+24 a b c^3 x^3 \text{Li}_2(-i c x)-24 a b c^3 x^3 \text{Li}_2(i c x)-160 a b c^3 x^3 \log (c x)-72 i a b c^3 x^3 \tan ^{-1}(c x)-72 i a b c^2 x^2+144 a b c^2 x^2 \tan ^{-1}(c x)+80 a b c^3 x^3 \log \left(c^2 x^2+1\right)-8 a b c x-72 i a b c x \tan ^{-1}(c x)-16 a b \tan ^{-1}(c x)+24 b^2 c^3 x^3 \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+24 b^2 c^3 x^3 \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+80 i b^2 c^3 x^3 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)-12 i b^2 c^3 x^3 \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)+12 i b^2 c^3 x^3 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)-\pi ^3 b^2 c^3 x^3+16 b^2 c^3 x^3 \tan ^{-1}(c x)^3+44 i b^2 c^3 x^3 \tan ^{-1}(c x)^2-8 b^2 c^3 x^3 \tan ^{-1}(c x)-24 i b^2 c^3 x^3 \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-160 b^2 c^3 x^3 \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+24 i b^2 c^3 x^3 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-8 b^2 c^2 x^2+72 b^2 c^2 x^2 \tan ^{-1}(c x)^2-72 i b^2 c^2 x^2 \tan ^{-1}(c x)+72 i b^2 c^3 x^3 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)-36 i b^2 c x \tan ^{-1}(c x)^2-8 b^2 c x \tan ^{-1}(c x)-8 b^2 \tan ^{-1}(c x)^2\right)}{24 x^3}","-b c^3 d^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)+b c^3 d^3 \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)+\frac{11}{6} i c^3 d^3 \left(a+b \tan ^{-1}(c x)\right)^2-\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 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{d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 x^3}-\frac{3 i c d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{2 x^2}-\frac{b c d^3 \left(a+b \tan ^{-1}(c x)\right)}{3 x^2}+\frac{10}{3} i b^2 c^3 d^3 \text{Li}_2\left(\frac{2}{1-i c x}-1\right)+\frac{1}{2} i b^2 c^3 d^3 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)-\frac{1}{2} i b^2 c^3 d^3 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)+3 i b^2 c^3 d^3 \log (x)-\frac{1}{3} b^2 c^3 d^3 \tan ^{-1}(c x)-\frac{b^2 c^2 d^3}{3 x}-\frac{3}{2} i b^2 c^3 d^3 \log \left(c^2 x^2+1\right)",1,"(d^3*(-8*a^2 - (36*I)*a^2*c*x - 8*a*b*c*x + 72*a^2*c^2*x^2 - (72*I)*a*b*c^2*x^2 - 8*b^2*c^2*x^2 - b^2*c^3*Pi^3*x^3 - 16*a*b*ArcTan[c*x] - (72*I)*a*b*c*x*ArcTan[c*x] - 8*b^2*c*x*ArcTan[c*x] + 144*a*b*c^2*x^2*ArcTan[c*x] - (72*I)*b^2*c^2*x^2*ArcTan[c*x] - (72*I)*a*b*c^3*x^3*ArcTan[c*x] - 8*b^2*c^3*x^3*ArcTan[c*x] - 8*b^2*ArcTan[c*x]^2 - (36*I)*b^2*c*x*ArcTan[c*x]^2 + 72*b^2*c^2*x^2*ArcTan[c*x]^2 + (44*I)*b^2*c^3*x^3*ArcTan[c*x]^2 + 16*b^2*c^3*x^3*ArcTan[c*x]^3 - (24*I)*b^2*c^3*x^3*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 160*b^2*c^3*x^3*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + (24*I)*b^2*c^3*x^3*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - (24*I)*a^2*c^3*x^3*Log[x] - 160*a*b*c^3*x^3*Log[c*x] + (72*I)*b^2*c^3*x^3*Log[(c*x)/Sqrt[1 + c^2*x^2]] + 80*a*b*c^3*x^3*Log[1 + c^2*x^2] + 24*b^2*c^3*x^3*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + 24*b^2*c^3*x^3*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (80*I)*b^2*c^3*x^3*PolyLog[2, E^((2*I)*ArcTan[c*x])] + 24*a*b*c^3*x^3*PolyLog[2, (-I)*c*x] - 24*a*b*c^3*x^3*PolyLog[2, I*c*x] - (12*I)*b^2*c^3*x^3*PolyLog[3, E^((-2*I)*ArcTan[c*x])] + (12*I)*b^2*c^3*x^3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/(24*x^3)","A",0
92,1,322,293,0.8985126,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x^5} \, dx","Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^5,x]","\frac{d^3 \left(12 i a^2 c^3 x^3+18 a^2 c^2 x^2-12 i a^2 c x-3 a^2-48 i a b c^4 x^4 \log (c x)+42 a b c^3 x^3-12 i a b c^2 x^2+24 i a b c^4 x^4 \log \left(c^2 x^2+1\right)+2 b \tan ^{-1}(c x) \left(3 a \left(7 c^4 x^4+4 i c^3 x^3+6 c^2 x^2-4 i c x-1\right)-24 i b c^4 x^4 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+b c x \left(-6 i c^3 x^3+21 c^2 x^2-6 i c x-1\right)\right)-2 a b c x-24 b^2 c^4 x^4 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)-b^2 c^4 x^4-12 i b^2 c^3 x^3-b^2 c^2 x^2-44 b^2 c^4 x^4 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)-3 b^2 (c x-i)^4 \tan ^{-1}(c x)^2\right)}{12 x^4}","-4 i a b c^4 d^3 \log (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{7 b c^3 d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x}-\frac{i b c^2 d^3 \left(a+b \tan ^{-1}(c x)\right)}{x^2}-\frac{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)^2}{4 x^4}-\frac{b c d^3 \left(a+b \tan ^{-1}(c x)\right)}{6 x^3}+2 b^2 c^4 d^3 \text{Li}_2(-i c x)-2 b^2 c^4 d^3 \text{Li}_2(i c x)-2 b^2 c^4 d^3 \text{Li}_2\left(1-\frac{2}{1-i c x}\right)-\frac{11}{3} b^2 c^4 d^3 \log (x)-i b^2 c^4 d^3 \tan ^{-1}(c x)-\frac{i b^2 c^3 d^3}{x}-\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)",1,"(d^3*(-3*a^2 - (12*I)*a^2*c*x - 2*a*b*c*x + 18*a^2*c^2*x^2 - (12*I)*a*b*c^2*x^2 - b^2*c^2*x^2 + (12*I)*a^2*c^3*x^3 + 42*a*b*c^3*x^3 - (12*I)*b^2*c^3*x^3 - b^2*c^4*x^4 - 3*b^2*(-I + c*x)^4*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b*c*x*(-1 - (6*I)*c*x + 21*c^2*x^2 - (6*I)*c^3*x^3) + 3*a*(-1 - (4*I)*c*x + 6*c^2*x^2 + (4*I)*c^3*x^3 + 7*c^4*x^4) - (24*I)*b*c^4*x^4*Log[1 - E^((2*I)*ArcTan[c*x])]) - (48*I)*a*b*c^4*x^4*Log[c*x] - 44*b^2*c^4*x^4*Log[(c*x)/Sqrt[1 + c^2*x^2]] + (24*I)*a*b*c^4*x^4*Log[1 + c^2*x^2] - 24*b^2*c^4*x^4*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/(12*x^4)","A",0
93,1,363,384,1.401265,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x^6} \, dx","Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^6,x]","\frac{d^3 \left(30 i a^2 c^3 x^3+60 a^2 c^2 x^2-45 i a^2 c x-12 a^2+144 a b c^5 x^5 \log (c x)+150 i a b c^4 x^4+72 a b c^3 x^3-30 i a b c^2 x^2-72 a b c^5 x^5 \log \left(c^2 x^2+1\right)+6 b \tan ^{-1}(c x) \left(a \left(25 i c^5 x^5+10 i c^3 x^3+20 c^2 x^2-15 i c x-4\right)+24 b c^5 x^5 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+b c x \left(13 c^4 x^4+25 i c^3 x^3+12 c^2 x^2-5 i c x-1\right)\right)-6 a b c x-72 i b^2 c^5 x^5 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)-15 i b^2 c^5 x^5+78 b^2 c^4 x^4-15 i b^2 c^3 x^3-2 b^2 c^2 x^2-180 i b^2 c^5 x^5 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+3 i b^2 (c x-i)^4 (c x+4 i) \tan ^{-1}(c x)^2\right)}{60 x^5}","\frac{12}{5} a b c^5 d^3 \log (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{5 i b c^4 d^3 \left(a+b \tan ^{-1}(c x)\right)}{2 x}+\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{d^3 (1+i c x)^4 \left(a+b \tan ^{-1}(c x)\right)^2}{5 x^5}+\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{6}{5} i b^2 c^5 d^3 \text{Li}_2(-i c x)-\frac{6}{5} i b^2 c^5 d^3 \text{Li}_2(i c x)-\frac{6}{5} i b^2 c^5 d^3 \text{Li}_2\left(1-\frac{2}{1-i c x}\right)-3 i b^2 c^5 d^3 \log (x)+\frac{13}{10} b^2 c^5 d^3 \tan ^{-1}(c x)+\frac{13 b^2 c^4 d^3}{10 x}-\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)",1,"(d^3*(-12*a^2 - (45*I)*a^2*c*x - 6*a*b*c*x + 60*a^2*c^2*x^2 - (30*I)*a*b*c^2*x^2 - 2*b^2*c^2*x^2 + (30*I)*a^2*c^3*x^3 + 72*a*b*c^3*x^3 - (15*I)*b^2*c^3*x^3 + (150*I)*a*b*c^4*x^4 + 78*b^2*c^4*x^4 - (15*I)*b^2*c^5*x^5 + (3*I)*b^2*(-I + c*x)^4*(4*I + c*x)*ArcTan[c*x]^2 + 6*b*ArcTan[c*x]*(b*c*x*(-1 - (5*I)*c*x + 12*c^2*x^2 + (25*I)*c^3*x^3 + 13*c^4*x^4) + a*(-4 - (15*I)*c*x + 20*c^2*x^2 + (10*I)*c^3*x^3 + (25*I)*c^5*x^5) + 24*b*c^5*x^5*Log[1 - E^((2*I)*ArcTan[c*x])]) + 144*a*b*c^5*x^5*Log[c*x] - (180*I)*b^2*c^5*x^5*Log[(c*x)/Sqrt[1 + c^2*x^2]] - 72*a*b*c^5*x^5*Log[1 + c^2*x^2] - (72*I)*b^2*c^5*x^5*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/(60*x^5)","A",0
94,1,401,513,1.6121432,"\int \frac{(d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{x^7} \, dx","Integrate[((d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2)/x^7,x]","\frac{d^3 \left(60 i a^2 c^3 x^3+135 a^2 c^2 x^2-108 i a^2 c x-30 a^2+336 i a b c^6 x^6 \log (c x)-330 a b c^5 x^5+168 i a b c^4 x^4+110 a b c^3 x^3-54 i a b c^2 x^2-168 i a b c^6 x^6 \log \left(c^2 x^2+1\right)+2 b \tan ^{-1}(c x) \left(-3 a \left(55 c^6 x^6-20 i c^3 x^3-45 c^2 x^2+36 i c x+10\right)+168 i b c^6 x^6 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+b c x \left(111 i c^5 x^5-165 c^4 x^4+84 i c^3 x^3+55 c^2 x^2-27 i c x-6\right)\right)-12 a b c x+168 b^2 c^6 x^6 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+64 b^2 c^6 x^6+222 i b^2 c^5 x^5+61 b^2 c^4 x^4-18 i b^2 c^3 x^3-3 b^2 c^2 x^2+3 b^2 (c x-i)^4 \left(c^2 x^2+4 i c x-10\right) \tan ^{-1}(c x)^2+452 b^2 c^6 x^6 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)\right)}{180 x^6}","\frac{28}{15} i a b c^6 d^3 \log (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{11 b c^5 d^3 \left(a+b \tan ^{-1}(c x)\right)}{6 x}+\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{d^3 \left(a+b \tan ^{-1}(c x)\right)^2}{6 x^6}-\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{14}{15} b^2 c^6 d^3 \text{Li}_2(-i c x)+\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(i c x)+\frac{37}{40} b^2 c^6 d^3 \text{Li}_2\left(1-\frac{2}{1-i c x}\right)-\frac{1}{120} b^2 c^6 d^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)+\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{37 i b^2 c^5 d^3}{30 x}+\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)",1,"(d^3*(-30*a^2 - (108*I)*a^2*c*x - 12*a*b*c*x + 135*a^2*c^2*x^2 - (54*I)*a*b*c^2*x^2 - 3*b^2*c^2*x^2 + (60*I)*a^2*c^3*x^3 + 110*a*b*c^3*x^3 - (18*I)*b^2*c^3*x^3 + (168*I)*a*b*c^4*x^4 + 61*b^2*c^4*x^4 - 330*a*b*c^5*x^5 + (222*I)*b^2*c^5*x^5 + 64*b^2*c^6*x^6 + 3*b^2*(-I + c*x)^4*(-10 + (4*I)*c*x + c^2*x^2)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(b*c*x*(-6 - (27*I)*c*x + 55*c^2*x^2 + (84*I)*c^3*x^3 - 165*c^4*x^4 + (111*I)*c^5*x^5) - 3*a*(10 + (36*I)*c*x - 45*c^2*x^2 - (20*I)*c^3*x^3 + 55*c^6*x^6) + (168*I)*b*c^6*x^6*Log[1 - E^((2*I)*ArcTan[c*x])]) + (336*I)*a*b*c^6*x^6*Log[c*x] + 452*b^2*c^6*x^6*Log[(c*x)/Sqrt[1 + c^2*x^2]] - (168*I)*a*b*c^6*x^6*Log[1 + c^2*x^2] + 168*b^2*c^6*x^6*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/(180*x^6)","A",0
95,1,421,356,1.0384963,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{d+i c d x} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x),x]","-\frac{i a^2 \tan ^{-1}(c x)}{c^4 d}+\frac{i a^2 x}{c^3 d}+\frac{a^2 x^2}{2 c^2 d}-\frac{a^2 \log \left(c^2 x^2+1\right)}{2 c^4 d}-\frac{i a^2 x^3}{3 c d}-\frac{i a b \left(-8 \log \left(\frac{1}{\sqrt{c^2 x^2+1}}\right)+\left(c^2 x^2+1\right) \left(2 c x \tan ^{-1}(c x)+3 i \tan ^{-1}(c x)-1\right)+3 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-3 i c x+6 \tan ^{-1}(c x)^2-8 c x \tan ^{-1}(c x)+6 i \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{3 c^4 d}-\frac{i b^2 \left(-6 i \log \left(\frac{1}{\sqrt{c^2 x^2+1}}\right)+2 c x \left(c^2 x^2+1\right) \tan ^{-1}(c x)^2+3 i \left(c^2 x^2+1\right) \tan ^{-1}(c x)^2-2 \left(c^2 x^2+1\right) \tan ^{-1}(c x)+\left(6 \tan ^{-1}(c x)+8 i\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+3 i \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+2 c x+4 \tan ^{-1}(c x)^3-8 c x \tan ^{-1}(c x)^2+8 i \tan ^{-1}(c x)^2-6 i c x \tan ^{-1}(c x)+6 i \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-16 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{6 c^4 d}","\frac{i b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 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{a b x}{c^3 d}+\frac{i x \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 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{i x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 c d}-\frac{4 b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{3 c^4 d}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 c^4 d}+\frac{i b^2 \tan ^{-1}(c x)}{3 c^4 d}-\frac{i b^2 x}{3 c^3 d}-\frac{b^2 x \tan ^{-1}(c x)}{c^3 d}+\frac{b^2 \log \left(c^2 x^2+1\right)}{2 c^4 d}",1,"(I*a^2*x)/(c^3*d) + (a^2*x^2)/(2*c^2*d) - ((I/3)*a^2*x^3)/(c*d) - (I*a^2*ArcTan[c*x])/(c^4*d) - (a^2*Log[1 + c^2*x^2])/(2*c^4*d) - ((I/3)*a*b*((-3*I)*c*x - 8*c*x*ArcTan[c*x] + 6*ArcTan[c*x]^2 + (1 + c^2*x^2)*(-1 + (3*I)*ArcTan[c*x] + 2*c*x*ArcTan[c*x]) + (6*I)*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - 8*Log[1/Sqrt[1 + c^2*x^2]] + 3*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(c^4*d) - ((I/6)*b^2*(2*c*x - (6*I)*c*x*ArcTan[c*x] - 2*(1 + c^2*x^2)*ArcTan[c*x] + (8*I)*ArcTan[c*x]^2 - 8*c*x*ArcTan[c*x]^2 + (3*I)*(1 + c^2*x^2)*ArcTan[c*x]^2 + 2*c*x*(1 + c^2*x^2)*ArcTan[c*x]^2 + 4*ArcTan[c*x]^3 - 16*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + (6*I)*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - (6*I)*Log[1/Sqrt[1 + c^2*x^2]] + (8*I + 6*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (3*I)*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/(c^4*d)","A",0
96,1,330,277,0.6020355,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{d+i c d x} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x),x]","-\frac{i \left(3 a^2 c^2 x^2-3 a^2 \log \left(c^2 x^2+1\right)+6 i a^2 c x-6 i a^2 \tan ^{-1}(c x)-6 i a b \log \left(c^2 x^2+1\right)+6 a b c^2 x^2 \tan ^{-1}(c x)+6 b \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right) \left(-i a-i b \tan ^{-1}(c x)+b\right)-6 a b c x-12 i a b \tan ^{-1}(c x)^2+6 a b \tan ^{-1}(c x)+12 i a b c x \tan ^{-1}(c x)+12 a b \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+3 b^2 \log \left(c^2 x^2+1\right)+3 b^2 c^2 x^2 \tan ^{-1}(c x)^2+3 b^2 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)-4 i b^2 \tan ^{-1}(c x)^3+9 b^2 \tan ^{-1}(c x)^2+6 i b^2 c x \tan ^{-1}(c x)^2-6 b^2 c x \tan ^{-1}(c x)+6 b^2 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+12 i b^2 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{6 c^3 d}","\frac{b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 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 a b x}{c^2 d}+\frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d}-\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c d}+\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c^3 d}-\frac{i b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 c^3 d}+\frac{i b^2 x \tan ^{-1}(c x)}{c^2 d}-\frac{i b^2 \log \left(c^2 x^2+1\right)}{2 c^3 d}",1,"((-1/6*I)*((6*I)*a^2*c*x - 6*a*b*c*x + 3*a^2*c^2*x^2 - (6*I)*a^2*ArcTan[c*x] + 6*a*b*ArcTan[c*x] + (12*I)*a*b*c*x*ArcTan[c*x] - 6*b^2*c*x*ArcTan[c*x] + 6*a*b*c^2*x^2*ArcTan[c*x] - (12*I)*a*b*ArcTan[c*x]^2 + 9*b^2*ArcTan[c*x]^2 + (6*I)*b^2*c*x*ArcTan[c*x]^2 + 3*b^2*c^2*x^2*ArcTan[c*x]^2 - (4*I)*b^2*ArcTan[c*x]^3 + 12*a*b*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + (12*I)*b^2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 6*b^2*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - 3*a^2*Log[1 + c^2*x^2] - (6*I)*a*b*Log[1 + c^2*x^2] + 3*b^2*Log[1 + c^2*x^2] + 6*b*((-I)*a + b - I*b*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 3*b^2*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/(c^3*d)","A",0
97,1,239,192,0.522076,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{d+i c d x} \, dx","Integrate[(x*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x),x]","-\frac{i \left(3 i a^2 \log \left(c^2 x^2+1\right)+6 a^2 c x-6 a^2 \tan ^{-1}(c x)-6 a b \log \left(c^2 x^2+1\right)-6 b \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right) \left(a+b \tan ^{-1}(c x)+i b\right)-12 a b \tan ^{-1}(c x)^2+12 a b c x \tan ^{-1}(c x)-12 i a b \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-3 i b^2 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)-4 b^2 \tan ^{-1}(c x)^3-6 i b^2 \tan ^{-1}(c x)^2+6 b^2 c x \tan ^{-1}(c x)^2-6 i b^2 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+12 b^2 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{6 c^2 d}","-\frac{i b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{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{b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c^2 d}-\frac{b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 c^2 d}",1,"((-1/6*I)*(6*a^2*c*x - 6*a^2*ArcTan[c*x] + 12*a*b*c*x*ArcTan[c*x] - 12*a*b*ArcTan[c*x]^2 - (6*I)*b^2*ArcTan[c*x]^2 + 6*b^2*c*x*ArcTan[c*x]^2 - 4*b^2*ArcTan[c*x]^3 - (12*I)*a*b*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 12*b^2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - (6*I)*b^2*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (3*I)*a^2*Log[1 + c^2*x^2] - 6*a*b*Log[1 + c^2*x^2] - 6*b*(a + I*b + b*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (3*I)*b^2*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/(c^2*d)","A",0
98,1,95,98,0.0377407,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d+i c d x} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(d + I*c*d*x),x]","\frac{i \left(2 \log \left(\frac{2 d}{d+i c d x}\right) \left(a+b \tan ^{-1}(c x)\right)^2+2 i b \text{Li}_2\left(\frac{c x+i}{c x-i}\right) \left(a+b \tan ^{-1}(c x)\right)+b^2 \text{Li}_3\left(\frac{c x+i}{c x-i}\right)\right)}{2 c d}","-\frac{b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{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{i b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 c d}",1,"((I/2)*(2*(a + b*ArcTan[c*x])^2*Log[(2*d)/(d + I*c*d*x)] + (2*I)*b*(a + b*ArcTan[c*x])*PolyLog[2, (I + c*x)/(-I + c*x)] + b^2*PolyLog[3, (I + c*x)/(-I + c*x)]))/(c*d)","A",1
99,1,113,88,0.1224422,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x (d+i c d x)} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x*(d + I*c*d*x)),x]","\frac{2 i b \text{Li}_2\left(\frac{c x+i}{i-c x}\right) \left(a+b \tan ^{-1}(c x)\right)+2 \left(\log \left(\frac{2 i}{-c x+i}\right)+2 \tanh ^{-1}\left(\frac{c x+i}{c x-i}\right)\right) \left(a+b \tan ^{-1}(c x)\right)^2+b^2 \text{Li}_3\left(\frac{c x+i}{i-c x}\right)}{2 d}","\frac{i b \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{\log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{d}+\frac{b^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)}{2 d}",1,"(2*(a + b*ArcTan[c*x])^2*(2*ArcTanh[(I + c*x)/(-I + c*x)] + Log[(2*I)/(I - c*x)]) + (2*I)*b*(a + b*ArcTan[c*x])*PolyLog[2, (I + c*x)/(I - c*x)] + b^2*PolyLog[3, (I + c*x)/(I - c*x)])/(2*d)","A",0
100,1,265,186,0.9241684,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^2 (d+i c d x)} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x^2*(d + I*c*d*x)),x]","-\frac{-i a^2 c \log \left(c^2 x^2+1\right)+2 i a^2 c \log (x)+2 a^2 c \tan ^{-1}(c x)+\frac{2 a^2}{x}+2 a b c \left(2 \left(-\log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+\tan ^{-1}(c x)^2+\tan ^{-1}(c x) \left(\frac{1}{c x}+i \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)\right)\right)+\text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)\right)+2 i b^2 c \left(i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+\text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-\frac{i \tan ^{-1}(c x)^2}{c x}+\tan ^{-1}(c x)^2+\tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)+2 i \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)-\frac{i \pi ^3}{24}\right)}{2 d}","\frac{b c \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{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{i b^2 c \text{Li}_2\left(\frac{2}{1-i c x}-1\right)}{d}-\frac{i b^2 c \text{Li}_3\left(\frac{2}{i c x+1}-1\right)}{2 d}",1,"-1/2*((2*a^2)/x + 2*a^2*c*ArcTan[c*x] + (2*I)*a^2*c*Log[x] - I*a^2*c*Log[1 + c^2*x^2] + 2*a*b*c*(2*(ArcTan[c*x]^2 + ArcTan[c*x]*(1/(c*x) + I*Log[1 - E^((2*I)*ArcTan[c*x])]) - Log[(c*x)/Sqrt[1 + c^2*x^2]]) + PolyLog[2, E^((2*I)*ArcTan[c*x])]) + (2*I)*b^2*c*((-1/24*I)*Pi^3 + ArcTan[c*x]^2 - (I*ArcTan[c*x]^2)/(c*x) + ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + (2*I)*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + PolyLog[2, E^((2*I)*ArcTan[c*x])] + PolyLog[3, E^((-2*I)*ArcTan[c*x])]/2))/d","A",0
101,1,372,273,1.2093354,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^3 (d+i c d x)} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x^3*(d + I*c*d*x)),x]","\frac{a^2 c^2 \log \left(c^2 x^2+1\right)-2 a^2 c^2 \log (x)+2 i a^2 c^2 \tan ^{-1}(c x)+\frac{2 i a^2 c}{x}-\frac{a^2}{x^2}+\frac{2 i a b \left(c^2 x^2 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+c x \left(-2 c x \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+i\right)+2 c^2 x^2 \tan ^{-1}(c x)^2+\tan ^{-1}(c x) \left(i c^2 x^2+2 i c^2 x^2 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+2 c x+i\right)\right)}{x^2}+2 b^2 c^2 \left(\log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)-\frac{\tan ^{-1}(c x)^2}{2 c^2 x^2}-i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)-\text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)+\frac{i \tan ^{-1}(c x)^2}{c x}-\frac{3}{2} \tan ^{-1}(c x)^2-\frac{\tan ^{-1}(c x)}{c x}+\tan ^{-1}(c x)^2 \left(-\log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)\right)-2 i \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+\frac{i \pi ^3}{24}\right)}{2 d}","-\frac{i b c^2 \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{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 \text{Li}_2\left(\frac{2}{1-i c x}-1\right)}{d}-\frac{b^2 c^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)}{2 d}-\frac{b^2 c^2 \log \left(c^2 x^2+1\right)}{2 d}+\frac{b^2 c^2 \log (x)}{d}",1,"(-(a^2/x^2) + ((2*I)*a^2*c)/x + (2*I)*a^2*c^2*ArcTan[c*x] - 2*a^2*c^2*Log[x] + a^2*c^2*Log[1 + c^2*x^2] + ((2*I)*a*b*(2*c^2*x^2*ArcTan[c*x]^2 + ArcTan[c*x]*(I + 2*c*x + I*c^2*x^2 + (2*I)*c^2*x^2*Log[1 - E^((2*I)*ArcTan[c*x])]) + c*x*(I - 2*c*x*Log[(c*x)/Sqrt[1 + c^2*x^2]]) + c^2*x^2*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/x^2 + 2*b^2*c^2*((I/24)*Pi^3 - ArcTan[c*x]/(c*x) - (3*ArcTan[c*x]^2)/2 - ArcTan[c*x]^2/(2*c^2*x^2) + (I*ArcTan[c*x]^2)/(c*x) - ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - (2*I)*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + Log[(c*x)/Sqrt[1 + c^2*x^2]] - I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - PolyLog[2, E^((2*I)*ArcTan[c*x])] - PolyLog[3, E^((-2*I)*ArcTan[c*x])]/2))/(2*d)","A",0
102,1,535,365,1.2099761,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^4 (d+i c d x)} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x^4*(d + I*c*d*x)),x]","\frac{i a^2 c^3 \log (x)}{d}+\frac{a^2 c^3 \tan ^{-1}(c x)}{d}+\frac{a^2 c^2}{d x}-\frac{i a^2 c^3 \log \left(c^2 x^2+1\right)}{2 d}+\frac{i a^2 c}{2 d x^2}-\frac{a^2}{3 d x^3}-\frac{2 i a b c^3 \left(-\frac{i \left(c^2 x^2+1\right)}{6 c^2 x^2}-\frac{4}{3} i \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)-\frac{\left(c^2 x^2+1\right) \tan ^{-1}(c x)}{2 c^2 x^2}-\frac{i \left(c^2 x^2+1\right) \tan ^{-1}(c x)}{3 c^3 x^3}+\frac{1}{2} i \left(\tan ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)\right)-\frac{1}{2 c x}+\frac{1}{2} i \tan ^{-1}(c x)^2+\frac{4 i \tan ^{-1}(c x)}{3 c x}-\tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)\right)}{d}+\frac{b^2 c^3 \left(-24 i \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+\frac{12 i \left(c^2 x^2+1\right) \tan ^{-1}(c x)^2}{c^2 x^2}-\frac{8 \left(c^2 x^2+1\right) \tan ^{-1}(c x)}{c^2 x^2}-\frac{8 \left(c^2 x^2+1\right) \tan ^{-1}(c x)^2}{c^3 x^3}-24 \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+32 i \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+12 i \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-\frac{8}{c x}+\frac{32 \tan ^{-1}(c x)^2}{c x}+32 i \tan ^{-1}(c x)^2+\frac{24 i \tan ^{-1}(c x)}{c x}+24 i \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-64 \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+\pi ^3\right)}{24 d}","-\frac{b c^3 \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{11 i c^3 \left(a+b \tan ^{-1}(c x)\right)^2}{6 d}-\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{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{\left(a+b \tan ^{-1}(c x)\right)^2}{3 d x^3}+\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{4 i b^2 c^3 \text{Li}_2\left(\frac{2}{1-i c x}-1\right)}{3 d}+\frac{i b^2 c^3 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)}{2 d}-\frac{i b^2 c^3 \log (x)}{d}-\frac{b^2 c^3 \tan ^{-1}(c x)}{3 d}-\frac{b^2 c^2}{3 d x}+\frac{i b^2 c^3 \log \left(c^2 x^2+1\right)}{2 d}",1,"-1/3*a^2/(d*x^3) + ((I/2)*a^2*c)/(d*x^2) + (a^2*c^2)/(d*x) + (a^2*c^3*ArcTan[c*x])/d + (I*a^2*c^3*Log[x])/d - ((I/2)*a^2*c^3*Log[1 + c^2*x^2])/d - ((2*I)*a*b*c^3*(-1/2*1/(c*x) - ((I/6)*(1 + c^2*x^2))/(c^2*x^2) + (((4*I)/3)*ArcTan[c*x])/(c*x) - ((I/3)*(1 + c^2*x^2)*ArcTan[c*x])/(c^3*x^3) - ((1 + c^2*x^2)*ArcTan[c*x])/(2*c^2*x^2) + (I/2)*ArcTan[c*x]^2 - ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - ((4*I)/3)*Log[(c*x)/Sqrt[1 + c^2*x^2]] + (I/2)*(ArcTan[c*x]^2 + PolyLog[2, E^((2*I)*ArcTan[c*x])])))/d + (b^2*c^3*(Pi^3 - 8/(c*x) + ((24*I)*ArcTan[c*x])/(c*x) - (8*(1 + c^2*x^2)*ArcTan[c*x])/(c^2*x^2) + (32*I)*ArcTan[c*x]^2 + (32*ArcTan[c*x]^2)/(c*x) - (8*(1 + c^2*x^2)*ArcTan[c*x]^2)/(c^3*x^3) + ((12*I)*(1 + c^2*x^2)*ArcTan[c*x]^2)/(c^2*x^2) + (24*I)*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 64*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - (24*I)*Log[(c*x)/Sqrt[1 + c^2*x^2]] - 24*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + (32*I)*PolyLog[2, E^((2*I)*ArcTan[c*x])] + (12*I)*PolyLog[3, E^((-2*I)*ArcTan[c*x])]))/(24*d)","A",0
103,1,502,433,2.5442099,"\int \frac{x^4 \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^2} \, dx","Integrate[(x^4*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^2,x]","-\frac{4 a^2 c^3 x^3+12 i a^2 c^2 x^2-24 i a^2 \log \left(c^2 x^2+1\right)-36 a^2 c x-\frac{12 a^2}{c x-i}+48 a^2 \tan ^{-1}(c x)+2 a b \left(-2 c^2 x^2+20 \log \left(c^2 x^2+1\right)+2 \tan ^{-1}(c x) \left(2 c^3 x^3+6 i c^2 x^2-18 c x+24 i \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-3 \sin \left(2 \tan ^{-1}(c x)\right)-3 i \cos \left(2 \tan ^{-1}(c x)\right)+6 i\right)+24 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-12 i c x+48 \tan ^{-1}(c x)^2+3 i \sin \left(2 \tan ^{-1}(c x)\right)-3 \cos \left(2 \tan ^{-1}(c x)\right)-2\right)+b^2 \left(4 c^3 x^3 \tan ^{-1}(c x)^2+12 i \log \left(c^2 x^2+1\right)+12 i c^2 x^2 \tan ^{-1}(c x)^2-4 c^2 x^2 \tan ^{-1}(c x)+8 \left(6 \tan ^{-1}(c x)+5 i\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+24 i \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+4 c x-36 c x \tan ^{-1}(c x)^2-24 i c x \tan ^{-1}(c x)+32 \tan ^{-1}(c x)^3+52 i \tan ^{-1}(c x)^2-4 \tan ^{-1}(c x)+48 i \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-80 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-6 \tan ^{-1}(c x)^2 \sin \left(2 \tan ^{-1}(c x)\right)+6 i \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)+3 \sin \left(2 \tan ^{-1}(c x)\right)-6 i \tan ^{-1}(c x)^2 \cos \left(2 \tan ^{-1}(c x)\right)-6 \tan ^{-1}(c x) \cos \left(2 \tan ^{-1}(c x)\right)+3 i \cos \left(2 \tan ^{-1}(c x)\right)\right)}{12 c^5 d^2}","\frac{4 b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^5 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{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{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{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^2 d^2}+\frac{10 i b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{3 c^5 d^2}-\frac{2 i b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{c^5 d^2}+\frac{b^2}{2 c^5 d^2 (-c x+i)}-\frac{b^2 \tan ^{-1}(c x)}{6 c^5 d^2}-\frac{b^2 x}{3 c^4 d^2}+\frac{2 i b^2 x \tan ^{-1}(c x)}{c^4 d^2}-\frac{i b^2 \log \left(c^2 x^2+1\right)}{c^5 d^2}",1,"-1/12*(-36*a^2*c*x + (12*I)*a^2*c^2*x^2 + 4*a^2*c^3*x^3 - (12*a^2)/(-I + c*x) + 48*a^2*ArcTan[c*x] - (24*I)*a^2*Log[1 + c^2*x^2] + 2*a*b*(-2 - (12*I)*c*x - 2*c^2*x^2 + 48*ArcTan[c*x]^2 - 3*Cos[2*ArcTan[c*x]] + 20*Log[1 + c^2*x^2] + 24*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 2*ArcTan[c*x]*(6*I - 18*c*x + (6*I)*c^2*x^2 + 2*c^3*x^3 - (3*I)*Cos[2*ArcTan[c*x]] + (24*I)*Log[1 + E^((2*I)*ArcTan[c*x])] - 3*Sin[2*ArcTan[c*x]]) + (3*I)*Sin[2*ArcTan[c*x]]) + b^2*(4*c*x - 4*ArcTan[c*x] - (24*I)*c*x*ArcTan[c*x] - 4*c^2*x^2*ArcTan[c*x] + (52*I)*ArcTan[c*x]^2 - 36*c*x*ArcTan[c*x]^2 + (12*I)*c^2*x^2*ArcTan[c*x]^2 + 4*c^3*x^3*ArcTan[c*x]^2 + 32*ArcTan[c*x]^3 + (3*I)*Cos[2*ArcTan[c*x]] - 6*ArcTan[c*x]*Cos[2*ArcTan[c*x]] - (6*I)*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] - 80*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + (48*I)*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (12*I)*Log[1 + c^2*x^2] + 8*(5*I + 6*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (24*I)*PolyLog[3, -E^((2*I)*ArcTan[c*x])] + 3*Sin[2*ArcTan[c*x]] + (6*I)*ArcTan[c*x]*Sin[2*ArcTan[c*x]] - 6*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]]))/(c^5*d^2)","A",0
104,1,429,364,1.9791184,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^2} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^2,x]","-\frac{2 a^2 c^2 x^2-6 a^2 \log \left(c^2 x^2+1\right)+8 i a^2 c x+\frac{4 i a^2}{c x-i}-12 i a^2 \tan ^{-1}(c x)+2 a b \left(-4 i \log \left(c^2 x^2+1\right)+2 \tan ^{-1}(c x) \left(c^2 x^2+4 i c x+6 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+i \sin \left(2 \tan ^{-1}(c x)\right)-\cos \left(2 \tan ^{-1}(c x)\right)+1\right)-6 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-2 c x-12 i \tan ^{-1}(c x)^2+\sin \left(2 \tan ^{-1}(c x)\right)+i \cos \left(2 \tan ^{-1}(c x)\right)\right)+b^2 \left(2 \log \left(c^2 x^2+1\right)+2 c^2 x^2 \tan ^{-1}(c x)^2+4 \left(2-3 i \tan ^{-1}(c x)\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+6 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)-8 i \tan ^{-1}(c x)^3+8 i c x \tan ^{-1}(c x)^2+10 \tan ^{-1}(c x)^2-4 c x \tan ^{-1}(c x)+12 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+16 i \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+2 i \tan ^{-1}(c x)^2 \sin \left(2 \tan ^{-1}(c x)\right)+2 \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)-i \sin \left(2 \tan ^{-1}(c x)\right)-2 \tan ^{-1}(c x)^2 \cos \left(2 \tan ^{-1}(c x)\right)+2 i \tan ^{-1}(c x) \cos \left(2 \tan ^{-1}(c x)\right)+\cos \left(2 \tan ^{-1}(c x)\right)\right)}{4 c^4 d^2}","-\frac{3 i b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^2}+\frac{b \left(a+b \tan ^{-1}(c x)\right)}{c^4 d^2 (-c x+i)}+\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{a b x}{c^3 d^2}-\frac{2 i x \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^2}-\frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 d^2}+\frac{2 b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c^4 d^2}-\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 c^4 d^2}-\frac{i b^2}{2 c^4 d^2 (-c x+i)}+\frac{i b^2 \tan ^{-1}(c x)}{2 c^4 d^2}+\frac{b^2 x \tan ^{-1}(c x)}{c^3 d^2}-\frac{b^2 \log \left(c^2 x^2+1\right)}{2 c^4 d^2}",1,"-1/4*((8*I)*a^2*c*x + 2*a^2*c^2*x^2 + ((4*I)*a^2)/(-I + c*x) - (12*I)*a^2*ArcTan[c*x] - 6*a^2*Log[1 + c^2*x^2] + 2*a*b*(-2*c*x - (12*I)*ArcTan[c*x]^2 + I*Cos[2*ArcTan[c*x]] - (4*I)*Log[1 + c^2*x^2] - (6*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 2*ArcTan[c*x]*(1 + (4*I)*c*x + c^2*x^2 - Cos[2*ArcTan[c*x]] + 6*Log[1 + E^((2*I)*ArcTan[c*x])] + I*Sin[2*ArcTan[c*x]]) + Sin[2*ArcTan[c*x]]) + b^2*(-4*c*x*ArcTan[c*x] + 10*ArcTan[c*x]^2 + (8*I)*c*x*ArcTan[c*x]^2 + 2*c^2*x^2*ArcTan[c*x]^2 - (8*I)*ArcTan[c*x]^3 + Cos[2*ArcTan[c*x]] + (2*I)*ArcTan[c*x]*Cos[2*ArcTan[c*x]] - 2*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + (16*I)*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 12*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + 2*Log[1 + c^2*x^2] + 4*(2 - (3*I)*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 6*PolyLog[3, -E^((2*I)*ArcTan[c*x])] - I*Sin[2*ArcTan[c*x]] + 2*ArcTan[c*x]*Sin[2*ArcTan[c*x]] + (2*I)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]]))/(c^4*d^2)","A",0
105,1,362,292,1.3845819,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^2} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^2,x]","-\frac{12 i a^2 \log \left(c^2 x^2+1\right)+12 a^2 c x+\frac{12 a^2}{c x-i}-24 a^2 \tan ^{-1}(c x)+6 a b \left(-2 \log \left(c^2 x^2+1\right)-4 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-8 \tan ^{-1}(c x)^2-i \sin \left(2 \tan ^{-1}(c x)\right)+\cos \left(2 \tan ^{-1}(c x)\right)+2 \tan ^{-1}(c x) \left(2 c x-4 i \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+\sin \left(2 \tan ^{-1}(c x)\right)+i \cos \left(2 \tan ^{-1}(c x)\right)\right)\right)+b^2 \left(-12 \left(2 \tan ^{-1}(c x)+i\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-12 i \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)-16 \tan ^{-1}(c x)^3+12 c x \tan ^{-1}(c x)^2-12 i \tan ^{-1}(c x)^2-24 i \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+24 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+6 \tan ^{-1}(c x)^2 \sin \left(2 \tan ^{-1}(c x)\right)-6 i \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)-3 \sin \left(2 \tan ^{-1}(c x)\right)+6 i \tan ^{-1}(c x)^2 \cos \left(2 \tan ^{-1}(c x)\right)+6 \tan ^{-1}(c x) \cos \left(2 \tan ^{-1}(c x)\right)-3 i \cos \left(2 \tan ^{-1}(c x)\right)\right)}{12 c^3 d^2}","-\frac{2 b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(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{\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{x \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d^2}-\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c^3 d^2}+\frac{i b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{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,"-1/12*(12*a^2*c*x + (12*a^2)/(-I + c*x) - 24*a^2*ArcTan[c*x] + (12*I)*a^2*Log[1 + c^2*x^2] + b^2*((-12*I)*ArcTan[c*x]^2 + 12*c*x*ArcTan[c*x]^2 - 16*ArcTan[c*x]^3 - (3*I)*Cos[2*ArcTan[c*x]] + 6*ArcTan[c*x]*Cos[2*ArcTan[c*x]] + (6*I)*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + 24*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - (24*I)*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - 12*(I + 2*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (12*I)*PolyLog[3, -E^((2*I)*ArcTan[c*x])] - 3*Sin[2*ArcTan[c*x]] - (6*I)*ArcTan[c*x]*Sin[2*ArcTan[c*x]] + 6*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]]) + 6*a*b*(-8*ArcTan[c*x]^2 + Cos[2*ArcTan[c*x]] - 2*Log[1 + c^2*x^2] - 4*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - I*Sin[2*ArcTan[c*x]] + 2*ArcTan[c*x]*(2*c*x + I*Cos[2*ArcTan[c*x]] - (4*I)*Log[1 + E^((2*I)*ArcTan[c*x])] + Sin[2*ArcTan[c*x]])))/(c^3*d^2)","A",0
106,1,300,216,0.8480101,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^2} \, dx","Integrate[(x*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^2,x]","\frac{-6 a^2 \log \left(c^2 x^2+1\right)+\frac{12 i a^2}{c x-i}-12 i a^2 \tan ^{-1}(c x)-6 i a b \left(2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+4 \tan ^{-1}(c x)^2+i \sin \left(2 \tan ^{-1}(c x)\right)-\cos \left(2 \tan ^{-1}(c x)\right)-2 i \tan ^{-1}(c x) \left(-2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-i \sin \left(2 \tan ^{-1}(c x)\right)+\cos \left(2 \tan ^{-1}(c x)\right)\right)\right)+b^2 \left(-12 i \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+6 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)-8 i \tan ^{-1}(c x)^3+12 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+6 i \tan ^{-1}(c x)^2 \sin \left(2 \tan ^{-1}(c x)\right)+6 \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)-3 i \sin \left(2 \tan ^{-1}(c x)\right)-6 \tan ^{-1}(c x)^2 \cos \left(2 \tan ^{-1}(c x)\right)+6 i \tan ^{-1}(c x) \cos \left(2 \tan ^{-1}(c x)\right)+3 \cos \left(2 \tan ^{-1}(c x)\right)\right)}{12 c^2 d^2}","\frac{i b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{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{b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\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,"(((12*I)*a^2)/(-I + c*x) - (12*I)*a^2*ArcTan[c*x] - 6*a^2*Log[1 + c^2*x^2] - (6*I)*a*b*(4*ArcTan[c*x]^2 - Cos[2*ArcTan[c*x]] + 2*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (2*I)*ArcTan[c*x]*(Cos[2*ArcTan[c*x]] - 2*Log[1 + E^((2*I)*ArcTan[c*x])] - I*Sin[2*ArcTan[c*x]]) + I*Sin[2*ArcTan[c*x]]) + b^2*((-8*I)*ArcTan[c*x]^3 + 3*Cos[2*ArcTan[c*x]] + (6*I)*ArcTan[c*x]*Cos[2*ArcTan[c*x]] - 6*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + 12*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - (12*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 6*PolyLog[3, -E^((2*I)*ArcTan[c*x])] - (3*I)*Sin[2*ArcTan[c*x]] + 6*ArcTan[c*x]*Sin[2*ArcTan[c*x]] + (6*I)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]]))/(12*c^2*d^2)","A",0
107,1,72,122,0.197591,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^2} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(d + I*c*d*x)^2,x]","-\frac{-2 a^2+b (b+2 i a) (c x+i) \tan ^{-1}(c x)+2 i a b+b^2 (-1+i c x) \tan ^{-1}(c x)^2+b^2}{2 c d^2 (c x-i)}","\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,"-1/2*(-2*a^2 + (2*I)*a*b + b^2 + b*((2*I)*a + b)*(I + c*x)*ArcTan[c*x] + b^2*(-1 + I*c*x)*ArcTan[c*x]^2)/(c*d^2*(-I + c*x))","A",1
108,1,299,221,1.0428181,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x (d+i c d x)^2} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x*(d + I*c*d*x)^2),x]","\frac{-12 a^2 \log \left(c^2 x^2+1\right)-\frac{24 i a^2}{c x-i}+24 a^2 \log (c x)-24 i a^2 \tan ^{-1}(c x)-12 a b \left(2 i \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+4 i \tan ^{-1}(c x)^2+\sin \left(2 \tan ^{-1}(c x)\right)+i \cos \left(2 \tan ^{-1}(c x)\right)-2 \tan ^{-1}(c x) \left(2 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)-i \sin \left(2 \tan ^{-1}(c x)\right)+\cos \left(2 \tan ^{-1}(c x)\right)\right)\right)+b^2 \left(24 i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+12 \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)+24 \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-12 i \tan ^{-1}(c x)^2 \sin \left(2 \tan ^{-1}(c x)\right)-12 \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)+6 i \sin \left(2 \tan ^{-1}(c x)\right)+12 \tan ^{-1}(c x)^2 \cos \left(2 \tan ^{-1}(c x)\right)-12 i \tan ^{-1}(c x) \cos \left(2 \tan ^{-1}(c x)\right)-6 \cos \left(2 \tan ^{-1}(c x)\right)-i \pi ^3\right)}{24 d^2}","\frac{i b \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{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{b^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\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,"(((-24*I)*a^2)/(-I + c*x) - (24*I)*a^2*ArcTan[c*x] + 24*a^2*Log[c*x] - 12*a^2*Log[1 + c^2*x^2] - 12*a*b*((4*I)*ArcTan[c*x]^2 + I*Cos[2*ArcTan[c*x]] + (2*I)*PolyLog[2, E^((2*I)*ArcTan[c*x])] - 2*ArcTan[c*x]*(Cos[2*ArcTan[c*x]] + 2*Log[1 - E^((2*I)*ArcTan[c*x])] - I*Sin[2*ArcTan[c*x]]) + Sin[2*ArcTan[c*x]]) + b^2*((-I)*Pi^3 - 6*Cos[2*ArcTan[c*x]] - (12*I)*ArcTan[c*x]*Cos[2*ArcTan[c*x]] + 12*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + 24*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + (24*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[c*x])] + (6*I)*Sin[2*ArcTan[c*x]] - 12*ArcTan[c*x]*Sin[2*ArcTan[c*x]] - (12*I)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]]))/(24*d^2)","A",0
109,1,398,306,2.7043432,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^2 (d+i c d x)^2} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x^2*(d + I*c*d*x)^2),x]","-\frac{-12 i a^2 c \log \left(c^2 x^2+1\right)+\frac{12 a^2 c}{c x-i}+24 i a^2 c \log (c x)+24 a^2 c \tan ^{-1}(c x)+\frac{12 a^2}{x}+6 a b c \left(-4 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+4 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+8 \tan ^{-1}(c x)^2-i \sin \left(2 \tan ^{-1}(c x)\right)+\cos \left(2 \tan ^{-1}(c x)\right)+\tan ^{-1}(c x) \left(\frac{4}{c x}+8 i \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+2 \sin \left(2 \tan ^{-1}(c x)\right)+2 i \cos \left(2 \tan ^{-1}(c x)\right)\right)\right)+b^2 c \left(-24 \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+12 i \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+12 i \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)+\frac{12 \tan ^{-1}(c x)^2}{c x}+12 i \tan ^{-1}(c x)^2+24 i \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-24 \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+6 \tan ^{-1}(c x)^2 \sin \left(2 \tan ^{-1}(c x)\right)-6 i \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)-3 \sin \left(2 \tan ^{-1}(c x)\right)+6 i \tan ^{-1}(c x)^2 \cos \left(2 \tan ^{-1}(c x)\right)+6 \tan ^{-1}(c x) \cos \left(2 \tan ^{-1}(c x)\right)-3 i \cos \left(2 \tan ^{-1}(c x)\right)+\pi ^3\right)}{12 d^2}","\frac{2 b c \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(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{i b^2 c \text{Li}_2\left(\frac{2}{1-i c x}-1\right)}{d^2}-\frac{i b^2 c \text{Li}_3\left(\frac{2}{i c x+1}-1\right)}{d^2}-\frac{b^2 c}{2 d^2 (-c x+i)}+\frac{b^2 c \tan ^{-1}(c x)}{2 d^2}",1,"-1/12*((12*a^2)/x + (12*a^2*c)/(-I + c*x) + 24*a^2*c*ArcTan[c*x] + (24*I)*a^2*c*Log[c*x] - (12*I)*a^2*c*Log[1 + c^2*x^2] + b^2*c*(Pi^3 + (12*I)*ArcTan[c*x]^2 + (12*ArcTan[c*x]^2)/(c*x) - (3*I)*Cos[2*ArcTan[c*x]] + 6*ArcTan[c*x]*Cos[2*ArcTan[c*x]] + (6*I)*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + (24*I)*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 24*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - 24*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + (12*I)*PolyLog[2, E^((2*I)*ArcTan[c*x])] + (12*I)*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - 3*Sin[2*ArcTan[c*x]] - (6*I)*ArcTan[c*x]*Sin[2*ArcTan[c*x]] + 6*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]]) + 6*a*b*c*(8*ArcTan[c*x]^2 + Cos[2*ArcTan[c*x]] - 4*Log[(c*x)/Sqrt[1 + c^2*x^2]] + 4*PolyLog[2, E^((2*I)*ArcTan[c*x])] - I*Sin[2*ArcTan[c*x]] + ArcTan[c*x]*(4/(c*x) + (2*I)*Cos[2*ArcTan[c*x]] + (8*I)*Log[1 - E^((2*I)*ArcTan[c*x])] + 2*Sin[2*ArcTan[c*x]])))/d^2","A",0
110,1,491,403,3.1361967,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^3 (d+i c d x)^2} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x^3*(d + I*c*d*x)^2),x]","\frac{12 a^2 c^2 \log \left(c^2 x^2+1\right)+\frac{8 i a^2 c^2}{c x-i}-24 a^2 c^2 \log (x)+24 i a^2 c^2 \tan ^{-1}(c x)+\frac{16 i a^2 c}{x}-\frac{4 a^2}{x^2}+4 i a b c^2 \left(-8 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+2 \tan ^{-1}(c x) \left(\frac{i}{c^2 x^2}+\frac{4}{c x}+6 i \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+\sin \left(2 \tan ^{-1}(c x)\right)+i \cos \left(2 \tan ^{-1}(c x)\right)+i\right)+6 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+\frac{2 i}{c x}+12 \tan ^{-1}(c x)^2-i \sin \left(2 \tan ^{-1}(c x)\right)+\cos \left(2 \tan ^{-1}(c x)\right)\right)-b^2 c^2 \left(-8 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+\frac{4 \tan ^{-1}(c x)^2}{c^2 x^2}+24 i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+16 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+12 \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-\frac{16 i \tan ^{-1}(c x)^2}{c x}+20 \tan ^{-1}(c x)^2+\frac{8 \tan ^{-1}(c x)}{c x}+24 \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)+32 i \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)-4 i \tan ^{-1}(c x)^2 \sin \left(2 \tan ^{-1}(c x)\right)-4 \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)+2 i \sin \left(2 \tan ^{-1}(c x)\right)+4 \tan ^{-1}(c x)^2 \cos \left(2 \tan ^{-1}(c x)\right)-4 i \tan ^{-1}(c x) \cos \left(2 \tan ^{-1}(c x)\right)-2 \cos \left(2 \tan ^{-1}(c x)\right)-i \pi ^3\right)}{8 d^2}","-\frac{3 i b c^2 \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{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{2 b^2 c^2 \text{Li}_2\left(\frac{2}{1-i c x}-1\right)}{d^2}-\frac{3 b^2 c^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)}{2 d^2}-\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,"((-4*a^2)/x^2 + ((16*I)*a^2*c)/x + ((8*I)*a^2*c^2)/(-I + c*x) + (24*I)*a^2*c^2*ArcTan[c*x] - 24*a^2*c^2*Log[x] + 12*a^2*c^2*Log[1 + c^2*x^2] - b^2*c^2*((-I)*Pi^3 + (8*ArcTan[c*x])/(c*x) + 20*ArcTan[c*x]^2 + (4*ArcTan[c*x]^2)/(c^2*x^2) - ((16*I)*ArcTan[c*x]^2)/(c*x) - 2*Cos[2*ArcTan[c*x]] - (4*I)*ArcTan[c*x]*Cos[2*ArcTan[c*x]] + 4*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + 24*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + (32*I)*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - 8*Log[(c*x)/Sqrt[1 + c^2*x^2]] + (24*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + 16*PolyLog[2, E^((2*I)*ArcTan[c*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[c*x])] + (2*I)*Sin[2*ArcTan[c*x]] - 4*ArcTan[c*x]*Sin[2*ArcTan[c*x]] - (4*I)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]]) + (4*I)*a*b*c^2*((2*I)/(c*x) + 12*ArcTan[c*x]^2 + Cos[2*ArcTan[c*x]] - 8*Log[(c*x)/Sqrt[1 + c^2*x^2]] + 6*PolyLog[2, E^((2*I)*ArcTan[c*x])] - I*Sin[2*ArcTan[c*x]] + 2*ArcTan[c*x]*(I + I/(c^2*x^2) + 4/(c*x) + I*Cos[2*ArcTan[c*x]] + (6*I)*Log[1 - E^((2*I)*ArcTan[c*x])] + Sin[2*ArcTan[c*x]])))/(8*d^2)","A",0
111,1,578,462,2.6687311,"\int \frac{x^4 \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^3} \, dx","Integrate[(x^4*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3,x]","\frac{8 i a^2 c^2 x^2-48 i a^2 \log \left(c^2 x^2+1\right)-48 a^2 c x-\frac{64 a^2}{c x-i}-\frac{8 i a^2}{(c x-i)^2}+96 a^2 \tan ^{-1}(c x)+a b \left(48 \log \left(c^2 x^2+1\right)+4 i \tan ^{-1}(c x) \left(4 c^2 x^2+24 i c x+48 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+14 i \sin \left(2 \tan ^{-1}(c x)\right)-i \sin \left(4 \tan ^{-1}(c x)\right)-14 \cos \left(2 \tan ^{-1}(c x)\right)+\cos \left(4 \tan ^{-1}(c x)\right)+4\right)+96 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-16 i c x+192 \tan ^{-1}(c x)^2+28 i \sin \left(2 \tan ^{-1}(c x)\right)-i \sin \left(4 \tan ^{-1}(c x)\right)-28 \cos \left(2 \tan ^{-1}(c x)\right)+\cos \left(4 \tan ^{-1}(c x)\right)\right)+16 i b^2 \left(\frac{1}{2} \log \left(c^2 x^2+1\right)+\frac{1}{2} \left(c^2 x^2+1\right) \tan ^{-1}(c x)^2+\left(3-6 i \tan ^{-1}(c x)\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+3 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)-4 i \tan ^{-1}(c x)^3+3 i c x \tan ^{-1}(c x)^2+3 \tan ^{-1}(c x)^2-c x \tan ^{-1}(c x)+6 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+6 i \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+\frac{7}{4} i \tan ^{-1}(c x)^2 \sin \left(2 \tan ^{-1}(c x)\right)-\frac{1}{8} i \tan ^{-1}(c x)^2 \sin \left(4 \tan ^{-1}(c x)\right)+\frac{7}{4} \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)-\frac{1}{16} \tan ^{-1}(c x) \sin \left(4 \tan ^{-1}(c x)\right)-\frac{7}{8} i \sin \left(2 \tan ^{-1}(c x)\right)+\frac{1}{64} i \sin \left(4 \tan ^{-1}(c x)\right)+\frac{1}{8} \tan ^{-1}(c x)^2 \cos \left(4 \tan ^{-1}(c x)\right)-\frac{1}{16} i \tan ^{-1}(c x) \cos \left(4 \tan ^{-1}(c x)\right)-\frac{7}{8} \left(2 \tan ^{-1}(c x)^2-2 i \tan ^{-1}(c x)-1\right) \cos \left(2 \tan ^{-1}(c x)\right)-\frac{1}{64} \cos \left(4 \tan ^{-1}(c x)\right)\right)}{16 c^5 d^3}","-\frac{6 b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^5 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{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 a b x}{c^4 d^3}-\frac{3 x \left(a+b \tan ^{-1}(c x)\right)^2}{c^4 d^3}+\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 d^3}-\frac{3 i b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c^5 d^3}+\frac{3 i b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{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{29 b^2 \tan ^{-1}(c x)}{16 c^5 d^3}-\frac{i b^2 x \tan ^{-1}(c x)}{c^4 d^3}+\frac{i b^2 \log \left(c^2 x^2+1\right)}{2 c^5 d^3}",1,"(-48*a^2*c*x + (8*I)*a^2*c^2*x^2 - ((8*I)*a^2)/(-I + c*x)^2 - (64*a^2)/(-I + c*x) + 96*a^2*ArcTan[c*x] - (48*I)*a^2*Log[1 + c^2*x^2] + a*b*((-16*I)*c*x + 192*ArcTan[c*x]^2 - 28*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] + 48*Log[1 + c^2*x^2] + 96*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (28*I)*Sin[2*ArcTan[c*x]] + (4*I)*ArcTan[c*x]*(4 + (24*I)*c*x + 4*c^2*x^2 - 14*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] + 48*Log[1 + E^((2*I)*ArcTan[c*x])] + (14*I)*Sin[2*ArcTan[c*x]] - I*Sin[4*ArcTan[c*x]]) - I*Sin[4*ArcTan[c*x]]) + (16*I)*b^2*(-(c*x*ArcTan[c*x]) + 3*ArcTan[c*x]^2 + (3*I)*c*x*ArcTan[c*x]^2 + ((1 + c^2*x^2)*ArcTan[c*x]^2)/2 - (4*I)*ArcTan[c*x]^3 - (7*(-1 - (2*I)*ArcTan[c*x] + 2*ArcTan[c*x]^2)*Cos[2*ArcTan[c*x]])/8 - Cos[4*ArcTan[c*x]]/64 - (I/16)*ArcTan[c*x]*Cos[4*ArcTan[c*x]] + (ArcTan[c*x]^2*Cos[4*ArcTan[c*x]])/8 + (6*I)*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 6*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + Log[1 + c^2*x^2]/2 + (3 - (6*I)*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])] - ((7*I)/8)*Sin[2*ArcTan[c*x]] + (7*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/4 + ((7*I)/4)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]] + (I/64)*Sin[4*ArcTan[c*x]] - (ArcTan[c*x]*Sin[4*ArcTan[c*x]])/16 - (I/8)*ArcTan[c*x]^2*Sin[4*ArcTan[c*x]]))/(16*c^5*d^3)","A",0
112,1,507,383,1.6574551,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^3} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3,x]","\frac{-96 a^2 \log \left(c^2 x^2+1\right)+64 i a^2 c x+\frac{192 i a^2}{c x-i}-\frac{32 a^2}{(c x-i)^2}-192 i a^2 \tan ^{-1}(c x)+4 i a b \left(-16 \log \left(c^2 x^2+1\right)-48 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-96 \tan ^{-1}(c x)^2-20 i \sin \left(2 \tan ^{-1}(c x)\right)+i \sin \left(4 \tan ^{-1}(c x)\right)+20 \cos \left(2 \tan ^{-1}(c x)\right)-\cos \left(4 \tan ^{-1}(c x)\right)+4 \tan ^{-1}(c x) \left(8 c x-24 i \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+10 \sin \left(2 \tan ^{-1}(c x)\right)-\sin \left(4 \tan ^{-1}(c x)\right)+10 i \cos \left(2 \tan ^{-1}(c x)\right)-i \cos \left(4 \tan ^{-1}(c x)\right)\right)\right)+i b^2 \left(-64 \left(3 \tan ^{-1}(c x)+i\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-96 i \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)-128 \tan ^{-1}(c x)^3+64 c x \tan ^{-1}(c x)^2-64 i \tan ^{-1}(c x)^2-192 i \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+128 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+80 \tan ^{-1}(c x)^2 \sin \left(2 \tan ^{-1}(c x)\right)-8 \tan ^{-1}(c x)^2 \sin \left(4 \tan ^{-1}(c x)\right)-80 i \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)+4 i \tan ^{-1}(c x) \sin \left(4 \tan ^{-1}(c x)\right)-40 \sin \left(2 \tan ^{-1}(c x)\right)+\sin \left(4 \tan ^{-1}(c x)\right)+80 i \tan ^{-1}(c x)^2 \cos \left(2 \tan ^{-1}(c x)\right)-8 i \tan ^{-1}(c x)^2 \cos \left(4 \tan ^{-1}(c x)\right)+80 \tan ^{-1}(c x) \cos \left(2 \tan ^{-1}(c x)\right)-4 \tan ^{-1}(c x) \cos \left(4 \tan ^{-1}(c x)\right)-40 i \cos \left(2 \tan ^{-1}(c x)\right)+i \cos \left(4 \tan ^{-1}(c x)\right)\right)}{64 c^4 d^3}","\frac{3 i b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{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{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{i x \left(a+b \tan ^{-1}(c x)\right)^2}{c^3 d^3}-\frac{b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c^4 d^3}+\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\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,"((64*I)*a^2*c*x - (32*a^2)/(-I + c*x)^2 + ((192*I)*a^2)/(-I + c*x) - (192*I)*a^2*ArcTan[c*x] - 96*a^2*Log[1 + c^2*x^2] + (4*I)*a*b*(-96*ArcTan[c*x]^2 + 20*Cos[2*ArcTan[c*x]] - Cos[4*ArcTan[c*x]] - 16*Log[1 + c^2*x^2] - 48*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (20*I)*Sin[2*ArcTan[c*x]] + 4*ArcTan[c*x]*(8*c*x + (10*I)*Cos[2*ArcTan[c*x]] - I*Cos[4*ArcTan[c*x]] - (24*I)*Log[1 + E^((2*I)*ArcTan[c*x])] + 10*Sin[2*ArcTan[c*x]] - Sin[4*ArcTan[c*x]]) + I*Sin[4*ArcTan[c*x]]) + I*b^2*((-64*I)*ArcTan[c*x]^2 + 64*c*x*ArcTan[c*x]^2 - 128*ArcTan[c*x]^3 - (40*I)*Cos[2*ArcTan[c*x]] + 80*ArcTan[c*x]*Cos[2*ArcTan[c*x]] + (80*I)*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + I*Cos[4*ArcTan[c*x]] - 4*ArcTan[c*x]*Cos[4*ArcTan[c*x]] - (8*I)*ArcTan[c*x]^2*Cos[4*ArcTan[c*x]] + 128*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - (192*I)*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - 64*(I + 3*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (96*I)*PolyLog[3, -E^((2*I)*ArcTan[c*x])] - 40*Sin[2*ArcTan[c*x]] - (80*I)*ArcTan[c*x]*Sin[2*ArcTan[c*x]] + 80*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]] + Sin[4*ArcTan[c*x]] + (4*I)*ArcTan[c*x]*Sin[4*ArcTan[c*x]] - 8*ArcTan[c*x]^2*Sin[4*ArcTan[c*x]]))/(64*c^4*d^3)","A",0
113,1,431,304,1.2607612,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^3} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3,x]","\frac{96 i a^2 \log \left(c^2 x^2+1\right)+\frac{384 a^2}{c x-i}+\frac{96 i a^2}{(c x-i)^2}-192 a^2 \tan ^{-1}(c x)-12 a b \left(16 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+32 \tan ^{-1}(c x)^2+12 i \sin \left(2 \tan ^{-1}(c x)\right)-i \sin \left(4 \tan ^{-1}(c x)\right)-12 \cos \left(2 \tan ^{-1}(c x)\right)+\cos \left(4 \tan ^{-1}(c x)\right)+4 \tan ^{-1}(c x) \left(8 i \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-6 \sin \left(2 \tan ^{-1}(c x)\right)+\sin \left(4 \tan ^{-1}(c x)\right)-6 i \cos \left(2 \tan ^{-1}(c x)\right)+i \cos \left(4 \tan ^{-1}(c x)\right)\right)\right)-b^2 \left(192 \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+96 i \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+128 \tan ^{-1}(c x)^3+192 i \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-144 \tan ^{-1}(c x)^2 \sin \left(2 \tan ^{-1}(c x)\right)+24 \tan ^{-1}(c x)^2 \sin \left(4 \tan ^{-1}(c x)\right)+144 i \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)-12 i \tan ^{-1}(c x) \sin \left(4 \tan ^{-1}(c x)\right)+72 \sin \left(2 \tan ^{-1}(c x)\right)-3 \sin \left(4 \tan ^{-1}(c x)\right)-144 i \tan ^{-1}(c x)^2 \cos \left(2 \tan ^{-1}(c x)\right)+24 i \tan ^{-1}(c x)^2 \cos \left(4 \tan ^{-1}(c x)\right)-144 \tan ^{-1}(c x) \cos \left(2 \tan ^{-1}(c x)\right)+12 \tan ^{-1}(c x) \cos \left(4 \tan ^{-1}(c x)\right)+72 i \cos \left(2 \tan ^{-1}(c x)\right)-3 i \cos \left(4 \tan ^{-1}(c x)\right)\right)}{192 c^3 d^3}","\frac{b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{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{i b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\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,"(((96*I)*a^2)/(-I + c*x)^2 + (384*a^2)/(-I + c*x) - 192*a^2*ArcTan[c*x] + (96*I)*a^2*Log[1 + c^2*x^2] - b^2*(128*ArcTan[c*x]^3 + (72*I)*Cos[2*ArcTan[c*x]] - 144*ArcTan[c*x]*Cos[2*ArcTan[c*x]] - (144*I)*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] - (3*I)*Cos[4*ArcTan[c*x]] + 12*ArcTan[c*x]*Cos[4*ArcTan[c*x]] + (24*I)*ArcTan[c*x]^2*Cos[4*ArcTan[c*x]] + (192*I)*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + 192*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (96*I)*PolyLog[3, -E^((2*I)*ArcTan[c*x])] + 72*Sin[2*ArcTan[c*x]] + (144*I)*ArcTan[c*x]*Sin[2*ArcTan[c*x]] - 144*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]] - 3*Sin[4*ArcTan[c*x]] - (12*I)*ArcTan[c*x]*Sin[4*ArcTan[c*x]] + 24*ArcTan[c*x]^2*Sin[4*ArcTan[c*x]]) - 12*a*b*(32*ArcTan[c*x]^2 - 12*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] + 16*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (12*I)*Sin[2*ArcTan[c*x]] - I*Sin[4*ArcTan[c*x]] + 4*ArcTan[c*x]*((-6*I)*Cos[2*ArcTan[c*x]] + I*Cos[4*ArcTan[c*x]] + (8*I)*Log[1 + E^((2*I)*ArcTan[c*x])] - 6*Sin[2*ArcTan[c*x]] + Sin[4*ArcTan[c*x]])))/(192*c^3*d^3)","A",0
114,1,117,178,0.3918494,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^3} \, dx","Integrate[(x*(a + b*ArcTan[c*x])^2)/(d + I*c*d*x)^3,x]","\frac{a^2 (-8-16 i c x)+4 a b (-3 c x+2 i)+b (c x+i) \tan ^{-1}(c x) (a (-12 c x+4 i)+b (3+5 i c x))-2 b^2 \left(3 c^2 x^2+2 i c x+1\right) \tan ^{-1}(c x)^2+b^2 (4+5 i c x)}{16 c^2 d^3 (c x-i)^2}","\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,"(4*a*b*(2*I - 3*c*x) + b^2*(4 + (5*I)*c*x) + a^2*(-8 - (16*I)*c*x) + b*(I + c*x)*(a*(4*I - 12*c*x) + b*(3 + (5*I)*c*x))*ArcTan[c*x] - 2*b^2*(1 + (2*I)*c*x + 3*c^2*x^2)*ArcTan[c*x]^2)/(16*c^2*d^3*(-I + c*x)^2)","A",1
115,1,110,180,0.1924295,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{(d+i c d x)^3} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(d + I*c*d*x)^3,x]","-\frac{i \left(8 a^2+4 a b (c x-2 i)+b (c x+i) \tan ^{-1}(c x) (4 a (c x-3 i)+b (-5-3 i c x))+2 b^2 \left(c^2 x^2-2 i c x+3\right) \tan ^{-1}(c x)^2+b^2 (-4-3 i c x)\right)}{16 c d^3 (c x-i)^2}","\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,"((-1/16*I)*(8*a^2 + b^2*(-4 - (3*I)*c*x) + 4*a*b*(-2*I + c*x) + b*(I + c*x)*(b*(-5 - (3*I)*c*x) + 4*a*(-3*I + c*x))*ArcTan[c*x] + 2*b^2*(3 - (2*I)*c*x + c^2*x^2)*ArcTan[c*x]^2))/(c*d^3*(-I + c*x)^2)","A",1
116,1,435,299,1.7014937,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x (d+i c d x)^3} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x*(d + I*c*d*x)^3),x]","\frac{-96 a^2 \log \left(c^2 x^2+1\right)-\frac{192 i a^2}{c x-i}-\frac{96 a^2}{(c x-i)^2}+192 a^2 \log (c x)-192 i a^2 \tan ^{-1}(c x)+12 i a b \left(-16 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)-32 \tan ^{-1}(c x)^2+12 i \sin \left(2 \tan ^{-1}(c x)\right)+i \sin \left(4 \tan ^{-1}(c x)\right)-12 \cos \left(2 \tan ^{-1}(c x)\right)-\cos \left(4 \tan ^{-1}(c x)\right)-4 i \tan ^{-1}(c x) \left(8 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)-6 i \sin \left(2 \tan ^{-1}(c x)\right)-i \sin \left(4 \tan ^{-1}(c x)\right)+6 \cos \left(2 \tan ^{-1}(c x)\right)+\cos \left(4 \tan ^{-1}(c x)\right)\right)\right)+b^2 \left(192 i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+96 \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)+192 \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-144 i \tan ^{-1}(c x)^2 \sin \left(2 \tan ^{-1}(c x)\right)-24 i \tan ^{-1}(c x)^2 \sin \left(4 \tan ^{-1}(c x)\right)-144 \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)-12 \tan ^{-1}(c x) \sin \left(4 \tan ^{-1}(c x)\right)+72 i \sin \left(2 \tan ^{-1}(c x)\right)+3 i \sin \left(4 \tan ^{-1}(c x)\right)+144 \tan ^{-1}(c x)^2 \cos \left(2 \tan ^{-1}(c x)\right)+24 \tan ^{-1}(c x)^2 \cos \left(4 \tan ^{-1}(c x)\right)-144 i \tan ^{-1}(c x) \cos \left(2 \tan ^{-1}(c x)\right)-12 i \tan ^{-1}(c x) \cos \left(4 \tan ^{-1}(c x)\right)-72 \cos \left(2 \tan ^{-1}(c x)\right)-3 \cos \left(4 \tan ^{-1}(c x)\right)-8 i \pi ^3\right)}{192 d^3}","\frac{i b \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{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{b^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\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,"((-96*a^2)/(-I + c*x)^2 - ((192*I)*a^2)/(-I + c*x) - (192*I)*a^2*ArcTan[c*x] + 192*a^2*Log[c*x] - 96*a^2*Log[1 + c^2*x^2] + (12*I)*a*b*(-32*ArcTan[c*x]^2 - 12*Cos[2*ArcTan[c*x]] - Cos[4*ArcTan[c*x]] - 16*PolyLog[2, E^((2*I)*ArcTan[c*x])] + (12*I)*Sin[2*ArcTan[c*x]] - (4*I)*ArcTan[c*x]*(6*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] + 8*Log[1 - E^((2*I)*ArcTan[c*x])] - (6*I)*Sin[2*ArcTan[c*x]] - I*Sin[4*ArcTan[c*x]]) + I*Sin[4*ArcTan[c*x]]) + b^2*((-8*I)*Pi^3 - 72*Cos[2*ArcTan[c*x]] - (144*I)*ArcTan[c*x]*Cos[2*ArcTan[c*x]] + 144*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] - 3*Cos[4*ArcTan[c*x]] - (12*I)*ArcTan[c*x]*Cos[4*ArcTan[c*x]] + 24*ArcTan[c*x]^2*Cos[4*ArcTan[c*x]] + 192*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + (192*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + 96*PolyLog[3, E^((-2*I)*ArcTan[c*x])] + (72*I)*Sin[2*ArcTan[c*x]] - 144*ArcTan[c*x]*Sin[2*ArcTan[c*x]] - (144*I)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]] + (3*I)*Sin[4*ArcTan[c*x]] - 12*ArcTan[c*x]*Sin[4*ArcTan[c*x]] - (24*I)*ArcTan[c*x]^2*Sin[4*ArcTan[c*x]]))/(192*d^3)","A",0
117,1,549,391,3.8503094,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^2 (d+i c d x)^3} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x^2*(d + I*c*d*x)^3),x]","-\frac{-96 i a^2 c \log \left(c^2 x^2+1\right)+\frac{128 a^2 c}{c x-i}-\frac{32 i a^2 c}{(c x-i)^2}+192 i a^2 c \log (x)+192 a^2 c \tan ^{-1}(c x)+\frac{64 a^2}{x}+\frac{4 a b \left(c x \left(-32 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)-20 i \sin \left(2 \tan ^{-1}(c x)\right)-i \sin \left(4 \tan ^{-1}(c x)\right)+20 \cos \left(2 \tan ^{-1}(c x)\right)+\cos \left(4 \tan ^{-1}(c x)\right)\right)+48 c x \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+96 c x \tan ^{-1}(c x)^2+4 \tan ^{-1}(c x) \left(24 i c x \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+10 c x \sin \left(2 \tan ^{-1}(c x)\right)+c x \sin \left(4 \tan ^{-1}(c x)\right)+10 i c x \cos \left(2 \tan ^{-1}(c x)\right)+i c x \cos \left(4 \tan ^{-1}(c x)\right)+8\right)\right)}{x}-i b^2 c \left(-192 i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)-64 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)-96 \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)+\frac{64 i \tan ^{-1}(c x)^2}{c x}-64 \tan ^{-1}(c x)^2-192 \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-128 i \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+80 i \tan ^{-1}(c x)^2 \sin \left(2 \tan ^{-1}(c x)\right)+8 i \tan ^{-1}(c x)^2 \sin \left(4 \tan ^{-1}(c x)\right)+80 \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)+4 \tan ^{-1}(c x) \sin \left(4 \tan ^{-1}(c x)\right)-40 i \sin \left(2 \tan ^{-1}(c x)\right)-i \sin \left(4 \tan ^{-1}(c x)\right)-80 \tan ^{-1}(c x)^2 \cos \left(2 \tan ^{-1}(c x)\right)-8 \tan ^{-1}(c x)^2 \cos \left(4 \tan ^{-1}(c x)\right)+80 i \tan ^{-1}(c x) \cos \left(2 \tan ^{-1}(c x)\right)+4 i \tan ^{-1}(c x) \cos \left(4 \tan ^{-1}(c x)\right)+40 \cos \left(2 \tan ^{-1}(c x)\right)+\cos \left(4 \tan ^{-1}(c x)\right)+8 i \pi ^3\right)}{64 d^3}","\frac{3 b c \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{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{i b^2 c \text{Li}_2\left(\frac{2}{1-i c x}-1\right)}{d^3}-\frac{3 i b^2 c \text{Li}_3\left(\frac{2}{i c x+1}-1\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,"-1/64*((64*a^2)/x - ((32*I)*a^2*c)/(-I + c*x)^2 + (128*a^2*c)/(-I + c*x) + 192*a^2*c*ArcTan[c*x] + (192*I)*a^2*c*Log[x] - (96*I)*a^2*c*Log[1 + c^2*x^2] - I*b^2*c*((8*I)*Pi^3 - 64*ArcTan[c*x]^2 + ((64*I)*ArcTan[c*x]^2)/(c*x) + 40*Cos[2*ArcTan[c*x]] + (80*I)*ArcTan[c*x]*Cos[2*ArcTan[c*x]] - 80*ArcTan[c*x]^2*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] + (4*I)*ArcTan[c*x]*Cos[4*ArcTan[c*x]] - 8*ArcTan[c*x]^2*Cos[4*ArcTan[c*x]] - 192*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - (128*I)*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - (192*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - 64*PolyLog[2, E^((2*I)*ArcTan[c*x])] - 96*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - (40*I)*Sin[2*ArcTan[c*x]] + 80*ArcTan[c*x]*Sin[2*ArcTan[c*x]] + (80*I)*ArcTan[c*x]^2*Sin[2*ArcTan[c*x]] - I*Sin[4*ArcTan[c*x]] + 4*ArcTan[c*x]*Sin[4*ArcTan[c*x]] + (8*I)*ArcTan[c*x]^2*Sin[4*ArcTan[c*x]]) + (4*a*b*(96*c*x*ArcTan[c*x]^2 + 48*c*x*PolyLog[2, E^((2*I)*ArcTan[c*x])] + c*x*(20*Cos[2*ArcTan[c*x]] + Cos[4*ArcTan[c*x]] - 32*Log[(c*x)/Sqrt[1 + c^2*x^2]] - (20*I)*Sin[2*ArcTan[c*x]] - I*Sin[4*ArcTan[c*x]]) + 4*ArcTan[c*x]*(8 + (10*I)*c*x*Cos[2*ArcTan[c*x]] + I*c*x*Cos[4*ArcTan[c*x]] + (24*I)*c*x*Log[1 - E^((2*I)*ArcTan[c*x])] + 10*c*x*Sin[2*ArcTan[c*x]] + c*x*Sin[4*ArcTan[c*x]])))/x)/d^3","A",0
118,1,155,207,0.2304256,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{(1+i c x)^4} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(1 + I*c*x)^4,x]","-\frac{144 a^2+12 a b \left(3 i c^2 x^2+9 c x-10 i\right)+3 b (c x+i) \tan ^{-1}(c x) \left(12 a \left(i c^2 x^2+4 c x-7 i\right)+b \left(11 c^2 x^2-32 i c x-29\right)\right)+b^2 \left(33 c^2 x^2-81 i c x-56\right)+18 b^2 \left(i c^3 x^3+3 c^2 x^2-3 i c x+7\right) \tan ^{-1}(c x)^2}{432 c (c x-i)^3}","\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,"-1/432*(144*a^2 + 12*a*b*(-10*I + 9*c*x + (3*I)*c^2*x^2) + b^2*(-56 - (81*I)*c*x + 33*c^2*x^2) + 3*b*(I + c*x)*(12*a*(-7*I + 4*c*x + I*c^2*x^2) + b*(-29 - (32*I)*c*x + 11*c^2*x^2))*ArcTan[c*x] + 18*b^2*(7 - (3*I)*c*x + 3*c^2*x^2 + I*c^3*x^3)*ArcTan[c*x]^2)/(c*(-I + c*x)^3)","A",1
119,1,82,76,0.2816468,"\int \frac{\tan ^{-1}(a x)^2}{c x-i a c x^2} \, dx","Integrate[ArcTan[a*x]^2/(c*x - I*a*c*x^2),x]","\frac{24 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+12 \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)+16 i \tan ^{-1}(a x)^3+24 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-i \pi ^3}{24 c}","\frac{\text{Li}_3\left(\frac{2}{1-i a x}-1\right)}{2 c}-\frac{i \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{c}+\frac{\log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c}",1,"((-I)*Pi^3 + (16*I)*ArcTan[a*x]^3 + 24*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (24*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/(24*c)","A",0
120,1,693,382,1.8012193,"\int (d+i c d x)^3 \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Integrate[(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^3,x]","-\frac{i d^3 \left(a^3 c^4 x^4-4 i a^3 c^3 x^3-6 a^3 c^2 x^2+4 i a^3 c x+3 a^2 b c^4 x^4 \tan ^{-1}(c x)-a^2 b c^3 x^3-12 i a^2 b c^3 x^3 \tan ^{-1}(c x)+6 i a^2 b c^2 x^2-12 i a^2 b \log \left(c^2 x^2+1\right)-18 a^2 b c^2 x^2 \tan ^{-1}(c x)+21 a^2 b c x+12 i a^2 b c x \tan ^{-1}(c x)-21 a^2 b \tan ^{-1}(c x)+3 a b^2 c^4 x^4 \tan ^{-1}(c x)^2-12 i a b^2 c^3 x^3 \tan ^{-1}(c x)^2-2 a b^2 c^3 x^3 \tan ^{-1}(c x)+a b^2 c^2 x^2-22 a b^2 \log \left(c^2 x^2+1\right)-18 a b^2 c^2 x^2 \tan ^{-1}(c x)^2+12 i a b^2 c^2 x^2 \tan ^{-1}(c x)+2 b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right) \left(12 a+12 b \tan ^{-1}(c x)-11 i b\right)-12 i a b^2 c x+12 i a b^2 c x \tan ^{-1}(c x)^2+42 a b^2 c x \tan ^{-1}(c x)+3 a b^2 \tan ^{-1}(c x)^2+12 i a b^2 \tan ^{-1}(c x)+48 i a b^2 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+a b^2+b^3 c^4 x^4 \tan ^{-1}(c x)^3-4 i b^3 c^3 x^3 \tan ^{-1}(c x)^3-b^3 c^3 x^3 \tan ^{-1}(c x)^2+6 i b^3 \log \left(c^2 x^2+1\right)-6 b^3 c^2 x^2 \tan ^{-1}(c x)^3+6 i b^3 c^2 x^2 \tan ^{-1}(c x)^2+b^3 c^2 x^2 \tan ^{-1}(c x)+12 i b^3 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)-b^3 c x+4 i b^3 c x \tan ^{-1}(c x)^3+21 b^3 c x \tan ^{-1}(c x)^2-12 i b^3 c x \tan ^{-1}(c x)+b^3 \tan ^{-1}(c x)^3-16 i b^3 \tan ^{-1}(c x)^2+b^3 \tan ^{-1}(c x)+24 i b^3 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+44 b^3 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{4 c}","-\frac{6 i b^2 d^3 \text{Li}_2\left(1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(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{11 b^3 d^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 c}+\frac{3 b^3 d^3 \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{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,"((-1/4*I)*d^3*(a*b^2 + (4*I)*a^3*c*x + 21*a^2*b*c*x - (12*I)*a*b^2*c*x - b^3*c*x - 6*a^3*c^2*x^2 + (6*I)*a^2*b*c^2*x^2 + a*b^2*c^2*x^2 - (4*I)*a^3*c^3*x^3 - a^2*b*c^3*x^3 + a^3*c^4*x^4 - 21*a^2*b*ArcTan[c*x] + (12*I)*a*b^2*ArcTan[c*x] + b^3*ArcTan[c*x] + (12*I)*a^2*b*c*x*ArcTan[c*x] + 42*a*b^2*c*x*ArcTan[c*x] - (12*I)*b^3*c*x*ArcTan[c*x] - 18*a^2*b*c^2*x^2*ArcTan[c*x] + (12*I)*a*b^2*c^2*x^2*ArcTan[c*x] + b^3*c^2*x^2*ArcTan[c*x] - (12*I)*a^2*b*c^3*x^3*ArcTan[c*x] - 2*a*b^2*c^3*x^3*ArcTan[c*x] + 3*a^2*b*c^4*x^4*ArcTan[c*x] + 3*a*b^2*ArcTan[c*x]^2 - (16*I)*b^3*ArcTan[c*x]^2 + (12*I)*a*b^2*c*x*ArcTan[c*x]^2 + 21*b^3*c*x*ArcTan[c*x]^2 - 18*a*b^2*c^2*x^2*ArcTan[c*x]^2 + (6*I)*b^3*c^2*x^2*ArcTan[c*x]^2 - (12*I)*a*b^2*c^3*x^3*ArcTan[c*x]^2 - b^3*c^3*x^3*ArcTan[c*x]^2 + 3*a*b^2*c^4*x^4*ArcTan[c*x]^2 + b^3*ArcTan[c*x]^3 + (4*I)*b^3*c*x*ArcTan[c*x]^3 - 6*b^3*c^2*x^2*ArcTan[c*x]^3 - (4*I)*b^3*c^3*x^3*ArcTan[c*x]^3 + b^3*c^4*x^4*ArcTan[c*x]^3 + (48*I)*a*b^2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 44*b^3*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*b^3*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - (12*I)*a^2*b*Log[1 + c^2*x^2] - 22*a*b^2*Log[1 + c^2*x^2] + (6*I)*b^3*Log[1 + c^2*x^2] + 2*b^2*(12*a - (11*I)*b + 12*b*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (12*I)*b^3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/c","A",0
121,1,528,298,1.0669907,"\int (d+i c d x)^2 \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Integrate[(d + I*c*d*x)^2*(a + b*ArcTan[c*x])^3,x]","-\frac{d^2 \left(2 a^3 c^3 x^3-6 i a^3 c^2 x^2-6 a^3 c x+6 a^2 b c^3 x^3 \tan ^{-1}(c x)-3 a^2 b c^2 x^2+12 a^2 b \log \left(c^2 x^2+1\right)-18 i a^2 b c^2 x^2 \tan ^{-1}(c x)+18 i a^2 b c x-18 i a^2 b \tan ^{-1}(c x)-18 a^2 b c x \tan ^{-1}(c x)+6 a b^2 c^3 x^3 \tan ^{-1}(c x)^2-18 i a b^2 \log \left(c^2 x^2+1\right)-18 i a b^2 c^2 x^2 \tan ^{-1}(c x)^2-6 a b^2 c^2 x^2 \tan ^{-1}(c x)+6 b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right) \left(4 i a+4 i b \tan ^{-1}(c x)+3 b\right)+6 a b^2 c x+6 i a b^2 \tan ^{-1}(c x)^2-18 a b^2 c x \tan ^{-1}(c x)^2-6 a b^2 \tan ^{-1}(c x)+36 i a b^2 c x \tan ^{-1}(c x)-48 a b^2 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+2 b^3 c^3 x^3 \tan ^{-1}(c x)^3-3 b^3 \log \left(c^2 x^2+1\right)-6 i b^3 c^2 x^2 \tan ^{-1}(c x)^3-3 b^3 c^2 x^2 \tan ^{-1}(c x)^2-12 b^3 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+2 i b^3 \tan ^{-1}(c x)^3-6 b^3 c x \tan ^{-1}(c x)^3+15 b^3 \tan ^{-1}(c x)^2+18 i b^3 c x \tan ^{-1}(c x)^2+6 b^3 c x \tan ^{-1}(c x)-24 b^3 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+36 i b^3 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{6 c}","-\frac{4 i b^2 d^2 \text{Li}_2\left(1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(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}+\frac{3 b^3 d^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c}+\frac{2 b^3 d^2 \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{c}+b^3 \left(-d^2\right) x \tan ^{-1}(c x)",1,"-1/6*(d^2*(-6*a^3*c*x + (18*I)*a^2*b*c*x + 6*a*b^2*c*x - (6*I)*a^3*c^2*x^2 - 3*a^2*b*c^2*x^2 + 2*a^3*c^3*x^3 - (18*I)*a^2*b*ArcTan[c*x] - 6*a*b^2*ArcTan[c*x] - 18*a^2*b*c*x*ArcTan[c*x] + (36*I)*a*b^2*c*x*ArcTan[c*x] + 6*b^3*c*x*ArcTan[c*x] - (18*I)*a^2*b*c^2*x^2*ArcTan[c*x] - 6*a*b^2*c^2*x^2*ArcTan[c*x] + 6*a^2*b*c^3*x^3*ArcTan[c*x] + (6*I)*a*b^2*ArcTan[c*x]^2 + 15*b^3*ArcTan[c*x]^2 - 18*a*b^2*c*x*ArcTan[c*x]^2 + (18*I)*b^3*c*x*ArcTan[c*x]^2 - (18*I)*a*b^2*c^2*x^2*ArcTan[c*x]^2 - 3*b^3*c^2*x^2*ArcTan[c*x]^2 + 6*a*b^2*c^3*x^3*ArcTan[c*x]^2 + (2*I)*b^3*ArcTan[c*x]^3 - 6*b^3*c*x*ArcTan[c*x]^3 - (6*I)*b^3*c^2*x^2*ArcTan[c*x]^3 + 2*b^3*c^3*x^3*ArcTan[c*x]^3 - 48*a*b^2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + (36*I)*b^3*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - 24*b^3*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + 12*a^2*b*Log[1 + c^2*x^2] - (18*I)*a*b^2*Log[1 + c^2*x^2] - 3*b^3*Log[1 + c^2*x^2] + 6*b^2*((4*I)*a + 3*b + (4*I)*b*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - 12*b^3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/c","A",0
122,1,367,220,0.4941046,"\int (d+i c d x) \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Integrate[(d + I*c*d*x)*(a + b*ArcTan[c*x])^3,x]","\frac{i d \left(a^3 c^2 x^2-2 i a^3 c x+3 i a^2 b \log \left(c^2 x^2+1\right)+3 a^2 b c^2 x^2 \tan ^{-1}(c x)-3 a^2 b c x+3 a^2 b \tan ^{-1}(c x)-6 i a^2 b c x \tan ^{-1}(c x)+3 a b^2 \log \left(c^2 x^2+1\right)+3 a b^2 c^2 x^2 \tan ^{-1}(c x)^2-3 b^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right) \left(2 a+2 b \tan ^{-1}(c x)-i b\right)-3 a b^2 \tan ^{-1}(c x)^2-6 i a b^2 c x \tan ^{-1}(c x)^2-6 a b^2 c x \tan ^{-1}(c x)-12 i a b^2 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+b^3 c^2 x^2 \tan ^{-1}(c x)^3-3 i b^3 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)-b^3 \tan ^{-1}(c x)^3-2 i b^3 c x \tan ^{-1}(c x)^3+3 i b^3 \tan ^{-1}(c x)^2-3 b^3 c x \tan ^{-1}(c x)^2-6 i b^3 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-6 b^3 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{2 c}","-\frac{3 i b^2 d \text{Li}_2\left(1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{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 b^3 d \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 c}+\frac{3 b^3 d \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{2 c}",1,"((I/2)*d*((-2*I)*a^3*c*x - 3*a^2*b*c*x + a^3*c^2*x^2 + 3*a^2*b*ArcTan[c*x] - (6*I)*a^2*b*c*x*ArcTan[c*x] - 6*a*b^2*c*x*ArcTan[c*x] + 3*a^2*b*c^2*x^2*ArcTan[c*x] - 3*a*b^2*ArcTan[c*x]^2 + (3*I)*b^3*ArcTan[c*x]^2 - (6*I)*a*b^2*c*x*ArcTan[c*x]^2 - 3*b^3*c*x*ArcTan[c*x]^2 + 3*a*b^2*c^2*x^2*ArcTan[c*x]^2 - b^3*ArcTan[c*x]^3 - (2*I)*b^3*c*x*ArcTan[c*x]^3 + b^3*c^2*x^2*ArcTan[c*x]^3 - (12*I)*a*b^2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - 6*b^3*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - (6*I)*b^3*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (3*I)*a^2*b*Log[1 + c^2*x^2] + 3*a*b^2*Log[1 + c^2*x^2] - 3*b^2*(2*a - I*b + 2*b*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (3*I)*b^3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/c","A",0
123,1,133,139,0.0856097,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{d+i c d x} \, dx","Integrate[(a + b*ArcTan[c*x])^3/(d + I*c*d*x),x]","\frac{i \left(4 \log \left(\frac{2 d}{d+i c d x}\right) \left(a+b \tan ^{-1}(c x)\right)^3+3 i b \left(2 \text{Li}_2\left(\frac{c x+i}{c x-i}\right) \left(a+b \tan ^{-1}(c x)\right)^2-b \left(2 i \text{Li}_3\left(\frac{c x+i}{c x-i}\right) \left(a+b \tan ^{-1}(c x)\right)+b \text{Li}_4\left(\frac{c x+i}{c x-i}\right)\right)\right)\right)}{4 c d}","\frac{3 i b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c d}-\frac{3 b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 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 b^3 \text{Li}_4\left(1-\frac{2}{i c x+1}\right)}{4 c d}",1,"((I/4)*(4*(a + b*ArcTan[c*x])^3*Log[(2*d)/(d + I*c*d*x)] + (3*I)*b*(2*(a + b*ArcTan[c*x])^2*PolyLog[2, (I + c*x)/(-I + c*x)] - b*((2*I)*(a + b*ArcTan[c*x])*PolyLog[3, (I + c*x)/(-I + c*x)] + b*PolyLog[4, (I + c*x)/(-I + c*x)]))))/(c*d)","A",1
124,1,121,182,0.2379674,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{(d+i c d x)^2} \, dx","Integrate[(a + b*ArcTan[c*x])^3/(d + I*c*d*x)^2,x]","\frac{4 a^3+3 i b \left(-2 a^2+2 i a b+b^2\right) (c x+i) \tan ^{-1}(c x)-6 i a^2 b-3 b^2 (b+2 i a) (c x+i) \tan ^{-1}(c x)^2-6 a b^2+2 b^3 (1-i c x) \tan ^{-1}(c x)^3+3 i b^3}{4 c d^2 (c x-i)}","\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,"(4*a^3 - (6*I)*a^2*b - 6*a*b^2 + (3*I)*b^3 + (3*I)*b*(-2*a^2 + (2*I)*a*b + b^2)*(I + c*x)*ArcTan[c*x] - 3*b^2*((2*I)*a + b)*(I + c*x)*ArcTan[c*x]^2 + 2*b^3*(1 - I*c*x)*ArcTan[c*x]^3)/(4*c*d^2*(-I + c*x))","A",1
125,1,183,271,0.2986658,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{(d+i c d x)^3} \, dx","Integrate[(a + b*ArcTan[c*x])^3/(d + I*c*d*x)^3,x]","-\frac{i \left(32 a^3+3 b (c x+i) \tan ^{-1}(c x) \left(8 a^2 (c x-3 i)+4 a b (-5-3 i c x)+b^2 (-7 c x+9 i)\right)+24 a^2 b (c x-2 i)+12 a b^2 (-4-3 i c x)+6 b^2 (c x+i) \tan ^{-1}(c x)^2 (4 a (c x-3 i)+b (-5-3 i c x))+8 b^3 \left(c^2 x^2-2 i c x+3\right) \tan ^{-1}(c x)^3+3 b^3 (-7 c x+8 i)\right)}{64 c d^3 (c x-i)^2}","\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,"((-1/64*I)*(32*a^3 + 3*b^3*(8*I - 7*c*x) + 12*a*b^2*(-4 - (3*I)*c*x) + 24*a^2*b*(-2*I + c*x) + 3*b*(I + c*x)*(b^2*(9*I - 7*c*x) + 4*a*b*(-5 - (3*I)*c*x) + 8*a^2*(-3*I + c*x))*ArcTan[c*x] + 6*b^2*(I + c*x)*(b*(-5 - (3*I)*c*x) + 4*a*(-3*I + c*x))*ArcTan[c*x]^2 + 8*b^3*(3 - (2*I)*c*x + c^2*x^2)*ArcTan[c*x]^3))/(c*d^3*(-I + c*x)^2)","A",1
126,1,269,360,0.3183885,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{(d+i c d x)^4} \, dx","Integrate[(a + b*ArcTan[c*x])^3/(d + I*c*d*x)^4,x]","\frac{-576 a^3+3 b (c x+i) \tan ^{-1}(c x) \left(-72 i a^2 \left(c^2 x^2-4 i c x-7\right)+12 a b \left(-11 c^2 x^2+32 i c x+29\right)+b^2 \left(85 i c^2 x^2+208 c x-139 i\right)\right)-72 i a^2 b \left(3 c^2 x^2-9 i c x-10\right)+12 a b^2 \left(-33 c^2 x^2+81 i c x+56\right)-18 i b^2 (c x+i) \tan ^{-1}(c x)^2 \left(12 a \left(c^2 x^2-4 i c x-7\right)+b \left(-11 i c^2 x^2-32 c x+29 i\right)\right)+b^3 \left(255 i c^2 x^2+567 c x-328 i\right)-72 i b^3 \left(c^3 x^3-3 i c^2 x^2-3 c x-7 i\right) \tan ^{-1}(c x)^3}{1728 c d^4 (c x-i)^3}","\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,"(-576*a^3 + 12*a*b^2*(56 + (81*I)*c*x - 33*c^2*x^2) + b^3*(-328*I + 567*c*x + (255*I)*c^2*x^2) - (72*I)*a^2*b*(-10 - (9*I)*c*x + 3*c^2*x^2) + 3*b*(I + c*x)*(12*a*b*(29 + (32*I)*c*x - 11*c^2*x^2) + b^2*(-139*I + 208*c*x + (85*I)*c^2*x^2) - (72*I)*a^2*(-7 - (4*I)*c*x + c^2*x^2))*ArcTan[c*x] - (18*I)*b^2*(I + c*x)*(b*(29*I - 32*c*x - (11*I)*c^2*x^2) + 12*a*(-7 - (4*I)*c*x + c^2*x^2))*ArcTan[c*x]^2 - (72*I)*b^3*(-7*I - 3*c*x - (3*I)*c^2*x^2 + c^3*x^3)*ArcTan[c*x]^3)/(1728*c*d^4*(-I + c*x)^3)","A",1
127,1,541,410,1.0310123,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^3}{d+i c d x} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x])^3)/(d + I*c*d*x),x]","-\frac{i \left(2 a^3 c^2 x^2-2 a^3 \log \left(c^2 x^2+1\right)+4 i a^3 c x-4 i a^3 \tan ^{-1}(c x)-6 i a^2 b \log \left(c^2 x^2+1\right)+6 a^2 b c^2 x^2 \tan ^{-1}(c x)-6 a^2 b c x-12 i a^2 b \tan ^{-1}(c x)^2+6 a^2 b \tan ^{-1}(c x)+12 i a^2 b c x \tan ^{-1}(c x)+12 a^2 b \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+6 a b^2 \log \left(c^2 x^2+1\right)+6 a b^2 c^2 x^2 \tan ^{-1}(c x)^2+6 b^2 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right) \left(a+b \tan ^{-1}(c x)+i b\right)-8 i a b^2 \tan ^{-1}(c x)^3+18 a b^2 \tan ^{-1}(c x)^2+12 i a b^2 c x \tan ^{-1}(c x)^2-12 a b^2 c x \tan ^{-1}(c x)+12 a b^2 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+24 i a b^2 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-6 i b \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right) \left(a+b \tan ^{-1}(c x)+i b\right)^2+2 b^3 c^2 x^2 \tan ^{-1}(c x)^3+3 i b^3 \text{Li}_4\left(-e^{2 i \tan ^{-1}(c x)}\right)-2 i b^3 \tan ^{-1}(c x)^4+6 b^3 \tan ^{-1}(c x)^3+4 i b^3 c x \tan ^{-1}(c x)^3+6 i b^3 \tan ^{-1}(c x)^2-6 b^3 c x \tan ^{-1}(c x)^2+4 b^3 \tan ^{-1}(c x)^3 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+12 i b^3 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-12 b^3 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{4 c^3 d}","\frac{3 i b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3 d}-\frac{3 i b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 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 b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 d}-\frac{3 b \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3 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{3 i b x \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 d}+\frac{x \left(a+b \tan ^{-1}(c x)\right)^3}{c^2 d}-\frac{i x^2 \left(a+b \tan ^{-1}(c x)\right)^3}{2 c d}-\frac{3 b^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 c^3 d}+\frac{3 b^3 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 c^3 d}-\frac{3 b^3 \text{Li}_4\left(1-\frac{2}{i c x+1}\right)}{4 c^3 d}",1,"((-1/4*I)*((4*I)*a^3*c*x - 6*a^2*b*c*x + 2*a^3*c^2*x^2 - (4*I)*a^3*ArcTan[c*x] + 6*a^2*b*ArcTan[c*x] + (12*I)*a^2*b*c*x*ArcTan[c*x] - 12*a*b^2*c*x*ArcTan[c*x] + 6*a^2*b*c^2*x^2*ArcTan[c*x] - (12*I)*a^2*b*ArcTan[c*x]^2 + 18*a*b^2*ArcTan[c*x]^2 + (6*I)*b^3*ArcTan[c*x]^2 + (12*I)*a*b^2*c*x*ArcTan[c*x]^2 - 6*b^3*c*x*ArcTan[c*x]^2 + 6*a*b^2*c^2*x^2*ArcTan[c*x]^2 - (8*I)*a*b^2*ArcTan[c*x]^3 + 6*b^3*ArcTan[c*x]^3 + (4*I)*b^3*c*x*ArcTan[c*x]^3 + 2*b^3*c^2*x^2*ArcTan[c*x]^3 - (2*I)*b^3*ArcTan[c*x]^4 + 12*a^2*b*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*a*b^2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - 12*b^3*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 12*a*b^2*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (12*I)*b^3*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + 4*b^3*ArcTan[c*x]^3*Log[1 + E^((2*I)*ArcTan[c*x])] - 2*a^3*Log[1 + c^2*x^2] - (6*I)*a^2*b*Log[1 + c^2*x^2] + 6*a*b^2*Log[1 + c^2*x^2] - (6*I)*b*(a + I*b + b*ArcTan[c*x])^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 6*b^2*(a + I*b + b*ArcTan[c*x])*PolyLog[3, -E^((2*I)*ArcTan[c*x])] + (3*I)*b^3*PolyLog[4, -E^((2*I)*ArcTan[c*x])]))/(c^3*d)","A",0
128,1,393,277,0.7977675,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)^3}{d+i c d x} \, dx","Integrate[(x*(a + b*ArcTan[c*x])^3)/(d + I*c*d*x),x]","-\frac{i \left(2 i a^3 \log \left(c^2 x^2+1\right)+4 a^3 c x-4 a^3 \tan ^{-1}(c x)-6 a^2 b \log \left(c^2 x^2+1\right)-12 a^2 b \tan ^{-1}(c x)^2+12 a^2 b c x \tan ^{-1}(c x)-12 i a^2 b \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-6 b \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right) \left(2 b (a+i b) \tan ^{-1}(c x)+a (a+2 i b)+b^2 \tan ^{-1}(c x)^2\right)+6 b^2 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right) \left(-i a-i b \tan ^{-1}(c x)+b\right)-8 a b^2 \tan ^{-1}(c x)^3-12 i a b^2 \tan ^{-1}(c x)^2+12 a b^2 c x \tan ^{-1}(c x)^2-12 i a b^2 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+24 a b^2 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+3 b^3 \text{Li}_4\left(-e^{2 i \tan ^{-1}(c x)}\right)-2 b^3 \tan ^{-1}(c x)^4-4 i b^3 \tan ^{-1}(c x)^3+4 b^3 c x \tan ^{-1}(c x)^3-4 i b^3 \tan ^{-1}(c x)^3 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+12 b^3 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{4 c^2 d}","\frac{3 b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d}-\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c^2 d}-\frac{3 i b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 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 i b^3 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 c^2 d}+\frac{3 i b^3 \text{Li}_4\left(1-\frac{2}{i c x+1}\right)}{4 c^2 d}",1,"((-1/4*I)*(4*a^3*c*x - 4*a^3*ArcTan[c*x] + 12*a^2*b*c*x*ArcTan[c*x] - 12*a^2*b*ArcTan[c*x]^2 - (12*I)*a*b^2*ArcTan[c*x]^2 + 12*a*b^2*c*x*ArcTan[c*x]^2 - 8*a*b^2*ArcTan[c*x]^3 - (4*I)*b^3*ArcTan[c*x]^3 + 4*b^3*c*x*ArcTan[c*x]^3 - 2*b^3*ArcTan[c*x]^4 - (12*I)*a^2*b*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 24*a*b^2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - (12*I)*a*b^2*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + 12*b^3*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - (4*I)*b^3*ArcTan[c*x]^3*Log[1 + E^((2*I)*ArcTan[c*x])] + (2*I)*a^3*Log[1 + c^2*x^2] - 6*a^2*b*Log[1 + c^2*x^2] - 6*b*(a*(a + (2*I)*b) + 2*(a + I*b)*b*ArcTan[c*x] + b^2*ArcTan[c*x]^2)*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 6*b^2*((-I)*a + b - I*b*ArcTan[c*x])*PolyLog[3, -E^((2*I)*ArcTan[c*x])] + 3*b^3*PolyLog[4, -E^((2*I)*ArcTan[c*x])]))/(c^2*d)","A",0
129,1,133,139,0.0792463,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{d+i c d x} \, dx","Integrate[(a + b*ArcTan[c*x])^3/(d + I*c*d*x),x]","\frac{i \left(4 \log \left(\frac{2 d}{d+i c d x}\right) \left(a+b \tan ^{-1}(c x)\right)^3+3 i b \left(2 \text{Li}_2\left(\frac{c x+i}{c x-i}\right) \left(a+b \tan ^{-1}(c x)\right)^2-b \left(2 i \text{Li}_3\left(\frac{c x+i}{c x-i}\right) \left(a+b \tan ^{-1}(c x)\right)+b \text{Li}_4\left(\frac{c x+i}{c x-i}\right)\right)\right)\right)}{4 c d}","\frac{3 i b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c d}-\frac{3 b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 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 b^3 \text{Li}_4\left(1-\frac{2}{i c x+1}\right)}{4 c d}",1,"((I/4)*(4*(a + b*ArcTan[c*x])^3*Log[(2*d)/(d + I*c*d*x)] + (3*I)*b*(2*(a + b*ArcTan[c*x])^2*PolyLog[2, (I + c*x)/(-I + c*x)] - b*((2*I)*(a + b*ArcTan[c*x])*PolyLog[3, (I + c*x)/(-I + c*x)] + b*PolyLog[4, (I + c*x)/(-I + c*x)]))))/(c*d)","A",1
130,1,311,128,0.1818731,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{x (d+i c d x)} \, dx","Integrate[(a + b*ArcTan[c*x])^3/(x*(d + I*c*d*x)),x]","\frac{4 a^3 \log \left(\frac{2 i}{-c x+i}\right)+8 a^3 \tanh ^{-1}\left(\frac{c x+i}{c x-i}\right)+12 a^2 b \log \left(\frac{2 i}{-c x+i}\right) \tan ^{-1}(c x)+24 a^2 b \tan ^{-1}(c x) \tanh ^{-1}\left(\frac{c x+i}{c x-i}\right)+6 b^2 \text{Li}_3\left(\frac{c x+i}{i-c x}\right) \left(a+b \tan ^{-1}(c x)\right)+12 a b^2 \log \left(\frac{2 i}{-c x+i}\right) \tan ^{-1}(c x)^2+24 a b^2 \tan ^{-1}(c x)^2 \tanh ^{-1}\left(\frac{c x+i}{c x-i}\right)+6 i b \text{Li}_2\left(\frac{c x+i}{i-c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2-3 i b^3 \text{Li}_4\left(\frac{c x+i}{i-c x}\right)+4 b^3 \log \left(\frac{2 i}{-c x+i}\right) \tan ^{-1}(c x)^3+8 b^3 \tan ^{-1}(c x)^3 \tanh ^{-1}\left(\frac{c x+i}{c x-i}\right)}{4 d}","\frac{3 b^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{2 d}+\frac{3 i b \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}+\frac{\log \left(2-\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{d}-\frac{3 i b^3 \text{Li}_4\left(\frac{2}{i c x+1}-1\right)}{4 d}",1,"(8*a^3*ArcTanh[(I + c*x)/(-I + c*x)] + 24*a^2*b*ArcTan[c*x]*ArcTanh[(I + c*x)/(-I + c*x)] + 24*a*b^2*ArcTan[c*x]^2*ArcTanh[(I + c*x)/(-I + c*x)] + 8*b^3*ArcTan[c*x]^3*ArcTanh[(I + c*x)/(-I + c*x)] + 4*a^3*Log[(2*I)/(I - c*x)] + 12*a^2*b*ArcTan[c*x]*Log[(2*I)/(I - c*x)] + 12*a*b^2*ArcTan[c*x]^2*Log[(2*I)/(I - c*x)] + 4*b^3*ArcTan[c*x]^3*Log[(2*I)/(I - c*x)] + (6*I)*b*(a + b*ArcTan[c*x])^2*PolyLog[2, (I + c*x)/(I - c*x)] + 6*b^2*(a + b*ArcTan[c*x])*PolyLog[3, (I + c*x)/(I - c*x)] - (3*I)*b^3*PolyLog[4, (I + c*x)/(I - c*x)])/(4*d)","B",0
131,1,436,263,1.4882892,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{x^2 (d+i c d x)} \, dx","Integrate[(a + b*ArcTan[c*x])^3/(x^2*(d + I*c*d*x)),x]","-\frac{-i a^3 c \log \left(c^2 x^2+1\right)+2 i a^3 c \log (x)+2 a^3 c \tan ^{-1}(c x)+\frac{2 a^3}{x}+3 a^2 b c \left(2 \left(-\log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+\tan ^{-1}(c x)^2+\tan ^{-1}(c x) \left(\frac{1}{c x}+i \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)\right)\right)+\text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)\right)+6 i a b^2 c \left(i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+\text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-\frac{i \tan ^{-1}(c x)^2}{c x}+\tan ^{-1}(c x)^2+\tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)+2 i \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)-\frac{i \pi ^3}{24}\right)+2 i b^3 c \left(\frac{3}{2} i \left(\tan ^{-1}(c x)+2 i\right) \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+\frac{3}{2} \left(\tan ^{-1}(c x)+i\right) \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-\frac{3}{4} i \text{Li}_4\left(e^{-2 i \tan ^{-1}(c x)}\right)-\frac{i \tan ^{-1}(c x)^3}{c x}-\tan ^{-1}(c x)^3+\tan ^{-1}(c x)^3 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)+3 i \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-\frac{i \pi ^4}{64}+\frac{\pi ^3}{8}\right)}{2 d}","-\frac{3 i b^2 c \text{Li}_2\left(\frac{2}{1-i c x}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{3 i b^2 c \text{Li}_3\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{2 d}+\frac{3 b c \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 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 b^3 c \text{Li}_3\left(\frac{2}{1-i c x}-1\right)}{2 d}-\frac{3 b^3 c \text{Li}_4\left(\frac{2}{i c x+1}-1\right)}{4 d}",1,"-1/2*((2*a^3)/x + 2*a^3*c*ArcTan[c*x] + (2*I)*a^3*c*Log[x] - I*a^3*c*Log[1 + c^2*x^2] + 3*a^2*b*c*(2*(ArcTan[c*x]^2 + ArcTan[c*x]*(1/(c*x) + I*Log[1 - E^((2*I)*ArcTan[c*x])]) - Log[(c*x)/Sqrt[1 + c^2*x^2]]) + PolyLog[2, E^((2*I)*ArcTan[c*x])]) + (6*I)*a*b^2*c*((-1/24*I)*Pi^3 + ArcTan[c*x]^2 - (I*ArcTan[c*x]^2)/(c*x) + ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + (2*I)*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + PolyLog[2, E^((2*I)*ArcTan[c*x])] + PolyLog[3, E^((-2*I)*ArcTan[c*x])]/2) + (2*I)*b^3*c*(Pi^3/8 - (I/64)*Pi^4 - ArcTan[c*x]^3 - (I*ArcTan[c*x]^3)/(c*x) + (3*I)*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + ArcTan[c*x]^3*Log[1 - E^((-2*I)*ArcTan[c*x])] + ((3*I)/2)*ArcTan[c*x]*(2*I + ArcTan[c*x])*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + (3*(I + ArcTan[c*x])*PolyLog[3, E^((-2*I)*ArcTan[c*x])])/2 - ((3*I)/4)*PolyLog[4, E^((-2*I)*ArcTan[c*x])]))/d","A",0
132,1,634,414,2.6409874,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{x^3 (d+i c d x)} \, dx","Integrate[(a + b*ArcTan[c*x])^3/(x^3*(d + I*c*d*x)),x]","\frac{a^3 c^2 \log \left(c^2 x^2+1\right)-2 a^3 c^2 \log (x)+2 i a^3 c^2 \tan ^{-1}(c x)+\frac{2 i a^3 c}{x}-\frac{a^3}{x^2}+\frac{3 i a^2 b \left(c^2 x^2 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+c x \left(-2 c x \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+i\right)+2 c^2 x^2 \tan ^{-1}(c x)^2+\tan ^{-1}(c x) \left(i c^2 x^2+2 i c^2 x^2 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+2 c x+i\right)\right)}{x^2}+6 a b^2 c^2 \left(\log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)-\frac{\tan ^{-1}(c x)^2}{2 c^2 x^2}-i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)-\text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)+\frac{i \tan ^{-1}(c x)^2}{c x}-\frac{3}{2} \tan ^{-1}(c x)^2-\frac{\tan ^{-1}(c x)}{c x}+\tan ^{-1}(c x)^2 \left(-\log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)\right)-2 i \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+\frac{i \pi ^3}{24}\right)+2 b^3 c^2 \left(-\frac{\left(c^2 x^2+1\right) \tan ^{-1}(c x)^3}{2 c^2 x^2}+\frac{3}{2} \left(2-i \tan ^{-1}(c x)\right) \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)-\frac{3}{2} i \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)-\frac{3}{2} \left(\tan ^{-1}(c x)+i\right) \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)+\frac{3}{4} i \text{Li}_4\left(e^{-2 i \tan ^{-1}(c x)}\right)+\frac{i \tan ^{-1}(c x)^3}{c x}+\tan ^{-1}(c x)^3-\frac{3 \tan ^{-1}(c x)^2}{2 c x}-\frac{3}{2} i \tan ^{-1}(c x)^2-\tan ^{-1}(c x)^3 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-3 i \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)+3 \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)+\frac{i \pi ^4}{64}-\frac{\pi ^3}{8}\right)}{2 d}","-\frac{3 b^2 c^2 \text{Li}_2\left(\frac{2}{1-i c x}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{3 b^2 c^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{2 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 \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 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 i b^3 c^2 \text{Li}_2\left(\frac{2}{1-i c x}-1\right)}{2 d}-\frac{3 i b^3 c^2 \text{Li}_3\left(\frac{2}{1-i c x}-1\right)}{2 d}+\frac{3 i b^3 c^2 \text{Li}_4\left(\frac{2}{i c x+1}-1\right)}{4 d}",1,"(-(a^3/x^2) + ((2*I)*a^3*c)/x + (2*I)*a^3*c^2*ArcTan[c*x] - 2*a^3*c^2*Log[x] + a^3*c^2*Log[1 + c^2*x^2] + ((3*I)*a^2*b*(2*c^2*x^2*ArcTan[c*x]^2 + ArcTan[c*x]*(I + 2*c*x + I*c^2*x^2 + (2*I)*c^2*x^2*Log[1 - E^((2*I)*ArcTan[c*x])]) + c*x*(I - 2*c*x*Log[(c*x)/Sqrt[1 + c^2*x^2]]) + c^2*x^2*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/x^2 + 6*a*b^2*c^2*((I/24)*Pi^3 - ArcTan[c*x]/(c*x) - (3*ArcTan[c*x]^2)/2 - ArcTan[c*x]^2/(2*c^2*x^2) + (I*ArcTan[c*x]^2)/(c*x) - ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - (2*I)*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + Log[(c*x)/Sqrt[1 + c^2*x^2]] - I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - PolyLog[2, E^((2*I)*ArcTan[c*x])] - PolyLog[3, E^((-2*I)*ArcTan[c*x])]/2) + 2*b^3*c^2*(-1/8*Pi^3 + (I/64)*Pi^4 - ((3*I)/2)*ArcTan[c*x]^2 - (3*ArcTan[c*x]^2)/(2*c*x) + ArcTan[c*x]^3 + (I*ArcTan[c*x]^3)/(c*x) - ((1 + c^2*x^2)*ArcTan[c*x]^3)/(2*c^2*x^2) - (3*I)*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - ArcTan[c*x]^3*Log[1 - E^((-2*I)*ArcTan[c*x])] + 3*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + (3*(2 - I*ArcTan[c*x])*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])])/2 - ((3*I)/2)*PolyLog[2, E^((2*I)*ArcTan[c*x])] - (3*(I + ArcTan[c*x])*PolyLog[3, E^((-2*I)*ArcTan[c*x])])/2 + ((3*I)/4)*PolyLog[4, E^((-2*I)*ArcTan[c*x])]))/(2*d)","A",0
133,0,0,25,3.4046645,"\int \frac{1}{(d+i c d x) \left(a+b \tan ^{-1}(c x)\right)} \, dx","Integrate[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,"Integrate[1/((d + I*c*d*x)*(a + b*ArcTan[c*x])), x]","A",-1
134,1,484,297,3.5741367,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{d+e x} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x]))/(d + e*x),x]","-\frac{6 a d^3 \log (d+e x)-6 a d^2 e x+3 a d e^2 x^2-2 a e^3 x^3+\frac{b e^3}{c^3}+\frac{3}{2} \pi  b d^3 \log \left(c^2 x^2+1\right)-\frac{3 b d^2 e \sqrt{\frac{c^2 d^2}{e^2}+1} \tan ^{-1}(c x)^2 e^{i \tan ^{-1}\left(\frac{c d}{e}\right)}}{c}+\frac{3 b d^2 e \log \left(c^2 x^2+1\right)}{c}+\frac{3 b d e^2 \tan ^{-1}(c x)}{c^2}-\frac{b e^3 \log \left(c^2 x^2+1\right)}{c^3}-3 i b d^3 \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)-6 i b d^3 \tan ^{-1}(c x) \tan ^{-1}\left(\frac{c d}{e}\right)+6 b d^3 \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)+6 b d^3 \tan ^{-1}(c x) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)-6 b d^3 \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right)+3 i b d^3 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+3 i b d^3 \tan ^{-1}(c x)^2+3 i \pi  b d^3 \tan ^{-1}(c x)+3 \pi  b d^3 \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right)-6 b d^3 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+\frac{3 b d^2 e \tan ^{-1}(c x)^2}{c}-6 b d^2 e x \tan ^{-1}(c x)+3 b d e^2 x^2 \tan ^{-1}(c x)-\frac{3 b d e^2 x}{c}-2 b e^3 x^3 \tan ^{-1}(c x)+\frac{b e^3 x^2}{c}}{6 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{i b d^3 \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 e^4}+\frac{i b d^3 \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 e^4}+\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,"-1/6*((b*e^3)/c^3 - 6*a*d^2*e*x - (3*b*d*e^2*x)/c + 3*a*d*e^2*x^2 + (b*e^3*x^2)/c - 2*a*e^3*x^3 + (3*b*d*e^2*ArcTan[c*x])/c^2 + (3*I)*b*d^3*Pi*ArcTan[c*x] - 6*b*d^2*e*x*ArcTan[c*x] + 3*b*d*e^2*x^2*ArcTan[c*x] - 2*b*e^3*x^3*ArcTan[c*x] - (6*I)*b*d^3*ArcTan[(c*d)/e]*ArcTan[c*x] + (3*I)*b*d^3*ArcTan[c*x]^2 + (3*b*d^2*e*ArcTan[c*x]^2)/c - (3*b*d^2*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^2)/c + 3*b*d^3*Pi*Log[1 + E^((-2*I)*ArcTan[c*x])] - 6*b*d^3*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 6*b*d^3*ArcTan[(c*d)/e]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 6*b*d^3*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 6*a*d^3*Log[d + e*x] + (3*b*d^2*e*Log[1 + c^2*x^2])/c - (b*e^3*Log[1 + c^2*x^2])/c^3 + (3*b*d^3*Pi*Log[1 + c^2*x^2])/2 - 6*b*d^3*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + (3*I)*b*d^3*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (3*I)*b*d^3*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))])/e^4","A",0
135,1,404,237,1.6781256,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{d+e x} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x]))/(d + e*x),x]","\frac{2 a d^2 \log (d+e x)-2 a d e x+a e^2 x^2-\frac{b d e \sqrt{\frac{c^2 d^2}{e^2}+1} \tan ^{-1}(c x)^2 e^{i \tan ^{-1}\left(\frac{c d}{e}\right)}}{c}+\frac{1}{2} \pi  b d^2 \log \left(c^2 x^2+1\right)+\frac{b d e \log \left(c^2 x^2+1\right)}{c}+\frac{b e^2 \tan ^{-1}(c x)}{c^2}-i b d^2 \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)-2 i b d^2 \tan ^{-1}(c x) \tan ^{-1}\left(\frac{c d}{e}\right)+2 b d^2 \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)+2 b d^2 \tan ^{-1}(c x) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)-2 b d^2 \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right)+i b d^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+i b d^2 \tan ^{-1}(c x)^2+i \pi  b d^2 \tan ^{-1}(c x)+\pi  b d^2 \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right)-2 b d^2 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+\frac{b d e \tan ^{-1}(c x)^2}{c}-2 b d e x \tan ^{-1}(c x)+b e^2 x^2 \tan ^{-1}(c x)-\frac{b e^2 x}{c}}{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{i b d^2 \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 e^3}-\frac{i b d^2 \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 e^3}-\frac{b d x \tan ^{-1}(c x)}{e^2}-\frac{b x}{2 c e}",1,"(-2*a*d*e*x - (b*e^2*x)/c + a*e^2*x^2 + (b*e^2*ArcTan[c*x])/c^2 + I*b*d^2*Pi*ArcTan[c*x] - 2*b*d*e*x*ArcTan[c*x] + b*e^2*x^2*ArcTan[c*x] - (2*I)*b*d^2*ArcTan[(c*d)/e]*ArcTan[c*x] + I*b*d^2*ArcTan[c*x]^2 + (b*d*e*ArcTan[c*x]^2)/c - (b*d*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^2)/c + b*d^2*Pi*Log[1 + E^((-2*I)*ArcTan[c*x])] - 2*b*d^2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 2*b*d^2*ArcTan[(c*d)/e]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 2*b*d^2*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 2*a*d^2*Log[d + e*x] + (b*d*e*Log[1 + c^2*x^2])/c + (b*d^2*Pi*Log[1 + c^2*x^2])/2 - 2*b*d^2*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + I*b*d^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - I*b*d^2*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))])/(2*e^3)","A",0
136,1,329,179,1.5853103,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{d+e x} \, dx","Integrate[(x*(a + b*ArcTan[c*x]))/(d + e*x),x]","\frac{-2 a d \log (d+e x)+2 a e x+\frac{b \left(e \sqrt{\frac{c^2 d^2}{e^2}+1} \tan ^{-1}(c x)^2 e^{i \tan ^{-1}\left(\frac{c d}{e}\right)}-\frac{1}{2} \pi  c d \log \left(c^2 x^2+1\right)-e \log \left(c^2 x^2+1\right)+i c d \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)+2 i c d \tan ^{-1}(c x) \tan ^{-1}\left(\frac{c d}{e}\right)-2 c d \tan ^{-1}(c x) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)-2 c d \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)+2 c d \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right)-i c d \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-i c d \tan ^{-1}(c x)^2-i \pi  c d \tan ^{-1}(c x)+2 c d \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-\pi  c d \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right)-e \tan ^{-1}(c x)^2+2 c e x \tan ^{-1}(c x)\right)}{c}}{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{i b d \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 e^2}+\frac{i b d \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 e^2}+\frac{b x \tan ^{-1}(c x)}{e}",1,"(2*a*e*x - 2*a*d*Log[d + e*x] + (b*((-I)*c*d*Pi*ArcTan[c*x] + 2*c*e*x*ArcTan[c*x] + (2*I)*c*d*ArcTan[(c*d)/e]*ArcTan[c*x] - I*c*d*ArcTan[c*x]^2 - e*ArcTan[c*x]^2 + Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^2 - c*d*Pi*Log[1 + E^((-2*I)*ArcTan[c*x])] + 2*c*d*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - 2*c*d*ArcTan[(c*d)/e]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - 2*c*d*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - e*Log[1 + c^2*x^2] - (c*d*Pi*Log[1 + c^2*x^2])/2 + 2*c*d*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] - I*c*d*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + I*c*d*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/c)/(2*e^2)","A",0
137,1,138,138,0.0615195,"\int \frac{a+b \tan ^{-1}(c x)}{d+e x} \, dx","Integrate[(a + b*ArcTan[c*x])/(d + e*x),x]","\frac{2 a \log (d+e x)+i b \text{Li}_2\left(\frac{e (1-i c x)}{i c d+e}\right)-i b \text{Li}_2\left(-\frac{e (c x-i)}{c d+i e}\right)+i b \log (1-i c x) \log \left(\frac{c (d+e x)}{c d-i e}\right)-i b \log (1+i c x) \log \left(\frac{c (d+e x)}{c d+i e}\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{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 e}+\frac{i b \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 e}",1,"(2*a*Log[d + e*x] + I*b*Log[1 - I*c*x]*Log[(c*(d + e*x))/(c*d - I*e)] - I*b*Log[1 + I*c*x]*Log[(c*(d + e*x))/(c*d + I*e)] + I*b*PolyLog[2, (e*(1 - I*c*x))/(I*c*d + e)] - I*b*PolyLog[2, -((e*(-I + c*x))/(c*d + I*e))])/(2*e)","A",0
138,1,169,181,0.1000622,"\int \frac{a+b \tan ^{-1}(c x)}{x (d+e x)} \, dx","Integrate[(a + b*ArcTan[c*x])/(x*(d + e*x)),x]","\frac{-2 a \log (d+e x)+2 a \log (x)-i b \text{Li}_2\left(\frac{e (1-i c x)}{i c d+e}\right)+i b \text{Li}_2\left(-\frac{e (c x-i)}{c d+i e}\right)-i b \log (1-i c x) \log \left(\frac{c (d+e x)}{c d-i e}\right)+i b \log (1+i c x) \log \left(\frac{c (d+e x)}{c d+i e}\right)+i b \text{Li}_2(-i c x)-i b \text{Li}_2(i c x)}{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{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 d}+\frac{i b \text{Li}_2(-i c x)}{2 d}-\frac{i b \text{Li}_2(i c x)}{2 d}-\frac{i b \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 d}",1,"(2*a*Log[x] - 2*a*Log[d + e*x] - I*b*Log[1 - I*c*x]*Log[(c*(d + e*x))/(c*d - I*e)] + I*b*Log[1 + I*c*x]*Log[(c*(d + e*x))/(c*d + I*e)] + I*b*PolyLog[2, (-I)*c*x] - I*b*PolyLog[2, I*c*x] - I*b*PolyLog[2, (e*(1 - I*c*x))/(I*c*d + e)] + I*b*PolyLog[2, -((e*(-I + c*x))/(c*d + I*e))])/(2*d)","A",0
139,1,223,232,0.1546634,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 (d+e x)} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^2*(d + e*x)),x]","-\frac{-2 a e x \log (d+e x)+2 a d+2 a e x \log (x)+b c d x \log \left(c^2 x^2+1\right)-i b e x \text{Li}_2\left(\frac{e (1-i c x)}{i c d+e}\right)+i b e x \text{Li}_2\left(-\frac{e (c x-i)}{c d+i e}\right)-i b e x \log (1-i c x) \log \left(\frac{c (d+e x)}{c d-i e}\right)+i b e x \log (1+i c x) \log \left(\frac{c (d+e x)}{c d+i e}\right)-2 b c d x \log (x)+2 b d \tan ^{-1}(c x)+i b e x \text{Li}_2(-i c x)-i b e x \text{Li}_2(i c x)}{2 d^2 x}","-\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{i b e \text{Li}_2(-i c x)}{2 d^2}+\frac{i b e \text{Li}_2(i c x)}{2 d^2}+\frac{i b e \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 d^2}-\frac{i b e \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 d^2}+\frac{b c \log (x)}{d}",1,"-1/2*(2*a*d + 2*b*d*ArcTan[c*x] - 2*b*c*d*x*Log[x] + 2*a*e*x*Log[x] - 2*a*e*x*Log[d + e*x] - I*b*e*x*Log[1 - I*c*x]*Log[(c*(d + e*x))/(c*d - I*e)] + I*b*e*x*Log[1 + I*c*x]*Log[(c*(d + e*x))/(c*d + I*e)] + b*c*d*x*Log[1 + c^2*x^2] + I*b*e*x*PolyLog[2, (-I)*c*x] - I*b*e*x*PolyLog[2, I*c*x] - I*b*e*x*PolyLog[2, (e*(1 - I*c*x))/(I*c*d + e)] + I*b*e*x*PolyLog[2, -((e*(-I + c*x))/(c*d + I*e))])/(d^2*x)","A",0
140,1,298,293,0.1861479,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 (d+e x)} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^3*(d + e*x)),x]","\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{a e^2 \log (d+e x)}{d^3}-\frac{b c e \left(2 \log (x)-\log \left(c^2 x^2+1\right)\right)}{2 d^2}-\frac{b c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)}{2 d x}+\frac{i b e^2 \text{Li}_2(-i c x)}{2 d^3}-\frac{i b e^2 \text{Li}_2(i c x)}{2 d^3}-\frac{i b \left(e^2 \text{Li}_2\left(\frac{e (1-i c x)}{i c d+e}\right)+e^2 \log (1-i c x) \log \left(\frac{c (d+e x)}{c d-i e}\right)\right)}{2 d^3}+\frac{i b \left(e^2 \text{Li}_2\left(-\frac{e (i c x+1)}{i c d-e}\right)+e^2 \log (1+i c x) \log \left(\frac{c (d+e x)}{c d+i e}\right)\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{i b e^2 \text{Li}_2(-i c x)}{2 d^3}-\frac{i b e^2 \text{Li}_2(i c x)}{2 d^3}-\frac{i b e^2 \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 d^3}+\frac{i b e^2 \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 d^3}-\frac{b c e \log (x)}{d^2}-\frac{b c}{2 d x}",1,"-1/2*(a + b*ArcTan[c*x])/(d*x^2) + (e*(a + b*ArcTan[c*x]))/(d^2*x) - (b*c*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/(2*d*x) + (a*e^2*Log[x])/d^3 - (a*e^2*Log[d + e*x])/d^3 - (b*c*e*(2*Log[x] - 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*Log[1 - I*c*x]*Log[(c*(d + e*x))/(c*d - I*e)] + e^2*PolyLog[2, (e*(1 - I*c*x))/(I*c*d + e)]))/d^3 + ((I/2)*b*(e^2*Log[1 + I*c*x]*Log[(c*(d + e*x))/(c*d + I*e)] + e^2*PolyLog[2, -((e*(1 + I*c*x))/(I*c*d - e))]))/d^3","C",0
141,1,1413,598,22.327387,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{d+e x} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x])^2)/(d + e*x),x]","-\frac{-2 a^2 e^3 x^3 c^3+4 i b^2 d^3 \tan ^{-1}(c x)^3 c^3+3 a^2 d e^2 x^2 c^3+6 i a b d^3 \tan ^{-1}(c x)^2 c^3-2 b^2 e^3 x^3 \tan ^{-1}(c x)^2 c^3+3 b^2 d e^2 x^2 \tan ^{-1}(c x)^2 c^3-6 b^2 d^2 e x \tan ^{-1}(c x)^2 c^3-6 a^2 d^2 e x c^3-4 a b e^3 x^3 \tan ^{-1}(c x) c^3+6 a b d e^2 x^2 \tan ^{-1}(c x) c^3-12 a b d^2 e x \tan ^{-1}(c x) c^3-12 i a b d^3 \tan ^{-1}\left(\frac{c d}{e}\right) \tan ^{-1}(c x) c^3+6 i a b d^3 \pi  \tan ^{-1}(c x) c^3+6 b^2 d^3 \pi  \tan ^{-1}(c x) \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right) c^3+6 a b d^3 \pi  \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right) c^3-6 b^2 d^3 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^3-12 a b d^3 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^3+12 b^2 d^3 \tan ^{-1}(c x)^2 \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) c^3+12 a b d^3 \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) c^3+12 a b d^3 \tan ^{-1}(c x) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) c^3+12 b^2 d^3 \tan ^{-1}\left(\frac{c d}{e}\right) \tan ^{-1}(c x) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) c^3-6 b^2 d^3 \pi  \tan ^{-1}(c x) \log \left(-\frac{2 i}{c x-i}\right) c^3+6 a^2 d^3 \log (d+e x) c^3+3 a b d^3 \pi  \log \left(c^2 x^2+1\right) c^3+12 b^2 d^3 \tan ^{-1}\left(\frac{c d}{e}\right) \tan ^{-1}(c x) \log \left(\frac{e^{-i \tan ^{-1}\left(\frac{c d}{e}\right)} \left(c x+e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} (c x-i)+i\right)}{2 \sqrt{c^2 x^2+1}}\right) c^3-6 b^2 d^3 \tan ^{-1}(c x)^2 \log \left(-e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \cos \left(2 \tan ^{-1}(c x)\right)-i e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \sin \left(2 \tan ^{-1}(c x)\right)+1\right) c^3-12 b^2 d^3 \tan ^{-1}\left(\frac{c d}{e}\right) \tan ^{-1}(c x) \log \left(-e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \cos \left(2 \tan ^{-1}(c x)\right)-i e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \sin \left(2 \tan ^{-1}(c x)\right)+1\right) c^3-12 a b d^3 \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right) c^3-12 b^2 d^3 \tan ^{-1}\left(\frac{c d}{e}\right) \tan ^{-1}(c x) \log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right) c^3-6 i b d^3 \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) c^3-3 b^2 d^3 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right) c^3+3 b^2 d^3 \text{Li}_3\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) c^3-4 b^2 d^2 \sqrt{\frac{c^2 d^2}{e^2}+1} e e^{i \tan ^{-1}\left(\frac{c d}{e}\right)} \tan ^{-1}(c x)^3 c^2+4 b^2 d^2 e \tan ^{-1}(c x)^3 c^2+2 a b e^3 x^2 c^2-6 a b d^2 \sqrt{\frac{c^2 d^2}{e^2}+1} e e^{i \tan ^{-1}\left(\frac{c d}{e}\right)} \tan ^{-1}(c x)^2 c^2+6 i b^2 d^2 e \tan ^{-1}(c x)^2 c^2+6 a b d^2 e \tan ^{-1}(c x)^2 c^2-6 a b d e^2 x c^2+2 b^2 e^3 x^2 \tan ^{-1}(c x) c^2-6 b^2 d e^2 x \tan ^{-1}(c x) c^2-12 b^2 d^2 e \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2+6 a b d^2 e \log \left(c^2 x^2+1\right) c^2+3 b^2 d e^2 \tan ^{-1}(c x)^2 c-2 b^2 e^3 x c+6 a b d e^2 \tan ^{-1}(c x) c+3 b^2 d e^2 \log \left(c^2 x^2+1\right) c+2 a b e^3-2 i b^2 e^3 \tan ^{-1}(c x)^2+2 b^2 e^3 \tan ^{-1}(c x)+4 b^2 e^3 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-2 a b e^3 \log \left(c^2 x^2+1\right)+2 i b \left(3 a c^3 d^3+3 b c^3 \tan ^{-1}(c x) d^3+3 b c^2 e d^2-b e^3\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)}{6 c^3 e^4}","-\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 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 e^2}-\frac{i b d^3 \text{Li}_2\left(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{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{e^4}+\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{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{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{i b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{3 c^3 e}-\frac{b^2 \tan ^{-1}(c x)}{3 c^3 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 d^3 \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{2 e^4}-\frac{b^2 d^3 \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 e^4}+\frac{i b^2 d^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c e^3}+\frac{b^2 d x \tan ^{-1}(c x)}{c e^2}",1,"-1/6*(2*a*b*e^3 - 6*a^2*c^3*d^2*e*x - 6*a*b*c^2*d*e^2*x - 2*b^2*c*e^3*x + 3*a^2*c^3*d*e^2*x^2 + 2*a*b*c^2*e^3*x^2 - 2*a^2*c^3*e^3*x^3 + 6*a*b*c*d*e^2*ArcTan[c*x] + 2*b^2*e^3*ArcTan[c*x] + (6*I)*a*b*c^3*d^3*Pi*ArcTan[c*x] - 12*a*b*c^3*d^2*e*x*ArcTan[c*x] - 6*b^2*c^2*d*e^2*x*ArcTan[c*x] + 6*a*b*c^3*d*e^2*x^2*ArcTan[c*x] + 2*b^2*c^2*e^3*x^2*ArcTan[c*x] - 4*a*b*c^3*e^3*x^3*ArcTan[c*x] - (12*I)*a*b*c^3*d^3*ArcTan[(c*d)/e]*ArcTan[c*x] + (6*I)*a*b*c^3*d^3*ArcTan[c*x]^2 + 6*a*b*c^2*d^2*e*ArcTan[c*x]^2 + (6*I)*b^2*c^2*d^2*e*ArcTan[c*x]^2 + 3*b^2*c*d*e^2*ArcTan[c*x]^2 - (2*I)*b^2*e^3*ArcTan[c*x]^2 - 6*a*b*c^2*d^2*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^2 - 6*b^2*c^3*d^2*e*x*ArcTan[c*x]^2 + 3*b^2*c^3*d*e^2*x^2*ArcTan[c*x]^2 - 2*b^2*c^3*e^3*x^3*ArcTan[c*x]^2 + (4*I)*b^2*c^3*d^3*ArcTan[c*x]^3 + 4*b^2*c^2*d^2*e*ArcTan[c*x]^3 - 4*b^2*c^2*d^2*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^3 + 6*a*b*c^3*d^3*Pi*Log[1 + E^((-2*I)*ArcTan[c*x])] + 6*b^2*c^3*d^3*Pi*ArcTan[c*x]*Log[1 + E^((-2*I)*ArcTan[c*x])] - 12*a*b*c^3*d^3*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - 12*b^2*c^2*d^2*e*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 4*b^2*e^3*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - 6*b^2*c^3*d^3*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + 12*a*b*c^3*d^3*ArcTan[(c*d)/e]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 12*a*b*c^3*d^3*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 12*b^2*c^3*d^3*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 12*b^2*c^3*d^3*ArcTan[c*x]^2*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - 6*b^2*c^3*d^3*Pi*ArcTan[c*x]*Log[(-2*I)/(-I + c*x)] + 6*a^2*c^3*d^3*Log[d + e*x] + 6*a*b*c^2*d^2*e*Log[1 + c^2*x^2] + 3*b^2*c*d*e^2*Log[1 + c^2*x^2] - 2*a*b*e^3*Log[1 + c^2*x^2] + 3*a*b*c^3*d^3*Pi*Log[1 + c^2*x^2] + 12*b^2*c^3*d^3*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[(I + c*x + E^((2*I)*ArcTan[(c*d)/e])*(-I + c*x))/(2*E^(I*ArcTan[(c*d)/e])*Sqrt[1 + c^2*x^2])] - 12*b^2*c^3*d^3*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]] - 6*b^2*c^3*d^3*ArcTan[c*x]^2*Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]] - 12*a*b*c^3*d^3*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] - 12*b^2*c^3*d^3*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + (2*I)*b*(3*a*c^3*d^3 + 3*b*c^2*d^2*e - b*e^3 + 3*b*c^3*d^3*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (6*I)*b*c^3*d^3*(a + b*ArcTan[c*x])*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - 3*b^2*c^3*d^3*PolyLog[3, -E^((2*I)*ArcTan[c*x])] + 3*b^2*c^3*d^3*PolyLog[3, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))])/(c^3*e^4)","B",0
142,1,1035,430,17.9789868,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{d+e x} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x])^2)/(d + e*x),x]","-\frac{\left(-2 i c^2 d^2 \tan ^{-1}(c x)^3-2 c d e \tan ^{-1}(c x)^3-6 i c d e \tan ^{-1}(c x)^2+6 c^2 d e x \tan ^{-1}(c x)^2-3 e^2 \left(c^2 x^2+1\right) \tan ^{-1}(c x)^2+6 c^2 d^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)^2+6 c e^2 x \tan ^{-1}(c x)+12 c d e \log \left(1+e^{2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)-3 e^2 \log \left(c^2 x^2+1\right)-6 i c d \left(e+c d \tan ^{-1}(c x)\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+3 c^2 d^2 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+c d \left(2 \tan ^{-1}(c x) \left(\left(-i c d+2 \sqrt{\frac{c^2 d^2}{e^2}+1} e e^{i \tan ^{-1}\left(\frac{c d}{e}\right)}-e\right) \tan ^{-1}(c x)^2-3 c d \left(2 \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)-\log \left(-e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \cos \left(2 \tan ^{-1}(c x)\right)-i e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \sin \left(2 \tan ^{-1}(c x)\right)+1\right)\right) \tan ^{-1}(c x)+3 c d \left(\pi  \left(\log \left(-\frac{2 i}{c x-i}\right)-\log \left(1+e^{-2 i \tan ^{-1}(c x)}\right)\right)-2 \tan ^{-1}\left(\frac{c d}{e}\right) \left(\log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)+\log \left(\frac{e^{-i \tan ^{-1}\left(\frac{c d}{e}\right)} \left(c x+e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} (c x-i)+i\right)}{2 \sqrt{c^2 x^2+1}}\right)-\log \left(-e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \cos \left(2 \tan ^{-1}(c x)\right)-i e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \sin \left(2 \tan ^{-1}(c x)\right)+1\right)-\log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right)\right)\right)\right)+6 i c d \tan ^{-1}(c x) \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)-3 c d \text{Li}_3\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)\right)\right) b^2+6 a \left(-i d^2 \tan ^{-1}(c x)^2 c^2+2 d e x \tan ^{-1}(c x) c^2+2 i d^2 \tan ^{-1}\left(\frac{c d}{e}\right) \tan ^{-1}(c x) c^2-i d^2 \pi  \tan ^{-1}(c x) c^2-d^2 \pi  \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right) c^2+2 d^2 \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 d^2 \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) c^2-2 d^2 \tan ^{-1}(c x) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) c^2-\frac{1}{2} d^2 \pi  \log \left(c^2 x^2+1\right) c^2+2 d^2 \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right) c^2-i d^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right) c^2+i d^2 \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) c^2+d \sqrt{\frac{c^2 d^2}{e^2}+1} e e^{i \tan ^{-1}\left(\frac{c d}{e}\right)} \tan ^{-1}(c x)^2 c-d e \tan ^{-1}(c x)^2 c+e^2 x c-d e \log \left(c^2 x^2+1\right) c-e^2 \left(c^2 x^2+1\right) \tan ^{-1}(c x)\right) b-3 a^2 c^2 e^2 x^2+6 a^2 c^2 d e x-6 a^2 c^2 d^2 \log (d+e x)}{6 c^2 e^3}","\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 e}+\frac{i b d^2 \text{Li}_2\left(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{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{e^3}-\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 d^2 \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{2 e^3}+\frac{b^2 d^2 \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 e^3}-\frac{i b^2 d \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c e^2}-\frac{b^2 x \tan ^{-1}(c x)}{c e}",1,"-1/6*(6*a^2*c^2*d*e*x - 3*a^2*c^2*e^2*x^2 - 6*a^2*c^2*d^2*Log[d + e*x] + 6*a*b*(c*e^2*x - I*c^2*d^2*Pi*ArcTan[c*x] + 2*c^2*d*e*x*ArcTan[c*x] - e^2*(1 + c^2*x^2)*ArcTan[c*x] + (2*I)*c^2*d^2*ArcTan[(c*d)/e]*ArcTan[c*x] - I*c^2*d^2*ArcTan[c*x]^2 - c*d*e*ArcTan[c*x]^2 + c*d*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^2 - c^2*d^2*Pi*Log[1 + E^((-2*I)*ArcTan[c*x])] + 2*c^2*d^2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - 2*c^2*d^2*ArcTan[(c*d)/e]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - 2*c^2*d^2*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - c*d*e*Log[1 + c^2*x^2] - (c^2*d^2*Pi*Log[1 + c^2*x^2])/2 + 2*c^2*d^2*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] - I*c^2*d^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + I*c^2*d^2*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]) + b^2*(6*c*e^2*x*ArcTan[c*x] - (6*I)*c*d*e*ArcTan[c*x]^2 + 6*c^2*d*e*x*ArcTan[c*x]^2 - 3*e^2*(1 + c^2*x^2)*ArcTan[c*x]^2 - (2*I)*c^2*d^2*ArcTan[c*x]^3 - 2*c*d*e*ArcTan[c*x]^3 + 12*c*d*e*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 6*c^2*d^2*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - 3*e^2*Log[1 + c^2*x^2] - (6*I)*c*d*(e + c*d*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 3*c^2*d^2*PolyLog[3, -E^((2*I)*ArcTan[c*x])] + c*d*(2*ArcTan[c*x]*(((-I)*c*d - e + 2*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e]))*ArcTan[c*x]^2 - 3*c*d*ArcTan[c*x]*(2*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]]) + 3*c*d*(Pi*(-Log[1 + E^((-2*I)*ArcTan[c*x])] + Log[(-2*I)/(-I + c*x)]) - 2*ArcTan[(c*d)/e]*(Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + Log[(I + c*x + E^((2*I)*ArcTan[(c*d)/e])*(-I + c*x))/(2*E^(I*ArcTan[(c*d)/e])*Sqrt[1 + c^2*x^2])] - Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]] - Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]]))) + (6*I)*c*d*ArcTan[c*x]*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - 3*c*d*PolyLog[3, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))])))/(c^2*e^3)","B",0
143,1,940,323,15.9319608,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{d+e x} \, dx","Integrate[(x*(a + b*ArcTan[c*x])^2)/(d + e*x),x]","-\frac{-4 b^2 \sqrt{\frac{c^2 d^2}{e^2}+1} e e^{i \tan ^{-1}\left(\frac{c d}{e}\right)} \tan ^{-1}(c x)^3+4 i b^2 c d \tan ^{-1}(c x)^3+4 b^2 e \tan ^{-1}(c x)^3-6 a b \sqrt{\frac{c^2 d^2}{e^2}+1} e e^{i \tan ^{-1}\left(\frac{c d}{e}\right)} \tan ^{-1}(c x)^2+6 i a b c d \tan ^{-1}(c x)^2+6 i b^2 e \tan ^{-1}(c x)^2+6 a b e \tan ^{-1}(c x)^2-6 b^2 c e x \tan ^{-1}(c x)^2-6 b^2 c d \log \left(1+e^{2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)^2+12 b^2 c d \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) \tan ^{-1}(c x)^2-6 b^2 c d \log \left(-e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \cos \left(2 \tan ^{-1}(c x)\right)-i e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \sin \left(2 \tan ^{-1}(c x)\right)+1\right) \tan ^{-1}(c x)^2-12 a b c e x \tan ^{-1}(c x)-12 i a b c d \tan ^{-1}\left(\frac{c d}{e}\right) \tan ^{-1}(c x)+6 b^2 c d \pi  \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)-12 a b c d \log \left(1+e^{2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)-12 b^2 e \log \left(1+e^{2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)+12 a b c d \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) \tan ^{-1}(c x)+12 b^2 c d \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) \tan ^{-1}(c x)-6 b^2 c d \pi  \log \left(-\frac{2 i}{c x-i}\right) \tan ^{-1}(c x)+12 b^2 c d \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(\frac{e^{-i \tan ^{-1}\left(\frac{c d}{e}\right)} \left(c x+e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} (c x-i)+i\right)}{2 \sqrt{c^2 x^2+1}}\right) \tan ^{-1}(c x)-12 b^2 c d \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(-e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \cos \left(2 \tan ^{-1}(c x)\right)-i e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \sin \left(2 \tan ^{-1}(c x)\right)+1\right) \tan ^{-1}(c x)-12 b^2 c d \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right) \tan ^{-1}(c x)+6 i a b c d \pi  \tan ^{-1}(c x)-6 a^2 c e x+6 a b c d \pi  \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right)+12 a b c d \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)+6 a^2 c d \log (d+e x)+6 a b e \log \left(c^2 x^2+1\right)+3 a b c d \pi  \log \left(c^2 x^2+1\right)-12 a b c d \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right)+6 i b \left(a c d+b c \tan ^{-1}(c x) d+b e\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-6 i b c d \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)-3 b^2 c d \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+3 b^2 c d \text{Li}_3\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)}{6 c e^2}","-\frac{i b d \text{Li}_2\left(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{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{e^2}+\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{b^2 d \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{2 e^2}-\frac{b^2 d \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 e^2}+\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c e}",1,"-1/6*(-6*a^2*c*e*x + (6*I)*a*b*c*d*Pi*ArcTan[c*x] - 12*a*b*c*e*x*ArcTan[c*x] - (12*I)*a*b*c*d*ArcTan[(c*d)/e]*ArcTan[c*x] + (6*I)*a*b*c*d*ArcTan[c*x]^2 + 6*a*b*e*ArcTan[c*x]^2 + (6*I)*b^2*e*ArcTan[c*x]^2 - 6*a*b*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^2 - 6*b^2*c*e*x*ArcTan[c*x]^2 + (4*I)*b^2*c*d*ArcTan[c*x]^3 + 4*b^2*e*ArcTan[c*x]^3 - 4*b^2*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^3 + 6*a*b*c*d*Pi*Log[1 + E^((-2*I)*ArcTan[c*x])] + 6*b^2*c*d*Pi*ArcTan[c*x]*Log[1 + E^((-2*I)*ArcTan[c*x])] - 12*a*b*c*d*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - 12*b^2*e*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - 6*b^2*c*d*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + 12*a*b*c*d*ArcTan[(c*d)/e]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 12*a*b*c*d*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 12*b^2*c*d*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 12*b^2*c*d*ArcTan[c*x]^2*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - 6*b^2*c*d*Pi*ArcTan[c*x]*Log[(-2*I)/(-I + c*x)] + 6*a^2*c*d*Log[d + e*x] + 6*a*b*e*Log[1 + c^2*x^2] + 3*a*b*c*d*Pi*Log[1 + c^2*x^2] + 12*b^2*c*d*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[(I + c*x + E^((2*I)*ArcTan[(c*d)/e])*(-I + c*x))/(2*E^(I*ArcTan[(c*d)/e])*Sqrt[1 + c^2*x^2])] - 12*b^2*c*d*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]] - 6*b^2*c*d*ArcTan[c*x]^2*Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]] - 12*a*b*c*d*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] - 12*b^2*c*d*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + (6*I)*b*(a*c*d + b*e + b*c*d*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (6*I)*b*c*d*(a + b*ArcTan[c*x])*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - 3*b^2*c*d*PolyLog[3, -E^((2*I)*ArcTan[c*x])] + 3*b^2*c*d*PolyLog[3, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))])/(c*e^2)","B",0
144,1,741,223,14.9683967,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d+e x} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(d + e*x),x]","\frac{6 a^2 c d \log (d+e x)+12 a b c d \left(\tan ^{-1}(c x) \left(\frac{1}{2} \log \left(c^2 x^2+1\right)+\log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right)\right)+\frac{1}{2} \left(-\log \left(\frac{2}{\sqrt{c^2 x^2+1}}\right) \left(\pi -2 \tan ^{-1}(c x)\right)-i \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)-i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)^2+2 \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)-2 \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right) \log \left(2 \sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right)-i \text{Li}_2\left(-e^{-2 i \tan ^{-1}(c x)}\right)-\frac{1}{4} i \left(\pi -2 \tan ^{-1}(c x)\right)^2+\left(\pi -2 \tan ^{-1}(c x)\right) \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right)\right)\right)+b^2 \left(-2 \tan ^{-1}(c x) \left(\tan ^{-1}(c x)^2 \left(2 e \sqrt{\frac{c^2 d^2}{e^2}+1} e^{i \tan ^{-1}\left(\frac{c d}{e}\right)}-i c d-e\right)+3 c d \left(\pi  \left(\log \left(-\frac{2 i}{c x-i}\right)-\log \left(1+e^{-2 i \tan ^{-1}(c x)}\right)\right)-2 \tan ^{-1}\left(\frac{c d}{e}\right) \left(\log \left(\frac{e^{-i \tan ^{-1}\left(\frac{c d}{e}\right)} \left((c x-i) e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)}+c x+i\right)}{2 \sqrt{c^2 x^2+1}}\right)+\log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)-\log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right)-\log \left(-i e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \sin \left(2 \tan ^{-1}(c x)\right)-e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \cos \left(2 \tan ^{-1}(c x)\right)+1\right)\right)\right)-3 c d \tan ^{-1}(c x) \left(2 \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)-\log \left(-i e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \sin \left(2 \tan ^{-1}(c x)\right)-e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \cos \left(2 \tan ^{-1}(c x)\right)+1\right)\right)\right)-6 i c d \tan ^{-1}(c x) \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)+3 c d \text{Li}_3\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)+2 \tan ^{-1}(c x)^2 \left((e+i c d) \tan ^{-1}(c x)-3 c d \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+6 i c d \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-3 c d \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)\right)}{6 c d e}","-\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{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{i b \text{Li}_2\left(1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{e}+\frac{b^2 \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 e}-\frac{b^2 \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{2 e}",1,"(6*a^2*c*d*Log[d + e*x] + 12*a*b*c*d*(ArcTan[c*x]*(Log[1 + c^2*x^2]/2 + Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]]) + ((-1/4*I)*(Pi - 2*ArcTan[c*x])^2 - I*(ArcTan[(c*d)/e] + ArcTan[c*x])^2 + (Pi - 2*ArcTan[c*x])*Log[1 + E^((-2*I)*ArcTan[c*x])] + 2*(ArcTan[(c*d)/e] + ArcTan[c*x])*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - (Pi - 2*ArcTan[c*x])*Log[2/Sqrt[1 + c^2*x^2]] - 2*(ArcTan[(c*d)/e] + ArcTan[c*x])*Log[2*Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] - I*PolyLog[2, -E^((-2*I)*ArcTan[c*x])] - I*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))])/2) + b^2*(2*ArcTan[c*x]^2*((I*c*d + e)*ArcTan[c*x] - 3*c*d*Log[1 + E^((2*I)*ArcTan[c*x])]) - 2*ArcTan[c*x]*(((-I)*c*d - e + 2*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e]))*ArcTan[c*x]^2 - 3*c*d*ArcTan[c*x]*(2*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]]) + 3*c*d*(Pi*(-Log[1 + E^((-2*I)*ArcTan[c*x])] + Log[(-2*I)/(-I + c*x)]) - 2*ArcTan[(c*d)/e]*(Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + Log[(I + c*x + E^((2*I)*ArcTan[(c*d)/e])*(-I + c*x))/(2*E^(I*ArcTan[(c*d)/e])*Sqrt[1 + c^2*x^2])] - Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]] - Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]]))) + (6*I)*c*d*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (6*I)*c*d*ArcTan[c*x]*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - 3*c*d*PolyLog[3, -E^((2*I)*ArcTan[c*x])] + 3*c*d*PolyLog[3, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/(6*c*d*e)","B",0
145,1,835,369,14.9559597,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x (d+e x)} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x*(d + e*x)),x]","\frac{24 c d \log (x) a^2-24 c d \log (d+e x) a^2-24 b \left(-\sqrt{\frac{c^2 d^2}{e^2}+1} e e^{i \tan ^{-1}\left(\frac{c d}{e}\right)} \tan ^{-1}(c x)^2+i c d \tan ^{-1}(c x)^2+e \tan ^{-1}(c x)^2-2 i c d \tan ^{-1}\left(\frac{c d}{e}\right) \tan ^{-1}(c x)-2 c d \log \left(1-e^{2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)+2 c d \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) \tan ^{-1}(c x)+i c d \pi  \tan ^{-1}(c x)+c d \pi  \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right)+2 c d \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)+\frac{1}{2} c d \pi  \log \left(c^2 x^2+1\right)-2 c d \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right)+i c d \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)-i c d \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)\right) a+b^2 \left(16 \sqrt{\frac{c^2 d^2}{e^2}+1} e e^{i \tan ^{-1}\left(\frac{c d}{e}\right)} \tan ^{-1}(c x)^3-16 e \tan ^{-1}(c x)^3+24 c d \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)^2-48 c d \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) \tan ^{-1}(c x)^2+24 c d \log \left(-e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \cos \left(2 \tan ^{-1}(c x)\right)-i e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \sin \left(2 \tan ^{-1}(c x)\right)+1\right) \tan ^{-1}(c x)^2-24 c d \pi  \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)-48 c d \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) \tan ^{-1}(c x)+24 c d \pi  \log \left(-\frac{2 i}{c x-i}\right) \tan ^{-1}(c x)-48 c d \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(\frac{e^{-i \tan ^{-1}\left(\frac{c d}{e}\right)} \left(c x+e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} (c x-i)+i\right)}{2 \sqrt{c^2 x^2+1}}\right) \tan ^{-1}(c x)+48 c d \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(-e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \cos \left(2 \tan ^{-1}(c x)\right)-i e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \sin \left(2 \tan ^{-1}(c x)\right)+1\right) \tan ^{-1}(c x)+48 c d \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right) \tan ^{-1}(c x)+24 i c d \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)+24 i c d \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) \tan ^{-1}(c x)+12 c d \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-12 c d \text{Li}_3\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)-i c d \pi ^3\right)}{24 c d^2}","\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{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{i b \text{Li}_2\left(1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{i b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{i b \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\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{b^2 \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 d}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{2 d}-\frac{b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 d}+\frac{b^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)}{2 d}",1,"(24*a^2*c*d*Log[x] - 24*a^2*c*d*Log[d + e*x] - 24*a*b*(I*c*d*Pi*ArcTan[c*x] - (2*I)*c*d*ArcTan[(c*d)/e]*ArcTan[c*x] + I*c*d*ArcTan[c*x]^2 + e*ArcTan[c*x]^2 - Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^2 + c*d*Pi*Log[1 + E^((-2*I)*ArcTan[c*x])] - 2*c*d*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + 2*c*d*ArcTan[(c*d)/e]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 2*c*d*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + (c*d*Pi*Log[1 + c^2*x^2])/2 - 2*c*d*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + I*c*d*PolyLog[2, E^((2*I)*ArcTan[c*x])] - I*c*d*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]) + b^2*((-I)*c*d*Pi^3 - 16*e*ArcTan[c*x]^3 + 16*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^3 + 24*c*d*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 24*c*d*Pi*ArcTan[c*x]*Log[1 + E^((-2*I)*ArcTan[c*x])] - 48*c*d*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - 48*c*d*ArcTan[c*x]^2*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 24*c*d*Pi*ArcTan[c*x]*Log[(-2*I)/(-I + c*x)] - 48*c*d*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[(I + c*x + E^((2*I)*ArcTan[(c*d)/e])*(-I + c*x))/(2*E^(I*ArcTan[(c*d)/e])*Sqrt[1 + c^2*x^2])] + 48*c*d*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]] + 24*c*d*ArcTan[c*x]^2*Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]] + 48*c*d*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + (24*I)*c*d*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + (24*I)*c*d*ArcTan[c*x]*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 12*c*d*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - 12*c*d*PolyLog[3, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/(24*c*d^2)","B",0
146,1,968,473,17.8500533,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^2 (d+e x)} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x^2*(d + e*x)),x]","\frac{-24 d e \log (x) a^2+24 d e \log (d+e x) a^2-\frac{24 d^2 a^2}{x}+\frac{24 b \left(-\frac{2 c \tan ^{-1}(c x) d^2}{x}+2 c^2 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right) d^2+i c e \tan ^{-1}(c x)^2 d-2 i c e \tan ^{-1}\left(\frac{c d}{e}\right) \tan ^{-1}(c x) d+i c e \pi  \tan ^{-1}(c x) d+c e \pi  \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right) d-2 c e \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right) d+2 c e \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) d+2 c e \tan ^{-1}(c x) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) d+\frac{1}{2} c e \pi  \log \left(c^2 x^2+1\right) d-2 c e \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right) d+i c e \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right) d-i c e \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) d-\sqrt{\frac{c^2 d^2}{e^2}+1} e^2 e^{i \tan ^{-1}\left(\frac{c d}{e}\right)} \tan ^{-1}(c x)^2+e^2 \tan ^{-1}(c x)^2\right) a}{c}+\frac{i b^2 \left(-8 i e^2 \tan ^{-1}(c x)^3-8 c d e \tan ^{-1}(c x)^3-24 c^2 d^2 \tan ^{-1}(c x)^2+24 i c d e \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)^2+\frac{24 i c d^2 \tan ^{-1}(c x)^2}{x}-48 i c^2 d^2 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)-24 c d e \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)-24 c^2 d^2 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)+12 i c d e \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)+c d e \pi ^3\right)}{c}+\frac{4 b^2 e \left(2 \tan ^{-1}(c x) \left(\left(i c d-2 \sqrt{\frac{c^2 d^2}{e^2}+1} e e^{i \tan ^{-1}\left(\frac{c d}{e}\right)}+e\right) \tan ^{-1}(c x)^2+3 c d \left(2 \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)-\log \left(-e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \cos \left(2 \tan ^{-1}(c x)\right)-i e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \sin \left(2 \tan ^{-1}(c x)\right)+1\right)\right) \tan ^{-1}(c x)+3 c d \left(\pi  \left(\log \left(1+e^{-2 i \tan ^{-1}(c x)}\right)-\log \left(-\frac{2 i}{c x-i}\right)\right)+2 \tan ^{-1}\left(\frac{c d}{e}\right) \left(\log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)+\log \left(\frac{e^{-i \tan ^{-1}\left(\frac{c d}{e}\right)} \left(c x+e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} (c x-i)+i\right)}{2 \sqrt{c^2 x^2+1}}\right)-\log \left(-e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \cos \left(2 \tan ^{-1}(c x)\right)-i e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \sin \left(2 \tan ^{-1}(c x)\right)+1\right)-\log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right)\right)\right)\right)-6 i c d \tan ^{-1}(c x) \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)+3 c d \text{Li}_3\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right)\right)}{c}}{24 d^3}","\frac{i b e \text{Li}_2\left(1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{i b e \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{i b e \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{i b e \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{d^2}-\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{b^2 e \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{2 d^2}+\frac{b^2 e \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 d^2}-\frac{b^2 e \text{Li}_3\left(\frac{2}{i c x+1}-1\right)}{2 d^2}+\frac{b^2 e \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 d^2}-\frac{i b^2 c \text{Li}_2\left(\frac{2}{1-i c x}-1\right)}{d}",1,"((-24*a^2*d^2)/x - 24*a^2*d*e*Log[x] + 24*a^2*d*e*Log[d + e*x] + (24*a*b*(I*c*d*e*Pi*ArcTan[c*x] - (2*c*d^2*ArcTan[c*x])/x - (2*I)*c*d*e*ArcTan[(c*d)/e]*ArcTan[c*x] + I*c*d*e*ArcTan[c*x]^2 + e^2*ArcTan[c*x]^2 - Sqrt[1 + (c^2*d^2)/e^2]*e^2*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^2 + c*d*e*Pi*Log[1 + E^((-2*I)*ArcTan[c*x])] - 2*c*d*e*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + 2*c*d*e*ArcTan[(c*d)/e]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 2*c*d*e*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 2*c^2*d^2*Log[(c*x)/Sqrt[1 + c^2*x^2]] + (c*d*e*Pi*Log[1 + c^2*x^2])/2 - 2*c*d*e*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + I*c*d*e*PolyLog[2, E^((2*I)*ArcTan[c*x])] - I*c*d*e*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/c + (I*b^2*(c*d*e*Pi^3 - 24*c^2*d^2*ArcTan[c*x]^2 + ((24*I)*c*d^2*ArcTan[c*x]^2)/x - 8*c*d*e*ArcTan[c*x]^3 - (8*I)*e^2*ArcTan[c*x]^3 + (24*I)*c*d*e*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - (48*I)*c^2*d^2*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - 24*c*d*e*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] - 24*c^2*d^2*PolyLog[2, E^((2*I)*ArcTan[c*x])] + (12*I)*c*d*e*PolyLog[3, E^((-2*I)*ArcTan[c*x])]))/c + (4*b^2*e*(2*ArcTan[c*x]*((I*c*d + e - 2*Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e]))*ArcTan[c*x]^2 + 3*c*d*ArcTan[c*x]*(2*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]]) + 3*c*d*(Pi*(Log[1 + E^((-2*I)*ArcTan[c*x])] - Log[(-2*I)/(-I + c*x)]) + 2*ArcTan[(c*d)/e]*(Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + Log[(I + c*x + E^((2*I)*ArcTan[(c*d)/e])*(-I + c*x))/(2*E^(I*ArcTan[(c*d)/e])*Sqrt[1 + c^2*x^2])] - Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]] - Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]]))) - (6*I)*c*d*ArcTan[c*x]*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 3*c*d*PolyLog[3, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/c)/(24*d^3)","B",0
147,1,1173,591,21.8790511,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^3 (d+e x)} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x^3*(d + e*x)),x]","\frac{e^2 \log (x) a^2}{d^3}-\frac{e^2 \log (d+e x) a^2}{d^3}+\frac{e a^2}{d^2 x}-\frac{a^2}{2 d x^2}-\frac{b \left(\frac{c \left(c^2 x^2+1\right) \tan ^{-1}(c x) d^3}{x^2}+\frac{c^2 d^3}{x}-\frac{2 c e \tan ^{-1}(c x) d^2}{x}+2 c^2 e \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right) d^2+i c e^2 \tan ^{-1}(c x)^2 d-2 i c e^2 \tan ^{-1}\left(\frac{c d}{e}\right) \tan ^{-1}(c x) d+i c e^2 \pi  \tan ^{-1}(c x) d+c e^2 \pi  \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right) d-2 c e^2 \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right) d+2 c e^2 \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) d+2 c e^2 \tan ^{-1}(c x) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) d+\frac{1}{2} c e^2 \pi  \log \left(c^2 x^2+1\right) d-2 c e^2 \tan ^{-1}\left(\frac{c d}{e}\right) \log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right) d+i c e^2 \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right) d-i c e^2 \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) d-\sqrt{\frac{c^2 d^2}{e^2}+1} e^3 e^{i \tan ^{-1}\left(\frac{c d}{e}\right)} \tan ^{-1}(c x)^2+e^3 \tan ^{-1}(c x)^2\right) a}{c d^4}+\frac{b^2 \left(-\frac{12 c \left(c^2 x^2+1\right) \tan ^{-1}(c x)^2 d^3}{x^2}-\frac{24 c^2 \tan ^{-1}(c x) d^3}{x}+24 c^3 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right) d^3+24 i c^2 e \tan ^{-1}(c x)^2 d^2+\frac{24 c e \tan ^{-1}(c x)^2 d^2}{x}-48 c^2 e \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right) d^2+24 i c^2 e \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right) d^2+24 c e^2 \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right) d-24 c e^2 \pi  \tan ^{-1}(c x) \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right) d-48 c e^2 \tan ^{-1}(c x)^2 \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) d-48 c e^2 \tan ^{-1}\left(\frac{c d}{e}\right) \tan ^{-1}(c x) \log \left(1-e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) d+24 c e^2 \pi  \tan ^{-1}(c x) \log \left(-\frac{2 i}{c x-i}\right) d-48 c e^2 \tan ^{-1}\left(\frac{c d}{e}\right) \tan ^{-1}(c x) \log \left(\frac{e^{-i \tan ^{-1}\left(\frac{c d}{e}\right)} \left(c x+e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} (c x-i)+i\right)}{2 \sqrt{c^2 x^2+1}}\right) d+24 c e^2 \tan ^{-1}(c x)^2 \log \left(-e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \cos \left(2 \tan ^{-1}(c x)\right)-i e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \sin \left(2 \tan ^{-1}(c x)\right)+1\right) d+48 c e^2 \tan ^{-1}\left(\frac{c d}{e}\right) \tan ^{-1}(c x) \log \left(-e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \cos \left(2 \tan ^{-1}(c x)\right)-i e^{2 i \tan ^{-1}\left(\frac{c d}{e}\right)} \sin \left(2 \tan ^{-1}(c x)\right)+1\right) d+48 c e^2 \tan ^{-1}\left(\frac{c d}{e}\right) \tan ^{-1}(c x) \log \left(\sin \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)\right) d+24 i c e^2 \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right) d+24 i c e^2 \tan ^{-1}(c x) \text{Li}_2\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) d+12 c e^2 \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right) d-12 c e^2 \text{Li}_3\left(e^{2 i \left(\tan ^{-1}\left(\frac{c d}{e}\right)+\tan ^{-1}(c x)\right)}\right) d-i c e^2 \pi ^3 d+16 \sqrt{\frac{c^2 d^2}{e^2}+1} e^3 e^{i \tan ^{-1}\left(\frac{c d}{e}\right)} \tan ^{-1}(c x)^3-16 e^3 \tan ^{-1}(c x)^3\right)}{24 c d^4}","-\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}-\frac{i b e^2 \text{Li}_2\left(1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}-\frac{i b e^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^3}+\frac{i b e^2 \text{Li}_2\left(\frac{2}{i c x+1}-1\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{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{d^3}+\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{b^2 e^2 \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{2 d^3}-\frac{b^2 e^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 d^3}+\frac{b^2 e^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)}{2 d^3}-\frac{b^2 e^2 \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 d^3}+\frac{i b^2 c e \text{Li}_2\left(\frac{2}{1-i c x}-1\right)}{d^2}",1,"-1/2*a^2/(d*x^2) + (a^2*e)/(d^2*x) + (a^2*e^2*Log[x])/d^3 - (a^2*e^2*Log[d + e*x])/d^3 - (a*b*((c^2*d^3)/x + I*c*d*e^2*Pi*ArcTan[c*x] - (2*c*d^2*e*ArcTan[c*x])/x + (c*d^3*(1 + c^2*x^2)*ArcTan[c*x])/x^2 - (2*I)*c*d*e^2*ArcTan[(c*d)/e]*ArcTan[c*x] + I*c*d*e^2*ArcTan[c*x]^2 + e^3*ArcTan[c*x]^2 - Sqrt[1 + (c^2*d^2)/e^2]*e^3*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^2 + c*d*e^2*Pi*Log[1 + E^((-2*I)*ArcTan[c*x])] - 2*c*d*e^2*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + 2*c*d*e^2*ArcTan[(c*d)/e]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 2*c*d*e^2*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 2*c^2*d^2*e*Log[(c*x)/Sqrt[1 + c^2*x^2]] + (c*d*e^2*Pi*Log[1 + c^2*x^2])/2 - 2*c*d*e^2*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + I*c*d*e^2*PolyLog[2, E^((2*I)*ArcTan[c*x])] - I*c*d*e^2*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/(c*d^4) + (b^2*((-I)*c*d*e^2*Pi^3 - (24*c^2*d^3*ArcTan[c*x])/x + (24*I)*c^2*d^2*e*ArcTan[c*x]^2 + (24*c*d^2*e*ArcTan[c*x]^2)/x - (12*c*d^3*(1 + c^2*x^2)*ArcTan[c*x]^2)/x^2 - 16*e^3*ArcTan[c*x]^3 + 16*Sqrt[1 + (c^2*d^2)/e^2]*e^3*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^3 + 24*c*d*e^2*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 24*c*d*e^2*Pi*ArcTan[c*x]*Log[1 + E^((-2*I)*ArcTan[c*x])] - 48*c^2*d^2*e*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] - 48*c*d*e^2*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - 48*c*d*e^2*ArcTan[c*x]^2*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 24*c*d*e^2*Pi*ArcTan[c*x]*Log[(-2*I)/(-I + c*x)] + 24*c^3*d^3*Log[(c*x)/Sqrt[1 + c^2*x^2]] - 48*c*d*e^2*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[(I + c*x + E^((2*I)*ArcTan[(c*d)/e])*(-I + c*x))/(2*E^(I*ArcTan[(c*d)/e])*Sqrt[1 + c^2*x^2])] + 48*c*d*e^2*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]] + 24*c*d*e^2*ArcTan[c*x]^2*Log[1 - E^((2*I)*ArcTan[(c*d)/e])*Cos[2*ArcTan[c*x]] - I*E^((2*I)*ArcTan[(c*d)/e])*Sin[2*ArcTan[c*x]]] + 48*c*d*e^2*ArcTan[(c*d)/e]*ArcTan[c*x]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + (24*I)*c*d*e^2*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + (24*I)*c^2*d^2*e*PolyLog[2, E^((2*I)*ArcTan[c*x])] + (24*I)*c*d*e^2*ArcTan[c*x]*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] + 12*c*d*e^2*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - 12*c*d*e^2*PolyLog[3, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/(24*c*d^4)","A",0
148,0,0,21,0.5300621,"\int \frac{1}{(d+e x) \left(a+b \tan ^{-1}(c x)\right)} \, dx","Integrate[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,"Integrate[1/((d + e*x)*(a + b*ArcTan[c*x])), x]","A",-1
149,1,69,69,0.0060111,"\int x^3 \left(c+a^2 c x^2\right) \tan ^{-1}(a x) \, dx","Integrate[x^3*(c + a^2*c*x^2)*ArcTan[a*x],x]","-\frac{c \tan ^{-1}(a x)}{12 a^4}+\frac{c x}{12 a^3}+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)-\frac{1}{30} a c x^5+\frac{1}{4} c x^4 \tan ^{-1}(a x)-\frac{c x^3}{36 a}","-\frac{c \tan ^{-1}(a x)}{12 a^4}+\frac{c x}{12 a^3}+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)-\frac{1}{30} a c x^5+\frac{1}{4} c x^4 \tan ^{-1}(a x)-\frac{c x^3}{36 a}",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",1
150,1,66,66,0.0221274,"\int x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x) \, dx","Integrate[x^2*(c + a^2*c*x^2)*ArcTan[a*x],x]","\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)+\frac{c \log \left(a^2 x^2+1\right)}{15 a^3}-\frac{1}{20} a c x^4+\frac{1}{3} c x^3 \tan ^{-1}(a x)-\frac{c x^2}{15 a}","\frac{1}{5} a^2 c x^5 \tan ^{-1}(a x)+\frac{c \log \left(a^2 x^2+1\right)}{15 a^3}-\frac{1}{20} a c x^4+\frac{1}{3} c x^3 \tan ^{-1}(a x)-\frac{c x^2}{15 a}",1,"-1/15*(c*x^2)/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",1
151,1,58,42,0.0042868,"\int x \left(c+a^2 c x^2\right) \tan ^{-1}(a x) \, dx","Integrate[x*(c + a^2*c*x^2)*ArcTan[a*x],x]","\frac{1}{4} a^2 c x^4 \tan ^{-1}(a x)+\frac{c \tan ^{-1}(a x)}{4 a^2}-\frac{1}{12} a c x^3+\frac{1}{2} c x^2 \tan ^{-1}(a x)-\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,"-1/4*(c*x)/a - (a*c*x^3)/12 + (c*ArcTan[a*x])/(4*a^2) + (c*x^2*ArcTan[a*x])/2 + (a^2*c*x^4*ArcTan[a*x])/4","A",1
152,1,50,50,0.0098761,"\int \left(c+a^2 c x^2\right) \tan ^{-1}(a x) \, dx","Integrate[(c + a^2*c*x^2)*ArcTan[a*x],x]","\frac{1}{3} a^2 c x^3 \tan ^{-1}(a x)-\frac{c \log \left(a^2 x^2+1\right)}{3 a}-\frac{1}{6} a c x^2+c x \tan ^{-1}(a x)","\frac{1}{3} a^2 c x^3 \tan ^{-1}(a x)-\frac{c \log \left(a^2 x^2+1\right)}{3 a}-\frac{1}{6} a c x^2+c x \tan ^{-1}(a x)",1,"-1/6*(a*c*x^2) + c*x*ArcTan[a*x] + (a^2*c*x^3*ArcTan[a*x])/3 - (c*Log[1 + a^2*x^2])/(3*a)","A",1
153,1,62,62,0.0036628,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)}{x} \, dx","Integrate[((c + a^2*c*x^2)*ArcTan[a*x])/x,x]","\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)+\frac{1}{2} i c \text{Li}_2(-i a x)-\frac{1}{2} i c \text{Li}_2(i a x)-\frac{a c x}{2}+\frac{1}{2} c \tan ^{-1}(a x)","\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)+\frac{1}{2} i c \text{Li}_2(-i a x)-\frac{1}{2} i c \text{Li}_2(i a x)-\frac{a c x}{2}+\frac{1}{2} c \tan ^{-1}(a x)",1,"-1/2*(a*c*x) + (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",1
154,1,40,40,0.0048946,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)}{x^2} \, dx","Integrate[((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",1
155,1,74,70,0.0050308,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)}{x^3} \, dx","Integrate[((c + a^2*c*x^2)*ArcTan[a*x])/x^3,x]","-\frac{a c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-a^2 x^2\right)}{2 x}+\frac{1}{2} i a^2 c \text{Li}_2(-i a x)-\frac{1}{2} i a^2 c \text{Li}_2(i a x)-\frac{c \tan ^{-1}(a x)}{2 x^2}","\frac{1}{2} i a^2 c \text{Li}_2(-i a x)-\frac{1}{2} i a^2 c \text{Li}_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,"-1/2*(c*ArcTan[a*x])/x^2 - (a*c*Hypergeometric2F1[-1/2, 1, 1/2, -(a^2*x^2)])/(2*x) + (I/2)*a^2*c*PolyLog[2, (-I)*a*x] - (I/2)*a^2*c*PolyLog[2, I*a*x]","C",0
156,1,58,63,0.0204814,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)}{x^4} \, dx","Integrate[((c + a^2*c*x^2)*ArcTan[a*x])/x^4,x]","\frac{c \left(a x \left(4 a^2 x^2 \log (x)-2 a^2 x^2 \log \left(a^2 x^2+1\right)-1\right)-2 \left(3 a^2 x^2+1\right) \tan ^{-1}(a x)\right)}{6 x^3}","\frac{2}{3} a^3 c \log (x)-\frac{a^2 c \tan ^{-1}(a x)}{x}-\frac{1}{3} a^3 c \log \left(a^2 x^2+1\right)-\frac{c \tan ^{-1}(a x)}{3 x^3}-\frac{a c}{6 x^2}",1,"(c*(-2*(1 + 3*a^2*x^2)*ArcTan[a*x] + a*x*(-1 + 4*a^2*x^2*Log[x] - 2*a^2*x^2*Log[1 + a^2*x^2])))/(6*x^3)","A",1
157,1,111,111,0.09299,"\int x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x) \, dx","Integrate[x^3*(c + a^2*c*x^2)^2*ArcTan[a*x],x]","\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)-\frac{c^2 \tan ^{-1}(a x)}{24 a^4}-\frac{1}{56} a^3 c^2 x^7+\frac{c^2 x}{24 a^3}+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)-\frac{1}{24} a c^2 x^5+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)-\frac{c^2 x^3}{72 a}","\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)-\frac{c^2 \tan ^{-1}(a x)}{24 a^4}-\frac{1}{56} a^3 c^2 x^7+\frac{c^2 x}{24 a^3}+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)-\frac{1}{24} a c^2 x^5+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)-\frac{c^2 x^3}{72 a}",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",1
158,1,106,106,0.0649557,"\int x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x) \, dx","Integrate[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x],x]","\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)-\frac{1}{42} a^3 c^2 x^6+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac{4 c^2 \log \left(a^2 x^2+1\right)}{105 a^3}-\frac{9}{140} a c^2 x^4+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)-\frac{4 c^2 x^2}{105 a}","\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)-\frac{1}{42} a^3 c^2 x^6+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac{4 c^2 \log \left(a^2 x^2+1\right)}{105 a^3}-\frac{9}{140} a c^2 x^4+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)-\frac{4 c^2 x^2}{105 a}",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",1
159,1,98,61,0.0497484,"\int x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x) \, dx","Integrate[x*(c + a^2*c*x^2)^2*ArcTan[a*x],x]","\frac{1}{6} a^4 c^2 x^6 \tan ^{-1}(a x)-\frac{1}{30} a^3 c^2 x^5+\frac{1}{2} a^2 c^2 x^4 \tan ^{-1}(a x)+\frac{c^2 \tan ^{-1}(a x)}{6 a^2}-\frac{1}{9} a c^2 x^3+\frac{1}{2} c^2 x^2 \tan ^{-1}(a x)-\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,"-1/6*(c^2*x)/a - (a*c^2*x^3)/9 - (a^3*c^2*x^5)/30 + (c^2*ArcTan[a*x])/(6*a^2) + (c^2*x^2*ArcTan[a*x])/2 + (a^2*c^2*x^4*ArcTan[a*x])/2 + (a^4*c^2*x^6*ArcTan[a*x])/6","A",1
160,1,65,117,0.0592516,"\int \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x) \, dx","Integrate[(c + a^2*c*x^2)^2*ArcTan[a*x],x]","\frac{c^2 \left(-3 a^4 x^4-14 a^2 x^2-16 \log \left(a^2 x^2+1\right)+4 a x \left(3 a^4 x^4+10 a^2 x^2+15\right) \tan ^{-1}(a x)\right)}{60 a}","-\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,"(c^2*(-14*a^2*x^2 - 3*a^4*x^4 + 4*a*x*(15 + 10*a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x] - 16*Log[1 + a^2*x^2]))/(60*a)","A",1
161,1,99,99,0.0370081,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)}{x} \, dx","Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x])/x,x]","\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)-\frac{1}{12} a^3 c^2 x^3+a^2 c^2 x^2 \tan ^{-1}(a x)+\frac{1}{2} i c^2 \text{Li}_2(-i a x)-\frac{1}{2} i c^2 \text{Li}_2(i a x)-\frac{3}{4} a c^2 x+\frac{3}{4} c^2 \tan ^{-1}(a x)","\frac{1}{4} a^4 c^2 x^4 \tan ^{-1}(a x)-\frac{1}{12} a^3 c^2 x^3+a^2 c^2 x^2 \tan ^{-1}(a x)+\frac{1}{2} i c^2 \text{Li}_2(-i a x)-\frac{1}{2} i c^2 \text{Li}_2(i 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",1
162,1,62,81,0.0521938,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)}{x^2} \, dx","Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x])/x^2,x]","\frac{c^2 \left(2 \left(a^4 x^4+6 a^2 x^2-3\right) \tan ^{-1}(a x)-a x \left(a^2 x^2+8 \log \left(a^2 x^2+1\right)-6 \log (x)\right)\right)}{6 x}","\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)-\frac{1}{6} a^3 c^2 x^2-\frac{4}{3} a c^2 \log \left(a^2 x^2+1\right)+2 a^2 c^2 x \tan ^{-1}(a x)+a c^2 \log (x)-\frac{c^2 \tan ^{-1}(a x)}{x}",1,"(c^2*(2*(-3 + 6*a^2*x^2 + a^4*x^4)*ArcTan[a*x] - a*x*(a^2*x^2 - 6*Log[x] + 8*Log[1 + a^2*x^2])))/(6*x)","A",1
163,1,103,90,0.0472397,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)}{x^3} \, dx","Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x])/x^3,x]","\frac{c^2 \left(a^4 x^4 \tan ^{-1}(a x)-a^3 x^3-a x \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-a^2 x^2\right)+2 i a^2 x^2 \text{Li}_2(-i a x)-2 i a^2 x^2 \text{Li}_2(i a x)+a^2 x^2 \tan ^{-1}(a x)-\tan ^{-1}(a x)\right)}{2 x^2}","\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)-\frac{1}{2} a^3 c^2 x+i a^2 c^2 \text{Li}_2(-i a x)-i a^2 c^2 \text{Li}_2(i a x)-\frac{c^2 \tan ^{-1}(a x)}{2 x^2}-\frac{a c^2}{2 x}",1,"(c^2*(-(a^3*x^3) - ArcTan[a*x] + a^2*x^2*ArcTan[a*x] + a^4*x^4*ArcTan[a*x] - a*x*Hypergeometric2F1[-1/2, 1, 1/2, -(a^2*x^2)] + (2*I)*a^2*x^2*PolyLog[2, (-I)*a*x] - (2*I)*a^2*x^2*PolyLog[2, I*a*x]))/(2*x^2)","C",0
164,1,68,85,0.0570527,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)}{x^4} \, dx","Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x])/x^4,x]","\frac{c^2 \left(a x \left(10 a^2 x^2 \log (x)-8 a^2 x^2 \log \left(a^2 x^2+1\right)-1\right)+2 \left(3 a^4 x^4-6 a^2 x^2-1\right) \tan ^{-1}(a x)\right)}{6 x^3}","a^4 c^2 x \tan ^{-1}(a x)+\frac{5}{3} a^3 c^2 \log (x)-\frac{2 a^2 c^2 \tan ^{-1}(a x)}{x}-\frac{4}{3} a^3 c^2 \log \left(a^2 x^2+1\right)-\frac{c^2 \tan ^{-1}(a x)}{3 x^3}-\frac{a c^2}{6 x^2}",1,"(c^2*(2*(-1 - 6*a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x] + a*x*(-1 + 10*a^2*x^2*Log[x] - 8*a^2*x^2*Log[1 + a^2*x^2])))/(6*x^3)","A",1
165,1,141,141,0.1445516,"\int x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x) \, dx","Integrate[x^3*(c + a^2*c*x^2)^3*ArcTan[a*x],x]","\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)-\frac{1}{90} a^5 c^3 x^9+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)}{40 a^4}-\frac{11}{280} a^3 c^3 x^7+\frac{c^3 x}{40 a^3}+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)-\frac{9}{200} a c^3 x^5+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)-\frac{c^3 x^3}{120 a}","\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)-\frac{1}{90} a^5 c^3 x^9+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)}{40 a^4}-\frac{11}{280} a^3 c^3 x^7+\frac{c^3 x}{40 a^3}+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)-\frac{9}{200} a c^3 x^5+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)-\frac{c^3 x^3}{120 a}",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",1
166,1,136,136,0.087506,"\int x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x) \, dx","Integrate[x^2*(c + a^2*c*x^2)^3*ArcTan[a*x],x]","\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)-\frac{1}{72} a^5 c^3 x^8+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)-\frac{10}{189} a^3 c^3 x^6+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac{8 c^3 \log \left(a^2 x^2+1\right)}{315 a^3}-\frac{89 a c^3 x^4}{1260}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)-\frac{8 c^3 x^2}{315 a}","\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)-\frac{1}{72} a^5 c^3 x^8+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)-\frac{10}{189} a^3 c^3 x^6+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac{8 c^3 \log \left(a^2 x^2+1\right)}{315 a^3}-\frac{89 a c^3 x^4}{1260}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)-\frac{8 c^3 x^2}{315 a}",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",1
167,1,128,74,0.0880938,"\int x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x) \, dx","Integrate[x*(c + a^2*c*x^2)^3*ArcTan[a*x],x]","\frac{1}{8} a^6 c^3 x^8 \tan ^{-1}(a x)-\frac{1}{56} a^5 c^3 x^7+\frac{1}{2} a^4 c^3 x^6 \tan ^{-1}(a x)-\frac{3}{40} a^3 c^3 x^5+\frac{3}{4} a^2 c^3 x^4 \tan ^{-1}(a x)+\frac{c^3 \tan ^{-1}(a x)}{8 a^2}-\frac{1}{8} a c^3 x^3+\frac{1}{2} c^3 x^2 \tan ^{-1}(a x)-\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,"-1/8*(c^3*x)/a - (a*c^3*x^3)/8 - (3*a^3*c^3*x^5)/40 - (a^5*c^3*x^7)/56 + (c^3*ArcTan[a*x])/(8*a^2) + (c^3*x^2*ArcTan[a*x])/2 + (3*a^2*c^3*x^4*ArcTan[a*x])/4 + (a^4*c^3*x^6*ArcTan[a*x])/2 + (a^6*c^3*x^8*ArcTan[a*x])/8","A",1
168,1,83,161,0.0788261,"\int \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x) \, dx","Integrate[(c + a^2*c*x^2)^3*ArcTan[a*x],x]","\frac{c^3 \left(-48 \log \left(a^2 x^2+1\right)-a^2 x^2 \left(5 a^4 x^4+24 a^2 x^2+57\right)+6 a x \left(5 a^6 x^6+21 a^4 x^4+35 a^2 x^2+35\right) \tan ^{-1}(a x)\right)}{210 a}","-\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,"(c^3*(-(a^2*x^2*(57 + 24*a^2*x^2 + 5*a^4*x^4)) + 6*a*x*(35 + 35*a^2*x^2 + 21*a^4*x^4 + 5*a^6*x^6)*ArcTan[a*x] - 48*Log[1 + a^2*x^2]))/(210*a)","A",1
169,1,132,132,0.0547999,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)}{x} \, dx","Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x])/x,x]","\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)-\frac{1}{30} a^5 c^3 x^5+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)-\frac{7}{36} a^3 c^3 x^3+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)+\frac{1}{2} i c^3 \text{Li}_2(-i a x)-\frac{1}{2} i c^3 \text{Li}_2(i a x)-\frac{11}{12} a c^3 x+\frac{11}{12} c^3 \tan ^{-1}(a x)","\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)-\frac{1}{30} a^5 c^3 x^5+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)-\frac{7}{36} a^3 c^3 x^3+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)+\frac{1}{2} i c^3 \text{Li}_2(-i a x)-\frac{1}{2} i c^3 \text{Li}_2(i 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",1
170,1,78,108,0.0739223,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)}{x^2} \, dx","Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x])/x^2,x]","\frac{c^3 \left(4 \left(a^6 x^6+5 a^4 x^4+15 a^2 x^2-5\right) \tan ^{-1}(a x)-a x \left(a^4 x^4+8 a^2 x^2+32 \log \left(a^2 x^2+1\right)-20 \log (x)\right)\right)}{20 x}","\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)-\frac{1}{20} a^5 c^3 x^4+a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{2}{5} a^3 c^3 x^2-\frac{8}{5} a c^3 \log \left(a^2 x^2+1\right)+3 a^2 c^3 x \tan ^{-1}(a x)+a c^3 \log (x)-\frac{c^3 \tan ^{-1}(a x)}{x}",1,"(c^3*(4*(-5 + 15*a^2*x^2 + 5*a^4*x^4 + a^6*x^6)*ArcTan[a*x] - a*x*(8*a^2*x^2 + a^4*x^4 - 20*Log[x] + 32*Log[1 + a^2*x^2])))/(20*x)","A",1
171,1,154,138,0.0445829,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)}{x^3} \, dx","Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x])/x^3,x]","\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)-\frac{1}{12} a^5 c^3 x^3+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac{5}{4} a^3 c^3 x-\frac{a c^3 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-a^2 x^2\right)}{2 x}+\frac{3}{2} i a^2 c^3 \text{Li}_2(-i a x)-\frac{3}{2} i a^2 c^3 \text{Li}_2(i a x)+\frac{5}{4} a^2 c^3 \tan ^{-1}(a x)-\frac{c^3 \tan ^{-1}(a x)}{2 x^2}","\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)-\frac{1}{12} a^5 c^3 x^3+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac{5}{4} a^3 c^3 x+\frac{3}{2} i a^2 c^3 \text{Li}_2(-i a x)-\frac{3}{2} i a^2 c^3 \text{Li}_2(i a 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,"(-5*a^3*c^3*x)/4 - (a^5*c^3*x^3)/12 + (5*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 - (a*c^3*Hypergeometric2F1[-1/2, 1, 1/2, -(a^2*x^2)])/(2*x) + ((3*I)/2)*a^2*c^3*PolyLog[2, (-I)*a*x] - ((3*I)/2)*a^2*c^3*PolyLog[2, I*a*x]","C",0
172,1,83,116,0.0701185,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)}{x^4} \, dx","Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x])/x^4,x]","\frac{c^3 \left(2 \left(a^6 x^6+9 a^4 x^4-9 a^2 x^2-1\right) \tan ^{-1}(a x)-a x \left(a^4 x^4-16 a^2 x^2 \log (x)+16 a^2 x^2 \log \left(a^2 x^2+1\right)+1\right)\right)}{6 x^3}","\frac{1}{3} a^6 c^3 x^3 \tan ^{-1}(a x)-\frac{1}{6} a^5 c^3 x^2+3 a^4 c^3 x \tan ^{-1}(a x)+\frac{8}{3} a^3 c^3 \log (x)-\frac{3 a^2 c^3 \tan ^{-1}(a x)}{x}-\frac{8}{3} a^3 c^3 \log \left(a^2 x^2+1\right)-\frac{c^3 \tan ^{-1}(a x)}{3 x^3}-\frac{a c^3}{6 x^2}",1,"(c^3*(2*(-1 - 9*a^2*x^2 + 9*a^4*x^4 + a^6*x^6)*ArcTan[a*x] - a*x*(1 + a^4*x^4 - 16*a^2*x^2*Log[x] + 16*a^2*x^2*Log[1 + a^2*x^2])))/(6*x^3)","A",1
173,1,56,80,0.0584967,"\int \frac{x^4 \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx","Integrate[(x^4*ArcTan[a*x])/(c + a^2*c*x^2),x]","\frac{-a^2 x^2+4 \log \left(a^2 x^2+1\right)+2 a x \left(a^2 x^2-3\right) \tan ^{-1}(a x)+3 \tan ^{-1}(a x)^2}{6 a^5 c}","\frac{\tan ^{-1}(a x)^2}{2 a^5 c}-\frac{x \tan ^{-1}(a x)}{a^4 c}-\frac{x^2}{6 a^3 c}+\frac{x^3 \tan ^{-1}(a x)}{3 a^2 c}+\frac{2 \log \left(a^2 x^2+1\right)}{3 a^5 c}",1,"(-(a^2*x^2) + 2*a*x*(-3 + a^2*x^2)*ArcTan[a*x] + 3*ArcTan[a*x]^2 + 4*Log[1 + a^2*x^2])/(6*a^5*c)","A",1
174,1,120,113,0.0349354,"\int \frac{x^3 \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx","Integrate[(x^3*ArcTan[a*x])/(c + a^2*c*x^2),x]","\frac{i \text{Li}_2\left(-\frac{a x+i}{i-a x}\right)}{2 a^4 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 i}{-a x+i}\right) \tan ^{-1}(a x)}{a^4 c}-\frac{x}{2 a^3 c}+\frac{x^2 \tan ^{-1}(a x)}{2 a^2 c}","\frac{i \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{2 a^4 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{x}{2 a^3 c}+\frac{x^2 \tan ^{-1}(a x)}{2 a^2 c}",1,"-1/2*x/(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*I)/(I - a*x)])/(a^4*c) + ((I/2)*PolyLog[2, -((I + a*x)/(I - a*x))])/(a^4*c)","A",1
175,1,49,49,0.0287827,"\int \frac{x^2 \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx","Integrate[(x^2*ArcTan[a*x])/(c + a^2*c*x^2),x]","-\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}-\frac{\log \left(a^2 x^2+1\right)}{2 a^3 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",1
176,1,77,72,0.0058696,"\int \frac{x \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx","Integrate[(x*ArcTan[a*x])/(c + a^2*c*x^2),x]","-\frac{i \text{Li}_2\left(\frac{a x+i}{a x-i}\right)}{2 a^2 c}-\frac{i \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{\log \left(\frac{2 i}{-a x+i}\right) \tan ^{-1}(a x)}{a^2 c}","-\frac{i \text{Li}_2\left(1-\frac{2}{i a x+1}\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,"((-1/2*I)*ArcTan[a*x]^2)/(a^2*c) - (ArcTan[a*x]*Log[(2*I)/(I - a*x)])/(a^2*c) - ((I/2)*PolyLog[2, (I + a*x)/(-I + a*x)])/(a^2*c)","A",1
177,1,16,16,0.0035975,"\int \frac{\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx","Integrate[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
178,1,103,64,0.0265488,"\int \frac{\tan ^{-1}(a x)}{x \left(c+a^2 c x^2\right)} \, dx","Integrate[ArcTan[a*x]/(x*(c + a^2*c*x^2)),x]","\frac{i \text{Li}_2(-i a x)}{2 c}-\frac{i \text{Li}_2(i a x)}{2 c}+\frac{i \text{Li}_2\left(-\frac{a x+i}{i-a x}\right)}{2 c}+\frac{i \tan ^{-1}(a x)^2}{2 c}+\frac{\log \left(\frac{2 i}{-a x+i}\right) \tan ^{-1}(a x)}{c}","-\frac{i \text{Li}_2\left(\frac{2}{1-i a x}-1\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*I)/(I - a*x)])/c + ((I/2)*PolyLog[2, (-I)*a*x])/c - ((I/2)*PolyLog[2, I*a*x])/c + ((I/2)*PolyLog[2, -((I + a*x)/(I - a*x))])/c","A",0
179,1,52,52,0.0079977,"\int \frac{\tan ^{-1}(a x)}{x^2 \left(c+a^2 c x^2\right)} \, dx","Integrate[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",1
180,1,142,113,0.0644875,"\int \frac{\tan ^{-1}(a x)}{x^3 \left(c+a^2 c x^2\right)} \, dx","Integrate[ArcTan[a*x]/(x^3*(c + a^2*c*x^2)),x]","-\frac{a \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-a^2 x^2\right)}{2 c x}-\frac{a^2 \left(\frac{1}{2} i \text{Li}_2(-i a x)-\frac{1}{2} i \text{Li}_2(i a x)+\frac{1}{2} \left(i \text{Li}_2\left(-\frac{a x+i}{i-a x}\right)+2 \log \left(\frac{2 i}{-a x+i}\right) \tan ^{-1}(a x)\right)+\frac{1}{2} i \tan ^{-1}(a x)^2\right)}{c}-\frac{\tan ^{-1}(a x)}{2 c x^2}","\frac{i a^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\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,"-1/2*ArcTan[a*x]/(c*x^2) - (a*Hypergeometric2F1[-1/2, 1, 1/2, -(a^2*x^2)])/(2*c*x) - (a^2*((I/2)*ArcTan[a*x]^2 + (I/2)*PolyLog[2, (-I)*a*x] - (I/2)*PolyLog[2, I*a*x] + (2*ArcTan[a*x]*Log[(2*I)/(I - a*x)] + I*PolyLog[2, -((I + a*x)/(I - a*x))])/2))/c","C",0
181,1,88,88,0.0176352,"\int \frac{\tan ^{-1}(a x)}{x^4 \left(c+a^2 c x^2\right)} \, dx","Integrate[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)),x]","-\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{2 a^3 \log \left(a^2 x^2+1\right)}{3 c}-\frac{\tan ^{-1}(a x)}{3 c x^3}-\frac{a}{6 c x^2}","-\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{2 a^3 \log \left(a^2 x^2+1\right)}{3 c}-\frac{\tan ^{-1}(a x)}{3 c x^3}-\frac{a}{6 c x^2}",1,"-1/6*a/(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",1
182,1,90,157,0.2448897,"\int \frac{x^5 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x^5*ArcTan[a*x])/(c + a^2*c*x^2)^2,x]","\frac{2 \tan ^{-1}(a x) \left(2 a^2 x^2+8 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-\cos \left(2 \tan ^{-1}(a x)\right)+2\right)-8 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-4 a x-8 i \tan ^{-1}(a x)^2+\sin \left(2 \tan ^{-1}(a x)\right)}{8 a^6 c^2}","\frac{i \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{a^6 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{x}{2 a^5 c^2}+\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}{4 a^5 c^2 \left(a^2 x^2+1\right)}",1,"(-4*a*x - (8*I)*ArcTan[a*x]^2 + 2*ArcTan[a*x]*(2 + 2*a^2*x^2 - Cos[2*ArcTan[a*x]] + 8*Log[1 + E^((2*I)*ArcTan[a*x])]) - (8*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + Sin[2*ArcTan[a*x]])/(8*a^6*c^2)","A",0
183,1,79,96,0.0658711,"\int \frac{x^4 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x^4*ArcTan[a*x])/(c + a^2*c*x^2)^2,x]","\frac{\left(4 a^3 x^3+6 a x\right) \tan ^{-1}(a x)-2 \left(a^2 x^2+1\right) \log \left(a^2 x^2+1\right)-3 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2+1}{4 a^5 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)}",1,"(1 + (6*a*x + 4*a^3*x^3)*ArcTan[a*x] - 3*(1 + a^2*x^2)*ArcTan[a*x]^2 - 2*(1 + a^2*x^2)*Log[1 + a^2*x^2])/(4*a^5*c^2*(1 + a^2*x^2))","A",1
184,1,77,133,0.1249518,"\int \frac{x^3 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x^3*ArcTan[a*x])/(c + a^2*c*x^2)^2,x]","\frac{4 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+4 i \tan ^{-1}(a x)^2-\sin \left(2 \tan ^{-1}(a x)\right)+2 \tan ^{-1}(a x) \left(\cos \left(2 \tan ^{-1}(a x)\right)-4 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)\right)}{8 a^4 c^2}","-\frac{i \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{2 a^4 c^2}-\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{\tan ^{-1}(a x)}{2 a^4 c^2 \left(a^2 x^2+1\right)}-\frac{x}{4 a^3 c^2 \left(a^2 x^2+1\right)}",1,"((4*I)*ArcTan[a*x]^2 + 2*ArcTan[a*x]*(Cos[2*ArcTan[a*x]] - 4*Log[1 + E^((2*I)*ArcTan[a*x])]) + (4*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - Sin[2*ArcTan[a*x]])/(8*a^4*c^2)","A",0
185,1,47,64,0.0441712,"\int \frac{x^2 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x^2*ArcTan[a*x])/(c + a^2*c*x^2)^2,x]","\frac{\left(a^2 x^2+1\right) \tan ^{-1}(a x)^2-2 a x \tan ^{-1}(a x)-1}{4 a^3 c^2 \left(a^2 x^2+1\right)}","\frac{\tan ^{-1}(a x)^2}{4 a^3 c^2}-\frac{x \tan ^{-1}(a x)}{2 a^2 c^2 \left(a^2 x^2+1\right)}-\frac{1}{4 a^3 c^2 \left(a^2 x^2+1\right)}",1,"(-1 - 2*a*x*ArcTan[a*x] + (1 + a^2*x^2)*ArcTan[a*x]^2)/(4*a^3*c^2*(1 + a^2*x^2))","A",1
186,1,39,62,0.0293851,"\int \frac{x \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x*ArcTan[a*x])/(c + a^2*c*x^2)^2,x]","\frac{\left(a^2 x^2-1\right) \tan ^{-1}(a x)+a x}{4 a^2 c^2 \left(a^2 x^2+1\right)}","\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,"(a*x + (-1 + a^2*x^2)*ArcTan[a*x])/(4*a^2*c^2*(1 + a^2*x^2))","A",1
187,1,44,61,0.0219261,"\int \frac{\tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]/(c + a^2*c*x^2)^2,x]","\frac{\left(a^2 x^2+1\right) \tan ^{-1}(a x)^2+2 a x \tan ^{-1}(a x)+1}{4 c^2 \left(a^3 x^2+a\right)}","\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 + 2*a*x*ArcTan[a*x] + (1 + a^2*x^2)*ArcTan[a*x]^2)/(4*c^2*(a + a^3*x^2))","A",1
188,1,72,117,0.1697152,"\int \frac{\tan ^{-1}(a x)}{x \left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]/(x*(c + a^2*c*x^2)^2),x]","-\frac{4 i \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)+4 i \tan ^{-1}(a x)^2+\sin \left(2 \tan ^{-1}(a x)\right)-2 \tan ^{-1}(a x) \left(\cos \left(2 \tan ^{-1}(a x)\right)+4 \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)\right)}{8 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 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)}{2 c^2}-\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,"-1/8*((4*I)*ArcTan[a*x]^2 - 2*ArcTan[a*x]*(Cos[2*ArcTan[a*x]] + 4*Log[1 - E^((2*I)*ArcTan[a*x])]) + (4*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] + Sin[2*ArcTan[a*x]])/c^2","A",0
189,1,94,97,0.0756482,"\int \frac{\tan ^{-1}(a x)}{x^2 \left(c+a^2 c x^2\right)^2} \, dx","Integrate[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{\left(3 a^2 x^2+2\right) \tan ^{-1}(a x)}{2 c^2 x \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{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,"-1/4*a/(c^2*(1 + a^2*x^2)) - ((2 + 3*a^2*x^2)*ArcTan[a*x])/(2*c^2*x*(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",1
190,1,93,156,0.4228369,"\int \frac{\tan ^{-1}(a x)}{x^3 \left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]/(x^3*(c + a^2*c*x^2)^2),x]","\frac{a^2 \left(\tan ^{-1}(a x) \left(-\frac{4}{a^2 x^2}-16 \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)-2 \cos \left(2 \tan ^{-1}(a x)\right)-4\right)+8 i \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)-\frac{4}{a x}+8 i \tan ^{-1}(a x)^2+\sin \left(2 \tan ^{-1}(a x)\right)\right)}{8 c^2}","\frac{i a^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)}{c^2}-\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{a^3 x}{4 c^2 \left(a^2 x^2+1\right)}-\frac{\tan ^{-1}(a x)}{2 c^2 x^2}-\frac{a}{2 c^2 x}",1,"(a^2*(-4/(a*x) + (8*I)*ArcTan[a*x]^2 + ArcTan[a*x]*(-4 - 4/(a^2*x^2) - 2*Cos[2*ArcTan[a*x]] - 16*Log[1 - E^((2*I)*ArcTan[a*x])]) + (8*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] + Sin[2*ArcTan[a*x]]))/(8*c^2)","A",0
191,1,124,136,0.1011824,"\int \frac{\tan ^{-1}(a x)}{x^4 \left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^2),x]","-\frac{7 a^3 \log (x)}{3 c^2}+\frac{5 a^3 \tan ^{-1}(a x)^2}{4 c^2}+\frac{\left(15 a^4 x^4+10 a^2 x^2-2\right) \tan ^{-1}(a x)}{6 c^2 x^3 \left(a^2 x^2+1\right)}+\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}{6 c^2 x^2}","-\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^4 x \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}+\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{\tan ^{-1}(a x)}{3 c^2 x^3}-\frac{a}{6 c^2 x^2}",1,"-1/6*a/(c^2*x^2) + a^3/(4*c^2*(1 + a^2*x^2)) + ((-2 + 10*a^2*x^2 + 15*a^4*x^4)*ArcTan[a*x])/(6*c^2*x^3*(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",1
192,1,58,86,0.1617206,"\int \frac{x^3 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^3*ArcTan[a*x])/(c + a^2*c*x^2)^3,x]","\frac{a x \left(5 a^2 x^2+3\right)+\left(5 a^4 x^4-6 a^2 x^2-3\right) \tan ^{-1}(a x)}{32 a^4 c^3 \left(a^2 x^2+1\right)^2}","-\frac{3 \tan ^{-1}(a x)}{32 a^4 c^3}+\frac{x^4 \tan ^{-1}(a x)}{4 c^3 \left(a^2 x^2+1\right)^2}+\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)}",1,"(a*x*(3 + 5*a^2*x^2) + (-3 - 6*a^2*x^2 + 5*a^4*x^4)*ArcTan[a*x])/(32*a^4*c^3*(1 + a^2*x^2)^2)","A",1
193,1,64,111,0.0565671,"\int \frac{x^2 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^2*ArcTan[a*x])/(c + a^2*c*x^2)^3,x]","\frac{a^2 x^2+2 a x \left(a^2 x^2-1\right) \tan ^{-1}(a x)+\left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}{16 a^3 c^3 \left(a^2 x^2+1\right)^2}","\frac{\tan ^{-1}(a x)^2}{16 a^3 c^3}+\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{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}",1,"(a^2*x^2 + 2*a*x*(-1 + a^2*x^2)*ArcTan[a*x] + (1 + a^2*x^2)^2*ArcTan[a*x]^2)/(16*a^3*c^3*(1 + a^2*x^2)^2)","A",1
194,1,55,84,0.0510047,"\int \frac{x \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x*ArcTan[a*x])/(c + a^2*c*x^2)^3,x]","\frac{a x \left(3 a^2 x^2+5\right)+\left(3 a^4 x^4+6 a^2 x^2-5\right) \tan ^{-1}(a x)}{32 c^3 \left(a^3 x^2+a\right)^2}","\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,"(a*x*(5 + 3*a^2*x^2) + (-5 + 6*a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x])/(32*c^3*(a + a^3*x^2)^2)","A",1
195,1,68,105,0.0293463,"\int \frac{\tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]/(c + a^2*c*x^2)^3,x]","\frac{3 a^2 x^2+2 a x \left(3 a^2 x^2+5\right) \tan ^{-1}(a x)+3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2+4}{16 a c^3 \left(a^2 x^2+1\right)^2}","\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,"(4 + 3*a^2*x^2 + 2*a*x*(5 + 3*a^2*x^2)*ArcTan[a*x] + 3*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(16*a*c^3*(1 + a^2*x^2)^2)","A",1
196,1,90,159,0.2382025,"\int \frac{\tan ^{-1}(a x)}{x \left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]/(x*(c + a^2*c*x^2)^3),x]","-\frac{64 i \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)+64 i \tan ^{-1}(a x)^2+24 \sin \left(2 \tan ^{-1}(a x)\right)+\sin \left(4 \tan ^{-1}(a x)\right)-4 \tan ^{-1}(a x) \left(32 \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)+12 \cos \left(2 \tan ^{-1}(a x)\right)+\cos \left(4 \tan ^{-1}(a x)\right)\right)}{128 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 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)}{2 c^3}-\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,"-1/128*((64*I)*ArcTan[a*x]^2 - 4*ArcTan[a*x]*(12*Cos[2*ArcTan[a*x]] + Cos[4*ArcTan[a*x]] + 32*Log[1 - E^((2*I)*ArcTan[a*x])]) + (64*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] + 24*Sin[2*ArcTan[a*x]] + Sin[4*ArcTan[a*x]])/c^3","A",0
197,1,118,142,0.1016164,"\int \frac{\tan ^{-1}(a x)}{x^2 \left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]/(x^2*(c + a^2*c*x^2)^3),x]","\frac{a x \left(-7 a^2 x^2+16 \left(a^2 x^2+1\right)^2 \log (x)-8 \left(a^2 x^2+1\right)^2 \log \left(a^2 x^2+1\right)-8\right)-15 a x \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2-2 \left(15 a^4 x^4+25 a^2 x^2+8\right) \tan ^{-1}(a x)}{16 c^3 x \left(a^2 x^2+1\right)^2}","-\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,"(-2*(8 + 25*a^2*x^2 + 15*a^4*x^4)*ArcTan[a*x] - 15*a*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2 + a*x*(-8 - 7*a^2*x^2 + 16*(1 + a^2*x^2)^2*Log[x] - 8*(1 + a^2*x^2)^2*Log[1 + a^2*x^2]))/(16*c^3*x*(1 + a^2*x^2)^2)","A",1
198,1,111,205,0.6298762,"\int \frac{\tan ^{-1}(a x)}{x^3 \left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]/(x^3*(c + a^2*c*x^2)^3),x]","\frac{a^2 \left(\tan ^{-1}(a x) \left(-\frac{64}{a^2 x^2}-384 \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)-80 \cos \left(2 \tan ^{-1}(a x)\right)-4 \cos \left(4 \tan ^{-1}(a x)\right)-64\right)+192 i \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)-\frac{64}{a x}+192 i \tan ^{-1}(a x)^2+40 \sin \left(2 \tan ^{-1}(a x)\right)+\sin \left(4 \tan ^{-1}(a x)\right)\right)}{128 c^3}","\frac{3 i a^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)}{2 c^3}-\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{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{\tan ^{-1}(a x)}{2 c^3 x^2}-\frac{a}{2 c^3 x}",1,"(a^2*(-64/(a*x) + (192*I)*ArcTan[a*x]^2 + ArcTan[a*x]*(-64 - 64/(a^2*x^2) - 80*Cos[2*ArcTan[a*x]] - 4*Cos[4*ArcTan[a*x]] - 384*Log[1 - E^((2*I)*ArcTan[a*x])]) + (192*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] + 40*Sin[2*ArcTan[a*x]] + Sin[4*ArcTan[a*x]]))/(128*c^3)","A",0
199,1,142,183,0.1320929,"\int \frac{\tan ^{-1}(a x)}{x^4 \left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^3),x]","\frac{105 a^3 x^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2+2 \left(105 a^6 x^6+175 a^4 x^4+56 a^2 x^2-8\right) \tan ^{-1}(a x)+a x \left(25 a^4 x^4-160 \left(a^3 x^3+a x\right)^2 \log (x)+20 a^2 x^2+80 \left(a^3 x^3+a x\right)^2 \log \left(a^2 x^2+1\right)-8\right)}{48 c^3 x^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{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{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{\tan ^{-1}(a x)}{3 c^3 x^3}-\frac{a}{6 c^3 x^2}",1,"(2*(-8 + 56*a^2*x^2 + 175*a^4*x^4 + 105*a^6*x^6)*ArcTan[a*x] + 105*a^3*x^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2 + a*x*(-8 + 20*a^2*x^2 + 25*a^4*x^4 - 160*(a*x + a^3*x^3)^2*Log[x] + 80*(a*x + a^3*x^3)^2*Log[1 + a^2*x^2]))/(48*c^3*x^3*(1 + a^2*x^2)^2)","A",1
200,1,105,160,0.1304982,"\int x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx","Integrate[x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x],x]","\frac{a x \left(5-6 a^2 x^2\right) \sqrt{a^2 c x^2+c}+11 \sqrt{c} \log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+a c x\right)+8 \left(3 a^4 x^4+a^2 x^2-2\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{120 a^4}","\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{15 a^2}+\frac{1}{5} x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{x^3 \sqrt{a^2 c x^2+c}}{20 a}-\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 \sqrt{a^2 c x^2+c}}{24 a^3}",1,"(a*x*(5 - 6*a^2*x^2)*Sqrt[c + a^2*c*x^2] + 8*Sqrt[c + a^2*c*x^2]*(-2 + a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x] + 11*Sqrt[c]*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(120*a^4)","A",1
201,1,278,298,3.0065308,"\int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx","Integrate[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x],x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(-\frac{1}{4} \left(a^2 x^2+1\right)^2 \left(-\frac{2}{\sqrt{a^2 x^2+1}}+3 \tan ^{-1}(a x) \left(-\frac{14 a x}{\sqrt{a^2 x^2+1}}+3 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)-3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+2 \sin \left(3 \tan ^{-1}(a x)\right)+4 \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right) \cos \left(2 \tan ^{-1}(a x)\right)+\left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right) \cos \left(4 \tan ^{-1}(a x)\right)\right)-6 \cos \left(3 \tan ^{-1}(a x)\right)\right)-6 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+6 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)\right)}{48 a^3 \sqrt{a^2 x^2+1}}","\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{8 a^2}+\frac{1}{4} x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{i c \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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{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*(1 + a^2*x^2)]*((-6*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (6*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - ((1 + a^2*x^2)^2*(-2/Sqrt[1 + a^2*x^2] - 6*Cos[3*ArcTan[a*x]] + 3*ArcTan[a*x]*((-14*a*x)/Sqrt[1 + a^2*x^2] + 3*Log[1 - I*E^(I*ArcTan[a*x])] + 4*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) - 3*Log[1 + I*E^(I*ArcTan[a*x])] + 2*Sin[3*ArcTan[a*x]])))/4))/(48*a^3*Sqrt[1 + a^2*x^2])","A",0
202,1,86,86,0.107665,"\int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx","Integrate[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x],x]","-\frac{a x \sqrt{a^2 c x^2+c}+\sqrt{c} \log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+a c x\right)-2 \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{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,"-1/6*(a*x*Sqrt[c + a^2*c*x^2] - 2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + Sqrt[c]*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/a^2","A",1
203,1,141,244,0.6089186,"\int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx","Integrate[Sqrt[c + a^2*c*x^2]*ArcTan[a*x],x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(\sqrt{a^2 x^2+1} \left(a x \tan ^{-1}(a x)-1\right)+i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)\right)}{2 a \sqrt{a^2 x^2+1}}","\frac{i c \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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*(1 + a^2*x^2)]*(Sqrt[1 + a^2*x^2]*(-1 + a*x*ArcTan[a*x]) + ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + I*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - I*PolyLog[2, I*E^(I*ArcTan[a*x])]))/(2*a*Sqrt[1 + a^2*x^2])","A",0
204,1,164,229,0.1896604,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x} \, dx","Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x,x]","\frac{\sqrt{a^2 c x^2+c} \left(\sqrt{a^2 x^2+1} \tan ^{-1}(a x)+i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)-\tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)+\log \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-\log \left(\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)+\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)\right)}{\sqrt{a^2 x^2+1}}","\frac{i c \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i c \sqrt{a^2 x^2+1} \text{Li}_2\left(\frac{\sqrt{i a x+1}}{\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]*(Sqrt[1 + a^2*x^2]*ArcTan[a*x] + ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + Log[Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]] - Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + I*PolyLog[2, -E^(I*ArcTan[a*x])] - I*PolyLog[2, E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2]","A",1
205,1,163,242,0.4579217,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^2} \, dx","Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x^2,x]","-\frac{a \sqrt{c \left(a^2 x^2+1\right)} \left(\frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x)}{a x}-i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x) \left(-\log \left(1-i e^{i \tan ^{-1}(a x)}\right)\right)+\tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-\log \left(\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)+\log \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)\right)}{\sqrt{a^2 x^2+1}}","\frac{i a c \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i a c \sqrt{a^2 x^2+1} \text{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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,"-((a*Sqrt[c*(1 + a^2*x^2)]*((Sqrt[1 + a^2*x^2]*ArcTan[a*x])/(a*x) - ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] + ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] + Log[Cos[ArcTan[a*x]/2]] - Log[Sin[ArcTan[a*x]/2]] - I*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + I*PolyLog[2, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2])","A",0
206,1,165,240,1.1244388,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^3} \, dx","Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x^3,x]","\frac{a^2 \sqrt{c \left(a^2 x^2+1\right)} \left(4 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-4 i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-2 \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)+4 \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)-4 \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)-2 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\tan ^{-1}(a x) \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+\tan ^{-1}(a x) \sec ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}{8 \sqrt{a^2 x^2+1}}","\frac{i a^2 c \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\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^2*Sqrt[c*(1 + a^2*x^2)]*(-2*Cot[ArcTan[a*x]/2] - ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 + 4*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 4*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + (4*I)*PolyLog[2, -E^(I*ArcTan[a*x])] - (4*I)*PolyLog[2, E^(I*ArcTan[a*x])] + ArcTan[a*x]*Sec[ArcTan[a*x]/2]^2 - 2*Tan[ArcTan[a*x]/2]))/(8*Sqrt[1 + a^2*x^2])","A",0
207,1,105,84,0.1236573,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^4} \, dx","Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x^4,x]","\frac{a^3 \sqrt{c} x^3 \log (x)-a x \left(\sqrt{a^2 c x^2+c}+a^2 \sqrt{c} x^2 \log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+c\right)\right)-2 \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{6 x^3}","-\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,"(-2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + a^3*Sqrt[c]*x^3*Log[x] - a*x*(Sqrt[c + a^2*c*x^2] + a^2*Sqrt[c]*x^2*Log[c + Sqrt[c]*Sqrt[c + a^2*c*x^2]]))/(6*x^3)","A",1
208,1,119,217,0.1720846,"\int x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x) \, dx","Integrate[x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x],x]","\frac{51 c^{3/2} \log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+a c x\right)+48 c \left(5 a^2 x^2-2\right) \left(a^2 x^2+1\right)^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+a c x \left(-40 a^4 x^4-46 a^2 x^2+45\right) \sqrt{a^2 c x^2+c}}{1680 a^4}","\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{35 a^2}+\frac{1}{7} a^2 c x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{1}{42} a c x^5 \sqrt{a^2 c x^2+c}+\frac{8}{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}}{840 a}+\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{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{35 a^4}+\frac{3 c x \sqrt{a^2 c x^2+c}}{112 a^3}",1,"(a*c*x*Sqrt[c + a^2*c*x^2]*(45 - 46*a^2*x^2 - 40*a^4*x^4) + 48*c*(1 + a^2*x^2)^2*(-2 + 5*a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + 51*c^(3/2)*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(1680*a^4)","A",1
209,1,576,357,6.3373465,"\int x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x) \, dx","Integrate[x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x],x]","\frac{c \sqrt{a^2 c x^2+c} \left(\frac{3}{4} \left(a^2 x^2+1\right)^{5/2}+\frac{55}{8} \left(a^2 x^2+1\right)^3 \cos \left(3 \tan ^{-1}(a x)\right)-\frac{45}{8} \left(a^2 x^2+1\right)^3 \cos \left(5 \tan ^{-1}(a x)\right)+\frac{15}{16} \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x) \left(\frac{156 a x}{\sqrt{a^2 x^2+1}}+30 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)-30 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-94 \sin \left(3 \tan ^{-1}(a x)\right)+6 \sin \left(5 \tan ^{-1}(a x)\right)+3 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \cos \left(6 \tan ^{-1}(a x)\right)+45 \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right) \cos \left(2 \tan ^{-1}(a x)\right)+18 \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right) \cos \left(4 \tan ^{-1}(a x)\right)-3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \cos \left(6 \tan ^{-1}(a x)\right)\right)-\frac{15}{2} \left(a^2 x^2+1\right)^2 \left(-\frac{2}{\sqrt{a^2 x^2+1}}+3 \tan ^{-1}(a x) \left(-\frac{14 a x}{\sqrt{a^2 x^2+1}}+3 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)-3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+2 \sin \left(3 \tan ^{-1}(a x)\right)+4 \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right) \cos \left(2 \tan ^{-1}(a x)\right)+\left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right) \cos \left(4 \tan ^{-1}(a x)\right)\right)-6 \cos \left(3 \tan ^{-1}(a x)\right)\right)-90 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+90 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)\right)}{1440 a^3 \sqrt{a^2 x^2+1}}","\frac{c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{16 a^2}+\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{i c^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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}",1,"(c*Sqrt[c + a^2*c*x^2]*((3*(1 + a^2*x^2)^(5/2))/4 + (55*(1 + a^2*x^2)^3*Cos[3*ArcTan[a*x]])/8 - (45*(1 + a^2*x^2)^3*Cos[5*ArcTan[a*x]])/8 - (90*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (90*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - (15*(1 + a^2*x^2)^2*(-2/Sqrt[1 + a^2*x^2] - 6*Cos[3*ArcTan[a*x]] + 3*ArcTan[a*x]*((-14*a*x)/Sqrt[1 + a^2*x^2] + 3*Log[1 - I*E^(I*ArcTan[a*x])] + 4*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) - 3*Log[1 + I*E^(I*ArcTan[a*x])] + 2*Sin[3*ArcTan[a*x]])))/2 + (15*(1 + a^2*x^2)^3*ArcTan[a*x]*((156*a*x)/Sqrt[1 + a^2*x^2] + 30*Log[1 - I*E^(I*ArcTan[a*x])] + 3*Cos[6*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + 45*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + 18*Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) - 30*Log[1 + I*E^(I*ArcTan[a*x])] - 3*Cos[6*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - 94*Sin[3*ArcTan[a*x]] + 6*Sin[5*ArcTan[a*x]]))/16))/(1440*a^3*Sqrt[1 + a^2*x^2])","A",0
210,1,101,109,0.1754147,"\int x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x) \, dx","Integrate[x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x],x]","-\frac{3 c^{3/2} \log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+a c x\right)+a c x \left(2 a^2 x^2+5\right) \sqrt{a^2 c x^2+c}-8 c \left(a^2 x^2+1\right)^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{40 a^2}","-\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,"-1/40*(a*c*x*(5 + 2*a^2*x^2)*Sqrt[c + a^2*c*x^2] - 8*c*(1 + a^2*x^2)^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + 3*c^(3/2)*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/a^2","A",1
211,1,351,298,2.7335301,"\int \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x) \, dx","Integrate[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x],x]","\frac{c \sqrt{a^2 c x^2+c} \left(2 \left(a^2 x^2+1\right)^{3/2}+96 \sqrt{a^2 x^2+1} \left(a x \tan ^{-1}(a x)-1\right)+6 \left(a^2 x^2+1\right)^2 \cos \left(3 \tan ^{-1}(a x)\right)-3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x) \left(-\frac{14 a x}{\sqrt{a^2 x^2+1}}+3 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)-3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+2 \sin \left(3 \tan ^{-1}(a x)\right)+4 \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right) \cos \left(2 \tan ^{-1}(a x)\right)+\left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right) \cos \left(4 \tan ^{-1}(a x)\right)\right)+72 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-72 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+96 \tan ^{-1}(a x) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)\right)}{192 a \sqrt{a^2 x^2+1}}","\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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,"(c*Sqrt[c + a^2*c*x^2]*(2*(1 + a^2*x^2)^(3/2) + 96*Sqrt[1 + a^2*x^2]*(-1 + a*x*ArcTan[a*x]) + 6*(1 + a^2*x^2)^2*Cos[3*ArcTan[a*x]] + 96*ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + (72*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (72*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - 3*(1 + a^2*x^2)^2*ArcTan[a*x]*((-14*a*x)/Sqrt[1 + a^2*x^2] + 3*Log[1 - I*E^(I*ArcTan[a*x])] + 4*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) - 3*Log[1 + I*E^(I*ArcTan[a*x])] + 2*Sin[3*ArcTan[a*x]])))/(192*a*Sqrt[1 + a^2*x^2])","A",0
212,1,220,281,0.2585939,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)}{x} \, dx","Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x,x]","\frac{c \sqrt{a^2 c x^2+c} \left(-a x \sqrt{a^2 x^2+1}+2 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+8 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+6 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-6 i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-\sinh ^{-1}(a x)+6 \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)-6 \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)+6 \log \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-6 \log \left(\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)+\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)\right)}{6 \sqrt{a^2 x^2+1}}","-\frac{7}{6} c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)+\frac{i c^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i c^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\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,"(c*Sqrt[c + a^2*c*x^2]*(-(a*x*Sqrt[1 + a^2*x^2]) - ArcSinh[a*x] + 8*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 2*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 6*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 6*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + 6*Log[Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]] - 6*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + (6*I)*PolyLog[2, -E^(I*ArcTan[a*x])] - (6*I)*PolyLog[2, E^(I*ArcTan[a*x])]))/(6*Sqrt[1 + a^2*x^2])","A",1
213,1,218,300,0.9527005,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)}{x^2} \, dx","Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x^2,x]","\frac{c \sqrt{a^2 c x^2+c} \left(-a x \sqrt{a^2 x^2+1}+a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)-2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+3 i a x \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-3 i a x \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+3 a x \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)-3 a x \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+2 a x \log \left(\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-2 a x \log \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)\right)}{2 x \sqrt{a^2 x^2+1}}","-a c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)+\frac{3 i a c^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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}}-\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,"(c*Sqrt[c + a^2*c*x^2]*(-(a*x*Sqrt[1 + a^2*x^2]) - 2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 3*a*x*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] - 3*a*x*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] - 2*a*x*Log[Cos[ArcTan[a*x]/2]] + 2*a*x*Log[Sin[ArcTan[a*x]/2]] + (3*I)*a*x*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (3*I)*a*x*PolyLog[2, I*E^(I*ArcTan[a*x])]))/(2*x*Sqrt[1 + a^2*x^2])","A",0
214,1,301,304,1.7027651,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)}{x^3} \, dx","Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x^3,x]","\frac{a^2 c \sqrt{a^2 c x^2+c} \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right) \left(12 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-12 i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-2 \cot ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+4 a x \tan ^{-1}(a x) \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+12 \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-12 \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\tan ^{-1}(a x) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right) \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+\tan ^{-1}(a x) \csc \left(\frac{1}{2} \tan ^{-1}(a x)\right) \sec \left(\frac{1}{2} \tan ^{-1}(a x)\right)+8 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right) \log \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-8 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right) \log \left(\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)+\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-2\right)}{8 \sqrt{a^2 x^2+1}}","-a^2 c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)+\frac{3 i a^2 c^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\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^2*c*Sqrt[c + a^2*c*x^2]*(-2 - 2*Cot[ArcTan[a*x]/2]^2 + 4*a*x*ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 - ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Csc[ArcTan[a*x]/2]^2 + 12*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 - E^(I*ArcTan[a*x])] - 12*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 + E^(I*ArcTan[a*x])] + 8*Cot[ArcTan[a*x]/2]*Log[Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]] - 8*Cot[ArcTan[a*x]/2]*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + (12*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, -E^(I*ArcTan[a*x])] - (12*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, E^(I*ArcTan[a*x])] + ArcTan[a*x]*Csc[ArcTan[a*x]/2]*Sec[ArcTan[a*x]/2])*Tan[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2])","A",0
215,1,263,310,0.5029371,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)}{x^4} \, dx","Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/x^4,x]","-\frac{c \sqrt{a^2 c x^2+c} \left(-6 i a^3 x^3 \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+6 i a^3 x^3 \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-6 a^3 x^3 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)+6 a^3 x^3 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-6 a^3 x^3 \log \left(\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)+6 a^3 x^3 \log \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)+a x \sqrt{a^2 x^2+1}+8 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+a^3 x^3 \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)\right)}{6 x^3 \sqrt{a^2 x^2+1}}","-\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{7}{6} a^3 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)+\frac{i a^3 c^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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}}",1,"-1/6*(c*Sqrt[c + a^2*c*x^2]*(a*x*Sqrt[1 + a^2*x^2] + 2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 8*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + a^3*x^3*ArcTanh[Sqrt[1 + a^2*x^2]] - 6*a^3*x^3*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] + 6*a^3*x^3*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] + 6*a^3*x^3*Log[Cos[ArcTan[a*x]/2]] - 6*a^3*x^3*Log[Sin[ArcTan[a*x]/2]] - (6*I)*a^3*x^3*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (6*I)*a^3*x^3*PolyLog[2, I*E^(I*ArcTan[a*x])]))/(x^3*Sqrt[1 + a^2*x^2])","A",0
216,1,129,289,0.2513365,"\int x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x) \, dx","Integrate[x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x],x]","\frac{c^2 \left(345 \sqrt{c} \log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+a c x\right)+384 \left(7 a^2 x^2-2\right) \left(a^2 x^2+1\right)^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-a x \left(336 a^6 x^6+824 a^4 x^4+410 a^2 x^2-423\right) \sqrt{a^2 c x^2+c}\right)}{24192 a^4}","\frac{c^2 x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{63 a^2}+\frac{19}{63} a^2 c^2 x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{103 a c^2 x^5 \sqrt{a^2 c x^2+c}}{3024}+\frac{5}{21} c^2 x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{205 c^2 x^3 \sqrt{a^2 c x^2+c}}{12096 a}+\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{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{63 a^4}+\frac{1}{9} a^4 c^2 x^8 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+\frac{47 c^2 x \sqrt{a^2 c x^2+c}}{2688 a^3}-\frac{1}{72} a^3 c^2 x^7 \sqrt{a^2 c x^2+c}",1,"(c^2*(-(a*x*Sqrt[c + a^2*c*x^2]*(-423 + 410*a^2*x^2 + 824*a^4*x^4 + 336*a^6*x^6)) + 384*(1 + a^2*x^2)^3*(-2 + 7*a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + 345*Sqrt[c]*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]]))/(24192*a^4)","A",1
217,1,1059,418,15.2771015,"\int x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x) \, dx","Integrate[x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x],x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(-\frac{1}{4} \left(-6 \cos \left(3 \tan ^{-1}(a x)\right)+3 \tan ^{-1}(a x) \left(-\frac{14 a x}{\sqrt{a^2 x^2+1}}+3 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)+4 \cos \left(2 \tan ^{-1}(a x)\right) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)+\cos \left(4 \tan ^{-1}(a x)\right) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)-3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+2 \sin \left(3 \tan ^{-1}(a x)\right)\right)-\frac{2}{\sqrt{a^2 x^2+1}}\right) \left(a^2 x^2+1\right)^2-6 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+6 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)\right) c^2}{48 a^3 \sqrt{a^2 x^2+1}}+\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(\frac{1}{16} \left(110 \cos \left(3 \tan ^{-1}(a x)\right)-90 \cos \left(5 \tan ^{-1}(a x)\right)+15 \tan ^{-1}(a x) \left(\frac{156 a x}{\sqrt{a^2 x^2+1}}+3 \cos \left(6 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)+30 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)+45 \cos \left(2 \tan ^{-1}(a x)\right) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)+18 \cos \left(4 \tan ^{-1}(a x)\right) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)-3 \cos \left(6 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-30 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-94 \sin \left(3 \tan ^{-1}(a x)\right)+6 \sin \left(5 \tan ^{-1}(a x)\right)\right)+\frac{12}{\sqrt{a^2 x^2+1}}\right) \left(a^2 x^2+1\right)^3+90 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-90 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)\right) c^2}{720 a^3 \sqrt{a^2 x^2+1}}+\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(-\frac{1}{64} \left(7658 \cos \left(3 \tan ^{-1}(a x)\right)+35 \left(314 \cos \left(5 \tan ^{-1}(a x)\right)-90 \cos \left(7 \tan ^{-1}(a x)\right)+3 \tan ^{-1}(a x) \left(-\frac{3530 a x}{\sqrt{a^2 x^2+1}}+120 \cos \left(6 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)+15 \cos \left(8 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)+525 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)+840 \cos \left(2 \tan ^{-1}(a x)\right) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)+420 \cos \left(4 \tan ^{-1}(a x)\right) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)-120 \cos \left(6 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-15 \cos \left(8 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-525 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+1790 \sin \left(3 \tan ^{-1}(a x)\right)-794 \sin \left(5 \tan ^{-1}(a x)\right)+30 \sin \left(7 \tan ^{-1}(a x)\right)\right)\right)+\frac{38134}{\sqrt{a^2 x^2+1}}\right) \left(a^2 x^2+1\right)^4-3150 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+3150 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)\right) c^2}{80640 a^3 \sqrt{a^2 x^2+1}}","\frac{5 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{128 a^2}+\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{1}{8} a^4 c^2 x^7 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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} \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{5 c^2 \sqrt{a^2 c x^2+c}}{128 a^3}-\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,"(c^2*Sqrt[c*(1 + a^2*x^2)]*((-6*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (6*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - ((1 + a^2*x^2)^2*(-2/Sqrt[1 + a^2*x^2] - 6*Cos[3*ArcTan[a*x]] + 3*ArcTan[a*x]*((-14*a*x)/Sqrt[1 + a^2*x^2] + 3*Log[1 - I*E^(I*ArcTan[a*x])] + 4*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) - 3*Log[1 + I*E^(I*ArcTan[a*x])] + 2*Sin[3*ArcTan[a*x]])))/4))/(48*a^3*Sqrt[1 + a^2*x^2]) + (c^2*Sqrt[c*(1 + a^2*x^2)]*((90*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (90*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] + ((1 + a^2*x^2)^3*(12/Sqrt[1 + a^2*x^2] + 110*Cos[3*ArcTan[a*x]] - 90*Cos[5*ArcTan[a*x]] + 15*ArcTan[a*x]*((156*a*x)/Sqrt[1 + a^2*x^2] + 30*Log[1 - I*E^(I*ArcTan[a*x])] + 3*Cos[6*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + 45*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + 18*Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) - 30*Log[1 + I*E^(I*ArcTan[a*x])] - 3*Cos[6*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - 94*Sin[3*ArcTan[a*x]] + 6*Sin[5*ArcTan[a*x]])))/16))/(720*a^3*Sqrt[1 + a^2*x^2]) + (c^2*Sqrt[c*(1 + a^2*x^2)]*((-3150*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (3150*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - ((1 + a^2*x^2)^4*(38134/Sqrt[1 + a^2*x^2] + 7658*Cos[3*ArcTan[a*x]] + 35*(314*Cos[5*ArcTan[a*x]] - 90*Cos[7*ArcTan[a*x]] + 3*ArcTan[a*x]*((-3530*a*x)/Sqrt[1 + a^2*x^2] + 525*Log[1 - I*E^(I*ArcTan[a*x])] + 120*Cos[6*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + 15*Cos[8*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + 840*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + 420*Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) - 525*Log[1 + I*E^(I*ArcTan[a*x])] - 120*Cos[6*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - 15*Cos[8*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 1790*Sin[3*ArcTan[a*x]] - 794*Sin[5*ArcTan[a*x]] + 30*Sin[7*ArcTan[a*x]]))))/64))/(80640*a^3*Sqrt[1 + a^2*x^2])","B",0
218,1,111,134,0.2170043,"\int x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x) \, dx","Integrate[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x],x]","\frac{c^2 \left(-15 \sqrt{c} \log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+a c x\right)+48 \left(a^2 x^2+1\right)^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-a x \left(8 a^4 x^4+26 a^2 x^2+33\right) \sqrt{a^2 c x^2+c}\right)}{336 a^2}","-\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{5 c^2 x \sqrt{a^2 c x^2+c}}{112 a}-\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,"(c^2*(-(a*x*Sqrt[c + a^2*c*x^2]*(33 + 26*a^2*x^2 + 8*a^4*x^4)) + 48*(1 + a^2*x^2)^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - 15*Sqrt[c]*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]]))/(336*a^2)","A",1
219,1,643,348,6.4972853,"\int \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x) \, dx","Integrate[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x],x]","\frac{c^2 \sqrt{a^2 c x^2+c} \left(\frac{3}{4} \left(a^2 x^2+1\right)^{5/2}+720 \sqrt{a^2 x^2+1} \left(a x \tan ^{-1}(a x)-1\right)+\frac{55}{8} \left(a^2 x^2+1\right)^3 \cos \left(3 \tan ^{-1}(a x)\right)-\frac{45}{8} \left(a^2 x^2+1\right)^3 \cos \left(5 \tan ^{-1}(a x)\right)+\frac{15}{16} \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x) \left(\frac{156 a x}{\sqrt{a^2 x^2+1}}+30 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)-30 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-94 \sin \left(3 \tan ^{-1}(a x)\right)+6 \sin \left(5 \tan ^{-1}(a x)\right)+3 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \cos \left(6 \tan ^{-1}(a x)\right)+45 \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right) \cos \left(2 \tan ^{-1}(a x)\right)+18 \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right) \cos \left(4 \tan ^{-1}(a x)\right)-3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \cos \left(6 \tan ^{-1}(a x)\right)\right)-15 \left(a^2 x^2+1\right)^2 \left(-\frac{2}{\sqrt{a^2 x^2+1}}+3 \tan ^{-1}(a x) \left(-\frac{14 a x}{\sqrt{a^2 x^2+1}}+3 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)-3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+2 \sin \left(3 \tan ^{-1}(a x)\right)+4 \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right) \cos \left(2 \tan ^{-1}(a x)\right)+\left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right) \cos \left(4 \tan ^{-1}(a x)\right)\right)-6 \cos \left(3 \tan ^{-1}(a x)\right)\right)+450 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-450 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+720 \tan ^{-1}(a x) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)\right)}{1440 a \sqrt{a^2 x^2+1}}","\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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} \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 c^2 \sqrt{a^2 c x^2+c}}{16 a}+\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,"(c^2*Sqrt[c + a^2*c*x^2]*((3*(1 + a^2*x^2)^(5/2))/4 + 720*Sqrt[1 + a^2*x^2]*(-1 + a*x*ArcTan[a*x]) + (55*(1 + a^2*x^2)^3*Cos[3*ArcTan[a*x]])/8 - (45*(1 + a^2*x^2)^3*Cos[5*ArcTan[a*x]])/8 + 720*ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + (450*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (450*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - 15*(1 + a^2*x^2)^2*(-2/Sqrt[1 + a^2*x^2] - 6*Cos[3*ArcTan[a*x]] + 3*ArcTan[a*x]*((-14*a*x)/Sqrt[1 + a^2*x^2] + 3*Log[1 - I*E^(I*ArcTan[a*x])] + 4*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) - 3*Log[1 + I*E^(I*ArcTan[a*x])] + 2*Sin[3*ArcTan[a*x]])) + (15*(1 + a^2*x^2)^3*ArcTan[a*x]*((156*a*x)/Sqrt[1 + a^2*x^2] + 30*Log[1 - I*E^(I*ArcTan[a*x])] + 3*Cos[6*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + 45*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + 18*Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) - 30*Log[1 + I*E^(I*ArcTan[a*x])] - 3*Cos[6*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - 94*Sin[3*ArcTan[a*x]] + 6*Sin[5*ArcTan[a*x]]))/16))/(1440*a*Sqrt[1 + a^2*x^2])","A",0
220,1,268,329,0.3518437,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)}{x} \, dx","Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x,x]","\frac{c^2 \sqrt{a^2 c x^2+c} \left(-35 a x \sqrt{a^2 x^2+1}+88 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+184 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+24 a^4 x^4 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)-6 a^3 x^3 \sqrt{a^2 x^2+1}+120 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-120 i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-29 \sinh ^{-1}(a x)+120 \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)-120 \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)+120 \log \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-120 \log \left(\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)+\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)\right)}{120 \sqrt{a^2 x^2+1}}","-\frac{149}{120} c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)+\frac{i c^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i c^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\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{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{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,"(c^2*Sqrt[c + a^2*c*x^2]*(-35*a*x*Sqrt[1 + a^2*x^2] - 6*a^3*x^3*Sqrt[1 + a^2*x^2] - 29*ArcSinh[a*x] + 184*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 88*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 24*a^4*x^4*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 120*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 120*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + 120*Log[Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]] - 120*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + (120*I)*PolyLog[2, -E^(I*ArcTan[a*x])] - (120*I)*PolyLog[2, E^(I*ArcTan[a*x])]))/(120*Sqrt[1 + a^2*x^2])","A",1
221,1,491,355,4.1558135,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)}{x^2} \, dx","Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x^2,x]","\frac{a c^2 \sqrt{a^2 c x^2+c} \left(-48 \left(\frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x)}{a x}-i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x) \left(-\log \left(1-i e^{i \tan ^{-1}(a x)}\right)\right)+\tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-\log \left(\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)+\log \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)\right)+\frac{1}{2} \left(a^2 x^2+1\right)^{3/2}+48 \sqrt{a^2 x^2+1} \left(a x \tan ^{-1}(a x)-1\right)+\frac{3}{2} \left(a^2 x^2+1\right)^2 \cos \left(3 \tan ^{-1}(a x)\right)-\frac{3}{4} \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x) \left(-\frac{14 a x}{\sqrt{a^2 x^2+1}}+3 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)-3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+2 \sin \left(3 \tan ^{-1}(a x)\right)+4 \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right) \cos \left(2 \tan ^{-1}(a x)\right)+\left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right) \cos \left(4 \tan ^{-1}(a x)\right)\right)+42 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-42 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+48 \tan ^{-1}(a x) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)\right)}{48 \sqrt{a^2 x^2+1}}","-a c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)+\frac{15 i a c^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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} \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 c^2 \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}-\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,"(a*c^2*Sqrt[c + a^2*c*x^2]*((1 + a^2*x^2)^(3/2)/2 + 48*Sqrt[1 + a^2*x^2]*(-1 + a*x*ArcTan[a*x]) + (3*(1 + a^2*x^2)^2*Cos[3*ArcTan[a*x]])/2 + 48*ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + (42*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - 48*((Sqrt[1 + a^2*x^2]*ArcTan[a*x])/(a*x) - ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] + ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] + Log[Cos[ArcTan[a*x]/2]] - Log[Sin[ArcTan[a*x]/2]] - I*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + I*PolyLog[2, I*E^(I*ArcTan[a*x])]) - (42*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - (3*(1 + a^2*x^2)^2*ArcTan[a*x]*((-14*a*x)/Sqrt[1 + a^2*x^2] + 3*Log[1 - I*E^(I*ArcTan[a*x])] + 4*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) - 3*Log[1 + I*E^(I*ArcTan[a*x])] + 2*Sin[3*ArcTan[a*x]]))/4))/(48*Sqrt[1 + a^2*x^2])","A",0
222,1,361,364,2.0775529,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)}{x^3} \, dx","Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x^3,x]","\frac{a^2 c^2 \sqrt{a^2 c x^2+c} \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right) \left(4 a^3 x^3 \tan ^{-1}(a x) \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)-2 a^2 x^2 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+60 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-60 i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-6 \cot ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+28 a x \tan ^{-1}(a x) \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+60 \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-60 \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-3 \tan ^{-1}(a x) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right) \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)-4 \sinh ^{-1}(a x) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+3 \tan ^{-1}(a x) \csc \left(\frac{1}{2} \tan ^{-1}(a x)\right) \sec \left(\frac{1}{2} \tan ^{-1}(a x)\right)+48 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right) \log \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-48 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right) \log \left(\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)+\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-6\right)}{24 \sqrt{a^2 x^2+1}}","-\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 i a^2 c^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{2 \sqrt{a^2 c x^2+c}}-\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{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{1}{3} a^2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)-\frac{1}{6} a^3 c^2 x \sqrt{a^2 c x^2+c}",1,"(a^2*c^2*Sqrt[c + a^2*c*x^2]*(-6 - 4*ArcSinh[a*x]*Cot[ArcTan[a*x]/2] - 6*Cot[ArcTan[a*x]/2]^2 - 2*a^2*x^2*Csc[ArcTan[a*x]/2]^2 + 28*a*x*ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 + 4*a^3*x^3*ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 - 3*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Csc[ArcTan[a*x]/2]^2 + 60*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 - E^(I*ArcTan[a*x])] - 60*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 + E^(I*ArcTan[a*x])] + 48*Cot[ArcTan[a*x]/2]*Log[Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]] - 48*Cot[ArcTan[a*x]/2]*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + (60*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, -E^(I*ArcTan[a*x])] - (60*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, E^(I*ArcTan[a*x])] + 3*ArcTan[a*x]*Csc[ArcTan[a*x]/2]*Sec[ArcTan[a*x]/2])*Tan[ArcTan[a*x]/2])/(24*Sqrt[1 + a^2*x^2])","A",0
223,1,313,372,1.0361045,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)}{x^4} \, dx","Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/x^4,x]","\frac{c^2 \sqrt{a^2 c x^2+c} \left(15 i a^3 x^3 \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-15 i a^3 x^3 \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+15 a^3 x^3 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)-15 a^3 x^3 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+12 a^3 x^3 \log \left(\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-12 a^3 x^3 \log \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-a x \sqrt{a^2 x^2+1}-14 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)-2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+3 a^4 x^4 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)-3 a^3 x^3 \sqrt{a^2 x^2+1}-a^3 x^3 \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)\right)}{6 x^3 \sqrt{a^2 x^2+1}}","-\frac{a c^2 \sqrt{a^2 c x^2+c}}{6 x^2}-\frac{2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}{3 x^3}+\frac{1}{2} a^4 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{13}{6} a^3 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)+\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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} \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{1}{2} a^3 c^2 \sqrt{a^2 c x^2+c}",1,"(c^2*Sqrt[c + a^2*c*x^2]*(-(a*x*Sqrt[1 + a^2*x^2]) - 3*a^3*x^3*Sqrt[1 + a^2*x^2] - 2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] - 14*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 3*a^4*x^4*Sqrt[1 + a^2*x^2]*ArcTan[a*x] - a^3*x^3*ArcTanh[Sqrt[1 + a^2*x^2]] + 15*a^3*x^3*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] - 15*a^3*x^3*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] - 12*a^3*x^3*Log[Cos[ArcTan[a*x]/2]] + 12*a^3*x^3*Log[Sin[ArcTan[a*x]/2]] + (15*I)*a^3*x^3*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (15*I)*a^3*x^3*PolyLog[2, I*E^(I*ArcTan[a*x])]))/(6*x^3*Sqrt[1 + a^2*x^2])","A",0
224,1,91,120,0.1255063,"\int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(x^3*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","\frac{-a x \sqrt{a^2 c x^2+c}+5 \sqrt{c} \log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+a c x\right)+2 \left(a^2 x^2-2\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{6 a^4 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}",1,"(-(a*x*Sqrt[c + a^2*c*x^2]) + 2*(-2 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] + 5*Sqrt[c]*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(6*a^4*c)","A",1
225,1,158,250,0.6189811,"\int \frac{x^2 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(x^2*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","-\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(\sqrt{a^2 x^2+1}-a x \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)}{2 a^3 c \sqrt{a^2 x^2+1}}","\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{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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}}",1,"-1/2*(Sqrt[c*(1 + a^2*x^2)]*(Sqrt[1 + a^2*x^2] - a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] - ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] + I*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - I*PolyLog[2, I*E^(I*ArcTan[a*x])]))/(a^3*c*Sqrt[1 + a^2*x^2])","A",0
226,1,60,59,0.064173,"\int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(x*ArcTan[a*x])/Sqrt[c + a^2*c*x^2],x]","\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\sqrt{c} \log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+a c x\right)}{a^2 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] - Sqrt[c]*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(a^2*c)","A",1
227,1,118,193,0.1125642,"\int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]/Sqrt[c + a^2*c*x^2],x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)\right)}{a c \sqrt{a^2 x^2+1}}","\frac{i \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{a \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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,"(Sqrt[c*(1 + a^2*x^2)]*(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + I*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - I*PolyLog[2, I*E^(I*ArcTan[a*x])]))/(a*c*Sqrt[1 + a^2*x^2])","A",0
228,1,100,177,0.1436432,"\int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]/(x*Sqrt[c + a^2*c*x^2]),x]","\frac{\sqrt{a^2 x^2+1} \left(i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x) \left(\log \left(1-e^{i \tan ^{-1}(a x)}\right)-\log \left(1+e^{i \tan ^{-1}(a x)}\right)\right)\right)}{\sqrt{c \left(a^2 x^2+1\right)}}","\frac{i \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{Li}_2\left(\frac{\sqrt{i a x+1}}{\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,"(Sqrt[1 + a^2*x^2]*(ArcTan[a*x]*(Log[1 - E^(I*ArcTan[a*x])] - Log[1 + E^(I*ArcTan[a*x])]) + I*PolyLog[2, -E^(I*ArcTan[a*x])] - I*PolyLog[2, E^(I*ArcTan[a*x])]))/Sqrt[c*(1 + a^2*x^2)]","A",0
229,1,62,56,0.0963364,"\int \frac{\tan ^{-1}(a x)}{x^2 \sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]/(x^2*Sqrt[c + a^2*c*x^2]),x]","\frac{a \left(\log (x)-\log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+c\right)\right)}{\sqrt{c}}-\frac{\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)}{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*(Log[x] - Log[c + Sqrt[c]*Sqrt[c + a^2*c*x^2]]))/Sqrt[c]","A",1
230,1,165,242,0.7223751,"\int \frac{\tan ^{-1}(a x)}{x^3 \sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]/(x^3*Sqrt[c + a^2*c*x^2]),x]","\frac{a^2 \sqrt{a^2 x^2+1} \left(-4 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+4 i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-2 \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)-4 \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)+4 \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)-2 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\tan ^{-1}(a x) \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+\tan ^{-1}(a x) \sec ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}{8 \sqrt{c \left(a^2 x^2+1\right)}}","-\frac{i a^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\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^2*Sqrt[1 + a^2*x^2]*(-2*Cot[ArcTan[a*x]/2] - ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 - 4*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] + 4*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] - (4*I)*PolyLog[2, -E^(I*ArcTan[a*x])] + (4*I)*PolyLog[2, E^(I*ArcTan[a*x])] + ArcTan[a*x]*Sec[ArcTan[a*x]/2]^2 - 2*Tan[ArcTan[a*x]/2]))/(8*Sqrt[c*(1 + a^2*x^2)])","A",0
231,1,110,118,0.124619,"\int \frac{\tan ^{-1}(a x)}{x^4 \sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]/(x^4*Sqrt[c + a^2*c*x^2]),x]","\frac{-5 a^3 \sqrt{c} x^3 \log (x)-a x \sqrt{a^2 c x^2+c}+2 \left(2 a^2 x^2-1\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)+5 a^3 \sqrt{c} x^3 \log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+c\right)}{6 c x^3}","-\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*x*Sqrt[c + a^2*c*x^2]) + 2*(-1 + 2*a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - 5*a^3*Sqrt[c]*x^3*Log[x] + 5*a^3*Sqrt[c]*x^3*Log[c + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(6*c*x^3)","A",1
232,1,107,107,0.1337301,"\int \frac{x^3 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(x^3*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]","\frac{-a x \sqrt{a^2 c x^2+c}-\sqrt{c} \left(a^2 x^2+1\right) \log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+a c x\right)+\left(a^2 x^2+2\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^4 c^2 \left(a^2 x^2+1\right)}","-\frac{\tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a^4 c^{3/2}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^4 c^2}+\frac{\tan ^{-1}(a x)}{a^4 c \sqrt{a^2 c x^2+c}}-\frac{x}{a^3 c \sqrt{a^2 c x^2+c}}",1,"(-(a*x*Sqrt[c + a^2*c*x^2]) + (2 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - Sqrt[c]*(1 + a^2*x^2)*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(a^4*c^2*(1 + a^2*x^2))","A",1
233,1,155,251,0.2542235,"\int \frac{x^2 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(x^2*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]","-\frac{\sqrt{a^2 x^2+1} \left(\frac{1}{\sqrt{a^2 x^2+1}}+\frac{a x \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x) \left(-\log \left(1-i e^{i \tan ^{-1}(a x)}\right)\right)+\tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)}{a^3 c \sqrt{c \left(a^2 x^2+1\right)}}","-\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{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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}}",1,"-((Sqrt[1 + a^2*x^2]*(1/Sqrt[1 + a^2*x^2] + (a*x*ArcTan[a*x])/Sqrt[1 + a^2*x^2] - ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] + ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] - I*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + I*PolyLog[2, I*E^(I*ArcTan[a*x])]))/(a^3*c*Sqrt[c*(1 + a^2*x^2)]))","A",0
234,1,42,49,0.0533466,"\int \frac{x \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(x*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(a x-\tan ^{-1}(a x)\right)}{a^2 c^2 \left(a^2 x^2+1\right)}","\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,"(Sqrt[c + a^2*c*x^2]*(a*x - ArcTan[a*x]))/(a^2*c^2*(1 + a^2*x^2))","A",1
235,1,38,45,0.0482366,"\int \frac{\tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]/(c + a^2*c*x^2)^(3/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(a x \tan ^{-1}(a x)+1\right)}{c^2 \left(a^3 x^2+a\right)}","\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,"(Sqrt[c + a^2*c*x^2]*(1 + a*x*ArcTan[a*x]))/(c^2*(a + a^3*x^2))","A",1
236,1,141,229,0.2296447,"\int \frac{\tan ^{-1}(a x)}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]/(x*(c + a^2*c*x^2)^(3/2)),x]","\frac{\sqrt{a^2 x^2+1} \left(-\frac{a x}{\sqrt{a^2 x^2+1}}+\frac{\tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)-\tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)\right)}{c \sqrt{c \left(a^2 x^2+1\right)}}","\frac{i \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{Li}_2\left(\frac{\sqrt{i a x+1}}{\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,"(Sqrt[1 + a^2*x^2]*(-((a*x)/Sqrt[1 + a^2*x^2]) + ArcTan[a*x]/Sqrt[1 + a^2*x^2] + ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + I*PolyLog[2, -E^(I*ArcTan[a*x])] - I*PolyLog[2, E^(I*ArcTan[a*x])]))/(c*Sqrt[c*(1 + a^2*x^2)])","A",0
237,1,122,103,0.1881586,"\int \frac{\tan ^{-1}(a x)}{x^2 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]/(x^2*(c + a^2*c*x^2)^(3/2)),x]","-\frac{a \log \left(\sqrt{c} \sqrt{c \left(a^2 x^2+1\right)}+c\right)}{c^{3/2}}-\frac{a \sqrt{c \left(a^2 x^2+1\right)}}{c^2 \left(a^2 x^2+1\right)}-\frac{\left(2 a^2 x^2+1\right) \sqrt{c \left(a^2 x^2+1\right)} \tan ^{-1}(a x)}{c^2 x \left(a^2 x^2+1\right)}+\frac{a \log (x)}{c^{3/2}}","-\frac{a \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{c^{3/2}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c^2 x}-\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*Sqrt[c*(1 + a^2*x^2)])/(c^2*(1 + a^2*x^2))) - (Sqrt[c*(1 + a^2*x^2)]*(1 + 2*a^2*x^2)*ArcTan[a*x])/(c^2*x*(1 + a^2*x^2)) + (a*Log[x])/c^(3/2) - (a*Log[c + Sqrt[c]*Sqrt[c*(1 + a^2*x^2)]])/c^(3/2)","A",1
238,1,258,300,1.3154732,"\int \frac{\tan ^{-1}(a x)}{x^3 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]/(x^3*(c + a^2*c*x^2)^(3/2)),x]","-\frac{a^2 \left(12 i \sqrt{a^2 x^2+1} \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-12 i \sqrt{a^2 x^2+1} \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+2 \sqrt{a^2 x^2+1} \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)+12 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)-12 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)+\sqrt{a^2 x^2+1} \tan ^{-1}(a x) \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)-\sqrt{a^2 x^2+1} \tan ^{-1}(a x) \sec ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)-8 a x+8 \tan ^{-1}(a x)+a x \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}{8 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{3 i a^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{2 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{a^3 x}{c \sqrt{a^2 c x^2+c}}",1,"-1/8*(a^2*(-8*a*x + 8*ArcTan[a*x] + a*x*Csc[ArcTan[a*x]/2]^2 + Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 + 12*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 12*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + (12*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, -E^(I*ArcTan[a*x])] - (12*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, E^(I*ArcTan[a*x])] - Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Sec[ArcTan[a*x]/2]^2 + 2*Sqrt[1 + a^2*x^2]*Tan[ArcTan[a*x]/2]))/(c*Sqrt[c + a^2*c*x^2])","A",0
239,1,143,165,0.3385602,"\int \frac{\tan ^{-1}(a x)}{x^4 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^(3/2)),x]","\frac{-11 a^3 \sqrt{c} \log (x)+\frac{a \left(5 a^2 x^2-1\right) \sqrt{a^2 c x^2+c}}{a^2 x^4+x^2}+\frac{2 \left(8 a^4 x^4+4 a^2 x^2-1\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^2 x^5+x^3}+11 a^3 \sqrt{c} \log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+c\right)}{6 c^2}","-\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{a^4 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}+\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}}",1,"((a*(-1 + 5*a^2*x^2)*Sqrt[c + a^2*c*x^2])/(x^2 + a^2*x^4) + (2*Sqrt[c + a^2*c*x^2]*(-1 + 4*a^2*x^2 + 8*a^4*x^4)*ArcTan[a*x])/(x^3 + a^2*x^5) - 11*a^3*Sqrt[c]*Log[x] + 11*a^3*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(6*c^2)","A",1
240,1,131,170,0.2113907,"\int \frac{x^5 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^5*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2),x]","-\frac{a x \left(16 a^2 x^2+15\right) \sqrt{a^2 c x^2+c}+9 \sqrt{c} \left(a^2 x^2+1\right)^2 \log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+a c x\right)-3 \left(3 a^4 x^4+12 a^2 x^2+8\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{9 a^6 c^3 \left(a^2 x^2+1\right)^2}","-\frac{\tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{a^6 c^{5/2}}+\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{5 x}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^2 \tan ^{-1}(a x)}{3 a^4 c \left(a^2 c x^2+c\right)^{3/2}}-\frac{x^3}{9 a^3 c \left(a^2 c x^2+c\right)^{3/2}}",1,"-1/9*(a*x*(15 + 16*a^2*x^2)*Sqrt[c + a^2*c*x^2] - 3*Sqrt[c + a^2*c*x^2]*(8 + 12*a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x] + 9*Sqrt[c]*(1 + a^2*x^2)^2*Log[a*c*x + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/(a^6*c^3*(1 + a^2*x^2)^2)","A",1
241,1,177,308,0.3996001,"\int \frac{x^4 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^4*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(-\frac{45}{\sqrt{a^2 x^2+1}}-\frac{45 a x \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+36 i \left(\text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)\right)+36 \tan ^{-1}(a x) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)+3 \tan ^{-1}(a x) \sin \left(3 \tan ^{-1}(a x)\right)+\cos \left(3 \tan ^{-1}(a x)\right)\right)}{36 a^5 c^3 \sqrt{a^2 x^2+1}}","-\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{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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{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 \tan ^{-1}(a x)}{a^4 c^2 \sqrt{a^2 c x^2+c}}",1,"(Sqrt[c*(1 + a^2*x^2)]*(-45/Sqrt[1 + a^2*x^2] - (45*a*x*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + Cos[3*ArcTan[a*x]] + 36*ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + (36*I)*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])]) + 3*ArcTan[a*x]*Sin[3*ArcTan[a*x]]))/(36*a^5*c^3*Sqrt[1 + a^2*x^2])","A",0
242,1,65,112,0.0792599,"\int \frac{x^3 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^3*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(a x \left(7 a^2 x^2+6\right)-3 \left(3 a^2 x^2+2\right) \tan ^{-1}(a x)\right)}{9 a^4 c^3 \left(a^2 x^2+1\right)^2}","-\frac{x^2 \tan ^{-1}(a x)}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^3}{9 a c \left(a^2 c x^2+c\right)^{3/2}}-\frac{2 \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}",1,"(Sqrt[c + a^2*c*x^2]*(a*x*(6 + 7*a^2*x^2) - 3*(2 + 3*a^2*x^2)*ArcTan[a*x]))/(9*a^4*c^3*(1 + a^2*x^2)^2)","A",1
243,1,57,77,0.0649152,"\int \frac{x^2 \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^2*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(3 a^3 x^3 \tan ^{-1}(a x)+3 a^2 x^2+2\right)}{9 a^3 c^3 \left(a^2 x^2+1\right)^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}}",1,"(Sqrt[c + a^2*c*x^2]*(2 + 3*a^2*x^2 + 3*a^3*x^3*ArcTan[a*x]))/(9*a^3*c^3*(1 + a^2*x^2)^2)","A",1
244,1,51,79,0.0564562,"\int \frac{x \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x*ArcTan[a*x])/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(2 a^3 x^3+3 a x-3 \tan ^{-1}(a x)\right)}{9 c^3 \left(a^3 x^2+a\right)^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,"(Sqrt[c + a^2*c*x^2]*(3*a*x + 2*a^3*x^3 - 3*ArcTan[a*x]))/(9*c^3*(a + a^3*x^2)^2)","A",1
245,1,63,101,0.0527463,"\int \frac{\tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcTan[a*x]/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(\left(6 a^3 x^3+9 a x\right) \tan ^{-1}(a x)+6 a^2 x^2+7\right)}{9 a c^3 \left(a^2 x^2+1\right)^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,"(Sqrt[c + a^2*c*x^2]*(7 + 6*a^2*x^2 + (9*a*x + 6*a^3*x^3)*ArcTan[a*x]))/(9*a*c^3*(1 + a^2*x^2)^2)","A",1
246,1,168,279,0.3800556,"\int \frac{\tan ^{-1}(a x)}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcTan[a*x]/(x*(c + a^2*c*x^2)^(5/2)),x]","\frac{\left(a^2 x^2+1\right)^{3/2} \left(-\frac{45 a x}{\sqrt{a^2 x^2+1}}+\frac{45 \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+36 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-36 i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+36 \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)-36 \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)-\sin \left(3 \tan ^{-1}(a x)\right)+3 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)\right)}{36 c \left(c \left(a^2 x^2+1\right)\right)^{3/2}}","\frac{i \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{i \sqrt{a^2 x^2+1} \text{Li}_2\left(\frac{\sqrt{i a x+1}}{\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,"((1 + a^2*x^2)^(3/2)*((-45*a*x)/Sqrt[1 + a^2*x^2] + (45*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + 3*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 36*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 36*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + (36*I)*PolyLog[2, -E^(I*ArcTan[a*x])] - (36*I)*PolyLog[2, E^(I*ArcTan[a*x])] - Sin[3*ArcTan[a*x]]))/(36*c*(c*(1 + a^2*x^2))^(3/2))","A",0
247,1,151,158,0.2547725,"\int \frac{\tan ^{-1}(a x)}{x^2 \left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcTan[a*x]/(x^2*(c + a^2*c*x^2)^(5/2)),x]","\frac{a x \left(-\left(15 a^2 x^2+16\right) \sqrt{a^2 c x^2+c}+9 \sqrt{c} \left(a^2 x^2+1\right)^2 \log (x)-9 \sqrt{c} \left(a^2 x^2+1\right)^2 \log \left(\sqrt{c} \sqrt{a^2 c x^2+c}+c\right)\right)-3 \left(8 a^4 x^4+12 a^2 x^2+3\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{9 c^3 x \left(a^2 x^2+1\right)^2}","-\frac{a \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{c^{5/2}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{c^3 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{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,"(-3*Sqrt[c + a^2*c*x^2]*(3 + 12*a^2*x^2 + 8*a^4*x^4)*ArcTan[a*x] + a*x*(-((16 + 15*a^2*x^2)*Sqrt[c + a^2*c*x^2]) + 9*Sqrt[c]*(1 + a^2*x^2)^2*Log[x] - 9*Sqrt[c]*(1 + a^2*x^2)^2*Log[c + Sqrt[c]*Sqrt[c + a^2*c*x^2]]))/(9*c^3*x*(1 + a^2*x^2)^2)","A",1
248,1,234,270,0.35858,"\int x^m \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x) \, dx","Integrate[x^m*(c + a^2*c*x^2)^3*ArcTan[a*x],x]","c^3 x^{m+1} \left(\frac{a^6 x^6 \tan ^{-1}(a x)}{m+7}+\frac{3 a^4 x^4 \tan ^{-1}(a x)}{m+5}-\frac{a x \, _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^2 x^2 \tan ^{-1}(a x)}{m+3}-\frac{a^7 x^7 \, _2F_1\left(1,\frac{m}{2}+4;\frac{m}{2}+5;-a^2 x^2\right)}{(m+7) (m+8)}-\frac{3 a^5 x^5 \, _2F_1\left(1,\frac{m+6}{2};\frac{m+8}{2};-a^2 x^2\right)}{(m+5) (m+6)}-\frac{3 a^3 x^3 \, _2F_1\left(1,\frac{m+4}{2};\frac{m+6}{2};-a^2 x^2\right)}{m^2+7 m+12}+\frac{\tan ^{-1}(a x)}{m+1}\right)","\frac{a^6 c^3 x^{m+7} \tan ^{-1}(a x)}{m+7}+\frac{3 a^4 c^3 x^{m+5} \tan ^{-1}(a x)}{m+5}-\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^2 c^3 x^{m+3} \tan ^{-1}(a x)}{m+3}-\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^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{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{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*x^2*ArcTan[a*x])/(3 + m) + (3*a^4*x^4*ArcTan[a*x])/(5 + m) + (a^6*x^6*ArcTan[a*x])/(7 + m) - (a^7*x^7*Hypergeometric2F1[1, 4 + m/2, 5 + m/2, -(a^2*x^2)])/((7 + m)*(8 + m)) - (a*x*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + 3*m + m^2) - (3*a^3*x^3*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, -(a^2*x^2)])/(12 + 7*m + m^2) - (3*a^5*x^5*Hypergeometric2F1[1, (6 + m)/2, (8 + m)/2, -(a^2*x^2)])/((5 + m)*(6 + m)))","A",1
249,1,175,201,0.1435686,"\int x^m \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x) \, dx","Integrate[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x],x]","c^2 x^{m+1} \left(\frac{a^4 x^4 \tan ^{-1}(a x)}{m+5}-\frac{a x \, _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^2 x^2 \tan ^{-1}(a x)}{m+3}-\frac{a^5 x^5 \, _2F_1\left(1,\frac{m+6}{2};\frac{m+8}{2};-a^2 x^2\right)}{(m+5) (m+6)}-\frac{2 a^3 x^3 \, _2F_1\left(1,\frac{m+4}{2};\frac{m+6}{2};-a^2 x^2\right)}{m^2+7 m+12}+\frac{\tan ^{-1}(a x)}{m+1}\right)","\frac{a^4 c^2 x^{m+5} \tan ^{-1}(a x)}{m+5}-\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^2 c^2 x^{m+3} \tan ^{-1}(a x)}{m+3}-\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^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{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*x^2*ArcTan[a*x])/(3 + m) + (a^4*x^4*ArcTan[a*x])/(5 + m) - (a*x*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + 3*m + m^2) - (2*a^3*x^3*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, -(a^2*x^2)])/(12 + 7*m + m^2) - (a^5*x^5*Hypergeometric2F1[1, (6 + m)/2, (8 + m)/2, -(a^2*x^2)])/((5 + m)*(6 + m)))","A",1
250,1,111,124,0.1245646,"\int x^m \left(c+a^2 c x^2\right) \tan ^{-1}(a x) \, dx","Integrate[x^m*(c + a^2*c*x^2)*ArcTan[a*x],x]","c x^{m+1} \left(-\frac{a x \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-a^2 x^2\right)}{m^2+3 m+2}+\left(\frac{a^2 x^2}{m+3}+\frac{1}{m+1}\right) \tan ^{-1}(a x)-\frac{a^3 x^3 \, _2F_1\left(1,\frac{m+4}{2};\frac{m+6}{2};-a^2 x^2\right)}{m^2+7 m+12}\right)","-\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^2 c x^{m+3} \tan ^{-1}(a x)}{m+3}-\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{c x^{m+1} \tan ^{-1}(a x)}{m+1}",1,"c*x^(1 + m)*(((1 + m)^(-1) + (a^2*x^2)/(3 + m))*ArcTan[a*x] - (a*x*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(a^2*x^2)])/(2 + 3*m + m^2) - (a^3*x^3*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, -(a^2*x^2)])/(12 + 7*m + m^2))","A",1
251,0,0,23,0.913036,"\int \frac{x^m \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x])/(c + a^2*c*x^2), x]","A",-1
252,0,0,23,0.6480787,"\int \frac{x^m \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x])/(c + a^2*c*x^2)^2, x]","A",-1
253,0,0,25,0.9860051,"\int x^m \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x) \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x], x]","A",-1
254,0,0,25,0.5250859,"\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x) \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x], x]","A",-1
255,0,0,113,0.1092312,"\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx","Integrate[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,"Integrate[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x], x]","A",-1
256,0,0,25,0.4704165,"\int \frac{x^m \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x])/Sqrt[c + a^2*c*x^2], x]","A",-1
257,0,0,25,0.5319288,"\int \frac{x^m \tan ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2), x]","A",-1
258,1,89,124,0.0401768,"\int x^3 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2 \, dx","Integrate[x^3*(c + a^2*c*x^2)*ArcTan[a*x]^2,x]","\frac{c \left(3 a^4 x^4-a^2 x^2-14 \log \left(a^2 x^2+1\right)+15 \left(2 a^6 x^6+3 a^4 x^4-1\right) \tan ^{-1}(a x)^2-2 a x \left(6 a^4 x^4+5 a^2 x^2-15\right) \tan ^{-1}(a x)\right)}{180 a^4}","-\frac{c \tan ^{-1}(a x)^2}{12 a^4}+\frac{c x \tan ^{-1}(a x)}{6 a^3}+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^2-\frac{c x^2}{180 a^2}-\frac{7 c \log \left(a^2 x^2+1\right)}{90 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*(-(a^2*x^2) + 3*a^4*x^4 - 2*a*x*(-15 + 5*a^2*x^2 + 6*a^4*x^4)*ArcTan[a*x] + 15*(-1 + 3*a^4*x^4 + 2*a^6*x^6)*ArcTan[a*x]^2 - 14*Log[1 + a^2*x^2]))/(180*a^4)","A",1
259,1,104,156,0.6436482,"\int x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2 \, dx","Integrate[x^2*(c + a^2*c*x^2)*ArcTan[a*x]^2,x]","\frac{c \left(a^3 x^3+2 \left(3 a^5 x^5+5 a^3 x^3+2 i\right) \tan ^{-1}(a x)^2-\tan ^{-1}(a x) \left(3 a^4 x^4+4 a^2 x^2+8 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)+1\right)+4 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+a x\right)}{30 a^3}","-\frac{2 i c \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{15 a^3}-\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}{5} a^2 c x^5 \tan ^{-1}(a x)^2+\frac{c x}{30 a^2}-\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*(a*x + a^3*x^3 + 2*(2*I + 5*a^3*x^3 + 3*a^5*x^5)*ArcTan[a*x]^2 - ArcTan[a*x]*(1 + 4*a^2*x^2 + 3*a^4*x^4 + 8*Log[1 + E^((2*I)*ArcTan[a*x])]) + (4*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(30*a^3)","A",0
260,1,64,96,0.0297269,"\int x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2 \, dx","Integrate[x*(c + a^2*c*x^2)*ArcTan[a*x]^2,x]","\frac{c \left(a^2 x^2+2 \log \left(a^2 x^2+1\right)-2 a x \left(a^2 x^2+3\right) \tan ^{-1}(a x)+3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2\right)}{12 a^2}","\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*(a^2*x^2 - 2*a*x*(3 + a^2*x^2)*ArcTan[a*x] + 3*(1 + a^2*x^2)^2*ArcTan[a*x]^2 + 2*Log[1 + a^2*x^2]))/(12*a^2)","A",1
261,1,82,128,0.0530275,"\int \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2 \, dx","Integrate[(c + a^2*c*x^2)*ArcTan[a*x]^2,x]","\frac{c \left(\left(a^3 x^3+3 a x-2 i\right) \tan ^{-1}(a x)^2-\tan ^{-1}(a x) \left(a^2 x^2-4 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)+1\right)-2 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+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 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{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*(a*x + (-2*I + 3*a*x + a^3*x^3)*ArcTan[a*x]^2 - ArcTan[a*x]*(1 + a^2*x^2 - 4*Log[1 + E^((2*I)*ArcTan[a*x])]) - (2*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(3*a)","A",0
262,1,177,169,0.0460268,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2}{x} \, dx","Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^2)/x,x]","\frac{1}{2} c \log \left(a^2 x^2+1\right)+\frac{1}{2} c \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2+\frac{1}{2} c \text{Li}_3\left(\frac{-a x-i}{a x-i}\right)-\frac{1}{2} c \text{Li}_3\left(\frac{a x+i}{a x-i}\right)+i c \text{Li}_2\left(\frac{-a x-i}{a x-i}\right) \tan ^{-1}(a x)-i c \text{Li}_2\left(\frac{a x+i}{a x-i}\right) \tan ^{-1}(a x)-a c x \tan ^{-1}(a x)+2 c \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2 i}{-a x+i}\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 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)+\frac{1}{2} c \text{Li}_3\left(\frac{2}{i a x+1}-1\right)-i c \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)+i c \text{Li}_2\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)+\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*(1 + a^2*x^2)*ArcTan[a*x]^2)/2 + 2*c*ArcTan[a*x]^2*ArcTanh[1 - (2*I)/(I - a*x)] + (c*Log[1 + a^2*x^2])/2 + I*c*ArcTan[a*x]*PolyLog[2, (-I - a*x)/(-I + a*x)] - I*c*ArcTan[a*x]*PolyLog[2, (I + a*x)/(-I + a*x)] + (c*PolyLog[3, (-I - a*x)/(-I + a*x)])/2 - (c*PolyLog[3, (I + a*x)/(-I + a*x)])/2","A",1
263,1,94,113,0.1560618,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2}{x^2} \, dx","Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^2)/x^2,x]","c \left(-i a \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-i a \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)+\frac{(a x-i)^2 \tan ^{-1}(a x)^2}{x}+2 a \tan ^{-1}(a x) \left(\log \left(1-e^{2 i \tan ^{-1}(a x)}\right)+\log \left(1+e^{2 i \tan ^{-1}(a x)}\right)\right)\right)","a^2 c x \tan ^{-1}(a x)^2-i a c \text{Li}_2\left(\frac{2}{1-i a x}-1\right)+i a c \text{Li}_2\left(1-\frac{2}{i a x+1}\right)-\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*(((-I + a*x)^2*ArcTan[a*x]^2)/x + 2*a*ArcTan[a*x]*(Log[1 - E^((2*I)*ArcTan[a*x])] + Log[1 + E^((2*I)*ArcTan[a*x])]) - I*a*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - I*a*PolyLog[2, E^((2*I)*ArcTan[a*x])])","A",0
264,1,208,196,0.0884268,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2}{x^3} \, dx","Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^2)/x^3,x]","\frac{1}{2} a^2 c \text{Li}_3\left(\frac{-a x-i}{a x-i}\right)-\frac{1}{2} a^2 c \text{Li}_3\left(\frac{a x+i}{a x-i}\right)+i a^2 c \text{Li}_2\left(\frac{-a x-i}{a x-i}\right) \tan ^{-1}(a x)-i a^2 c \text{Li}_2\left(\frac{a x+i}{a x-i}\right) \tan ^{-1}(a x)-\frac{1}{2} a^2 c \log \left(a^2 x^2+1\right)+\frac{c \left(-a^2 x^2-1\right) \tan ^{-1}(a x)^2}{2 x^2}+a^2 c \log (x)+2 a^2 c \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2 i}{-a x+i}\right)-\frac{a c \tan ^{-1}(a x)}{x}","-\frac{1}{2} a^2 c \text{Li}_3\left(1-\frac{2}{i a x+1}\right)+\frac{1}{2} a^2 c \text{Li}_3\left(\frac{2}{i a x+1}-1\right)-i a^2 c \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)+i a^2 c \text{Li}_2\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)-\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) + (c*(-1 - a^2*x^2)*ArcTan[a*x]^2)/(2*x^2) + 2*a^2*c*ArcTan[a*x]^2*ArcTanh[1 - (2*I)/(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, (-I - a*x)/(-I + a*x)] - I*a^2*c*ArcTan[a*x]*PolyLog[2, (I + a*x)/(-I + a*x)] + (a^2*c*PolyLog[3, (-I - a*x)/(-I + a*x)])/2 - (a^2*c*PolyLog[3, (I + a*x)/(-I + a*x)])/2","A",1
265,1,103,135,0.6268697,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2}{x^4} \, dx","Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^2)/x^4,x]","\frac{c \left(-2 i a^3 x^3 \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)-a^2 x^2+a x \tan ^{-1}(a x) \left(-a^2 x^2+4 a^2 x^2 \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)-1\right)+(1-2 i a x) (a x-i)^2 \tan ^{-1}(a x)^2\right)}{3 x^3}","-\frac{2}{3} i a^3 c \text{Li}_2\left(\frac{2}{1-i a x}-1\right)-\frac{2}{3} i a^3 c \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c \tan ^{-1}(a x)+\frac{4}{3} a^3 c \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)-\frac{a^2 c}{3 x}-\frac{a^2 c \tan ^{-1}(a x)^2}{x}-\frac{c \tan ^{-1}(a x)^2}{3 x^3}-\frac{a c \tan ^{-1}(a x)}{3 x^2}",1,"(c*(-(a^2*x^2) + (1 - (2*I)*a*x)*(-I + a*x)^2*ArcTan[a*x]^2 + a*x*ArcTan[a*x]*(-1 - a^2*x^2 + 4*a^2*x^2*Log[1 - E^((2*I)*ArcTan[a*x])]) - (2*I)*a^3*x^3*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/(3*x^3)","A",0
266,1,110,191,0.0841344,"\int x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2 \, dx","Integrate[x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]","\frac{c^2 \left(3 a^6 x^6+6 a^4 x^4-5 a^2 x^2-16 \log \left(a^2 x^2+1\right)+21 \left(a^2 x^2+1\right)^3 \left(3 a^2 x^2-1\right) \tan ^{-1}(a x)^2-2 a x \left(9 a^6 x^6+21 a^4 x^4+7 a^2 x^2-21\right) \tan ^{-1}(a x)\right)}{504 a^4}","\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^2}{24 a^4}-\frac{1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)+\frac{c^2 x \tan ^{-1}(a x)}{12 a^3}+\frac{1}{168} a^2 c^2 x^6+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2-\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}{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,"(c^2*(-5*a^2*x^2 + 6*a^4*x^4 + 3*a^6*x^6 - 2*a*x*(-21 + 7*a^2*x^2 + 21*a^4*x^4 + 9*a^6*x^6)*ArcTan[a*x] + 21*(1 + a^2*x^2)^3*(-1 + 3*a^2*x^2)*ArcTan[a*x]^2 - 16*Log[1 + a^2*x^2]))/(504*a^4)","A",1
267,1,133,225,1.295128,"\int x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2 \, dx","Integrate[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]","\frac{c^2 \left(a x \left(6 a^4 x^4+17 a^2 x^2-3\right)+6 \left(15 a^7 x^7+42 a^5 x^5+35 a^3 x^3+8 i\right) \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \left(10 a^6 x^6+27 a^4 x^4+16 a^2 x^2+32 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-1\right)+48 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)\right)}{630 a^3}","\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{8 i c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{105 a^3}-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\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{1}{105} a^2 c^2 x^5+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2-\frac{c^2 x}{210 a^2}-\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*(a*x*(-3 + 17*a^2*x^2 + 6*a^4*x^4) + 6*(8*I + 35*a^3*x^3 + 42*a^5*x^5 + 15*a^7*x^7)*ArcTan[a*x]^2 - 3*ArcTan[a*x]*(-1 + 16*a^2*x^2 + 27*a^4*x^4 + 10*a^6*x^6 + 32*Log[1 + E^((2*I)*ArcTan[a*x])]) + (48*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(630*a^3)","A",0
268,1,84,153,0.0657077,"\int x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2 \, dx","Integrate[x*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]","\frac{c^2 \left(3 a^4 x^4+14 a^2 x^2+16 \log \left(a^2 x^2+1\right)+30 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^2-4 a x \left(3 a^4 x^4+10 a^2 x^2+15\right) \tan ^{-1}(a x)\right)}{180 a^2}","\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,"(c^2*(14*a^2*x^2 + 3*a^4*x^4 - 4*a*x*(15 + 10*a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x] + 30*(1 + a^2*x^2)^3*ArcTan[a*x]^2 + 16*Log[1 + a^2*x^2]))/(180*a^2)","A",1
269,1,112,205,0.6941317,"\int \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2 \, dx","Integrate[(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]","\frac{c^2 \left(a x \left(a^2 x^2+11\right)+2 \left(3 a^5 x^5+10 a^3 x^3+15 a x-8 i\right) \tan ^{-1}(a x)^2-\tan ^{-1}(a x) \left(3 a^4 x^4+14 a^2 x^2-32 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)+11\right)-16 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)\right)}{30 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 i c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{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,"(c^2*(a*x*(11 + a^2*x^2) + 2*(-8*I + 15*a*x + 10*a^3*x^3 + 3*a^5*x^5)*ArcTan[a*x]^2 - ArcTan[a*x]*(11 + 14*a^2*x^2 + 3*a^4*x^4 - 32*Log[1 + E^((2*I)*ArcTan[a*x])]) - (16*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(30*a)","A",0
270,1,218,235,0.3358459,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2}{x} \, dx","Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x]^2)/x,x]","\frac{1}{24} c^2 \left(6 a^4 x^4 \tan ^{-1}(a x)^2-4 a^3 x^3 \tan ^{-1}(a x)+2 a^2 x^2+16 \log \left(a^2 x^2+1\right)+24 a^2 x^2 \tan ^{-1}(a x)^2+24 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+24 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+12 \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-12 \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-36 a x \tan ^{-1}(a x)+16 i \tan ^{-1}(a x)^3+18 \tan ^{-1}(a x)^2+24 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-24 \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-i \pi ^3+2\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)+\frac{1}{12} a^2 c^2 x^2+\frac{2}{3} c^2 \log \left(a^2 x^2+1\right)+a^2 c^2 x^2 \tan ^{-1}(a x)^2-\frac{1}{2} c^2 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)+\frac{1}{2} c^2 \text{Li}_3\left(\frac{2}{i a x+1}-1\right)-i c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)+i c^2 \text{Li}_2\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)-\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,"(c^2*(2 - I*Pi^3 + 2*a^2*x^2 - 36*a*x*ArcTan[a*x] - 4*a^3*x^3*ArcTan[a*x] + 18*ArcTan[a*x]^2 + 24*a^2*x^2*ArcTan[a*x]^2 + 6*a^4*x^4*ArcTan[a*x]^2 + (16*I)*ArcTan[a*x]^3 + 24*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] - 24*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 16*Log[1 + a^2*x^2] + (24*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (24*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 12*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/24","A",0
271,1,167,205,0.3790575,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2}{x^2} \, dx","Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x]^2)/x^2,x]","\frac{c^2 \left(a^4 x^4 \tan ^{-1}(a x)^2-a^3 x^3 \tan ^{-1}(a x)+a^2 x^2+6 a^2 x^2 \tan ^{-1}(a x)^2-5 i a x \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-3 i a x \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)-8 i a x \tan ^{-1}(a x)^2-a x \tan ^{-1}(a x)-3 \tan ^{-1}(a x)^2+6 a x \tan ^{-1}(a x) \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)+10 a x \tan ^{-1}(a x) \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)\right)}{3 x}","\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-i a c^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)+\frac{5}{3} i a c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)+\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,"(c^2*(a^2*x^2 - a*x*ArcTan[a*x] - a^3*x^3*ArcTan[a*x] - 3*ArcTan[a*x]^2 - (8*I)*a*x*ArcTan[a*x]^2 + 6*a^2*x^2*ArcTan[a*x]^2 + a^4*x^4*ArcTan[a*x]^2 + 6*a*x*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] + 10*a*x*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - (5*I)*a*x*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - (3*I)*a*x*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/(3*x)","A",0
272,1,226,207,0.3320299,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2}{x^3} \, dx","Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x]^2)/x^3,x]","a^2 c^2 \left(\log \left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+\frac{1}{2} \log \left(a^2 x^2+1\right)+\frac{1}{2} a^2 x^2 \tan ^{-1}(a x)^2-\frac{\tan ^{-1}(a x)^2}{2 a^2 x^2}+2 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+2 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+\text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-\text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)+\frac{4}{3} i \tan ^{-1}(a x)^3-a x \tan ^{-1}(a x)-\frac{\tan ^{-1}(a x)}{a x}+2 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-2 \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-\frac{i \pi ^3}{12}\right)","\frac{1}{2} a^4 c^2 x^2 \tan ^{-1}(a x)^2-a^3 c^2 x \tan ^{-1}(a x)-a^2 c^2 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)+a^2 c^2 \text{Li}_3\left(\frac{2}{i a x+1}-1\right)-2 i a^2 c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)+2 i a^2 c^2 \text{Li}_2\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)+a^2 c^2 \log (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^2*c^2*((-1/12*I)*Pi^3 - ArcTan[a*x]/(a*x) - a*x*ArcTan[a*x] - ArcTan[a*x]^2/(2*a^2*x^2) + (a^2*x^2*ArcTan[a*x]^2)/2 + ((4*I)/3)*ArcTan[a*x]^3 + 2*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] - 2*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + Log[(a*x)/Sqrt[1 + a^2*x^2]] + Log[1 + a^2*x^2]/2 + (2*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (2*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + PolyLog[3, E^((-2*I)*ArcTan[a*x])] - PolyLog[3, -E^((2*I)*ArcTan[a*x])])","A",0
273,1,189,216,0.4119418,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2}{x^4} \, dx","Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x]^2)/x^4,x]","\frac{c^2 \left(3 a^4 x^4 \tan ^{-1}(a x)^2-3 i a^3 x^3 \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-5 i a^3 x^3 \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)-8 i a^3 x^3 \tan ^{-1}(a x)^2-a^3 x^3 \tan ^{-1}(a x)+10 a^3 x^3 \tan ^{-1}(a x) \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)+6 a^3 x^3 \tan ^{-1}(a x) \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-a^2 x^2-6 a^2 x^2 \tan ^{-1}(a x)^2-a x \tan ^{-1}(a x)-\tan ^{-1}(a x)^2\right)}{3 x^3}","a^4 c^2 x \tan ^{-1}(a x)^2-\frac{5}{3} i a^3 c^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)+i a^3 c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)-\frac{2}{3} i a^3 c^2 \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c^2 \tan ^{-1}(a 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^2 c^2}{3 x}-\frac{2 a^2 c^2 \tan ^{-1}(a x)^2}{x}-\frac{c^2 \tan ^{-1}(a x)^2}{3 x^3}-\frac{a c^2 \tan ^{-1}(a x)}{3 x^2}",1,"(c^2*(-(a^2*x^2) - a*x*ArcTan[a*x] - a^3*x^3*ArcTan[a*x] - ArcTan[a*x]^2 - 6*a^2*x^2*ArcTan[a*x]^2 - (8*I)*a^3*x^3*ArcTan[a*x]^2 + 3*a^4*x^4*ArcTan[a*x]^2 + 10*a^3*x^3*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] + 6*a^3*x^3*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - (3*I)*a^3*x^3*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - (5*I)*a^3*x^3*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/(3*x^3)","A",0
274,1,126,240,0.1012506,"\int x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2 \, dx","Integrate[x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^2,x]","\frac{c^3 \left(105 a^8 x^8+355 a^6 x^6+318 a^4 x^4-321 a^2 x^2-624 \log \left(a^2 x^2+1\right)+945 \left(a^2 x^2+1\right)^4 \left(4 a^2 x^2-1\right) \tan ^{-1}(a x)^2-6 a x \left(140 a^8 x^8+495 a^6 x^6+567 a^4 x^4+105 a^2 x^2-315\right) \tan ^{-1}(a x)\right)}{37800 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{1}{360} a^4 c^3 x^8+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^2}{40 a^4}-\frac{11}{140} a^3 c^3 x^7 \tan ^{-1}(a x)+\frac{c^3 x \tan ^{-1}(a x)}{20 a^3}+\frac{71 a^2 c^3 x^6}{7560}+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^2-\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{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,"(c^3*(-321*a^2*x^2 + 318*a^4*x^4 + 355*a^6*x^6 + 105*a^8*x^8 - 6*a*x*(-315 + 105*a^2*x^2 + 567*a^4*x^4 + 495*a^6*x^6 + 140*a^8*x^8)*ArcTan[a*x] + 945*(1 + a^2*x^2)^4*(-1 + 4*a^2*x^2)*ArcTan[a*x]^2 - 624*Log[1 + a^2*x^2]))/(37800*a^4)","A",1
275,1,157,274,2.2629815,"\int x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2 \, dx","Integrate[x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^2,x]","\frac{c^3 \left(a x \left(45 a^6 x^6+177 a^4 x^4+239 a^2 x^2-141\right)+36 \left(35 a^9 x^9+135 a^7 x^7+189 a^5 x^5+105 a^3 x^3+16 i\right) \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \left(105 a^8 x^8+400 a^6 x^6+534 a^4 x^4+192 a^2 x^2+384 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-47\right)+576 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)\right)}{11340 a^3}","\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{1}{252} a^4 c^3 x^7+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2-\frac{16 i c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{315 a^3}-\frac{20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)-\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{59 a^2 c^3 x^5}{3780}+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2-\frac{47 c^3 x}{3780 a^2}-\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,"(c^3*(a*x*(-141 + 239*a^2*x^2 + 177*a^4*x^4 + 45*a^6*x^6) + 36*(16*I + 105*a^3*x^3 + 189*a^5*x^5 + 135*a^7*x^7 + 35*a^9*x^9)*ArcTan[a*x]^2 - 3*ArcTan[a*x]*(-47 + 192*a^2*x^2 + 534*a^4*x^4 + 400*a^6*x^6 + 105*a^8*x^8 + 384*Log[1 + E^((2*I)*ArcTan[a*x])]) + (576*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(11340*a^3)","A",0
276,1,100,200,0.0856291,"\int x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2 \, dx","Integrate[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^2,x]","\frac{c^3 \left(5 a^6 x^6+24 a^4 x^4+57 a^2 x^2+48 \log \left(a^2 x^2+1\right)+105 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x)^2-6 a x \left(5 a^6 x^6+21 a^4 x^4+35 a^2 x^2+35\right) \tan ^{-1}(a x)\right)}{840 a^2}","\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*(57*a^2*x^2 + 24*a^4*x^4 + 5*a^6*x^6 - 6*a*x*(35 + 35*a^2*x^2 + 21*a^4*x^4 + 5*a^6*x^6)*ArcTan[a*x] + 105*(1 + a^2*x^2)^4*ArcTan[a*x]^2 + 48*Log[1 + a^2*x^2]))/(840*a^2)","A",1
277,1,137,268,1.2641118,"\int \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2 \, dx","Integrate[(c + a^2*c*x^2)^3*ArcTan[a*x]^2,x]","\frac{c^3 \left(a x \left(3 a^4 x^4+19 a^2 x^2+114\right)+9 \left(5 a^7 x^7+21 a^5 x^5+35 a^3 x^3+35 a x-16 i\right) \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \left(5 a^6 x^6+24 a^4 x^4+57 a^2 x^2-96 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)+38\right)-144 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)\right)}{315 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 i c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{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,"(c^3*(a*x*(114 + 19*a^2*x^2 + 3*a^4*x^4) + 9*(-16*I + 35*a*x + 35*a^3*x^3 + 21*a^5*x^5 + 5*a^7*x^7)*ArcTan[a*x]^2 - 3*ArcTan[a*x]*(38 + 57*a^2*x^2 + 24*a^4*x^4 + 5*a^6*x^6 - 96*Log[1 + E^((2*I)*ArcTan[a*x])]) - (144*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(315*a)","A",0
278,1,252,287,0.5672886,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2}{x} \, dx","Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x]^2)/x,x]","\frac{1}{360} c^3 \left(60 a^6 x^6 \tan ^{-1}(a x)^2-24 a^5 x^5 \tan ^{-1}(a x)+6 a^4 x^4+270 a^4 x^4 \tan ^{-1}(a x)^2-140 a^3 x^3 \tan ^{-1}(a x)+58 a^2 x^2+272 \log \left(a^2 x^2+1\right)+540 a^2 x^2 \tan ^{-1}(a x)^2+360 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+360 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+180 \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-180 \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-660 a x \tan ^{-1}(a x)+240 i \tan ^{-1}(a x)^3+330 \tan ^{-1}(a x)^2+360 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-360 \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-15 i \pi ^3+52\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{1}{60} a^4 c^3 x^4+\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{29}{180} a^2 c^3 x^2+\frac{34}{45} c^3 \log \left(a^2 x^2+1\right)+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2-\frac{1}{2} c^3 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)+\frac{1}{2} c^3 \text{Li}_3\left(\frac{2}{i a x+1}-1\right)-i c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)+i c^3 \text{Li}_2\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)-\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,"(c^3*(52 - (15*I)*Pi^3 + 58*a^2*x^2 + 6*a^4*x^4 - 660*a*x*ArcTan[a*x] - 140*a^3*x^3*ArcTan[a*x] - 24*a^5*x^5*ArcTan[a*x] + 330*ArcTan[a*x]^2 + 540*a^2*x^2*ArcTan[a*x]^2 + 270*a^4*x^4*ArcTan[a*x]^2 + 60*a^6*x^6*ArcTan[a*x]^2 + (240*I)*ArcTan[a*x]^3 + 360*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] - 360*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 272*Log[1 + a^2*x^2] + (360*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (360*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 180*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 180*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/360","A",0
279,1,202,251,0.7615669,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2}{x^2} \, dx","Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x]^2)/x^2,x]","\frac{c^3 \left(6 a^6 x^6 \tan ^{-1}(a x)^2-3 a^5 x^5 \tan ^{-1}(a x)+a^4 x^4+30 a^4 x^4 \tan ^{-1}(a x)^2-24 a^3 x^3 \tan ^{-1}(a x)+21 a^2 x^2+90 a^2 x^2 \tan ^{-1}(a x)^2-66 i a x \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-30 i a x \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)-96 i a x \tan ^{-1}(a x)^2-21 a x \tan ^{-1}(a x)-30 \tan ^{-1}(a x)^2+60 a x \tan ^{-1}(a x) \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)+132 a x \tan ^{-1}(a x) \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)\right)}{30 x}","\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)+\frac{1}{30} a^4 c^3 x^3+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-i a c^3 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)+\frac{11}{5} i a c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)+\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,"(c^3*(21*a^2*x^2 + a^4*x^4 - 21*a*x*ArcTan[a*x] - 24*a^3*x^3*ArcTan[a*x] - 3*a^5*x^5*ArcTan[a*x] - 30*ArcTan[a*x]^2 - (96*I)*a*x*ArcTan[a*x]^2 + 90*a^2*x^2*ArcTan[a*x]^2 + 30*a^4*x^4*ArcTan[a*x]^2 + 6*a^6*x^6*ArcTan[a*x]^2 + 60*a*x*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] + 132*a*x*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - (66*I)*a*x*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - (30*I)*a*x*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/(30*x)","A",0
280,1,333,299,0.4087312,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2}{x^3} \, dx","Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x]^2)/x^3,x]","\frac{c^3 \left(6 a^6 x^6 \tan ^{-1}(a x)^2-4 a^5 x^5 \tan ^{-1}(a x)+2 a^4 x^4+36 a^4 x^4 \tan ^{-1}(a x)^2-60 a^3 x^3 \tan ^{-1}(a x)+72 i a^2 x^2 \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+72 i a^2 x^2 \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+36 a^2 x^2 \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-36 a^2 x^2 \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)+2 a^2 x^2-3 i \pi ^3 a^2 x^2+24 a^2 x^2 \log \left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+28 a^2 x^2 \log \left(a^2 x^2+1\right)+48 i a^2 x^2 \tan ^{-1}(a x)^3+18 a^2 x^2 \tan ^{-1}(a x)^2+72 a^2 x^2 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-72 a^2 x^2 \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-24 a x \tan ^{-1}(a x)-12 \tan ^{-1}(a x)^2\right)}{24 x^2}","\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{1}{12} a^4 c^3 x^2+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^2-\frac{5}{2} a^3 c^3 x \tan ^{-1}(a x)-\frac{3}{2} a^2 c^3 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)+\frac{3}{2} a^2 c^3 \text{Li}_3\left(\frac{2}{i a x+1}-1\right)-3 i a^2 c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)+3 i a^2 c^3 \text{Li}_2\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)+\frac{2}{3} a^2 c^3 \log \left(a^2 x^2+1\right)+a^2 c^3 \log (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,"(c^3*(2*a^2*x^2 - (3*I)*a^2*Pi^3*x^2 + 2*a^4*x^4 - 24*a*x*ArcTan[a*x] - 60*a^3*x^3*ArcTan[a*x] - 4*a^5*x^5*ArcTan[a*x] - 12*ArcTan[a*x]^2 + 18*a^2*x^2*ArcTan[a*x]^2 + 36*a^4*x^4*ArcTan[a*x]^2 + 6*a^6*x^6*ArcTan[a*x]^2 + (48*I)*a^2*x^2*ArcTan[a*x]^3 + 72*a^2*x^2*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] - 72*a^2*x^2*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 24*a^2*x^2*Log[(a*x)/Sqrt[1 + a^2*x^2]] + 28*a^2*x^2*Log[1 + a^2*x^2] + (72*I)*a^2*x^2*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (72*I)*a^2*x^2*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 36*a^2*x^2*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 36*a^2*x^2*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(24*x^2)","A",0
281,1,221,250,0.6191921,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2}{x^4} \, dx","Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x]^2)/x^4,x]","\frac{c^3 \left(a^6 x^6 \tan ^{-1}(a x)^2-a^5 x^5 \tan ^{-1}(a x)+a^4 x^4+9 a^4 x^4 \tan ^{-1}(a x)^2-8 i a^3 x^3 \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-8 i a^3 x^3 \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)-16 i a^3 x^3 \tan ^{-1}(a x)^2-2 a^3 x^3 \tan ^{-1}(a x)+16 a^3 x^3 \tan ^{-1}(a x) \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)+16 a^3 x^3 \tan ^{-1}(a x) \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-a^2 x^2-9 a^2 x^2 \tan ^{-1}(a x)^2-a x \tan ^{-1}(a x)-\tan ^{-1}(a x)^2\right)}{3 x^3}","\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+3 a^4 c^3 x \tan ^{-1}(a x)^2-\frac{8}{3} i a^3 c^3 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)+\frac{8}{3} i a^3 c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)-\frac{2}{3} a^3 c^3 \tan ^{-1}(a 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^2 c^3}{3 x}-\frac{3 a^2 c^3 \tan ^{-1}(a x)^2}{x}-\frac{c^3 \tan ^{-1}(a x)^2}{3 x^3}-\frac{a c^3 \tan ^{-1}(a x)}{3 x^2}",1,"(c^3*(-(a^2*x^2) + a^4*x^4 - a*x*ArcTan[a*x] - 2*a^3*x^3*ArcTan[a*x] - a^5*x^5*ArcTan[a*x] - ArcTan[a*x]^2 - 9*a^2*x^2*ArcTan[a*x]^2 - (16*I)*a^3*x^3*ArcTan[a*x]^2 + 9*a^4*x^4*ArcTan[a*x]^2 + a^6*x^6*ArcTan[a*x]^2 + 16*a^3*x^3*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] + 16*a^3*x^3*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - (8*I)*a^3*x^3*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - (8*I)*a^3*x^3*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/(3*x^3)","A",0
282,1,90,166,0.3089566,"\int \frac{x^4 \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx","Integrate[(x^4*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]","\frac{\left(a^3 x^3-3 a x+4 i\right) \tan ^{-1}(a x)^2-\tan ^{-1}(a x) \left(a^2 x^2+8 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)+1\right)+4 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+a x+\tan ^{-1}(a x)^3}{3 a^5 c}","-\frac{4 i \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{3 a^5 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{x}{3 a^4 c}-\frac{x \tan ^{-1}(a x)^2}{a^4 c}-\frac{x^2 \tan ^{-1}(a x)}{3 a^3 c}+\frac{x^3 \tan ^{-1}(a x)^2}{3 a^2 c}",1,"(a*x + (4*I - 3*a*x + a^3*x^3)*ArcTan[a*x]^2 + ArcTan[a*x]^3 - ArcTan[a*x]*(1 + a^2*x^2 + 8*Log[1 + E^((2*I)*ArcTan[a*x])]) + (4*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])])/(3*a^5*c)","A",0
283,1,123,169,0.1176777,"\int \frac{x^3 \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx","Integrate[(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]","\frac{-\log \left(\frac{1}{\sqrt{a^2 x^2+1}}\right)+\frac{1}{2} \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2-i \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-\frac{1}{3} i \tan ^{-1}(a x)^3-a x \tan ^{-1}(a x)+\tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)}{a^4 c}","\frac{\text{Li}_3\left(1-\frac{2}{i a x+1}\right)}{2 a^4 c}+\frac{i \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)}{a^4 c}+\frac{i \tan ^{-1}(a x)^3}{3 a^4 c}+\frac{\tan ^{-1}(a x)^2}{2 a^4 c}+\frac{\log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{a^4 c}-\frac{x \tan ^{-1}(a x)}{a^3 c}+\frac{x^2 \tan ^{-1}(a x)^2}{2 a^2 c}+\frac{\log \left(a^2 x^2+1\right)}{2 a^4 c}",1,"(-(a*x*ArcTan[a*x]) + ((1 + a^2*x^2)*ArcTan[a*x]^2)/2 - (I/3)*ArcTan[a*x]^3 + ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - Log[1/Sqrt[1 + a^2*x^2]] - I*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + PolyLog[3, -E^((2*I)*ArcTan[a*x])]/2)/(a^4*c)","A",0
284,1,69,98,0.1803025,"\int \frac{x^2 \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx","Integrate[(x^2*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]","\frac{-i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-\frac{1}{3} \tan ^{-1}(a x) \left(\tan ^{-1}(a x)^2+(-3 a x+3 i) \tan ^{-1}(a x)-6 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)\right)}{a^3 c}","\frac{i \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{a^3 c}-\frac{\tan ^{-1}(a x)^3}{3 a^3 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{x \tan ^{-1}(a x)^2}{a^2 c}",1,"(-1/3*(ArcTan[a*x]*((3*I - 3*a*x)*ArcTan[a*x] + ArcTan[a*x]^2 - 6*Log[1 + E^((2*I)*ArcTan[a*x])])) - I*PolyLog[2, -E^((2*I)*ArcTan[a*x])])/(a^3*c)","A",0
285,1,110,102,0.0101564,"\int \frac{x \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx","Integrate[(x*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]","-\frac{\text{Li}_3\left(\frac{a x+i}{a x-i}\right)}{2 a^2 c}-\frac{i \text{Li}_2\left(\frac{a x+i}{a x-i}\right) \tan ^{-1}(a x)}{a^2 c}-\frac{i \tan ^{-1}(a x)^3}{3 a^2 c}-\frac{\log \left(\frac{2 i}{-a x+i}\right) \tan ^{-1}(a x)^2}{a^2 c}","-\frac{\text{Li}_3\left(1-\frac{2}{i a x+1}\right)}{2 a^2 c}-\frac{i \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)}{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,"((-1/3*I)*ArcTan[a*x]^3)/(a^2*c) - (ArcTan[a*x]^2*Log[(2*I)/(I - a*x)])/(a^2*c) - (I*ArcTan[a*x]*PolyLog[2, (I + a*x)/(-I + a*x)])/(a^2*c) - PolyLog[3, (I + a*x)/(-I + a*x)]/(2*a^2*c)","A",1
286,1,16,16,0.0035484,"\int \frac{\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx","Integrate[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
287,1,243,91,0.0506034,"\int \frac{\tan ^{-1}(a x)^2}{x \left(c+a^2 c x^2\right)} \, dx","Integrate[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)),x]","\frac{\text{Li}_3\left(\frac{-a x-i}{a x-i}\right)}{2 c}+\frac{\text{Li}_3\left(-\frac{a x+i}{i-a x}\right)}{2 c}-\frac{\text{Li}_3\left(\frac{a x+i}{a x-i}\right)}{2 c}+\frac{i \text{Li}_2\left(\frac{-a x-i}{a x-i}\right) \tan ^{-1}(a x)}{c}+\frac{i \text{Li}_2\left(-\frac{a x+i}{i-a x}\right) \tan ^{-1}(a x)}{c}-\frac{i \text{Li}_2\left(\frac{a x+i}{a x-i}\right) \tan ^{-1}(a x)}{c}+\frac{i \tan ^{-1}(a x)^3}{3 c}+\frac{\log \left(\frac{2 i}{-a x+i}\right) \tan ^{-1}(a x)^2}{c}+\frac{2 \tan ^{-1}(a x)^2 \tanh ^{-1}\left(1-\frac{2 i}{-a x+i}\right)}{c}","\frac{\text{Li}_3\left(\frac{2}{1-i a x}-1\right)}{2 c}-\frac{i \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{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 + (2*ArcTan[a*x]^2*ArcTanh[1 - (2*I)/(I - a*x)])/c + (ArcTan[a*x]^2*Log[(2*I)/(I - a*x)])/c + (I*ArcTan[a*x]*PolyLog[2, (-I - a*x)/(-I + a*x)])/c + (I*ArcTan[a*x]*PolyLog[2, -((I + a*x)/(I - a*x))])/c - (I*ArcTan[a*x]*PolyLog[2, (I + a*x)/(-I + a*x)])/c + PolyLog[3, (-I - a*x)/(-I + a*x)]/(2*c) + PolyLog[3, -((I + a*x)/(I - a*x))]/(2*c) - PolyLog[3, (I + a*x)/(-I + a*x)]/(2*c)","B",0
288,1,73,92,0.1952861,"\int \frac{\tan ^{-1}(a x)^2}{x^2 \left(c+a^2 c x^2\right)} \, dx","Integrate[ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)),x]","\frac{a \left(-i \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)-\frac{1}{3} \tan ^{-1}(a x) \left(\left(\tan ^{-1}(a x)+3 i\right) \tan ^{-1}(a x)+\frac{3 \tan ^{-1}(a x)}{a x}-6 \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)\right)\right)}{c}","-\frac{i a \text{Li}_2\left(\frac{2}{1-i a x}-1\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,"(a*(-1/3*(ArcTan[a*x]*((3*ArcTan[a*x])/(a*x) + ArcTan[a*x]*(3*I + ArcTan[a*x]) - 6*Log[1 - E^((2*I)*ArcTan[a*x])])) - I*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/c","A",0
289,1,142,178,0.3309191,"\int \frac{\tan ^{-1}(a x)^2}{x^3 \left(c+a^2 c x^2\right)} \, dx","Integrate[ArcTan[a*x]^2/(x^3*(c + a^2*c*x^2)),x]","\frac{a^2 \left(\log \left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-\frac{\left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{2 a^2 x^2}-i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-\frac{1}{3} i \tan ^{-1}(a x)^3-\frac{\tan ^{-1}(a x)}{a x}-\tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)+\frac{i \pi ^3}{24}\right)}{c}","-\frac{a^2 \text{Li}_3\left(\frac{2}{1-i a x}-1\right)}{2 c}+\frac{i a^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{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^2*((I/24)*Pi^3 - ArcTan[a*x]/(a*x) - ((1 + a^2*x^2)*ArcTan[a*x]^2)/(2*a^2*x^2) - (I/3)*ArcTan[a*x]^3 - ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + Log[(a*x)/Sqrt[1 + a^2*x^2]] - I*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - PolyLog[3, E^((-2*I)*ArcTan[a*x])]/2))/c","A",0
290,1,120,166,0.3846091,"\int \frac{\tan ^{-1}(a x)^2}{x^4 \left(c+a^2 c x^2\right)} \, dx","Integrate[ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)),x]","\frac{a^3 \left(-\frac{\frac{\left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{a^2 x^2}-4 \tan ^{-1}(a x)^2+1}{a x}+\tan ^{-1}(a x) \left(-\frac{a^2 x^2+1}{a^2 x^2}+\tan ^{-1}(a x) \left(\tan ^{-1}(a x)+4 i\right)-8 \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)\right)+4 i \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)\right)}{3 c}","\frac{4 i a^3 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)}{3 c}+\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{8 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{3 c}-\frac{a^2}{3 c x}+\frac{a^2 \tan ^{-1}(a x)^2}{c x}-\frac{\tan ^{-1}(a x)^2}{3 c x^3}-\frac{a \tan ^{-1}(a x)}{3 c x^2}",1,"(a^3*(-((1 - 4*ArcTan[a*x]^2 + ((1 + a^2*x^2)*ArcTan[a*x]^2)/(a^2*x^2))/(a*x)) + ArcTan[a*x]*(-((1 + a^2*x^2)/(a^2*x^2)) + ArcTan[a*x]*(4*I + ArcTan[a*x]) - 8*Log[1 - E^((2*I)*ArcTan[a*x])]) + (4*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/(3*c)","A",0
291,1,117,192,0.1880489,"\int \frac{x^3 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2,x]","\frac{i \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-\frac{1}{2} \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)+\frac{1}{3} i \tan ^{-1}(a x)^3-\tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-\frac{1}{4} \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)+\frac{1}{8} \left(2 \tan ^{-1}(a x)^2-1\right) \cos \left(2 \tan ^{-1}(a x)\right)}{a^4 c^2}","-\frac{\text{Li}_3\left(1-\frac{2}{i a x+1}\right)}{2 a^4 c^2}-\frac{i \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)}{a^4 c^2}-\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{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)}",1,"((I/3)*ArcTan[a*x]^3 + ((-1 + 2*ArcTan[a*x]^2)*Cos[2*ArcTan[a*x]])/8 - ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + I*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - PolyLog[3, -E^((2*I)*ArcTan[a*x])]/2 - (ArcTan[a*x]*Sin[2*ArcTan[a*x]])/4)/(a^4*c^2)","A",0
292,1,68,106,0.0940775,"\int \frac{x^2 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x^2*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2,x]","\frac{2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3+3 \left(a^2 x^2-1\right) \tan ^{-1}(a x)+3 a x-6 a x \tan ^{-1}(a x)^2}{12 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)}",1,"(3*a*x + 3*(-1 + a^2*x^2)*ArcTan[a*x] - 6*a*x*ArcTan[a*x]^2 + 2*(1 + a^2*x^2)*ArcTan[a*x]^3)/(12*a^3*c^2*(1 + a^2*x^2))","A",1
293,1,47,91,0.0345222,"\int \frac{x \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2,x]","\frac{\left(a^2 x^2-1\right) \tan ^{-1}(a x)^2+2 a x \tan ^{-1}(a x)+1}{4 a^2 c^2 \left(a^2 x^2+1\right)}","\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 + 2*a*x*ArcTan[a*x] + (-1 + a^2*x^2)*ArcTan[a*x]^2)/(4*a^2*c^2*(1 + a^2*x^2))","A",1
294,1,65,100,0.0414878,"\int \frac{\tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]^2/(c + a^2*c*x^2)^2,x]","\frac{2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3+\left(3-3 a^2 x^2\right) \tan ^{-1}(a x)-3 a x+6 a x \tan ^{-1}(a x)^2}{12 c^2 \left(a^3 x^2+a\right)}","-\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,"(-3*a*x + (3 - 3*a^2*x^2)*ArcTan[a*x] + 6*a*x*ArcTan[a*x]^2 + 2*(1 + a^2*x^2)*ArcTan[a*x]^3)/(12*c^2*(a + a^3*x^2))","A",1
295,1,119,170,0.2076072,"\int \frac{\tan ^{-1}(a x)^2}{x \left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)^2),x]","\frac{24 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+12 \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)+8 i \tan ^{-1}(a x)^3+24 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-6 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)+6 \tan ^{-1}(a x)^2 \cos \left(2 \tan ^{-1}(a x)\right)-3 \cos \left(2 \tan ^{-1}(a x)\right)-i \pi ^3}{24 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{\text{Li}_3\left(\frac{2}{1-i a x}-1\right)}{2 c^2}-\frac{i \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{c^2}-\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,"((-I)*Pi^3 + (8*I)*ArcTan[a*x]^3 - 3*Cos[2*ArcTan[a*x]] + 6*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 24*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (24*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 6*ArcTan[a*x]*Sin[2*ArcTan[a*x]])/(24*c^2)","A",0
296,1,109,177,0.3392124,"\int \frac{\tan ^{-1}(a x)^2}{x^2 \left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)^2),x]","-\frac{8 i a x \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)+4 a x \tan ^{-1}(a x)^3+2 \tan ^{-1}(a x)^2 \left(4 i a x+a x \sin \left(2 \tan ^{-1}(a x)\right)+4\right)-a x \sin \left(2 \tan ^{-1}(a x)\right)+2 a x \tan ^{-1}(a x) \left(\cos \left(2 \tan ^{-1}(a x)\right)-8 \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)\right)}{8 c^2 x}","\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{i a \text{Li}_2\left(\frac{2}{1-i a x}-1\right)}{c^2}-\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,"-1/8*(4*a*x*ArcTan[a*x]^3 + 2*a*x*ArcTan[a*x]*(Cos[2*ArcTan[a*x]] - 8*Log[1 - E^((2*I)*ArcTan[a*x])]) + (8*I)*a*x*PolyLog[2, E^((2*I)*ArcTan[a*x])] - a*x*Sin[2*ArcTan[a*x]] + 2*ArcTan[a*x]^2*(4 + (4*I)*a*x + a*x*Sin[2*ArcTan[a*x]]))/(c^2*x)","A",0
297,1,183,250,0.6719432,"\int \frac{\tan ^{-1}(a x)^2}{x^3 \left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]^2/(x^3*(c + a^2*c*x^2)^2),x]","\frac{a^2 \left(\log \left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-\frac{\left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{2 a^2 x^2}-2 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)-\text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-\frac{2}{3} i \tan ^{-1}(a x)^3-\frac{\tan ^{-1}(a x)}{a x}-2 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)+\frac{1}{4} \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)-\frac{1}{4} \tan ^{-1}(a x)^2 \cos \left(2 \tan ^{-1}(a x)\right)+\frac{1}{8} \cos \left(2 \tan ^{-1}(a x)\right)+\frac{i \pi ^3}{12}\right)}{c^2}","-\frac{a^2 \text{Li}_3\left(\frac{2}{1-i a x}-1\right)}{c^2}+\frac{2 i a^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{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^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{a^3 x \tan ^{-1}(a x)}{2 c^2 \left(a^2 x^2+1\right)}-\frac{\tan ^{-1}(a x)^2}{2 c^2 x^2}-\frac{a \tan ^{-1}(a x)}{c^2 x}",1,"(a^2*((I/12)*Pi^3 - ArcTan[a*x]/(a*x) - ((1 + a^2*x^2)*ArcTan[a*x]^2)/(2*a^2*x^2) - ((2*I)/3)*ArcTan[a*x]^3 + Cos[2*ArcTan[a*x]]/8 - (ArcTan[a*x]^2*Cos[2*ArcTan[a*x]])/4 - 2*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + Log[(a*x)/Sqrt[1 + a^2*x^2]] - (2*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - PolyLog[3, E^((-2*I)*ArcTan[a*x])] + (ArcTan[a*x]*Sin[2*ArcTan[a*x]])/4))/c^2","A",0
298,1,166,242,0.4469587,"\int \frac{\tan ^{-1}(a x)^2}{x^4 \left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)^2),x]","\frac{56 i a^3 x^3 \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)+20 a^3 x^3 \tan ^{-1}(a x)^3-a^2 x^2 \left(3 a x \sin \left(2 \tan ^{-1}(a x)\right)+8\right)+2 a x \tan ^{-1}(a x) \left(-4 a^2 x^2-56 a^2 x^2 \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)+3 a^2 x^2 \cos \left(2 \tan ^{-1}(a x)\right)-4\right)+\tan ^{-1}(a x)^2 \left(56 i a^3 x^3+6 a^3 x^3 \sin \left(2 \tan ^{-1}(a x)\right)+48 a^2 x^2-8\right)}{24 c^2 x^3}","\frac{7 i a^3 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)}{3 c^2}+\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{14 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{3 c^2}-\frac{a^2}{3 c^2 x}+\frac{2 a^2 \tan ^{-1}(a x)^2}{c^2 x}-\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{\tan ^{-1}(a x)^2}{3 c^2 x^3}-\frac{a \tan ^{-1}(a x)}{3 c^2 x^2}",1,"(20*a^3*x^3*ArcTan[a*x]^3 + 2*a*x*ArcTan[a*x]*(-4 - 4*a^2*x^2 + 3*a^2*x^2*Cos[2*ArcTan[a*x]] - 56*a^2*x^2*Log[1 - E^((2*I)*ArcTan[a*x])]) + (56*I)*a^3*x^3*PolyLog[2, E^((2*I)*ArcTan[a*x])] - a^2*x^2*(8 + 3*a*x*Sin[2*ArcTan[a*x]]) + ArcTan[a*x]^2*(-8 + 48*a^2*x^2 + (56*I)*a^3*x^3 + 6*a^3*x^3*Sin[2*ArcTan[a*x]]))/(24*c^2*x^3)","A",0
299,1,74,140,0.0910198,"\int \frac{x^3 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2)^3,x]","\frac{5 a^2 x^2+2 a x \left(5 a^2 x^2+3\right) \tan ^{-1}(a x)+\left(5 a^4 x^4-6 a^2 x^2-3\right) \tan ^{-1}(a x)^2+4}{32 a^4 c^3 \left(a^2 x^2+1\right)^2}","-\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{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}{32 a^4 c^3 \left(a^2 x^2+1\right)}+\frac{3 x \tan ^{-1}(a x)}{16 a^3 c^3 \left(a^2 x^2+1\right)}",1,"(4 + 5*a^2*x^2 + 2*a*x*(3 + 5*a^2*x^2)*ArcTan[a*x] + (-3 - 6*a^2*x^2 + 5*a^4*x^4)*ArcTan[a*x]^2)/(32*a^4*c^3*(1 + a^2*x^2)^2)","A",1
300,1,95,181,0.116691,"\int \frac{x^2 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^2*ArcTan[a*x]^2)/(c + a^2*c*x^2)^3,x]","\frac{-3 a^3 x^3+24 a x \left(a^2 x^2-1\right) \tan ^{-1}(a x)^2+8 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^3-3 \left(a^4 x^4-6 a^2 x^2+1\right) \tan ^{-1}(a x)+3 a x}{192 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}",1,"(3*a*x - 3*a^3*x^3 - 3*(1 - 6*a^2*x^2 + a^4*x^4)*ArcTan[a*x] + 24*a*x*(-1 + a^2*x^2)*ArcTan[a*x]^2 + 8*(1 + a^2*x^2)^2*ArcTan[a*x]^3)/(192*a^3*c^3*(1 + a^2*x^2)^2)","A",1
301,1,71,138,0.042604,"\int \frac{x \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^3,x]","\frac{3 a^2 x^2+2 a x \left(3 a^2 x^2+5\right) \tan ^{-1}(a x)+\left(3 a^4 x^4+6 a^2 x^2-5\right) \tan ^{-1}(a x)^2+4}{32 c^3 \left(a^3 x^2+a\right)^2}","\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,"(4 + 3*a^2*x^2 + 2*a*x*(5 + 3*a^2*x^2)*ArcTan[a*x] + (-5 + 6*a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x]^2)/(32*c^3*(a + a^3*x^2)^2)","A",1
302,1,98,169,0.0502324,"\int \frac{\tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]^2/(c + a^2*c*x^2)^3,x]","\frac{-a x \left(15 a^2 x^2+17\right)+8 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^3+8 a x \left(3 a^2 x^2+5\right) \tan ^{-1}(a x)^2+\left(-15 a^4 x^4-6 a^2 x^2+17\right) \tan ^{-1}(a x)}{64 a c^3 \left(a^2 x^2+1\right)^2}","-\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,"(-(a*x*(17 + 15*a^2*x^2)) + (17 - 6*a^2*x^2 - 15*a^4*x^4)*ArcTan[a*x] + 8*a*x*(5 + 3*a^2*x^2)*ArcTan[a*x]^2 + 8*(1 + a^2*x^2)^2*ArcTan[a*x]^3)/(64*a*c^3*(1 + a^2*x^2)^2)","A",1
303,1,156,236,0.2737818,"\int \frac{\tan ^{-1}(a x)^2}{x \left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)^3),x]","\frac{768 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+384 \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)+256 i \tan ^{-1}(a x)^3+768 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-288 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)-12 \tan ^{-1}(a x) \sin \left(4 \tan ^{-1}(a x)\right)+288 \tan ^{-1}(a x)^2 \cos \left(2 \tan ^{-1}(a x)\right)+24 \tan ^{-1}(a x)^2 \cos \left(4 \tan ^{-1}(a x)\right)-144 \cos \left(2 \tan ^{-1}(a x)\right)-3 \cos \left(4 \tan ^{-1}(a x)\right)-32 i \pi ^3}{768 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{\text{Li}_3\left(\frac{2}{1-i a x}-1\right)}{2 c^3}-\frac{i \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{c^3}-\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,"((-32*I)*Pi^3 + (256*I)*ArcTan[a*x]^3 - 144*Cos[2*ArcTan[a*x]] + 288*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] - 3*Cos[4*ArcTan[a*x]] + 24*ArcTan[a*x]^2*Cos[4*ArcTan[a*x]] + 768*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (768*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + 384*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 288*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 12*ArcTan[a*x]*Sin[4*ArcTan[a*x]])/(768*c^3)","A",0
304,1,139,250,0.4243893,"\int \frac{\tan ^{-1}(a x)^2}{x^2 \left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)^3),x]","-\frac{256 i a x \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)+160 a x \tan ^{-1}(a x)^3+8 \tan ^{-1}(a x)^2 \left(32 i a x+16 a x \sin \left(2 \tan ^{-1}(a x)\right)+a x \sin \left(4 \tan ^{-1}(a x)\right)+32\right)-a x \left(64 \sin \left(2 \tan ^{-1}(a x)\right)+\sin \left(4 \tan ^{-1}(a x)\right)\right)+4 a x \tan ^{-1}(a x) \left(-128 \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)+32 \cos \left(2 \tan ^{-1}(a x)\right)+\cos \left(4 \tan ^{-1}(a x)\right)\right)}{256 c^3 x}","\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{i a \text{Li}_2\left(\frac{2}{1-i a x}-1\right)}{c^3}-\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,"-1/256*(160*a*x*ArcTan[a*x]^3 + 4*a*x*ArcTan[a*x]*(32*Cos[2*ArcTan[a*x]] + Cos[4*ArcTan[a*x]] - 128*Log[1 - E^((2*I)*ArcTan[a*x])]) + (256*I)*a*x*PolyLog[2, E^((2*I)*ArcTan[a*x])] - a*x*(64*Sin[2*ArcTan[a*x]] + Sin[4*ArcTan[a*x]]) + 8*ArcTan[a*x]^2*(32 + (32*I)*a*x + 16*a*x*Sin[2*ArcTan[a*x]] + a*x*Sin[4*ArcTan[a*x]]))/(c^3*x)","A",0
305,1,226,322,0.8194792,"\int \frac{\tan ^{-1}(a x)^2}{x^3 \left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]^2/(x^3*(c + a^2*c*x^2)^3),x]","\frac{a^2 \left(\log \left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-\frac{\left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{2 a^2 x^2}-3 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)-\frac{3}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-i \tan ^{-1}(a x)^3-\frac{\tan ^{-1}(a x)}{a x}-3 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)+\frac{5}{8} \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)+\frac{1}{64} \tan ^{-1}(a x) \sin \left(4 \tan ^{-1}(a x)\right)-\frac{5}{8} \tan ^{-1}(a x)^2 \cos \left(2 \tan ^{-1}(a x)\right)-\frac{1}{32} \tan ^{-1}(a x)^2 \cos \left(4 \tan ^{-1}(a x)\right)+\frac{5}{16} \cos \left(2 \tan ^{-1}(a x)\right)+\frac{1}{256} \cos \left(4 \tan ^{-1}(a x)\right)+\frac{i \pi ^3}{8}\right)}{c^3}","-\frac{3 a^2 \text{Li}_3\left(\frac{2}{1-i a x}-1\right)}{2 c^3}+\frac{3 i a^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{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{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{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{\tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{a \tan ^{-1}(a x)}{c^3 x}",1,"(a^2*((I/8)*Pi^3 - ArcTan[a*x]/(a*x) - ((1 + a^2*x^2)*ArcTan[a*x]^2)/(2*a^2*x^2) - I*ArcTan[a*x]^3 + (5*Cos[2*ArcTan[a*x]])/16 - (5*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]])/8 + Cos[4*ArcTan[a*x]]/256 - (ArcTan[a*x]^2*Cos[4*ArcTan[a*x]])/32 - 3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + Log[(a*x)/Sqrt[1 + a^2*x^2]] - (3*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (3*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/2 + (5*ArcTan[a*x]*Sin[2*ArcTan[a*x]])/8 + (ArcTan[a*x]*Sin[4*ArcTan[a*x]])/64))/c^3","A",0
306,1,189,317,0.8164601,"\int \frac{\tan ^{-1}(a x)^2}{x^4 \left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)^3),x]","\frac{a^3 \left(-\frac{256 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}{a^2 x^2}-\frac{256 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}{a^3 x^3}+2560 i \left(\tan ^{-1}(a x)^2+\text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)\right)+1120 \tan ^{-1}(a x)^3+\frac{256 \left(10 \tan ^{-1}(a x)^2-1\right)}{a x}-5120 \tan ^{-1}(a x) \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)+288 \left(2 \tan ^{-1}(a x)^2-1\right) \sin \left(2 \tan ^{-1}(a x)\right)+3 \left(8 \tan ^{-1}(a x)^2-1\right) \sin \left(4 \tan ^{-1}(a x)\right)+576 \tan ^{-1}(a x) \cos \left(2 \tan ^{-1}(a x)\right)+12 \tan ^{-1}(a x) \cos \left(4 \tan ^{-1}(a x)\right)\right)}{768 c^3}","\frac{10 i a^3 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)}{3 c^3}+\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{20 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)}{3 c^3}-\frac{a^2}{3 c^3 x}+\frac{3 a^2 \tan ^{-1}(a x)^2}{c^3 x}-\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{\tan ^{-1}(a x)^2}{3 c^3 x^3}-\frac{a \tan ^{-1}(a x)}{3 c^3 x^2}",1,"(a^3*((-256*(1 + a^2*x^2)*ArcTan[a*x])/(a^2*x^2) - (256*(1 + a^2*x^2)*ArcTan[a*x]^2)/(a^3*x^3) + 1120*ArcTan[a*x]^3 + (256*(-1 + 10*ArcTan[a*x]^2))/(a*x) + 576*ArcTan[a*x]*Cos[2*ArcTan[a*x]] + 12*ArcTan[a*x]*Cos[4*ArcTan[a*x]] - 5120*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] + (2560*I)*(ArcTan[a*x]^2 + PolyLog[2, E^((2*I)*ArcTan[a*x])]) + 288*(-1 + 2*ArcTan[a*x]^2)*Sin[2*ArcTan[a*x]] + 3*(-1 + 8*ArcTan[a*x]^2)*Sin[4*ArcTan[a*x]]))/(768*c^3)","A",0
307,1,360,385,1.2179808,"\int x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx","Integrate[x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]","-\frac{\left(a^2 x^2+1\right)^2 \sqrt{c \left(a^2 x^2+1\right)} \left(-\frac{176 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{5/2}}+\frac{176 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{5/2}}-\frac{110 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{110 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-32 \tan ^{-1}(a x)^2+4 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)-22 \tan ^{-1}(a x) \sin \left(4 \tan ^{-1}(a x)\right)+160 \tan ^{-1}(a x)^2 \cos \left(2 \tan ^{-1}(a x)\right)+72 \cos \left(2 \tan ^{-1}(a x)\right)+22 \cos \left(4 \tan ^{-1}(a x)\right)-55 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \cos \left(3 \tan ^{-1}(a x)\right)-11 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \cos \left(5 \tan ^{-1}(a x)\right)+55 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \cos \left(3 \tan ^{-1}(a x)\right)+11 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \cos \left(5 \tan ^{-1}(a x)\right)+50\right)}{960 a^4}","\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{15 a^2}+\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{11 i c \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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{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{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{12 a^3}",1,"-1/960*((1 + a^2*x^2)^2*Sqrt[c*(1 + a^2*x^2)]*(50 - 32*ArcTan[a*x]^2 + 72*Cos[2*ArcTan[a*x]] + 160*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 22*Cos[4*ArcTan[a*x]] - (110*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (110*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((176*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + ((176*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + 4*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 22*ArcTan[a*x]*Sin[4*ArcTan[a*x]]))/a^4","A",0
308,1,267,436,1.2851681,"\int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx","Integrate[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]","\frac{\sqrt{a^2 c x^2+c} \left(\left(a^2 x^2+1\right)^{3/2} \left(-3 \tan ^{-1}(a x)^2 \left(\sqrt{a^2 x^2+1} \sin \left(3 \tan ^{-1}(a x)\right)-7 a x\right)+2 \left(\sqrt{a^2 x^2+1} \sin \left(3 \tan ^{-1}(a x)\right)+a x\right)+\tan ^{-1}(a x) \left(6 \sqrt{a^2 x^2+1} \cos \left(3 \tan ^{-1}(a x)\right)+2\right)\right)+8 \left(-2 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-3 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+3 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+3 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-3 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+3 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)\right)}{96 a^3 \sqrt{a^2 x^2+1}}","\frac{x \sqrt{a^2 c x^2+c}}{12 a^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{1}{4} x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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}\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,"(Sqrt[c + a^2*c*x^2]*(8*((3*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 2*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - (3*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (3*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 3*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 3*PolyLog[3, I*E^(I*ArcTan[a*x])]) + (1 + a^2*x^2)^(3/2)*(ArcTan[a*x]*(2 + 6*Sqrt[1 + a^2*x^2]*Cos[3*ArcTan[a*x]]) - 3*ArcTan[a*x]^2*(-7*a*x + Sqrt[1 + a^2*x^2]*Sin[3*ArcTan[a*x]]) + 2*(a*x + Sqrt[1 + a^2*x^2]*Sin[3*ArcTan[a*x]]))))/(96*a^3*Sqrt[1 + a^2*x^2])","A",0
309,1,260,279,0.6220322,"\int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx","Integrate[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]","\frac{\left(a^2 x^2+1\right) \sqrt{c \left(a^2 x^2+1\right)} \left(-\frac{4 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}+\frac{4 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}-\frac{3 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{3 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+4 \tan ^{-1}(a x)^2-2 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)+2 \cos \left(2 \tan ^{-1}(a x)\right)-\tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \cos \left(3 \tan ^{-1}(a x)\right)+\tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \cos \left(3 \tan ^{-1}(a x)\right)+2\right)}{12 a^2}","-\frac{i c \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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,"((1 + a^2*x^2)*Sqrt[c*(1 + a^2*x^2)]*(2 + 4*ArcTan[a*x]^2 + 2*Cos[2*ArcTan[a*x]] - (3*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (3*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((4*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) + ((4*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) - 2*ArcTan[a*x]*Sin[2*ArcTan[a*x]]))/(12*a^2)","A",0
310,1,201,340,0.4162995,"\int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx","Integrate[Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(a x \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2-2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+2 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+2 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-2 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-2 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+2 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-2 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)}{2 a \sqrt{a^2 x^2+1}}","\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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*(1 + a^2*x^2)]*(-2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 - (2*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 2*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (2*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (2*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 2*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 2*PolyLog[3, I*E^(I*ArcTan[a*x])]))/(2*a*Sqrt[1 + a^2*x^2])","A",0
311,1,250,439,0.2669536,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x} \, dx","Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x,x]","\frac{\sqrt{a^2 c x^2+c} \left(\sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+2 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-2 i \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-2 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+2 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-2 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+2 \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)-\tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)-2 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)+2 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)}{\sqrt{a^2 x^2+1}}","-\frac{2 i c \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 i c \sqrt{a^2 x^2+1} \text{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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{Li}_2\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} \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 c \sqrt{a^2 x^2+1} \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 c \sqrt{a^2 x^2+1} \text{Li}_3\left(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]*(Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] - 2*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] - ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + (2*I)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - (2*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (2*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - (2*I)*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - 2*PolyLog[3, -E^(I*ArcTan[a*x])] + 2*PolyLog[3, E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2]","A",0
312,1,265,458,0.7954871,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^2} \, dx","Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x^2,x]","-\frac{a \sqrt{c \left(a^2 x^2+1\right)} \left(\frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2}{a x}-2 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+2 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-2 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+2 i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+2 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-2 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x)^2 \left(-\log \left(1-i e^{i \tan ^{-1}(a x)}\right)\right)+\tan ^{-1}(a x)^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-2 \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)+2 \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)\right)}{\sqrt{a^2 x^2+1}}","\frac{2 i a c \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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,"-((a*Sqrt[c*(1 + a^2*x^2)]*((Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2)/(a*x) - 2*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] + ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] - (2*I)*PolyLog[2, -E^(I*ArcTan[a*x])] - (2*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (2*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + (2*I)*PolyLog[2, E^(I*ArcTan[a*x])] + 2*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 2*PolyLog[3, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2])","A",0
313,1,222,328,1.9291372,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^3} \, dx","Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x^3,x]","\frac{a^2 \sqrt{c \left(a^2 x^2+1\right)} \left(8 i \tan ^{-1}(a x) \left(\text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-\text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)\right)+8 \left(\text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)-\text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)\right)-4 \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+4 \tan ^{-1}(a x)^2 \left(\log \left(1-e^{i \tan ^{-1}(a x)}\right)-\log \left(1+e^{i \tan ^{-1}(a x)}\right)\right)+8 \log \left(\tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-4 \tan ^{-1}(a x) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+\tan ^{-1}(a x)^2 \left(-\csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)+\tan ^{-1}(a x)^2 \sec ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}{8 \sqrt{a^2 x^2+1}}","\frac{i a^2 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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^2*Sqrt[c*(1 + a^2*x^2)]*(-4*ArcTan[a*x]*Cot[ArcTan[a*x]/2] - ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 + 4*ArcTan[a*x]^2*(Log[1 - E^(I*ArcTan[a*x])] - Log[1 + E^(I*ArcTan[a*x])]) + 8*Log[Tan[ArcTan[a*x]/2]] + (8*I)*ArcTan[a*x]*(PolyLog[2, -E^(I*ArcTan[a*x])] - PolyLog[2, E^(I*ArcTan[a*x])]) + 8*(-PolyLog[3, -E^(I*ArcTan[a*x])] + PolyLog[3, E^(I*ArcTan[a*x])]) + ArcTan[a*x]^2*Sec[ArcTan[a*x]/2]^2 - 4*ArcTan[a*x]*Tan[ArcTan[a*x]/2]))/(8*Sqrt[1 + a^2*x^2])","A",0
314,1,239,275,1.7197213,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^4} \, dx","Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/x^4,x]","-\frac{c \sqrt{a^2 x^2+1} \left(-4 i a^3 x^3 \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+4 i a^3 x^3 \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+\sqrt{a^2 x^2+1} \left(4 a^2 x^2+4 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2+\tan ^{-1}(a x) \left(a x \left(-3 \sqrt{a^2 x^2+1} \log \left(1-e^{i \tan ^{-1}(a x)}\right)+3 \sqrt{a^2 x^2+1} \log \left(1+e^{i \tan ^{-1}(a x)}\right)+4\right)+\left(a^2 x^2+1\right) \left(\log \left(1-e^{i \tan ^{-1}(a x)}\right)-\log \left(1+e^{i \tan ^{-1}(a x)}\right)\right) \sin \left(3 \tan ^{-1}(a x)\right)\right)\right)\right)}{12 x^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{i a^3 c \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}-\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,"-1/12*(c*Sqrt[1 + a^2*x^2]*((-4*I)*a^3*x^3*PolyLog[2, -E^(I*ArcTan[a*x])] + (4*I)*a^3*x^3*PolyLog[2, E^(I*ArcTan[a*x])] + Sqrt[1 + a^2*x^2]*(4*a^2*x^2 + 4*(1 + a^2*x^2)*ArcTan[a*x]^2 + ArcTan[a*x]*(a*x*(4 - 3*Sqrt[1 + a^2*x^2]*Log[1 - E^(I*ArcTan[a*x])] + 3*Sqrt[1 + a^2*x^2]*Log[1 + E^(I*ArcTan[a*x])]) + (1 + a^2*x^2)*(Log[1 - E^(I*ArcTan[a*x])] - Log[1 + E^(I*ArcTan[a*x])])*Sin[3*ArcTan[a*x]]))))/(x^3*Sqrt[c + a^2*c*x^2])","A",0
315,1,797,476,4.8118208,"\int x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2 \, dx","Integrate[x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2,x]","\frac{c \left(a^2 x^2+1\right)^2 \sqrt{a^2 c x^2+c} \left(\left(a^2 x^2+1\right) \left(-5376 \cos \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2+6720 \cos \left(4 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2+10944 \tan ^{-1}(a x)^2-6489 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-2163 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-309 \cos \left(7 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-\frac{10815 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+6489 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+2163 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+309 \cos \left(7 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+\frac{10815 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-1266 \sin \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+360 \sin \left(4 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)-618 \sin \left(6 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+6262 \cos \left(2 \tan ^{-1}(a x)\right)+2764 \cos \left(4 \tan ^{-1}(a x)\right)+618 \cos \left(6 \tan ^{-1}(a x)\right)-\frac{19776 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{7/2}}+\frac{19776 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{7/2}}+4116\right)-168 \left(160 \cos \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2-32 \tan ^{-1}(a x)^2-55 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-11 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-\frac{110 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+55 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+11 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+\frac{110 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+4 \sin \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)-22 \sin \left(4 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+72 \cos \left(2 \tan ^{-1}(a x)\right)+22 \cos \left(4 \tan ^{-1}(a x)\right)-\frac{176 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{5/2}}+\frac{176 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{5/2}}+50\right)\right)}{161280 a^4}","\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{35 a^2}+\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{17 i c^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{35 a^4}+\frac{3 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{56 a^3}",1,"(c*(1 + a^2*x^2)^2*Sqrt[c + a^2*c*x^2]*(-168*(50 - 32*ArcTan[a*x]^2 + 72*Cos[2*ArcTan[a*x]] + 160*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 22*Cos[4*ArcTan[a*x]] - (110*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (110*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((176*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + ((176*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + 4*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 22*ArcTan[a*x]*Sin[4*ArcTan[a*x]]) + (1 + a^2*x^2)*(4116 + 10944*ArcTan[a*x]^2 + 6262*Cos[2*ArcTan[a*x]] - 5376*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 2764*Cos[4*ArcTan[a*x]] + 6720*ArcTan[a*x]^2*Cos[4*ArcTan[a*x]] + 618*Cos[6*ArcTan[a*x]] - (10815*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 6489*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 2163*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 309*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (10815*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 6489*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 2163*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 309*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((19776*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(7/2) + ((19776*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(7/2) - 1266*ArcTan[a*x]*Sin[2*ArcTan[a*x]] + 360*ArcTan[a*x]*Sin[4*ArcTan[a*x]] - 618*ArcTan[a*x]*Sin[6*ArcTan[a*x]])))/(161280*a^4)","A",0
316,1,527,531,3.6729751,"\int x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2 \, dx","Integrate[x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2,x]","\frac{c \sqrt{a^2 c x^2+c} \left(960 \left(-2 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-3 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+3 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+3 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-3 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+3 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)+32 \left(19 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+45 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-45 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-45 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+45 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-45 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)+\left(a^2 x^2+1\right)^3 \left(-\frac{56 a x}{\sqrt{a^2 x^2+1}}+15 \tan ^{-1}(a x)^2 \left(\frac{78 a x}{\sqrt{a^2 x^2+1}}-47 \sin \left(3 \tan ^{-1}(a x)\right)+3 \sin \left(5 \tan ^{-1}(a x)\right)\right)+\tan ^{-1}(a x) \left(\frac{12}{\sqrt{a^2 x^2+1}}+110 \cos \left(3 \tan ^{-1}(a x)\right)-90 \cos \left(5 \tan ^{-1}(a x)\right)\right)-108 \sin \left(3 \tan ^{-1}(a x)\right)-52 \sin \left(5 \tan ^{-1}(a x)\right)\right)+120 \left(a^2 x^2+1\right)^{3/2} \left(-3 \tan ^{-1}(a x)^2 \left(\sqrt{a^2 x^2+1} \sin \left(3 \tan ^{-1}(a x)\right)-7 a x\right)+2 \left(\sqrt{a^2 x^2+1} \sin \left(3 \tan ^{-1}(a x)\right)+a x\right)+\tan ^{-1}(a x) \left(6 \sqrt{a^2 x^2+1} \cos \left(3 \tan ^{-1}(a x)\right)+2\right)\right)\right)}{11520 a^3 \sqrt{a^2 x^2+1}}","\frac{c x \sqrt{a^2 c x^2+c}}{36 a^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{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{1}{60} c x^3 \sqrt{a^2 c x^2+c}+\frac{7}{24} c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\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{i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{31 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{360 a^3}",1,"(c*Sqrt[c + a^2*c*x^2]*(960*((3*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 2*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - (3*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (3*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 3*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 3*PolyLog[3, I*E^(I*ArcTan[a*x])]) + 32*((-45*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 19*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (45*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (45*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 45*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 45*PolyLog[3, I*E^(I*ArcTan[a*x])]) + 120*(1 + a^2*x^2)^(3/2)*(ArcTan[a*x]*(2 + 6*Sqrt[1 + a^2*x^2]*Cos[3*ArcTan[a*x]]) - 3*ArcTan[a*x]^2*(-7*a*x + Sqrt[1 + a^2*x^2]*Sin[3*ArcTan[a*x]]) + 2*(a*x + Sqrt[1 + a^2*x^2]*Sin[3*ArcTan[a*x]])) + (1 + a^2*x^2)^3*((-56*a*x)/Sqrt[1 + a^2*x^2] + ArcTan[a*x]*(12/Sqrt[1 + a^2*x^2] + 110*Cos[3*ArcTan[a*x]] - 90*Cos[5*ArcTan[a*x]]) - 108*Sin[3*ArcTan[a*x]] - 52*Sin[5*ArcTan[a*x]] + 15*ArcTan[a*x]^2*((78*a*x)/Sqrt[1 + a^2*x^2] - 47*Sin[3*ArcTan[a*x]] + 3*Sin[5*ArcTan[a*x]]))))/(11520*a^3*Sqrt[1 + a^2*x^2])","A",0
317,1,601,334,4.3059724,"\int x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2 \, dx","Integrate[x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2,x]","\frac{c \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \left(80 \left(-\frac{4 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}+\frac{4 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}-\frac{3 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{3 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+4 \tan ^{-1}(a x)^2-2 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)+2 \cos \left(2 \tan ^{-1}(a x)\right)-\tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \cos \left(3 \tan ^{-1}(a x)\right)+\tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \cos \left(3 \tan ^{-1}(a x)\right)+2\right)-\left(a^2 x^2+1\right) \left(-\frac{176 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{5/2}}+\frac{176 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{5/2}}-\frac{110 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{110 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-32 \tan ^{-1}(a x)^2+4 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)-22 \tan ^{-1}(a x) \sin \left(4 \tan ^{-1}(a x)\right)+160 \tan ^{-1}(a x)^2 \cos \left(2 \tan ^{-1}(a x)\right)+72 \cos \left(2 \tan ^{-1}(a x)\right)+22 \cos \left(4 \tan ^{-1}(a x)\right)-55 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \cos \left(3 \tan ^{-1}(a x)\right)-11 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \cos \left(5 \tan ^{-1}(a x)\right)+55 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \cos \left(3 \tan ^{-1}(a x)\right)+11 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \cos \left(5 \tan ^{-1}(a x)\right)+50\right)\right)}{960 a^2}","-\frac{3 i c^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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,"(c*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*(80*(2 + 4*ArcTan[a*x]^2 + 2*Cos[2*ArcTan[a*x]] - (3*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (3*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((4*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) + ((4*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) - 2*ArcTan[a*x]*Sin[2*ArcTan[a*x]]) - (1 + a^2*x^2)*(50 - 32*ArcTan[a*x]^2 + 72*Cos[2*ArcTan[a*x]] + 160*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 22*Cos[4*ArcTan[a*x]] - (110*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (110*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((176*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + ((176*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + 4*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 22*ArcTan[a*x]*Sin[4*ArcTan[a*x]])))/(960*a^2)","A",0
318,1,439,438,0.9849571,"\int \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2 \, dx","Integrate[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2,x]","\frac{c \sqrt{a^2 c x^2+c} \left(2 a^4 x^4 \sin \left(3 \tan ^{-1}(a x)\right)-3 a^4 x^4 \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)+6 a^4 x^4 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)+2 a x \sqrt{a^2 x^2+1}+2 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+69 a x \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2-94 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+80 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+4 a^2 x^2 \sin \left(3 \tan ^{-1}(a x)\right)-6 a^2 x^2 \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)+12 a^2 x^2 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)+2 a^3 x^3 \sqrt{a^2 x^2+1}+21 a^3 x^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+72 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-72 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-72 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+72 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-72 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)+2 \sin \left(3 \tan ^{-1}(a x)\right)+6 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)\right)}{96 a \sqrt{a^2 x^2+1}}","\frac{5 c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{6 a}+\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{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*Sqrt[c + a^2*c*x^2]*(2*a*x*Sqrt[1 + a^2*x^2] + 2*a^3*x^3*Sqrt[1 + a^2*x^2] - 94*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 2*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 69*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + 21*a^3*x^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 - (72*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 80*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + 6*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 12*a^2*x^2*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 6*a^4*x^4*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + (72*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (72*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 72*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 72*PolyLog[3, I*E^(I*ArcTan[a*x])] + 2*Sin[3*ArcTan[a*x]] + 4*a^2*x^2*Sin[3*ArcTan[a*x]] + 2*a^4*x^4*Sin[3*ArcTan[a*x]] - 3*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 6*a^2*x^2*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 3*a^4*x^4*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]]))/(96*a*Sqrt[1 + a^2*x^2])","A",0
319,1,496,530,3.3947141,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2}{x} \, dx","Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/x,x]","\frac{1}{12} c \sqrt{a^2 c x^2+c} \left(\frac{12 \left(\sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+2 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-2 i \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-2 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+2 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-2 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+2 \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)-\tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)-2 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)+2 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)}{\sqrt{a^2 x^2+1}}+\left(a^2 x^2+1\right) \left(-\frac{4 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}+\frac{4 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}-\frac{3 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{3 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+4 \tan ^{-1}(a x)^2-2 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)+2 \cos \left(2 \tan ^{-1}(a x)\right)-\tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \cos \left(3 \tan ^{-1}(a x)\right)+\tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \cos \left(3 \tan ^{-1}(a x)\right)+2\right)\right)","-\frac{7 i c^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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]*((12*(Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] - 2*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] - ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + (2*I)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - (2*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (2*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - (2*I)*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - 2*PolyLog[3, -E^(I*ArcTan[a*x])] + 2*PolyLog[3, E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] + (1 + a^2*x^2)*(2 + 4*ArcTan[a*x]^2 + 2*Cos[2*ArcTan[a*x]] - (3*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (3*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((4*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) + ((4*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) - 2*ArcTan[a*x]*Sin[2*ArcTan[a*x]])))/12","A",0
320,1,376,556,1.0899654,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2}{x^2} \, dx","Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/x^2,x]","\frac{c \sqrt{a^2 c x^2+c} \left(a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2-2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2-2 a x \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+2 a x \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+6 i a x \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-6 i a x \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+4 i a x \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-4 i a x \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-6 a x \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+6 a x \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-2 i a x \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2+2 a x \tan ^{-1}(a x)^2 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)-2 a x \tan ^{-1}(a x)^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+4 a x \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)-4 a x \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)\right)}{2 x \sqrt{a^2 x^2+1}}","a c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)+\frac{2 i a c^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\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} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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}}-\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,"(c*Sqrt[c + a^2*c*x^2]*(-2*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x] - 2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 - (2*I)*a*x*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 2*a*x*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + 4*a*x*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] + 2*a*x*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] - 2*a*x*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] - 4*a*x*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + (4*I)*a*x*PolyLog[2, -E^(I*ArcTan[a*x])] + (6*I)*a*x*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (6*I)*a*x*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - (4*I)*a*x*PolyLog[2, E^(I*ArcTan[a*x])] - 6*a*x*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 6*a*x*PolyLog[3, I*E^(I*ArcTan[a*x])]))/(2*x*Sqrt[1 + a^2*x^2])","A",0
321,1,455,567,3.1895458,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2}{x^3} \, dx","Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/x^3,x]","\frac{a^2 c \sqrt{a^2 c x^2+c} \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right) \left(24 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-24 i \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-16 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+16 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-24 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+24 \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-4 \tan ^{-1}(a x)-4 \tan ^{-1}(a x) \cot ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+4 a x \tan ^{-1}(a x)^2 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+12 \tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-12 \tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-16 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+16 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+8 \log \left(\tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\tan ^{-1}(a x)^2 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right) \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+\tan ^{-1}(a x)^2 \csc \left(\frac{1}{2} \tan ^{-1}(a x)\right) \sec \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}{8 \sqrt{a^2 x^2+1}}","-a^2 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{2 i a^2 c^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \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} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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}}-\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^2*c*Sqrt[c + a^2*c*x^2]*(-4*ArcTan[a*x] - 4*ArcTan[a*x]*Cot[ArcTan[a*x]/2]^2 + 4*a*x*ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 - ArcTan[a*x]^2*Cot[ArcTan[a*x]/2]*Csc[ArcTan[a*x]/2]^2 + 12*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2]*Log[1 - E^(I*ArcTan[a*x])] - 16*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 - I*E^(I*ArcTan[a*x])] + 16*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 + I*E^(I*ArcTan[a*x])] - 12*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2]*Log[1 + E^(I*ArcTan[a*x])] + 8*Cot[ArcTan[a*x]/2]*Log[Tan[ArcTan[a*x]/2]] + (24*I)*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*PolyLog[2, -E^(I*ArcTan[a*x])] - (16*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (16*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, I*E^(I*ArcTan[a*x])] - (24*I)*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*PolyLog[2, E^(I*ArcTan[a*x])] - 24*Cot[ArcTan[a*x]/2]*PolyLog[3, -E^(I*ArcTan[a*x])] + 24*Cot[ArcTan[a*x]/2]*PolyLog[3, E^(I*ArcTan[a*x])] + ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]*Sec[ArcTan[a*x]/2])*Tan[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2])","A",0
322,1,537,579,7.3820314,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2}{x^4} \, dx","Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/x^4,x]","\frac{a^3 c^2 \sqrt{a^2 x^2+1} \left(8 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-\frac{2 \left(a^2 x^2+1\right)^{3/2} \left(-\frac{3 a x \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{3 a x \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{4 i a^3 x^3 \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}+4 \tan ^{-1}(a x)^2+2 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)-2 \cos \left(2 \tan ^{-1}(a x)\right)+\tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right) \sin \left(3 \tan ^{-1}(a x)\right)-\tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right) \sin \left(3 \tan ^{-1}(a x)\right)+2\right)}{a^3 x^3}\right)}{24 \sqrt{c \left(a^2 x^2+1\right)}}-\frac{a^3 c \sqrt{c \left(a^2 x^2+1\right)} \left(\frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2}{a x}-2 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+2 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-2 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+2 i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+2 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-2 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x)^2 \left(-\log \left(1-i e^{i \tan ^{-1}(a x)}\right)\right)+\tan ^{-1}(a x)^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-2 \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)+2 \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)\right)}{\sqrt{a^2 x^2+1}}","-\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{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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}}",1,"-((a^3*c*Sqrt[c*(1 + a^2*x^2)]*((Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2)/(a*x) - 2*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] + ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] - (2*I)*PolyLog[2, -E^(I*ArcTan[a*x])] - (2*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (2*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + (2*I)*PolyLog[2, E^(I*ArcTan[a*x])] + 2*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 2*PolyLog[3, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2]) + (a^3*c^2*Sqrt[1 + a^2*x^2]*((8*I)*PolyLog[2, -E^(I*ArcTan[a*x])] - (2*(1 + a^2*x^2)^(3/2)*(2 + 4*ArcTan[a*x]^2 - 2*Cos[2*ArcTan[a*x]] - (3*a*x*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (3*a*x*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((4*I)*a^3*x^3*PolyLog[2, E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) + 2*ArcTan[a*x]*Sin[2*ArcTan[a*x]] + ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])]*Sin[3*ArcTan[a*x]] - ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])]*Sin[3*ArcTan[a*x]]))/(a^3*x^3)))/(24*Sqrt[c*(1 + a^2*x^2)])","A",0
323,1,1320,578,8.7569701,"\int x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2 \, dx","Integrate[x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]","\frac{\left(a^2 c x^2+c\right)^{5/2} \left(-\left(\left(3276288 \cos \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2-580608 \cos \left(4 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2+483840 \cos \left(6 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2-820224 \tan ^{-1}(a x)^2-1378188 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-590652 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-147663 \cos \left(7 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-16407 \cos \left(9 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-\frac{2067282 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+1378188 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+590652 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+147663 \cos \left(7 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+16407 \cos \left(9 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+\frac{2067282 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+78444 \sin \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)-160452 \sin \left(4 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+38172 \sin \left(6 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)-32814 \sin \left(8 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+1083168 \cos \left(2 \tan ^{-1}(a x)\right)+576936 \cos \left(4 \tan ^{-1}(a x)\right)+184160 \cos \left(6 \tan ^{-1}(a x)\right)+32814 \cos \left(8 \tan ^{-1}(a x)\right)-\frac{4200192 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{9/2}}+\frac{4200192 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{9/2}}+657578\right) \left(a^2 x^2+1\right)^2\right)+576 \left(-5376 \cos \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2+6720 \cos \left(4 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2+10944 \tan ^{-1}(a x)^2-6489 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-2163 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-309 \cos \left(7 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-\frac{10815 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+6489 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+2163 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+309 \cos \left(7 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+\frac{10815 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-1266 \sin \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+360 \sin \left(4 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)-618 \sin \left(6 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+6262 \cos \left(2 \tan ^{-1}(a x)\right)+2764 \cos \left(4 \tan ^{-1}(a x)\right)+618 \cos \left(6 \tan ^{-1}(a x)\right)-\frac{19776 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{7/2}}+\frac{19776 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{7/2}}+4116\right) \left(a^2 x^2+1\right)-48384 \left(160 \cos \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2-32 \tan ^{-1}(a x)^2-55 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-11 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-\frac{110 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+55 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+11 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+\frac{110 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+4 \sin \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)-22 \sin \left(4 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+72 \cos \left(2 \tan ^{-1}(a x)\right)+22 \cos \left(4 \tan ^{-1}(a x)\right)-\frac{176 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{5/2}}+\frac{176 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{5/2}}+50\right)\right)}{46448640 a^4}","\frac{c^2 x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{63 a^2}+\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{115 i c^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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} \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{115 c^2 \sqrt{a^2 c x^2+c}}{4032 a^4}-\frac{2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{63 a^4}+\frac{1}{9} a^4 c^2 x^8 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+\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{47 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{1344 a^3}-\frac{1}{36} a^3 c^2 x^7 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)",1,"((c + a^2*c*x^2)^(5/2)*(-48384*(50 - 32*ArcTan[a*x]^2 + 72*Cos[2*ArcTan[a*x]] + 160*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 22*Cos[4*ArcTan[a*x]] - (110*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (110*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((176*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + ((176*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + 4*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 22*ArcTan[a*x]*Sin[4*ArcTan[a*x]]) + 576*(1 + a^2*x^2)*(4116 + 10944*ArcTan[a*x]^2 + 6262*Cos[2*ArcTan[a*x]] - 5376*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 2764*Cos[4*ArcTan[a*x]] + 6720*ArcTan[a*x]^2*Cos[4*ArcTan[a*x]] + 618*Cos[6*ArcTan[a*x]] - (10815*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 6489*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 2163*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 309*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (10815*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 6489*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 2163*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 309*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((19776*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(7/2) + ((19776*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(7/2) - 1266*ArcTan[a*x]*Sin[2*ArcTan[a*x]] + 360*ArcTan[a*x]*Sin[4*ArcTan[a*x]] - 618*ArcTan[a*x]*Sin[6*ArcTan[a*x]]) - (1 + a^2*x^2)^2*(657578 - 820224*ArcTan[a*x]^2 + 1083168*Cos[2*ArcTan[a*x]] + 3276288*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 576936*Cos[4*ArcTan[a*x]] - 580608*ArcTan[a*x]^2*Cos[4*ArcTan[a*x]] + 184160*Cos[6*ArcTan[a*x]] + 483840*ArcTan[a*x]^2*Cos[6*ArcTan[a*x]] + 32814*Cos[8*ArcTan[a*x]] - (2067282*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 1378188*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 590652*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 147663*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 16407*ArcTan[a*x]*Cos[9*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (2067282*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 1378188*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 590652*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 147663*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 16407*ArcTan[a*x]*Cos[9*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((4200192*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(9/2) + ((4200192*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(9/2) + 78444*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 160452*ArcTan[a*x]*Sin[4*ArcTan[a*x]] + 38172*ArcTan[a*x]*Sin[6*ArcTan[a*x]] - 32814*ArcTan[a*x]*Sin[8*ArcTan[a*x]])))/(46448640*a^4)","B",0
324,1,759,638,4.8411667,"\int x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2 \, dx","Integrate[x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]","\frac{c^2 \sqrt{a^2 c x^2+c} \left(7006 a x \left(a^2 x^2+1\right)^{7/2}-25088 a x \left(a^2 x^2+1\right)^{5/2}+53760 a x \left(a^2 x^2+1\right)^{3/2}+185325 a x \left(a^2 x^2+1\right)^{7/2} \tan ^{-1}(a x)^2-38134 \left(a^2 x^2+1\right)^{7/2} \tan ^{-1}(a x)+524160 a x \left(a^2 x^2+1\right)^{5/2} \tan ^{-1}(a x)^2+5376 \left(a^2 x^2+1\right)^{5/2} \tan ^{-1}(a x)+564480 a x \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^2+53760 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)-203264 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-93975 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)+12246 \left(a^2 x^2+1\right)^4 \sin \left(3 \tan ^{-1}(a x)\right)+41685 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x)^2 \sin \left(5 \tan ^{-1}(a x)\right)+7678 \left(a^2 x^2+1\right)^4 \sin \left(5 \tan ^{-1}(a x)\right)-1575 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x)^2 \sin \left(7 \tan ^{-1}(a x)\right)+2438 \left(a^2 x^2+1\right)^4 \sin \left(7 \tan ^{-1}(a x)\right)-315840 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)-48384 \left(a^2 x^2+1\right)^3 \sin \left(3 \tan ^{-1}(a x)\right)+20160 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^2 \sin \left(5 \tan ^{-1}(a x)\right)-23296 \left(a^2 x^2+1\right)^3 \sin \left(5 \tan ^{-1}(a x)\right)-80640 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)+53760 \left(a^2 x^2+1\right)^2 \sin \left(3 \tan ^{-1}(a x)\right)-7658 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)-10990 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x) \cos \left(5 \tan ^{-1}(a x)\right)+3150 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x) \cos \left(7 \tan ^{-1}(a x)\right)+49280 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)-40320 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x) \cos \left(5 \tan ^{-1}(a x)\right)+161280 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)-201600 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+201600 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+201600 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-201600 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+201600 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)}{2580480 a^3 \sqrt{a^2 x^2+1}}","\frac{43 c^2 x \sqrt{a^2 c x^2+c}}{4032 a^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{1}{168} a^2 c^2 x^5 \sqrt{a^2 c x^2+c}+\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{29 c^2 x^3 \sqrt{a^2 c x^2+c}}{1680}+\frac{59}{192} c^2 x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+\frac{1}{8} a^4 c^2 x^7 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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}\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{1}{28} a^3 c^2 x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)",1,"(c^2*Sqrt[c + a^2*c*x^2]*(53760*a*x*(1 + a^2*x^2)^(3/2) - 25088*a*x*(1 + a^2*x^2)^(5/2) + 7006*a*x*(1 + a^2*x^2)^(7/2) + 53760*(1 + a^2*x^2)^(3/2)*ArcTan[a*x] + 5376*(1 + a^2*x^2)^(5/2)*ArcTan[a*x] - 38134*(1 + a^2*x^2)^(7/2)*ArcTan[a*x] + 564480*a*x*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2 + 524160*a*x*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]^2 + 185325*a*x*(1 + a^2*x^2)^(7/2)*ArcTan[a*x]^2 + (201600*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 203264*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + 161280*(1 + a^2*x^2)^2*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 49280*(1 + a^2*x^2)^3*ArcTan[a*x]*Cos[3*ArcTan[a*x]] - 7658*(1 + a^2*x^2)^4*ArcTan[a*x]*Cos[3*ArcTan[a*x]] - 40320*(1 + a^2*x^2)^3*ArcTan[a*x]*Cos[5*ArcTan[a*x]] - 10990*(1 + a^2*x^2)^4*ArcTan[a*x]*Cos[5*ArcTan[a*x]] + 3150*(1 + a^2*x^2)^4*ArcTan[a*x]*Cos[7*ArcTan[a*x]] - (201600*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (201600*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 201600*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 201600*PolyLog[3, I*E^(I*ArcTan[a*x])] + 53760*(1 + a^2*x^2)^2*Sin[3*ArcTan[a*x]] - 48384*(1 + a^2*x^2)^3*Sin[3*ArcTan[a*x]] + 12246*(1 + a^2*x^2)^4*Sin[3*ArcTan[a*x]] - 80640*(1 + a^2*x^2)^2*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 315840*(1 + a^2*x^2)^3*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 93975*(1 + a^2*x^2)^4*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 23296*(1 + a^2*x^2)^3*Sin[5*ArcTan[a*x]] + 7678*(1 + a^2*x^2)^4*Sin[5*ArcTan[a*x]] + 20160*(1 + a^2*x^2)^3*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + 41685*(1 + a^2*x^2)^4*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + 2438*(1 + a^2*x^2)^4*Sin[7*ArcTan[a*x]] - 1575*(1 + a^2*x^2)^4*ArcTan[a*x]^2*Sin[7*ArcTan[a*x]]))/(2580480*a^3*Sqrt[1 + a^2*x^2])","A",0
325,1,1087,387,7.8785081,"\int x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2 \, dx","Integrate[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]","\frac{c^2 \sqrt{c \left(a^2 x^2+1\right)} \left(-5376 \cos \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2+6720 \cos \left(4 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2+10944 \tan ^{-1}(a x)^2-6489 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-2163 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-309 \cos \left(7 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-\frac{10815 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+6489 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+2163 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+309 \cos \left(7 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+\frac{10815 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-1266 \sin \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+360 \sin \left(4 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)-618 \sin \left(6 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+6262 \cos \left(2 \tan ^{-1}(a x)\right)+2764 \cos \left(4 \tan ^{-1}(a x)\right)+618 \cos \left(6 \tan ^{-1}(a x)\right)-\frac{19776 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{7/2}}+\frac{19776 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{7/2}}+4116\right) \left(a^2 x^2+1\right)^3}{161280 a^2}-\frac{c^2 \sqrt{c \left(a^2 x^2+1\right)} \left(160 \cos \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2-32 \tan ^{-1}(a x)^2-55 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-11 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-\frac{110 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+55 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+11 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+\frac{110 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+4 \sin \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)-22 \sin \left(4 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+72 \cos \left(2 \tan ^{-1}(a x)\right)+22 \cos \left(4 \tan ^{-1}(a x)\right)-\frac{176 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{5/2}}+\frac{176 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{5/2}}+50\right) \left(a^2 x^2+1\right)^2}{480 a^2}+\frac{c^2 \sqrt{c \left(a^2 x^2+1\right)} \left(4 \tan ^{-1}(a x)^2-\cos \left(3 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-\frac{3 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\cos \left(3 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+\frac{3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-2 \sin \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+2 \cos \left(2 \tan ^{-1}(a x)\right)-\frac{4 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}+\frac{4 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}+2\right) \left(a^2 x^2+1\right)}{12 a^2}","-\frac{5 i c^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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} \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{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{\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,"(c^2*(1 + a^2*x^2)*Sqrt[c*(1 + a^2*x^2)]*(2 + 4*ArcTan[a*x]^2 + 2*Cos[2*ArcTan[a*x]] - (3*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (3*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((4*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) + ((4*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) - 2*ArcTan[a*x]*Sin[2*ArcTan[a*x]]))/(12*a^2) - (c^2*(1 + a^2*x^2)^2*Sqrt[c*(1 + a^2*x^2)]*(50 - 32*ArcTan[a*x]^2 + 72*Cos[2*ArcTan[a*x]] + 160*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 22*Cos[4*ArcTan[a*x]] - (110*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (110*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((176*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + ((176*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + 4*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 22*ArcTan[a*x]*Sin[4*ArcTan[a*x]]))/(480*a^2) + (c^2*(1 + a^2*x^2)^3*Sqrt[c*(1 + a^2*x^2)]*(4116 + 10944*ArcTan[a*x]^2 + 6262*Cos[2*ArcTan[a*x]] - 5376*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 2764*Cos[4*ArcTan[a*x]] + 6720*ArcTan[a*x]^2*Cos[4*ArcTan[a*x]] + 618*Cos[6*ArcTan[a*x]] - (10815*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 6489*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 2163*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 309*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (10815*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 6489*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 2163*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 309*ArcTan[a*x]*Cos[7*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((19776*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(7/2) + ((19776*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(7/2) - 1266*ArcTan[a*x]*Sin[2*ArcTan[a*x]] + 360*ArcTan[a*x]*Sin[4*ArcTan[a*x]] - 618*ArcTan[a*x]*Sin[6*ArcTan[a*x]]))/(161280*a^2)","B",0
326,1,771,516,1.6556652,"\int \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2 \, dx","Integrate[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]","\frac{c^2 \sqrt{a^2 c x^2+c} \left(-108 a^6 x^6 \sin \left(3 \tan ^{-1}(a x)\right)-705 a^6 x^6 \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)-52 a^6 x^6 \sin \left(5 \tan ^{-1}(a x)\right)+45 a^6 x^6 \tan ^{-1}(a x)^2 \sin \left(5 \tan ^{-1}(a x)\right)+110 a^6 x^6 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)-90 a^6 x^6 \tan ^{-1}(a x) \cos \left(5 \tan ^{-1}(a x)\right)+156 a^4 x^4 \sin \left(3 \tan ^{-1}(a x)\right)-2835 a^4 x^4 \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)-156 a^4 x^4 \sin \left(5 \tan ^{-1}(a x)\right)+135 a^4 x^4 \tan ^{-1}(a x)^2 \sin \left(5 \tan ^{-1}(a x)\right)+1770 a^4 x^4 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)-270 a^4 x^4 \tan ^{-1}(a x) \cos \left(5 \tan ^{-1}(a x)\right)+424 a x \sqrt{a^2 x^2+1}+504 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+11970 a x \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2-11028 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+8288 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+636 a^2 x^2 \sin \left(3 \tan ^{-1}(a x)\right)-3555 a^2 x^2 \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)-156 a^2 x^2 \sin \left(5 \tan ^{-1}(a x)\right)+135 a^2 x^2 \tan ^{-1}(a x)^2 \sin \left(5 \tan ^{-1}(a x)\right)+3210 a^2 x^2 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)-270 a^2 x^2 \tan ^{-1}(a x) \cos \left(5 \tan ^{-1}(a x)\right)-56 a^5 x^5 \sqrt{a^2 x^2+1}+1170 a^5 x^5 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+12 a^4 x^4 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+368 a^3 x^3 \sqrt{a^2 x^2+1}+7380 a^3 x^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+7200 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-7200 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-7200 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+7200 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-7200 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2-1425 \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)+372 \sin \left(3 \tan ^{-1}(a x)\right)+45 \tan ^{-1}(a x)^2 \sin \left(5 \tan ^{-1}(a x)\right)-52 \sin \left(5 \tan ^{-1}(a x)\right)+1550 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)-90 \tan ^{-1}(a x) \cos \left(5 \tan ^{-1}(a x)\right)\right)}{11520 a \sqrt{a^2 x^2+1}}","\frac{259 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{360 a}+\frac{5 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{8 a \sqrt{a^2 c x^2+c}}+\frac{17}{180} c^2 x \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{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,"(c^2*Sqrt[c + a^2*c*x^2]*(424*a*x*Sqrt[1 + a^2*x^2] + 368*a^3*x^3*Sqrt[1 + a^2*x^2] - 56*a^5*x^5*Sqrt[1 + a^2*x^2] - 11028*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 504*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 12*a^4*x^4*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 11970*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + 7380*a^3*x^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + 1170*a^5*x^5*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 - (7200*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 8288*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + 1550*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 3210*a^2*x^2*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 1770*a^4*x^4*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 110*a^6*x^6*ArcTan[a*x]*Cos[3*ArcTan[a*x]] - 90*ArcTan[a*x]*Cos[5*ArcTan[a*x]] - 270*a^2*x^2*ArcTan[a*x]*Cos[5*ArcTan[a*x]] - 270*a^4*x^4*ArcTan[a*x]*Cos[5*ArcTan[a*x]] - 90*a^6*x^6*ArcTan[a*x]*Cos[5*ArcTan[a*x]] + (7200*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (7200*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 7200*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 7200*PolyLog[3, I*E^(I*ArcTan[a*x])] + 372*Sin[3*ArcTan[a*x]] + 636*a^2*x^2*Sin[3*ArcTan[a*x]] + 156*a^4*x^4*Sin[3*ArcTan[a*x]] - 108*a^6*x^6*Sin[3*ArcTan[a*x]] - 1425*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 3555*a^2*x^2*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 2835*a^4*x^4*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 705*a^6*x^6*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 52*Sin[5*ArcTan[a*x]] - 156*a^2*x^2*Sin[5*ArcTan[a*x]] - 156*a^4*x^4*Sin[5*ArcTan[a*x]] - 52*a^6*x^6*Sin[5*ArcTan[a*x]] + 45*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + 135*a^2*x^2*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + 135*a^4*x^4*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + 45*a^6*x^6*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]]))/(11520*a*Sqrt[1 + a^2*x^2])","A",0
327,1,889,605,7.1359222,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2}{x} \, dx","Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/x,x]","\sqrt{c \left(a^2 x^2+1\right)} \left(\frac{\left(\log \left(1-e^{i \tan ^{-1}(a x)}\right)-\log \left(1+e^{i \tan ^{-1}(a x)}\right)\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\tan ^{-1}(a x)^2+\frac{2 i \left(\text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-\text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{2 \left(\tan ^{-1}(a x) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)+i \left(\text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)\right)\right)}{\sqrt{a^2 x^2+1}}+\frac{2 \left(\text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)-\text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)\right)}{\sqrt{a^2 x^2+1}}\right) c^2+\frac{1}{6} \left(a^2 x^2+1\right) \sqrt{c \left(a^2 x^2+1\right)} \left(4 \tan ^{-1}(a x)^2-\cos \left(3 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-\frac{3 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\cos \left(3 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+\frac{3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-2 \sin \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+2 \cos \left(2 \tan ^{-1}(a x)\right)-\frac{4 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}+\frac{4 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}+2\right) c^2-\frac{1}{960} \left(a^2 x^2+1\right)^2 \sqrt{c \left(a^2 x^2+1\right)} \left(160 \cos \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2-32 \tan ^{-1}(a x)^2-55 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-11 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-\frac{110 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+55 \cos \left(3 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+11 \cos \left(5 \tan ^{-1}(a x)\right) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+\frac{110 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+4 \sin \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)-22 \sin \left(4 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+72 \cos \left(2 \tan ^{-1}(a x)\right)+22 \cos \left(4 \tan ^{-1}(a x)\right)-\frac{176 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{5/2}}+\frac{176 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{5/2}}+50\right) c^2","-\frac{149 i c^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)}{\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}}-\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{29}{60} c^2 \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{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,"c^2*Sqrt[c*(1 + a^2*x^2)]*(ArcTan[a*x]^2 + (ArcTan[a*x]^2*(Log[1 - E^(I*ArcTan[a*x])] - Log[1 + E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] - (2*(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])])))/Sqrt[1 + a^2*x^2] + ((2*I)*ArcTan[a*x]*(PolyLog[2, -E^(I*ArcTan[a*x])] - PolyLog[2, E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] + (2*(-PolyLog[3, -E^(I*ArcTan[a*x])] + PolyLog[3, E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2]) + (c^2*(1 + a^2*x^2)*Sqrt[c*(1 + a^2*x^2)]*(2 + 4*ArcTan[a*x]^2 + 2*Cos[2*ArcTan[a*x]] - (3*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (3*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((4*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) + ((4*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) - 2*ArcTan[a*x]*Sin[2*ArcTan[a*x]]))/6 - (c^2*(1 + a^2*x^2)^2*Sqrt[c*(1 + a^2*x^2)]*(50 - 32*ArcTan[a*x]^2 + 72*Cos[2*ArcTan[a*x]] + 160*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 22*Cos[4*ArcTan[a*x]] - (110*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (110*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 11*ArcTan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((176*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + ((176*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(5/2) + 4*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 22*ArcTan[a*x]*Sin[4*ArcTan[a*x]]))/960","A",0
328,1,626,655,1.7935738,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2}{x^2} \, dx","Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/x^2,x]","\frac{c^2 \sqrt{a^2 c x^2+c} \left(2 a^5 x^5 \sin \left(3 \tan ^{-1}(a x)\right)-3 a^5 x^5 \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)+6 a^5 x^5 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)+4 a^3 x^3 \sin \left(3 \tan ^{-1}(a x)\right)-6 a^3 x^3 \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)+12 a^3 x^3 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)+2 a^2 x^2 \sqrt{a^2 x^2+1}+117 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2-190 a x \sqrt{a^2 x^2+1} \tan ^{-1}(a x)-96 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+176 a x \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+2 a^4 x^4 \sqrt{a^2 x^2+1}+21 a^4 x^4 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+2 a^3 x^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+192 i a x \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+360 i a x \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-360 i a x \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-192 i a x \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-360 a x \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+360 a x \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-168 i a x \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2+192 a x \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)+96 a x \tan ^{-1}(a x)^2 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)-96 a x \tan ^{-1}(a x)^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-192 a x \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)-3 a x \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)+2 a x \sin \left(3 \tan ^{-1}(a x)\right)+6 a x \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)\right)}{96 x \sqrt{a^2 x^2+1}}","\frac{11}{6} a c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)+\frac{2 i a c^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{4 \sqrt{a^2 c x^2+c}}-\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}{12} a^2 c^2 x \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{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,"(c^2*Sqrt[c + a^2*c*x^2]*(2*a^2*x^2*Sqrt[1 + a^2*x^2] + 2*a^4*x^4*Sqrt[1 + a^2*x^2] - 190*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 2*a^3*x^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x] - 96*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + 117*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + 21*a^4*x^4*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 - (168*I)*a*x*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 176*a*x*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + 6*a*x*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 12*a^3*x^3*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 6*a^5*x^5*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 192*a*x*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] + 96*a*x*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] - 96*a*x*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] - 192*a*x*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + (192*I)*a*x*PolyLog[2, -E^(I*ArcTan[a*x])] + (360*I)*a*x*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (360*I)*a*x*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - (192*I)*a*x*PolyLog[2, E^(I*ArcTan[a*x])] - 360*a*x*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 360*a*x*PolyLog[3, I*E^(I*ArcTan[a*x])] + 2*a*x*Sin[3*ArcTan[a*x]] + 4*a^3*x^3*Sin[3*ArcTan[a*x]] + 2*a^5*x^5*Sin[3*ArcTan[a*x]] - 3*a*x*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 6*a^3*x^3*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 3*a^5*x^5*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]]))/(96*x*Sqrt[1 + a^2*x^2])","A",0
329,1,761,661,7.5891975,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2}{x^3} \, dx","Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/x^3,x]","2 a^2 c^2 \sqrt{c \left(a^2 x^2+1\right)} \left(\frac{2 i \tan ^{-1}(a x) \left(\text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-\text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)\right)}{\sqrt{a^2 x^2+1}}+\frac{2 \left(\text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)-\text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)\right)}{\sqrt{a^2 x^2+1}}-\frac{2 \left(i \left(\text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)\right)+\tan ^{-1}(a x) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)\right)}{\sqrt{a^2 x^2+1}}+\frac{\tan ^{-1}(a x)^2 \left(\log \left(1-e^{i \tan ^{-1}(a x)}\right)-\log \left(1+e^{i \tan ^{-1}(a x)}\right)\right)}{\sqrt{a^2 x^2+1}}+\tan ^{-1}(a x)^2\right)+\frac{1}{12} a^2 c^2 \left(a^2 x^2+1\right) \sqrt{c \left(a^2 x^2+1\right)} \left(-\frac{4 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}+\frac{4 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}-\frac{3 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{3 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+4 \tan ^{-1}(a x)^2-2 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)+2 \cos \left(2 \tan ^{-1}(a x)\right)-\tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \cos \left(3 \tan ^{-1}(a x)\right)+\tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \cos \left(3 \tan ^{-1}(a x)\right)+2\right)+\frac{a^2 c^2 \sqrt{c \left(a^2 x^2+1\right)} \left(8 i \tan ^{-1}(a x) \left(\text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-\text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)\right)+8 \left(\text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)-\text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)\right)-4 \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+4 \tan ^{-1}(a x)^2 \left(\log \left(1-e^{i \tan ^{-1}(a x)}\right)-\log \left(1+e^{i \tan ^{-1}(a x)}\right)\right)+8 \log \left(\tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-4 \tan ^{-1}(a x) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+\tan ^{-1}(a x)^2 \left(-\csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)+\tan ^{-1}(a x)^2 \sec ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}{8 \sqrt{a^2 x^2+1}}","-a^2 c^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)-\frac{13 i a^2 c^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\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{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^2 \sqrt{a^2 c x^2+c}+2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\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}+\frac{1}{3} a^2 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2-\frac{1}{3} a^3 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)",1,"2*a^2*c^2*Sqrt[c*(1 + a^2*x^2)]*(ArcTan[a*x]^2 + (ArcTan[a*x]^2*(Log[1 - E^(I*ArcTan[a*x])] - Log[1 + E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] - (2*(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])])))/Sqrt[1 + a^2*x^2] + ((2*I)*ArcTan[a*x]*(PolyLog[2, -E^(I*ArcTan[a*x])] - PolyLog[2, E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] + (2*(-PolyLog[3, -E^(I*ArcTan[a*x])] + PolyLog[3, E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2]) + (a^2*c^2*(1 + a^2*x^2)*Sqrt[c*(1 + a^2*x^2)]*(2 + 4*ArcTan[a*x]^2 + 2*Cos[2*ArcTan[a*x]] - (3*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (3*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((4*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) + ((4*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) - 2*ArcTan[a*x]*Sin[2*ArcTan[a*x]]))/12 + (a^2*c^2*Sqrt[c*(1 + a^2*x^2)]*(-4*ArcTan[a*x]*Cot[ArcTan[a*x]/2] - ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 + 4*ArcTan[a*x]^2*(Log[1 - E^(I*ArcTan[a*x])] - Log[1 + E^(I*ArcTan[a*x])]) + 8*Log[Tan[ArcTan[a*x]/2]] + (8*I)*ArcTan[a*x]*(PolyLog[2, -E^(I*ArcTan[a*x])] - PolyLog[2, E^(I*ArcTan[a*x])]) + 8*(-PolyLog[3, -E^(I*ArcTan[a*x])] + PolyLog[3, E^(I*ArcTan[a*x])]) + ArcTan[a*x]^2*Sec[ArcTan[a*x]/2]^2 - 4*ArcTan[a*x]*Tan[ArcTan[a*x]/2]))/(8*Sqrt[1 + a^2*x^2])","A",0
330,1,644,675,4.7572859,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2}{x^4} \, dx","Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2)/x^4,x]","-\frac{c^3 \sqrt{a^2 x^2+1} \left(-52 i a^3 x^3 \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-60 i a^3 x^3 \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+60 i a^3 x^3 \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+52 i a^3 x^3 \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+60 a^3 x^3 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-60 a^3 x^3 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+12 i a^3 x^3 \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2-51 a^3 x^3 \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)-24 a^3 x^3 \tan ^{-1}(a x)^2 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)+24 a^3 x^3 \tan ^{-1}(a x)^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+51 a^3 x^3 \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)+2 \left(a^2 x^2+1\right)^{3/2}+24 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+4 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^2+2 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)-2 \left(a^2 x^2+1\right)^{3/2} \cos \left(2 \tan ^{-1}(a x)\right)+\left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right) \sin \left(3 \tan ^{-1}(a x)\right)-\left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right) \sin \left(3 \tan ^{-1}(a x)\right)-6 a^4 x^4 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+12 a^3 x^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)-12 a^3 x^3 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-3 a x \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)+3 a x \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)\right)}{12 x^3 \sqrt{a^2 c x^2+c}}","-\frac{a^2 c^2 \sqrt{a^2 c x^2+c}}{3 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}-\frac{c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}{3 x^3}+\frac{1}{2} a^4 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+a^3 c^{5/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)+\frac{13 i a^3 c^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 c x^2+c}}-\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}}-a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)",1,"-1/12*(c^3*Sqrt[1 + a^2*x^2]*(2*(1 + a^2*x^2)^(3/2) + 12*a^3*x^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 24*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 - 6*a^4*x^4*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + 4*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2 + (12*I)*a^3*x^3*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 12*a^3*x^3*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - 2*(1 + a^2*x^2)^(3/2)*Cos[2*ArcTan[a*x]] - 3*a*x*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 51*a^3*x^3*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 24*a^3*x^3*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] + 24*a^3*x^3*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] + 3*a*x*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + 51*a^3*x^3*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] - (52*I)*a^3*x^3*PolyLog[2, -E^(I*ArcTan[a*x])] - (60*I)*a^3*x^3*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (60*I)*a^3*x^3*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + (52*I)*a^3*x^3*PolyLog[2, E^(I*ArcTan[a*x])] + 60*a^3*x^3*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 60*a^3*x^3*PolyLog[3, I*E^(I*ArcTan[a*x])] + 2*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]*Sin[2*ArcTan[a*x]] + (1 + a^2*x^2)^(3/2)*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])]*Sin[3*ArcTan[a*x]] - (1 + a^2*x^2)^(3/2)*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])]*Sin[3*ArcTan[a*x]]))/(x^3*Sqrt[c + a^2*c*x^2])","A",0
331,1,279,315,0.6995704,"\int \frac{x^3 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(x^3*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2],x]","\frac{\left(a^2 x^2+1\right) \sqrt{c \left(a^2 x^2+1\right)} \left(\frac{20 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}-\frac{20 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}+\frac{15 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{15 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-2 \tan ^{-1}(a x)^2-2 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)-6 \tan ^{-1}(a x)^2 \cos \left(2 \tan ^{-1}(a x)\right)+2 \cos \left(2 \tan ^{-1}(a x)\right)+5 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \cos \left(3 \tan ^{-1}(a x)\right)-5 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \cos \left(3 \tan ^{-1}(a x)\right)+2\right)}{12 a^4 c}","\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 a^2 c}+\frac{5 i \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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{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,"((1 + a^2*x^2)*Sqrt[c*(1 + a^2*x^2)]*(2 - 2*ArcTan[a*x]^2 + 2*Cos[2*ArcTan[a*x]] - 6*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + (15*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 5*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - (15*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 5*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + ((20*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) - ((20*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) - 2*ArcTan[a*x]*Sin[2*ArcTan[a*x]]))/(12*a^4*c)","A",0
332,1,175,344,0.3586599,"\int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(x^2*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2],x]","\frac{\sqrt{a^2 c x^2+c} \left(\tan ^{-1}(a x) \left(a x \tan ^{-1}(a x)-2\right)+\frac{2 \left(\tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+\text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)}{\sqrt{a^2 x^2+1}}\right)}{2 a^3 c}","\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{\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]*(-2 + a*x*ArcTan[a*x]) + (2*(I*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + I*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2]))/(2*a^3*c)","A",0
333,1,126,220,0.2484861,"\int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(x*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2],x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(\tan ^{-1}(a x)^2-\frac{2 \left(i \left(\text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)\right)+\tan ^{-1}(a x) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)\right)}{\sqrt{a^2 x^2+1}}\right)}{a^2 c}","-\frac{2 i \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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*(1 + a^2*x^2)]*(ArcTan[a*x]^2 - (2*(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])])))/Sqrt[1 + a^2*x^2]))/(a^2*c)","A",0
334,1,140,256,0.1179767,"\int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]^2/Sqrt[c + a^2*c*x^2],x]","\frac{2 \sqrt{c \left(a^2 x^2+1\right)} \left(i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+\text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)}{a c \sqrt{a^2 x^2+1}}","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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*Sqrt[c*(1 + a^2*x^2)]*((-I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - I*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + PolyLog[3, I*E^(I*ArcTan[a*x])]))/(a*c*Sqrt[1 + a^2*x^2])","A",1
335,1,145,227,0.169403,"\int \frac{\tan ^{-1}(a x)^2}{x \sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]^2/(x*Sqrt[c + a^2*c*x^2]),x]","\frac{\sqrt{a^2 x^2+1} \left(2 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-2 i \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-2 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+2 \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)-\tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)\right)}{\sqrt{c \left(a^2 x^2+1\right)}}","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\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{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{Li}_3\left(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,"(Sqrt[1 + a^2*x^2]*(ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] - ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + (2*I)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - (2*I)*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - 2*PolyLog[3, -E^(I*ArcTan[a*x])] + 2*PolyLog[3, E^(I*ArcTan[a*x])]))/Sqrt[c*(1 + a^2*x^2)]","A",0
336,1,128,208,0.430151,"\int \frac{\tan ^{-1}(a x)^2}{x^2 \sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]^2/(x^2*Sqrt[c + a^2*c*x^2]),x]","-\frac{a \sqrt{a^2 x^2+1} \left(\tan ^{-1}(a x) \left(\frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x)}{a x}-2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)+2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)\right)-2 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+2 i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)\right)}{\sqrt{c \left(a^2 x^2+1\right)}}","\frac{2 i a \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{\sqrt{a^2 c x^2+c}}-\frac{2 i a \sqrt{a^2 x^2+1} \text{Li}_2\left(\frac{\sqrt{i a x+1}}{\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,"-((a*Sqrt[1 + a^2*x^2]*(ArcTan[a*x]*((Sqrt[1 + a^2*x^2]*ArcTan[a*x])/(a*x) - 2*Log[1 - E^(I*ArcTan[a*x])] + 2*Log[1 + E^(I*ArcTan[a*x])]) - (2*I)*PolyLog[2, -E^(I*ArcTan[a*x])] + (2*I)*PolyLog[2, E^(I*ArcTan[a*x])]))/Sqrt[c*(1 + a^2*x^2)])","A",0
337,1,231,328,1.2341561,"\int \frac{\tan ^{-1}(a x)^2}{x^3 \sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]^2/(x^3*Sqrt[c + a^2*c*x^2]),x]","\frac{a^2 \sqrt{a^2 x^2+1} \left(-8 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+8 i \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+8 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)-8 \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)-4 \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)-4 \tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)+4 \tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)+8 \log \left(\tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-4 \tan ^{-1}(a x) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+\tan ^{-1}(a x)^2 \left(-\csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)+\tan ^{-1}(a x)^2 \sec ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}{8 \sqrt{c \left(a^2 x^2+1\right)}}","-\frac{i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\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{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{a^2 \sqrt{a^2 x^2+1} \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 x^2+1} \text{Li}_3\left(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^2*Sqrt[1 + a^2*x^2]*(-4*ArcTan[a*x]*Cot[ArcTan[a*x]/2] - ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 - 4*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] + 4*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 8*Log[Tan[ArcTan[a*x]/2]] - (8*I)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] + (8*I)*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] + 8*PolyLog[3, -E^(I*ArcTan[a*x])] - 8*PolyLog[3, E^(I*ArcTan[a*x])] + ArcTan[a*x]^2*Sec[ArcTan[a*x]/2]^2 - 4*ArcTan[a*x]*Tan[ArcTan[a*x]/2]))/(8*Sqrt[c*(1 + a^2*x^2)])","A",0
338,1,228,311,2.7005721,"\int \frac{\tan ^{-1}(a x)^2}{x^4 \sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]^2/(x^4*Sqrt[c + a^2*c*x^2]),x]","\frac{a^3 \sqrt{a^2 c x^2+c} \left(\frac{\left(a^2 x^2+1\right)^{3/2} \left(\tan ^{-1}(a x) \left(-2 \sin \left(2 \tan ^{-1}(a x)\right)+\frac{5 \left(\log \left(1-e^{i \tan ^{-1}(a x)}\right)-\log \left(1+e^{i \tan ^{-1}(a x)}\right)\right) \left(\sqrt{a^2 x^2+1} \sin \left(3 \tan ^{-1}(a x)\right)-3 a x\right)}{\sqrt{a^2 x^2+1}}\right)+\frac{20 i a^3 x^3 \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}+\tan ^{-1}(a x)^2 \left(2-6 \cos \left(2 \tan ^{-1}(a x)\right)\right)+2 \left(\cos \left(2 \tan ^{-1}(a x)\right)-1\right)\right)}{a^3 x^3}-20 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)\right)}{12 c \sqrt{a^2 x^2+1}}","-\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{5 i a^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{3 \sqrt{a^2 c x^2+c}}+\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^3*Sqrt[c + a^2*c*x^2]*((-20*I)*PolyLog[2, -E^(I*ArcTan[a*x])] + ((1 + a^2*x^2)^(3/2)*(ArcTan[a*x]^2*(2 - 6*Cos[2*ArcTan[a*x]]) + 2*(-1 + Cos[2*ArcTan[a*x]]) + ((20*I)*a^3*x^3*PolyLog[2, E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) + ArcTan[a*x]*(-2*Sin[2*ArcTan[a*x]] + (5*(Log[1 - E^(I*ArcTan[a*x])] - Log[1 + E^(I*ArcTan[a*x])])*(-3*a*x + Sqrt[1 + a^2*x^2]*Sin[3*ArcTan[a*x]]))/Sqrt[1 + a^2*x^2])))/(a^3*x^3)))/(12*c*Sqrt[1 + a^2*x^2])","A",0
339,1,209,305,0.8668394,"\int \frac{x^3 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2),x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(-\frac{4 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{4 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{4 \tan ^{-1}(a x) \log \left(1-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{4 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+3 \tan ^{-1}(a x)^2-2 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)+\tan ^{-1}(a x)^2 \cos \left(2 \tan ^{-1}(a x)\right)-2 \cos \left(2 \tan ^{-1}(a x)\right)-2\right)}{2 a^4 c^2}","\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{a^4 c^2}-\frac{2 i \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{a^4 c \sqrt{a^2 c x^2+c}}-\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,"(Sqrt[c*(1 + a^2*x^2)]*(-2 + 3*ArcTan[a*x]^2 - 2*Cos[2*ArcTan[a*x]] + ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] - (4*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (4*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((4*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((4*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 2*ArcTan[a*x]*Sin[2*ArcTan[a*x]]))/(2*a^4*c^2)","A",0
340,1,228,349,0.3608084,"\int \frac{x^2 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(x^2*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2),x]","-\frac{\sqrt{a^2 x^2+1} \left(-\frac{2 a x}{\sqrt{a^2 x^2+1}}+\frac{a x \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{2 \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-2 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+2 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+2 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-2 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x)^2 \left(-\log \left(1-i e^{i \tan ^{-1}(a x)}\right)\right)+\tan ^{-1}(a x)^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)}{a^3 c \sqrt{c \left(a^2 x^2+1\right)}}","\frac{2 x}{a^2 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 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{a^3 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,"-((Sqrt[1 + a^2*x^2]*((-2*a*x)/Sqrt[1 + a^2*x^2] + (2*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + (a*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] - ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] + ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] - (2*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (2*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 2*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 2*PolyLog[3, I*E^(I*ArcTan[a*x])]))/(a^3*c*Sqrt[c*(1 + a^2*x^2)]))","A",0
341,1,50,78,0.0778516,"\int \frac{x \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(-\tan ^{-1}(a x)^2+2 a x \tan ^{-1}(a x)+2\right)}{a^2 c^2 \left(a^2 x^2+1\right)}","\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,"(Sqrt[c + a^2*c*x^2]*(2 + 2*a*x*ArcTan[a*x] - ArcTan[a*x]^2))/(a^2*c^2*(1 + a^2*x^2))","A",1
342,1,49,72,0.0617921,"\int \frac{\tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]^2/(c + a^2*c*x^2)^(3/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(-2 a x+a x \tan ^{-1}(a x)^2+2 \tan ^{-1}(a x)\right)}{c^2 \left(a^3 x^2+a\right)}","-\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,"(Sqrt[c + a^2*c*x^2]*(-2*a*x + 2*ArcTan[a*x] + a*x*ArcTan[a*x]^2))/(c^2*(a + a^3*x^2))","A",1
343,1,204,310,0.3124773,"\int \frac{\tan ^{-1}(a x)^2}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)^(3/2)),x]","\frac{\sqrt{a^2 x^2+1} \left(-\frac{2}{\sqrt{a^2 x^2+1}}+\frac{\tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{2 a x \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+2 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-2 i \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-2 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+2 \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)-\tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)\right)}{c \sqrt{c \left(a^2 x^2+1\right)}}","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\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{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \text{Li}_3\left(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,"(Sqrt[1 + a^2*x^2]*(-2/Sqrt[1 + a^2*x^2] - (2*a*x*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + ArcTan[a*x]^2/Sqrt[1 + a^2*x^2] + ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] - ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + (2*I)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - (2*I)*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - 2*PolyLog[3, -E^(I*ArcTan[a*x])] + 2*PolyLog[3, E^(I*ArcTan[a*x])]))/(c*Sqrt[c*(1 + a^2*x^2)])","A",0
344,1,226,293,1.1164691,"\int \frac{\tan ^{-1}(a x)^2}{x^2 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)^(3/2)),x]","\frac{a \left(4 i \sqrt{a^2 x^2+1} \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-4 i \sqrt{a^2 x^2+1} \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+4 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)-4 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)-\frac{2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2 \sin ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)}{a x}+4 a x-2 a x \tan ^{-1}(a x)^2-4 \tan ^{-1}(a x)-\frac{1}{2} a x \tan ^{-1}(a x)^2 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}{2 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 i a \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{c \sqrt{a^2 c x^2+c}}+\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,"(a*(4*a*x - 4*ArcTan[a*x] - 2*a*x*ArcTan[a*x]^2 - (a*x*ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2)/2 + 4*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 4*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + (4*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, -E^(I*ArcTan[a*x])] - (4*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, E^(I*ArcTan[a*x])] - (2*(1 + a^2*x^2)*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]^2)/(a*x)))/(2*c*Sqrt[c + a^2*c*x^2])","A",0
345,1,371,422,2.3854669,"\int \frac{\tan ^{-1}(a x)^2}{x^3 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]^2/(x^3*(c + a^2*c*x^2)^(3/2)),x]","\frac{a^2 \left(-24 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+24 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+24 \sqrt{a^2 x^2+1} \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)-24 \sqrt{a^2 x^2+1} \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)-4 \sqrt{a^2 x^2+1} \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)-12 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)+12 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)+8 \sqrt{a^2 x^2+1} \log \left(\tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-\sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+\sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \sec ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)-8 \tan ^{-1}(a x)^2+16 a x \tan ^{-1}(a x)-2 a x \tan ^{-1}(a x) \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+16\right)}{8 c \sqrt{a^2 c x^2+c}}","-\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)}{c^{3/2}}-\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{3 i a^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\frac{2 a^2}{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{2 a^3 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}",1,"(a^2*(16 + 16*a*x*ArcTan[a*x] - 8*ArcTan[a*x]^2 - 2*a*x*ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 - Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 - 12*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] + 12*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 8*Sqrt[1 + a^2*x^2]*Log[Tan[ArcTan[a*x]/2]] - (24*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] + (24*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] + 24*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])] - 24*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])] + Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Sec[ArcTan[a*x]/2]^2 - 4*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Tan[ArcTan[a*x]/2]))/(8*c*Sqrt[c + a^2*c*x^2])","A",0
346,1,270,397,3.5981402,"\int \frac{\tan ^{-1}(a x)^2}{x^4 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)^(3/2)),x]","\frac{a^3 \sqrt{a^2 x^2+1} \left(\frac{\left(a^2 x^2+1\right)^{3/2} \left(\tan ^{-1}(a x) \left(\frac{66 a x \left(\log \left(1+e^{i \tan ^{-1}(a x)}\right)-\log \left(1-e^{i \tan ^{-1}(a x)}\right)\right)}{\sqrt{a^2 x^2+1}}+8 \sin \left(2 \tan ^{-1}(a x)\right)-6 \sin \left(4 \tan ^{-1}(a x)\right)+22 \left(\log \left(1-e^{i \tan ^{-1}(a x)}\right)-\log \left(1+e^{i \tan ^{-1}(a x)}\right)\right) \sin \left(3 \tan ^{-1}(a x)\right)\right)+\frac{88 i a^3 x^3 \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)}{\left(a^2 x^2+1\right)^{3/2}}+28 \cos \left(2 \tan ^{-1}(a x)\right)-6 \cos \left(4 \tan ^{-1}(a x)\right)+\tan ^{-1}(a x)^2 \left(-36 \cos \left(2 \tan ^{-1}(a x)\right)+3 \cos \left(4 \tan ^{-1}(a x)\right)+25\right)-22\right)}{a^3 x^3}-88 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)\right)}{24 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{11 i a^3 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right)}{3 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,"(a^3*Sqrt[1 + a^2*x^2]*((-88*I)*PolyLog[2, -E^(I*ArcTan[a*x])] + ((1 + a^2*x^2)^(3/2)*(-22 + 28*Cos[2*ArcTan[a*x]] - 6*Cos[4*ArcTan[a*x]] + ArcTan[a*x]^2*(25 - 36*Cos[2*ArcTan[a*x]] + 3*Cos[4*ArcTan[a*x]]) + ((88*I)*a^3*x^3*PolyLog[2, E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) + ArcTan[a*x]*((66*a*x*(-Log[1 - E^(I*ArcTan[a*x])] + Log[1 + E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] + 8*Sin[2*ArcTan[a*x]] + 22*(Log[1 - E^(I*ArcTan[a*x])] - Log[1 + E^(I*ArcTan[a*x])])*Sin[3*ArcTan[a*x]] - 6*Sin[4*ArcTan[a*x]])))/(a^3*x^3)))/(24*c*Sqrt[c + a^2*c*x^2])","A",0
347,1,229,400,1.3437937,"\int \frac{x^5 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^5*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2),x]","\frac{-432 i \sqrt{a^2 x^2+1} \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+432 i \sqrt{a^2 x^2+1} \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-9 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2 \left(-20 \cos \left(2 \tan ^{-1}(a x)\right)+\cos \left(4 \tan ^{-1}(a x)\right)-45\right)+6 \tan ^{-1}(a x) \left(-72 \sqrt{a^2 x^2+1} \log \left(1-i e^{i \tan ^{-1}(a x)}\right)+72 \sqrt{a^2 x^2+1} \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+\left(a^2 x^2+1\right) \sin \left(4 \tan ^{-1}(a x)\right)-124 a x\right)+8 \left(\cos \left(2 \tan ^{-1}(a x)\right)-95\right)}{216 a^6 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{2 i \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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{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{10 x \tan ^{-1}(a x)}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 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}}",1,"(8*(-95 + Cos[2*ArcTan[a*x]]) - 9*(1 + a^2*x^2)*ArcTan[a*x]^2*(-45 - 20*Cos[2*ArcTan[a*x]] + Cos[4*ArcTan[a*x]]) - (432*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (432*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 6*ArcTan[a*x]*(-124*a*x - 72*Sqrt[1 + a^2*x^2]*Log[1 - I*E^(I*ArcTan[a*x])] + 72*Sqrt[1 + a^2*x^2]*Log[1 + I*E^(I*ArcTan[a*x])] + (1 + a^2*x^2)*Sin[4*ArcTan[a*x]]))/(216*a^6*c^2*Sqrt[c + a^2*c*x^2])","A",0
348,1,239,444,0.5860831,"\int \frac{x^4 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^4*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(-\frac{270 \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{135 a x \left(\tan ^{-1}(a x)^2-2\right)}{\sqrt{a^2 x^2+1}}+216 i \tan ^{-1}(a x) \left(\text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)\right)-216 \left(\text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)\right)+108 \tan ^{-1}(a x)^2 \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)+\left(9 \tan ^{-1}(a x)^2-2\right) \sin \left(3 \tan ^{-1}(a x)\right)+6 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)\right)}{108 a^5 c^3 \sqrt{a^2 x^2+1}}","\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 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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}\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{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 x^2 \tan ^{-1}(a x)}{9 a^3 c \left(a^2 c x^2+c\right)^{3/2}}",1,"(Sqrt[c*(1 + a^2*x^2)]*((-270*ArcTan[a*x])/Sqrt[1 + a^2*x^2] - (135*a*x*(-2 + ArcTan[a*x]^2))/Sqrt[1 + a^2*x^2] + 6*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 108*ArcTan[a*x]^2*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + (216*I)*ArcTan[a*x]*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])]) - 216*(PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])]) + (-2 + 9*ArcTan[a*x]^2)*Sin[3*ArcTan[a*x]]))/(108*a^5*c^3*Sqrt[1 + a^2*x^2])","A",0
349,1,81,172,0.1195413,"\int \frac{x^3 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^3*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(42 a^2 x^2+6 a x \left(7 a^2 x^2+6\right) \tan ^{-1}(a x)-9 \left(3 a^2 x^2+2\right) \tan ^{-1}(a x)^2+40\right)}{27 a^4 c^3 \left(a^2 x^2+1\right)^2}","-\frac{x^2 \tan ^{-1}(a x)^2}{3 a^2 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{14}{9 a^4 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{4 x \tan ^{-1}(a x)}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}",1,"(Sqrt[c + a^2*c*x^2]*(40 + 42*a^2*x^2 + 6*a*x*(6 + 7*a^2*x^2)*ArcTan[a*x] - 9*(2 + 3*a^2*x^2)*ArcTan[a*x]^2))/(27*a^4*c^3*(1 + a^2*x^2)^2)","A",1
350,1,80,139,0.0899363,"\int \frac{x^2 \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^2*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(9 a^3 x^3 \tan ^{-1}(a x)^2-2 a x \left(7 a^2 x^2+6\right)+6 \left(3 a^2 x^2+2\right) \tan ^{-1}(a x)\right)}{27 a^3 c^3 \left(a^2 x^2+1\right)^2}","-\frac{4 x}{9 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x^2 \tan ^{-1}(a x)}{9 a c \left(a^2 c x^2+c\right)^{3/2}}-\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{4 \tan ^{-1}(a x)}{9 a^3 c^2 \sqrt{a^2 c x^2+c}}",1,"(Sqrt[c + a^2*c*x^2]*(-2*a*x*(6 + 7*a^2*x^2) + 6*(2 + 3*a^2*x^2)*ArcTan[a*x] + 9*a^3*x^3*ArcTan[a*x]^2))/(27*a^3*c^3*(1 + a^2*x^2)^2)","A",1
351,1,71,137,0.0858494,"\int \frac{x \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(2 \left(6 a^2 x^2+7\right)+6 a x \left(2 a^2 x^2+3\right) \tan ^{-1}(a x)-9 \tan ^{-1}(a x)^2\right)}{27 c^3 \left(a^3 x^2+a\right)^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,"(Sqrt[c + a^2*c*x^2]*(2*(7 + 6*a^2*x^2) + 6*a*x*(3 + 2*a^2*x^2)*ArcTan[a*x] - 9*ArcTan[a*x]^2))/(27*c^3*(a + a^3*x^2)^2)","A",1
352,1,86,157,0.0757326,"\int \frac{\tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcTan[a*x]^2/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(-2 a x \left(20 a^2 x^2+21\right)+9 a x \left(2 a^2 x^2+3\right) \tan ^{-1}(a x)^2+6 \left(6 a^2 x^2+7\right) \tan ^{-1}(a x)\right)}{27 a c^3 \left(a^2 x^2+1\right)^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,"(Sqrt[c + a^2*c*x^2]*(-2*a*x*(21 + 20*a^2*x^2) + 6*(7 + 6*a^2*x^2)*ArcTan[a*x] + 9*a*x*(3 + 2*a^2*x^2)*ArcTan[a*x]^2))/(27*a*c^3*(1 + a^2*x^2)^2)","A",1
353,1,246,389,0.5156296,"\int \frac{\tan ^{-1}(a x)^2}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcTan[a*x]^2/(x*(c + a^2*c*x^2)^(5/2)),x]","\frac{\left(a^2 x^2+1\right)^{3/2} \left(-\frac{270}{\sqrt{a^2 x^2+1}}+\frac{135 \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{270 a x \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+216 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-216 i \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-216 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+216 \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)+108 \tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)-108 \tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)-6 \tan ^{-1}(a x) \sin \left(3 \tan ^{-1}(a x)\right)+9 \tan ^{-1}(a x)^2 \cos \left(3 \tan ^{-1}(a x)\right)-2 \cos \left(3 \tan ^{-1}(a x)\right)\right)}{108 c \left(c \left(a^2 x^2+1\right)\right)^{3/2}}","\frac{2 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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,"((1 + a^2*x^2)^(3/2)*(-270/Sqrt[1 + a^2*x^2] - (270*a*x*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + (135*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] - 2*Cos[3*ArcTan[a*x]] + 9*ArcTan[a*x]^2*Cos[3*ArcTan[a*x]] + 108*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] - 108*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + (216*I)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - (216*I)*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - 216*PolyLog[3, -E^(I*ArcTan[a*x])] + 216*PolyLog[3, E^(I*ArcTan[a*x])] - 6*ArcTan[a*x]*Sin[3*ArcTan[a*x]]))/(108*c*(c*(1 + a^2*x^2))^(3/2))","A",0
354,1,296,381,1.6963858,"\int \frac{\tan ^{-1}(a x)^2}{x^2 \left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)^(5/2)),x]","-\frac{a \left(-216 i \sqrt{a^2 x^2+1} \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+216 i \sqrt{a^2 x^2+1} \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+54 \sqrt{a^2 x^2+1} \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2-216 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)+216 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)+9 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)-2 \sqrt{a^2 x^2+1} \sin \left(3 \tan ^{-1}(a x)\right)+6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)-378 a x+189 a x \tan ^{-1}(a x)^2+378 \tan ^{-1}(a x)+27 a x \tan ^{-1}(a x)^2 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}{108 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{2 i a \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\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{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,"-1/108*(a*(-378*a*x + 378*ArcTan[a*x] + 189*a*x*ArcTan[a*x]^2 + 6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 27*a*x*ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 - 216*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] + 216*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] - (216*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, -E^(I*ArcTan[a*x])] + (216*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, E^(I*ArcTan[a*x])] - 2*Sqrt[1 + a^2*x^2]*Sin[3*ArcTan[a*x]] + 9*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] + 54*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Tan[ArcTan[a*x]/2]))/(c^2*Sqrt[c + a^2*c*x^2])","A",0
355,0,0,25,1.8947857,"\int x^m \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2 \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^2, x]","A",-1
356,0,0,23,0.9639371,"\int x^m \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2 \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)*ArcTan[a*x]^2, x]","A",-1
357,0,0,25,0.913508,"\int \frac{x^m \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2), x]","A",-1
358,0,0,25,0.699818,"\int \frac{x^m \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2, x]","A",-1
359,0,0,27,1.0407723,"\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2 \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2, x]","A",-1
360,0,0,27,0.1593128,"\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx","Integrate[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,"Integrate[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2, x]","A",-1
361,0,0,27,0.5292009,"\int \frac{x^m \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2], x]","A",-1
362,0,0,27,0.7096726,"\int \frac{x^m \tan ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2), x]","A",-1
363,1,135,219,0.6692094,"\int x^3 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3 \, dx","Integrate[x^3*(c + a^2*c*x^2)*ArcTan[a*x]^3,x]","\frac{c \left(-a^3 x^3+5 \left(2 a^6 x^6+3 a^4 x^4-1\right) \tan ^{-1}(a x)^3-\left(6 a^5 x^5+5 a^3 x^3-15 a x+14 i\right) \tan ^{-1}(a x)^2+\tan ^{-1}(a x) \left(3 a^4 x^4-a^2 x^2+28 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-4\right)-14 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+4 a x\right)}{60 a^4}","\frac{7 i c \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{30 a^4}-\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}{15 a^3}+\frac{c x \tan ^{-1}(a x)^2}{4 a^3}+\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{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}{60 a}-\frac{c x^3 \tan ^{-1}(a x)^2}{12 a}",1,"(c*(4*a*x - a^3*x^3 - (14*I - 15*a*x + 5*a^3*x^3 + 6*a^5*x^5)*ArcTan[a*x]^2 + 5*(-1 + 3*a^4*x^4 + 2*a^6*x^6)*ArcTan[a*x]^3 + ArcTan[a*x]*(-4 - a^2*x^2 + 3*a^4*x^4 + 28*Log[1 + E^((2*I)*ArcTan[a*x])]) - (14*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(60*a^4)","A",0
364,1,171,211,0.5688665,"\int x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3 \, dx","Integrate[x^2*(c + a^2*c*x^2)*ArcTan[a*x]^3,x]","\frac{c \left(12 a^5 x^5 \tan ^{-1}(a x)^3-9 a^4 x^4 \tan ^{-1}(a x)^2+20 a^3 x^3 \tan ^{-1}(a x)^3+6 a^3 x^3 \tan ^{-1}(a x)-3 a^2 x^2-12 a^2 x^2 \tan ^{-1}(a x)^2+24 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-12 \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)+6 a x \tan ^{-1}(a x)+8 i \tan ^{-1}(a x)^3-3 \tan ^{-1}(a x)^2-24 \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-3\right)}{60 a^3}","-\frac{c \text{Li}_3\left(1-\frac{2}{i a x+1}\right)}{5 a^3}-\frac{2 i c \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)}{5 a^3}-\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{1}{5} a^2 c x^5 \tan ^{-1}(a x)^3+\frac{c x \tan ^{-1}(a x)}{10 a^2}-\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}{20 a}-\frac{c x^2 \tan ^{-1}(a x)^2}{5 a}",1,"(c*(-3 - 3*a^2*x^2 + 6*a*x*ArcTan[a*x] + 6*a^3*x^3*ArcTan[a*x] - 3*ArcTan[a*x]^2 - 12*a^2*x^2*ArcTan[a*x]^2 - 9*a^4*x^4*ArcTan[a*x]^2 + (8*I)*ArcTan[a*x]^3 + 20*a^3*x^3*ArcTan[a*x]^3 + 12*a^5*x^5*ArcTan[a*x]^3 - 24*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + (24*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - 12*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(60*a^3)","A",0
365,1,101,160,0.0701386,"\int x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3 \, dx","Integrate[x*(c + a^2*c*x^2)*ArcTan[a*x]^3,x]","\frac{c \left(-\left(a^3 x^3+3 a x-2 i\right) \tan ^{-1}(a x)^2+\left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^3+\tan ^{-1}(a x) \left(a^2 x^2-4 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)+1\right)+2 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-a x\right)}{4 a^2}","-\frac{i c \text{Li}_2\left(1-\frac{2}{i a x+1}\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*(-(a*x) - (-2*I + 3*a*x + a^3*x^3)*ArcTan[a*x]^2 + (1 + a^2*x^2)^2*ArcTan[a*x]^3 + ArcTan[a*x]*(1 + a^2*x^2 - 4*Log[1 + E^((2*I)*ArcTan[a*x])]) + (2*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(4*a^2)","A",0
366,1,144,172,0.0483473,"\int \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3 \, dx","Integrate[(c + a^2*c*x^2)*ArcTan[a*x]^3,x]","\frac{c \left(2 a^3 x^3 \tan ^{-1}(a x)^3-3 \log \left(a^2 x^2+1\right)-3 a^2 x^2 \tan ^{-1}(a x)^2-12 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+6 \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)+6 a x \tan ^{-1}(a x)^3-4 i \tan ^{-1}(a x)^3-3 \tan ^{-1}(a x)^2+6 a x \tan ^{-1}(a x)+12 \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)\right)}{6 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{c \text{Li}_3\left(1-\frac{2}{i a x+1}\right)}{a}+\frac{2 i c \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)}{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*(6*a*x*ArcTan[a*x] - 3*ArcTan[a*x]^2 - 3*a^2*x^2*ArcTan[a*x]^2 - (4*I)*ArcTan[a*x]^3 + 6*a*x*ArcTan[a*x]^3 + 2*a^3*x^3*ArcTan[a*x]^3 + 12*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 3*Log[1 + a^2*x^2] - (12*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 6*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(6*a)","A",0
367,1,284,276,0.0712588,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3}{x} \, dx","Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x,x]","\frac{1}{2} c \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3-\frac{3}{4} i c \text{Li}_4\left(\frac{-a x-i}{a x-i}\right)+\frac{3}{4} i c \text{Li}_4\left(\frac{a x+i}{a x-i}\right)+\frac{3}{2} i c \text{Li}_2\left(\frac{-a x-i}{a x-i}\right) \tan ^{-1}(a x)^2-\frac{3}{2} i c \text{Li}_2\left(\frac{a x+i}{a x-i}\right) \tan ^{-1}(a x)^2+\frac{3}{2} c \text{Li}_3\left(\frac{-a x-i}{a x-i}\right) \tan ^{-1}(a x)-\frac{3}{2} c \text{Li}_3\left(\frac{a x+i}{a x-i}\right) \tan ^{-1}(a x)-\frac{3}{2} c \left(-i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+a x \tan ^{-1}(a x)^2-i \tan ^{-1}(a x)^2+2 \tan ^{-1}(a x) \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)\right)+2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2 i}{-a x+i}\right)","\frac{1}{2} a^2 c x^2 \tan ^{-1}(a x)^3-\frac{3}{2} i c \text{Li}_2\left(1-\frac{2}{i a x+1}\right)+\frac{3}{4} i c \text{Li}_4\left(1-\frac{2}{i a x+1}\right)-\frac{3}{4} i c \text{Li}_4\left(\frac{2}{i a x+1}-1\right)-\frac{3}{2} i c \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)^2+\frac{3}{2} i c \text{Li}_2\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)^2-\frac{3}{2} c \text{Li}_3\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)+\frac{3}{2} c \text{Li}_3\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)+\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,"(c*(1 + a^2*x^2)*ArcTan[a*x]^3)/2 + 2*c*ArcTan[a*x]^3*ArcTanh[1 - (2*I)/(I - a*x)] - (3*c*((-I)*ArcTan[a*x]^2 + a*x*ArcTan[a*x]^2 + 2*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - I*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/2 + ((3*I)/2)*c*ArcTan[a*x]^2*PolyLog[2, (-I - a*x)/(-I + a*x)] - ((3*I)/2)*c*ArcTan[a*x]^2*PolyLog[2, (I + a*x)/(-I + a*x)] + (3*c*ArcTan[a*x]*PolyLog[3, (-I - a*x)/(-I + a*x)])/2 - (3*c*ArcTan[a*x]*PolyLog[3, (I + a*x)/(-I + a*x)])/2 - ((3*I)/4)*c*PolyLog[4, (-I - a*x)/(-I + a*x)] + ((3*I)/4)*c*PolyLog[4, (I + a*x)/(-I + a*x)]","A",0
368,1,188,169,0.1808183,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3}{x^2} \, dx","Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^2,x]","a^2 c x \tan ^{-1}(a x)^3-3 i a c \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+\frac{3}{2} a c \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)+a c \left(3 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+\frac{3}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-\frac{\tan ^{-1}(a x)^3}{a x}+i \tan ^{-1}(a x)^3+3 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-\frac{i \pi ^3}{8}\right)-i a c \tan ^{-1}(a x)^3+3 a c \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)","a^2 c x \tan ^{-1}(a x)^3+\frac{3}{2} a c \text{Li}_3\left(\frac{2}{1-i a x}-1\right)+\frac{3}{2} a c \text{Li}_3\left(1-\frac{2}{i a x+1}\right)-3 i a c \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)+3 i a c \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)-\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,"(-I)*a*c*ArcTan[a*x]^3 + a^2*c*x*ArcTan[a*x]^3 + 3*a*c*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - (3*I)*a*c*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + a*c*((-1/8*I)*Pi^3 + I*ArcTan[a*x]^3 - ArcTan[a*x]^3/(a*x) + 3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (3*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (3*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/2) + (3*a*c*PolyLog[3, -E^((2*I)*ArcTan[a*x])])/2","A",0
369,1,337,310,0.2498444,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3}{x^3} \, dx","Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^3,x]","-\frac{3}{4} i a^2 c \text{Li}_4\left(\frac{-a x-i}{a x-i}\right)+\frac{3}{4} i a^2 c \text{Li}_4\left(\frac{a x+i}{a x-i}\right)+\frac{3}{2} i a^2 c \text{Li}_2\left(\frac{-a x-i}{a x-i}\right) \tan ^{-1}(a x)^2-\frac{3}{2} i a^2 c \text{Li}_2\left(\frac{a x+i}{a x-i}\right) \tan ^{-1}(a x)^2+\frac{3}{2} a^2 c \text{Li}_3\left(\frac{-a x-i}{a x-i}\right) \tan ^{-1}(a x)-\frac{3}{2} a^2 c \text{Li}_3\left(\frac{a x+i}{a x-i}\right) \tan ^{-1}(a x)+\frac{3}{2} a^2 c \left(-i \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)-\frac{1}{3} \tan ^{-1}(a x) \left(\left(\tan ^{-1}(a x)+3 i\right) \tan ^{-1}(a x)+\frac{3 \tan ^{-1}(a x)}{a x}-6 \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)\right)\right)+\frac{c \left(-a^2 x^2-1\right) \tan ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} a^2 c \tan ^{-1}(a x)^3+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2 i}{-a x+i}\right)","-\frac{3}{2} i a^2 c \text{Li}_2\left(\frac{2}{1-i a x}-1\right)+\frac{3}{4} i a^2 c \text{Li}_4\left(1-\frac{2}{i a x+1}\right)-\frac{3}{4} i a^2 c \text{Li}_4\left(\frac{2}{i a x+1}-1\right)-\frac{3}{2} i a^2 c \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)^2+\frac{3}{2} i a^2 c \text{Li}_2\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)^2-\frac{3}{2} a^2 c \text{Li}_3\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)+\frac{3}{2} a^2 c \text{Li}_3\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)-\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,"(a^2*c*ArcTan[a*x]^3)/2 + (c*(-1 - a^2*x^2)*ArcTan[a*x]^3)/(2*x^2) + 2*a^2*c*ArcTan[a*x]^3*ArcTanh[1 - (2*I)/(I - a*x)] + (3*a^2*c*(-1/3*(ArcTan[a*x]*((3*ArcTan[a*x])/(a*x) + ArcTan[a*x]*(3*I + ArcTan[a*x]) - 6*Log[1 - E^((2*I)*ArcTan[a*x])])) - I*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/2 + ((3*I)/2)*a^2*c*ArcTan[a*x]^2*PolyLog[2, (-I - a*x)/(-I + a*x)] - ((3*I)/2)*a^2*c*ArcTan[a*x]^2*PolyLog[2, (I + a*x)/(-I + a*x)] + (3*a^2*c*ArcTan[a*x]*PolyLog[3, (-I - a*x)/(-I + a*x)])/2 - (3*a^2*c*ArcTan[a*x]*PolyLog[3, (I + a*x)/(-I + a*x)])/2 - ((3*I)/4)*a^2*c*PolyLog[4, (-I - a*x)/(-I + a*x)] + ((3*I)/4)*a^2*c*PolyLog[4, (I + a*x)/(-I + a*x)]","A",0
370,1,177,189,0.393933,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3}{x^4} \, dx","Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^4,x]","\frac{1}{12} c \left(24 i a^3 \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+12 a^3 \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)+8 i a^3 \tan ^{-1}(a x)^3-6 a^3 \tan ^{-1}(a x)^2+24 a^3 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-i \pi ^3 a^3-\frac{12 a^2 \tan ^{-1}(a x)^3}{x}-\frac{12 a^2 \tan ^{-1}(a x)}{x}+12 a^3 \log \left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-\frac{4 \tan ^{-1}(a x)^3}{x^3}-\frac{6 a \tan ^{-1}(a x)^2}{x^2}\right)","a^3 c \text{Li}_3\left(\frac{2}{1-i a x}-1\right)-2 i a^3 c \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)+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+2 a^3 c \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2-\frac{a^2 c \tan ^{-1}(a x)^3}{x}-\frac{a^2 c \tan ^{-1}(a x)}{x}-\frac{1}{2} a^3 c \log \left(a^2 x^2+1\right)-\frac{c \tan ^{-1}(a x)^3}{3 x^3}-\frac{a c \tan ^{-1}(a x)^2}{2 x^2}",1,"(c*((-I)*a^3*Pi^3 - (12*a^2*ArcTan[a*x])/x - 6*a^3*ArcTan[a*x]^2 - (6*a*ArcTan[a*x]^2)/x^2 + (8*I)*a^3*ArcTan[a*x]^3 - (4*ArcTan[a*x]^3)/x^3 - (12*a^2*ArcTan[a*x]^3)/x + 24*a^3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + 12*a^3*Log[(a*x)/Sqrt[1 + a^2*x^2]] + (24*I)*a^3*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + 12*a^3*PolyLog[3, E^((-2*I)*ArcTan[a*x])]))/12","A",0
371,1,165,313,1.3265579,"\int x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3 \, dx","Integrate[x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^3,x]","\frac{c^2 \left(-3 a^5 x^5-5 a^3 x^3+35 \left(a^2 x^2+1\right)^3 \left(3 a^2 x^2-1\right) \tan ^{-1}(a x)^3-5 \left(9 a^7 x^7+21 a^5 x^5+7 a^3 x^3-21 a x+16 i\right) \tan ^{-1}(a x)^2+5 \tan ^{-1}(a x) \left(3 a^6 x^6+6 a^4 x^4-5 a^2 x^2+32 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-8\right)-80 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+40 a x\right)}{840 a^4}","\frac{2 i c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{21 a^4}+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^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{3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2+\frac{c^2 x}{21 a^3}+\frac{c^2 x \tan ^{-1}(a x)^2}{8 a^3}+\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{1}{280} a c^2 x^5-\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}{168 a}-\frac{c^2 x^3 \tan ^{-1}(a x)^2}{24 a}",1,"(c^2*(40*a*x - 5*a^3*x^3 - 3*a^5*x^5 - 5*(16*I - 21*a*x + 7*a^3*x^3 + 21*a^5*x^5 + 9*a^7*x^7)*ArcTan[a*x]^2 + 35*(1 + a^2*x^2)^3*(-1 + 3*a^2*x^2)*ArcTan[a*x]^3 + 5*ArcTan[a*x]*(-8 - 5*a^2*x^2 + 6*a^4*x^4 + 3*a^6*x^6 + 32*Log[1 + E^((2*I)*ArcTan[a*x])]) - (80*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(840*a^4)","A",0
372,1,233,321,1.1921153,"\int x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3 \, dx","Integrate[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^3,x]","\frac{c^2 \left(60 a^7 x^7 \tan ^{-1}(a x)^3-30 a^6 x^6 \tan ^{-1}(a x)^2+168 a^5 x^5 \tan ^{-1}(a x)^3+12 a^5 x^5 \tan ^{-1}(a x)-3 a^4 x^4-81 a^4 x^4 \tan ^{-1}(a x)^2+140 a^3 x^3 \tan ^{-1}(a x)^3+34 a^3 x^3 \tan ^{-1}(a x)-11 a^2 x^2+14 \log \left(a^2 x^2+1\right)-48 a^2 x^2 \tan ^{-1}(a x)^2+96 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-48 \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-6 a x \tan ^{-1}(a x)+32 i \tan ^{-1}(a x)^3+3 \tan ^{-1}(a x)^2-96 \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-8\right)}{420 a^3}","\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^3-\frac{4 c^2 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)}{35 a^3}-\frac{8 i c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)}{35 a^3}-\frac{1}{14} a^3 c^2 x^6 \tan ^{-1}(a x)^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{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{c^2 \log \left(a^2 x^2+1\right)}{30 a^3}-\frac{1}{140} a c^2 x^4-\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{11 c^2 x^2}{420 a}-\frac{4 c^2 x^2 \tan ^{-1}(a x)^2}{35 a}",1,"(c^2*(-8 - 11*a^2*x^2 - 3*a^4*x^4 - 6*a*x*ArcTan[a*x] + 34*a^3*x^3*ArcTan[a*x] + 12*a^5*x^5*ArcTan[a*x] + 3*ArcTan[a*x]^2 - 48*a^2*x^2*ArcTan[a*x]^2 - 81*a^4*x^4*ArcTan[a*x]^2 - 30*a^6*x^6*ArcTan[a*x]^2 + (32*I)*ArcTan[a*x]^3 + 140*a^3*x^3*ArcTan[a*x]^3 + 168*a^5*x^5*ArcTan[a*x]^3 + 60*a^7*x^7*ArcTan[a*x]^3 - 96*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 14*Log[1 + a^2*x^2] + (96*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - 48*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(420*a^3)","A",0
373,1,131,242,0.7874086,"\int x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3 \, dx","Integrate[x*(c + a^2*c*x^2)^2*ArcTan[a*x]^3,x]","\frac{c^2 \left(-a x \left(a^2 x^2+11\right)+10 \left(a^2 x^2+1\right)^3 \tan ^{-1}(a x)^3-2 \left(3 a^5 x^5+10 a^3 x^3+15 a x-8 i\right) \tan ^{-1}(a x)^2+\tan ^{-1}(a x) \left(3 a^4 x^4+14 a^2 x^2-32 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)+11\right)+16 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)\right)}{60 a^2}","-\frac{4 i c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\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,"(c^2*(-(a*x*(11 + a^2*x^2)) - 2*(-8*I + 15*a*x + 10*a^3*x^3 + 3*a^5*x^5)*ArcTan[a*x]^2 + 10*(1 + a^2*x^2)^3*ArcTan[a*x]^3 + ArcTan[a*x]*(11 + 14*a^2*x^2 + 3*a^4*x^4 - 32*Log[1 + E^((2*I)*ArcTan[a*x])]) + (16*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(60*a^2)","A",0
374,1,195,289,0.6046298,"\int \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3 \, dx","Integrate[(c + a^2*c*x^2)^2*ArcTan[a*x]^3,x]","\frac{c^2 \left(12 a^5 x^5 \tan ^{-1}(a x)^3-9 a^4 x^4 \tan ^{-1}(a x)^2+40 a^3 x^3 \tan ^{-1}(a x)^3+6 a^3 x^3 \tan ^{-1}(a x)-3 a^2 x^2-30 \log \left(a^2 x^2+1\right)-42 a^2 x^2 \tan ^{-1}(a x)^2-96 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+48 \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)+60 a x \tan ^{-1}(a x)^3+66 a x \tan ^{-1}(a x)-32 i \tan ^{-1}(a x)^3-33 \tan ^{-1}(a x)^2+96 \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-3\right)}{60 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{4 c^2 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)}{5 a}+\frac{8 i c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)}{5 a}+\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*(-3 - 3*a^2*x^2 + 66*a*x*ArcTan[a*x] + 6*a^3*x^3*ArcTan[a*x] - 33*ArcTan[a*x]^2 - 42*a^2*x^2*ArcTan[a*x]^2 - 9*a^4*x^4*ArcTan[a*x]^2 - (32*I)*ArcTan[a*x]^3 + 60*a*x*ArcTan[a*x]^3 + 40*a^3*x^3*ArcTan[a*x]^3 + 12*a^5*x^5*ArcTan[a*x]^3 + 96*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 30*Log[1 + a^2*x^2] - (96*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 48*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(60*a)","A",0
375,1,302,370,0.576645,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3}{x} \, dx","Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x]^3)/x,x]","\frac{1}{64} c^2 \left(16 a^4 x^4 \tan ^{-1}(a x)^3-16 a^3 x^3 \tan ^{-1}(a x)^2+64 a^2 x^2 \tan ^{-1}(a x)^3+16 a^2 x^2 \tan ^{-1}(a x)+96 i \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+32 i \left(3 \tan ^{-1}(a x)^2+4\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+96 \tan ^{-1}(a x) \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-96 \tan ^{-1}(a x) \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-48 i \text{Li}_4\left(e^{-2 i \tan ^{-1}(a x)}\right)-48 i \text{Li}_4\left(-e^{2 i \tan ^{-1}(a x)}\right)-16 a x-144 a x \tan ^{-1}(a x)^2+32 i \tan ^{-1}(a x)^4+48 \tan ^{-1}(a x)^3+128 i \tan ^{-1}(a x)^2+16 \tan ^{-1}(a x)+64 \tan ^{-1}(a x)^3 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-64 \tan ^{-1}(a x)^3 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-256 \tan ^{-1}(a x) \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-i \pi ^4\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)-2 i c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)+\frac{3}{4} i c^2 \text{Li}_4\left(1-\frac{2}{i a x+1}\right)-\frac{3}{4} i c^2 \text{Li}_4\left(\frac{2}{i a x+1}-1\right)-\frac{3}{2} i c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)^2+\frac{3}{2} i c^2 \text{Li}_2\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)^2-\frac{3}{2} c^2 \text{Li}_3\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)+\frac{3}{2} c^2 \text{Li}_3\left(\frac{2}{i a x+1}-1\right) \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,"(c^2*((-I)*Pi^4 - 16*a*x + 16*ArcTan[a*x] + 16*a^2*x^2*ArcTan[a*x] + (128*I)*ArcTan[a*x]^2 - 144*a*x*ArcTan[a*x]^2 - 16*a^3*x^3*ArcTan[a*x]^2 + 48*ArcTan[a*x]^3 + 64*a^2*x^2*ArcTan[a*x]^3 + 16*a^4*x^4*ArcTan[a*x]^3 + (32*I)*ArcTan[a*x]^4 + 64*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] - 256*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - 64*ArcTan[a*x]^3*Log[1 + E^((2*I)*ArcTan[a*x])] + (96*I)*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (32*I)*(4 + 3*ArcTan[a*x]^2)*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 96*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 96*ArcTan[a*x]*PolyLog[3, -E^((2*I)*ArcTan[a*x])] - (48*I)*PolyLog[4, E^((-2*I)*ArcTan[a*x])] - (48*I)*PolyLog[4, -E^((2*I)*ArcTan[a*x])]))/64","A",0
376,1,246,284,0.401379,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3}{x^2} \, dx","Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x]^3)/x^2,x]","\frac{c^2 \left(8 a^4 x^4 \tan ^{-1}(a x)^3-12 a^3 x^3 \tan ^{-1}(a x)^2-12 a x \log \left(a^2 x^2+1\right)+48 a^2 x^2 \tan ^{-1}(a x)^3+24 a^2 x^2 \tan ^{-1}(a x)+72 i a x \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)-120 i a x \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+36 a x \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)+60 a x \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-3 i \pi ^3 a x-16 i a x \tan ^{-1}(a x)^3-12 a x \tan ^{-1}(a x)^2-24 \tan ^{-1}(a x)^3+72 a x \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)+120 a x \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)\right)}{24 x}","\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-\frac{1}{2} a c^2 \log \left(a^2 x^2+1\right)+2 a^2 c^2 x \tan ^{-1}(a x)^3+a^2 c^2 x \tan ^{-1}(a x)+\frac{3}{2} a c^2 \text{Li}_3\left(\frac{2}{1-i a x}-1\right)+\frac{5}{2} a c^2 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)-3 i a c^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)+5 i a c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \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,"(c^2*((-3*I)*a*Pi^3*x + 24*a^2*x^2*ArcTan[a*x] - 12*a*x*ArcTan[a*x]^2 - 12*a^3*x^3*ArcTan[a*x]^2 - 24*ArcTan[a*x]^3 - (16*I)*a*x*ArcTan[a*x]^3 + 48*a^2*x^2*ArcTan[a*x]^3 + 8*a^4*x^4*ArcTan[a*x]^3 + 72*a*x*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + 120*a*x*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 12*a*x*Log[1 + a^2*x^2] + (72*I)*a*x*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (120*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 36*a*x*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 60*a*x*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(24*x)","A",0
377,1,302,399,0.6057281,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3}{x^3} \, dx","Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x]^3)/x^3,x]","\frac{1}{32} a^2 c^2 \left(16 a^2 x^2 \tan ^{-1}(a x)^3-\frac{16 \tan ^{-1}(a x)^3}{a^2 x^2}+96 i \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+96 \tan ^{-1}(a x) \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-96 \tan ^{-1}(a x) \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)+48 i \left(2 \tan ^{-1}(a x)^2+1\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-48 i \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)-48 i \text{Li}_4\left(e^{-2 i \tan ^{-1}(a x)}\right)-48 i \text{Li}_4\left(-e^{2 i \tan ^{-1}(a x)}\right)+32 i \tan ^{-1}(a x)^4-48 a x \tan ^{-1}(a x)^2-\frac{48 \tan ^{-1}(a x)^2}{a x}+64 \tan ^{-1}(a x)^3 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-64 \tan ^{-1}(a x)^3 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)+96 \tan ^{-1}(a x) \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)-96 \tan ^{-1}(a x) \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-i \pi ^4\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-\frac{3}{2} i a^2 c^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)-\frac{3}{2} i a^2 c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)+\frac{3}{2} i a^2 c^2 \text{Li}_4\left(1-\frac{2}{i a x+1}\right)-\frac{3}{2} i a^2 c^2 \text{Li}_4\left(\frac{2}{i a x+1}-1\right)-3 i a^2 c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)^2+3 i a^2 c^2 \text{Li}_2\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)^2-3 a^2 c^2 \text{Li}_3\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)+3 a^2 c^2 \text{Li}_3\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)-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,"(a^2*c^2*((-I)*Pi^4 - (48*ArcTan[a*x]^2)/(a*x) - 48*a*x*ArcTan[a*x]^2 - (16*ArcTan[a*x]^3)/(a^2*x^2) + 16*a^2*x^2*ArcTan[a*x]^3 + (32*I)*ArcTan[a*x]^4 + 64*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] + 96*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] - 96*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - 64*ArcTan[a*x]^3*Log[1 + E^((2*I)*ArcTan[a*x])] + (96*I)*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (48*I)*(1 + 2*ArcTan[a*x]^2)*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - (48*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] + 96*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 96*ArcTan[a*x]*PolyLog[3, -E^((2*I)*ArcTan[a*x])] - (48*I)*PolyLog[4, E^((-2*I)*ArcTan[a*x])] - (48*I)*PolyLog[4, -E^((2*I)*ArcTan[a*x])]))/32","A",0
378,1,289,311,0.5189982,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3}{x^4} \, dx","Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x]^3)/x^4,x]","\frac{c^2 \left(24 a^4 x^4 \tan ^{-1}(a x)^3+120 i a^3 x^3 \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)-72 i a^3 x^3 \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+60 a^3 x^3 \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)+36 a^3 x^3 \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-5 i \pi ^3 a^3 x^3+16 i a^3 x^3 \tan ^{-1}(a x)^3-12 a^3 x^3 \tan ^{-1}(a x)^2+120 a^3 x^3 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)+72 a^3 x^3 \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-48 a^2 x^2 \tan ^{-1}(a x)^3-24 a^2 x^2 \tan ^{-1}(a x)+24 a^3 x^3 \log \left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-12 a x \tan ^{-1}(a x)^2-8 \tan ^{-1}(a x)^3\right)}{24 x^3}","a^4 c^2 x \tan ^{-1}(a x)^3+\frac{5}{2} a^3 c^2 \text{Li}_3\left(\frac{2}{1-i a x}-1\right)+\frac{3}{2} a^3 c^2 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)-5 i a^3 c^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)+3 i a^3 c^2 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)+a^3 c^2 \log (x)-\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+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{2 a^2 c^2 \tan ^{-1}(a x)^3}{x}-\frac{a^2 c^2 \tan ^{-1}(a x)}{x}-\frac{1}{2} a^3 c^2 \log \left(a^2 x^2+1\right)-\frac{c^2 \tan ^{-1}(a x)^3}{3 x^3}-\frac{a c^2 \tan ^{-1}(a x)^2}{2 x^2}",1,"(c^2*((-5*I)*a^3*Pi^3*x^3 - 24*a^2*x^2*ArcTan[a*x] - 12*a*x*ArcTan[a*x]^2 - 12*a^3*x^3*ArcTan[a*x]^2 - 8*ArcTan[a*x]^3 - 48*a^2*x^2*ArcTan[a*x]^3 + (16*I)*a^3*x^3*ArcTan[a*x]^3 + 24*a^4*x^4*ArcTan[a*x]^3 + 120*a^3*x^3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + 72*a^3*x^3*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 24*a^3*x^3*Log[(a*x)/Sqrt[1 + a^2*x^2]] + (120*I)*a^3*x^3*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (72*I)*a^3*x^3*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 60*a^3*x^3*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 36*a^3*x^3*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(24*x^3)","A",0
379,1,191,381,2.1819401,"\int x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3 \, dx","Integrate[x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^3,x]","\frac{c^3 \left(945 \left(4 a^2 x^2-1\right) \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x)^3-a x \left(45 a^6 x^6+150 a^4 x^4+68 a^2 x^2-1167\right)-9 \left(140 a^9 x^9+495 a^7 x^7+567 a^5 x^5+105 a^3 x^3-315 a x+208 i\right) \tan ^{-1}(a x)^2+3 \tan ^{-1}(a x) \left(105 a^8 x^8+355 a^6 x^6+318 a^4 x^4-321 a^2 x^2+1248 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-389\right)-1872 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)\right)}{37800 a^4}","\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{26 i c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{525 a^4}+\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{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}{840} a^3 c^3 x^7-\frac{33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2+\frac{389 c^3 x}{12600 a^3}+\frac{3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}+\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{1}{252} a c^3 x^5-\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{17 c^3 x^3}{9450 a}-\frac{c^3 x^3 \tan ^{-1}(a x)^2}{40 a}",1,"(c^3*(-(a*x*(-1167 + 68*a^2*x^2 + 150*a^4*x^4 + 45*a^6*x^6)) - 9*(208*I - 315*a*x + 105*a^3*x^3 + 567*a^5*x^5 + 495*a^7*x^7 + 140*a^9*x^9)*ArcTan[a*x]^2 + 945*(1 + a^2*x^2)^4*(-1 + 4*a^2*x^2)*ArcTan[a*x]^3 + 3*ArcTan[a*x]*(-389 - 321*a^2*x^2 + 318*a^4*x^4 + 355*a^6*x^6 + 105*a^8*x^8 + 1248*Log[1 + E^((2*I)*ArcTan[a*x])]) - (1872*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(37800*a^4)","A",0
380,1,281,389,2.021404,"\int x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3 \, dx","Integrate[x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^3,x]","\frac{c^3 \left(840 a^9 x^9 \tan ^{-1}(a x)^3-315 a^8 x^8 \tan ^{-1}(a x)^2+3240 a^7 x^7 \tan ^{-1}(a x)^3+90 a^7 x^7 \tan ^{-1}(a x)-15 a^6 x^6-1200 a^6 x^6 \tan ^{-1}(a x)^2+4536 a^5 x^5 \tan ^{-1}(a x)^3+354 a^5 x^5 \tan ^{-1}(a x)-66 a^4 x^4-1602 a^4 x^4 \tan ^{-1}(a x)^2+2520 a^3 x^3 \tan ^{-1}(a x)^3+478 a^3 x^3 \tan ^{-1}(a x)-107 a^2 x^2+248 \log \left(a^2 x^2+1\right)-576 a^2 x^2 \tan ^{-1}(a x)^2+1152 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-576 \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-282 a x \tan ^{-1}(a x)+384 i \tan ^{-1}(a x)^3+141 \tan ^{-1}(a x)^2-1152 \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-56\right)}{7560 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{8 c^3 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)}{105 a^3}-\frac{16 i c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)}{105 a^3}-\frac{1}{504} a^3 c^3 x^6-\frac{10}{63} a^3 c^3 x^6 \tan ^{-1}(a x)^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{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{31 c^3 \log \left(a^2 x^2+1\right)}{945 a^3}-\frac{11 a c^3 x^4}{1260}-\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{107 c^3 x^2}{7560 a}-\frac{8 c^3 x^2 \tan ^{-1}(a x)^2}{105 a}",1,"(c^3*(-56 - 107*a^2*x^2 - 66*a^4*x^4 - 15*a^6*x^6 - 282*a*x*ArcTan[a*x] + 478*a^3*x^3*ArcTan[a*x] + 354*a^5*x^5*ArcTan[a*x] + 90*a^7*x^7*ArcTan[a*x] + 141*ArcTan[a*x]^2 - 576*a^2*x^2*ArcTan[a*x]^2 - 1602*a^4*x^4*ArcTan[a*x]^2 - 1200*a^6*x^6*ArcTan[a*x]^2 - 315*a^8*x^8*ArcTan[a*x]^2 + (384*I)*ArcTan[a*x]^3 + 2520*a^3*x^3*ArcTan[a*x]^3 + 4536*a^5*x^5*ArcTan[a*x]^3 + 3240*a^7*x^7*ArcTan[a*x]^3 + 840*a^9*x^9*ArcTan[a*x]^3 - 1152*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 248*Log[1 + a^2*x^2] + (1152*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - 576*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(7560*a^3)","A",0
381,1,157,308,1.4415621,"\int x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3 \, dx","Integrate[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^3,x]","\frac{c^3 \left(105 \left(a^2 x^2+1\right)^4 \tan ^{-1}(a x)^3-a x \left(3 a^4 x^4+19 a^2 x^2+114\right)-9 \left(5 a^7 x^7+21 a^5 x^5+35 a^3 x^3+35 a x-16 i\right) \tan ^{-1}(a x)^2+3 \tan ^{-1}(a x) \left(5 a^6 x^6+24 a^4 x^4+57 a^2 x^2-96 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)+38\right)+144 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)\right)}{840 a^2}","-\frac{1}{280} a^3 c^3 x^5-\frac{6 i c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{35 a^2}-\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,"(c^3*(-(a*x*(114 + 19*a^2*x^2 + 3*a^4*x^4)) - 9*(-16*I + 35*a*x + 35*a^3*x^3 + 21*a^5*x^5 + 5*a^7*x^7)*ArcTan[a*x]^2 + 105*(1 + a^2*x^2)^4*ArcTan[a*x]^3 + 3*ArcTan[a*x]*(38 + 57*a^2*x^2 + 24*a^4*x^4 + 5*a^6*x^6 - 96*Log[1 + E^((2*I)*ArcTan[a*x])]) + (144*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/(840*a^2)","A",0
382,1,243,388,1.2173736,"\int \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3 \, dx","Integrate[(c + a^2*c*x^2)^3*ArcTan[a*x]^3,x]","\frac{c^3 \left(60 a^7 x^7 \tan ^{-1}(a x)^3-30 a^6 x^6 \tan ^{-1}(a x)^2+252 a^5 x^5 \tan ^{-1}(a x)^3+12 a^5 x^5 \tan ^{-1}(a x)-3 a^4 x^4-144 a^4 x^4 \tan ^{-1}(a x)^2+420 a^3 x^3 \tan ^{-1}(a x)^3+76 a^3 x^3 \tan ^{-1}(a x)-32 a^2 x^2-196 \log \left(a^2 x^2+1\right)-342 a^2 x^2 \tan ^{-1}(a x)^2-576 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+288 \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)+420 a x \tan ^{-1}(a x)^3+456 a x \tan ^{-1}(a x)-192 i \tan ^{-1}(a x)^3-228 \tan ^{-1}(a x)^2+576 \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-29\right)}{420 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{24 c^3 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)}{35 a}+\frac{48 i c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)}{35 a}+\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,"(c^3*(-29 - 32*a^2*x^2 - 3*a^4*x^4 + 456*a*x*ArcTan[a*x] + 76*a^3*x^3*ArcTan[a*x] + 12*a^5*x^5*ArcTan[a*x] - 228*ArcTan[a*x]^2 - 342*a^2*x^2*ArcTan[a*x]^2 - 144*a^4*x^4*ArcTan[a*x]^2 - 30*a^6*x^6*ArcTan[a*x]^2 - (192*I)*ArcTan[a*x]^3 + 420*a*x*ArcTan[a*x]^3 + 420*a^3*x^3*ArcTan[a*x]^3 + 252*a^5*x^5*ArcTan[a*x]^3 + 60*a^7*x^7*ArcTan[a*x]^3 + 576*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 196*Log[1 + a^2*x^2] - (576*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 288*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(420*a)","A",0
383,1,350,447,1.0677085,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3}{x} \, dx","Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x]^3)/x,x]","\frac{1}{960} c^3 \left(160 a^6 x^6 \tan ^{-1}(a x)^3-96 a^5 x^5 \tan ^{-1}(a x)^2+720 a^4 x^4 \tan ^{-1}(a x)^3+48 a^4 x^4 \tan ^{-1}(a x)-16 a^3 x^3-560 a^3 x^3 \tan ^{-1}(a x)^2+1440 a^2 x^2 \tan ^{-1}(a x)^3+464 a^2 x^2 \tan ^{-1}(a x)+1440 i \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+32 i \left(45 \tan ^{-1}(a x)^2+68\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+1440 \tan ^{-1}(a x) \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-1440 \tan ^{-1}(a x) \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-720 i \text{Li}_4\left(e^{-2 i \tan ^{-1}(a x)}\right)-720 i \text{Li}_4\left(-e^{2 i \tan ^{-1}(a x)}\right)-416 a x-2640 a x \tan ^{-1}(a x)^2+480 i \tan ^{-1}(a x)^4+880 \tan ^{-1}(a x)^3+2176 i \tan ^{-1}(a x)^2+416 \tan ^{-1}(a x)+960 \tan ^{-1}(a x)^3 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-960 \tan ^{-1}(a x)^3 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-4352 \tan ^{-1}(a x) \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-15 i \pi ^4\right)","\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{1}{60} a^3 c^3 x^3-\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{34}{15} i c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)+\frac{3}{4} i c^3 \text{Li}_4\left(1-\frac{2}{i a x+1}\right)-\frac{3}{4} i c^3 \text{Li}_4\left(\frac{2}{i a x+1}-1\right)-\frac{3}{2} i c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)^2+\frac{3}{2} i c^3 \text{Li}_2\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)^2-\frac{3}{2} c^3 \text{Li}_3\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)+\frac{3}{2} c^3 \text{Li}_3\left(\frac{2}{i a x+1}-1\right) \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,"(c^3*((-15*I)*Pi^4 - 416*a*x - 16*a^3*x^3 + 416*ArcTan[a*x] + 464*a^2*x^2*ArcTan[a*x] + 48*a^4*x^4*ArcTan[a*x] + (2176*I)*ArcTan[a*x]^2 - 2640*a*x*ArcTan[a*x]^2 - 560*a^3*x^3*ArcTan[a*x]^2 - 96*a^5*x^5*ArcTan[a*x]^2 + 880*ArcTan[a*x]^3 + 1440*a^2*x^2*ArcTan[a*x]^3 + 720*a^4*x^4*ArcTan[a*x]^3 + 160*a^6*x^6*ArcTan[a*x]^3 + (480*I)*ArcTan[a*x]^4 + 960*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] - 4352*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - 960*ArcTan[a*x]^3*Log[1 + E^((2*I)*ArcTan[a*x])] + (1440*I)*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (32*I)*(68 + 45*ArcTan[a*x]^2)*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 1440*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 1440*ArcTan[a*x]*PolyLog[3, -E^((2*I)*ArcTan[a*x])] - (720*I)*PolyLog[4, E^((-2*I)*ArcTan[a*x])] - (720*I)*PolyLog[4, -E^((2*I)*ArcTan[a*x])]))/960","A",0
384,1,298,354,0.7662847,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3}{x^2} \, dx","Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x]^3)/x^2,x]","\frac{c^3 \left(8 a^6 x^6 \tan ^{-1}(a x)^3-6 a^5 x^5 \tan ^{-1}(a x)^2+40 a^4 x^4 \tan ^{-1}(a x)^3+4 a^4 x^4 \tan ^{-1}(a x)-2 a^3 x^3-48 a^3 x^3 \tan ^{-1}(a x)^2-40 a x \log \left(a^2 x^2+1\right)+120 a^2 x^2 \tan ^{-1}(a x)^3+84 a^2 x^2 \tan ^{-1}(a x)+120 i a x \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)-264 i a x \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+60 a x \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)+132 a x \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-2 a x-5 i \pi ^3 a x-48 i a x \tan ^{-1}(a x)^3-42 a x \tan ^{-1}(a x)^2-40 \tan ^{-1}(a x)^3+120 a x \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)+264 a x \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)\right)}{40 x}","\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{1}{20} a^3 c^3 x^2-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2-a c^3 \log \left(a^2 x^2+1\right)+3 a^2 c^3 x \tan ^{-1}(a x)^3+\frac{21}{10} a^2 c^3 x \tan ^{-1}(a x)+\frac{3}{2} a c^3 \text{Li}_3\left(\frac{2}{1-i a x}-1\right)+\frac{33}{10} a c^3 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)-3 i a c^3 \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)+\frac{33}{5} i a c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \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,"(c^3*(-2*a*x - (5*I)*a*Pi^3*x - 2*a^3*x^3 + 84*a^2*x^2*ArcTan[a*x] + 4*a^4*x^4*ArcTan[a*x] - 42*a*x*ArcTan[a*x]^2 - 48*a^3*x^3*ArcTan[a*x]^2 - 6*a^5*x^5*ArcTan[a*x]^2 - 40*ArcTan[a*x]^3 - (48*I)*a*x*ArcTan[a*x]^3 + 120*a^2*x^2*ArcTan[a*x]^3 + 40*a^4*x^4*ArcTan[a*x]^3 + 8*a^6*x^6*ArcTan[a*x]^3 + 120*a*x*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + 264*a*x*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 40*a*x*Log[1 + a^2*x^2] + (120*I)*a*x*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (264*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 60*a*x*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 132*a*x*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(40*x)","A",0
385,1,464,503,0.7782472,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3}{x^3} \, dx","Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x]^3)/x^3,x]","\frac{c^3 \left(16 a^6 x^6 \tan ^{-1}(a x)^3-16 a^5 x^5 \tan ^{-1}(a x)^2+96 a^4 x^4 \tan ^{-1}(a x)^3+16 a^4 x^4 \tan ^{-1}(a x)-16 a^3 x^3-240 a^3 x^3 \tan ^{-1}(a x)^2+288 i a^2 x^2 \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+32 i a^2 x^2 \left(9 \tan ^{-1}(a x)^2+7\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-96 i a^2 x^2 \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)+288 a^2 x^2 \tan ^{-1}(a x) \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-288 a^2 x^2 \tan ^{-1}(a x) \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-144 i a^2 x^2 \text{Li}_4\left(e^{-2 i \tan ^{-1}(a x)}\right)-144 i a^2 x^2 \text{Li}_4\left(-e^{2 i \tan ^{-1}(a x)}\right)-3 i \pi ^4 a^2 x^2+96 i a^2 x^2 \tan ^{-1}(a x)^4+48 a^2 x^2 \tan ^{-1}(a x)^3+128 i a^2 x^2 \tan ^{-1}(a x)^2+16 a^2 x^2 \tan ^{-1}(a x)+192 a^2 x^2 \tan ^{-1}(a x)^3 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)+192 a^2 x^2 \tan ^{-1}(a x) \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)-192 a^2 x^2 \tan ^{-1}(a x)^3 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-448 a^2 x^2 \tan ^{-1}(a x) \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-96 a x \tan ^{-1}(a x)^2-32 \tan ^{-1}(a x)^3\right)}{64 x^2}","\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}{2} i a^2 c^3 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)-\frac{7}{2} i a^2 c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right)+\frac{9}{4} i a^2 c^3 \text{Li}_4\left(1-\frac{2}{i a x+1}\right)-\frac{9}{4} i a^2 c^3 \text{Li}_4\left(\frac{2}{i a x+1}-1\right)-\frac{9}{2} i a^2 c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)^2+\frac{9}{2} i a^2 c^3 \text{Li}_2\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)^2-\frac{9}{2} a^2 c^3 \text{Li}_3\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)+\frac{9}{2} a^2 c^3 \text{Li}_3\left(\frac{2}{i a x+1}-1\right) \tan ^{-1}(a x)+\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,"(c^3*((-3*I)*a^2*Pi^4*x^2 - 16*a^3*x^3 + 16*a^2*x^2*ArcTan[a*x] + 16*a^4*x^4*ArcTan[a*x] - 96*a*x*ArcTan[a*x]^2 + (128*I)*a^2*x^2*ArcTan[a*x]^2 - 240*a^3*x^3*ArcTan[a*x]^2 - 16*a^5*x^5*ArcTan[a*x]^2 - 32*ArcTan[a*x]^3 + 48*a^2*x^2*ArcTan[a*x]^3 + 96*a^4*x^4*ArcTan[a*x]^3 + 16*a^6*x^6*ArcTan[a*x]^3 + (96*I)*a^2*x^2*ArcTan[a*x]^4 + 192*a^2*x^2*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] + 192*a^2*x^2*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] - 448*a^2*x^2*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] - 192*a^2*x^2*ArcTan[a*x]^3*Log[1 + E^((2*I)*ArcTan[a*x])] + (288*I)*a^2*x^2*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (32*I)*a^2*x^2*(7 + 9*ArcTan[a*x]^2)*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - (96*I)*a^2*x^2*PolyLog[2, E^((2*I)*ArcTan[a*x])] + 288*a^2*x^2*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - 288*a^2*x^2*ArcTan[a*x]*PolyLog[3, -E^((2*I)*ArcTan[a*x])] - (144*I)*a^2*x^2*PolyLog[4, E^((-2*I)*ArcTan[a*x])] - (144*I)*a^2*x^2*PolyLog[4, -E^((2*I)*ArcTan[a*x])]))/(64*x^2)","A",0
386,1,331,336,0.735561,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3}{x^4} \, dx","Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x]^3)/x^4,x]","\frac{c^3 \left(2 a^6 x^6 \tan ^{-1}(a x)^3-3 a^5 x^5 \tan ^{-1}(a x)^2+18 a^4 x^4 \tan ^{-1}(a x)^3+6 a^4 x^4 \tan ^{-1}(a x)+48 i a^3 x^3 \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)-48 i a^3 x^3 \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+24 a^3 x^3 \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)+24 a^3 x^3 \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-2 i \pi ^3 a^3 x^3-6 a^3 x^3 \tan ^{-1}(a x)^2+48 a^3 x^3 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)+48 a^3 x^3 \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-18 a^2 x^2 \tan ^{-1}(a x)^3-6 a^2 x^2 \tan ^{-1}(a x)+6 a^3 x^3 \log \left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-3 a^3 x^3 \log \left(a^2 x^2+1\right)-3 a x \tan ^{-1}(a x)^2-2 \tan ^{-1}(a x)^3\right)}{6 x^3}","\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+3 a^4 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x \tan ^{-1}(a x)+4 a^3 c^3 \text{Li}_3\left(\frac{2}{1-i a x}-1\right)+4 a^3 c^3 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)-8 i a^3 c^3 \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)+8 i a^3 c^3 \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)+a^3 c^3 \log (x)-a^3 c^3 \tan ^{-1}(a x)^2+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{3 a^2 c^3 \tan ^{-1}(a x)^3}{x}-\frac{a^2 c^3 \tan ^{-1}(a x)}{x}-a^3 c^3 \log \left(a^2 x^2+1\right)-\frac{c^3 \tan ^{-1}(a x)^3}{3 x^3}-\frac{a c^3 \tan ^{-1}(a x)^2}{2 x^2}",1,"(c^3*((-2*I)*a^3*Pi^3*x^3 - 6*a^2*x^2*ArcTan[a*x] + 6*a^4*x^4*ArcTan[a*x] - 3*a*x*ArcTan[a*x]^2 - 6*a^3*x^3*ArcTan[a*x]^2 - 3*a^5*x^5*ArcTan[a*x]^2 - 2*ArcTan[a*x]^3 - 18*a^2*x^2*ArcTan[a*x]^3 + 18*a^4*x^4*ArcTan[a*x]^3 + 2*a^6*x^6*ArcTan[a*x]^3 + 48*a^3*x^3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + 48*a^3*x^3*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] + 6*a^3*x^3*Log[(a*x)/Sqrt[1 + a^2*x^2]] - 3*a^3*x^3*Log[1 + a^2*x^2] + (48*I)*a^3*x^3*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (48*I)*a^3*x^3*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 24*a^3*x^3*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 24*a^3*x^3*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(6*x^3)","A",0
387,1,154,217,0.2603002,"\int \frac{x^4 \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx","Integrate[(x^4*ArcTan[a*x]^3)/(c + a^2*c*x^2),x]","\frac{4 a^3 x^3 \tan ^{-1}(a x)^3-6 \log \left(a^2 x^2+1\right)-6 a^2 x^2 \tan ^{-1}(a x)^2+48 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-24 \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)+3 \tan ^{-1}(a x)^4-12 a x \tan ^{-1}(a x)^3+16 i \tan ^{-1}(a x)^3-6 \tan ^{-1}(a x)^2+12 a x \tan ^{-1}(a x)-48 \tan ^{-1}(a x)^2 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)}{12 a^5 c}","-\frac{2 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)}{a^5 c}-\frac{4 i \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)}{a^5 c}+\frac{\tan ^{-1}(a x)^4}{4 a^5 c}-\frac{4 i \tan ^{-1}(a x)^3}{3 a^5 c}-\frac{\tan ^{-1}(a x)^2}{2 a^5 c}-\frac{4 \log \left(\frac{2}{1+i a x}\right) \tan ^{-1}(a x)^2}{a^5 c}-\frac{x \tan ^{-1}(a x)^3}{a^4 c}+\frac{x \tan ^{-1}(a x)}{a^4 c}-\frac{x^2 \tan ^{-1}(a x)^2}{2 a^3 c}+\frac{x^3 \tan ^{-1}(a x)^3}{3 a^2 c}-\frac{\log \left(a^2 x^2+1\right)}{2 a^5 c}",1,"(12*a*x*ArcTan[a*x] - 6*ArcTan[a*x]^2 - 6*a^2*x^2*ArcTan[a*x]^2 + (16*I)*ArcTan[a*x]^3 - 12*a*x*ArcTan[a*x]^3 + 4*a^3*x^3*ArcTan[a*x]^3 + 3*ArcTan[a*x]^4 - 48*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 6*Log[1 + a^2*x^2] + (48*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - 24*PolyLog[3, -E^((2*I)*ArcTan[a*x])])/(12*a^5*c)","A",0
388,1,162,260,0.2927278,"\int \frac{x^3 \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx","Integrate[(x^3*ArcTan[a*x]^3)/(c + a^2*c*x^2),x]","\frac{2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3+6 \tan ^{-1}(a x) \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-6 i \left(\tan ^{-1}(a x)^2-1\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+3 i \text{Li}_4\left(-e^{2 i \tan ^{-1}(a x)}\right)-i \tan ^{-1}(a x)^4-6 a x \tan ^{-1}(a x)^2+6 i \tan ^{-1}(a x)^2+4 \tan ^{-1}(a x)^3 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-12 \tan ^{-1}(a x) \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)}{4 a^4 c}","-\frac{3 i \text{Li}_2\left(1-\frac{2}{i a x+1}\right)}{2 a^4 c}-\frac{3 i \text{Li}_4\left(1-\frac{2}{i a x+1}\right)}{4 a^4 c}+\frac{3 i \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)^2}{2 a^4 c}+\frac{3 \text{Li}_3\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)}{2 a^4 c}+\frac{i \tan ^{-1}(a x)^4}{4 a^4 c}+\frac{\tan ^{-1}(a x)^3}{2 a^4 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 x \tan ^{-1}(a x)^2}{2 a^3 c}+\frac{x^2 \tan ^{-1}(a x)^3}{2 a^2 c}",1,"((6*I)*ArcTan[a*x]^2 - 6*a*x*ArcTan[a*x]^2 + 2*(1 + a^2*x^2)*ArcTan[a*x]^3 - I*ArcTan[a*x]^4 - 12*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] + 4*ArcTan[a*x]^3*Log[1 + E^((2*I)*ArcTan[a*x])] - (6*I)*(-1 + ArcTan[a*x]^2)*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 6*ArcTan[a*x]*PolyLog[3, -E^((2*I)*ArcTan[a*x])] + (3*I)*PolyLog[4, -E^((2*I)*ArcTan[a*x])])/(4*a^4*c)","A",0
389,1,93,130,0.2163799,"\int \frac{x^2 \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx","Integrate[(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2),x]","\frac{-3 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)+\frac{3}{2} \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-\frac{1}{4} \tan ^{-1}(a x)^2 \left(\tan ^{-1}(a x)^2+(-4 a x+4 i) \tan ^{-1}(a x)-12 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)\right)}{a^3 c}","\frac{3 \text{Li}_3\left(1-\frac{2}{i a x+1}\right)}{2 a^3 c}+\frac{3 i \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)}{a^3 c}-\frac{\tan ^{-1}(a x)^4}{4 a^3 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{x \tan ^{-1}(a x)^3}{a^2 c}",1,"(-1/4*(ArcTan[a*x]^2*((4*I - 4*a*x)*ArcTan[a*x] + ArcTan[a*x]^2 - 12*Log[1 + E^((2*I)*ArcTan[a*x])])) - (3*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + (3*PolyLog[3, -E^((2*I)*ArcTan[a*x])])/2)/(a^3*c)","A",0
390,1,149,138,0.0119763,"\int \frac{x \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx","Integrate[(x*ArcTan[a*x]^3)/(c + a^2*c*x^2),x]","\frac{3 i \text{Li}_4\left(\frac{a x+i}{a x-i}\right)}{4 a^2 c}-\frac{3 i \text{Li}_2\left(\frac{a x+i}{a x-i}\right) \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{3 \text{Li}_3\left(\frac{a x+i}{a x-i}\right) \tan ^{-1}(a x)}{2 a^2 c}-\frac{i \tan ^{-1}(a x)^4}{4 a^2 c}-\frac{\log \left(\frac{2 i}{-a x+i}\right) \tan ^{-1}(a x)^3}{a^2 c}","\frac{3 i \text{Li}_4\left(1-\frac{2}{i a x+1}\right)}{4 a^2 c}-\frac{3 i \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)^2}{2 a^2 c}-\frac{3 \text{Li}_3\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)}{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,"((-1/4*I)*ArcTan[a*x]^4)/(a^2*c) - (ArcTan[a*x]^3*Log[(2*I)/(I - a*x)])/(a^2*c) - (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, (I + a*x)/(-I + a*x)])/(a^2*c) - (3*ArcTan[a*x]*PolyLog[3, (I + a*x)/(-I + a*x)])/(2*a^2*c) + (((3*I)/4)*PolyLog[4, (I + a*x)/(-I + a*x)])/(a^2*c)","A",1
391,1,16,16,0.003657,"\int \frac{\tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx","Integrate[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
392,1,354,124,0.0589468,"\int \frac{\tan ^{-1}(a x)^3}{x \left(c+a^2 c x^2\right)} \, dx","Integrate[ArcTan[a*x]^3/(x*(c + a^2*c*x^2)),x]","-\frac{3 i \text{Li}_4\left(\frac{-a x-i}{a x-i}\right)}{4 c}-\frac{3 i \text{Li}_4\left(-\frac{a x+i}{i-a x}\right)}{4 c}+\frac{3 i \text{Li}_4\left(\frac{a x+i}{a x-i}\right)}{4 c}+\frac{3 i \text{Li}_2\left(\frac{-a x-i}{a x-i}\right) \tan ^{-1}(a x)^2}{2 c}+\frac{3 i \text{Li}_2\left(-\frac{a x+i}{i-a x}\right) \tan ^{-1}(a x)^2}{2 c}-\frac{3 i \text{Li}_2\left(\frac{a x+i}{a x-i}\right) \tan ^{-1}(a x)^2}{2 c}+\frac{3 \text{Li}_3\left(\frac{-a x-i}{a x-i}\right) \tan ^{-1}(a x)}{2 c}+\frac{3 \text{Li}_3\left(-\frac{a x+i}{i-a x}\right) \tan ^{-1}(a x)}{2 c}-\frac{3 \text{Li}_3\left(\frac{a x+i}{a x-i}\right) \tan ^{-1}(a x)}{2 c}+\frac{i \tan ^{-1}(a x)^4}{4 c}+\frac{\log \left(\frac{2 i}{-a x+i}\right) \tan ^{-1}(a x)^3}{c}+\frac{2 \tan ^{-1}(a x)^3 \tanh ^{-1}\left(1-\frac{2 i}{-a x+i}\right)}{c}","\frac{3 i \text{Li}_4\left(\frac{2}{1-i a x}-1\right)}{4 c}-\frac{3 i \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)^2}{2 c}+\frac{3 \text{Li}_3\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{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 + (2*ArcTan[a*x]^3*ArcTanh[1 - (2*I)/(I - a*x)])/c + (ArcTan[a*x]^3*Log[(2*I)/(I - a*x)])/c + (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, (-I - a*x)/(-I + a*x)])/c + (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, -((I + a*x)/(I - a*x))])/c - (((3*I)/2)*ArcTan[a*x]^2*PolyLog[2, (I + a*x)/(-I + a*x)])/c + (3*ArcTan[a*x]*PolyLog[3, (-I - a*x)/(-I + a*x)])/(2*c) + (3*ArcTan[a*x]*PolyLog[3, -((I + a*x)/(I - a*x))])/(2*c) - (3*ArcTan[a*x]*PolyLog[3, (I + a*x)/(-I + a*x)])/(2*c) - (((3*I)/4)*PolyLog[4, (-I - a*x)/(-I + a*x)])/c - (((3*I)/4)*PolyLog[4, -((I + a*x)/(I - a*x))])/c + (((3*I)/4)*PolyLog[4, (I + a*x)/(-I + a*x)])/c","B",0
393,1,108,122,0.1777853,"\int \frac{\tan ^{-1}(a x)^3}{x^2 \left(c+a^2 c x^2\right)} \, dx","Integrate[ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)),x]","\frac{a \left(3 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+\frac{3}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-\frac{1}{4} \tan ^{-1}(a x)^4-\frac{\tan ^{-1}(a x)^3}{a x}+i \tan ^{-1}(a x)^3+3 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-\frac{i \pi ^3}{8}\right)}{c}","\frac{3 a \text{Li}_3\left(\frac{2}{1-i a x}-1\right)}{2 c}-\frac{3 i a \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{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,"(a*((-1/8*I)*Pi^3 + I*ArcTan[a*x]^3 - ArcTan[a*x]^3/(a*x) - ArcTan[a*x]^4/4 + 3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (3*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (3*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/2))/c","A",0
394,1,189,262,0.4052686,"\int \frac{\tan ^{-1}(a x)^3}{x^3 \left(c+a^2 c x^2\right)} \, dx","Integrate[ArcTan[a*x]^3/(x^3*(c + a^2*c*x^2)),x]","\frac{i a^2 \left(\frac{32 i \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3}{a^2 x^2}-96 \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+96 i \tan ^{-1}(a x) \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-96 \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)+48 \text{Li}_4\left(e^{-2 i \tan ^{-1}(a x)}\right)-16 \tan ^{-1}(a x)^4+\frac{96 i \tan ^{-1}(a x)^2}{a x}-96 \tan ^{-1}(a x)^2+64 i \tan ^{-1}(a x)^3 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-192 i \tan ^{-1}(a x) \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)+\pi ^4\right)}{64 c}","-\frac{3 i a^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)}{2 c}-\frac{3 i a^2 \text{Li}_4\left(\frac{2}{1-i a x}-1\right)}{4 c}+\frac{3 i a^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)^2}{2 c}-\frac{3 a^2 \text{Li}_3\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{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,"((I/64)*a^2*(Pi^4 - 96*ArcTan[a*x]^2 + ((96*I)*ArcTan[a*x]^2)/(a*x) + ((32*I)*(1 + a^2*x^2)*ArcTan[a*x]^3)/(a^2*x^2) - 16*ArcTan[a*x]^4 + (64*I)*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] - (192*I)*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] - 96*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - 96*PolyLog[2, E^((2*I)*ArcTan[a*x])] + (96*I)*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 48*PolyLog[4, E^((-2*I)*ArcTan[a*x])]))/c","A",0
395,1,180,227,0.5112354,"\int \frac{\tan ^{-1}(a x)^3}{x^4 \left(c+a^2 c x^2\right)} \, dx","Integrate[ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)),x]","\frac{a^3 \left(-\frac{\tan ^{-1}(a x)^3}{3 a^3 x^3}+\log \left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-\frac{\tan ^{-1}(a x)^2}{2 a^2 x^2}-4 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)-2 \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)+\frac{1}{4} \tan ^{-1}(a x)^4+\frac{\tan ^{-1}(a x)^3}{a x}-\frac{4}{3} i \tan ^{-1}(a x)^3-\frac{1}{2} \tan ^{-1}(a x)^2-\frac{\tan ^{-1}(a x)}{a x}-4 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)+\frac{i \pi ^3}{6}\right)}{c}","-\frac{2 a^3 \text{Li}_3\left(\frac{2}{1-i a x}-1\right)}{c}+\frac{4 i a^3 \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{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^3 \tan ^{-1}(a x)^2}{2 c}-\frac{4 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{c}+\frac{a^2 \tan ^{-1}(a x)^3}{c x}-\frac{a^2 \tan ^{-1}(a x)}{c x}-\frac{a^3 \log \left(a^2 x^2+1\right)}{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^3*((I/6)*Pi^3 - ArcTan[a*x]/(a*x) - ArcTan[a*x]^2/2 - ArcTan[a*x]^2/(2*a^2*x^2) - ((4*I)/3)*ArcTan[a*x]^3 - ArcTan[a*x]^3/(3*a^3*x^3) + ArcTan[a*x]^3/(a*x) + ArcTan[a*x]^4/4 - 4*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + Log[(a*x)/Sqrt[1 + a^2*x^2]] - (4*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - 2*PolyLog[3, E^((-2*I)*ArcTan[a*x])]))/c","A",0
396,1,156,270,0.197399,"\int \frac{x^3 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x^3*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2,x]","\frac{24 i \tan ^{-1}(a x)^2 \text{Li}_2\left(-e^{2 i \tan ^{-1}(a x)}\right)-24 \tan ^{-1}(a x) \text{Li}_3\left(-e^{2 i \tan ^{-1}(a x)}\right)-12 i \text{Li}_4\left(-e^{2 i \tan ^{-1}(a x)}\right)+4 i \tan ^{-1}(a x)^4-16 \tan ^{-1}(a x)^3 \log \left(1+e^{2 i \tan ^{-1}(a x)}\right)-6 \tan ^{-1}(a x)^2 \sin \left(2 \tan ^{-1}(a x)\right)+3 \sin \left(2 \tan ^{-1}(a x)\right)+4 \tan ^{-1}(a x)^3 \cos \left(2 \tan ^{-1}(a x)\right)-6 \tan ^{-1}(a x) \cos \left(2 \tan ^{-1}(a x)\right)}{16 a^4 c^2}","\frac{3 i \text{Li}_4\left(1-\frac{2}{i a x+1}\right)}{4 a^4 c^2}-\frac{3 i \text{Li}_2\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac{3 \text{Li}_3\left(1-\frac{2}{i a x+1}\right) \tan ^{-1}(a x)}{2 a^4 c^2}-\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{\tan ^{-1}(a x)^3}{2 a^4 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{3 x}{8 a^3 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)}",1,"((4*I)*ArcTan[a*x]^4 - 6*ArcTan[a*x]*Cos[2*ArcTan[a*x]] + 4*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]] - 16*ArcTan[a*x]^3*Log[1 + E^((2*I)*ArcTan[a*x])] + (24*I)*ArcTan[a*x]^2*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - 24*ArcTan[a*x]*PolyLog[3, -E^((2*I)*ArcTan[a*x])] - (12*I)*PolyLog[4, -E^((2*I)*ArcTan[a*x])] + 3*Sin[2*ArcTan[a*x]] - 6*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]])/(16*a^4*c^2)","A",0
397,1,74,135,0.0593568,"\int \frac{x^2 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2,x]","\frac{\left(a^2 x^2+1\right) \tan ^{-1}(a x)^4+3 \left(a^2 x^2-1\right) \tan ^{-1}(a x)^2-4 a x \tan ^{-1}(a x)^3+6 a x \tan ^{-1}(a x)+3}{8 a^3 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{x \tan ^{-1}(a x)^3}{2 a^2 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{3}{8 a^3 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)}",1,"(3 + 6*a*x*ArcTan[a*x] + 3*(-1 + a^2*x^2)*ArcTan[a*x]^2 - 4*a*x*ArcTan[a*x]^3 + (1 + a^2*x^2)*ArcTan[a*x]^4)/(8*a^3*c^2*(1 + a^2*x^2))","A",1
398,1,68,133,0.0490443,"\int \frac{x \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2,x]","\frac{2 \left(a^2 x^2-1\right) \tan ^{-1}(a x)^3+\left(3-3 a^2 x^2\right) \tan ^{-1}(a x)-3 a x+6 a x \tan ^{-1}(a x)^2}{8 a^2 c^2 \left(a^2 x^2+1\right)}","-\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*a*x + (3 - 3*a^2*x^2)*ArcTan[a*x] + 6*a*x*ArcTan[a*x]^2 + 2*(-1 + a^2*x^2)*ArcTan[a*x]^3)/(8*a^2*c^2*(1 + a^2*x^2))","A",1
399,1,71,129,0.030864,"\int \frac{\tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]^3/(c + a^2*c*x^2)^2,x]","\frac{\left(a^2 x^2+1\right) \tan ^{-1}(a x)^4+\left(3-3 a^2 x^2\right) \tan ^{-1}(a x)^2+4 a x \tan ^{-1}(a x)^3-6 a x \tan ^{-1}(a x)-3}{8 c^2 \left(a^3 x^2+a\right)}","-\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 - 6*a*x*ArcTan[a*x] + (3 - 3*a^2*x^2)*ArcTan[a*x]^2 + 4*a*x*ArcTan[a*x]^3 + (1 + a^2*x^2)*ArcTan[a*x]^4)/(8*c^2*(a + a^3*x^2))","A",1
400,1,156,240,0.2338679,"\int \frac{\tan ^{-1}(a x)^3}{x \left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]^3/(x*(c + a^2*c*x^2)^2),x]","\frac{96 i \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+96 \tan ^{-1}(a x) \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-48 i \text{Li}_4\left(e^{-2 i \tan ^{-1}(a x)}\right)+16 i \tan ^{-1}(a x)^4+64 \tan ^{-1}(a x)^3 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-24 \tan ^{-1}(a x)^2 \sin \left(2 \tan ^{-1}(a x)\right)+12 \sin \left(2 \tan ^{-1}(a x)\right)+16 \tan ^{-1}(a x)^3 \cos \left(2 \tan ^{-1}(a x)\right)-24 \tan ^{-1}(a x) \cos \left(2 \tan ^{-1}(a x)\right)-i \pi ^4}{64 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{3 i \text{Li}_4\left(\frac{2}{1-i a x}-1\right)}{4 c^2}-\frac{3 i \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)^2}{2 c^2}+\frac{3 \text{Li}_3\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{2 c^2}-\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,"((-I)*Pi^4 + (16*I)*ArcTan[a*x]^4 - 24*ArcTan[a*x]*Cos[2*ArcTan[a*x]] + 16*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]] + 64*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] + (96*I)*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + 96*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - (48*I)*PolyLog[4, E^((-2*I)*ArcTan[a*x])] + 12*Sin[2*ArcTan[a*x]] - 24*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]])/(64*c^2)","A",0
401,1,157,234,0.3559949,"\int \frac{\tan ^{-1}(a x)^3}{x^2 \left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)^2),x]","\frac{a \left(48 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+24 \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-6 \tan ^{-1}(a x)^4-\frac{16 \tan ^{-1}(a x)^3}{a x}+16 i \tan ^{-1}(a x)^3+48 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-4 \tan ^{-1}(a x)^3 \sin \left(2 \tan ^{-1}(a x)\right)+6 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)-6 \tan ^{-1}(a x)^2 \cos \left(2 \tan ^{-1}(a x)\right)+3 \cos \left(2 \tan ^{-1}(a x)\right)-2 i \pi ^3\right)}{16 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 \text{Li}_3\left(\frac{2}{1-i a x}-1\right)}{2 c^2}-\frac{3 i a \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{c^2}-\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,"(a*((-2*I)*Pi^3 + (16*I)*ArcTan[a*x]^3 - (16*ArcTan[a*x]^3)/(a*x) - 6*ArcTan[a*x]^4 + 3*Cos[2*ArcTan[a*x]] - 6*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 48*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (48*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + 24*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 6*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 4*ArcTan[a*x]^3*Sin[2*ArcTan[a*x]]))/(16*c^2)","A",0
402,1,243,374,0.7375999,"\int \frac{\tan ^{-1}(a x)^3}{x^3 \left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]^3/(x^3*(c + a^2*c*x^2)^2),x]","\frac{a^2 \left(-\frac{16 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3}{a^2 x^2}-96 i \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)-96 \tan ^{-1}(a x) \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-48 i \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)+48 i \text{Li}_4\left(e^{-2 i \tan ^{-1}(a x)}\right)-16 i \tan ^{-1}(a x)^4-\frac{48 \tan ^{-1}(a x)^2}{a x}-48 i \tan ^{-1}(a x)^2-64 \tan ^{-1}(a x)^3 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)+96 \tan ^{-1}(a x) \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)+12 \tan ^{-1}(a x)^2 \sin \left(2 \tan ^{-1}(a x)\right)-6 \sin \left(2 \tan ^{-1}(a x)\right)-8 \tan ^{-1}(a x)^3 \cos \left(2 \tan ^{-1}(a x)\right)+12 \tan ^{-1}(a x) \cos \left(2 \tan ^{-1}(a x)\right)+i \pi ^4\right)}{32 c^2}","-\frac{3 i a^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)}{2 c^2}-\frac{3 i a^2 \text{Li}_4\left(\frac{2}{1-i a x}-1\right)}{2 c^2}+\frac{3 i a^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)^2}{c^2}-\frac{3 a^2 \text{Li}_3\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{c^2}-\frac{a^2 \tan ^{-1}(a x)^3}{2 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{3 a^3 x}{8 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{\tan ^{-1}(a x)^3}{2 c^2 x^2}-\frac{3 a \tan ^{-1}(a x)^2}{2 c^2 x}",1,"(a^2*(I*Pi^4 - (48*I)*ArcTan[a*x]^2 - (48*ArcTan[a*x]^2)/(a*x) - (16*(1 + a^2*x^2)*ArcTan[a*x]^3)/(a^2*x^2) - (16*I)*ArcTan[a*x]^4 + 12*ArcTan[a*x]*Cos[2*ArcTan[a*x]] - 8*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]] - 64*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] + 96*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] - (96*I)*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (48*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] - 96*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + (48*I)*PolyLog[4, E^((-2*I)*ArcTan[a*x])] - 6*Sin[2*ArcTan[a*x]] + 12*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]]))/(32*c^2)","A",0
403,1,243,332,0.934401,"\int \frac{\tan ^{-1}(a x)^3}{x^4 \left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)^2),x]","\frac{a^3 \left(-\frac{\tan ^{-1}(a x)^3}{3 a^3 x^3}+\log \left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-\frac{\tan ^{-1}(a x)^2}{2 a^2 x^2}-7 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)-\frac{7}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)+\frac{5}{8} \tan ^{-1}(a x)^4+\frac{2 \tan ^{-1}(a x)^3}{a x}-\frac{7}{3} i \tan ^{-1}(a x)^3-\frac{1}{2} \tan ^{-1}(a x)^2-\frac{\tan ^{-1}(a x)}{a x}-7 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)+\frac{1}{4} \tan ^{-1}(a x)^3 \sin \left(2 \tan ^{-1}(a x)\right)-\frac{3}{8} \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)+\frac{3}{8} \tan ^{-1}(a x)^2 \cos \left(2 \tan ^{-1}(a x)\right)-\frac{3}{16} \cos \left(2 \tan ^{-1}(a x)\right)+\frac{7 i \pi ^3}{24}\right)}{c^2}","-\frac{7 a^3 \text{Li}_3\left(\frac{2}{1-i a x}-1\right)}{2 c^2}+\frac{7 i a^3 \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{c^2}+\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{7 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{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{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}{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{3 a^3 \tan ^{-1}(a x)^2}{4 c^2 \left(a^2 x^2+1\right)}-\frac{\tan ^{-1}(a x)^3}{3 c^2 x^3}-\frac{a \tan ^{-1}(a x)^2}{2 c^2 x^2}",1,"(a^3*(((7*I)/24)*Pi^3 - ArcTan[a*x]/(a*x) - ArcTan[a*x]^2/2 - ArcTan[a*x]^2/(2*a^2*x^2) - ((7*I)/3)*ArcTan[a*x]^3 - ArcTan[a*x]^3/(3*a^3*x^3) + (2*ArcTan[a*x]^3)/(a*x) + (5*ArcTan[a*x]^4)/8 - (3*Cos[2*ArcTan[a*x]])/16 + (3*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]])/8 - 7*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + Log[(a*x)/Sqrt[1 + a^2*x^2]] - (7*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (7*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/2 - (3*ArcTan[a*x]*Sin[2*ArcTan[a*x]])/8 + (ArcTan[a*x]^3*Sin[2*ArcTan[a*x]])/4))/c^2","A",0
404,1,105,212,0.2695616,"\int \frac{x^3 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^3*ArcTan[a*x]^3)/(c + a^2*c*x^2)^3,x]","\frac{-3 a x \left(17 a^2 x^2+15\right)+24 a x \left(5 a^2 x^2+3\right) \tan ^{-1}(a x)^2+8 \left(5 a^4 x^4-6 a^2 x^2-3\right) \tan ^{-1}(a x)^3+\left(-51 a^4 x^4+18 a^2 x^2+45\right) \tan ^{-1}(a x)}{256 a^4 c^3 \left(a^2 x^2+1\right)^2}","-\frac{3 \tan ^{-1}(a x)^3}{32 a^4 c^3}-\frac{27 \tan ^{-1}(a x)}{256 a^4 c^3}+\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}{128 a 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 \tan ^{-1}(a x)}{32 a^4 c^3 \left(a^2 x^2+1\right)}-\frac{45 x}{256 a^3 c^3 \left(a^2 x^2+1\right)}+\frac{9 x \tan ^{-1}(a x)^2}{32 a^3 c^3 \left(a^2 x^2+1\right)}",1,"(-3*a*x*(15 + 17*a^2*x^2) + (45 + 18*a^2*x^2 - 51*a^4*x^4)*ArcTan[a*x] + 24*a*x*(3 + 5*a^2*x^2)*ArcTan[a*x]^2 + 8*(-3 - 6*a^2*x^2 + 5*a^4*x^4)*ArcTan[a*x]^3)/(256*a^4*c^3*(1 + a^2*x^2)^2)","A",1
405,1,111,237,0.0763498,"\int \frac{x^2 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^3,x]","\frac{\left(6 a x-6 a^3 x^3\right) \tan ^{-1}(a x)-3 a^2 x^2+4 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^4+16 a x \left(a^2 x^2-1\right) \tan ^{-1}(a x)^3-3 \left(a^4 x^4-6 a^2 x^2+1\right) \tan ^{-1}(a x)^2}{128 a^3 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{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 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{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{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}",1,"(-3*a^2*x^2 + (6*a*x - 6*a^3*x^3)*ArcTan[a*x] - 3*(1 - 6*a^2*x^2 + a^4*x^4)*ArcTan[a*x]^2 + 16*a*x*(-1 + a^2*x^2)*ArcTan[a*x]^3 + 4*(1 + a^2*x^2)^2*ArcTan[a*x]^4)/(128*a^3*c^3*(1 + a^2*x^2)^2)","A",1
406,1,103,208,0.0862676,"\int \frac{x \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x*ArcTan[a*x]^3)/(c + a^2*c*x^2)^3,x]","\frac{-3 a x \left(15 a^2 x^2+17\right)+24 a x \left(3 a^2 x^2+5\right) \tan ^{-1}(a x)^2+8 \left(3 a^4 x^4+6 a^2 x^2-5\right) \tan ^{-1}(a x)^3-3 \left(15 a^4 x^4+6 a^2 x^2-17\right) \tan ^{-1}(a x)}{256 c^3 \left(a^3 x^2+a\right)^2}","-\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*a*x*(17 + 15*a^2*x^2) - 3*(-17 + 6*a^2*x^2 + 15*a^4*x^4)*ArcTan[a*x] + 24*a*x*(5 + 3*a^2*x^2)*ArcTan[a*x]^2 + 8*(-5 + 6*a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x]^3)/(256*c^3*(a + a^3*x^2)^2)","A",1
407,1,114,225,0.0607102,"\int \frac{\tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]^3/(c + a^2*c*x^2)^3,x]","-\frac{45 a^2 x^2-12 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^4-16 a x \left(3 a^2 x^2+5\right) \tan ^{-1}(a x)^3+6 a x \left(15 a^2 x^2+17\right) \tan ^{-1}(a x)+3 \left(15 a^4 x^4+6 a^2 x^2-17\right) \tan ^{-1}(a x)^2+48}{128 a c^3 \left(a^2 x^2+1\right)^2}","-\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,"-1/128*(48 + 45*a^2*x^2 + 6*a*x*(17 + 15*a^2*x^2)*ArcTan[a*x] + 3*(-17 + 6*a^2*x^2 + 15*a^4*x^4)*ArcTan[a*x]^2 - 16*a*x*(5 + 3*a^2*x^2)*ArcTan[a*x]^3 - 12*(1 + a^2*x^2)^2*ArcTan[a*x]^4)/(a*c^3*(1 + a^2*x^2)^2)","A",1
408,1,208,332,0.3324509,"\int \frac{\tan ^{-1}(a x)^3}{x \left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]^3/(x*(c + a^2*c*x^2)^3),x]","\frac{1536 i \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+1536 \tan ^{-1}(a x) \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-768 i \text{Li}_4\left(e^{-2 i \tan ^{-1}(a x)}\right)+256 i \tan ^{-1}(a x)^4+1024 \tan ^{-1}(a x)^3 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-576 \tan ^{-1}(a x)^2 \sin \left(2 \tan ^{-1}(a x)\right)-24 \tan ^{-1}(a x)^2 \sin \left(4 \tan ^{-1}(a x)\right)+288 \sin \left(2 \tan ^{-1}(a x)\right)+3 \sin \left(4 \tan ^{-1}(a x)\right)+384 \tan ^{-1}(a x)^3 \cos \left(2 \tan ^{-1}(a x)\right)+32 \tan ^{-1}(a x)^3 \cos \left(4 \tan ^{-1}(a x)\right)-576 \tan ^{-1}(a x) \cos \left(2 \tan ^{-1}(a x)\right)-12 \tan ^{-1}(a x) \cos \left(4 \tan ^{-1}(a x)\right)-16 i \pi ^4}{1024 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{3 i \text{Li}_4\left(\frac{2}{1-i a x}-1\right)}{4 c^3}-\frac{3 i \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)^2}{2 c^3}+\frac{3 \text{Li}_3\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{2 c^3}-\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,"((-16*I)*Pi^4 + (256*I)*ArcTan[a*x]^4 - 576*ArcTan[a*x]*Cos[2*ArcTan[a*x]] + 384*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]] - 12*ArcTan[a*x]*Cos[4*ArcTan[a*x]] + 32*ArcTan[a*x]^3*Cos[4*ArcTan[a*x]] + 1024*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] + (1536*I)*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + 1536*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - (768*I)*PolyLog[4, E^((-2*I)*ArcTan[a*x])] + 288*Sin[2*ArcTan[a*x]] - 576*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]] + 3*Sin[4*ArcTan[a*x]] - 24*ArcTan[a*x]^2*Sin[4*ArcTan[a*x]])/(1024*c^3)","A",0
409,1,232,332,0.5658386,"\int \frac{\tan ^{-1}(a x)^3}{x^2 \left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)^3),x]","\frac{a \left(-\frac{a x \tan ^{-1}(a x)^3}{a^2 x^2+1}+3 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)+\frac{3}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-\frac{15}{32} \tan ^{-1}(a x)^4-\frac{\tan ^{-1}(a x)^3}{a x}+i \tan ^{-1}(a x)^3+3 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)-\frac{1}{32} \tan ^{-1}(a x)^3 \sin \left(4 \tan ^{-1}(a x)\right)+\frac{3}{4} \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)+\frac{3}{256} \tan ^{-1}(a x) \sin \left(4 \tan ^{-1}(a x)\right)-\frac{3}{4} \tan ^{-1}(a x)^2 \cos \left(2 \tan ^{-1}(a x)\right)-\frac{3}{128} \tan ^{-1}(a x)^2 \cos \left(4 \tan ^{-1}(a x)\right)+\frac{3}{8} \cos \left(2 \tan ^{-1}(a x)\right)+\frac{3 \cos \left(4 \tan ^{-1}(a x)\right)}{1024}-\frac{i \pi ^3}{8}\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{3 a \text{Li}_3\left(\frac{2}{1-i a x}-1\right)}{2 c^3}-\frac{3 i a \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{c^3}-\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,"(a*((-1/8*I)*Pi^3 + I*ArcTan[a*x]^3 - ArcTan[a*x]^3/(a*x) - (a*x*ArcTan[a*x]^3)/(1 + a^2*x^2) - (15*ArcTan[a*x]^4)/32 + (3*Cos[2*ArcTan[a*x]])/8 - (3*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]])/4 + (3*Cos[4*ArcTan[a*x]])/1024 - (3*ArcTan[a*x]^2*Cos[4*ArcTan[a*x]])/128 + 3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (3*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (3*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/2 + (3*ArcTan[a*x]*Sin[2*ArcTan[a*x]])/4 + (3*ArcTan[a*x]*Sin[4*ArcTan[a*x]])/256 - (ArcTan[a*x]^3*Sin[4*ArcTan[a*x]])/32))/c^3","A",0
410,1,295,478,1.0969377,"\int \frac{\tan ^{-1}(a x)^3}{x^3 \left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]^3/(x^3*(c + a^2*c*x^2)^3),x]","\frac{a^2 \left(-\frac{512 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3}{a^2 x^2}-4608 i \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)-4608 \tan ^{-1}(a x) \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)-1536 i \text{Li}_2\left(e^{2 i \tan ^{-1}(a x)}\right)+2304 i \text{Li}_4\left(e^{-2 i \tan ^{-1}(a x)}\right)-768 i \tan ^{-1}(a x)^4-\frac{1536 \tan ^{-1}(a x)^2}{a x}-1536 i \tan ^{-1}(a x)^2-3072 \tan ^{-1}(a x)^3 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)+3072 \tan ^{-1}(a x) \log \left(1-e^{2 i \tan ^{-1}(a x)}\right)+960 \tan ^{-1}(a x)^2 \sin \left(2 \tan ^{-1}(a x)\right)+24 \tan ^{-1}(a x)^2 \sin \left(4 \tan ^{-1}(a x)\right)-480 \sin \left(2 \tan ^{-1}(a x)\right)-3 \sin \left(4 \tan ^{-1}(a x)\right)-640 \tan ^{-1}(a x)^3 \cos \left(2 \tan ^{-1}(a x)\right)-32 \tan ^{-1}(a x)^3 \cos \left(4 \tan ^{-1}(a x)\right)+960 \tan ^{-1}(a x) \cos \left(2 \tan ^{-1}(a x)\right)+12 \tan ^{-1}(a x) \cos \left(4 \tan ^{-1}(a x)\right)+48 i \pi ^4\right)}{1024 c^3}","-\frac{3 i a^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right)}{2 c^3}-\frac{9 i a^2 \text{Li}_4\left(\frac{2}{1-i a x}-1\right)}{4 c^3}+\frac{9 i a^2 \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)^2}{2 c^3}-\frac{9 a^2 \text{Li}_3\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{2 c^3}-\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^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{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{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{\tan ^{-1}(a x)^3}{2 c^3 x^2}-\frac{3 a \tan ^{-1}(a x)^2}{2 c^3 x}",1,"(a^2*((48*I)*Pi^4 - (1536*I)*ArcTan[a*x]^2 - (1536*ArcTan[a*x]^2)/(a*x) - (512*(1 + a^2*x^2)*ArcTan[a*x]^3)/(a^2*x^2) - (768*I)*ArcTan[a*x]^4 + 960*ArcTan[a*x]*Cos[2*ArcTan[a*x]] - 640*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]] + 12*ArcTan[a*x]*Cos[4*ArcTan[a*x]] - 32*ArcTan[a*x]^3*Cos[4*ArcTan[a*x]] - 3072*ArcTan[a*x]^3*Log[1 - E^((-2*I)*ArcTan[a*x])] + 3072*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] - (4608*I)*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - (1536*I)*PolyLog[2, E^((2*I)*ArcTan[a*x])] - 4608*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + (2304*I)*PolyLog[4, E^((-2*I)*ArcTan[a*x])] - 480*Sin[2*ArcTan[a*x]] + 960*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]] - 3*Sin[4*ArcTan[a*x]] + 24*ArcTan[a*x]^2*Sin[4*ArcTan[a*x]]))/(1024*c^3)","A",0
411,1,301,432,1.2155586,"\int \frac{\tan ^{-1}(a x)^3}{x^4 \left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]^3/(x^4*(c + a^2*c*x^2)^3),x]","\frac{a^3 \left(-\frac{\tan ^{-1}(a x)^3}{3 a^3 x^3}+\log \left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-\frac{\tan ^{-1}(a x)^2}{2 a^2 x^2}-10 i \tan ^{-1}(a x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(a x)}\right)-5 \text{Li}_3\left(e^{-2 i \tan ^{-1}(a x)}\right)+\frac{35}{32} \tan ^{-1}(a x)^4+\frac{3 \tan ^{-1}(a x)^3}{a x}-\frac{10}{3} i \tan ^{-1}(a x)^3-\frac{1}{2} \tan ^{-1}(a x)^2-\frac{\tan ^{-1}(a x)}{a x}-10 \tan ^{-1}(a x)^2 \log \left(1-e^{-2 i \tan ^{-1}(a x)}\right)+\frac{3}{4} \tan ^{-1}(a x)^3 \sin \left(2 \tan ^{-1}(a x)\right)+\frac{1}{32} \tan ^{-1}(a x)^3 \sin \left(4 \tan ^{-1}(a x)\right)-\frac{9}{8} \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)-\frac{3}{256} \tan ^{-1}(a x) \sin \left(4 \tan ^{-1}(a x)\right)+\frac{9}{8} \tan ^{-1}(a x)^2 \cos \left(2 \tan ^{-1}(a x)\right)+\frac{3}{128} \tan ^{-1}(a x)^2 \cos \left(4 \tan ^{-1}(a x)\right)-\frac{9}{16} \cos \left(2 \tan ^{-1}(a x)\right)-\frac{3 \cos \left(4 \tan ^{-1}(a x)\right)}{1024}+\frac{5 i \pi ^3}{12}\right)}{c^3}","-\frac{5 a^3 \text{Li}_3\left(\frac{2}{1-i a x}-1\right)}{c^3}+\frac{10 i a^3 \text{Li}_2\left(\frac{2}{1-i a x}-1\right) \tan ^{-1}(a x)}{c^3}+\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{10 a^3 \log \left(2-\frac{2}{1-i a x}\right) \tan ^{-1}(a x)^2}{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{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{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{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{\tan ^{-1}(a x)^3}{3 c^3 x^3}-\frac{a \tan ^{-1}(a x)^2}{2 c^3 x^2}",1,"(a^3*(((5*I)/12)*Pi^3 - ArcTan[a*x]/(a*x) - ArcTan[a*x]^2/2 - ArcTan[a*x]^2/(2*a^2*x^2) - ((10*I)/3)*ArcTan[a*x]^3 - ArcTan[a*x]^3/(3*a^3*x^3) + (3*ArcTan[a*x]^3)/(a*x) + (35*ArcTan[a*x]^4)/32 - (9*Cos[2*ArcTan[a*x]])/16 + (9*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]])/8 - (3*Cos[4*ArcTan[a*x]])/1024 + (3*ArcTan[a*x]^2*Cos[4*ArcTan[a*x]])/128 - 10*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + Log[(a*x)/Sqrt[1 + a^2*x^2]] - (10*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - 5*PolyLog[3, E^((-2*I)*ArcTan[a*x])] - (9*ArcTan[a*x]*Sin[2*ArcTan[a*x]])/8 + (3*ArcTan[a*x]^3*Sin[2*ArcTan[a*x]])/4 - (3*ArcTan[a*x]*Sin[4*ArcTan[a*x]])/256 + (ArcTan[a*x]^3*Sin[4*ArcTan[a*x]])/32))/c^3","A",0
412,1,262,523,1.2235031,"\int x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx","Integrate[x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]","\frac{\sqrt{a^2 c x^2+c} \left(-\left(a^2 x^2+1\right)^2 \left(\frac{48 a x}{\left(a^2 x^2+1\right)^2}+\tan ^{-1}(a x)^2 \left(6 \sin \left(2 \tan ^{-1}(a x)\right)-33 \sin \left(4 \tan ^{-1}(a x)\right)\right)+32 \tan ^{-1}(a x)^3 \left(5 \cos \left(2 \tan ^{-1}(a x)\right)-1\right)+6 \tan ^{-1}(a x) \left(36 \cos \left(2 \tan ^{-1}(a x)\right)+11 \cos \left(4 \tan ^{-1}(a x)\right)+25\right)\right)+\frac{48 \left(10 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+11 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-11 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-11 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+11 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-11 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)}{\sqrt{a^2 x^2+1}}\right)}{960 a^4}","\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{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{11 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)}{20 a^4 \sqrt{a^2 c x^2+c}}-\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{x \sqrt{a^2 c x^2+c}}{20 a^3}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^3}",1,"(Sqrt[c + a^2*c*x^2]*((48*((-11*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 10*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (11*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (11*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 11*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 11*PolyLog[3, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] - (1 + a^2*x^2)^2*((48*a*x)/(1 + a^2*x^2)^2 + 32*ArcTan[a*x]^3*(-1 + 5*Cos[2*ArcTan[a*x]]) + 6*ArcTan[a*x]*(25 + 36*Cos[2*ArcTan[a*x]] + 11*Cos[4*ArcTan[a*x]]) + ArcTan[a*x]^2*(6*Sin[2*ArcTan[a*x]] - 33*Sin[4*ArcTan[a*x]]))))/(960*a^4)","A",0
413,1,1844,747,12.1335861,"\int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx","Integrate[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]","\frac{\frac{\sqrt{c \left(a^2 x^2+1\right)} \tan ^{-1}(a x)^3}{16 \sqrt{a^2 x^2+1} \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)^4}-\frac{\sqrt{c \left(a^2 x^2+1\right)} \tan ^{-1}(a x)^3}{16 \sqrt{a^2 x^2+1} \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)+\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)^4}-\frac{\sqrt{c \left(a^2 x^2+1\right)} \sin \left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2}{8 \sqrt{a^2 x^2+1} \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)^3}+\frac{\sqrt{c \left(a^2 x^2+1\right)} \sin \left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2}{8 \sqrt{a^2 x^2+1} \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)+\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)^3}+\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(\tan ^{-1}(a x)^2-1\right)}{4 \sqrt{a^2 x^2+1}}+\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(-\tan ^{-1}(a x) \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)-i \left(\text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)\right)\right)}{2 \sqrt{a^2 x^2+1}}+\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(-\frac{1}{8} \pi ^3 \log \left(\cot \left(\frac{1}{2} \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)\right)\right)-\frac{3}{4} \pi ^2 \left(\left(\frac{\pi }{2}-\tan ^{-1}(a x)\right) \left(\log \left(1-e^{i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)}\right)-\log \left(1+e^{i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)}\right)\right)+i \left(\text{Li}_2\left(-e^{i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)}\right)-\text{Li}_2\left(e^{i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)}\right)\right)\right)+\frac{3}{2} \pi  \left(\left(\log \left(1-e^{i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)}\right)-\log \left(1+e^{i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)}\right)\right) \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)^2+2 i \left(\text{Li}_2\left(-e^{i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)}\right)-\text{Li}_2\left(e^{i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)}\right)\right) \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)+2 \left(\text{Li}_3\left(e^{i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)}\right)-\text{Li}_3\left(-e^{i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)}\right)\right)\right)-8 \left(\frac{1}{64} i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)^4-\frac{1}{8} \log \left(1+e^{i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)}\right) \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)^3+\frac{3}{8} i \text{Li}_2\left(-e^{i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)}\right) \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)^2-\frac{3}{4} \text{Li}_3\left(-e^{i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)}\right) \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)+\frac{1}{4} i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^4-\frac{1}{8} \pi ^3 \left(i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)-\log \left(1+e^{2 i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)\right)-\left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^3 \log \left(1+e^{2 i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)+\frac{3}{4} \pi ^2 \left(\frac{1}{2} i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^2-\log \left(1+e^{2 i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right) \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)+\frac{1}{2} i \text{Li}_2\left(-e^{2 i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)\right)+\frac{3}{2} i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^2 \text{Li}_2\left(-e^{2 i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)-\frac{3}{2} \pi  \left(\frac{1}{3} i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^3-\log \left(1+e^{2 i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right) \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^2+i \text{Li}_2\left(-e^{2 i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right) \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)-\frac{1}{2} \text{Li}_3\left(-e^{2 i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)\right)-\frac{3}{2} \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right) \text{Li}_3\left(-e^{2 i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)-\frac{3}{4} i \text{Li}_4\left(-e^{i \left(\frac{\pi }{2}-\tan ^{-1}(a x)\right)}\right)-\frac{3}{4} i \text{Li}_4\left(-e^{2 i \left(\frac{1}{2} \left(\tan ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)\right)\right)}{8 \sqrt{a^2 x^2+1}}+\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\tan ^{-1}(a x)^2 \sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}{4 \sqrt{a^2 x^2+1} \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)+\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}+\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(\tan ^{-1}(a x)^2 \sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}{4 \sqrt{a^2 x^2+1} \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}+\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(-\tan ^{-1}(a x)^3-\tan ^{-1}(a x)^2+2 \tan ^{-1}(a x)\right)}{16 \sqrt{a^2 x^2+1} \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)^2}+\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(\tan ^{-1}(a x)^3-\tan ^{-1}(a x)^2-2 \tan ^{-1}(a x)\right)}{16 \sqrt{a^2 x^2+1} \left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)+\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)^2}}{a^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{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{4 a^2}+\frac{1}{4} x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{i c \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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{\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}}",1,"((Sqrt[c*(1 + a^2*x^2)]*(-1 + ArcTan[a*x]^2))/(4*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])])) - I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])])))/(2*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-1/8*(Pi^3*Log[Cot[(Pi/2 - ArcTan[a*x])/2]]) - (3*Pi^2*((Pi/2 - ArcTan[a*x])*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))])))/4 + (3*Pi*((Pi/2 - ArcTan[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + (2*I)*(Pi/2 - ArcTan[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcTan[a*x]))])))/2 - 8*((I/64)*(Pi/2 - ArcTan[a*x])^4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^4 - ((Pi/2 - ArcTan[a*x])^3*Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/8 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + ((3*I)/8)*(Pi/2 - ArcTan[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + (I/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - (3*(Pi/2 - ArcTan[a*x])*PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))])/4 - (3*Pi*((I/3)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]/2))/2 - (3*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])/2 - ((3*I)/4)*PolyLog[4, -E^(I*(Pi/2 - ArcTan[a*x]))] - ((3*I)/4)*PolyLog[4, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])))/(8*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*(2*ArcTan[a*x] - ArcTan[a*x]^2 - ArcTan[a*x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^2) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^3) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^3) + (Sqrt[c*(1 + a^2*x^2)]*(-2*ArcTan[a*x] - ArcTan[a*x]^2 + ArcTan[a*x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^2) + (Sqrt[c*(1 + a^2*x^2)]*(Sin[ArcTan[a*x]/2] - ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])) + (Sqrt[c*(1 + a^2*x^2)]*(-Sin[ArcTan[a*x]/2] + ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])))/a^3","B",0
414,1,206,373,0.5855794,"\int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx","Integrate[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]","\frac{\sqrt{a^2 c x^2+c} \left(\left(a^2 x^2+1\right) \tan ^{-1}(a x) \left(4 \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)+6 \cos \left(2 \tan ^{-1}(a x)\right)+6\right)+\frac{12 \left(-\tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+\text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)}{\sqrt{a^2 x^2+1}}\right)}{12 a^2}","-\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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]*((12*(I*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + I*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] + (1 + a^2*x^2)*ArcTan[a*x]*(6 + 4*ArcTan[a*x]^2 + 6*Cos[2*ArcTan[a*x]] - 3*ArcTan[a*x]*Sin[2*ArcTan[a*x]])))/(12*a^2)","A",0
415,1,258,626,0.8415563,"\int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx","Integrate[Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]","-\frac{i \sqrt{c \left(a^2 x^2+1\right)} \left(i a x \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3-3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2-6 i \tan ^{-1}(a x) \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+6 i \tan ^{-1}(a x) \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-3 \left(\tan ^{-1}(a x)^2+2\right) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+3 \left(\tan ^{-1}(a x)^2+2\right) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+6 \text{Li}_4\left(-i e^{i \tan ^{-1}(a x)}\right)-6 \text{Li}_4\left(i e^{i \tan ^{-1}(a x)}\right)+2 \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3+12 \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)\right)}{2 a \sqrt{a^2 x^2+1}}","\frac{3 i c \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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,"((-1/2*I)*Sqrt[c*(1 + a^2*x^2)]*(12*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x] - (3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + I*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + 2*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 - 3*(2 + ArcTan[a*x]^2)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + 3*(2 + ArcTan[a*x]^2)*PolyLog[2, I*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (6*I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + 6*PolyLog[4, (-I)*E^(I*ArcTan[a*x])] - 6*PolyLog[4, I*E^(I*ArcTan[a*x])]))/(a*Sqrt[1 + a^2*x^2])","A",0
416,1,366,600,0.6299522,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x} \, dx","Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x,x]","\frac{\sqrt{a^2 c x^2+c} \left(8 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3+24 i \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-i \tan ^{-1}(a x)}\right)+24 i \tan ^{-1}(a x)^2 \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-48 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+48 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+48 \tan ^{-1}(a x) \text{Li}_3\left(e^{-i \tan ^{-1}(a x)}\right)-48 \tan ^{-1}(a x) \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+48 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-48 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-48 i \text{Li}_4\left(e^{-i \tan ^{-1}(a x)}\right)-48 i \text{Li}_4\left(-e^{i \tan ^{-1}(a x)}\right)+2 i \tan ^{-1}(a x)^4+8 \tan ^{-1}(a x)^3 \log \left(1-e^{-i \tan ^{-1}(a x)}\right)-8 \tan ^{-1}(a x)^3 \log \left(1+e^{i \tan ^{-1}(a x)}\right)-24 \tan ^{-1}(a x)^2 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)+24 \tan ^{-1}(a x)^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-i \pi ^4\right)}{8 \sqrt{a^2 x^2+1}}","\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{Li}_2\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{Li}_2\left(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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 c \sqrt{a^2 x^2+1} \text{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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,"(Sqrt[c + a^2*c*x^2]*((-I)*Pi^4 + 8*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + (2*I)*ArcTan[a*x]^4 + 8*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])] - 24*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] + 24*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] - 8*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] + (24*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])] + (24*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])] - (48*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (48*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 48*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])] - 48*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] + 48*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 48*PolyLog[3, I*E^(I*ArcTan[a*x])] - (48*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] - (48*I)*PolyLog[4, -E^(I*ArcTan[a*x])]))/(8*Sqrt[1 + a^2*x^2])","A",0
417,1,768,622,3.5131171,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x^2} \, dx","Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x^2,x]","\frac{a \sqrt{a^2 c x^2+c} \csc \left(\frac{1}{2} \tan ^{-1}(a x)\right) \sec \left(\frac{1}{2} \tan ^{-1}(a x)\right) \left(-64 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3+192 i a x \tan ^{-1}(a x)^2 \text{Li}_2\left(-i e^{-i \tan ^{-1}(a x)}\right)+192 i a x \tan ^{-1}(a x)^2 \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+384 i a x \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-192 i \pi  a x \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-384 i a x \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+384 a x \tan ^{-1}(a x) \text{Li}_3\left(-i e^{-i \tan ^{-1}(a x)}\right)-384 a x \tan ^{-1}(a x) \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+48 i \pi  a x \left(\pi -4 \tan ^{-1}(a x)\right) \text{Li}_2\left(i e^{-i \tan ^{-1}(a x)}\right)+48 i \pi ^2 a x \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-192 \pi  a x \text{Li}_3\left(i e^{-i \tan ^{-1}(a x)}\right)-384 a x \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+192 \pi  a x \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+384 a x \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)-384 i a x \text{Li}_4\left(-i e^{-i \tan ^{-1}(a x)}\right)-384 i a x \text{Li}_4\left(-i e^{i \tan ^{-1}(a x)}\right)-7 i \pi ^4 a x+16 i a x \tan ^{-1}(a x)^4-32 i \pi  a x \tan ^{-1}(a x)^3+24 i \pi ^2 a x \tan ^{-1}(a x)^2-8 i \pi ^3 a x \tan ^{-1}(a x)+64 a x \tan ^{-1}(a x)^3 \log \left(1+i e^{-i \tan ^{-1}(a x)}\right)-64 a x \tan ^{-1}(a x)^3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-96 \pi  a x \tan ^{-1}(a x)^2 \log \left(1-i e^{-i \tan ^{-1}(a x)}\right)+192 a x \tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)+96 \pi  a x \tan ^{-1}(a x)^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-192 a x \tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)+48 \pi ^2 a x \tan ^{-1}(a x) \log \left(1-i e^{-i \tan ^{-1}(a x)}\right)-48 \pi ^2 a x \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-8 \pi ^3 a x \log \left(1+i e^{-i \tan ^{-1}(a x)}\right)+8 \pi ^3 a x \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+8 \pi ^3 a x \log \left(\tan \left(\frac{1}{4} \left(2 \tan ^{-1}(a x)+\pi \right)\right)\right)\right)}{128 \left(a^2 x^2+1\right)}","\frac{3 i a c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{Li}_2\left(-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{Li}_2\left(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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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,"(a*Sqrt[c + a^2*c*x^2]*Csc[ArcTan[a*x]/2]*((-7*I)*a*Pi^4*x - (8*I)*a*Pi^3*x*ArcTan[a*x] + (24*I)*a*Pi^2*x*ArcTan[a*x]^2 - (32*I)*a*Pi*x*ArcTan[a*x]^3 - 64*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + (16*I)*a*x*ArcTan[a*x]^4 + 48*a*Pi^2*x*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])] - 96*a*Pi*x*ArcTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])] - 8*a*Pi^3*x*Log[1 + I/E^(I*ArcTan[a*x])] + 64*a*x*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])] + 192*a*x*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] + 8*a*Pi^3*x*Log[1 + I*E^(I*ArcTan[a*x])] - 48*a*Pi^2*x*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] + 96*a*Pi*x*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] - 64*a*x*ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])] - 192*a*x*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 8*a*Pi^3*x*Log[Tan[(Pi + 2*ArcTan[a*x])/4]] + (192*I)*a*x*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])] + (48*I)*a*Pi*x*(Pi - 4*ArcTan[a*x])*PolyLog[2, I/E^(I*ArcTan[a*x])] + (384*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] + (48*I)*a*Pi^2*x*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (192*I)*a*Pi*x*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (192*I)*a*x*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (384*I)*a*x*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] + 384*a*x*ArcTan[a*x]*PolyLog[3, (-I)/E^(I*ArcTan[a*x])] - 192*a*Pi*x*PolyLog[3, I/E^(I*ArcTan[a*x])] - 384*a*x*PolyLog[3, -E^(I*ArcTan[a*x])] + 192*a*Pi*x*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 384*a*x*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 384*a*x*PolyLog[3, E^(I*ArcTan[a*x])] - (384*I)*a*x*PolyLog[4, (-I)/E^(I*ArcTan[a*x])] - (384*I)*a*x*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])*Sec[ArcTan[a*x]/2])/(128*(1 + a^2*x^2))","A",1
418,1,345,602,5.9421779,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x^3} \, dx","Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x^3,x]","\frac{a^2 \sqrt{c \left(a^2 x^2+1\right)} \left(24 i \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-i \tan ^{-1}(a x)}\right)+48 \tan ^{-1}(a x) \text{Li}_3\left(e^{-i \tan ^{-1}(a x)}\right)-48 \tan ^{-1}(a x) \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+24 i \left(\tan ^{-1}(a x)^2+2\right) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-48 i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-48 i \text{Li}_4\left(e^{-i \tan ^{-1}(a x)}\right)-48 i \text{Li}_4\left(-e^{i \tan ^{-1}(a x)}\right)+2 i \tan ^{-1}(a x)^4-12 \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2+8 \tan ^{-1}(a x)^3 \log \left(1-e^{-i \tan ^{-1}(a x)}\right)-8 \tan ^{-1}(a x)^3 \log \left(1+e^{i \tan ^{-1}(a x)}\right)+48 \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)-48 \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)-12 \tan ^{-1}(a x)^2 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-2 \tan ^{-1}(a x)^3 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+2 \tan ^{-1}(a x)^3 \sec ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)-i \pi ^4\right)}{16 \sqrt{a^2 x^2+1}}","\frac{3 i a^2 c \sqrt{a^2 x^2+1} \text{Li}_2\left(-\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 \sqrt{a^2 x^2+1} \text{Li}_2\left(\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 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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,"(a^2*Sqrt[c*(1 + a^2*x^2)]*((-I)*Pi^4 + (2*I)*ArcTan[a*x]^4 - 12*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2] - 2*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^2 + 8*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])] + 48*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 48*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] - 8*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] + (24*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])] + (24*I)*(2 + ArcTan[a*x]^2)*PolyLog[2, -E^(I*ArcTan[a*x])] - (48*I)*PolyLog[2, E^(I*ArcTan[a*x])] + 48*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])] - 48*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] - (48*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] - (48*I)*PolyLog[4, -E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]^3*Sec[ArcTan[a*x]/2]^2 - 12*ArcTan[a*x]^2*Tan[ArcTan[a*x]/2]))/(16*Sqrt[1 + a^2*x^2])","A",0
419,1,341,361,3.9249608,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x^4} \, dx","Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x^4,x]","\frac{a^3 c \sqrt{a^2 x^2+1} \left(-\frac{a x \tan ^{-1}(a x)^3 \csc ^4\left(\frac{1}{2} \tan ^{-1}(a x)\right)}{2 \sqrt{a^2 x^2+1}}-\frac{8 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^3 \sin ^4\left(\frac{1}{2} \tan ^{-1}(a x)\right)}{a^3 x^3}+24 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-24 i \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-24 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+24 \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)-2 \tan ^{-1}(a x)^3 \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)-12 \tan ^{-1}(a x) \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)+12 \tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)-12 \tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)+24 \log \left(\tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)-2 \tan ^{-1}(a x)^3 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-12 \tan ^{-1}(a x) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-3 \tan ^{-1}(a x)^2 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+3 \tan ^{-1}(a x)^2 \sec ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}{24 \sqrt{c \left(a^2 x^2+1\right)}}","-\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}+\frac{i a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-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^3*c*Sqrt[1 + a^2*x^2]*(-12*ArcTan[a*x]*Cot[ArcTan[a*x]/2] - 2*ArcTan[a*x]^3*Cot[ArcTan[a*x]/2] - 3*ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 - (a*x*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^4)/(2*Sqrt[1 + a^2*x^2]) + 12*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] - 12*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 24*Log[Tan[ArcTan[a*x]/2]] + (24*I)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - (24*I)*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - 24*PolyLog[3, -E^(I*ArcTan[a*x])] + 24*PolyLog[3, E^(I*ArcTan[a*x])] + 3*ArcTan[a*x]^2*Sec[ArcTan[a*x]/2]^2 - (8*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^3*Sin[ArcTan[a*x]/2]^4)/(a^3*x^3) - 12*ArcTan[a*x]*Tan[ArcTan[a*x]/2] - 2*ArcTan[a*x]^3*Tan[ArcTan[a*x]/2]))/(24*Sqrt[c*(1 + a^2*x^2)])","A",0
420,1,538,652,3.6890582,"\int x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3 \, dx","Integrate[x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3,x]","\frac{c \sqrt{a^2 c x^2+c} \left(64 \left(-259 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-309 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+309 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+309 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-309 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+309 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)+2688 \left(10 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+11 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-11 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-11 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+11 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-11 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)+\left(a^2 x^2+1\right)^{7/2} \left(\frac{8 \tan ^{-1}(a x) \left(764 \cos \left(2 \tan ^{-1}(a x)\right)+309 \cos \left(4 \tan ^{-1}(a x)\right)+647\right)}{a^2 x^2+1}-3 \tan ^{-1}(a x)^2 \left(211 \sin \left(2 \tan ^{-1}(a x)\right)-60 \sin \left(4 \tan ^{-1}(a x)\right)+103 \sin \left(6 \tan ^{-1}(a x)\right)\right)+4 \left(101 \sin \left(2 \tan ^{-1}(a x)\right)+88 \sin \left(4 \tan ^{-1}(a x)\right)+25 \sin \left(6 \tan ^{-1}(a x)\right)\right)+64 \tan ^{-1}(a x)^3 \left(-28 \cos \left(2 \tan ^{-1}(a x)\right)+35 \cos \left(4 \tan ^{-1}(a x)\right)+57\right)\right)-56 \left(a^2 x^2+1\right)^{5/2} \left(\frac{48 a x}{\left(a^2 x^2+1\right)^2}+\tan ^{-1}(a x)^2 \left(6 \sin \left(2 \tan ^{-1}(a x)\right)-33 \sin \left(4 \tan ^{-1}(a x)\right)\right)+32 \tan ^{-1}(a x)^3 \left(5 \cos \left(2 \tan ^{-1}(a x)\right)-1\right)+6 \tan ^{-1}(a x) \left(36 \cos \left(2 \tan ^{-1}(a x)\right)+11 \cos \left(4 \tan ^{-1}(a x)\right)+25\right)\right)\right)}{53760 a^4 \sqrt{a^2 x^2+1}}","\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{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{c x^3 \sqrt{a^2 c x^2+c}}{140 a}-\frac{23 c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{280 a}+\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{51 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{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{c x \sqrt{a^2 c x^2+c}}{420 a^3}+\frac{9 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{112 a^3}",1,"(c*Sqrt[c + a^2*c*x^2]*(64*((309*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 259*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - (309*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (309*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 309*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 309*PolyLog[3, I*E^(I*ArcTan[a*x])]) + 2688*((-11*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 10*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (11*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (11*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 11*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 11*PolyLog[3, I*E^(I*ArcTan[a*x])]) - 56*(1 + a^2*x^2)^(5/2)*((48*a*x)/(1 + a^2*x^2)^2 + 32*ArcTan[a*x]^3*(-1 + 5*Cos[2*ArcTan[a*x]]) + 6*ArcTan[a*x]*(25 + 36*Cos[2*ArcTan[a*x]] + 11*Cos[4*ArcTan[a*x]]) + ArcTan[a*x]^2*(6*Sin[2*ArcTan[a*x]] - 33*Sin[4*ArcTan[a*x]])) + (1 + a^2*x^2)^(7/2)*(64*ArcTan[a*x]^3*(57 - 28*Cos[2*ArcTan[a*x]] + 35*Cos[4*ArcTan[a*x]]) + (8*ArcTan[a*x]*(647 + 764*Cos[2*ArcTan[a*x]] + 309*Cos[4*ArcTan[a*x]]))/(1 + a^2*x^2) + 4*(101*Sin[2*ArcTan[a*x]] + 88*Sin[4*ArcTan[a*x]] + 25*Sin[6*ArcTan[a*x]]) - 3*ArcTan[a*x]^2*(211*Sin[2*ArcTan[a*x]] - 60*Sin[4*ArcTan[a*x]] + 103*Sin[6*ArcTan[a*x]]))))/(53760*a^4*Sqrt[1 + a^2*x^2])","A",0
421,1,4015,882,18.2719427,"\int x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3 \, dx","Integrate[x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3,x]","\text{Result too large to show}","\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{Li}_2\left(-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{Li}_2\left(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{Li}_2\left(-\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{Li}_2\left(\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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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*(1 + a^2*x^2)]*(-1 + ArcTan[a*x]^2))/(4*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])])) - I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])])))/(2*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-1/8*(Pi^3*Log[Cot[(Pi/2 - ArcTan[a*x])/2]]) - (3*Pi^2*((Pi/2 - ArcTan[a*x])*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))])))/4 + (3*Pi*((Pi/2 - ArcTan[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + (2*I)*(Pi/2 - ArcTan[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcTan[a*x]))])))/2 - 8*((I/64)*(Pi/2 - ArcTan[a*x])^4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^4 - ((Pi/2 - ArcTan[a*x])^3*Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/8 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + ((3*I)/8)*(Pi/2 - ArcTan[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + (I/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - (3*(Pi/2 - ArcTan[a*x])*PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))])/4 - (3*Pi*((I/3)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]/2))/2 - (3*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])/2 - ((3*I)/4)*PolyLog[4, -E^(I*(Pi/2 - ArcTan[a*x]))] - ((3*I)/4)*PolyLog[4, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])))/(8*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*(2*ArcTan[a*x] - ArcTan[a*x]^2 - ArcTan[a*x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^2) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^3) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^3) + (Sqrt[c*(1 + a^2*x^2)]*(-2*ArcTan[a*x] - ArcTan[a*x]^2 + ArcTan[a*x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^2) + (Sqrt[c*(1 + a^2*x^2)]*(Sin[ArcTan[a*x]/2] - ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])) + (Sqrt[c*(1 + a^2*x^2)]*(-Sin[ArcTan[a*x]/2] + ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]))))/a^3 + (c*((Sqrt[c*(1 + a^2*x^2)]*(50 - 19*ArcTan[a*x]^2))/(240*Sqrt[1 + a^2*x^2]) + (19*Sqrt[c*(1 + a^2*x^2)]*(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])])))/(120*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*((Pi^3*Log[Cot[(Pi/2 - ArcTan[a*x])/2]])/8 + (3*Pi^2*((Pi/2 - ArcTan[a*x])*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))])))/4 - (3*Pi*((Pi/2 - ArcTan[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + (2*I)*(Pi/2 - ArcTan[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcTan[a*x]))])))/2 + 8*((I/64)*(Pi/2 - ArcTan[a*x])^4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^4 - ((Pi/2 - ArcTan[a*x])^3*Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/8 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + ((3*I)/8)*(Pi/2 - ArcTan[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + (I/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - (3*(Pi/2 - ArcTan[a*x])*PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))])/4 - (3*Pi*((I/3)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]/2))/2 - (3*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])/2 - ((3*I)/4)*PolyLog[4, -E^(I*(Pi/2 - ArcTan[a*x]))] - ((3*I)/4)*PolyLog[4, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])))/(16*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(48*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^6) + (Sqrt[c*(1 + a^2*x^2)]*(ArcTan[a*x] - ArcTan[a*x]^2 - 5*ArcTan[a*x]^3))/(80*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*(-2 - 52*ArcTan[a*x] + 26*ArcTan[a*x]^2 + 15*ArcTan[a*x]^3))/(480*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^2) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(40*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^5) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(48*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^6) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(40*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^5) + (Sqrt[c*(1 + a^2*x^2)]*(-ArcTan[a*x] - ArcTan[a*x]^2 + 5*ArcTan[a*x]^3))/(80*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*(-2 + 52*ArcTan[a*x] + 26*ArcTan[a*x]^2 - 15*ArcTan[a*x]^3))/(480*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^2) + (Sqrt[c*(1 + a^2*x^2)]*(50*Sin[ArcTan[a*x]/2] - 19*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(240*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])) + (Sqrt[c*(1 + a^2*x^2)]*(Sin[ArcTan[a*x]/2] - 13*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(120*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^3) + (Sqrt[c*(1 + a^2*x^2)]*(-Sin[ArcTan[a*x]/2] + 13*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(120*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^3) + (Sqrt[c*(1 + a^2*x^2)]*(-50*Sin[ArcTan[a*x]/2] + 19*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(240*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]))))/a^3","B",0
422,1,441,477,3.9941238,"\int x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3 \, dx","Integrate[x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3,x]","\frac{c \sqrt{a^2 c x^2+c} \left(960 \left(-\tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+\text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)+48 \left(10 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+11 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-11 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-11 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+11 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-11 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)-\left(\left(a^2 x^2+1\right)^{5/2} \left(\frac{48 a x}{\left(a^2 x^2+1\right)^2}+\tan ^{-1}(a x)^2 \left(6 \sin \left(2 \tan ^{-1}(a x)\right)-33 \sin \left(4 \tan ^{-1}(a x)\right)\right)+32 \tan ^{-1}(a x)^3 \left(5 \cos \left(2 \tan ^{-1}(a x)\right)-1\right)+6 \tan ^{-1}(a x) \left(36 \cos \left(2 \tan ^{-1}(a x)\right)+11 \cos \left(4 \tan ^{-1}(a x)\right)+25\right)\right)\right)+80 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x) \left(4 \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)+6 \cos \left(2 \tan ^{-1}(a x)\right)+6\right)\right)}{960 a^2 \sqrt{a^2 x^2+1}}","-\frac{c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)}{2 a^2}-\frac{9 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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 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*Sqrt[c + a^2*c*x^2]*(960*(I*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + I*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])]) + 48*((-11*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 10*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (11*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (11*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 11*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 11*PolyLog[3, I*E^(I*ArcTan[a*x])]) + 80*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]*(6 + 4*ArcTan[a*x]^2 + 6*Cos[2*ArcTan[a*x]] - 3*ArcTan[a*x]*Sin[2*ArcTan[a*x]]) - (1 + a^2*x^2)^(5/2)*((48*a*x)/(1 + a^2*x^2)^2 + 32*ArcTan[a*x]^3*(-1 + 5*Cos[2*ArcTan[a*x]]) + 6*ArcTan[a*x]*(25 + 36*Cos[2*ArcTan[a*x]] + 11*Cos[4*ArcTan[a*x]]) + ArcTan[a*x]^2*(6*Sin[2*ArcTan[a*x]] - 33*Sin[4*ArcTan[a*x]]))))/(960*a^2*Sqrt[1 + a^2*x^2])","A",0
423,1,2105,760,12.843109,"\int \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3 \, dx","Integrate[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3,x]","\text{Result too large to show}","\frac{5 i c^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(\frac{i \sqrt{i a x+1}}{\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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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,"((-1/2*I)*c*Sqrt[c*(1 + a^2*x^2)]*(12*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x] - (3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + I*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + 2*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 - 3*(2 + ArcTan[a*x]^2)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + 3*(2 + ArcTan[a*x]^2)*PolyLog[2, I*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (6*I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + 6*PolyLog[4, (-I)*E^(I*ArcTan[a*x])] - 6*PolyLog[4, I*E^(I*ArcTan[a*x])]))/(a*Sqrt[1 + a^2*x^2]) + (c*((Sqrt[c*(1 + a^2*x^2)]*(-1 + ArcTan[a*x]^2))/(4*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])])) - I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])])))/(2*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-1/8*(Pi^3*Log[Cot[(Pi/2 - ArcTan[a*x])/2]]) - (3*Pi^2*((Pi/2 - ArcTan[a*x])*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))])))/4 + (3*Pi*((Pi/2 - ArcTan[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + (2*I)*(Pi/2 - ArcTan[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcTan[a*x]))])))/2 - 8*((I/64)*(Pi/2 - ArcTan[a*x])^4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^4 - ((Pi/2 - ArcTan[a*x])^3*Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/8 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + ((3*I)/8)*(Pi/2 - ArcTan[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + (I/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - (3*(Pi/2 - ArcTan[a*x])*PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))])/4 - (3*Pi*((I/3)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]/2))/2 - (3*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])/2 - ((3*I)/4)*PolyLog[4, -E^(I*(Pi/2 - ArcTan[a*x]))] - ((3*I)/4)*PolyLog[4, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])))/(8*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*(2*ArcTan[a*x] - ArcTan[a*x]^2 - ArcTan[a*x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^2) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^3) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^3) + (Sqrt[c*(1 + a^2*x^2)]*(-2*ArcTan[a*x] - ArcTan[a*x]^2 + ArcTan[a*x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^2) + (Sqrt[c*(1 + a^2*x^2)]*(Sin[ArcTan[a*x]/2] - ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])) + (Sqrt[c*(1 + a^2*x^2)]*(-Sin[ArcTan[a*x]/2] + ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]))))/a","B",0
424,1,555,726,1.4900325,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3}{x} \, dx","Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x,x]","\frac{c \sqrt{a^2 c x^2+c} \left(\frac{40 a^2 x^2 \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}+\frac{24 a^2 x^2 \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{12 a x \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{32 \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}+\frac{24 \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-24 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+\frac{8 a^4 x^4 \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}-\frac{12 a^3 x^3 \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+72 i \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-i \tan ^{-1}(a x)}\right)+72 i \tan ^{-1}(a x)^2 \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-168 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+168 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+144 \tan ^{-1}(a x) \text{Li}_3\left(e^{-i \tan ^{-1}(a x)}\right)-144 \tan ^{-1}(a x) \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+168 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-168 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-144 i \text{Li}_4\left(e^{-i \tan ^{-1}(a x)}\right)-144 i \text{Li}_4\left(-e^{i \tan ^{-1}(a x)}\right)+6 i \tan ^{-1}(a x)^4+24 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2+24 \tan ^{-1}(a x)^3 \log \left(1-e^{-i \tan ^{-1}(a x)}\right)-72 \tan ^{-1}(a x)^2 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)+72 \tan ^{-1}(a x)^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-24 \tan ^{-1}(a x)^3 \log \left(1+e^{i \tan ^{-1}(a x)}\right)-3 i \pi ^4\right)}{24 \sqrt{a^2 x^2+1}}","-c^{3/2} \tanh ^{-1}\left(\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right)+\frac{3 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{Li}_2\left(-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{Li}_2\left(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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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}}-\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]*((-3*I)*Pi^4 + (24*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + (24*a^2*x^2*ArcTan[a*x])/Sqrt[1 + a^2*x^2] - (12*a*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] - (12*a^3*x^3*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] + (24*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + (32*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + (40*a^2*x^2*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + (8*a^4*x^4*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + (6*I)*ArcTan[a*x]^4 - 24*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + 24*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])] - 72*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] + 72*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] - 24*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] + (72*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])] + (72*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])] - (168*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (168*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 144*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])] - 144*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] + 168*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 168*PolyLog[3, I*E^(I*ArcTan[a*x])] - (144*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] - (144*I)*PolyLog[4, -E^(I*ArcTan[a*x])]))/(24*Sqrt[1 + a^2*x^2])","A",0
425,1,1387,901,7.0670136,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3}{x^2} \, dx","Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^2,x]","\frac{a c \sqrt{a^2 c x^2+c} \left(16 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^4+32 a^3 x^3 \tan ^{-1}(a x)^3-32 a x \tan ^{-1}(a x)^3-64 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3+64 \sqrt{a^2 x^2+1} \log \left(1+i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3-64 \sqrt{a^2 x^2+1} \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3-32 i \pi  \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3-\frac{64 \tan ^{-1}(a x)^3}{a x}-96 a^2 x^2 \tan ^{-1}(a x)^2-96 \pi  \sqrt{a^2 x^2+1} \log \left(1-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2+192 \sqrt{a^2 x^2+1} \log \left(1-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2+96 \pi  \sqrt{a^2 x^2+1} \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2-192 \sqrt{a^2 x^2+1} \log \left(1+e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2+192 i \sqrt{a^2 x^2+1} \text{Li}_2\left(-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2+288 i \sqrt{a^2 x^2+1} \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2-96 i \sqrt{a^2 x^2+1} \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2+24 i \pi ^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2-96 \tan ^{-1}(a x)^2-384 i \sqrt{a^2 x^2+1} \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+48 \pi ^2 \sqrt{a^2 x^2+1} \log \left(1-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-48 \pi ^2 \sqrt{a^2 x^2+1} \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+384 i \sqrt{a^2 x^2+1} \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-192 i \pi  \sqrt{a^2 x^2+1} \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-384 i \sqrt{a^2 x^2+1} \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+384 \sqrt{a^2 x^2+1} \text{Li}_3\left(-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-576 \sqrt{a^2 x^2+1} \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+192 \sqrt{a^2 x^2+1} \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-8 i \pi ^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)-8 \pi ^3 \sqrt{a^2 x^2+1} \log \left(1+i e^{-i \tan ^{-1}(a x)}\right)+8 \pi ^3 \sqrt{a^2 x^2+1} \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+8 \pi ^3 \sqrt{a^2 x^2+1} \log \left(2 \sqrt{a^2 x^2+1} \sin ^2\left(\frac{1}{4} \left(2 \tan ^{-1}(a x)+\pi \right)\right)\right)+48 i \pi  \sqrt{a^2 x^2+1} \left(\pi -4 \tan ^{-1}(a x)\right) \text{Li}_2\left(i e^{-i \tan ^{-1}(a x)}\right)+48 i \pi ^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+192 i \sqrt{a^2 x^2+1} \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-192 i \sqrt{a^2 x^2+1} \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-192 \pi  \sqrt{a^2 x^2+1} \text{Li}_3\left(i e^{-i \tan ^{-1}(a x)}\right)-384 \sqrt{a^2 x^2+1} \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+192 \pi  \sqrt{a^2 x^2+1} \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+384 \sqrt{a^2 x^2+1} \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)-384 i \sqrt{a^2 x^2+1} \text{Li}_4\left(-i e^{-i \tan ^{-1}(a x)}\right)-576 i \sqrt{a^2 x^2+1} \text{Li}_4\left(-i e^{i \tan ^{-1}(a x)}\right)+192 i \sqrt{a^2 x^2+1} \text{Li}_4\left(i e^{i \tan ^{-1}(a x)}\right)-7 i \pi ^4 \sqrt{a^2 x^2+1}\right)}{64 \left(a^2 x^2+1\right)}","-\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{Li}_2\left(-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{Li}_2\left(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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_2\left(-\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{Li}_2\left(\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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}",1,"(a*c*Sqrt[c + a^2*c*x^2]*((-7*I)*Pi^4*Sqrt[1 + a^2*x^2] - (8*I)*Pi^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x] - (384*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x] - 96*ArcTan[a*x]^2 - 96*a^2*x^2*ArcTan[a*x]^2 + (24*I)*Pi^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 - (64*ArcTan[a*x]^3)/(a*x) - 32*a*x*ArcTan[a*x]^3 + 32*a^3*x^3*ArcTan[a*x]^3 - (32*I)*Pi*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 - (64*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 + (16*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^4 + 48*Pi^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])] - 96*Pi*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])] - 8*Pi^3*Sqrt[1 + a^2*x^2]*Log[1 + I/E^(I*ArcTan[a*x])] + 64*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])] + 192*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] + 8*Pi^3*Sqrt[1 + a^2*x^2]*Log[1 + I*E^(I*ArcTan[a*x])] - 48*Pi^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] + 96*Pi*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] - 64*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])] - 192*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 8*Pi^3*Sqrt[1 + a^2*x^2]*Log[2*Sqrt[1 + a^2*x^2]*Sin[(Pi + 2*ArcTan[a*x])/4]^2] + (192*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])] + (48*I)*Pi*Sqrt[1 + a^2*x^2]*(Pi - 4*ArcTan[a*x])*PolyLog[2, I/E^(I*ArcTan[a*x])] + (384*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] + (192*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (48*I)*Pi^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (192*I)*Pi*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (288*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (192*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, I*E^(I*ArcTan[a*x])] - (96*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])] - (384*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] + 384*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)/E^(I*ArcTan[a*x])] - 192*Pi*Sqrt[1 + a^2*x^2]*PolyLog[3, I/E^(I*ArcTan[a*x])] - 384*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])] + 192*Pi*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 576*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 192*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + 384*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])] - (384*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)/E^(I*ArcTan[a*x])] - (576*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])] + (192*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])]))/(64*(1 + a^2*x^2))","A",0
426,1,691,919,9.5154285,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3}{x^3} \, dx","Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^3,x]","\frac{a^2 c \sqrt{a^2 c x^2+c} \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right) \left(72 i \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-96 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+96 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+144 \tan ^{-1}(a x) \text{Li}_3\left(e^{-i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-144 \tan ^{-1}(a x) \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+24 i \left(3 \tan ^{-1}(a x)^2+2\right) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-48 i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+96 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-96 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-144 i \text{Li}_4\left(e^{-i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-144 i \text{Li}_4\left(-e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-12 \tan ^{-1}(a x)^2-12 \tan ^{-1}(a x)^2 \cot ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+6 i \tan ^{-1}(a x)^4 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-3 i \pi ^4 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+8 a x \tan ^{-1}(a x)^3 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+24 \tan ^{-1}(a x)^3 \log \left(1-e^{-i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-24 \tan ^{-1}(a x)^3 \log \left(1+e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-48 \tan ^{-1}(a x)^2 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+48 \tan ^{-1}(a x)^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+48 \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-48 \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-2 \tan ^{-1}(a x)^3 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right) \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+2 \tan ^{-1}(a x)^3 \csc \left(\frac{1}{2} \tan ^{-1}(a x)\right) \sec \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}{16 \sqrt{a^2 x^2+1}}","-\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{Li}_2\left(-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{Li}_2\left(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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_2\left(-\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{Li}_2\left(\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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}",1,"(a^2*c*Sqrt[c + a^2*c*x^2]*(-12*ArcTan[a*x]^2 - (3*I)*Pi^4*Cot[ArcTan[a*x]/2] + (6*I)*ArcTan[a*x]^4*Cot[ArcTan[a*x]/2] - 12*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2]^2 + 8*a*x*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^2 - 2*ArcTan[a*x]^3*Cot[ArcTan[a*x]/2]*Csc[ArcTan[a*x]/2]^2 + 24*ArcTan[a*x]^3*Cot[ArcTan[a*x]/2]*Log[1 - E^((-I)*ArcTan[a*x])] + 48*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 - E^(I*ArcTan[a*x])] - 48*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2]*Log[1 - I*E^(I*ArcTan[a*x])] + 48*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2]*Log[1 + I*E^(I*ArcTan[a*x])] - 48*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 + E^(I*ArcTan[a*x])] - 24*ArcTan[a*x]^3*Cot[ArcTan[a*x]/2]*Log[1 + E^(I*ArcTan[a*x])] + (72*I)*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2]*PolyLog[2, E^((-I)*ArcTan[a*x])] + (24*I)*(2 + 3*ArcTan[a*x]^2)*Cot[ArcTan[a*x]/2]*PolyLog[2, -E^(I*ArcTan[a*x])] - (96*I)*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (96*I)*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*PolyLog[2, I*E^(I*ArcTan[a*x])] - (48*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, E^(I*ArcTan[a*x])] + 144*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*PolyLog[3, E^((-I)*ArcTan[a*x])] - 144*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*PolyLog[3, -E^(I*ArcTan[a*x])] + 96*Cot[ArcTan[a*x]/2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 96*Cot[ArcTan[a*x]/2]*PolyLog[3, I*E^(I*ArcTan[a*x])] - (144*I)*Cot[ArcTan[a*x]/2]*PolyLog[4, E^((-I)*ArcTan[a*x])] - (144*I)*Cot[ArcTan[a*x]/2]*PolyLog[4, -E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]*Sec[ArcTan[a*x]/2])*Tan[ArcTan[a*x]/2])/(16*Sqrt[1 + a^2*x^2])","A",0
427,1,1508,788,10.5350162,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3}{x^4} \, dx","Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^4,x]","\frac{c \sqrt{c \left(a^2 x^2+1\right)} \csc \left(\frac{1}{2} \tan ^{-1}(a x)\right) \left(\frac{16 i a x \tan ^{-1}(a x)^4}{\sqrt{a^2 x^2+1}}+\frac{64 a x \log \left(1+i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}-\frac{64 a x \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}-\frac{32 i a \pi  x \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}-64 \tan ^{-1}(a x)^3-\frac{96 a \pi  x \log \left(1-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{192 a x \log \left(1-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{96 a \pi  x \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{192 a x \log \left(1+e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{192 i a x \text{Li}_2\left(-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{192 i a x \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{24 i a \pi ^2 x \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{48 a \pi ^2 x \log \left(1-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{48 a \pi ^2 x \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\frac{384 i a x \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{192 i a \pi  x \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{384 i a x \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\frac{384 a x \text{Li}_3\left(-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{384 a x \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{8 i a \pi ^3 x \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{8 a \pi ^3 x \log \left(1+i e^{-i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{8 a \pi ^3 x \log \left(1+i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{8 a \pi ^3 x \log \left(\tan \left(\frac{1}{4} \left(2 \tan ^{-1}(a x)+\pi \right)\right)\right)}{\sqrt{a^2 x^2+1}}+\frac{48 i a \pi  x \left(\pi -4 \tan ^{-1}(a x)\right) \text{Li}_2\left(i e^{-i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{48 i a \pi ^2 x \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{192 a \pi  x \text{Li}_3\left(i e^{-i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{384 a x \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{192 a \pi  x \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{384 a x \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{384 i a x \text{Li}_4\left(-i e^{-i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{384 i a x \text{Li}_4\left(-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{7 i a \pi ^4 x}{\sqrt{a^2 x^2+1}}\right) \sec \left(\frac{1}{2} \tan ^{-1}(a x)\right) a^3}{128 \sqrt{a^2 x^2+1}}+\frac{c^2 \sqrt{a^2 x^2+1} \left(-\frac{a x \tan ^{-1}(a x)^3 \csc ^4\left(\frac{1}{2} \tan ^{-1}(a x)\right)}{2 \sqrt{a^2 x^2+1}}-3 \tan ^{-1}(a x)^2 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)-\frac{8 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^3 \sin ^4\left(\frac{1}{2} \tan ^{-1}(a x)\right)}{a^3 x^3}+3 \tan ^{-1}(a x)^2 \sec ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)-2 \tan ^{-1}(a x)^3 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-12 \tan ^{-1}(a x) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+12 \tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)-12 \tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)+24 \log \left(\tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)+24 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-24 i \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-24 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+24 \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)-2 \tan ^{-1}(a x)^3 \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)-12 \tan ^{-1}(a x) \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right) a^3}{24 \sqrt{c \left(a^2 x^2+1\right)}}","-\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}-a^3 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right)+\frac{7 i a^3 c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(-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{Li}_2\left(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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(-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{Li}_3\left(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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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}}-\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}}",1,"(a^3*c*Sqrt[c*(1 + a^2*x^2)]*Csc[ArcTan[a*x]/2]*(((-7*I)*a*Pi^4*x)/Sqrt[1 + a^2*x^2] - ((8*I)*a*Pi^3*x*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + ((24*I)*a*Pi^2*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] - 64*ArcTan[a*x]^3 - ((32*I)*a*Pi*x*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + ((16*I)*a*x*ArcTan[a*x]^4)/Sqrt[1 + a^2*x^2] + (48*a*Pi^2*x*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (96*a*Pi*x*ArcTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (8*a*Pi^3*x*Log[1 + I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (64*a*x*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (192*a*x*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (8*a*Pi^3*x*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (48*a*Pi^2*x*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (96*a*Pi*x*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (64*a*x*ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (192*a*x*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (8*a*Pi^3*x*Log[Tan[(Pi + 2*ArcTan[a*x])/4]])/Sqrt[1 + a^2*x^2] + ((192*I)*a*x*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((48*I)*a*Pi*x*(Pi - 4*ArcTan[a*x])*PolyLog[2, I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((384*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((48*I)*a*Pi^2*x*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((192*I)*a*Pi*x*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((192*I)*a*x*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((384*I)*a*x*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (384*a*x*ArcTan[a*x]*PolyLog[3, (-I)/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (192*a*Pi*x*PolyLog[3, I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (384*a*x*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (192*a*Pi*x*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (384*a*x*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (384*a*x*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((384*I)*a*x*PolyLog[4, (-I)/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((384*I)*a*x*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2])*Sec[ArcTan[a*x]/2])/(128*Sqrt[1 + a^2*x^2]) + (a^3*c^2*Sqrt[1 + a^2*x^2]*(-12*ArcTan[a*x]*Cot[ArcTan[a*x]/2] - 2*ArcTan[a*x]^3*Cot[ArcTan[a*x]/2] - 3*ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 - (a*x*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^4)/(2*Sqrt[1 + a^2*x^2]) + 12*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] - 12*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 24*Log[Tan[ArcTan[a*x]/2]] + (24*I)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - (24*I)*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - 24*PolyLog[3, -E^(I*ArcTan[a*x])] + 24*PolyLog[3, E^(I*ArcTan[a*x])] + 3*ArcTan[a*x]^2*Sec[ArcTan[a*x]/2]^2 - (8*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^3*Sin[ArcTan[a*x]/2]^4)/(a^3*x^3) - 12*ArcTan[a*x]*Tan[ArcTan[a*x]/2] - 2*ArcTan[a*x]^3*Tan[ArcTan[a*x]/2]))/(24*Sqrt[c*(1 + a^2*x^2)])","A",0
428,1,850,798,7.2718622,"\int x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3 \, dx","Integrate[x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]","\frac{c^2 \sqrt{a^2 c x^2+c} \left(-\left(\left(1536 \left(711 \cos \left(2 \tan ^{-1}(a x)\right)-126 \cos \left(4 \tan ^{-1}(a x)\right)+105 \cos \left(6 \tan ^{-1}(a x)\right)-178\right) \tan ^{-1}(a x)^3+3 \left(13074 \sin \left(2 \tan ^{-1}(a x)\right)-26742 \sin \left(4 \tan ^{-1}(a x)\right)+6362 \sin \left(6 \tan ^{-1}(a x)\right)-5469 \sin \left(8 \tan ^{-1}(a x)\right)\right) \tan ^{-1}(a x)^2+\frac{8 \left(153529 \cos \left(2 \tan ^{-1}(a x)\right)+59266 \cos \left(4 \tan ^{-1}(a x)\right)+16407 \cos \left(6 \tan ^{-1}(a x)\right)+87630\right) \tan ^{-1}(a x)}{a^2 x^2+1}+74932 \sin \left(2 \tan ^{-1}(a x)\right)+77908 \sin \left(4 \tan ^{-1}(a x)\right)+36612 \sin \left(6 \tan ^{-1}(a x)\right)+7238 \sin \left(8 \tan ^{-1}(a x)\right)\right) \left(a^2 x^2+1\right)^{9/2}\right)-16128 \left(32 \left(5 \cos \left(2 \tan ^{-1}(a x)\right)-1\right) \tan ^{-1}(a x)^3+\left(6 \sin \left(2 \tan ^{-1}(a x)\right)-33 \sin \left(4 \tan ^{-1}(a x)\right)\right) \tan ^{-1}(a x)^2+6 \left(36 \cos \left(2 \tan ^{-1}(a x)\right)+11 \cos \left(4 \tan ^{-1}(a x)\right)+25\right) \tan ^{-1}(a x)+\frac{48 a x}{\left(a^2 x^2+1\right)^2}\right) \left(a^2 x^2+1\right)^{5/2}+774144 \left(-11 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2+11 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-11 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+10 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-11 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+11 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)\right)+256 \left(-16407 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2+16407 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-16407 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+12788 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-16407 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+16407 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)\right)+576 \left(\left(64 \left(-28 \cos \left(2 \tan ^{-1}(a x)\right)+35 \cos \left(4 \tan ^{-1}(a x)\right)+57\right) \tan ^{-1}(a x)^3-3 \left(211 \sin \left(2 \tan ^{-1}(a x)\right)-60 \sin \left(4 \tan ^{-1}(a x)\right)+103 \sin \left(6 \tan ^{-1}(a x)\right)\right) \tan ^{-1}(a x)^2+\frac{8 \left(764 \cos \left(2 \tan ^{-1}(a x)\right)+309 \cos \left(4 \tan ^{-1}(a x)\right)+647\right) \tan ^{-1}(a x)}{a^2 x^2+1}+4 \left(101 \sin \left(2 \tan ^{-1}(a x)\right)+88 \sin \left(4 \tan ^{-1}(a x)\right)+25 \sin \left(6 \tan ^{-1}(a x)\right)\right)\right) \left(a^2 x^2+1\right)^{7/2}+64 \left(309 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2-309 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+309 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-259 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+309 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-309 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)\right)\right)\right)}{15482880 a^4 \sqrt{a^2 x^2+1}}","\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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)}{1344 a^4 \sqrt{a^2 c x^2+c}}",1,"(c^2*Sqrt[c + a^2*c*x^2]*(774144*((-11*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 10*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (11*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (11*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 11*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 11*PolyLog[3, I*E^(I*ArcTan[a*x])]) + 256*((-16407*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 12788*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (16407*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (16407*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 16407*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 16407*PolyLog[3, I*E^(I*ArcTan[a*x])]) - 16128*(1 + a^2*x^2)^(5/2)*((48*a*x)/(1 + a^2*x^2)^2 + 32*ArcTan[a*x]^3*(-1 + 5*Cos[2*ArcTan[a*x]]) + 6*ArcTan[a*x]*(25 + 36*Cos[2*ArcTan[a*x]] + 11*Cos[4*ArcTan[a*x]]) + ArcTan[a*x]^2*(6*Sin[2*ArcTan[a*x]] - 33*Sin[4*ArcTan[a*x]])) + 576*(64*((309*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 259*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - (309*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (309*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 309*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 309*PolyLog[3, I*E^(I*ArcTan[a*x])]) + (1 + a^2*x^2)^(7/2)*(64*ArcTan[a*x]^3*(57 - 28*Cos[2*ArcTan[a*x]] + 35*Cos[4*ArcTan[a*x]]) + (8*ArcTan[a*x]*(647 + 764*Cos[2*ArcTan[a*x]] + 309*Cos[4*ArcTan[a*x]]))/(1 + a^2*x^2) + 4*(101*Sin[2*ArcTan[a*x]] + 88*Sin[4*ArcTan[a*x]] + 25*Sin[6*ArcTan[a*x]]) - 3*ArcTan[a*x]^2*(211*Sin[2*ArcTan[a*x]] - 60*Sin[4*ArcTan[a*x]] + 103*Sin[6*ArcTan[a*x]]))) - (1 + a^2*x^2)^(9/2)*(1536*ArcTan[a*x]^3*(-178 + 711*Cos[2*ArcTan[a*x]] - 126*Cos[4*ArcTan[a*x]] + 105*Cos[6*ArcTan[a*x]]) + (8*ArcTan[a*x]*(87630 + 153529*Cos[2*ArcTan[a*x]] + 59266*Cos[4*ArcTan[a*x]] + 16407*Cos[6*ArcTan[a*x]]))/(1 + a^2*x^2) + 74932*Sin[2*ArcTan[a*x]] + 77908*Sin[4*ArcTan[a*x]] + 36612*Sin[6*ArcTan[a*x]] + 3*ArcTan[a*x]^2*(13074*Sin[2*ArcTan[a*x]] - 26742*Sin[4*ArcTan[a*x]] + 6362*Sin[6*ArcTan[a*x]] - 5469*Sin[8*ArcTan[a*x]]) + 7238*Sin[8*ArcTan[a*x]])))/(15482880*a^4*Sqrt[1 + a^2*x^2])","A",0
429,1,6517,1019,24.4314247,"\int x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3 \, dx","Integrate[x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]","\text{Result too large to show}","\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{Li}_2\left(-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{Li}_2\left(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{Li}_2\left(-\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{Li}_2\left(\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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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,"Result too large to show","B",0
430,1,718,561,6.0922461,"\int x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3 \, dx","Integrate[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]","\frac{c^2 \sqrt{a^2 c x^2+c} \left(64 \left(-259 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-309 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+309 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+309 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-309 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+309 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)+53760 \left(-\tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+\text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)-112 \left(\left(a^2 x^2+1\right)^{5/2} \left(\frac{48 a x}{\left(a^2 x^2+1\right)^2}+\tan ^{-1}(a x)^2 \left(6 \sin \left(2 \tan ^{-1}(a x)\right)-33 \sin \left(4 \tan ^{-1}(a x)\right)\right)+32 \tan ^{-1}(a x)^3 \left(5 \cos \left(2 \tan ^{-1}(a x)\right)-1\right)+6 \tan ^{-1}(a x) \left(36 \cos \left(2 \tan ^{-1}(a x)\right)+11 \cos \left(4 \tan ^{-1}(a x)\right)+25\right)\right)+48 \left(-10 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-11 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+11 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+11 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-11 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+11 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)\right)+\left(a^2 x^2+1\right)^{7/2} \left(\frac{8 \tan ^{-1}(a x) \left(764 \cos \left(2 \tan ^{-1}(a x)\right)+309 \cos \left(4 \tan ^{-1}(a x)\right)+647\right)}{a^2 x^2+1}-3 \tan ^{-1}(a x)^2 \left(211 \sin \left(2 \tan ^{-1}(a x)\right)-60 \sin \left(4 \tan ^{-1}(a x)\right)+103 \sin \left(6 \tan ^{-1}(a x)\right)\right)+4 \left(101 \sin \left(2 \tan ^{-1}(a x)\right)+88 \sin \left(4 \tan ^{-1}(a x)\right)+25 \sin \left(6 \tan ^{-1}(a x)\right)\right)+64 \tan ^{-1}(a x)^3 \left(-28 \cos \left(2 \tan ^{-1}(a x)\right)+35 \cos \left(4 \tan ^{-1}(a x)\right)+57\right)\right)+4480 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x) \left(4 \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)+6 \cos \left(2 \tan ^{-1}(a x)\right)+6\right)\right)}{53760 a^2 \sqrt{a^2 x^2+1}}","-\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{15 i c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{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{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,"(c^2*Sqrt[c + a^2*c*x^2]*(64*((309*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 259*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - (309*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (309*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 309*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 309*PolyLog[3, I*E^(I*ArcTan[a*x])]) + 53760*(I*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + I*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])]) + 4480*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]*(6 + 4*ArcTan[a*x]^2 + 6*Cos[2*ArcTan[a*x]] - 3*ArcTan[a*x]*Sin[2*ArcTan[a*x]]) - 112*(48*((11*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 10*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - (11*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (11*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 11*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 11*PolyLog[3, I*E^(I*ArcTan[a*x])]) + (1 + a^2*x^2)^(5/2)*((48*a*x)/(1 + a^2*x^2)^2 + 32*ArcTan[a*x]^3*(-1 + 5*Cos[2*ArcTan[a*x]]) + 6*ArcTan[a*x]*(25 + 36*Cos[2*ArcTan[a*x]] + 11*Cos[4*ArcTan[a*x]]) + ArcTan[a*x]^2*(6*Sin[2*ArcTan[a*x]] - 33*Sin[4*ArcTan[a*x]]))) + (1 + a^2*x^2)^(7/2)*(64*ArcTan[a*x]^3*(57 - 28*Cos[2*ArcTan[a*x]] + 35*Cos[4*ArcTan[a*x]]) + (8*ArcTan[a*x]*(647 + 764*Cos[2*ArcTan[a*x]] + 309*Cos[4*ArcTan[a*x]]))/(1 + a^2*x^2) + 4*(101*Sin[2*ArcTan[a*x]] + 88*Sin[4*ArcTan[a*x]] + 25*Sin[6*ArcTan[a*x]]) - 3*ArcTan[a*x]^2*(211*Sin[2*ArcTan[a*x]] - 60*Sin[4*ArcTan[a*x]] + 103*Sin[6*ArcTan[a*x]]))))/(53760*a^2*Sqrt[1 + a^2*x^2])","A",0
431,1,4281,870,18.9499956,"\int \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3 \, dx","Integrate[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]","\text{Result too large to show}","-\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{Li}_2\left(-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{Li}_2\left(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{Li}_2\left(-\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{Li}_2\left(\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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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,"((-1/2*I)*c^2*Sqrt[c*(1 + a^2*x^2)]*(12*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x] - (3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + I*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + 2*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 - 3*(2 + ArcTan[a*x]^2)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + 3*(2 + ArcTan[a*x]^2)*PolyLog[2, I*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (6*I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + 6*PolyLog[4, (-I)*E^(I*ArcTan[a*x])] - 6*PolyLog[4, I*E^(I*ArcTan[a*x])]))/(a*Sqrt[1 + a^2*x^2]) + (2*c^2*((Sqrt[c*(1 + a^2*x^2)]*(-1 + ArcTan[a*x]^2))/(4*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])])) - I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])])))/(2*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-1/8*(Pi^3*Log[Cot[(Pi/2 - ArcTan[a*x])/2]]) - (3*Pi^2*((Pi/2 - ArcTan[a*x])*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))])))/4 + (3*Pi*((Pi/2 - ArcTan[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + (2*I)*(Pi/2 - ArcTan[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcTan[a*x]))])))/2 - 8*((I/64)*(Pi/2 - ArcTan[a*x])^4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^4 - ((Pi/2 - ArcTan[a*x])^3*Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/8 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + ((3*I)/8)*(Pi/2 - ArcTan[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + (I/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - (3*(Pi/2 - ArcTan[a*x])*PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))])/4 - (3*Pi*((I/3)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]/2))/2 - (3*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])/2 - ((3*I)/4)*PolyLog[4, -E^(I*(Pi/2 - ArcTan[a*x]))] - ((3*I)/4)*PolyLog[4, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])))/(8*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*(2*ArcTan[a*x] - ArcTan[a*x]^2 - ArcTan[a*x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^2) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^3) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^3) + (Sqrt[c*(1 + a^2*x^2)]*(-2*ArcTan[a*x] - ArcTan[a*x]^2 + ArcTan[a*x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^2) + (Sqrt[c*(1 + a^2*x^2)]*(Sin[ArcTan[a*x]/2] - ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])) + (Sqrt[c*(1 + a^2*x^2)]*(-Sin[ArcTan[a*x]/2] + ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]))))/a + (c^2*((Sqrt[c*(1 + a^2*x^2)]*(50 - 19*ArcTan[a*x]^2))/(240*Sqrt[1 + a^2*x^2]) + (19*Sqrt[c*(1 + a^2*x^2)]*(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])])))/(120*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*((Pi^3*Log[Cot[(Pi/2 - ArcTan[a*x])/2]])/8 + (3*Pi^2*((Pi/2 - ArcTan[a*x])*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))])))/4 - (3*Pi*((Pi/2 - ArcTan[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + (2*I)*(Pi/2 - ArcTan[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcTan[a*x]))])))/2 + 8*((I/64)*(Pi/2 - ArcTan[a*x])^4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^4 - ((Pi/2 - ArcTan[a*x])^3*Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/8 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + ((3*I)/8)*(Pi/2 - ArcTan[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + (I/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - (3*(Pi/2 - ArcTan[a*x])*PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))])/4 - (3*Pi*((I/3)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]/2))/2 - (3*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])/2 - ((3*I)/4)*PolyLog[4, -E^(I*(Pi/2 - ArcTan[a*x]))] - ((3*I)/4)*PolyLog[4, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])))/(16*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(48*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^6) + (Sqrt[c*(1 + a^2*x^2)]*(ArcTan[a*x] - ArcTan[a*x]^2 - 5*ArcTan[a*x]^3))/(80*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*(-2 - 52*ArcTan[a*x] + 26*ArcTan[a*x]^2 + 15*ArcTan[a*x]^3))/(480*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^2) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(40*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^5) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(48*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^6) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(40*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^5) + (Sqrt[c*(1 + a^2*x^2)]*(-ArcTan[a*x] - ArcTan[a*x]^2 + 5*ArcTan[a*x]^3))/(80*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*(-2 + 52*ArcTan[a*x] + 26*ArcTan[a*x]^2 - 15*ArcTan[a*x]^3))/(480*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^2) + (Sqrt[c*(1 + a^2*x^2)]*(50*Sin[ArcTan[a*x]/2] - 19*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(240*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])) + (Sqrt[c*(1 + a^2*x^2)]*(Sin[ArcTan[a*x]/2] - 13*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(120*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^3) + (Sqrt[c*(1 + a^2*x^2)]*(-Sin[ArcTan[a*x]/2] + 13*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(120*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^3) + (Sqrt[c*(1 + a^2*x^2)]*(-50*Sin[ArcTan[a*x]/2] + 19*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(240*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]))))/a","B",0
432,1,1066,845,7.1681861,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3}{x} \, dx","Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x,x]","\frac{1}{8} \sqrt{c \left(a^2 x^2+1\right)} \left(\frac{2 i \tan ^{-1}(a x)^4}{\sqrt{a^2 x^2+1}}+\frac{8 \log \left(1-e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}-\frac{8 \log \left(1+e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}+8 \tan ^{-1}(a x)^3-\frac{24 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{24 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{24 i \text{Li}_2\left(e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{24 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{48 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\frac{48 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\frac{48 \text{Li}_3\left(e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{48 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\frac{48 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{48 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{48 i \text{Li}_4\left(e^{-i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{48 i \text{Li}_4\left(-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{i \pi ^4}{\sqrt{a^2 x^2+1}}\right) c^2+2 \left(\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2-i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-\tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+\text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)\right)}{\sqrt{a^2 x^2+1}}+\frac{1}{12} \left(a^2 x^2+1\right) \sqrt{c \left(a^2 x^2+1\right)} \tan ^{-1}(a x) \left(4 \tan ^{-1}(a x)^2-3 \sin \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+6 \cos \left(2 \tan ^{-1}(a x)\right)+6\right)\right) c^2+\left(\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(-11 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2+11 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-11 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+10 \tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)-11 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+11 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)\right)}{20 \sqrt{a^2 x^2+1}}-\frac{1}{960} \left(a^2 x^2+1\right)^2 \sqrt{c \left(a^2 x^2+1\right)} \left(-32 \tan ^{-1}(a x)^3+6 \sin \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2-33 \sin \left(4 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2+8 \left(20 \tan ^{-1}(a x)^2+27\right) \cos \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+66 \cos \left(4 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+150 \tan ^{-1}(a x)+12 \sin \left(2 \tan ^{-1}(a x)\right)+6 \sin \left(4 \tan ^{-1}(a x)\right)\right)\right) c^2","\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{Li}_2\left(-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{Li}_2\left(-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{Li}_2\left(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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(-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{Li}_3\left(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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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,"(c^2*Sqrt[c*(1 + a^2*x^2)]*(((-I)*Pi^4)/Sqrt[1 + a^2*x^2] + 8*ArcTan[a*x]^3 + ((2*I)*ArcTan[a*x]^4)/Sqrt[1 + a^2*x^2] + (8*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (24*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (24*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (8*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((24*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((24*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((48*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((48*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (48*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (48*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (48*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (48*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((48*I)*PolyLog[4, E^((-I)*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((48*I)*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2]))/8 + 2*c^2*((Sqrt[c*(1 + a^2*x^2)]*(I*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + I*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] + ((1 + a^2*x^2)*Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]*(6 + 4*ArcTan[a*x]^2 + 6*Cos[2*ArcTan[a*x]] - 3*ArcTan[a*x]*Sin[2*ArcTan[a*x]]))/12) + c^2*((Sqrt[c*(1 + a^2*x^2)]*((-11*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 10*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (11*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (11*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 11*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 11*PolyLog[3, I*E^(I*ArcTan[a*x])]))/(20*Sqrt[1 + a^2*x^2]) - ((1 + a^2*x^2)^2*Sqrt[c*(1 + a^2*x^2)]*(150*ArcTan[a*x] - 32*ArcTan[a*x]^3 + 8*ArcTan[a*x]*(27 + 20*ArcTan[a*x]^2)*Cos[2*ArcTan[a*x]] + 66*ArcTan[a*x]*Cos[4*ArcTan[a*x]] + 12*Sin[2*ArcTan[a*x]] + 6*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]] + 6*Sin[4*ArcTan[a*x]] - 33*ArcTan[a*x]^2*Sin[4*ArcTan[a*x]]))/960)","A",0
433,1,3267,1027,15.5451792,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3}{x^2} \, dx","Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x^2,x]","\text{Result too large to show}","-\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{Li}_2\left(-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{Li}_2\left(-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{Li}_2\left(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{Li}_2\left(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{Li}_2\left(-\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{Li}_2\left(\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{Li}_3\left(-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{Li}_3\left(-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{Li}_3\left(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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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,"((-I)*a*c^2*Sqrt[c*(1 + a^2*x^2)]*(12*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x] - (3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + I*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + 2*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 - 3*(2 + ArcTan[a*x]^2)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + 3*(2 + ArcTan[a*x]^2)*PolyLog[2, I*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (6*I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + 6*PolyLog[4, (-I)*E^(I*ArcTan[a*x])] - 6*PolyLog[4, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] + (a*c^2*Sqrt[c*(1 + a^2*x^2)]*Csc[ArcTan[a*x]/2]*(((-7*I)*a*Pi^4*x)/Sqrt[1 + a^2*x^2] - ((8*I)*a*Pi^3*x*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + ((24*I)*a*Pi^2*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] - 64*ArcTan[a*x]^3 - ((32*I)*a*Pi*x*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + ((16*I)*a*x*ArcTan[a*x]^4)/Sqrt[1 + a^2*x^2] + (48*a*Pi^2*x*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (96*a*Pi*x*ArcTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (8*a*Pi^3*x*Log[1 + I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (64*a*x*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (192*a*x*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (8*a*Pi^3*x*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (48*a*Pi^2*x*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (96*a*Pi*x*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (64*a*x*ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (192*a*x*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (8*a*Pi^3*x*Log[Tan[(Pi + 2*ArcTan[a*x])/4]])/Sqrt[1 + a^2*x^2] + ((192*I)*a*x*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((48*I)*a*Pi*x*(Pi - 4*ArcTan[a*x])*PolyLog[2, I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((384*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((48*I)*a*Pi^2*x*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((192*I)*a*Pi*x*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((192*I)*a*x*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((384*I)*a*x*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (384*a*x*ArcTan[a*x]*PolyLog[3, (-I)/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (192*a*Pi*x*PolyLog[3, I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (384*a*x*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (192*a*Pi*x*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (384*a*x*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (384*a*x*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((384*I)*a*x*PolyLog[4, (-I)/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((384*I)*a*x*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2])*Sec[ArcTan[a*x]/2])/(128*Sqrt[1 + a^2*x^2]) + a*c^2*((Sqrt[c*(1 + a^2*x^2)]*(-1 + ArcTan[a*x]^2))/(4*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])])) - I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])])))/(2*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-1/8*(Pi^3*Log[Cot[(Pi/2 - ArcTan[a*x])/2]]) - (3*Pi^2*((Pi/2 - ArcTan[a*x])*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))])))/4 + (3*Pi*((Pi/2 - ArcTan[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + (2*I)*(Pi/2 - ArcTan[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcTan[a*x]))])))/2 - 8*((I/64)*(Pi/2 - ArcTan[a*x])^4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^4 - ((Pi/2 - ArcTan[a*x])^3*Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/8 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + ((3*I)/8)*(Pi/2 - ArcTan[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + (I/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - (3*(Pi/2 - ArcTan[a*x])*PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))])/4 - (3*Pi*((I/3)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] + I*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))] - PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]/2))/2 - (3*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])/2 - ((3*I)/4)*PolyLog[4, -E^(I*(Pi/2 - ArcTan[a*x]))] - ((3*I)/4)*PolyLog[4, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))])))/(8*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*(2*ArcTan[a*x] - ArcTan[a*x]^2 - ArcTan[a*x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^2) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^3) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^3) + (Sqrt[c*(1 + a^2*x^2)]*(-2*ArcTan[a*x] - ArcTan[a*x]^2 + ArcTan[a*x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^2) + (Sqrt[c*(1 + a^2*x^2)]*(Sin[ArcTan[a*x]/2] - ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])) + (Sqrt[c*(1 + a^2*x^2)]*(-Sin[ArcTan[a*x]/2] + ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])))","B",0
434,1,1128,1043,10.5222238,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3}{x^3} \, dx","Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x^3,x]","\frac{1}{4} a^2 \sqrt{c \left(a^2 x^2+1\right)} \left(\frac{2 i \tan ^{-1}(a x)^4}{\sqrt{a^2 x^2+1}}+\frac{8 \log \left(1-e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}-\frac{8 \log \left(1+e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}+8 \tan ^{-1}(a x)^3-\frac{24 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{24 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{24 i \text{Li}_2\left(e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{24 i \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{48 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\frac{48 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\frac{48 \text{Li}_3\left(e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{48 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\frac{48 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{48 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{48 i \text{Li}_4\left(e^{-i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{48 i \text{Li}_4\left(-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{i \pi ^4}{\sqrt{a^2 x^2+1}}\right) c^2+a^2 \left(\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2-i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-\tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+\text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)\right)}{\sqrt{a^2 x^2+1}}+\frac{1}{12} \left(a^2 x^2+1\right) \sqrt{c \left(a^2 x^2+1\right)} \tan ^{-1}(a x) \left(4 \tan ^{-1}(a x)^2-3 \sin \left(2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)+6 \cos \left(2 \tan ^{-1}(a x)\right)+6\right)\right) c^2+\frac{a^2 \sqrt{c \left(a^2 x^2+1\right)} \left(2 i \tan ^{-1}(a x)^4-2 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^3+2 \sec ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^3+8 \log \left(1-e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3-8 \log \left(1+e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3-12 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2+24 i \text{Li}_2\left(e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2-12 \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2+48 \log \left(1-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-48 \log \left(1+e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+48 \text{Li}_3\left(e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-48 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+24 i \left(\tan ^{-1}(a x)^2+2\right) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-48 i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-48 i \text{Li}_4\left(e^{-i \tan ^{-1}(a x)}\right)-48 i \text{Li}_4\left(-e^{i \tan ^{-1}(a x)}\right)-i \pi ^4\right) c^2}{16 \sqrt{a^2 x^2+1}}","-\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{Li}_2\left(-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{Li}_2\left(-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{Li}_2\left(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{Li}_2\left(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{Li}_2\left(-\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{Li}_2\left(\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{Li}_3\left(-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{Li}_3\left(-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{Li}_3\left(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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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*(1 + a^2*x^2)]*(((-I)*Pi^4)/Sqrt[1 + a^2*x^2] + 8*ArcTan[a*x]^3 + ((2*I)*ArcTan[a*x]^4)/Sqrt[1 + a^2*x^2] + (8*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (24*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (24*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (8*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((24*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((24*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((48*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((48*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (48*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (48*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (48*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (48*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((48*I)*PolyLog[4, E^((-I)*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((48*I)*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2]))/4 + a^2*c^2*((Sqrt[c*(1 + a^2*x^2)]*(I*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + I*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] + ((1 + a^2*x^2)*Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]*(6 + 4*ArcTan[a*x]^2 + 6*Cos[2*ArcTan[a*x]] - 3*ArcTan[a*x]*Sin[2*ArcTan[a*x]]))/12) + (a^2*c^2*Sqrt[c*(1 + a^2*x^2)]*((-I)*Pi^4 + (2*I)*ArcTan[a*x]^4 - 12*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2] - 2*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^2 + 8*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])] + 48*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 48*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] - 8*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] + (24*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])] + (24*I)*(2 + ArcTan[a*x]^2)*PolyLog[2, -E^(I*ArcTan[a*x])] - (48*I)*PolyLog[2, E^(I*ArcTan[a*x])] + 48*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])] - 48*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] - (48*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] - (48*I)*PolyLog[4, -E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]^3*Sec[ArcTan[a*x]/2]^2 - 12*ArcTan[a*x]^2*Tan[ArcTan[a*x]/2]))/(16*Sqrt[1 + a^2*x^2])","A",0
435,1,1771,1061,11.5000813,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3}{x^4} \, dx","Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x^4,x]","-\frac{i c^2 \sqrt{c \left(a^2 x^2+1\right)} \left(2 \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3+i a x \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3-3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2+12 \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-6 i \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+6 i \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-3 \left(\tan ^{-1}(a x)^2+2\right) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+3 \left(\tan ^{-1}(a x)^2+2\right) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+6 \text{Li}_4\left(-i e^{i \tan ^{-1}(a x)}\right)-6 \text{Li}_4\left(i e^{i \tan ^{-1}(a x)}\right)\right) a^3}{2 \sqrt{a^2 x^2+1}}+\frac{c^2 \sqrt{c \left(a^2 x^2+1\right)} \csc \left(\frac{1}{2} \tan ^{-1}(a x)\right) \left(\frac{16 i a x \tan ^{-1}(a x)^4}{\sqrt{a^2 x^2+1}}+\frac{64 a x \log \left(1+i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}-\frac{64 a x \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}-\frac{32 i a \pi  x \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}-64 \tan ^{-1}(a x)^3-\frac{96 a \pi  x \log \left(1-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{192 a x \log \left(1-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{96 a \pi  x \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{192 a x \log \left(1+e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{192 i a x \text{Li}_2\left(-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{192 i a x \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{24 i a \pi ^2 x \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{48 a \pi ^2 x \log \left(1-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{48 a \pi ^2 x \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\frac{384 i a x \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{192 i a \pi  x \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{384 i a x \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+\frac{384 a x \text{Li}_3\left(-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{384 a x \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{8 i a \pi ^3 x \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-\frac{8 a \pi ^3 x \log \left(1+i e^{-i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{8 a \pi ^3 x \log \left(1+i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{8 a \pi ^3 x \log \left(\tan \left(\frac{1}{4} \left(2 \tan ^{-1}(a x)+\pi \right)\right)\right)}{\sqrt{a^2 x^2+1}}+\frac{48 i a \pi  x \left(\pi -4 \tan ^{-1}(a x)\right) \text{Li}_2\left(i e^{-i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{48 i a \pi ^2 x \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{192 a \pi  x \text{Li}_3\left(i e^{-i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{384 a x \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{192 a \pi  x \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{384 a x \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{384 i a x \text{Li}_4\left(-i e^{-i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{384 i a x \text{Li}_4\left(-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{7 i a \pi ^4 x}{\sqrt{a^2 x^2+1}}\right) \sec \left(\frac{1}{2} \tan ^{-1}(a x)\right) a^3}{64 \sqrt{a^2 x^2+1}}+\frac{c^3 \sqrt{a^2 x^2+1} \left(-\frac{a x \tan ^{-1}(a x)^3 \csc ^4\left(\frac{1}{2} \tan ^{-1}(a x)\right)}{2 \sqrt{a^2 x^2+1}}-3 \tan ^{-1}(a x)^2 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)-\frac{8 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^3 \sin ^4\left(\frac{1}{2} \tan ^{-1}(a x)\right)}{a^3 x^3}+3 \tan ^{-1}(a x)^2 \sec ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)-2 \tan ^{-1}(a x)^3 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-12 \tan ^{-1}(a x) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)+12 \tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)-12 \tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)+24 \log \left(\tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)+24 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-24 i \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-24 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+24 \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)-2 \tan ^{-1}(a x)^3 \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)-12 \tan ^{-1}(a x) \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right) a^3}{24 \sqrt{c \left(a^2 x^2+1\right)}}","\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{Li}_2\left(-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{Li}_2\left(-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{Li}_2\left(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{Li}_2\left(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{Li}_2\left(-\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{Li}_2\left(\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{Li}_3\left(-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{Li}_3\left(-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{Li}_3\left(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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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,"((-1/2*I)*a^3*c^2*Sqrt[c*(1 + a^2*x^2)]*(12*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x] - (3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + I*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + 2*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 - 3*(2 + ArcTan[a*x]^2)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + 3*(2 + ArcTan[a*x]^2)*PolyLog[2, I*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (6*I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + 6*PolyLog[4, (-I)*E^(I*ArcTan[a*x])] - 6*PolyLog[4, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] + (a^3*c^2*Sqrt[c*(1 + a^2*x^2)]*Csc[ArcTan[a*x]/2]*(((-7*I)*a*Pi^4*x)/Sqrt[1 + a^2*x^2] - ((8*I)*a*Pi^3*x*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + ((24*I)*a*Pi^2*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] - 64*ArcTan[a*x]^3 - ((32*I)*a*Pi*x*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + ((16*I)*a*x*ArcTan[a*x]^4)/Sqrt[1 + a^2*x^2] + (48*a*Pi^2*x*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (96*a*Pi*x*ArcTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (8*a*Pi^3*x*Log[1 + I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (64*a*x*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (192*a*x*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (8*a*Pi^3*x*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (48*a*Pi^2*x*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (96*a*Pi*x*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (64*a*x*ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (192*a*x*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (8*a*Pi^3*x*Log[Tan[(Pi + 2*ArcTan[a*x])/4]])/Sqrt[1 + a^2*x^2] + ((192*I)*a*x*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((48*I)*a*Pi*x*(Pi - 4*ArcTan[a*x])*PolyLog[2, I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((384*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((48*I)*a*Pi^2*x*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((192*I)*a*Pi*x*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((192*I)*a*x*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((384*I)*a*x*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (384*a*x*ArcTan[a*x]*PolyLog[3, (-I)/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (192*a*Pi*x*PolyLog[3, I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (384*a*x*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (192*a*Pi*x*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (384*a*x*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (384*a*x*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((384*I)*a*x*PolyLog[4, (-I)/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((384*I)*a*x*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2])*Sec[ArcTan[a*x]/2])/(64*Sqrt[1 + a^2*x^2]) + (a^3*c^3*Sqrt[1 + a^2*x^2]*(-12*ArcTan[a*x]*Cot[ArcTan[a*x]/2] - 2*ArcTan[a*x]^3*Cot[ArcTan[a*x]/2] - 3*ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 - (a*x*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^4)/(2*Sqrt[1 + a^2*x^2]) + 12*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] - 12*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 24*Log[Tan[ArcTan[a*x]/2]] + (24*I)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - (24*I)*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - 24*PolyLog[3, -E^(I*ArcTan[a*x])] + 24*PolyLog[3, E^(I*ArcTan[a*x])] + 3*ArcTan[a*x]^2*Sec[ArcTan[a*x]/2]^2 - (8*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^3*Sin[ArcTan[a*x]/2]^4)/(a^3*x^3) - 12*ArcTan[a*x]*Tan[ArcTan[a*x]/2] - 2*ArcTan[a*x]^3*Tan[ArcTan[a*x]/2]))/(24*Sqrt[c*(1 + a^2*x^2)])","A",0
436,1,220,408,0.8534795,"\int \frac{x^3 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(x^3*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2],x]","\frac{\sqrt{a^2 c x^2+c} \left(-\left(a^2 x^2+1\right) \tan ^{-1}(a x) \left(2 \tan ^{-1}(a x)^2+3 \tan ^{-1}(a x) \sin \left(2 \tan ^{-1}(a x)\right)+6 \left(\tan ^{-1}(a x)^2-1\right) \cos \left(2 \tan ^{-1}(a x)\right)-6\right)+\frac{12 \left(-\tanh ^{-1}\left(\frac{a x}{\sqrt{a^2 x^2+1}}\right)+5 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-5 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-5 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+5 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-5 i \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2\right)}{\sqrt{a^2 x^2+1}}\right)}{12 a^4 c}","\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{3 a^2 c}+\frac{5 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)}{a^4 \sqrt{a^2 c x^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{\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{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a^3 c}",1,"(Sqrt[c + a^2*c*x^2]*((12*((-5*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (5*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (5*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 5*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 5*PolyLog[3, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] - (1 + a^2*x^2)*ArcTan[a*x]*(-6 + 2*ArcTan[a*x]^2 + 6*(-1 + ArcTan[a*x]^2)*Cos[2*ArcTan[a*x]] + 3*ArcTan[a*x]*Sin[2*ArcTan[a*x]])))/(12*a^4*c)","A",0
437,1,812,625,7.8687433,"\int \frac{x^2 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(x^2*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2],x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(-\frac{1}{2} i \tan ^{-1}(a x)^4-2 \log \left(1+i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3+2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3+\frac{\tan ^{-1}(a x)^3}{\left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)^2}-\frac{\tan ^{-1}(a x)^3}{\left(\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)+\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)^2}+i \pi  \tan ^{-1}(a x)^3+3 \pi  \log \left(1-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2-3 \pi  \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2-6 i \text{Li}_2\left(-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2-6 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^2-\frac{6 \sin \left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2}{\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)}+\frac{6 \sin \left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2}{\cos \left(\frac{1}{2} \tan ^{-1}(a x)\right)+\sin \left(\frac{1}{2} \tan ^{-1}(a x)\right)}-\frac{3}{4} i \pi ^2 \tan ^{-1}(a x)^2-6 \tan ^{-1}(a x)^2-\frac{3}{2} \pi ^2 \log \left(1-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+12 \log \left(1-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+\frac{3}{2} \pi ^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-12 \log \left(1+i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+6 i \pi  \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)-12 \text{Li}_3\left(-i e^{-i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+12 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)+\frac{1}{4} i \pi ^3 \tan ^{-1}(a x)+\frac{1}{4} \pi ^3 \log \left(1+i e^{-i \tan ^{-1}(a x)}\right)-\frac{1}{4} \pi ^3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-\frac{1}{4} \pi ^3 \log \left(\tan \left(\frac{1}{4} \left(2 \tan ^{-1}(a x)+\pi \right)\right)\right)-\frac{3}{2} i \pi  \left(\pi -4 \tan ^{-1}(a x)\right) \text{Li}_2\left(i e^{-i \tan ^{-1}(a x)}\right)-\frac{3}{2} i \pi ^2 \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+12 i \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-12 i \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+6 \pi  \text{Li}_3\left(i e^{-i \tan ^{-1}(a x)}\right)-6 \pi  \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+12 i \text{Li}_4\left(-i e^{-i \tan ^{-1}(a x)}\right)+12 i \text{Li}_4\left(-i e^{i \tan ^{-1}(a x)}\right)+\frac{7 i \pi ^4}{32}\right)}{4 a^3 c \sqrt{a^2 x^2+1}}","\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac{3 i \sqrt{a^2 x^2+1} \text{Li}_2\left(-\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 \sqrt{a^2 x^2+1} \text{Li}_2\left(\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 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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{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,"(Sqrt[c*(1 + a^2*x^2)]*(((7*I)/32)*Pi^4 + (I/4)*Pi^3*ArcTan[a*x] - 6*ArcTan[a*x]^2 - ((3*I)/4)*Pi^2*ArcTan[a*x]^2 + I*Pi*ArcTan[a*x]^3 - (I/2)*ArcTan[a*x]^4 - (3*Pi^2*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])])/2 + 3*Pi*ArcTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])] + (Pi^3*Log[1 + I/E^(I*ArcTan[a*x])])/4 - 2*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])] + 12*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] - (Pi^3*Log[1 + I*E^(I*ArcTan[a*x])])/4 - 12*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] + (3*Pi^2*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/2 - 3*Pi*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])] - (Pi^3*Log[Tan[(Pi + 2*ArcTan[a*x])/4]])/4 - (6*I)*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])] - ((3*I)/2)*Pi*(Pi - 4*ArcTan[a*x])*PolyLog[2, I/E^(I*ArcTan[a*x])] + (12*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - ((3*I)/2)*Pi^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (6*I)*Pi*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (12*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - 12*ArcTan[a*x]*PolyLog[3, (-I)/E^(I*ArcTan[a*x])] + 6*Pi*PolyLog[3, I/E^(I*ArcTan[a*x])] - 6*Pi*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 12*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (12*I)*PolyLog[4, (-I)/E^(I*ArcTan[a*x])] + (12*I)*PolyLog[4, (-I)*E^(I*ArcTan[a*x])] + ArcTan[a*x]^3/(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^2 - (6*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]) - ArcTan[a*x]^3/(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^2 + (6*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])))/(4*a^3*c*Sqrt[1 + a^2*x^2])","A",0
438,1,168,283,0.2951198,"\int \frac{x \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(x*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2],x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(\tan ^{-1}(a x)^3-\frac{3 \left(2 i \tan ^{-1}(a x) \left(\text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)\right)-2 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+2 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+\tan ^{-1}(a x)^2 \left(\log \left(1-i e^{i \tan ^{-1}(a x)}\right)-\log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)\right)}{\sqrt{a^2 x^2+1}}\right)}{a^2 c}","-\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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,"(Sqrt[c*(1 + a^2*x^2)]*(ArcTan[a*x]^3 - (3*(ArcTan[a*x]^2*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) + (2*I)*ArcTan[a*x]*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2, I*E^(I*ArcTan[a*x])]) - 2*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 2*PolyLog[3, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2]))/(a^2*c)","A",0
439,1,190,368,0.1525788,"\int \frac{\tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]^3/Sqrt[c + a^2*c*x^2],x]","-\frac{i \sqrt{c \left(a^2 x^2+1\right)} \left(-3 \tan ^{-1}(a x)^2 \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+3 \tan ^{-1}(a x)^2 \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)-6 i \tan ^{-1}(a x) \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+6 i \tan ^{-1}(a x) \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)+6 \text{Li}_4\left(-i e^{i \tan ^{-1}(a x)}\right)-6 \text{Li}_4\left(i e^{i \tan ^{-1}(a x)}\right)+2 \tan ^{-1}\left(e^{i \tan ^{-1}(a x)}\right) \tan ^{-1}(a x)^3\right)}{a c \sqrt{a^2 x^2+1}}","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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,"((-I)*Sqrt[c*(1 + a^2*x^2)]*(2*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 - 3*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + 3*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (6*I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + 6*PolyLog[4, (-I)*E^(I*ArcTan[a*x])] - 6*PolyLog[4, I*E^(I*ArcTan[a*x])]))/(a*c*Sqrt[1 + a^2*x^2])","A",1
440,1,208,327,0.2368591,"\int \frac{\tan ^{-1}(a x)^3}{x \sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]^3/(x*Sqrt[c + a^2*c*x^2]),x]","-\frac{i \sqrt{a^2 x^2+1} \left(-24 \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-i \tan ^{-1}(a x)}\right)-24 \tan ^{-1}(a x)^2 \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+48 i \tan ^{-1}(a x) \text{Li}_3\left(e^{-i \tan ^{-1}(a x)}\right)-48 i \tan ^{-1}(a x) \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+48 \text{Li}_4\left(e^{-i \tan ^{-1}(a x)}\right)+48 \text{Li}_4\left(-e^{i \tan ^{-1}(a x)}\right)-2 \tan ^{-1}(a x)^4+8 i \tan ^{-1}(a x)^3 \log \left(1-e^{-i \tan ^{-1}(a x)}\right)-8 i \tan ^{-1}(a x)^3 \log \left(1+e^{i \tan ^{-1}(a x)}\right)+\pi ^4\right)}{8 \sqrt{c \left(a^2 x^2+1\right)}}","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{Li}_2\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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 i \sqrt{a^2 x^2+1} \text{Li}_4\left(-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 i \sqrt{a^2 x^2+1} \text{Li}_4\left(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,"((-1/8*I)*Sqrt[1 + a^2*x^2]*(Pi^4 - 2*ArcTan[a*x]^4 + (8*I)*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])] - (8*I)*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] - 24*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])] - 24*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])] + (48*I)*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])] - (48*I)*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] + 48*PolyLog[4, E^((-I)*ArcTan[a*x])] + 48*PolyLog[4, -E^(I*ArcTan[a*x])]))/Sqrt[c*(1 + a^2*x^2)]","A",0
441,1,174,260,0.3449539,"\int \frac{\tan ^{-1}(a x)^3}{x^2 \sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]^3/(x^2*Sqrt[c + a^2*c*x^2]),x]","-\frac{a \sqrt{a^2 x^2+1} \left(\frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3}{a x}-6 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+6 i \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+6 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)-6 \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)-3 \tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)+3 \tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)\right)}{\sqrt{c \left(a^2 x^2+1\right)}}","\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\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{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}+\frac{6 a \sqrt{a^2 x^2+1} \text{Li}_3\left(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,"-((a*Sqrt[1 + a^2*x^2]*((Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3)/(a*x) - 3*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] + 3*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] + (6*I)*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] + 6*PolyLog[3, -E^(I*ArcTan[a*x])] - 6*PolyLog[3, E^(I*ArcTan[a*x])]))/Sqrt[c*(1 + a^2*x^2)])","A",0
442,1,345,597,4.2135206,"\int \frac{\tan ^{-1}(a x)^3}{x^3 \sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]^3/(x^3*Sqrt[c + a^2*c*x^2]),x]","\frac{a^2 \sqrt{a^2 x^2+1} \left(-24 i \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-i \tan ^{-1}(a x)}\right)-48 \tan ^{-1}(a x) \text{Li}_3\left(e^{-i \tan ^{-1}(a x)}\right)+48 \tan ^{-1}(a x) \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)-24 i \left(\tan ^{-1}(a x)^2-2\right) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-48 i \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+48 i \text{Li}_4\left(e^{-i \tan ^{-1}(a x)}\right)+48 i \text{Li}_4\left(-e^{i \tan ^{-1}(a x)}\right)-2 i \tan ^{-1}(a x)^4-12 \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^2-8 \tan ^{-1}(a x)^3 \log \left(1-e^{-i \tan ^{-1}(a x)}\right)+8 \tan ^{-1}(a x)^3 \log \left(1+e^{i \tan ^{-1}(a x)}\right)+48 \tan ^{-1}(a x) \log \left(1-e^{i \tan ^{-1}(a x)}\right)-48 \tan ^{-1}(a x) \log \left(1+e^{i \tan ^{-1}(a x)}\right)-12 \tan ^{-1}(a x)^2 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-2 \tan ^{-1}(a x)^3 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+2 \tan ^{-1}(a x)^3 \sec ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+i \pi ^4\right)}{16 \sqrt{c \left(a^2 x^2+1\right)}}","\frac{3 i a^2 \sqrt{a^2 x^2+1} \text{Li}_2\left(-\frac{\sqrt{i a x+1}}{\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{Li}_2\left(\frac{\sqrt{i a x+1}}{\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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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,"(a^2*Sqrt[1 + a^2*x^2]*(I*Pi^4 - (2*I)*ArcTan[a*x]^4 - 12*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2] - 2*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^2 - 8*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])] + 48*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a*x])] - 48*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + 8*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] - (24*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])] - (24*I)*(-2 + ArcTan[a*x]^2)*PolyLog[2, -E^(I*ArcTan[a*x])] - (48*I)*PolyLog[2, E^(I*ArcTan[a*x])] - 48*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])] + 48*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] + (48*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] + (48*I)*PolyLog[4, -E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]^3*Sec[ArcTan[a*x]/2]^2 - 12*ArcTan[a*x]^2*Tan[ArcTan[a*x]/2]))/(16*Sqrt[c*(1 + a^2*x^2)])","A",0
443,1,343,396,6.5061215,"\int \frac{\tan ^{-1}(a x)^3}{x^4 \sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcTan[a*x]^3/(x^4*Sqrt[c + a^2*c*x^2]),x]","\frac{a^3 \sqrt{c \left(a^2 x^2+1\right)} \left(-\frac{a x \tan ^{-1}(a x)^3 \csc ^4\left(\frac{1}{2} \tan ^{-1}(a x)\right)}{2 \sqrt{a^2 x^2+1}}-\frac{8 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^3 \sin ^4\left(\frac{1}{2} \tan ^{-1}(a x)\right)}{a^3 x^3}-120 i \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+120 i \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+120 \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)-120 \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)+10 \tan ^{-1}(a x)^3 \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)-12 \tan ^{-1}(a x) \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)-60 \tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)+60 \tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)+24 \log \left(\tan \left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)+10 \tan ^{-1}(a x)^3 \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-12 \tan ^{-1}(a x) \cot \left(\frac{1}{2} \tan ^{-1}(a x)\right)-3 \tan ^{-1}(a x)^2 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+3 \tan ^{-1}(a x)^2 \sec ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)\right)}{24 c \sqrt{a^2 x^2+1}}","\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{5 i a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 c x^2+c}}-\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^3*Sqrt[c*(1 + a^2*x^2)]*(-12*ArcTan[a*x]*Cot[ArcTan[a*x]/2] + 10*ArcTan[a*x]^3*Cot[ArcTan[a*x]/2] - 3*ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 - (a*x*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^4)/(2*Sqrt[1 + a^2*x^2]) - 60*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] + 60*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 24*Log[Tan[ArcTan[a*x]/2]] - (120*I)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] + (120*I)*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] + 120*PolyLog[3, -E^(I*ArcTan[a*x])] - 120*PolyLog[3, E^(I*ArcTan[a*x])] + 3*ArcTan[a*x]^2*Sec[ArcTan[a*x]/2]^2 - (8*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^3*Sin[ArcTan[a*x]/2]^4)/(a^3*x^3) - 12*ArcTan[a*x]*Tan[ArcTan[a*x]/2] + 10*ArcTan[a*x]^3*Tan[ArcTan[a*x]/2]))/(24*c*Sqrt[1 + a^2*x^2])","A",0
444,1,308,403,0.8892513,"\int \frac{x^3 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(x^3*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(3/2),x]","\frac{\sqrt{a^2 x^2+1} \left(\frac{6 a x}{\sqrt{a^2 x^2+1}}+\frac{3}{2} \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3-\frac{3 a x \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)+\frac{1}{2} \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \cos \left(2 \tan ^{-1}(a x)\right)-3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \cos \left(2 \tan ^{-1}(a x)\right)-6 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+6 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)+6 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-6 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)-3 \tan ^{-1}(a x)^2 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)+3 \tan ^{-1}(a x)^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)\right)}{a^4 c \sqrt{c \left(a^2 x^2+1\right)}}","\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{a^4 c^2}-\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)}{a^4 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{6 \tan ^{-1}(a x)}{a^4 c \sqrt{a^2 c x^2+c}}+\frac{6 x}{a^3 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}}",1,"(Sqrt[1 + a^2*x^2]*((6*a*x)/Sqrt[1 + a^2*x^2] - 3*Sqrt[1 + a^2*x^2]*ArcTan[a*x] - (3*a*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] + (3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3)/2 - 3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Cos[2*ArcTan[a*x]] + (Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]])/2 - 3*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] + 3*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (6*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 6*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 6*PolyLog[3, I*E^(I*ArcTan[a*x])]))/(a^4*c*Sqrt[c*(1 + a^2*x^2)])","A",0
445,1,639,495,1.7508189,"\int \frac{x^2 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(3/2),x]","-\frac{\sqrt{a^2 x^2+1} \left(-\frac{384}{\sqrt{a^2 x^2+1}}+\frac{64 a x \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}+\frac{192 \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{384 a x \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-192 i \tan ^{-1}(a x)^2 \text{Li}_2\left(-i e^{-i \tan ^{-1}(a x)}\right)-192 i \tan ^{-1}(a x)^2 \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+192 i \pi  \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-384 \tan ^{-1}(a x) \text{Li}_3\left(-i e^{-i \tan ^{-1}(a x)}\right)+384 \tan ^{-1}(a x) \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-48 i \pi  \left(\pi -4 \tan ^{-1}(a x)\right) \text{Li}_2\left(i e^{-i \tan ^{-1}(a x)}\right)-48 i \pi ^2 \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+192 \pi  \text{Li}_3\left(i e^{-i \tan ^{-1}(a x)}\right)-192 \pi  \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+384 i \text{Li}_4\left(-i e^{-i \tan ^{-1}(a x)}\right)+384 i \text{Li}_4\left(-i e^{i \tan ^{-1}(a x)}\right)-16 i \tan ^{-1}(a x)^4+32 i \pi  \tan ^{-1}(a x)^3-24 i \pi ^2 \tan ^{-1}(a x)^2+8 i \pi ^3 \tan ^{-1}(a x)-64 \tan ^{-1}(a x)^3 \log \left(1+i e^{-i \tan ^{-1}(a x)}\right)+64 \tan ^{-1}(a x)^3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+96 \pi  \tan ^{-1}(a x)^2 \log \left(1-i e^{-i \tan ^{-1}(a x)}\right)-96 \pi  \tan ^{-1}(a x)^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-48 \pi ^2 \tan ^{-1}(a x) \log \left(1-i e^{-i \tan ^{-1}(a x)}\right)+48 \pi ^2 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+8 \pi ^3 \log \left(1+i e^{-i \tan ^{-1}(a x)}\right)-8 \pi ^3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-8 \pi ^3 \log \left(\tan \left(\frac{1}{4} \left(2 \tan ^{-1}(a x)+\pi \right)\right)\right)+7 i \pi ^4\right)}{64 a^3 c \sqrt{c \left(a^2 x^2+1\right)}}","-\frac{x \tan ^{-1}(a x)^3}{a^2 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{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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{3 \tan ^{-1}(a x)^2}{a^3 c \sqrt{a^2 c x^2+c}}",1,"-1/64*(Sqrt[1 + a^2*x^2]*((7*I)*Pi^4 - 384/Sqrt[1 + a^2*x^2] + (8*I)*Pi^3*ArcTan[a*x] - (384*a*x*ArcTan[a*x])/Sqrt[1 + a^2*x^2] - (24*I)*Pi^2*ArcTan[a*x]^2 + (192*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] + (32*I)*Pi*ArcTan[a*x]^3 + (64*a*x*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] - (16*I)*ArcTan[a*x]^4 - 48*Pi^2*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])] + 96*Pi*ArcTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])] + 8*Pi^3*Log[1 + I/E^(I*ArcTan[a*x])] - 64*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])] - 8*Pi^3*Log[1 + I*E^(I*ArcTan[a*x])] + 48*Pi^2*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] - 96*Pi*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] + 64*ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])] - 8*Pi^3*Log[Tan[(Pi + 2*ArcTan[a*x])/4]] - (192*I)*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])] - (48*I)*Pi*(Pi - 4*ArcTan[a*x])*PolyLog[2, I/E^(I*ArcTan[a*x])] - (48*I)*Pi^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (192*I)*Pi*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (192*I)*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - 384*ArcTan[a*x]*PolyLog[3, (-I)/E^(I*ArcTan[a*x])] + 192*Pi*PolyLog[3, I/E^(I*ArcTan[a*x])] - 192*Pi*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 384*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (384*I)*PolyLog[4, (-I)/E^(I*ArcTan[a*x])] + (384*I)*PolyLog[4, (-I)*E^(I*ArcTan[a*x])]))/(a^3*c*Sqrt[c*(1 + a^2*x^2)])","A",0
446,1,61,107,0.1018551,"\int \frac{x \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(x*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(3/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(-6 a x-\tan ^{-1}(a x)^3+3 a x \tan ^{-1}(a x)^2+6 \tan ^{-1}(a x)\right)}{a^2 c^2 \left(a^2 x^2+1\right)}","-\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,"(Sqrt[c + a^2*c*x^2]*(-6*a*x + 6*ArcTan[a*x] + 3*a*x*ArcTan[a*x]^2 - ArcTan[a*x]^3))/(a^2*c^2*(1 + a^2*x^2))","A",1
447,1,56,100,0.077016,"\int \frac{\tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]^3/(c + a^2*c*x^2)^(3/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(a x \tan ^{-1}(a x)^3+3 \tan ^{-1}(a x)^2-6 a x \tan ^{-1}(a x)-6\right)}{c^2 \left(a^3 x^2+a\right)}","-\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,"(Sqrt[c + a^2*c*x^2]*(-6 - 6*a*x*ArcTan[a*x] + 3*ArcTan[a*x]^2 + a*x*ArcTan[a*x]^3))/(c^2*(a + a^3*x^2))","A",1
448,1,295,443,0.470463,"\int \frac{\tan ^{-1}(a x)^3}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]^3/(x*(c + a^2*c*x^2)^(3/2)),x]","\frac{\sqrt{a^2 x^2+1} \left(\frac{48 a x}{\sqrt{a^2 x^2+1}}+\frac{8 \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}-\frac{24 a x \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{48 \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+24 i \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-i \tan ^{-1}(a x)}\right)+24 i \tan ^{-1}(a x)^2 \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+48 \tan ^{-1}(a x) \text{Li}_3\left(e^{-i \tan ^{-1}(a x)}\right)-48 \tan ^{-1}(a x) \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)-48 i \text{Li}_4\left(e^{-i \tan ^{-1}(a x)}\right)-48 i \text{Li}_4\left(-e^{i \tan ^{-1}(a x)}\right)+2 i \tan ^{-1}(a x)^4+8 \tan ^{-1}(a x)^3 \log \left(1-e^{-i \tan ^{-1}(a x)}\right)-8 \tan ^{-1}(a x)^3 \log \left(1+e^{i \tan ^{-1}(a x)}\right)-i \pi ^4\right)}{8 c \sqrt{c \left(a^2 x^2+1\right)}}","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{Li}_2\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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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,"(Sqrt[1 + a^2*x^2]*((-I)*Pi^4 + (48*a*x)/Sqrt[1 + a^2*x^2] - (48*ArcTan[a*x])/Sqrt[1 + a^2*x^2] - (24*a*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] + (8*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + (2*I)*ArcTan[a*x]^4 + 8*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])] - 8*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] + (24*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])] + (24*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])] + 48*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])] - 48*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] - (48*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] - (48*I)*PolyLog[4, -E^(I*ArcTan[a*x])]))/(8*c*Sqrt[c*(1 + a^2*x^2)])","A",0
449,1,301,377,1.6014803,"\int \frac{\tan ^{-1}(a x)^3}{x^2 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)^(3/2)),x]","\frac{a \left(12 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)-12 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)-12 \sqrt{a^2 x^2+1} \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)+12 \sqrt{a^2 x^2+1} \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)+6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)-6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)-\frac{2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3 \sin ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)}{a x}-2 a x \tan ^{-1}(a x)^3-6 \tan ^{-1}(a x)^2+12 a x \tan ^{-1}(a x)-\frac{1}{2} a x \tan ^{-1}(a x)^3 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)+12\right)}{2 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 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)}{c \sqrt{a^2 c x^2+c}}+\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,"(a*(12 + 12*a*x*ArcTan[a*x] - 6*ArcTan[a*x]^2 - 2*a*x*ArcTan[a*x]^3 - (a*x*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^2)/2 + 6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] - 6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + (12*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - (12*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - 12*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])] + 12*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])] - (2*(1 + a^2*x^2)*ArcTan[a*x]^3*Sin[ArcTan[a*x]/2]^2)/(a*x)))/(2*c*Sqrt[c + a^2*c*x^2])","A",0
450,1,367,534,2.8665208,"\int \frac{x^5 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^5*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]","-\frac{\left(a^2 x^2+1\right)^2 \left(\frac{1296 i \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{1296 i \tan ^{-1}(a x) \text{Li}_2\left(i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{1296 \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{1296 \text{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}+\frac{648 \tan ^{-1}(a x)^2 \log \left(1-i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-\frac{648 \tan ^{-1}(a x)^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)}{\sqrt{a^2 x^2+1}}-405 \tan ^{-1}(a x)^3+1134 \tan ^{-1}(a x)+558 \tan ^{-1}(a x)^2 \sin \left(2 \tan ^{-1}(a x)\right)-9 \tan ^{-1}(a x)^2 \sin \left(4 \tan ^{-1}(a x)\right)-1132 \sin \left(2 \tan ^{-1}(a x)\right)+2 \sin \left(4 \tan ^{-1}(a x)\right)-180 \tan ^{-1}(a x)^3 \cos \left(2 \tan ^{-1}(a x)\right)+9 \tan ^{-1}(a x)^3 \cos \left(4 \tan ^{-1}(a x)\right)+1128 \tan ^{-1}(a x) \cos \left(2 \tan ^{-1}(a x)\right)-6 \tan ^{-1}(a x) \cos \left(4 \tan ^{-1}(a x)\right)\right)}{216 a^6 c \left(c \left(a^2 x^2+1\right)\right)^{3/2}}","\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{a^6 c^3}-\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(i e^{i \tan ^{-1}(a x)}\right)}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\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{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{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{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}}",1,"-1/216*((1 + a^2*x^2)^2*(1134*ArcTan[a*x] - 405*ArcTan[a*x]^3 + 1128*ArcTan[a*x]*Cos[2*ArcTan[a*x]] - 180*ArcTan[a*x]^3*Cos[2*ArcTan[a*x]] - 6*ArcTan[a*x]*Cos[4*ArcTan[a*x]] + 9*ArcTan[a*x]^3*Cos[4*ArcTan[a*x]] + (648*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (648*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((1296*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((1296*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (1296*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (1296*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - 1132*Sin[2*ArcTan[a*x]] + 558*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]] + 2*Sin[4*ArcTan[a*x]] - 9*ArcTan[a*x]^2*Sin[4*ArcTan[a*x]]))/(a^6*c*(c*(1 + a^2*x^2))^(3/2))","A",0
451,1,691,622,2.9240697,"\int \frac{x^4 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^4*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]","-\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(-\frac{12960}{\sqrt{a^2 x^2+1}}+\frac{2160 a x \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}+\frac{6480 \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{12960 a x \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}-5184 i \tan ^{-1}(a x)^2 \text{Li}_2\left(-i e^{-i \tan ^{-1}(a x)}\right)-5184 i \tan ^{-1}(a x)^2 \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+5184 i \pi  \tan ^{-1}(a x) \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)-10368 \tan ^{-1}(a x) \text{Li}_3\left(-i e^{-i \tan ^{-1}(a x)}\right)+10368 \tan ^{-1}(a x) \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)-1296 i \pi  \left(\pi -4 \tan ^{-1}(a x)\right) \text{Li}_2\left(i e^{-i \tan ^{-1}(a x)}\right)-1296 i \pi ^2 \text{Li}_2\left(-i e^{i \tan ^{-1}(a x)}\right)+5184 \pi  \text{Li}_3\left(i e^{-i \tan ^{-1}(a x)}\right)-5184 \pi  \text{Li}_3\left(-i e^{i \tan ^{-1}(a x)}\right)+10368 i \text{Li}_4\left(-i e^{-i \tan ^{-1}(a x)}\right)+10368 i \text{Li}_4\left(-i e^{i \tan ^{-1}(a x)}\right)-432 i \tan ^{-1}(a x)^4+864 i \pi  \tan ^{-1}(a x)^3-648 i \pi ^2 \tan ^{-1}(a x)^2+216 i \pi ^3 \tan ^{-1}(a x)-1728 \tan ^{-1}(a x)^3 \log \left(1+i e^{-i \tan ^{-1}(a x)}\right)+1728 \tan ^{-1}(a x)^3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+2592 \pi  \tan ^{-1}(a x)^2 \log \left(1-i e^{-i \tan ^{-1}(a x)}\right)-2592 \pi  \tan ^{-1}(a x)^2 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-1296 \pi ^2 \tan ^{-1}(a x) \log \left(1-i e^{-i \tan ^{-1}(a x)}\right)+1296 \pi ^2 \tan ^{-1}(a x) \log \left(1+i e^{i \tan ^{-1}(a x)}\right)+216 \pi ^3 \log \left(1+i e^{-i \tan ^{-1}(a x)}\right)-216 \pi ^3 \log \left(1+i e^{i \tan ^{-1}(a x)}\right)-216 \pi ^3 \log \left(\tan \left(\frac{1}{4} \left(2 \tan ^{-1}(a x)+\pi \right)\right)\right)-144 \tan ^{-1}(a x)^3 \sin \left(3 \tan ^{-1}(a x)\right)+96 \tan ^{-1}(a x) \sin \left(3 \tan ^{-1}(a x)\right)-144 \tan ^{-1}(a x)^2 \cos \left(3 \tan ^{-1}(a x)\right)+32 \cos \left(3 \tan ^{-1}(a x)\right)+189 i \pi ^4\right)}{1728 a^5 c^3 \sqrt{a^2 x^2+1}}","-\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{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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{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 \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{x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left(a^2 c x^2+c\right)^{3/2}}",1,"-1/1728*(Sqrt[c*(1 + a^2*x^2)]*((189*I)*Pi^4 - 12960/Sqrt[1 + a^2*x^2] + (216*I)*Pi^3*ArcTan[a*x] - (12960*a*x*ArcTan[a*x])/Sqrt[1 + a^2*x^2] - (648*I)*Pi^2*ArcTan[a*x]^2 + (6480*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] + (864*I)*Pi*ArcTan[a*x]^3 + (2160*a*x*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] - (432*I)*ArcTan[a*x]^4 + 32*Cos[3*ArcTan[a*x]] - 144*ArcTan[a*x]^2*Cos[3*ArcTan[a*x]] - 1296*Pi^2*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])] + 2592*Pi*ArcTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])] + 216*Pi^3*Log[1 + I/E^(I*ArcTan[a*x])] - 1728*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])] - 216*Pi^3*Log[1 + I*E^(I*ArcTan[a*x])] + 1296*Pi^2*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] - 2592*Pi*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] + 1728*ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])] - 216*Pi^3*Log[Tan[(Pi + 2*ArcTan[a*x])/4]] - (5184*I)*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])] - (1296*I)*Pi*(Pi - 4*ArcTan[a*x])*PolyLog[2, I/E^(I*ArcTan[a*x])] - (1296*I)*Pi^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (5184*I)*Pi*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (5184*I)*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - 10368*ArcTan[a*x]*PolyLog[3, (-I)/E^(I*ArcTan[a*x])] + 5184*Pi*PolyLog[3, I/E^(I*ArcTan[a*x])] - 5184*Pi*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 10368*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (10368*I)*PolyLog[4, (-I)/E^(I*ArcTan[a*x])] + (10368*I)*PolyLog[4, (-I)*E^(I*ArcTan[a*x])] + 96*ArcTan[a*x]*Sin[3*ArcTan[a*x]] - 144*ArcTan[a*x]^3*Sin[3*ArcTan[a*x]]))/(a^5*c^3*Sqrt[1 + a^2*x^2])","A",0
452,1,104,237,0.1421557,"\int \frac{x^3 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^3*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(-2 a x \left(61 a^2 x^2+60\right)-9 \left(3 a^2 x^2+2\right) \tan ^{-1}(a x)^3+9 a x \left(7 a^2 x^2+6\right) \tan ^{-1}(a x)^2+6 \left(21 a^2 x^2+20\right) \tan ^{-1}(a x)\right)}{27 a^4 c^3 \left(a^2 x^2+1\right)^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{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{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{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}}",1,"(Sqrt[c + a^2*c*x^2]*(-2*a*x*(60 + 61*a^2*x^2) + 6*(20 + 21*a^2*x^2)*ArcTan[a*x] + 9*a*x*(6 + 7*a^2*x^2)*ArcTan[a*x]^2 - 9*(2 + 3*a^2*x^2)*ArcTan[a*x]^3))/(27*a^4*c^3*(1 + a^2*x^2)^2)","A",1
453,1,95,199,0.1129149,"\int \frac{x^2 \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^2*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(9 a^3 x^3 \tan ^{-1}(a x)^3-42 a^2 x^2+9 \left(3 a^2 x^2+2\right) \tan ^{-1}(a x)^2-6 a x \left(7 a^2 x^2+6\right) \tan ^{-1}(a x)-40\right)}{27 a^3 c^3 \left(a^2 x^2+1\right)^2}","-\frac{4 x \tan ^{-1}(a x)}{3 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a 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{14}{9 a^3 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}}",1,"(Sqrt[c + a^2*c*x^2]*(-40 - 42*a^2*x^2 - 6*a*x*(6 + 7*a^2*x^2)*ArcTan[a*x] + 9*(2 + 3*a^2*x^2)*ArcTan[a*x]^2 + 9*a^3*x^3*ArcTan[a*x]^3))/(27*a^3*c^3*(1 + a^2*x^2)^2)","A",1
454,1,91,199,0.107852,"\int \frac{x \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(-2 a x \left(20 a^2 x^2+21\right)+9 a x \left(2 a^2 x^2+3\right) \tan ^{-1}(a x)^2+6 \left(6 a^2 x^2+7\right) \tan ^{-1}(a x)-9 \tan ^{-1}(a x)^3\right)}{27 c^3 \left(a^3 x^2+a\right)^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,"(Sqrt[c + a^2*c*x^2]*(-2*a*x*(21 + 20*a^2*x^2) + 6*(7 + 6*a^2*x^2)*ArcTan[a*x] + 9*a*x*(3 + 2*a^2*x^2)*ArcTan[a*x]^2 - 9*ArcTan[a*x]^3))/(27*c^3*(a + a^3*x^2)^2)","A",1
455,1,104,215,0.0981116,"\int \frac{\tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcTan[a*x]^3/(c + a^2*c*x^2)^(5/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(-2 \left(60 a^2 x^2+61\right)+9 a x \left(2 a^2 x^2+3\right) \tan ^{-1}(a x)^3+9 \left(6 a^2 x^2+7\right) \tan ^{-1}(a x)^2-6 a x \left(20 a^2 x^2+21\right) \tan ^{-1}(a x)\right)}{27 a c^3 \left(a^2 x^2+1\right)^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,"(Sqrt[c + a^2*c*x^2]*(-2*(61 + 60*a^2*x^2) - 6*a*x*(21 + 20*a^2*x^2)*ArcTan[a*x] + 9*(7 + 6*a^2*x^2)*ArcTan[a*x]^2 + 9*a*x*(3 + 2*a^2*x^2)*ArcTan[a*x]^3))/(27*a*c^3*(1 + a^2*x^2)^2)","A",1
456,1,347,553,0.8627866,"\int \frac{\tan ^{-1}(a x)^3}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcTan[a*x]^3/(x*(c + a^2*c*x^2)^(5/2)),x]","\frac{\left(a^2 x^2+1\right)^{3/2} \left(\frac{1620 a x}{\sqrt{a^2 x^2+1}}+\frac{270 \tan ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}-\frac{810 a x \tan ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}-\frac{1620 \tan ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+648 i \tan ^{-1}(a x)^2 \text{Li}_2\left(e^{-i \tan ^{-1}(a x)}\right)+648 i \tan ^{-1}(a x)^2 \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+1296 \tan ^{-1}(a x) \text{Li}_3\left(e^{-i \tan ^{-1}(a x)}\right)-1296 \tan ^{-1}(a x) \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)-1296 i \text{Li}_4\left(e^{-i \tan ^{-1}(a x)}\right)-1296 i \text{Li}_4\left(-e^{i \tan ^{-1}(a x)}\right)+54 i \tan ^{-1}(a x)^4+216 \tan ^{-1}(a x)^3 \log \left(1-e^{-i \tan ^{-1}(a x)}\right)-216 \tan ^{-1}(a x)^3 \log \left(1+e^{i \tan ^{-1}(a x)}\right)-18 \tan ^{-1}(a x)^2 \sin \left(3 \tan ^{-1}(a x)\right)+4 \sin \left(3 \tan ^{-1}(a x)\right)+18 \tan ^{-1}(a x)^3 \cos \left(3 \tan ^{-1}(a x)\right)-12 \tan ^{-1}(a x) \cos \left(3 \tan ^{-1}(a x)\right)-27 i \pi ^4\right)}{216 c \left(c \left(a^2 x^2+1\right)\right)^{3/2}}","\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{Li}_4\left(-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{Li}_4\left(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,"((1 + a^2*x^2)^(3/2)*((-27*I)*Pi^4 + (1620*a*x)/Sqrt[1 + a^2*x^2] - (1620*ArcTan[a*x])/Sqrt[1 + a^2*x^2] - (810*a*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] + (270*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + (54*I)*ArcTan[a*x]^4 - 12*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 18*ArcTan[a*x]^3*Cos[3*ArcTan[a*x]] + 216*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])] - 216*ArcTan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])] + (648*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x])] + (648*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])] + 1296*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])] - 1296*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] - (1296*I)*PolyLog[4, E^((-I)*ArcTan[a*x])] - (1296*I)*PolyLog[4, -E^(I*ArcTan[a*x])] + 4*Sin[3*ArcTan[a*x]] - 18*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]]))/(216*c*(c*(1 + a^2*x^2))^(3/2))","A",0
457,1,399,493,2.6531528,"\int \frac{\tan ^{-1}(a x)^3}{x^2 \left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcTan[a*x]^3/(x^2*(c + a^2*c*x^2)^(5/2)),x]","-\frac{a \left(-648 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-e^{i \tan ^{-1}(a x)}\right)+648 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(e^{i \tan ^{-1}(a x)}\right)+648 \sqrt{a^2 x^2+1} \text{Li}_3\left(-e^{i \tan ^{-1}(a x)}\right)-648 \sqrt{a^2 x^2+1} \text{Li}_3\left(e^{i \tan ^{-1}(a x)}\right)+54 \sqrt{a^2 x^2+1} \tan \left(\frac{1}{2} \tan ^{-1}(a x)\right) \tan ^{-1}(a x)^3-324 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \log \left(1-e^{i \tan ^{-1}(a x)}\right)+324 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \log \left(1+e^{i \tan ^{-1}(a x)}\right)+9 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^3 \sin \left(3 \tan ^{-1}(a x)\right)-6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \sin \left(3 \tan ^{-1}(a x)\right)+9 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \cos \left(3 \tan ^{-1}(a x)\right)-2 \sqrt{a^2 x^2+1} \cos \left(3 \tan ^{-1}(a x)\right)+189 a x \tan ^{-1}(a x)^3+567 \tan ^{-1}(a x)^2-1134 a x \tan ^{-1}(a x)+27 a x \tan ^{-1}(a x)^3 \csc ^2\left(\frac{1}{2} \tan ^{-1}(a x)\right)-1134\right)}{108 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{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Li}_2\left(-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{Li}_2\left(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{Li}_3\left(-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{Li}_3\left(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{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,"-1/108*(a*(-1134 - 1134*a*x*ArcTan[a*x] + 567*ArcTan[a*x]^2 + 189*a*x*ArcTan[a*x]^3 - 2*Sqrt[1 + a^2*x^2]*Cos[3*ArcTan[a*x]] + 9*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Cos[3*ArcTan[a*x]] + 27*a*x*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^2 - 324*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] + 324*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] - (648*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] + (648*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] + 648*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])] - 648*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])] - 6*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Sin[3*ArcTan[a*x]] + 9*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*Sin[3*ArcTan[a*x]] + 54*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*Tan[ArcTan[a*x]/2]))/(c^2*Sqrt[c + a^2*c*x^2])","A",0
458,0,0,25,2.1170276,"\int x^m \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3 \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^3, x]","A",-1
459,0,0,23,1.0544026,"\int x^m \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3 \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)*ArcTan[a*x]^3, x]","A",-1
460,0,0,25,0.9758773,"\int \frac{x^m \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2), x]","A",-1
461,0,0,25,0.7522952,"\int \frac{x^m \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2)^2, x]","A",-1
462,0,0,27,1.1383604,"\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3 \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3, x]","A",-1
463,0,0,27,0.1764308,"\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx","Integrate[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,"Integrate[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3, x]","A",-1
464,0,0,27,0.5723385,"\int \frac{x^m \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2], x]","A",-1
465,0,0,27,0.7043434,"\int \frac{x^m \tan ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(3/2), x]","A",-1
466,0,0,21,0.4585063,"\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2))/ArcTan[a*x], x]","A",-1
467,0,0,20,0.4132272,"\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)/ArcTan[a*x], x]","A",-1
468,0,0,23,0.6763822,"\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)/(x*ArcTan[a*x]), x]","A",-1
469,0,0,23,0.6109277,"\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x], x]","A",-1
470,0,0,22,0.4744799,"\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2/ArcTan[a*x], x]","A",-1
471,0,0,25,0.846594,"\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]), x]","A",-1
472,0,0,23,0.5766346,"\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x], x]","A",-1
473,0,0,22,0.4933817,"\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3/ArcTan[a*x], x]","A",-1
474,0,0,25,0.8203715,"\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]), x]","A",-1
475,0,0,25,1.6199488,"\int \frac{x^2}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^2/((c + a^2*c*x^2)*ArcTan[a*x]), x]","A",-1
476,0,0,23,0.5698374,"\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x/((c + a^2*c*x^2)*ArcTan[a*x]), x]","A",-1
477,1,12,12,0.0153838,"\int \frac{1}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","Integrate[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
478,0,0,25,0.2030522,"\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]), x]","A",-1
479,0,0,25,0.2200409,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[1/(x^2*(c + a^2*c*x^2)*ArcTan[a*x]), x]","A",-1
480,0,0,25,4.2494989,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^4/((c + a^2*c*x^2)^2*ArcTan[a*x]), x]","A",-1
481,0,0,25,2.8601242,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]), x]","A",-1
482,1,25,33,0.124147,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Integrate[x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]),x]","\frac{\log \left(\tan ^{-1}(a x)\right)-\text{Ci}\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{Ci}\left(2 \tan ^{-1}(a x)\right)}{2 a^3 c^2}",1,"(-CosIntegral[2*ArcTan[a*x]] + Log[ArcTan[a*x]])/(2*a^3*c^2)","A",1
483,1,17,17,0.0425444,"\int \frac{x}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Integrate[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",1
484,1,23,33,0.0224802,"\int \frac{1}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Integrate[1/((c + a^2*c*x^2)^2*ArcTan[a*x]),x]","\frac{\text{Ci}\left(2 \tan ^{-1}(a x)\right)+\log \left(\tan ^{-1}(a x)\right)}{2 a c^2}","\frac{\text{Ci}\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]] + Log[ArcTan[a*x]])/(2*a*c^2)","A",1
485,0,0,25,0.8077848,"\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]), x]","A",-1
486,0,0,25,1.2342356,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]), x]","A",-1
487,0,0,25,6.904226,"\int \frac{x^6}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^6/((c + a^2*c*x^2)^3*ArcTan[a*x]), x]","A",-1
488,0,0,25,8.9132107,"\int \frac{x^5}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^5/((c + a^2*c*x^2)^3*ArcTan[a*x]), x]","A",-1
489,1,34,50,0.1639333,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Integrate[x^4/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]","\frac{-4 \text{Ci}\left(2 \tan ^{-1}(a x)\right)+\text{Ci}\left(4 \tan ^{-1}(a x)\right)+3 \log \left(\tan ^{-1}(a x)\right)}{8 a^5 c^3}","-\frac{\text{Ci}\left(2 \tan ^{-1}(a x)\right)}{2 a^5 c^3}+\frac{\text{Ci}\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,"(-4*CosIntegral[2*ArcTan[a*x]] + CosIntegral[4*ArcTan[a*x]] + 3*Log[ArcTan[a*x]])/(8*a^5*c^3)","A",1
490,1,27,35,0.1298136,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Integrate[x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]","-\frac{\text{Si}\left(4 \tan ^{-1}(a x)\right)-2 \text{Si}\left(2 \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,"-1/8*(-2*SinIntegral[2*ArcTan[a*x]] + SinIntegral[4*ArcTan[a*x]])/(a^4*c^3)","A",1
491,1,25,33,0.0719026,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Integrate[x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]","\frac{\log \left(\tan ^{-1}(a x)\right)-\text{Ci}\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{Ci}\left(4 \tan ^{-1}(a x)\right)}{8 a^3 c^3}",1,"(-CosIntegral[4*ArcTan[a*x]] + Log[ArcTan[a*x]])/(8*a^3*c^3)","A",1
492,1,27,35,0.0917135,"\int \frac{x}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Integrate[x/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]","\frac{2 \text{Si}\left(2 \tan ^{-1}(a x)\right)+\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,"(2*SinIntegral[2*ArcTan[a*x]] + SinIntegral[4*ArcTan[a*x]])/(8*a^2*c^3)","A",1
493,1,34,50,0.0265182,"\int \frac{1}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Integrate[1/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]","\frac{4 \text{Ci}\left(2 \tan ^{-1}(a x)\right)+\text{Ci}\left(4 \tan ^{-1}(a x)\right)+3 \log \left(\tan ^{-1}(a x)\right)}{8 a c^3}","\frac{\text{Ci}\left(2 \tan ^{-1}(a x)\right)}{2 a c^3}+\frac{\text{Ci}\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,"(4*CosIntegral[2*ArcTan[a*x]] + CosIntegral[4*ArcTan[a*x]] + 3*Log[ArcTan[a*x]])/(8*a*c^3)","A",1
494,0,0,25,0.9880546,"\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]), x]","A",-1
495,0,0,25,1.4021885,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]), x]","A",-1
496,0,0,25,2.2549266,"\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x], x]","A",-1
497,0,0,24,0.2903439,"\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]/ArcTan[a*x], x]","A",-1
498,0,0,27,1.3581552,"\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]), x]","A",-1
499,0,0,25,2.3377787,"\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x], x]","A",-1
500,0,0,24,0.4172558,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x], x]","A",-1
501,0,0,27,1.4738603,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]), x]","A",-1
502,0,0,25,2.3977295,"\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x], x]","A",-1
503,0,0,24,0.4470011,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x], x]","A",-1
504,0,0,27,1.4645567,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]), x]","A",-1
505,0,0,25,0.8417286,"\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]","A",-1
506,0,0,24,0.2074954,"\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]","A",-1
507,0,0,27,0.6462331,"\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]","A",-1
508,0,0,27,4.8681019,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]","A",-1
509,-1,0,27,180.0017594,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","Integrate[x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]),x]","\text{\$Aborted}","\text{Int}\left(\frac{x^2}{\left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)},x\right)",0,"$Aborted","F",-1
510,1,37,39,0.1180452,"\int \frac{x}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","Integrate[x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]),x]","\frac{\left(a^2 x^2+1\right)^{3/2} \text{Si}\left(\tan ^{-1}(a x)\right)}{a^2 \left(c \left(a^2 x^2+1\right)\right)^{3/2}}","\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,"((1 + a^2*x^2)^(3/2)*SinIntegral[ArcTan[a*x]])/(a^2*(c*(1 + a^2*x^2))^(3/2))","A",1
511,1,37,39,0.0456451,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","Integrate[1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]),x]","\frac{\left(a^2 x^2+1\right)^{3/2} \text{Ci}\left(\tan ^{-1}(a x)\right)}{a \left(c \left(a^2 x^2+1\right)\right)^{3/2}}","\frac{\sqrt{a^2 x^2+1} \text{Ci}\left(\tan ^{-1}(a x)\right)}{a c \sqrt{a^2 c x^2+c}}",1,"((1 + a^2*x^2)^(3/2)*CosIntegral[ArcTan[a*x]])/(a*(c*(1 + a^2*x^2))^(3/2))","A",1
512,0,0,27,1.1894077,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]","A",-1
513,0,0,27,1.1122143,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]","A",-1
514,0,0,27,5.8998392,"\int \frac{x^5}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^5/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]","A",-1
515,0,0,27,101.3162567,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]","A",-1
516,1,52,87,0.1546985,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Integrate[x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]","\frac{\left(a^2 x^2+1\right)^{5/2} \left(3 \text{Si}\left(\tan ^{-1}(a x)\right)-\text{Si}\left(3 \tan ^{-1}(a x)\right)\right)}{4 a^4 \left(c \left(a^2 x^2+1\right)\right)^{5/2}}","\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,"((1 + a^2*x^2)^(5/2)*(3*SinIntegral[ArcTan[a*x]] - SinIntegral[3*ArcTan[a*x]]))/(4*a^4*(c*(1 + a^2*x^2))^(5/2))","A",1
517,1,53,87,0.1170991,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Integrate[x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(\text{Ci}\left(\tan ^{-1}(a x)\right)-\text{Ci}\left(3 \tan ^{-1}(a x)\right)\right)}{4 a^3 c^3 \sqrt{a^2 x^2+1}}","\frac{\sqrt{a^2 x^2+1} \text{Ci}\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{Ci}\left(3 \tan ^{-1}(a x)\right)}{4 a^3 c^2 \sqrt{a^2 c x^2+c}}",1,"(Sqrt[c*(1 + a^2*x^2)]*(CosIntegral[ArcTan[a*x]] - CosIntegral[3*ArcTan[a*x]]))/(4*a^3*c^3*Sqrt[1 + a^2*x^2])","A",1
518,1,51,87,0.1354921,"\int \frac{x}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Integrate[x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(\text{Si}\left(\tan ^{-1}(a x)\right)+\text{Si}\left(3 \tan ^{-1}(a x)\right)\right)}{4 a^2 c^3 \sqrt{a^2 x^2+1}}","\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[c*(1 + a^2*x^2)]*(SinIntegral[ArcTan[a*x]] + SinIntegral[3*ArcTan[a*x]]))/(4*a^2*c^3*Sqrt[1 + a^2*x^2])","A",1
519,1,50,87,0.0482417,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Integrate[1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]),x]","\frac{\left(a^2 x^2+1\right)^{5/2} \left(3 \text{Ci}\left(\tan ^{-1}(a x)\right)+\text{Ci}\left(3 \tan ^{-1}(a x)\right)\right)}{4 a \left(c \left(a^2 x^2+1\right)\right)^{5/2}}","\frac{3 \sqrt{a^2 x^2+1} \text{Ci}\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{Ci}\left(3 \tan ^{-1}(a x)\right)}{4 a c^2 \sqrt{a^2 c x^2+c}}",1,"((1 + a^2*x^2)^(5/2)*(3*CosIntegral[ArcTan[a*x]] + CosIntegral[3*ArcTan[a*x]]))/(4*a*(c*(1 + a^2*x^2))^(5/2))","A",1
520,0,0,27,1.3206027,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]","A",-1
521,0,0,27,1.39619,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]","A",-1
522,0,0,25,0.7507684,"\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x], x]","A",-1
523,0,0,25,0.8608931,"\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x], x]","A",-1
524,0,0,23,0.5635238,"\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2))/ArcTan[a*x], x]","A",-1
525,0,0,25,0.3714379,"\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)*ArcTan[a*x]), x]","A",-1
526,0,0,25,0.4769248,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]), x]","A",-1
527,0,0,25,0.4998625,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]), x]","A",-1
528,0,0,27,1.5069309,"\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x], x]","A",-1
529,0,0,27,0.8089194,"\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x], x]","A",-1
530,0,0,27,0.2113306,"\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x], x]","A",-1
531,0,0,27,0.5071497,"\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]), x]","A",-1
532,0,0,27,0.5822794,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]","A",-1
533,0,0,27,0.631619,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]), x]","A",-1
534,0,0,21,0.7825773,"\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2))/ArcTan[a*x]^2, x]","A",-1
535,0,0,20,0.582134,"\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)/ArcTan[a*x]^2, x]","A",-1
536,0,0,23,0.9345942,"\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)/(x*ArcTan[a*x]^2), x]","A",-1
537,0,0,23,0.9762452,"\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^2, x]","A",-1
538,0,0,22,1.185153,"\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2/ArcTan[a*x]^2, x]","A",-1
539,0,0,25,1.1850963,"\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^2), x]","A",-1
540,0,0,23,1.0022511,"\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^2, x]","A",-1
541,0,0,22,0.7792772,"\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3/ArcTan[a*x]^2, x]","A",-1
542,0,0,25,1.2214034,"\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^2), x]","A",-1
543,0,0,39,1.0807331,"\int \frac{x^3}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[x^3/((c + a^2*c*x^2)*ArcTan[a*x]^2), x]","A",-1
544,0,0,37,0.6000974,"\int \frac{x^2}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[x^2/((c + a^2*c*x^2)*ArcTan[a*x]^2), x]","A",-1
545,0,0,32,0.3011201,"\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[x/((c + a^2*c*x^2)*ArcTan[a*x]^2), x]","A",-1
546,1,14,14,0.0040969,"\int \frac{1}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Integrate[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
547,0,0,39,0.53018,"\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^2), x]","A",-1
548,0,0,39,0.7023395,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[1/(x^2*(c + a^2*c*x^2)*ArcTan[a*x]^2), x]","A",-1
549,0,0,39,1.0408326,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[1/(x^3*(c + a^2*c*x^2)*ArcTan[a*x]^2), x]","A",-1
550,0,0,39,1.2480238,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[1/(x^4*(c + a^2*c*x^2)*ArcTan[a*x]^2), x]","A",-1
551,0,0,72,7.0911759,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Integrate[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{Ci}\left(2 \tan ^{-1}(a x)\right)}{a^4 c^2}-\frac{x}{a^3 c^2 \tan ^{-1}(a x)}+\frac{x}{a^3 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)}",0,"Integrate[x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^2), x]","A",-1
552,1,40,43,0.1210479,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Integrate[x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]","\frac{\text{Si}\left(2 \tan ^{-1}(a x)\right)-\frac{a^2 x^2}{\left(a^2 x^2+1\right) \tan ^{-1}(a x)}}{a^3 c^2}","\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,"(-((a^2*x^2)/((1 + a^2*x^2)*ArcTan[a*x])) + SinIntegral[2*ArcTan[a*x]])/(a^3*c^2)","A",1
553,1,36,41,0.0792307,"\int \frac{x}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Integrate[x/((c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]","\frac{\text{Ci}\left(2 \tan ^{-1}(a x)\right)-\frac{a x}{\left(a^2 x^2+1\right) \tan ^{-1}(a x)}}{a^2 c^2}","\frac{\text{Ci}\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,"(-((a*x)/((1 + a^2*x^2)*ArcTan[a*x])) + CosIntegral[2*ArcTan[a*x]])/(a^2*c^2)","A",1
554,1,34,41,0.0717432,"\int \frac{1}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Integrate[1/((c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]","-\frac{\frac{1}{a^2 x^2 \tan ^{-1}(a x)+\tan ^{-1}(a x)}+\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,"-(((ArcTan[a*x] + a^2*x^2*ArcTan[a*x])^(-1) + SinIntegral[2*ArcTan[a*x]])/(a*c^2))","A",1
555,0,0,74,1.6698317,"\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Integrate[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{Ci}\left(2 \tan ^{-1}(a x)\right)}{c^2}-\frac{1}{a c^2 x \tan ^{-1}(a x)}",0,"Integrate[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^2), x]","A",-1
556,0,0,73,2.7938658,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^2), x]","A",-1
557,0,0,111,3.4324555,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Integrate[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{3 \text{Int}\left(\frac{1}{x^4 \tan ^{-1}(a x)},x\right)}{a c^2}+\frac{a \text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)},x\right)}{c^2}+\frac{a^2 \text{Ci}\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,"Integrate[1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^2), x]","A",-1
558,0,0,112,3.3093177,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Integrate[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{4 \text{Int}\left(\frac{1}{x^5 \tan ^{-1}(a x)},x\right)}{a c^2}+\frac{2 a \text{Int}\left(\frac{1}{x^3 \tan ^{-1}(a x)},x\right)}{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{1}{a c^2 x^4 \tan ^{-1}(a x)}+\frac{a}{c^2 x^2 \tan ^{-1}(a x)}",0,"Integrate[1/(x^4*(c + a^2*c*x^2)^2*ArcTan[a*x]^2), x]","A",-1
559,1,83,86,0.1513126,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Integrate[x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]^2),x]","\frac{-2 a^3 x^3+\left(a^2 x^2+1\right)^2 \tan ^{-1}(a x) \text{Ci}\left(2 \tan ^{-1}(a x)\right)-\left(a^2 x^2+1\right)^2 \tan ^{-1}(a x) \text{Ci}\left(4 \tan ^{-1}(a x)\right)}{2 a^4 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}","\frac{\text{Ci}\left(2 \tan ^{-1}(a x)\right)}{2 a^4 c^3}-\frac{\text{Ci}\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,"(-2*a^3*x^3 + (1 + a^2*x^2)^2*ArcTan[a*x]*CosIntegral[2*ArcTan[a*x]] - (1 + a^2*x^2)^2*ArcTan[a*x]*CosIntegral[4*ArcTan[a*x]])/(2*a^4*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])","A",1
560,1,59,67,0.2016798,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Integrate[x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]^2),x]","\frac{\left(a^2 x^2+1\right)^2 \tan ^{-1}(a x) \text{Si}\left(4 \tan ^{-1}(a x)\right)-2 a^2 x^2}{2 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,"(-2*a^2*x^2 + (1 + a^2*x^2)^2*ArcTan[a*x]*SinIntegral[4*ArcTan[a*x]])/(2*a^3*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])","A",1
561,1,75,61,0.0815949,"\int \frac{x}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Integrate[x/((c + a^2*c*x^2)^3*ArcTan[a*x]^2),x]","\frac{\left(a^2 x^2+1\right)^2 \tan ^{-1}(a x) \text{Ci}\left(2 \tan ^{-1}(a x)\right)+\left(a^2 x^2+1\right)^2 \tan ^{-1}(a x) \text{Ci}\left(4 \tan ^{-1}(a x)\right)-2 a x}{2 c^3 \left(a^3 x^2+a\right)^2 \tan ^{-1}(a x)}","\frac{\text{Ci}\left(2 \tan ^{-1}(a x)\right)}{2 a^2 c^3}+\frac{\text{Ci}\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,"(-2*a*x + (1 + a^2*x^2)^2*ArcTan[a*x]*CosIntegral[2*ArcTan[a*x]] + (1 + a^2*x^2)^2*ArcTan[a*x]*CosIntegral[4*ArcTan[a*x]])/(2*c^3*(a + a^3*x^2)^2*ArcTan[a*x])","A",1
562,1,45,58,0.1227512,"\int \frac{1}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Integrate[1/((c + a^2*c*x^2)^3*ArcTan[a*x]^2),x]","-\frac{\frac{1}{\left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)}+\text{Si}\left(2 \tan ^{-1}(a x)\right)+\frac{1}{2} \text{Si}\left(4 \tan ^{-1}(a x)\right)}{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/((1 + a^2*x^2)^2*ArcTan[a*x]) + SinIntegral[2*ArcTan[a*x]] + SinIntegral[4*ArcTan[a*x]]/2)/(a*c^3))","A",1
563,0,0,113,1.7752554,"\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Integrate[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{Ci}\left(2 \tan ^{-1}(a x)\right)}{2 c^3}-\frac{\text{Ci}\left(4 \tan ^{-1}(a x)\right)}{2 c^3}-\frac{1}{a c^3 x \tan ^{-1}(a x)}",0,"Integrate[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]^2), x]","A",-1
564,0,0,111,2.3388204,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^2), x]","A",-1
565,0,0,159,3.2243187,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Integrate[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{3 \text{Int}\left(\frac{1}{x^4 \tan ^{-1}(a x)},x\right)}{a c^3}+\frac{2 a \text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)},x\right)}{c^3}+\frac{5 a^2 \text{Ci}\left(2 \tan ^{-1}(a x)\right)}{2 c^3}+\frac{a^2 \text{Ci}\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,"Integrate[1/(x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^2), x]","A",-1
566,0,0,155,3.9552722,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Integrate[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 \text{Int}\left(\frac{1}{x^5 \tan ^{-1}(a x)},x\right)}{a c^3}+\frac{4 a \text{Int}\left(\frac{1}{x^3 \tan ^{-1}(a x)},x\right)}{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{1}{a c^3 x^4 \tan ^{-1}(a x)}+\frac{2 a}{c^3 x^2 \tan ^{-1}(a x)}",0,"Integrate[1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^2), x]","A",-1
567,0,0,25,1.6780104,"\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^2, x]","A",-1
568,0,0,24,0.4739733,"\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^2, x]","A",-1
569,0,0,27,3.6433848,"\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^2), x]","A",-1
570,0,0,25,4.2026671,"\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^2, x]","A",-1
571,0,0,24,0.9897556,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^2, x]","A",-1
572,0,0,27,4.4046714,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^2), x]","A",-1
573,0,0,25,2.1131745,"\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^2, x]","A",-1
574,0,0,24,0.8812799,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^2, x]","A",-1
575,0,0,27,2.3960117,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^2), x]","A",-1
576,0,0,25,1.238889,"\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]","A",-1
577,0,0,24,0.6477574,"\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]","A",-1
578,0,0,64,1.3143875,"\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]","A",-1
579,0,0,101,9.9596547,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Integrate[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{Ci}\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,"Integrate[x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x]","A",-1
580,0,0,98,8.9136764,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x]","A",-1
581,1,55,69,0.0995801,"\int \frac{x}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Integrate[x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2),x]","\frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Ci}\left(\tan ^{-1}(a x)\right)-a x}{a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}","\frac{\sqrt{a^2 x^2+1} \text{Ci}\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,"(-(a*x) + Sqrt[1 + a^2*x^2]*ArcTan[a*x]*CosIntegral[ArcTan[a*x]])/(a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])","A",1
582,1,53,69,0.1096074,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Integrate[1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2),x]","-\frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{Si}\left(\tan ^{-1}(a x)\right)+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 + Sqrt[1 + a^2*x^2]*ArcTan[a*x]*SinIntegral[ArcTan[a*x]])/(a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]))","A",1
583,0,0,130,2.0527674,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Integrate[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{Ci}\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,"Integrate[1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x]","A",-1
584,0,0,94,3.7014663,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x]","A",-1
585,0,0,160,6.2481958,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Integrate[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{Ci}\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,"Integrate[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x]","A",-1
586,0,0,135,7.0071234,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[1/(x^4*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x]","A",-1
587,0,0,177,12.6091512,"\int \frac{x^5}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Integrate[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{Ci}\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{Ci}\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,"Integrate[x^5/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x]","A",-1
588,0,0,174,11.6718376,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x]","A",-1
589,1,82,118,0.225838,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Integrate[x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","\frac{3 c \sqrt{a^2 x^2+1} \left(\text{Ci}\left(\tan ^{-1}(a x)\right)-\text{Ci}\left(3 \tan ^{-1}(a x)\right)\right)-\frac{4 a^3 c x^3}{\left(a^2 x^2+1\right) \tan ^{-1}(a x)}}{4 a^4 c^3 \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{Ci}\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{Ci}\left(3 \tan ^{-1}(a x)\right)}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}",1,"((-4*a^3*c*x^3)/((1 + a^2*x^2)*ArcTan[a*x]) + 3*c*Sqrt[1 + a^2*x^2]*(CosIntegral[ArcTan[a*x]] - CosIntegral[3*ArcTan[a*x]]))/(4*a^4*c^3*Sqrt[c + a^2*c*x^2])","A",1
590,1,99,142,0.2904501,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Integrate[x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","-\frac{\left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x) \text{Si}\left(\tan ^{-1}(a x)\right)-3 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x) \text{Si}\left(3 \tan ^{-1}(a x)\right)+4 a^2 x^2}{4 a^3 c^2 \left(a^2 x^2+1\right) \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)}{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,"-1/4*(4*a^2*x^2 + (1 + a^2*x^2)^(3/2)*ArcTan[a*x]*SinIntegral[ArcTan[a*x]] - 3*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]*SinIntegral[3*ArcTan[a*x]])/(a^3*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])","A",1
591,1,95,116,0.200105,"\int \frac{x}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Integrate[x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","\frac{\left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x) \text{Ci}\left(\tan ^{-1}(a x)\right)+3 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x) \text{Ci}\left(3 \tan ^{-1}(a x)\right)-4 a x}{4 a^2 c^2 \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}","\frac{\sqrt{a^2 x^2+1} \text{Ci}\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{Ci}\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,"(-4*a*x + (1 + a^2*x^2)^(3/2)*ArcTan[a*x]*CosIntegral[ArcTan[a*x]] + 3*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]*CosIntegral[3*ArcTan[a*x]])/(4*a^2*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])","A",1
592,1,84,115,0.1166247,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Integrate[1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2),x]","-\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(\frac{4}{\left(a^2 x^2+1\right)^{3/2}}+3 \tan ^{-1}(a x) \text{Si}\left(\tan ^{-1}(a x)\right)+3 \tan ^{-1}(a x) \text{Si}\left(3 \tan ^{-1}(a x)\right)\right)}{4 a c^3 \sqrt{a^2 x^2+1} \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,"-1/4*(Sqrt[c*(1 + a^2*x^2)]*(4/(1 + a^2*x^2)^(3/2) + 3*ArcTan[a*x]*SinIntegral[ArcTan[a*x]] + 3*ArcTan[a*x]*SinIntegral[3*ArcTan[a*x]]))/(a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x])","A",1
593,0,0,199,2.4721577,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Integrate[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{\sqrt{a^2 c x^2+c}}{a c^3 x \tan ^{-1}(a x)}-\frac{5 \sqrt{a^2 x^2+1} \text{Ci}\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{Ci}\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{a x}{c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",0,"Integrate[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x]","A",-1
594,0,0,164,4.4003019,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x]","A",-1
595,0,0,238,7.9512053,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Integrate[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{2 a \sqrt{a^2 c x^2+c}}{c^3 x \tan ^{-1}(a x)}+\frac{9 a^2 \sqrt{a^2 x^2+1} \text{Ci}\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{Ci}\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{a^3 x}{c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",0,"Integrate[1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x]","A",-1
596,0,0,209,7.4844417,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x]","A",-1
597,0,0,33,31.5742565,"\int \frac{\sqrt{f x}}{\left(d+c^2 d x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2} \, dx","Integrate[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,"Integrate[Sqrt[f*x]/((d + c^2*d*x^2)^2*(a + b*ArcTan[c*x])^2), x]","A",-1
598,0,0,25,0.8356309,"\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^2, x]","A",-1
599,0,0,25,0.9098897,"\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^2, x]","A",-1
600,0,0,23,0.5957856,"\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^2, x]","A",-1
601,0,0,41,0.5465902,"\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)*ArcTan[a*x]^2), x]","A",-1
602,0,0,25,0.6803724,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^2), x]","A",-1
603,0,0,25,0.7039383,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^2), x]","A",-1
604,0,0,27,1.6779432,"\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^2, x]","A",-1
605,0,0,27,1.3366753,"\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^2, x]","A",-1
606,0,0,27,0.371948,"\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^2} \, dx","Integrate[(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,"Integrate[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^2, x]","A",-1
607,0,0,27,0.6933599,"\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]","A",-1
608,0,0,27,0.7624586,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2), x]","A",-1
609,0,0,27,0.8266736,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^2} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2), x]","A",-1
610,0,0,21,1.1817262,"\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2))/ArcTan[a*x]^3, x]","A",-1
611,0,0,20,1.2649393,"\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)/ArcTan[a*x]^3, x]","A",-1
612,0,0,23,1.6215817,"\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)/(x*ArcTan[a*x]^3), x]","A",-1
613,0,0,23,1.0561173,"\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^3, x]","A",-1
614,0,0,22,0.7519342,"\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2/ArcTan[a*x]^3, x]","A",-1
615,0,0,25,1.340959,"\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^3), x]","A",-1
616,0,0,23,1.1025316,"\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^3, x]","A",-1
617,0,0,22,1.2506895,"\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3/ArcTan[a*x]^3, x]","A",-1
618,0,0,25,1.3182191,"\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^3), x]","A",-1
619,0,0,43,0.9311181,"\int \frac{x^3}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[x^3/((c + a^2*c*x^2)*ArcTan[a*x]^3), x]","A",-1
620,0,0,38,0.6243903,"\int \frac{x^2}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[x^2/((c + a^2*c*x^2)*ArcTan[a*x]^3), x]","A",-1
621,0,0,37,0.4804676,"\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[x/((c + a^2*c*x^2)*ArcTan[a*x]^3), x]","A",-1
622,1,16,16,0.0038257,"\int \frac{1}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Integrate[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*1/(a*c*ArcTan[a*x]^2)","A",1
623,0,0,43,0.4790904,"\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^3), x]","A",-1
624,0,0,41,1.141289,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[1/(x^2*(c + a^2*c*x^2)*ArcTan[a*x]^3), x]","A",-1
625,0,0,43,1.3603149,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[1/(x^3*(c + a^2*c*x^2)*ArcTan[a*x]^3), x]","A",-1
626,0,0,41,2.5322116,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[1/(x^4*(c + a^2*c*x^2)*ArcTan[a*x]^3), x]","A",-1
627,0,0,116,10.4764789,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Integrate[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 \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 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}",0,"Integrate[x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^3), x]","A",-1
628,1,51,71,0.118171,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Integrate[x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^3),x]","\frac{2 \text{Ci}\left(2 \tan ^{-1}(a x)\right)-\frac{a x \left(a x+2 \tan ^{-1}(a x)\right)}{\left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}}{2 a^3 c^2}","\frac{\text{Ci}\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,"(-((a*x*(a*x + 2*ArcTan[a*x]))/((1 + a^2*x^2)*ArcTan[a*x]^2)) + 2*CosIntegral[2*ArcTan[a*x]])/(2*a^3*c^2)","A",1
629,1,70,81,0.0595205,"\int \frac{x}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Integrate[x/((c + a^2*c*x^2)^2*ArcTan[a*x]^3),x]","\frac{-2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2 \text{Si}\left(2 \tan ^{-1}(a x)\right)+\left(a^2 x^2-1\right) \tan ^{-1}(a x)-a x}{2 a^2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}","-\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,"(-(a*x) + (-1 + a^2*x^2)*ArcTan[a*x] - 2*(1 + a^2*x^2)*ArcTan[a*x]^2*SinIntegral[2*ArcTan[a*x]])/(2*a^2*c^2*(1 + a^2*x^2)*ArcTan[a*x]^2)","A",1
630,1,58,65,0.0718731,"\int \frac{1}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Integrate[1/((c + a^2*c*x^2)^2*ArcTan[a*x]^3),x]","\frac{-2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2 \text{Ci}\left(2 \tan ^{-1}(a x)\right)+2 a x \tan ^{-1}(a x)-1}{2 c^2 \left(a^3 x^2+a\right) \tan ^{-1}(a x)^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{Ci}\left(2 \tan ^{-1}(a x)\right)}{a c^2}",1,"(-1 + 2*a*x*ArcTan[a*x] - 2*(1 + a^2*x^2)*ArcTan[a*x]^2*CosIntegral[2*ArcTan[a*x]])/(2*c^2*(a + a^3*x^2)*ArcTan[a*x]^2)","A",1
631,0,0,114,1.9372786,"\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^3), x]","A",-1
632,0,0,104,2.8175201,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Integrate[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{Ci}\left(2 \tan ^{-1}(a x)\right)}{c^2}-\frac{1}{2 a c^2 x^2 \tan ^{-1}(a x)^2}",0,"Integrate[1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^3), x]","A",-1
633,0,0,161,2.303364,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Integrate[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{3 \text{Int}\left(\frac{1}{x^4 \tan ^{-1}(a x)^2},x\right)}{2 a c^2}+\frac{a \text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)^2},x\right)}{2 c^2}-\frac{a^2 \text{Si}\left(2 \tan ^{-1}(a x)\right)}{c^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{a^3 x}{2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2}-\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,"Integrate[1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^3), x]","A",-1
634,0,0,143,7.1199083,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Integrate[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{2 \text{Int}\left(\frac{1}{x^5 \tan ^{-1}(a x)^2},x\right)}{a c^2}+\frac{a \text{Int}\left(\frac{1}{x^3 \tan ^{-1}(a x)^2},x\right)}{c^2}-\frac{a^3 \text{Ci}\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{1}{2 a c^2 x^4 \tan ^{-1}(a x)^2}+\frac{a}{2 c^2 x^2 \tan ^{-1}(a x)^2}",0,"Integrate[1/(x^4*(c + a^2*c*x^2)^2*ArcTan[a*x]^3), x]","A",-1
635,1,72,177,0.2553401,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Integrate[x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]","\frac{\frac{a^2 x^2 \left(\left(a^2 x^2-3\right) \tan ^{-1}(a x)-a x\right)}{\left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}-\text{Si}\left(2 \tan ^{-1}(a x)\right)+2 \text{Si}\left(4 \tan ^{-1}(a x)\right)}{2 a^4 c^3}","-\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{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{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}",1,"((a^2*x^2*(-(a*x) + (-3 + a^2*x^2)*ArcTan[a*x]))/((1 + a^2*x^2)^2*ArcTan[a*x]^2) - SinIntegral[2*ArcTan[a*x]] + 2*SinIntegral[4*ArcTan[a*x]])/(2*a^4*c^3)","A",1
636,1,60,120,0.1500895,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Integrate[x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]","\frac{\frac{a x \left(2 \left(a^2 x^2-1\right) \tan ^{-1}(a x)-a x\right)}{\left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}+2 \text{Ci}\left(4 \tan ^{-1}(a x)\right)}{2 a^3 c^3}","\frac{\text{Ci}\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,"((a*x*(-(a*x) + 2*(-1 + a^2*x^2)*ArcTan[a*x]))/((1 + a^2*x^2)^2*ArcTan[a*x]^2) + 2*CosIntegral[4*ArcTan[a*x]])/(2*a^3*c^3)","A",1
637,1,98,113,0.1864522,"\int \frac{x}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Integrate[x/((c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]","-\frac{\left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2 \text{Si}\left(2 \tan ^{-1}(a x)\right)+2 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2 \text{Si}\left(4 \tan ^{-1}(a x)\right)-3 a^2 x^2 \tan ^{-1}(a x)+a x+\tan ^{-1}(a x)}{2 a^2 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}","-\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,"-1/2*(a*x + ArcTan[a*x] - 3*a^2*x^2*ArcTan[a*x] + (1 + a^2*x^2)^2*ArcTan[a*x]^2*SinIntegral[2*ArcTan[a*x]] + 2*(1 + a^2*x^2)^2*ArcTan[a*x]^2*SinIntegral[4*ArcTan[a*x]])/(a^2*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2)","A",1
638,1,89,81,0.1063781,"\int \frac{1}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Integrate[1/((c + a^2*c*x^2)^3*ArcTan[a*x]^3),x]","-\frac{2 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2 \text{Ci}\left(2 \tan ^{-1}(a x)\right)+2 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2 \text{Ci}\left(4 \tan ^{-1}(a x)\right)-4 a x \tan ^{-1}(a x)+1}{2 a c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2}","\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{Ci}\left(2 \tan ^{-1}(a x)\right)}{a c^3}-\frac{\text{Ci}\left(4 \tan ^{-1}(a x)\right)}{a c^3}",1,"-1/2*(1 - 4*a*x*ArcTan[a*x] + 2*(1 + a^2*x^2)^2*ArcTan[a*x]^2*CosIntegral[2*ArcTan[a*x]] + 2*(1 + a^2*x^2)^2*ArcTan[a*x]^2*CosIntegral[4*ArcTan[a*x]])/(a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2)","A",1
639,0,0,200,2.8333127,"\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]^3), x]","A",-1
640,0,0,168,4.0331069,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Integrate[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{Ci}\left(2 \tan ^{-1}(a x)\right)}{c^3}+\frac{a \text{Ci}\left(4 \tan ^{-1}(a x)\right)}{c^3}-\frac{1}{2 a c^3 x^2 \tan ^{-1}(a x)^2}",0,"Integrate[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^3), x]","A",-1
641,0,0,248,5.2868091,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Integrate[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{3 \text{Int}\left(\frac{1}{x^4 \tan ^{-1}(a x)^2},x\right)}{2 a c^3}+\frac{a \text{Int}\left(\frac{1}{x^2 \tan ^{-1}(a x)^2},x\right)}{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^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{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{1}{2 a c^3 x^3 \tan ^{-1}(a x)^2}+\frac{a}{c^3 x \tan ^{-1}(a x)^2}",0,"Integrate[1/(x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^3), x]","A",-1
642,0,0,208,9.9476194,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Integrate[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 \text{Int}\left(\frac{1}{x^5 \tan ^{-1}(a x)^2},x\right)}{a c^3}+\frac{2 a \text{Int}\left(\frac{1}{x^3 \tan ^{-1}(a x)^2},x\right)}{c^3}-\frac{3 a^3 \text{Ci}\left(2 \tan ^{-1}(a x)\right)}{c^3}-\frac{a^3 \text{Ci}\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{1}{2 a c^3 x^4 \tan ^{-1}(a x)^2}+\frac{a}{c^3 x^2 \tan ^{-1}(a x)^2}",0,"Integrate[1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^3), x]","A",-1
643,1,16,16,0.1547446,"\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","Integrate[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,"-1/2*x^3/(a*ArcTan[a*x]^2)","A",1
644,0,0,25,1.9351407,"\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^3, x]","A",-1
645,0,0,24,1.1171287,"\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^3, x]","A",-1
646,0,0,27,3.6087722,"\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^3), x]","A",-1
647,0,0,25,6.6053175,"\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^3, x]","A",-1
648,0,0,24,1.1435049,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^3, x]","A",-1
649,0,0,27,4.4571996,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^3), x]","A",-1
650,0,0,25,2.3071049,"\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^3, x]","A",-1
651,0,0,24,1.2782857,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^3, x]","A",-1
652,0,0,27,2.7792768,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^3), x]","A",-1
653,0,0,25,1.3921431,"\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]","A",-1
654,0,0,24,0.7397358,"\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]","A",-1
655,0,0,68,7.5176033,"\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]","A",-1
656,0,0,27,5.4493738,"\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]","A",-1
657,0,0,27,6.8472343,"\int \frac{1}{x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[1/(x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]","A",-1
658,0,0,136,2.4647358,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Integrate[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{1}{2 a^4 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{x}{2 a^3 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}",0,"Integrate[x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x]","A",-1
659,0,0,135,2.2279173,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Integrate[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{x}{2 a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{\sqrt{a^2 x^2+1} \text{Ci}\left(\tan ^{-1}(a x)\right)}{2 a^3 c \sqrt{a^2 c x^2+c}}+\frac{1}{2 a^3 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}",0,"Integrate[x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x]","A",-1
660,1,63,104,0.13267,"\int \frac{x}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Integrate[x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3),x]","-\frac{\sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{Si}\left(\tan ^{-1}(a x)\right)+a x+\tan ^{-1}(a x)}{2 a^2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}","-\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,"-1/2*(a*x + ArcTan[a*x] + Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*SinIntegral[ArcTan[a*x]])/(a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)","A",1
661,1,65,101,0.0817013,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Integrate[1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3),x]","\frac{-\sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{Ci}\left(\tan ^{-1}(a x)\right)+a x \tan ^{-1}(a x)-1}{2 a c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}","-\frac{\sqrt{a^2 x^2+1} \text{Ci}\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 + a*x*ArcTan[a*x] - Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*CosIntegral[ArcTan[a*x]])/(2*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)","A",1
662,0,0,166,2.6113545,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x]","A",-1
663,0,0,131,2.6021789,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Integrate[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{Ci}\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,"Integrate[1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x]","A",-1
664,0,0,201,3.3287668,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Integrate[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^2}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{a^3 x}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}",0,"Integrate[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x]","A",-1
665,0,0,170,8.1307816,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Integrate[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^4 x}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{a^3 \sqrt{a^2 x^2+1} \text{Ci}\left(\tan ^{-1}(a x)\right)}{2 c \sqrt{a^2 c x^2+c}}-\frac{a^3}{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}",0,"Integrate[1/(x^4*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x]","A",-1
666,0,0,241,8.1533251,"\int \frac{x^5}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Integrate[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{2}{a^6 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{3}{2 a^6 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}+\frac{x}{2 a^5 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}+\frac{x^3}{2 a^3 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}",0,"Integrate[x^5/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x]","A",-1
667,0,0,235,7.6733091,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Integrate[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{Ci}\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{Ci}\left(3 \tan ^{-1}(a x)\right)}{8 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{1}{a^5 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}-\frac{1}{2 a^5 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}-\frac{x}{a^4 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}+\frac{3 x}{2 a^4 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",0,"Integrate[x^4/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x]","A",-1
668,1,114,180,0.3299089,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Integrate[x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]","\frac{-3 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^2 \text{Si}\left(\tan ^{-1}(a x)\right)+9 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^2 \text{Si}\left(3 \tan ^{-1}(a x)\right)-4 a^2 x^2 \left(a x+3 \tan ^{-1}(a x)\right)}{8 a^4 c^2 \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}","-\frac{x^3}{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{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{3}{2 a^4 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)}",1,"(-4*a^2*x^2*(a*x + 3*ArcTan[a*x]) - 3*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*SinIntegral[ArcTan[a*x]] + 9*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*SinIntegral[3*ArcTan[a*x]])/(8*a^4*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)","A",1
669,1,119,209,0.2386961,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Integrate[x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]","\frac{-\left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^2 \text{Ci}\left(\tan ^{-1}(a x)\right)+9 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^2 \text{Ci}\left(3 \tan ^{-1}(a x)\right)+4 a x \left(\left(a^2 x^2-2\right) \tan ^{-1}(a x)-a x\right)}{8 a^3 c^2 \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}","\frac{x}{2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{3 x}{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{Ci}\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{Ci}\left(3 \tan ^{-1}(a x)\right)}{8 a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{1}{2 a^3 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}+\frac{1}{2 a^3 c \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^2}",1,"(4*a*x*(-(a*x) + (-2 + a^2*x^2)*ArcTan[a*x]) - (1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*CosIntegral[ArcTan[a*x]] + 9*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*CosIntegral[3*ArcTan[a*x]])/(8*a^3*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)","A",1
670,1,118,175,0.2951334,"\int \frac{x}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Integrate[x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]","\frac{-\left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^2 \text{Si}\left(\tan ^{-1}(a x)\right)-9 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^2 \text{Si}\left(3 \tan ^{-1}(a x)\right)+8 a^2 x^2 \tan ^{-1}(a x)-4 a x-4 \tan ^{-1}(a x)}{8 a^2 c^2 \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}","-\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,"(-4*a*x - 4*ArcTan[a*x] + 8*a^2*x^2*ArcTan[a*x] - (1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*SinIntegral[ArcTan[a*x]] - 9*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*SinIntegral[3*ArcTan[a*x]])/(8*a^2*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)","A",1
671,1,102,145,0.2205997,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Integrate[1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]","\frac{-3 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^2 \text{Ci}\left(\tan ^{-1}(a x)\right)-9 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^2 \text{Ci}\left(3 \tan ^{-1}(a x)\right)+12 a x \tan ^{-1}(a x)-4}{8 c^2 \left(a^3 x^2+a\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}","-\frac{3 \sqrt{a^2 x^2+1} \text{Ci}\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{Ci}\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,"(-4 + 12*a*x*ArcTan[a*x] - 3*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*CosIntegral[ArcTan[a*x]] - 9*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*CosIntegral[3*ArcTan[a*x]])/(8*c^2*(a + a^3*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)","A",1
672,0,0,263,3.8001358,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Integrate[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{\sqrt{a^2 c x^2+c}}{2 a c^3 x \tan ^{-1}(a x)^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{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,"Integrate[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x]","A",-1
673,0,0,232,5.5687267,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Integrate[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{Ci}\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{Ci}\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,"Integrate[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x]","A",-1
674,0,0,25,0.8642379,"\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^3, x]","A",-1
675,0,0,25,0.9582404,"\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^3, x]","A",-1
676,0,0,23,0.6293273,"\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^3, x]","A",-1
677,0,0,46,0.8827758,"\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)*ArcTan[a*x]^3), x]","A",-1
678,0,0,25,1.003381,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^3), x]","A",-1
679,0,0,25,1.0368252,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^3), x]","A",-1
680,0,0,27,1.6668276,"\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^3, x]","A",-1
681,0,0,27,1.3691943,"\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^3, x]","A",-1
682,0,0,27,0.7285624,"\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^3} \, dx","Integrate[(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,"Integrate[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^3, x]","A",-1
683,0,0,27,1.038795,"\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3), x]","A",-1
684,0,0,27,1.1036801,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3), x]","A",-1
685,0,0,27,1.1625206,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^3} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x]","A",-1
686,0,0,25,2.1803843,"\int x^m \left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x]","A",-1
687,0,0,57,2.486812,"\int x \left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x*(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x]","A",-1
688,0,0,22,3.4293413,"\int \left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]], x]","A",-1
689,0,0,25,1.7315798,"\int \frac{\left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)}}{x} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]])/x, x]","A",-1
690,0,0,27,1.3447831,"\int x^m \left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]], x]","A",-1
691,0,0,61,1.9674495,"\int x \left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]], x]","A",-1
692,0,0,24,1.845775,"\int \left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]], x]","A",-1
693,0,0,27,1.3786004,"\int \frac{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}}{x} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]])/x, x]","A",-1
694,0,0,27,0.893173,"\int x^m \left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]], x]","A",-1
695,0,0,61,1.9551815,"\int x \left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]], x]","A",-1
696,0,0,24,1.8847411,"\int \left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]], x]","A",-1
697,0,0,27,1.4095781,"\int \frac{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}}{x} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]])/x, x]","A",-1
698,0,0,27,0.8272082,"\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Integrate[(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,"Integrate[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2), x]","A",-1
699,0,0,61,2.8677948,"\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Integrate[(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{2 \text{Int}\left(\tan ^{-1}(a x)^{3/2},x\right)}{3 a^3 c}+\frac{\text{Int}\left(x \sqrt{\tan ^{-1}(a x)},x\right)}{a^2 c}-\frac{2 x \tan ^{-1}(a x)^{3/2}}{3 a^3 c}",0,"Integrate[(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2), x]","A",-1
700,0,0,37,1.2300047,"\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Integrate[(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,"Integrate[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2), x]","A",-1
701,0,0,41,1.0403086,"\int \frac{x \sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Integrate[(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,"Integrate[(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2), x]","A",-1
702,1,18,18,0.0030965,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx","Integrate[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
703,0,0,49,0.6300217,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)} \, dx","Integrate[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,"Integrate[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)), x]","A",-1
704,0,0,36,1.5992114,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \left(c+a^2 c x^2\right)} \, dx","Integrate[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,"Integrate[Sqrt[ArcTan[a*x]]/(x^2*(c + a^2*c*x^2)), x]","A",-1
705,0,0,74,2.3096448,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^3 \left(c+a^2 c x^2\right)} \, dx","Integrate[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,"Integrate[Sqrt[ArcTan[a*x]]/(x^3*(c + a^2*c*x^2)), x]","A",-1
706,0,0,61,6.5064055,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4 \left(c+a^2 c x^2\right)} \, dx","Integrate[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,"Integrate[Sqrt[ArcTan[a*x]]/(x^4*(c + a^2*c*x^2)), x]","A",-1
707,0,0,27,1.756216,"\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(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,"Integrate[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2, x]","A",-1
708,0,0,27,3.9129678,"\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(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,"Integrate[(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2, x]","A",-1
709,1,66,80,0.2104319,"\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2,x]","\frac{4 \sqrt{\tan ^{-1}(a x)} \left(2 \tan ^{-1}(a x)-\frac{3 a x}{a^2 x^2+1}\right)+3 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{24 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{\tan ^{-1}(a x)^{3/2}}{3 a^3 c^2}-\frac{x \sqrt{\tan ^{-1}(a x)}}{2 a^2 c^2 \left(a^2 x^2+1\right)}",1,"(4*Sqrt[ArcTan[a*x]]*((-3*a*x)/(1 + a^2*x^2) + 2*ArcTan[a*x]) + 3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(24*a^3*c^2)","A",1
710,1,136,79,0.277605,"\int \frac{x \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^2,x]","\frac{4 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)+\frac{\frac{16 \left(a^2 x^2-1\right) \tan ^{-1}(a x)}{a^2 x^2+1}-i \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+i \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)}{\sqrt{\tan ^{-1}(a x)}}}{64 a^2 c^2}","\frac{\sqrt{\pi } C\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,"(4*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + ((16*(-1 + a^2*x^2)*ArcTan[a*x])/(1 + a^2*x^2) - I*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + I*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]])/Sqrt[ArcTan[a*x]])/(64*a^2*c^2)","C",0
711,1,89,77,0.1720673,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[Sqrt[ArcTan[a*x]]/(c + a^2*c*x^2)^2,x]","\frac{16 \tan ^{-1}(a x)^2+3 \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{3}{2},-2 i \tan ^{-1}(a x)\right)+3 \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{3}{2},2 i \tan ^{-1}(a x)\right)}{48 a c^2 \sqrt{\tan ^{-1}(a 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}",1,"(16*ArcTan[a*x]^2 + 3*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[3/2, (-2*I)*ArcTan[a*x]] + 3*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[3/2, (2*I)*ArcTan[a*x]])/(48*a*c^2*Sqrt[ArcTan[a*x]])","C",0
712,0,0,27,1.8430879,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)^2} \, dx","Integrate[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,"Integrate[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^2), x]","A",-1
713,0,0,27,2.1945121,"\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(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,"Integrate[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3, x]","A",-1
714,0,0,27,5.4548777,"\int \frac{x^5 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(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,"Integrate[(x^5*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3, x]","A",-1
715,1,181,139,0.534108,"\int \frac{x^4 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^4*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3,x]","\frac{-\frac{96 a x \tan ^{-1}(a x)}{\left(a^2 x^2+1\right)^2}-\frac{160 a^3 x^3 \tan ^{-1}(a x)}{\left(a^2 x^2+1\right)^2}+64 \tan ^{-1}(a x)^2-8 \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)-8 \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)+\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{256 a^5 c^3 \sqrt{\tan ^{-1}(a 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}",1,"((-96*a*x*ArcTan[a*x])/(1 + a^2*x^2)^2 - (160*a^3*x^3*ArcTan[a*x])/(1 + a^2*x^2)^2 + 64*ArcTan[a*x]^2 - 8*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] - 8*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] + Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(256*a^5*c^3*Sqrt[ArcTan[a*x]])","C",0
716,1,230,118,0.7068184,"\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3,x]","\frac{\frac{\frac{64 \left(5 a^4 x^4-6 a^2 x^2-3\right) \tan ^{-1}(a x)}{\left(a^2 x^2+1\right)^2}-12 i \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+12 i \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)+3 i \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)-3 i \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{\sqrt{\tan ^{-1}(a x)}}-10 \sqrt{2 \pi } C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)+80 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2048 a^4 c^3}","-\frac{\sqrt{\frac{\pi }{2}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{64 a^4 c^3}+\frac{\sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{16 a^4 c^3}-\frac{3 \sqrt{\tan ^{-1}(a x)}}{32 a^4 c^3}+\frac{x^4 \sqrt{\tan ^{-1}(a x)}}{4 c^3 \left(a^2 x^2+1\right)^2}",1,"(-10*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + 80*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + ((64*(-3 - 6*a^2*x^2 + 5*a^4*x^4)*ArcTan[a*x])/(1 + a^2*x^2)^2 - (12*I)*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + (12*I)*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] + (3*I)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] - (3*I)*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/Sqrt[ArcTan[a*x]])/(2048*a^4*c^3)","C",0
717,1,141,83,0.4234946,"\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3,x]","\frac{32 \tan ^{-1}(a x) \left(3 a x \left(a^2 x^2-1\right)+2 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)\right)-3 \left(a^2 x^2+1\right)^2 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)-3 \left(a^2 x^2+1\right)^2 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{768 a^3 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)}{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,"(32*ArcTan[a*x]*(3*a*x*(-1 + a^2*x^2) + 2*(1 + a^2*x^2)^2*ArcTan[a*x]) - 3*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] - 3*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(768*a^3*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])","C",0
718,1,230,118,0.689166,"\int \frac{x \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^3,x]","\frac{\frac{\frac{64 \left(3 a^4 x^4+6 a^2 x^2-5\right) \tan ^{-1}(a x)}{\left(a^2 x^2+1\right)^2}-20 i \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+20 i \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)-11 i \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+11 i \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{\sqrt{\tan ^{-1}(a x)}}-6 \sqrt{2 \pi } C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)+48 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2048 a^2 c^3}","\frac{\sqrt{\frac{\pi }{2}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{64 a^2 c^3}+\frac{\sqrt{\pi } C\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,"(-6*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + 48*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + ((64*(-5 + 6*a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x])/(1 + a^2*x^2)^2 - (20*I)*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + (20*I)*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] - (11*I)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + (11*I)*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/Sqrt[ArcTan[a*x]])/(2048*a^2*c^3)","C",0
719,1,103,139,0.4616341,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[Sqrt[ArcTan[a*x]]/(c + a^2*c*x^2)^3,x]","-\frac{-16 \sqrt{\tan ^{-1}(a x)} \left(\frac{a x \left(3 a^2 x^2+5\right)}{\left(a^2 x^2+1\right)^2}+2 \tan ^{-1}(a x)\right)+\sqrt{2 \pi } S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)+16 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{128 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,"-1/128*(-16*Sqrt[ArcTan[a*x]]*((a*x*(5 + 3*a^2*x^2))/(1 + a^2*x^2)^2 + 2*ArcTan[a*x]) + Sqrt[2*Pi]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + 16*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a*c^3)","A",0
720,0,0,27,2.4607846,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)^3} \, dx","Integrate[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,"Integrate[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^3), x]","A",-1
721,0,0,29,1.0436757,"\int x^m \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^m*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]","A",-1
722,0,0,29,3.7690914,"\int x^2 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]","A",-1
723,0,0,66,7.3776318,"\int x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]","A",-1
724,0,0,26,0.7313918,"\int \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]], x]","A",-1
725,0,0,29,1.0790455,"\int x^m \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x]","A",-1
726,0,0,29,4.0336447,"\int x^2 \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^2*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x]","A",-1
727,0,0,66,8.1157671,"\int x \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x]","A",-1
728,0,0,26,1.7656836,"\int \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]], x]","A",-1
729,0,0,29,1.4761643,"\int x^m \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x]","A",-1
730,0,0,29,3.5559749,"\int x^2 \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x]","A",-1
731,0,0,66,7.8369532,"\int x \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[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,"Integrate[x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x]","A",-1
732,0,0,26,0.5144533,"\int \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]], x]","A",-1
733,0,0,29,0.9399731,"\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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,"Integrate[(x^m*Sqrt[ArcTan[a*x]])/Sqrt[c + a^2*c*x^2], x]","A",-1
734,0,0,137,4.4996605,"\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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{x^2}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)}{6 a}+\frac{\text{Int}\left(\frac{1}{\sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)}{3 a^3}+\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,"Integrate[(x^3*Sqrt[ArcTan[a*x]])/Sqrt[c + a^2*c*x^2], x]","A",-1
735,0,0,101,2.7554465,"\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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,"Integrate[(x^2*Sqrt[ArcTan[a*x]])/Sqrt[c + a^2*c*x^2], x]","A",-1
736,0,0,63,0.8845263,"\int \frac{x \sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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,"Integrate[(x*Sqrt[ArcTan[a*x]])/Sqrt[c + a^2*c*x^2], x]","A",-1
737,0,0,26,0.2241459,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[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,"Integrate[Sqrt[ArcTan[a*x]]/Sqrt[c + a^2*c*x^2], x]","A",-1
738,0,0,29,1.2190148,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \sqrt{c+a^2 c x^2}} \, dx","Integrate[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,"Integrate[Sqrt[ArcTan[a*x]]/(x*Sqrt[c + a^2*c*x^2]), x]","A",-1
739,0,0,65,1.8971051,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \sqrt{c+a^2 c x^2}} \, dx","Integrate[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,"Integrate[Sqrt[ArcTan[a*x]]/(x^2*Sqrt[c + a^2*c*x^2]), x]","A",-1
740,0,0,103,3.7407496,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^3 \sqrt{c+a^2 c x^2}} \, dx","Integrate[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,"Integrate[Sqrt[ArcTan[a*x]]/(x^3*Sqrt[c + a^2*c*x^2]), x]","A",-1
741,0,0,138,21.0057592,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^4 \sqrt{c+a^2 c x^2}} \, dx","Integrate[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}{6} a \text{Int}\left(\frac{1}{x^3 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right)-\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{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,"Integrate[Sqrt[ArcTan[a*x]]/(x^4*Sqrt[c + a^2*c*x^2]), x]","A",-1
742,0,0,29,1.1169561,"\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2), x]","A",-1
743,0,0,29,27.0335377,"\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2), x]","A",-1
744,0,0,29,3.6741534,"\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2), x]","A",-1
745,1,121,93,0.1628707,"\int \frac{x \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2),x]","\frac{-i \sqrt{a^2 x^2+1} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+i \sqrt{a^2 x^2+1} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)-4 \tan ^{-1}(a x)}{4 a^2 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} C\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,"(-4*ArcTan[a*x] - I*Sqrt[1 + a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + I*Sqrt[1 + a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]])/(4*a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])","C",0
746,1,94,91,0.1516405,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[Sqrt[ArcTan[a*x]]/(c + a^2*c*x^2)^(3/2),x]","\frac{\left(a^2 x^2+1\right)^{3/2} \left(\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{3}{2},-i \tan ^{-1}(a x)\right)+\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{3}{2},i \tan ^{-1}(a x)\right)\right)}{2 a \left(c \left(a^2 x^2+1\right)\right)^{3/2} \sqrt{\tan ^{-1}(a 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}}",1,"((1 + a^2*x^2)^(3/2)*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[3/2, (-I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[3/2, I*ArcTan[a*x]]))/(2*a*(c*(1 + a^2*x^2))^(3/2)*Sqrt[ArcTan[a*x]])","C",0
747,0,0,29,2.0390659,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[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,"Integrate[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^(3/2)), x]","A",-1
748,0,0,29,6.0057345,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x^2 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[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,"Integrate[Sqrt[ArcTan[a*x]]/(x^2*(c + a^2*c*x^2)^(3/2)), x]","A",-1
749,0,0,29,1.837434,"\int \frac{x^m \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(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,"Integrate[(x^m*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2), x]","A",-1
750,0,0,29,3.9191938,"\int \frac{x^4 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(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,"Integrate[(x^4*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2), x]","A",-1
751,1,324,215,0.5275475,"\int \frac{x^3 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^3*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2),x]","\frac{-144 a^2 x^2 \tan ^{-1}(a x)+i a^2 x^2 \sqrt{3 a^2 x^2+3} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)-i a^2 x^2 \sqrt{3 a^2 x^2+3} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)-27 i \left(a^2 x^2+1\right)^{3/2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+27 i \left(a^2 x^2+1\right)^{3/2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+i \sqrt{3 a^2 x^2+3} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)-i \sqrt{3 a^2 x^2+3} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)-96 \tan ^{-1}(a x)}{144 a^4 c^2 \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}","\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} C\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} C\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,"(-96*ArcTan[a*x] - 144*a^2*x^2*ArcTan[a*x] - (27*I)*(1 + a^2*x^2)^(3/2)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + (27*I)*(1 + a^2*x^2)^(3/2)*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + I*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + I*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] - I*Sqrt[3 + 3*a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]] - I*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])/(144*a^4*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])","C",0
752,1,133,163,0.2691694,"\int \frac{x^2 \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^2*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2),x]","\frac{24 a^3 x^3 \sqrt{\tan ^{-1}(a x)}-9 \sqrt{2 \pi } \left(a^2 x^2+1\right)^{3/2} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)+\sqrt{6 \pi } \left(a^2 x^2+1\right)^{3/2} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{72 a^3 c^2 \left(a^2 x^2+1\right) \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}}",1,"(24*a^3*x^3*Sqrt[ArcTan[a*x]] - 9*Sqrt[2*Pi]*(1 + a^2*x^2)^(3/2)*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + Sqrt[6*Pi]*(1 + a^2*x^2)^(3/2)*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(72*a^3*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2])","A",1
753,1,167,163,0.4390546,"\int \frac{x \sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(5/2),x]","\frac{-48 \tan ^{-1}(a x)-i \left(a^2 x^2+1\right)^{3/2} \left(9 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)-9 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+\sqrt{3} \left(\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)-\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)\right)\right)}{144 a^2 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} C\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} C\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,"(-48*ArcTan[a*x] - I*(1 + a^2*x^2)^(3/2)*(9*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] - 9*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] - Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/(144*a^2*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])","C",0
754,1,137,213,0.1694156,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[Sqrt[ArcTan[a*x]]/(c + a^2*c*x^2)^(5/2),x]","\frac{-27 \sqrt{2 \pi } \left(a^2 x^2+1\right)^{3/2} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)-\sqrt{6 \pi } \left(a^2 x^2+1\right)^{3/2} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)+24 a x \left(2 a^2 x^2+3\right) \sqrt{\tan ^{-1}(a x)}}{72 c^2 \left(a^3 x^2+a\right) \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,"(24*a*x*(3 + 2*a^2*x^2)*Sqrt[ArcTan[a*x]] - 27*Sqrt[2*Pi]*(1 + a^2*x^2)^(3/2)*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] - Sqrt[6*Pi]*(1 + a^2*x^2)^(3/2)*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(72*c^2*(a + a^3*x^2)*Sqrt[c + a^2*c*x^2])","A",0
755,0,0,29,2.3243415,"\int \frac{\sqrt{\tan ^{-1}(a x)}}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[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,"Integrate[Sqrt[ArcTan[a*x]]/(x*(c + a^2*c*x^2)^(5/2)), x]","A",-1
756,0,0,25,2.0426906,"\int x^m \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2), x]","A",-1
757,0,0,25,4.3454618,"\int x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x^2*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2), x]","A",-1
758,0,0,57,1.3623963,"\int x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2), x]","A",-1
759,0,0,81,4.3012836,"\int \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)*ArcTan[a*x]^(3/2), x]","A",-1
760,0,0,25,1.8869484,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}}{x} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^(3/2))/x, x]","A",-1
761,0,0,25,1.729832,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}}{x^2} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^(3/2))/x^2, x]","A",-1
762,0,0,27,1.3835189,"\int x^m \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2), x]","A",-1
763,0,0,27,3.3124677,"\int x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2), x]","A",-1
764,0,0,61,1.3958885,"\int x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2), x]","A",-1
765,0,0,172,2.3240105,"\int \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2), x]","A",-1
766,0,0,27,1.8561362,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}}{x} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2))/x, x]","A",-1
767,0,0,27,2.2774079,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}}{x^2} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2))/x^2, x]","A",-1
768,0,0,27,0.9077007,"\int x^m \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2), x]","A",-1
769,0,0,27,3.117077,"\int x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2), x]","A",-1
770,0,0,61,1.4624299,"\int x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2), x]","A",-1
771,0,0,259,2.4147533,"\int \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2), x]","A",-1
772,0,0,27,1.8054149,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}}{x} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2))/x, x]","A",-1
773,0,0,27,2.7767701,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}}{x^2} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2))/x^2, x]","A",-1
774,0,0,27,0.7493977,"\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2), x]","A",-1
775,0,0,61,4.2447715,"\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx","Integrate[(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{2 \text{Int}\left(\tan ^{-1}(a x)^{5/2},x\right)}{5 a^3 c}+\frac{\text{Int}\left(x \tan ^{-1}(a x)^{3/2},x\right)}{a^2 c}-\frac{2 x \tan ^{-1}(a x)^{5/2}}{5 a^3 c}",0,"Integrate[(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2), x]","A",-1
776,0,0,37,1.3124202,"\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx","Integrate[(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,"Integrate[(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2), x]","A",-1
777,0,0,41,1.2146563,"\int \frac{x \tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx","Integrate[(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,"Integrate[(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2), x]","A",-1
778,1,18,18,0.003372,"\int \frac{\tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx","Integrate[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
779,0,0,49,0.6645628,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)), x]","A",-1
780,0,0,36,1.3730566,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \left(c+a^2 c x^2\right)} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(3/2)/(x^2*(c + a^2*c*x^2)), x]","A",-1
781,0,0,74,1.9272917,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \left(c+a^2 c x^2\right)} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(3/2)/(x^3*(c + a^2*c*x^2)), x]","A",-1
782,0,0,61,4.0361358,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^4 \left(c+a^2 c x^2\right)} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(3/2)/(x^4*(c + a^2*c*x^2)), x]","A",-1
783,0,0,27,1.4496723,"\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2, x]","A",-1
784,0,0,27,4.1030592,"\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(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,"Integrate[(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2, x]","A",-1
785,1,187,127,0.3910594,"\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2,x]","\frac{\frac{16 \sqrt{\tan ^{-1}(a x)} \left(15 \left(a^2 x^2-1\right)+16 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2-40 a x \tan ^{-1}(a x)\right)}{a^2 x^2+1}+60 \left(\sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)-2 \sqrt{\tan ^{-1}(a x)}\right)+\frac{15 \left(8 \tan ^{-1}(a x)-i \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+i \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)\right)}{\sqrt{\tan ^{-1}(a x)}}}{1280 a^3 c^2}","\frac{3 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a^3 c^2}+\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{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)}",1,"((16*Sqrt[ArcTan[a*x]]*(15*(-1 + a^2*x^2) - 40*a*x*ArcTan[a*x] + 16*(1 + a^2*x^2)*ArcTan[a*x]^2))/(1 + a^2*x^2) + 60*(-2*Sqrt[ArcTan[a*x]] + Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]]) + (15*(8*ArcTan[a*x] - I*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + I*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]]))/Sqrt[ArcTan[a*x]])/(1280*a^3*c^2)","C",0
786,1,75,109,0.1000962,"\int \frac{x \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^2,x]","\frac{\frac{4 \sqrt{\tan ^{-1}(a x)} \left(2 \left(a^2 x^2-1\right) \tan ^{-1}(a x)+3 a x\right)}{a^2 x^2+1}-3 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 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,"((4*Sqrt[ArcTan[a*x]]*(3*a*x + 2*(-1 + a^2*x^2)*ArcTan[a*x]))/(1 + a^2*x^2) - 3*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(32*a^2*c^2)","A",1
787,1,90,124,0.1774525,"\int \frac{\tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]^(3/2)/(c + a^2*c*x^2)^2,x]","\frac{\frac{2 \sqrt{\tan ^{-1}(a x)} \left(-15 a^2 x^2+16 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2+40 a x \tan ^{-1}(a x)+15\right)}{a^2 x^2+1}-15 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{160 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 } C\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,"((2*Sqrt[ArcTan[a*x]]*(15 - 15*a^2*x^2 + 40*a*x*ArcTan[a*x] + 16*(1 + a^2*x^2)*ArcTan[a*x]^2))/(1 + a^2*x^2) - 15*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(160*a*c^2)","A",1
788,0,0,27,1.9315954,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)^2} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^2), x]","A",-1
789,0,0,27,1.8661764,"\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3, x]","A",-1
790,0,0,27,6.9590147,"\int \frac{x^5 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(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,"Integrate[(x^5*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3, x]","A",-1
791,1,355,230,0.8620766,"\int \frac{x^4 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^4*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3,x]","\frac{\frac{64 \sqrt{\tan ^{-1}(a x)} \left(192 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2-160 a x \left(5 a^2 x^2+3\right) \tan ^{-1}(a x)+15 \left(17 a^4 x^4-6 a^2 x^2-15\right)\right)}{\left(a^2 x^2+1\right)^2}-510 \left(\sqrt{2 \pi } C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)-8 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)+12 \sqrt{\tan ^{-1}(a x)}\right)+90 \sqrt{\tan ^{-1}(a x)} \left(\frac{\Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)}{\sqrt{-i \tan ^{-1}(a x)}}+\frac{\Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{\sqrt{i \tan ^{-1}(a x)}}+8\right)+\frac{225 \left(24 \tan ^{-1}(a x)-4 i \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+4 i \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)-i \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+i \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)\right)}{\sqrt{\tan ^{-1}(a x)}}}{81920 a^5 c^3}","-\frac{3 \sqrt{\frac{\pi }{2}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{512 a^5 c^3}+\frac{3 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a^5 c^3}+\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 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{9 \sqrt{\tan ^{-1}(a x)}}{32 a^5 c^3 \left(a^2 x^2+1\right)}-\frac{3 x \tan ^{-1}(a x)^{3/2}}{8 a^4 c^3 \left(a^2 x^2+1\right)}",1,"((64*Sqrt[ArcTan[a*x]]*(15*(-15 - 6*a^2*x^2 + 17*a^4*x^4) - 160*a*x*(3 + 5*a^2*x^2)*ArcTan[a*x] + 192*(1 + a^2*x^2)^2*ArcTan[a*x]^2))/(1 + a^2*x^2)^2 - 510*(12*Sqrt[ArcTan[a*x]] + Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] - 8*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]]) + 90*Sqrt[ArcTan[a*x]]*(8 + Gamma[1/2, (-4*I)*ArcTan[a*x]]/Sqrt[(-I)*ArcTan[a*x]] + Gamma[1/2, (4*I)*ArcTan[a*x]]/Sqrt[I*ArcTan[a*x]]) + (225*(24*ArcTan[a*x] - (4*I)*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + (4*I)*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] - I*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + I*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]]))/Sqrt[ArcTan[a*x]])/(81920*a^5*c^3)","C",0
792,1,350,168,0.3146934,"\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3,x]","-\frac{9 \left(-2 \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)-2 \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)-\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)-\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)\right)}{4096 a^4 c^3 \sqrt{\tan ^{-1}(a x)}}-\frac{15 \left(-2 \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)-2 \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)+\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)\right)}{4096 a^4 c^3 \sqrt{\tan ^{-1}(a x)}}+\sqrt{\tan ^{-1}(a x)} \left(\frac{\left(5 a^4 x^4-6 a^2 x^2-3\right) \tan ^{-1}(a x)}{32 a^4 c^3 \left(a^2 x^2+1\right)^2}+\frac{3 x \left(5 a^2 x^2+3\right)}{64 a^3 c^3 \left(a^2 x^2+1\right)^2}\right)","\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{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{x^4 \tan ^{-1}(a x)^{3/2}}{4 c^3 \left(a^2 x^2+1\right)^2}",1,"Sqrt[ArcTan[a*x]]*((3*x*(3 + 5*a^2*x^2))/(64*a^3*c^3*(1 + a^2*x^2)^2) + ((-3 - 6*a^2*x^2 + 5*a^4*x^4)*ArcTan[a*x])/(32*a^4*c^3*(1 + a^2*x^2)^2)) - (9*(-2*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] - 2*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] - Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] - Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]]))/(4096*a^4*c^3*Sqrt[ArcTan[a*x]]) - (15*(-2*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] - 2*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] + Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]]))/(4096*a^4*c^3*Sqrt[ArcTan[a*x]])","C",0
793,1,353,108,0.8206802,"\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3,x]","\frac{\frac{64 \sqrt{\tan ^{-1}(a x)} \left(64 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2+160 a x \left(a^2 x^2-1\right) \tan ^{-1}(a x)-15 \left(a^4 x^4-6 a^2 x^2+1\right)\right)}{\left(a^2 x^2+1\right)^2}+30 \left(\sqrt{2 \pi } C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)-8 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)+12 \sqrt{\tan ^{-1}(a x)}\right)-90 \sqrt{\tan ^{-1}(a x)} \left(\frac{\Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)}{\sqrt{-i \tan ^{-1}(a x)}}+\frac{\Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{\sqrt{i \tan ^{-1}(a x)}}+8\right)+\frac{15 \left(24 \tan ^{-1}(a x)-4 i \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+4 i \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)-i \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+i \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)\right)}{\sqrt{\tan ^{-1}(a x)}}}{81920 a^3 c^3}","\frac{3 \sqrt{\frac{\pi }{2}} C\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,"((64*Sqrt[ArcTan[a*x]]*(-15*(1 - 6*a^2*x^2 + a^4*x^4) + 160*a*x*(-1 + a^2*x^2)*ArcTan[a*x] + 64*(1 + a^2*x^2)^2*ArcTan[a*x]^2))/(1 + a^2*x^2)^2 + 30*(12*Sqrt[ArcTan[a*x]] + Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] - 8*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]]) - 90*Sqrt[ArcTan[a*x]]*(8 + Gamma[1/2, (-4*I)*ArcTan[a*x]]/Sqrt[(-I)*ArcTan[a*x]] + Gamma[1/2, (4*I)*ArcTan[a*x]]/Sqrt[I*ArcTan[a*x]]) + (15*(24*ArcTan[a*x] - (4*I)*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + (4*I)*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] - I*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + I*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]]))/Sqrt[ArcTan[a*x]])/(81920*a^3*c^3)","C",0
794,1,347,168,0.2546066,"\int \frac{x \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^3,x]","\frac{192 a^4 x^4 \tan ^{-1}(a x)^2+3 a^4 x^4 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+3 a^4 x^4 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)+288 a^3 x^3 \tan ^{-1}(a x)+384 a^2 x^2 \tan ^{-1}(a x)^2+6 a^2 x^2 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+6 a^2 x^2 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)+24 \sqrt{2} \left(a^2 x^2+1\right)^2 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+24 \sqrt{2} \left(a^2 x^2+1\right)^2 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)+480 a x \tan ^{-1}(a x)-320 \tan ^{-1}(a x)^2+3 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+3 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{2048 c^3 \left(a^3 x^2+a\right)^2 \sqrt{\tan ^{-1}(a 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}",1,"(480*a*x*ArcTan[a*x] + 288*a^3*x^3*ArcTan[a*x] - 320*ArcTan[a*x]^2 + 384*a^2*x^2*ArcTan[a*x]^2 + 192*a^4*x^4*ArcTan[a*x]^2 + 24*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + 24*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] + 3*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + 6*a^2*x^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + 3*a^4*x^4*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + 3*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]] + 6*a^2*x^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]] + 3*a^4*x^4*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(2048*c^3*(a + a^3*x^2)^2*Sqrt[ArcTan[a*x]])","C",0
795,1,142,219,0.3886853,"\int \frac{\tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]^(3/2)/(c + a^2*c*x^2)^3,x]","\frac{\frac{4 \sqrt{\tan ^{-1}(a x)} \left(192 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^2+160 a x \left(3 a^2 x^2+5\right) \tan ^{-1}(a x)-15 \left(15 a^4 x^4+6 a^2 x^2-17\right)\right)}{\left(a^2 x^2+1\right)^2}-15 \sqrt{2 \pi } C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)-480 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{5120 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}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{512 a c^3}-\frac{3 \sqrt{\pi } C\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,"((4*Sqrt[ArcTan[a*x]]*(-15*(-17 + 6*a^2*x^2 + 15*a^4*x^4) + 160*a*x*(5 + 3*a^2*x^2)*ArcTan[a*x] + 192*(1 + a^2*x^2)^2*ArcTan[a*x]^2))/(1 + a^2*x^2)^2 - 15*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] - 480*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(5120*a*c^3)","A",1
796,0,0,27,2.5782036,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)^3} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^3), x]","A",-1
797,0,0,29,0.7268564,"\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2), x]","A",-1
798,0,0,29,3.1635758,"\int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2), x]","A",-1
799,0,0,66,6.6278137,"\int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2), x]","A",-1
800,0,0,119,0.3106272,"\int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2), x]","A",-1
801,0,0,29,3.3589832,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}}{x} \, dx","Integrate[(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,"Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))/x, x]","A",-1
802,0,0,29,1.1068853,"\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2), x]","A",-1
803,0,0,29,3.8999099,"\int x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2), x]","A",-1
804,0,0,66,2.5593619,"\int x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2), x]","A",-1
805,0,0,212,1.4944063,"\int \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2), x]","A",-1
806,0,0,29,2.2510658,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}}{x} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))/x, x]","A",-1
807,0,0,29,1.4006781,"\int x^m \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2), x]","A",-1
808,0,0,29,3.307239,"\int x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2), x]","A",-1
809,0,0,66,7.3519668,"\int x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2} \, dx","Integrate[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,"Integrate[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2), x]","A",-1
810,0,0,307,0.4951101,"\int \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2), x]","A",-1
811,0,0,29,2.3537873,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}}{x} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2))/x, x]","A",-1
812,0,0,29,0.879784,"\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^2], x]","A",-1
813,0,0,168,3.7347158,"\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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{2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{3 a^4 c}-\frac{x \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{4 a^3 c}",0,"Integrate[(x^3*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^2], x]","A",-1
814,0,0,132,2.4722757,"\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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,"Integrate[(x^2*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^2], x]","A",-1
815,0,0,63,0.693054,"\int \frac{x \tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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,"Integrate[(x*ArcTan[a*x]^(3/2))/Sqrt[c + a^2*c*x^2], x]","A",-1
816,0,0,26,0.2181018,"\int \frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(3/2)/Sqrt[c + a^2*c*x^2], x]","A",-1
817,0,0,29,1.3287009,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x \sqrt{c+a^2 c x^2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(3/2)/(x*Sqrt[c + a^2*c*x^2]), x]","A",-1
818,0,0,65,1.3485463,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \sqrt{c+a^2 c x^2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(3/2)/(x^2*Sqrt[c + a^2*c*x^2]), x]","A",-1
819,0,0,138,4.2032128,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(3/2)/(x^3*Sqrt[c + a^2*c*x^2]), x]","A",-1
820,0,0,173,19.2386587,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^4 \sqrt{c+a^2 c x^2}} \, dx","Integrate[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{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{5}{4} a^3 \text{Int}\left(\frac{\sqrt{\tan ^{-1}(a x)}}{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)^{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,"Integrate[ArcTan[a*x]^(3/2)/(x^4*Sqrt[c + a^2*c*x^2]), x]","A",-1
821,0,0,29,1.0021894,"\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2), x]","A",-1
822,0,0,29,8.3783625,"\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2), x]","A",-1
823,0,0,29,3.7039279,"\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2), x]","A",-1
824,1,128,129,0.1751532,"\int \frac{x \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(3/2),x]","\frac{3 \sqrt{a^2 x^2+1} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+3 \sqrt{a^2 x^2+1} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+4 \left(3 a x-2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)}{8 a^2 c \sqrt{a^2 c x^2+c} \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)}{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,"(4*(3*a*x - 2*ArcTan[a*x])*ArcTan[a*x] + 3*Sqrt[1 + a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + 3*Sqrt[1 + a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]])/(8*a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])","C",0
825,1,104,125,0.1708124,"\int \frac{\tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]^(3/2)/(c + a^2*c*x^2)^(3/2),x]","\frac{\left(a^2 x^2+1\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \left(\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{5}{2},-i \tan ^{-1}(a x)\right)+\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{5}{2},i \tan ^{-1}(a x)\right)\right)}{2 a \left(c \left(a^2 x^2+1\right)\right)^{3/2} \sqrt{\tan ^{-1}(a x)^2}}","-\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} C\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,"((1 + a^2*x^2)^(3/2)*Sqrt[ArcTan[a*x]]*(Sqrt[I*ArcTan[a*x]]*Gamma[5/2, (-I)*ArcTan[a*x]] + Sqrt[(-I)*ArcTan[a*x]]*Gamma[5/2, I*ArcTan[a*x]]))/(2*a*(c*(1 + a^2*x^2))^(3/2)*Sqrt[ArcTan[a*x]^2])","C",0
826,0,0,29,2.2977064,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^(3/2)), x]","A",-1
827,0,0,29,7.5358298,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(3/2)/(x^2*(c + a^2*c*x^2)^(3/2)), x]","A",-1
828,0,0,29,1.4885537,"\int \frac{x^m \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x]","A",-1
829,0,0,29,9.3124173,"\int \frac{x^5 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(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,"Integrate[(x^5*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x]","A",-1
830,0,0,29,3.8782307,"\int \frac{x^4 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(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,"Integrate[(x^4*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2), x]","A",-1
831,1,272,263,1.095108,"\int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^3*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2),x]","\frac{-7 \sqrt{6 \pi } \left(a^2 x^2+1\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \left(3 \sqrt{3} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)-S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)\right)+24 \tan ^{-1}(a x) \left(a x \left(7 a^2 x^2+6\right)-2 \left(3 a^2 x^2+2\right) \tan ^{-1}(a x)\right)+3 \left(a^2 x^2+1\right)^{3/2} \left(3 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+3 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+\sqrt{3} \left(\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)+\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)\right)\right)}{144 a^4 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}","-\frac{x^2 \tan ^{-1}(a x)^{3/2}}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{x^3 \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} 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{2 \tan ^{-1}(a x)^{3/2}}{3 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}}",1,"(24*ArcTan[a*x]*(a*x*(6 + 7*a^2*x^2) - 2*(2 + 3*a^2*x^2)*ArcTan[a*x]) - 7*Sqrt[6*Pi]*(1 + a^2*x^2)^(3/2)*Sqrt[ArcTan[a*x]]*(3*Sqrt[3]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] - FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]]) + 3*(1 + a^2*x^2)^(3/2)*(3*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + 3*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/(144*a^4*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])","C",0
832,1,338,247,0.574837,"\int \frac{x^2 \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^2*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2),x]","\frac{96 a^3 x^3 \tan ^{-1}(a x)^2+144 a^2 x^2 \tan ^{-1}(a x)-i a^2 x^2 \sqrt{3 a^2 x^2+3} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)+i a^2 x^2 \sqrt{3 a^2 x^2+3} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)+27 i \left(a^2 x^2+1\right)^{3/2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)-27 i \left(a^2 x^2+1\right)^{3/2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)-i \sqrt{3 a^2 x^2+3} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)+i \sqrt{3 a^2 x^2+3} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)+96 \tan ^{-1}(a x)}{288 a^3 c^2 \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}","\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} C\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} C\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}}",1,"(96*ArcTan[a*x] + 144*a^2*x^2*ArcTan[a*x] + 96*a^3*x^3*ArcTan[a*x]^2 + (27*I)*(1 + a^2*x^2)^(3/2)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] - (27*I)*(1 + a^2*x^2)^(3/2)*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] - I*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] - I*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + I*Sqrt[3 + 3*a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]] + I*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])/(288*a^3*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])","C",0
833,1,261,248,1.0505126,"\int \frac{x \tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x*ArcTan[a*x]^(3/2))/(c + a^2*c*x^2)^(5/2),x]","\frac{48 \left(2 a^3 x^3+3 a x-2 \tan ^{-1}(a x)\right) \tan ^{-1}(a x)-4 \sqrt{6 \pi } \left(a^2 x^2+1\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \left(3 \sqrt{3} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)-S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)\right)+3 \left(a^2 x^2+1\right)^{3/2} \left(3 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+3 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+\sqrt{3} \left(\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)+\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)\right)\right)}{288 a^2 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)}{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,"(48*(3*a*x + 2*a^3*x^3 - 2*ArcTan[a*x])*ArcTan[a*x] - 4*Sqrt[6*Pi]*(1 + a^2*x^2)^(3/2)*Sqrt[ArcTan[a*x]]*(3*Sqrt[3]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] - FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]]) + 3*(1 + a^2*x^2)^(3/2)*(3*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + 3*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/(288*a^2*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])","C",0
834,1,153,252,0.2120402,"\int \frac{\tan ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcTan[a*x]^(3/2)/(c + a^2*c*x^2)^(5/2),x]","\frac{-81 \sqrt{2 \pi } \left(a^2 x^2+1\right)^{3/2} C\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)-\sqrt{6 \pi } \left(a^2 x^2+1\right)^{3/2} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)+24 \sqrt{\tan ^{-1}(a x)} \left(\left(4 a^3 x^3+6 a x\right) \tan ^{-1}(a x)+6 a^2 x^2+7\right)}{144 c^2 \left(a^3 x^2+a\right) \sqrt{a^2 c x^2+c}}","-\frac{9 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} C\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} C\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,"(24*Sqrt[ArcTan[a*x]]*(7 + 6*a^2*x^2 + (6*a*x + 4*a^3*x^3)*ArcTan[a*x]) - 81*Sqrt[2*Pi]*(1 + a^2*x^2)^(3/2)*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] - Sqrt[6*Pi]*(1 + a^2*x^2)^(3/2)*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(144*c^2*(a + a^3*x^2)*Sqrt[c + a^2*c*x^2])","A",1
835,0,0,29,2.5063629,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(3/2)/(x*(c + a^2*c*x^2)^(5/2)), x]","A",-1
836,0,0,29,8.6573194,"\int \frac{\tan ^{-1}(a x)^{3/2}}{x^2 \left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(3/2)/(x^2*(c + a^2*c*x^2)^(5/2)), x]","A",-1
837,0,0,25,2.0377794,"\int x^m \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2), x]","A",-1
838,0,0,25,3.6905308,"\int x^2 \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x^2*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2), x]","A",-1
839,0,0,117,2.0951542,"\int x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2), x]","A",-1
840,0,0,81,3.9894057,"\int \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)*ArcTan[a*x]^(5/2), x]","A",-1
841,0,0,25,3.1734079,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}}{x} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^(5/2))/x, x]","A",-1
842,0,0,25,1.8050506,"\int \frac{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}}{x^2} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^(5/2))/x^2, x]","A",-1
843,0,0,27,1.418607,"\int x^m \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2), x]","A",-1
844,0,0,27,2.7440666,"\int x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2), x]","A",-1
845,0,0,216,1.5707525,"\int x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2), x]","A",-1
846,0,0,172,2.4195144,"\int \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2), x]","A",-1
847,0,0,27,1.9820745,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}}{x} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2))/x, x]","A",-1
848,0,0,27,2.7774548,"\int \frac{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}}{x^2} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2))/x^2, x]","A",-1
849,0,0,27,0.9423385,"\int x^m \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2), x]","A",-1
850,0,0,27,2.6381746,"\int x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2), x]","A",-1
851,0,0,309,1.6463696,"\int x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2), x]","A",-1
852,0,0,259,2.4050904,"\int \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2), x]","A",-1
853,0,0,27,1.6551325,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}}{x} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2))/x, x]","A",-1
854,0,0,27,2.8181779,"\int \frac{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}}{x^2} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2))/x^2, x]","A",-1
855,0,0,27,0.7376553,"\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2), x]","A",-1
856,0,0,61,4.5595298,"\int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx","Integrate[(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{2 \text{Int}\left(\tan ^{-1}(a x)^{7/2},x\right)}{7 a^3 c}+\frac{\text{Int}\left(x \tan ^{-1}(a x)^{5/2},x\right)}{a^2 c}-\frac{2 x \tan ^{-1}(a x)^{7/2}}{7 a^3 c}",0,"Integrate[(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2), x]","A",-1
857,0,0,37,1.3167923,"\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx","Integrate[(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,"Integrate[(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2), x]","A",-1
858,0,0,41,1.2229682,"\int \frac{x \tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx","Integrate[(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,"Integrate[(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2), x]","A",-1
859,1,18,18,0.003507,"\int \frac{\tan ^{-1}(a x)^{5/2}}{c+a^2 c x^2} \, dx","Integrate[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
860,0,0,49,0.6575961,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)), x]","A",-1
861,0,0,36,1.3929832,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x^2 \left(c+a^2 c x^2\right)} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(5/2)/(x^2*(c + a^2*c*x^2)), x]","A",-1
862,0,0,74,2.1660534,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x^3 \left(c+a^2 c x^2\right)} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(5/2)/(x^3*(c + a^2*c*x^2)), x]","A",-1
863,0,0,61,3.7993468,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x^4 \left(c+a^2 c x^2\right)} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(5/2)/(x^4*(c + a^2*c*x^2)), x]","A",-1
864,0,0,27,1.4298585,"\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2, x]","A",-1
865,0,0,27,4.5010586,"\int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(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,"Integrate[(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2, x]","A",-1
866,1,111,157,0.2074793,"\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2,x]","\frac{4 \sqrt{\tan ^{-1}(a x)} \left(32 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^3+70 \left(a^2 x^2-1\right) \tan ^{-1}(a x)+105 a x-112 a x \tan ^{-1}(a x)^2\right)-105 \sqrt{\pi } \left(a^2 x^2+1\right) S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{896 a^3 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^3 c^2}+\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{x \tan ^{-1}(a x)^{5/2}}{2 a^2 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{5 \tan ^{-1}(a x)^{3/2}}{8 a^3 c^2 \left(a^2 x^2+1\right)}",1,"(4*Sqrt[ArcTan[a*x]]*(105*a*x + 70*(-1 + a^2*x^2)*ArcTan[a*x] - 112*a*x*ArcTan[a*x]^2 + 32*(1 + a^2*x^2)*ArcTan[a*x]^3) - 105*Sqrt[Pi]*(1 + a^2*x^2)*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(896*a^3*c^2*(1 + a^2*x^2))","A",1
867,1,234,156,0.1786767,"\int \frac{x \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^2,x]","\frac{-60 \sqrt{\pi } \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)} C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)+256 a^2 x^2 \tan ^{-1}(a x)^3-240 a^2 x^2 \tan ^{-1}(a x)+15 i \sqrt{2} \left(a^2 x^2+1\right) \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)-15 i \sqrt{2} a^2 x^2 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)-256 \tan ^{-1}(a x)^3+640 a x \tan ^{-1}(a x)^2+240 \tan ^{-1}(a x)-15 i \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)}{1024 a^2 c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}","-\frac{15 \sqrt{\pi } C\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,"(240*ArcTan[a*x] - 240*a^2*x^2*ArcTan[a*x] + 640*a*x*ArcTan[a*x]^2 - 256*ArcTan[a*x]^3 + 256*a^2*x^2*ArcTan[a*x]^3 - 60*Sqrt[Pi]*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + (15*I)*Sqrt[2]*(1 + a^2*x^2)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] - (15*I)*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] - (15*I)*Sqrt[2]*a^2*x^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]])/(1024*a^2*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])","C",0
868,1,85,151,0.2816931,"\int \frac{\tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^2,x]","\frac{105 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)+2 \sqrt{\tan ^{-1}(a x)} \left(64 \tan ^{-1}(a x)^3+7 \left(16 \tan ^{-1}(a x)^2-15\right) \sin \left(2 \tan ^{-1}(a x)\right)+140 \tan ^{-1}(a x) \cos \left(2 \tan ^{-1}(a x)\right)\right)}{896 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,"(105*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + 2*Sqrt[ArcTan[a*x]]*(64*ArcTan[a*x]^3 + 140*ArcTan[a*x]*Cos[2*ArcTan[a*x]] + 7*(-15 + 16*ArcTan[a*x]^2)*Sin[2*ArcTan[a*x]]))/(896*a*c^2)","A",1
869,0,0,27,2.1252042,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)^2} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^2), x]","A",-1
870,0,0,27,1.9656502,"\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3, x]","A",-1
871,0,0,27,8.4242435,"\int \frac{x^5 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(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,"Integrate[(x^5*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3, x]","A",-1
872,1,287,310,0.6528397,"\int \frac{x^4 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^4*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3,x]","\frac{38080 a^4 x^4 \tan ^{-1}(a x)^2-71680 a^3 x^3 \tan ^{-1}(a x)^3+57120 a^3 x^3 \tan ^{-1}(a x)-13440 a^2 x^2 \tan ^{-1}(a x)^2+12288 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^4+3360 \sqrt{2} \left(a^2 x^2+1\right)^2 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+3360 \sqrt{2} \left(a^2 x^2+1\right)^2 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)-105 \left(a^2 x^2+1\right)^2 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)-105 \left(a^2 x^2+1\right)^2 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)-43008 a x \tan ^{-1}(a x)^3+50400 a x \tan ^{-1}(a x)-33600 \tan ^{-1}(a x)^2}{114688 a^5 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a 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{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{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{15 \tan ^{-1}(a x)^{3/2}}{32 a^5 c^3 \left(a^2 x^2+1\right)}-\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)}",1,"(50400*a*x*ArcTan[a*x] + 57120*a^3*x^3*ArcTan[a*x] - 33600*ArcTan[a*x]^2 - 13440*a^2*x^2*ArcTan[a*x]^2 + 38080*a^4*x^4*ArcTan[a*x]^2 - 43008*a*x*ArcTan[a*x]^3 - 71680*a^3*x^3*ArcTan[a*x]^3 + 12288*(1 + a^2*x^2)^2*ArcTan[a*x]^4 + 3360*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + 3360*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] - 105*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] - 105*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(114688*a^5*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])","C",0
873,1,359,256,0.7599983,"\int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3,x]","\frac{510 \sqrt{2 \pi } C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)+\frac{20480 a^4 x^4 \tan ^{-1}(a x)^3-16320 a^4 x^4 \tan ^{-1}(a x)+51200 a^3 x^3 \tan ^{-1}(a x)^2-4080 \sqrt{\pi } \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)} C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)-24576 a^2 x^2 \tan ^{-1}(a x)^3+5760 a^2 x^2 \tan ^{-1}(a x)+900 i \sqrt{2} \left(a^2 x^2+1\right)^2 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)-900 i \sqrt{2} \left(a^2 x^2+1\right)^2 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)+135 i \left(a^2 x^2+1\right)^2 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)-135 i \left(a^2 x^2+1\right)^2 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)+30720 a x \tan ^{-1}(a x)^2-12288 \tan ^{-1}(a x)^3+14400 \tan ^{-1}(a x)}{\left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}}{131072 a^4 c^3}","\frac{15 \sqrt{\frac{\pi }{2}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4096 a^4 c^3}-\frac{15 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{256 a^4 c^3}-\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{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{45 \sqrt{\tan ^{-1}(a x)}}{256 a^4 c^3 \left(a^2 x^2+1\right)}+\frac{15 x \tan ^{-1}(a x)^{3/2}}{64 a^3 c^3 \left(a^2 x^2+1\right)}",1,"(510*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + (14400*ArcTan[a*x] + 5760*a^2*x^2*ArcTan[a*x] - 16320*a^4*x^4*ArcTan[a*x] + 30720*a*x*ArcTan[a*x]^2 + 51200*a^3*x^3*ArcTan[a*x]^2 - 12288*ArcTan[a*x]^3 - 24576*a^2*x^2*ArcTan[a*x]^3 + 20480*a^4*x^4*ArcTan[a*x]^3 - 4080*Sqrt[Pi]*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + (900*I)*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] - (900*I)*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] + (135*I)*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] - (135*I)*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/((1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]))/(131072*a^4*c^3)","C",0
874,1,185,133,0.4773001,"\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3,x]","\frac{105 \left(a^2 x^2+1\right)^2 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+105 \left(a^2 x^2+1\right)^2 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)+32 \tan ^{-1}(a x) \left(-105 a x \left(a^2 x^2-1\right)+128 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^3+448 a x \left(a^2 x^2-1\right) \tan ^{-1}(a x)^2-70 \left(a^4 x^4-6 a^2 x^2+1\right) \tan ^{-1}(a x)\right)}{114688 a^3 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a 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}",1,"(32*ArcTan[a*x]*(-105*a*x*(-1 + a^2*x^2) - 70*(1 - 6*a^2*x^2 + a^4*x^4)*ArcTan[a*x] + 448*a*x*(-1 + a^2*x^2)*ArcTan[a*x]^2 + 128*(1 + a^2*x^2)^2*ArcTan[a*x]^3) + 105*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + 105*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(114688*a^3*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])","C",0
875,1,359,254,0.6968703,"\int \frac{x \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^3,x]","\frac{450 \sqrt{2 \pi } C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)+\frac{12288 a^4 x^4 \tan ^{-1}(a x)^3-14400 a^4 x^4 \tan ^{-1}(a x)+30720 a^3 x^3 \tan ^{-1}(a x)^2-3600 \sqrt{\pi } \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)} C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)+24576 a^2 x^2 \tan ^{-1}(a x)^3-5760 a^2 x^2 \tan ^{-1}(a x)+1020 i \sqrt{2} \left(a^2 x^2+1\right)^2 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)-1020 i \sqrt{2} \left(a^2 x^2+1\right)^2 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)+345 i \left(a^2 x^2+1\right)^2 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)-345 i \left(a^2 x^2+1\right)^2 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)+51200 a x \tan ^{-1}(a x)^2-20480 \tan ^{-1}(a x)^3+16320 \tan ^{-1}(a x)}{\left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}}{131072 a^2 c^3}","-\frac{15 \sqrt{\frac{\pi }{2}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4096 a^2 c^3}-\frac{15 \sqrt{\pi } C\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,"(450*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + (16320*ArcTan[a*x] - 5760*a^2*x^2*ArcTan[a*x] - 14400*a^4*x^4*ArcTan[a*x] + 51200*a*x*ArcTan[a*x]^2 + 30720*a^3*x^3*ArcTan[a*x]^2 - 20480*ArcTan[a*x]^3 + 24576*a^2*x^2*ArcTan[a*x]^3 + 12288*a^4*x^4*ArcTan[a*x]^3 - 3600*Sqrt[Pi]*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + (1020*I)*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] - (1020*I)*Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] + (345*I)*(1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] - (345*I)*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/((1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]))/(131072*a^2*c^3)","C",0
876,1,162,296,0.3809179,"\int \frac{\tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^3,x]","\frac{\frac{16 \sqrt{\tan ^{-1}(a x)} \left(-105 a x \left(15 a^2 x^2+17\right)+384 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^3+448 a x \left(3 a^2 x^2+5\right) \tan ^{-1}(a x)^2-70 \left(15 a^4 x^4+6 a^2 x^2-17\right) \tan ^{-1}(a x)\right)}{\left(a^2 x^2+1\right)^2}+105 \sqrt{2 \pi } S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)+6720 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{57344 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,"((16*Sqrt[ArcTan[a*x]]*(-105*a*x*(17 + 15*a^2*x^2) - 70*(-17 + 6*a^2*x^2 + 15*a^4*x^4)*ArcTan[a*x] + 448*a*x*(5 + 3*a^2*x^2)*ArcTan[a*x]^2 + 384*(1 + a^2*x^2)^2*ArcTan[a*x]^3))/(1 + a^2*x^2)^2 + 105*Sqrt[2*Pi]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + 6720*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(57344*a*c^3)","A",0
877,0,0,27,3.4397633,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)^3} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^3), x]","A",-1
878,0,0,29,0.7120012,"\int x^m \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x^m*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2), x]","A",-1
879,0,0,29,3.3549421,"\int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2), x]","A",-1
880,0,0,160,5.0867671,"\int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2), x]","A",-1
881,0,0,119,0.4859802,"\int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2), x]","A",-1
882,0,0,29,2.4637249,"\int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}}{x} \, dx","Integrate[(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,"Integrate[(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2))/x, x]","A",-1
883,0,0,29,1.1126856,"\int x^m \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2), x]","A",-1
884,0,0,29,4.0449102,"\int x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2), x]","A",-1
885,0,0,259,2.8245634,"\int x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2), x]","A",-1
886,0,0,212,1.5893458,"\int \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2), x]","A",-1
887,0,0,29,2.2257064,"\int \frac{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}}{x} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2))/x, x]","A",-1
888,0,0,29,1.482984,"\int x^m \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x^m*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2), x]","A",-1
889,0,0,29,3.4840956,"\int x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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,"Integrate[x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2), x]","A",-1
890,0,0,360,6.0514358,"\int x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2} \, dx","Integrate[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^3 \text{Int}\left(\frac{\tan ^{-1}(a x)^{3/2}}{\sqrt{a^2 c x^2+c}},x\right)}{224 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{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,"Integrate[x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2), x]","A",-1
891,0,0,307,0.4985925,"\int \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2), x]","A",-1
892,0,0,29,2.481907,"\int \frac{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}}{x} \, dx","Integrate[((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,"Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2))/x, x]","A",-1
893,0,0,29,0.845832,"\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^(5/2))/Sqrt[c + a^2*c*x^2], x]","A",-1
894,0,0,199,4.074998,"\int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}{8 a^4 c}-\frac{5 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}{12 a^3 c}",0,"Integrate[(x^3*ArcTan[a*x]^(5/2))/Sqrt[c + a^2*c*x^2], x]","A",-1
895,0,0,132,2.6008086,"\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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,"Integrate[(x^2*ArcTan[a*x]^(5/2))/Sqrt[c + a^2*c*x^2], x]","A",-1
896,0,0,63,0.6963678,"\int \frac{x \tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[(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,"Integrate[(x*ArcTan[a*x]^(5/2))/Sqrt[c + a^2*c*x^2], x]","A",-1
897,0,0,26,0.2274053,"\int \frac{\tan ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(5/2)/Sqrt[c + a^2*c*x^2], x]","A",-1
898,0,0,29,1.3216715,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x \sqrt{c+a^2 c x^2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(5/2)/(x*Sqrt[c + a^2*c*x^2]), x]","A",-1
899,0,0,65,1.4527905,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x^2 \sqrt{c+a^2 c x^2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(5/2)/(x^2*Sqrt[c + a^2*c*x^2]), x]","A",-1
900,0,0,138,4.7650303,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x^3 \sqrt{c+a^2 c x^2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(5/2)/(x^3*Sqrt[c + a^2*c*x^2]), x]","A",-1
901,0,0,208,17.0240641,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x^4 \sqrt{c+a^2 c x^2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(5/2)/(x^4*Sqrt[c + a^2*c*x^2]), x]","A",-1
902,0,0,29,0.9765191,"\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(3/2), x]","A",-1
903,0,0,29,3.6719176,"\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(3/2), x]","A",-1
904,1,139,161,0.1891022,"\int \frac{x \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(3/2),x]","\frac{15 i \sqrt{a^2 x^2+1} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)-15 i \sqrt{a^2 x^2+1} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+4 \tan ^{-1}(a x) \left(-4 \tan ^{-1}(a x)^2+10 a x \tan ^{-1}(a x)+15\right)}{16 a^2 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}","-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} C\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,"(4*ArcTan[a*x]*(15 + 10*a*x*ArcTan[a*x] - 4*ArcTan[a*x]^2) + (15*I)*Sqrt[1 + a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] - (15*I)*Sqrt[1 + a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]])/(16*a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])","C",0
905,1,97,155,0.1001913,"\int \frac{\tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^(3/2),x]","\frac{15 \sqrt{2 \pi } \sqrt{a^2 x^2+1} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)+2 \sqrt{\tan ^{-1}(a x)} \left(-15 a x+4 a x \tan ^{-1}(a x)^2+10 \tan ^{-1}(a x)\right)}{8 a 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,"(2*Sqrt[ArcTan[a*x]]*(-15*a*x + 10*ArcTan[a*x] + 4*a*x*ArcTan[a*x]^2) + 15*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(8*a*c*Sqrt[c + a^2*c*x^2])","A",1
906,0,0,29,2.3035575,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^(3/2)), x]","A",-1
907,0,0,29,1.4892923,"\int \frac{x^m \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(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,"Integrate[(x^m*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2), x]","A",-1
908,0,0,29,4.1368422,"\int \frac{x^4 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(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,"Integrate[(x^4*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2), x]","A",-1
909,1,370,350,0.5696323,"\int \frac{x^3 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^3*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2),x]","\frac{3360 a^3 x^3 \tan ^{-1}(a x)^2-1728 a^2 x^2 \tan ^{-1}(a x)^3+5040 a^2 x^2 \tan ^{-1}(a x)-5 i a^2 x^2 \sqrt{3 a^2 x^2+3} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)+5 i a^2 x^2 \sqrt{3 a^2 x^2+3} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)+1215 i \left(a^2 x^2+1\right)^{3/2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)-1215 i \left(a^2 x^2+1\right)^{3/2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)-5 i \sqrt{3 a^2 x^2+3} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)+5 i \sqrt{3 a^2 x^2+3} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)+2880 a x \tan ^{-1}(a x)^2-1152 \tan ^{-1}(a x)^3+4800 \tan ^{-1}(a x)}{1728 a^4 c^2 \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}","-\frac{x^2 \tan ^{-1}(a x)^{5/2}}{3 a^2 c \left(a^2 c x^2+c\right)^{3/2}}+\frac{5 x^3 \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} C\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} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{144 a^4 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 \tan ^{-1}(a x)^{3/2}}{3 a^3 c^2 \sqrt{a^2 c x^2+c}}",1,"(4800*ArcTan[a*x] + 5040*a^2*x^2*ArcTan[a*x] + 2880*a*x*ArcTan[a*x]^2 + 3360*a^3*x^3*ArcTan[a*x]^2 - 1152*ArcTan[a*x]^3 - 1728*a^2*x^2*ArcTan[a*x]^3 + (1215*I)*(1 + a^2*x^2)^(3/2)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] - (1215*I)*(1 + a^2*x^2)^(3/2)*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] - (5*I)*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] - (5*I)*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + (5*I)*Sqrt[3 + 3*a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]] + (5*I)*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])/(1728*a^4*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])","C",0
910,1,287,295,1.1334072,"\int \frac{x^2 \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x^2*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2),x]","\frac{35 \sqrt{6 \pi } \left(a^2 x^2+1\right)^{3/2} \sqrt{\tan ^{-1}(a x)} \left(3 \sqrt{3} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)-S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)\right)-15 \left(a^2 x^2+1\right)^{3/2} \left(3 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+3 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+\sqrt{3} \left(\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)+\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)\right)\right)-24 \tan ^{-1}(a x) \left(-12 a^3 x^3 \tan ^{-1}(a x)^2+5 a x \left(7 a^2 x^2+6\right)-10 \left(3 a^2 x^2+2\right) \tan ^{-1}(a x)\right)}{864 a^3 c \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}","-\frac{5 x \sqrt{\tan ^{-1}(a x)}}{6 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{5 x^2 \tan ^{-1}(a x)^{3/2}}{18 a c \left(a^2 c x^2+c\right)^{3/2}}+\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{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 \tan ^{-1}(a x)^{3/2}}{9 a^3 c^2 \sqrt{a^2 c x^2+c}}",1,"(-24*ArcTan[a*x]*(5*a*x*(6 + 7*a^2*x^2) - 10*(2 + 3*a^2*x^2)*ArcTan[a*x] - 12*a^3*x^3*ArcTan[a*x]^2) + 35*Sqrt[6*Pi]*(1 + a^2*x^2)^(3/2)*Sqrt[ArcTan[a*x]]*(3*Sqrt[3]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] - FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]]) - 15*(1 + a^2*x^2)^(3/2)*(3*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + 3*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/(864*a^3*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])","C",0
911,1,356,293,0.5365051,"\int \frac{x \tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[(x*ArcTan[a*x]^(5/2))/(c + a^2*c*x^2)^(5/2),x]","\frac{960 a^3 x^3 \tan ^{-1}(a x)^2+1440 a^2 x^2 \tan ^{-1}(a x)+5 i a^2 x^2 \sqrt{3 a^2 x^2+3} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)-5 i a^2 x^2 \sqrt{3 a^2 x^2+3} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)+405 i \left(a^2 x^2+1\right)^{3/2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)-405 i \left(a^2 x^2+1\right)^{3/2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+5 i \sqrt{3 a^2 x^2+3} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)-5 i \sqrt{3 a^2 x^2+3} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)+1440 a x \tan ^{-1}(a x)^2-576 \tan ^{-1}(a x)^3+1680 \tan ^{-1}(a x)}{1728 a^2 c^2 \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}","-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} C\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} C\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,"(1680*ArcTan[a*x] + 1440*a^2*x^2*ArcTan[a*x] + 1440*a*x*ArcTan[a*x]^2 + 960*a^3*x^3*ArcTan[a*x]^2 - 576*ArcTan[a*x]^3 + (405*I)*(1 + a^2*x^2)^(3/2)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] - (405*I)*(1 + a^2*x^2)^(3/2)*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + (5*I)*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + (5*I)*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] - (5*I)*Sqrt[3 + 3*a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]] - (5*I)*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])/(1728*a^2*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])","C",0
912,1,176,337,0.281839,"\int \frac{\tan ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcTan[a*x]^(5/2)/(c + a^2*c*x^2)^(5/2),x]","\frac{1215 \sqrt{2 \pi } \left(a^2 x^2+1\right)^{3/2} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)+5 \sqrt{6 \pi } \left(a^2 x^2+1\right)^{3/2} S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)+24 \sqrt{\tan ^{-1}(a x)} \left(-5 a x \left(20 a^2 x^2+21\right)+12 a x \left(2 a^2 x^2+3\right) \tan ^{-1}(a x)^2+10 \left(6 a^2 x^2+7\right) \tan ^{-1}(a x)\right)}{864 c^2 \left(a^3 x^2+a\right) \sqrt{a^2 c x^2+c}}","\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,"(24*Sqrt[ArcTan[a*x]]*(-5*a*x*(21 + 20*a^2*x^2) + 10*(7 + 6*a^2*x^2)*ArcTan[a*x] + 12*a*x*(3 + 2*a^2*x^2)*ArcTan[a*x]^2) + 1215*Sqrt[2*Pi]*(1 + a^2*x^2)^(3/2)*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + 5*Sqrt[6*Pi]*(1 + a^2*x^2)^(3/2)*FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]])/(864*c^2*(a + a^3*x^2)*Sqrt[c + a^2*c*x^2])","A",0
913,0,0,29,2.6501407,"\int \frac{\tan ^{-1}(a x)^{5/2}}{x \left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[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,"Integrate[ArcTan[a*x]^(5/2)/(x*(c + a^2*c*x^2)^(5/2)), x]","A",-1
914,0,0,25,2.1100751,"\int \frac{x^m \left(c+a^2 c x^2\right)}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2))/Sqrt[ArcTan[a*x]], x]","A",-1
915,0,0,23,1.0710377,"\int \frac{x \left(c+a^2 c x^2\right)}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2))/Sqrt[ArcTan[a*x]], x]","A",-1
916,0,0,22,0.1133739,"\int \frac{c+a^2 c x^2}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)/Sqrt[ArcTan[a*x]], x]","A",-1
917,0,0,25,1.5102901,"\int \frac{c+a^2 c x^2}{x \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)/(x*Sqrt[ArcTan[a*x]]), x]","A",-1
918,0,0,27,1.3853127,"\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^2)/Sqrt[ArcTan[a*x]], x]","A",-1
919,0,0,25,1.2802991,"\int \frac{x \left(c+a^2 c x^2\right)^2}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^2)/Sqrt[ArcTan[a*x]], x]","A",-1
920,0,0,24,0.7809607,"\int \frac{\left(c+a^2 c x^2\right)^2}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2/Sqrt[ArcTan[a*x]], x]","A",-1
921,0,0,27,1.4159583,"\int \frac{\left(c+a^2 c x^2\right)^2}{x \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2/(x*Sqrt[ArcTan[a*x]]), x]","A",-1
922,0,0,27,0.8375258,"\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^3)/Sqrt[ArcTan[a*x]], x]","A",-1
923,0,0,25,1.3045547,"\int \frac{x \left(c+a^2 c x^2\right)^3}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^3)/Sqrt[ArcTan[a*x]], x]","A",-1
924,0,0,24,0.7942222,"\int \frac{\left(c+a^2 c x^2\right)^3}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3/Sqrt[ArcTan[a*x]], x]","A",-1
925,0,0,27,1.4466005,"\int \frac{\left(c+a^2 c x^2\right)^3}{x \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3/(x*Sqrt[ArcTan[a*x]]), x]","A",-1
926,0,0,27,0.6433698,"\int \frac{x^m}{\left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]), x]","A",-1
927,0,0,37,0.8915779,"\int \frac{x}{\left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[x/((c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]), x]","A",-1
928,1,16,16,0.0033995,"\int \frac{1}{\left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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
929,0,0,27,0.2666366,"\int \frac{1}{x \left(c+a^2 c x^2\right) \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)*Sqrt[ArcTan[a*x]]), x]","A",-1
930,0,0,27,1.5128288,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]","A",-1
931,0,0,27,3.673465,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[x^3/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]","A",-1
932,1,122,47,0.217794,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[x^2/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]),x]","\frac{-4 \sqrt{\pi } \sqrt{\tan ^{-1}(a x)} C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)+16 \tan ^{-1}(a x)+i \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)-i \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)}{16 a^3 c^2 \sqrt{\tan ^{-1}(a x)}}","\frac{\sqrt{\tan ^{-1}(a x)}}{a^3 c^2}-\frac{\sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a^3 c^2}",1,"(16*ArcTan[a*x] - 4*Sqrt[Pi]*Sqrt[ArcTan[a*x]]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + I*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] - I*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]])/(16*a^3*c^2*Sqrt[ArcTan[a*x]])","C",0
933,1,31,31,0.0540809,"\int \frac{x}{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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",1
934,1,43,47,0.1173081,"\int \frac{1}{\left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[1/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)+2 \sqrt{\tan ^{-1}(a x)}}{2 a c^2}","\frac{\sqrt{\pi } C\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,"(2*Sqrt[ArcTan[a*x]] + Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(2*a*c^2)","A",1
935,0,0,27,1.4646283,"\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]","A",-1
936,0,0,27,1.9778229,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]","A",-1
937,0,0,27,4.4658911,"\int \frac{x^5}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[x^5/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]","A",-1
938,1,230,89,0.5267641,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[x^4/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]","\frac{10 \sqrt{2 \pi } \sqrt{\tan ^{-1}(a x)^2} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)-80 \sqrt{\pi } \sqrt{\tan ^{-1}(a x)^2} C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)+3 \sqrt{\tan ^{-1}(a x)} \left(64 \sqrt{\tan ^{-1}(a x)^2}+4 \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+4 \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)-\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)-\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)\right)}{256 a^5 c^3 \sqrt{\tan ^{-1}(a x)^2}}","\frac{\sqrt{\frac{\pi }{2}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^5 c^3}-\frac{\sqrt{\pi } C\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,"(10*Sqrt[2*Pi]*Sqrt[ArcTan[a*x]^2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] - 80*Sqrt[Pi]*Sqrt[ArcTan[a*x]^2]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + 3*Sqrt[ArcTan[a*x]]*(64*Sqrt[ArcTan[a*x]^2] + 4*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + 4*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] - Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] - Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]]))/(256*a^5*c^3*Sqrt[ArcTan[a*x]^2])","C",0
939,1,131,71,0.1494293,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[x^3/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]","\frac{-2 \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)-2 \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)+\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{32 a^4 c^3 \sqrt{\tan ^{-1}(a 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}",1,"(-2*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] - 2*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] + Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(32*a^4*c^3*Sqrt[ArcTan[a*x]])","C",0
940,1,229,58,0.4832179,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[x^2/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]","\frac{-2 \sqrt{2 \pi } \sqrt{\tan ^{-1}(a x)^2} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)+16 \sqrt{\pi } \sqrt{\tan ^{-1}(a x)^2} C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)+\sqrt{\tan ^{-1}(a x)} \left(64 \sqrt{\tan ^{-1}(a x)^2}+4 \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+4 \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)+7 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+7 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)\right)}{256 a^3 c^3 \sqrt{\tan ^{-1}(a x)^2}}","\frac{\sqrt{\tan ^{-1}(a x)}}{4 a^3 c^3}-\frac{\sqrt{\frac{\pi }{2}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a^3 c^3}",1,"(-2*Sqrt[2*Pi]*Sqrt[ArcTan[a*x]^2]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + 16*Sqrt[Pi]*Sqrt[ArcTan[a*x]^2]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + Sqrt[ArcTan[a*x]]*(64*Sqrt[ArcTan[a*x]^2] + 4*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + 4*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] + 7*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + 7*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]]))/(256*a^3*c^3*Sqrt[ArcTan[a*x]^2])","C",0
941,1,133,71,0.1139347,"\int \frac{x}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[x/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]","\frac{-2 \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)-2 \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)-\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)-\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{32 a^2 c^3 \sqrt{\tan ^{-1}(a 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}",1,"(-2*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] - 2*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] - Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] - Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(32*a^2*c^3*Sqrt[ArcTan[a*x]])","C",0
942,1,147,89,0.278237,"\int \frac{1}{\left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[1/((c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]),x]","\frac{24 \tan ^{-1}(a x)-4 i \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+4 i \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)-i \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+i \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{32 a c^3 \sqrt{\tan ^{-1}(a x)}}","\frac{\sqrt{\frac{\pi }{2}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{8 a c^3}+\frac{\sqrt{\pi } C\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,"(24*ArcTan[a*x] - (4*I)*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + (4*I)*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] - I*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + I*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(32*a*c^3*Sqrt[ArcTan[a*x]])","C",0
943,0,0,27,1.7970122,"\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]","A",-1
944,0,0,29,0.8533622,"\int \frac{x^m \sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(x^m*Sqrt[c + a^2*c*x^2])/Sqrt[ArcTan[a*x]], x]","A",-1
945,0,0,27,1.9530602,"\int \frac{x \sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(x*Sqrt[c + a^2*c*x^2])/Sqrt[ArcTan[a*x]], x]","A",-1
946,0,0,26,0.2756516,"\int \frac{\sqrt{c+a^2 c x^2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]/Sqrt[ArcTan[a*x]], x]","A",-1
947,0,0,29,2.903791,"\int \frac{\sqrt{c+a^2 c x^2}}{x \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]/(x*Sqrt[ArcTan[a*x]]), x]","A",-1
948,0,0,29,1.0378517,"\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^(3/2))/Sqrt[ArcTan[a*x]], x]","A",-1
949,0,0,27,2.996261,"\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^(3/2))/Sqrt[ArcTan[a*x]], x]","A",-1
950,0,0,26,0.3869624,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)/Sqrt[ArcTan[a*x]], x]","A",-1
951,0,0,29,2.5047897,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)/(x*Sqrt[ArcTan[a*x]]), x]","A",-1
952,0,0,29,1.4139335,"\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^(5/2))/Sqrt[ArcTan[a*x]], x]","A",-1
953,0,0,27,2.6116088,"\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^(5/2))/Sqrt[ArcTan[a*x]], x]","A",-1
954,0,0,26,0.442144,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)/Sqrt[ArcTan[a*x]], x]","A",-1
955,0,0,29,2.5426866,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)/(x*Sqrt[ArcTan[a*x]]), x]","A",-1
956,0,0,29,0.9065256,"\int \frac{x^m}{\sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[x^m/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]","A",-1
957,0,0,27,1.1493852,"\int \frac{x}{\sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[x/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]","A",-1
958,0,0,26,0.2259469,"\int \frac{1}{\sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]","A",-1
959,0,0,29,1.0284083,"\int \frac{1}{x \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[1/(x*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]), x]","A",-1
960,0,0,29,1.0264426,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]","A",-1
961,0,0,29,3.4943958,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[x^2/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]","A",-1
962,1,97,60,0.1381208,"\int \frac{x}{\left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[x/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]),x]","-\frac{\sqrt{a^2 x^2+1} \left(\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)\right)}{2 a^2 c \sqrt{c \left(a^2 x^2+1\right)} \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^2 c \sqrt{a^2 c x^2+c}}",1,"-1/2*(Sqrt[1 + a^2*x^2]*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]]))/(a^2*c*Sqrt[c*(1 + a^2*x^2)]*Sqrt[ArcTan[a*x]])","C",0
963,1,61,60,0.0488666,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[1/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} C\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a c \sqrt{c \left(a^2 x^2+1\right)}}","\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} C\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*(1 + a^2*x^2)])","A",1
964,0,0,29,1.9439924,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x]","A",-1
965,0,0,29,1.625189,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]","A",-1
966,0,0,29,3.6636777,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[x^4/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]","A",-1
967,1,95,131,0.2140009,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[x^3/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]),x]","\frac{\sqrt{\frac{\pi }{6}} \left(a^2 x^2+1\right)^{3/2} \left(3 \sqrt{3} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)-S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)\right)}{2 a^4 c \left(c \left(a^2 x^2+1\right)\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)}{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,"(Sqrt[Pi/6]*(1 + a^2*x^2)^(3/2)*(3*Sqrt[3]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] - FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]]))/(2*a^4*c*(c*(1 + a^2*x^2))^(3/2))","A",1
968,1,159,131,0.2038151,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[x^2/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]),x]","-\frac{i \sqrt{a^2 x^2+1} \left(3 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)-3 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+\sqrt{3} \left(\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)-\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)\right)\right)}{24 a^3 c^2 \sqrt{c \left(a^2 x^2+1\right)} \sqrt{\tan ^{-1}(a x)}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} C\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} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a^3 c^2 \sqrt{a^2 c x^2+c}}",1,"((-1/24*I)*Sqrt[1 + a^2*x^2]*(3*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] - 3*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + Sqrt[3]*(-(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]]) + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/(a^3*c^2*Sqrt[c*(1 + a^2*x^2)]*Sqrt[ArcTan[a*x]])","C",0
969,1,156,131,0.2243778,"\int \frac{x}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[x/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]),x]","-\frac{\left(a^2 x^2+1\right)^{3/2} \left(3 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+3 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+\sqrt{3} \left(\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)+\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)\right)\right)}{24 a^2 c \left(c \left(a^2 x^2+1\right)\right)^{3/2} \sqrt{\tan ^{-1}(a 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}}",1,"-1/24*((1 + a^2*x^2)^(3/2)*(3*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + 3*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/(a^2*c*(c*(1 + a^2*x^2))^(3/2)*Sqrt[ArcTan[a*x]])","C",0
970,1,159,131,0.2292411,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[1/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]),x]","-\frac{i \sqrt{c \left(a^2 x^2+1\right)} \left(9 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)-9 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+\sqrt{3} \left(\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)-\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)\right)\right)}{24 a c^3 \sqrt{a^2 x^2+1} \sqrt{\tan ^{-1}(a x)}}","\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} C\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} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{2 a c^2 \sqrt{a^2 c x^2+c}}",1,"((-1/24*I)*Sqrt[c*(1 + a^2*x^2)]*(9*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] - 9*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] - Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/(a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[ArcTan[a*x]])","C",0
971,0,0,29,2.1617368,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x]","A",-1
972,0,0,25,2.152388,"\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^(3/2), x]","A",-1
973,0,0,23,1.6625129,"\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2))/ArcTan[a*x]^(3/2), x]","A",-1
974,0,0,22,1.5489857,"\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)/ArcTan[a*x]^(3/2), x]","A",-1
975,0,0,25,2.0918815,"\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)/(x*ArcTan[a*x]^(3/2)), x]","A",-1
976,0,0,27,1.4980747,"\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(3/2), x]","A",-1
977,0,0,25,1.7941507,"\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(3/2), x]","A",-1
978,0,0,24,1.6310067,"\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2/ArcTan[a*x]^(3/2), x]","A",-1
979,0,0,27,2.1969545,"\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^(3/2)), x]","A",-1
980,0,0,27,0.9283939,"\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(3/2), x]","A",-1
981,0,0,25,1.8232571,"\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(3/2), x]","A",-1
982,0,0,24,1.6794496,"\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3/ArcTan[a*x]^(3/2), x]","A",-1
983,0,0,27,2.2484873,"\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^(3/2)), x]","A",-1
984,0,0,46,0.7527633,"\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)*ArcTan[a*x]^(3/2)), x]","A",-1
985,0,0,37,1.1295946,"\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[x/((c + a^2*c*x^2)*ArcTan[a*x]^(3/2)), x]","A",-1
986,1,16,16,0.0046632,"\int \frac{1}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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
987,0,0,43,0.2135163,"\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^(3/2)), x]","A",-1
988,0,0,27,1.1851416,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x]","A",-1
989,0,0,93,4.3876899,"\int \frac{x^4}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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","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{8 \text{Int}\left(\frac{x^3}{\left(a^2 c x^2+c\right)^2 \sqrt{\tan ^{-1}(a x)}},x\right)}{a}-\frac{2 x^4}{a c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[x^4/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x]","A",-1
990,0,0,106,4.4861091,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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 } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^4 c^2}+\frac{6 \sqrt{\tan ^{-1}(a x)}}{a^4 c^2}-\frac{2 x^3}{a c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x]","A",-1
991,1,60,60,0.1877461,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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",1
992,1,158,138,0.1907734,"\int \frac{x}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[x/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)),x]","\frac{4 \sqrt{\pi } \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)} C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)-i \sqrt{2} \left(a^2 x^2+1\right) \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+i \sqrt{2} \left(a^2 x^2+1\right) \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)-8 a x}{4 a^2 c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}","\frac{2 \sqrt{\pi } C\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,"(-8*a*x + 4*Sqrt[Pi]*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] - I*Sqrt[2]*(1 + a^2*x^2)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + I*Sqrt[2]*(1 + a^2*x^2)*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]])/(4*a^2*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]])","C",0
993,1,52,57,0.1685829,"\int \frac{1}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[1/((c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)),x]","\frac{-\frac{2}{\left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}-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/((1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) - 2*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(a*c^2)","A",1
994,0,0,102,4.7699105,"\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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 } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{c^2}-\frac{6 \sqrt{\tan ^{-1}(a x)}}{c^2}",0,"Integrate[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x]","A",-1
995,0,0,93,4.7551957,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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","-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{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}-\frac{2}{a c^2 x^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x]","A",-1
996,0,0,93,7.0410686,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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","-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{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}-\frac{2}{a c^2 x^3 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x]","A",-1
997,0,0,93,7.2713141,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[1/(x^4*(c + a^2*c*x^2)^2*ArcTan[a*x]^(3/2)), x]","A",-1
998,0,0,27,1.4377346,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x]","A",-1
999,1,148,96,0.3745107,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]","\frac{\frac{-\frac{32 a^3 x^3}{\left(a^2 x^2+1\right)^2}+3 i \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)-3 i \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{\sqrt{\tan ^{-1}(a x)}}-2 \sqrt{2 \pi } C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)+16 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{16 a^4 c^3}","-\frac{\sqrt{\frac{\pi }{2}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^4 c^3}+\frac{\sqrt{\pi } C\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*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + 16*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + ((-32*a^3*x^3)/(1 + a^2*x^2)^2 + (3*I)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] - (3*I)*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/Sqrt[ArcTan[a*x]])/(16*a^4*c^3)","C",0
1000,1,112,67,0.4136736,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]","\frac{-8 a^2 x^2-\left(a^2 x^2+1\right)^2 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)-\left(a^2 x^2+1\right)^2 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{4 a^3 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,"(-8*a^2*x^2 - (1 + a^2*x^2)^2*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] - (1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(4*a^3*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]])","C",0
1001,1,156,93,0.2572634,"\int \frac{x}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[x/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]","\frac{-\frac{8 a x}{\left(a^2 x^2+1\right)^2}-i \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+i \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)-i \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+i \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{4 a^2 c^3 \sqrt{\tan ^{-1}(a x)}}","\frac{\sqrt{\frac{\pi }{2}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^2 c^3}+\frac{\sqrt{\pi } C\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,"((-8*a*x)/(1 + a^2*x^2)^2 - I*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + I*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] - I*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + I*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(4*a^2*c^3*Sqrt[ArcTan[a*x]])","C",0
1002,1,144,94,0.318186,"\int \frac{1}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[1/((c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)),x]","\frac{-\frac{8}{\left(a^2 x^2+1\right)^2}+2 \sqrt{2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+2 \sqrt{2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)+\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{4 a c^3 \sqrt{\tan ^{-1}(a 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}",1,"(-8/(1 + a^2*x^2)^2 + 2*Sqrt[2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + 2*Sqrt[2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (2*I)*ArcTan[a*x]] + Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-4*I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (4*I)*ArcTan[a*x]])/(4*a*c^3*Sqrt[ArcTan[a*x]])","C",0
1003,0,0,140,5.3477475,"\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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}} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{4 c^3}-\frac{5 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)}}{2 c^3}",0,"Integrate[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x]","A",-1
1004,0,0,93,6.5430682,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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","-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{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}-\frac{2}{a c^3 x^2 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x]","A",-1
1005,0,0,93,7.6859692,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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","-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{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}-\frac{2}{a c^3 x^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[1/(x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x]","A",-1
1006,0,0,93,7.5551629,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^(3/2)), x]","A",-1
1007,0,0,29,0.7324909,"\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(3/2), x]","A",-1
1008,0,0,27,1.8924856,"\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(3/2), x]","A",-1
1009,0,0,26,0.4615507,"\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^(3/2), x]","A",-1
1010,0,0,29,4.8903219,"\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^(3/2)), x]","A",-1
1011,0,0,29,1.1179756,"\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(3/2), x]","A",-1
1012,0,0,27,7.1920611,"\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(3/2), x]","A",-1
1013,0,0,26,1.4164187,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^(3/2), x]","A",-1
1014,0,0,29,9.7948359,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^(3/2)), x]","A",-1
1015,0,0,29,1.4884657,"\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(3/2), x]","A",-1
1016,0,0,27,2.7335529,"\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(3/2), x]","A",-1
1017,0,0,26,1.1692585,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^(3/2), x]","A",-1
1018,0,0,29,4.8523029,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^(3/2)), x]","A",-1
1019,0,0,29,0.8990566,"\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x]","A",-1
1020,0,0,27,1.0994817,"\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x]","A",-1
1021,0,0,26,0.6674532,"\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x]","A",-1
1022,0,0,68,5.4219582,"\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x]","A",-1
1023,0,0,29,10.8724117,"\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x]","A",-1
1024,0,0,29,1.046087,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x]","A",-1
1025,0,0,100,5.9607964,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x]","A",-1
1026,0,0,127,5.8020637,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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{2 x^2}{a 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)}{a^3 c \sqrt{a^2 c x^2+c}}",0,"Integrate[x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x]","A",-1
1027,1,116,93,0.1582725,"\int \frac{x}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)),x]","\frac{-i \sqrt{a^2 x^2+1} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+i \sqrt{a^2 x^2+1} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)-2 a x}{a^2 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}","\frac{2 \sqrt{2 \pi } \sqrt{a^2 x^2+1} C\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*a*x - I*Sqrt[1 + a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + I*Sqrt[1 + a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]])/(a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])","C",0
1028,1,107,92,0.1376834,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)),x]","\frac{\sqrt{a^2 x^2+1} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+\sqrt{a^2 x^2+1} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)-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 + Sqrt[1 + a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + Sqrt[1 + a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]])/(a*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])","C",0
1029,0,0,126,4.9166164,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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} C\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,"Integrate[1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x]","A",-1
1030,0,0,100,20.6572198,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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","-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{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}-\frac{2}{a c x^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x]","A",-1
1031,0,0,100,17.2318138,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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","-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{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}-\frac{2}{a c x^3 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x]","A",-1
1032,0,0,100,19.4737996,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[1/(x^4*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2)), x]","A",-1
1033,0,0,29,1.1572375,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x]","A",-1
1034,1,182,160,0.5469061,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]","\frac{-\frac{8 a^3 c x^3}{a^2 x^2+1}-i c \sqrt{a^2 x^2+1} \left(3 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)-3 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+\sqrt{3} \left(\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)-\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)\right)\right)}{4 a^4 c^3 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}","-\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} C\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} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^4 c^2 \sqrt{a^2 c x^2+c}}",1,"((-8*a^3*c*x^3)/(1 + a^2*x^2) - I*c*Sqrt[1 + a^2*x^2]*(3*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] - 3*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + Sqrt[3]*(-(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]]) + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/(4*a^4*c^3*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])","C",0
1035,1,241,281,0.5312458,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]","\frac{\sqrt{6 \pi } \left(a^2 x^2+1\right)^{3/2} \left(S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)-3 \sqrt{3} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)\right)-\frac{12 a^2 x^2}{\sqrt{\tan ^{-1}(a x)}}-\frac{\left(a^2 x^2+1\right)^{3/2} \left(3 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+3 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+\sqrt{3} \left(\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)+\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)\right)\right)}{\sqrt{\tan ^{-1}(a x)}}}{6 a^3 c \left(a^2 c x^2+c\right)^{3/2}}","-\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}}",1,"((-12*a^2*x^2)/Sqrt[ArcTan[a*x]] + Sqrt[6*Pi]*(1 + a^2*x^2)^(3/2)*(-3*Sqrt[3]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]]) - ((1 + a^2*x^2)^(3/2)*(3*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + 3*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/Sqrt[ArcTan[a*x]])/(6*a^3*c*(c + a^2*c*x^2)^(3/2))","C",0
1036,1,299,280,0.5030709,"\int \frac{x}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]","-\frac{i \left(a^2 x^2 \sqrt{3 a^2 x^2+3} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)-a^2 x^2 \sqrt{3 a^2 x^2+3} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)+\left(a^2 x^2+1\right)^{3/2} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)-\left(a^2 x^2+1\right)^{3/2} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+\sqrt{3 a^2 x^2+3} \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)-\sqrt{3 a^2 x^2+3} \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)-8 i a x\right)}{4 a^2 c^2 \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}","-\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} C\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} C\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} C\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} C\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,"((-1/4*I)*((-8*I)*a*x + (1 + a^2*x^2)^(3/2)*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] - (1 + a^2*x^2)^(3/2)*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + a^2*x^2*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] - Sqrt[3 + 3*a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]] - a^2*x^2*Sqrt[3 + 3*a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]]))/(a^2*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])","C",0
1037,1,158,157,0.4032795,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)),x]","\frac{-8+\left(a^2 x^2+1\right)^{3/2} \left(3 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+3 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+\sqrt{3} \left(\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)+\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)\right)\right)}{4 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,"(-8 + (1 + a^2*x^2)^(3/2)*(3*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + 3*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/(4*a*c*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]])","C",0
1038,0,0,186,5.7520615,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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} C\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} C\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,"Integrate[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x]","A",-1
1039,0,0,100,15.6327202,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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","-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{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}-\frac{2}{a c x^2 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x]","A",-1
1040,0,0,100,26.7798583,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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","-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{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}-\frac{2}{a c x^3 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x]","A",-1
1041,0,0,100,22.2650336,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{3/2}} \, dx","Integrate[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,"Integrate[1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(3/2)), x]","A",-1
1042,0,0,25,2.189246,"\int \frac{x^m \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2))/ArcTan[a*x]^(5/2), x]","A",-1
1043,0,0,23,2.7989034,"\int \frac{x \left(c+a^2 c x^2\right)}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2))/ArcTan[a*x]^(5/2), x]","A",-1
1044,0,0,22,2.1488183,"\int \frac{c+a^2 c x^2}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)/ArcTan[a*x]^(5/2), x]","A",-1
1045,0,0,25,5.9800725,"\int \frac{c+a^2 c x^2}{x \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)/(x*ArcTan[a*x]^(5/2)), x]","A",-1
1046,0,0,27,1.5414796,"\int \frac{x^m \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(5/2), x]","A",-1
1047,0,0,25,2.1704102,"\int \frac{x \left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^2)/ArcTan[a*x]^(5/2), x]","A",-1
1048,0,0,24,1.6192775,"\int \frac{\left(c+a^2 c x^2\right)^2}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2/ArcTan[a*x]^(5/2), x]","A",-1
1049,0,0,27,3.2396265,"\int \frac{\left(c+a^2 c x^2\right)^2}{x \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^2/(x*ArcTan[a*x]^(5/2)), x]","A",-1
1050,0,0,27,0.9351239,"\int \frac{x^m \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(5/2), x]","A",-1
1051,0,0,25,2.2339907,"\int \frac{x \left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^3)/ArcTan[a*x]^(5/2), x]","A",-1
1052,0,0,24,1.6666406,"\int \frac{\left(c+a^2 c x^2\right)^3}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3/ArcTan[a*x]^(5/2), x]","A",-1
1053,0,0,27,4.0712684,"\int \frac{\left(c+a^2 c x^2\right)^3}{x \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^3/(x*ArcTan[a*x]^(5/2)), x]","A",-1
1054,0,0,50,0.7810987,"\int \frac{x^m}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)*ArcTan[a*x]^(5/2)), x]","A",-1
1055,0,0,41,1.5521876,"\int \frac{x}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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,"Integrate[x/((c + a^2*c*x^2)*ArcTan[a*x]^(5/2)), x]","A",-1
1056,1,18,18,0.0060043,"\int \frac{1}{\left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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
1057,0,0,47,1.427045,"\int \frac{1}{x \left(c+a^2 c x^2\right) \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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,"Integrate[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2)), x]","A",-1
1058,0,0,27,1.2420227,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x]","A",-1
1059,0,0,186,4.8112987,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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{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{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{4 \sqrt{\pi } S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^4 c^2}-\frac{4 x^2}{a^2 c^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}-\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}}",0,"Integrate[x^3/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x]","A",-1
1060,1,162,180,0.3657803,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[x^2/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]","\frac{4 \sqrt{\pi } \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2} C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)+\sqrt{2} \left(a^2 x^2+1\right) \left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+\sqrt{2} \left(a^2 x^2+1\right) \left(i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)-2 a x \left(a x+4 \tan ^{-1}(a x)\right)}{3 a^3 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}","\frac{8 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a^3 c^2}+\frac{16 \sqrt{\tan ^{-1}(a x)}}{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)}",1,"(-2*a*x*(a*x + 4*ArcTan[a*x]) + 4*Sqrt[Pi]*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + Sqrt[2]*(1 + a^2*x^2)*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-2*I)*ArcTan[a*x]] + Sqrt[2]*(1 + a^2*x^2)*(I*ArcTan[a*x])^(3/2)*Gamma[1/2, (2*I)*ArcTan[a*x]])/(3*a^3*c^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))","C",0
1061,1,88,101,0.0801775,"\int \frac{x}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[x/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]","-\frac{2 \left(4 \sqrt{\pi } \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2} S\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)+\left(2-2 a^2 x^2\right) \tan ^{-1}(a x)+a x\right)}{3 a^2 c^2 \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2}}","-\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*(a*x + (2 - 2*a^2*x^2)*ArcTan[a*x] + 4*Sqrt[Pi]*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]]))/(3*a^2*c^2*(1 + a^2*x^2)*ArcTan[a*x]^(3/2))","A",1
1062,1,170,174,0.4378911,"\int \frac{1}{\left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[1/((c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)),x]","\frac{-4 \sqrt{\pi } \left(a^2 x^2+1\right) \tan ^{-1}(a x)^{3/2} C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)+\frac{\sqrt{2} \left(a^2 x^2+1\right) \tan ^{-1}(a x)^2 \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)}{\sqrt{i \tan ^{-1}(a x)}}+\sqrt{2} \left(a^2 x^2+1\right) \sqrt{i \tan ^{-1}(a x)} \sqrt{\tan ^{-1}(a x)^2} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+8 a x \tan ^{-1}(a x)-2}{3 c^2 \left(a^3 x^2+a\right) \tan ^{-1}(a x)^{3/2}}","\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 } C\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 + 8*a*x*ArcTan[a*x] - 4*Sqrt[Pi]*(1 + a^2*x^2)*ArcTan[a*x]^(3/2)*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]] + Sqrt[2]*(1 + a^2*x^2)*Sqrt[I*ArcTan[a*x]]*Sqrt[ArcTan[a*x]^2]*Gamma[1/2, (-2*I)*ArcTan[a*x]] + (Sqrt[2]*(1 + a^2*x^2)*ArcTan[a*x]^2*Gamma[1/2, (2*I)*ArcTan[a*x]])/Sqrt[I*ArcTan[a*x]])/(3*c^2*(a + a^3*x^2)*ArcTan[a*x]^(3/2))","C",0
1063,0,0,180,4.5374053,"\int \frac{1}{x \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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{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{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{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,"Integrate[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x]","A",-1
1064,0,0,198,6.8779976,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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 C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{c^2}+\frac{16 a \sqrt{\tan ^{-1}(a x)}}{c^2}",0,"Integrate[1/(x^2*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x]","A",-1
1065,0,0,189,5.6378592,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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{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{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{20}{3 c^2 x^2 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}+\frac{4}{a^2 c^2 x^4 \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}}",0,"Integrate[1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x]","A",-1
1066,0,0,189,11.3649566,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^2 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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{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{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{16}{3 a^2 c^2 x^5 \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{8}{c^2 x^3 \left(a^2 x^2+1\right) \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[1/(x^4*(c + a^2*c*x^2)^2*ArcTan[a*x]^(5/2)), x]","A",-1
1067,0,0,27,1.4468384,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x]","A",-1
1068,1,227,160,0.4695538,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[x^3/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)),x]","\frac{i \sqrt{2} \left(a^2 x^2+1\right)^2 \left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+\sqrt{2} \left(a^2 x^2+1\right)^2 \sqrt{i \tan ^{-1}(a x)} \tan ^{-1}(a x) \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)-2 \left(a^2 x^2 \left(\left(6-2 a^2 x^2\right) \tan ^{-1}(a x)+a x\right)+i \left(a^2 x^2+1\right)^2 \left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+\left(a^2 x^2+1\right)^2 \sqrt{i \tan ^{-1}(a x)} \tan ^{-1}(a x) \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)\right)}{3 a^4 c^3 \left(a^2 x^2+1\right)^2 \tan ^{-1}(a x)^{3/2}}","\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^2}{a^2 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}+\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}}",1,"(I*Sqrt[2]*(1 + a^2*x^2)^2*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-2*I)*ArcTan[a*x]] + Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*ArcTan[a*x]*Gamma[1/2, (2*I)*ArcTan[a*x]] - 2*(a^2*x^2*(a*x + (6 - 2*a^2*x^2)*ArcTan[a*x]) + I*(1 + a^2*x^2)^2*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-4*I)*ArcTan[a*x]] + (1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*ArcTan[a*x]*Gamma[1/2, (4*I)*ArcTan[a*x]]))/(3*a^4*c^3*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2))","C",0
1069,1,259,129,0.9517764,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[x^2/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)),x]","\frac{\frac{2 \sqrt{2 \pi } C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3}-\frac{16 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a^3}+\frac{-\frac{32 x \tan ^{-1}(a x)}{\left(a^3 x^2+a\right)^2}+\frac{4 \sqrt{2} \left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)}{a^3}+\frac{4 \sqrt{2} \left(i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)}{a^3}+\frac{7 \left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)}{a^3}+\frac{7 \left(i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{a^3}-\frac{8 x^2}{a \left(a^2 x^2+1\right)^2}+\frac{32 x^3 \tan ^{-1}(a x)}{\left(a^2 x^2+1\right)^2}}{\tan ^{-1}(a x)^{3/2}}}{12 c^3}","\frac{4 \sqrt{2 \pi } C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^3 c^3}-\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{8 x^3}{3 c^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",1,"((2*Sqrt[2*Pi]*FresnelC[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/a^3 - (16*Sqrt[Pi]*FresnelC[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/a^3 + ((-8*x^2)/(a*(1 + a^2*x^2)^2) + (32*x^3*ArcTan[a*x])/(1 + a^2*x^2)^2 - (32*x*ArcTan[a*x])/(a + a^3*x^2)^2 + (4*Sqrt[2]*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-2*I)*ArcTan[a*x]])/a^3 + (4*Sqrt[2]*(I*ArcTan[a*x])^(3/2)*Gamma[1/2, (2*I)*ArcTan[a*x]])/a^3 + (7*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-4*I)*ArcTan[a*x]])/a^3 + (7*(I*ArcTan[a*x])^(3/2)*Gamma[1/2, (4*I)*ArcTan[a*x]])/a^3)/ArcTan[a*x]^(3/2))/(12*c^3)","C",0
1070,1,220,155,0.397726,"\int \frac{x}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[x/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)),x]","\frac{i \sqrt{2} \left(a^2 x^2+1\right)^2 \left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)+\sqrt{2} \left(a^2 x^2+1\right)^2 \sqrt{i \tan ^{-1}(a x)} \tan ^{-1}(a x) \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)+2 \left(6 a^2 x^2 \tan ^{-1}(a x)+i \left(a^2 x^2+1\right)^2 \left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)+\left(a^2 x^2+1\right)^2 \sqrt{i \tan ^{-1}(a x)} \tan ^{-1}(a x) \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)-a x-2 \tan ^{-1}(a x)\right)}{3 c^3 \left(a^3 x^2+a\right)^2 \tan ^{-1}(a x)^{3/2}}","-\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,"(I*Sqrt[2]*(1 + a^2*x^2)^2*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-2*I)*ArcTan[a*x]] + Sqrt[2]*(1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*ArcTan[a*x]*Gamma[1/2, (2*I)*ArcTan[a*x]] + 2*(-(a*x) - 2*ArcTan[a*x] + 6*a^2*x^2*ArcTan[a*x] + I*(1 + a^2*x^2)^2*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-4*I)*ArcTan[a*x]] + (1 + a^2*x^2)^2*Sqrt[I*ArcTan[a*x]]*ArcTan[a*x]*Gamma[1/2, (4*I)*ArcTan[a*x]]))/(3*c^3*(a + a^3*x^2)^2*ArcTan[a*x]^(3/2))","C",0
1071,1,186,125,0.7235581,"\int \frac{1}{\left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[1/((c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)),x]","\frac{2 \left(-\frac{1}{a \left(a^2 x^2+1\right)^2}+\frac{8 x \tan ^{-1}(a x)}{\left(a^2 x^2+1\right)^2}+\frac{\sqrt{2} \tan ^{-1}(a x)^2 \Gamma \left(\frac{1}{2},2 i \tan ^{-1}(a x)\right)}{a \sqrt{i \tan ^{-1}(a x)}}+\frac{\tan ^{-1}(a x)^2 \Gamma \left(\frac{1}{2},4 i \tan ^{-1}(a x)\right)}{a \sqrt{i \tan ^{-1}(a x)}}-\frac{\sqrt{2} \left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-2 i \tan ^{-1}(a x)\right)}{a}-\frac{\left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-4 i \tan ^{-1}(a x)\right)}{a}\right)}{3 c^3 \tan ^{-1}(a x)^{3/2}}","\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 } C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a c^3}-\frac{8 \sqrt{\pi } C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a c^3}",1,"(2*(-(1/(a*(1 + a^2*x^2)^2)) + (8*x*ArcTan[a*x])/(1 + a^2*x^2)^2 - (Sqrt[2]*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-2*I)*ArcTan[a*x]])/a + (Sqrt[2]*ArcTan[a*x]^2*Gamma[1/2, (2*I)*ArcTan[a*x]])/(a*Sqrt[I*ArcTan[a*x]]) - (((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-4*I)*ArcTan[a*x]])/a + (ArcTan[a*x]^2*Gamma[1/2, (4*I)*ArcTan[a*x]])/(a*Sqrt[I*ArcTan[a*x]])))/(3*c^3*ArcTan[a*x]^(3/2))","C",0
1072,0,0,216,6.2343418,"\int \frac{1}{x \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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","8 \text{Int}\left(\frac{1}{x \left(a^2 c x^2+c\right)^3 \sqrt{\tan ^{-1}(a x)}},x\right)+\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}+\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,"Integrate[1/(x*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x]","A",-1
1073,0,0,231,8.468209,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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 C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{c^3}+\frac{20 \sqrt{\pi } a C\left(\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right)}{c^3}+\frac{30 a \sqrt{\tan ^{-1}(a x)}}{c^3}",0,"Integrate[1/(x^2*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x]","A",-1
1074,0,0,187,6.2497575,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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{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{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{28}{3 c^3 x^2 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}+\frac{4}{a^2 c^3 x^4 \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}}",0,"Integrate[1/(x^3*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x]","A",-1
1075,0,0,191,13.2358772,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^3 \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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)+\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}+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{16}{3 a^2 c^3 x^5 \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{32}{3 c^3 x^3 \left(a^2 x^2+1\right)^2 \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[1/(x^4*(c + a^2*c*x^2)^3*ArcTan[a*x]^(5/2)), x]","A",-1
1076,0,0,29,0.7449325,"\int \frac{x^m \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(x^m*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(5/2), x]","A",-1
1077,0,0,27,2.712724,"\int \frac{x \sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(x*Sqrt[c + a^2*c*x^2])/ArcTan[a*x]^(5/2), x]","A",-1
1078,0,0,26,1.5937435,"\int \frac{\sqrt{c+a^2 c x^2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]/ArcTan[a*x]^(5/2), x]","A",-1
1079,0,0,29,18.8159504,"\int \frac{\sqrt{c+a^2 c x^2}}{x \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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,"Integrate[Sqrt[c + a^2*c*x^2]/(x*ArcTan[a*x]^(5/2)), x]","A",-1
1080,0,0,29,1.1235945,"\int \frac{x^m \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(5/2), x]","A",-1
1081,0,0,27,2.7516367,"\int \frac{x \left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^(3/2))/ArcTan[a*x]^(5/2), x]","A",-1
1082,0,0,26,1.7088139,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)/ArcTan[a*x]^(5/2), x]","A",-1
1083,0,0,29,12.1126981,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{x \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(3/2)/(x*ArcTan[a*x]^(5/2)), x]","A",-1
1084,0,0,29,1.4980575,"\int \frac{x^m \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(x^m*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(5/2), x]","A",-1
1085,0,0,27,3.4312291,"\int \frac{x \left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(x*(c + a^2*c*x^2)^(5/2))/ArcTan[a*x]^(5/2), x]","A",-1
1086,0,0,26,1.8299794,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)/ArcTan[a*x]^(5/2), x]","A",-1
1087,0,0,29,11.6178598,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{x \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[(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,"Integrate[(c + a^2*c*x^2)^(5/2)/(x*ArcTan[a*x]^(5/2)), x]","A",-1
1088,0,0,29,0.9097955,"\int \frac{x^m}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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,"Integrate[x^m/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]","A",-1
1089,0,0,27,1.8490518,"\int \frac{x}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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,"Integrate[x/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]","A",-1
1090,0,0,26,0.8329753,"\int \frac{1}{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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,"Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]","A",-1
1091,0,0,72,3.6188647,"\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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,"Integrate[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]","A",-1
1092,0,0,29,20.5535782,"\int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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,"Integrate[1/(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(5/2)), x]","A",-1
1093,0,0,29,1.0741738,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x]","A",-1
1094,0,0,229,7.5637333,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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","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{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)-\frac{4 x^2}{a^2 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}-\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{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}}",0,"Integrate[x^3/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x]","A",-1
1095,0,0,231,6.7781426,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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{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)}}-\frac{4 x^3}{3 c \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}+\frac{8 \sqrt{2 \pi } \sqrt{a^2 x^2+1} C\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{3 a^3 c \sqrt{a^2 c x^2+c}}",0,"Integrate[x^2/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x]","A",-1
1096,1,124,129,0.1857965,"\int \frac{x}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[x/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]","-\frac{2 \left(-i \sqrt{a^2 x^2+1} \left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+i \sqrt{a^2 x^2+1} \left(i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+a x+2 \tan ^{-1}(a x)\right)}{3 a^2 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} 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*(a*x + 2*ArcTan[a*x] - I*Sqrt[1 + a^2*x^2]*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-I)*ArcTan[a*x]] + I*Sqrt[1 + a^2*x^2]*(I*ArcTan[a*x])^(3/2)*Gamma[1/2, I*ArcTan[a*x]]))/(3*a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))","C",0
1097,1,120,126,0.149007,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[1/((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)),x]","\frac{-2 \sqrt{a^2 x^2+1} \left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)-2 \sqrt{a^2 x^2+1} \left(i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+4 a x \tan ^{-1}(a x)-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} C\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 + 4*a*x*ArcTan[a*x] - 2*Sqrt[1 + a^2*x^2]*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-I)*ArcTan[a*x]] - 2*Sqrt[1 + a^2*x^2]*(I*ArcTan[a*x])^(3/2)*Gamma[1/2, I*ArcTan[a*x]])/(3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))","C",0
1098,0,0,227,12.3350473,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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","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 \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}+\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,"Integrate[1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x]","A",-1
1099,0,0,227,13.9517292,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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{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 \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{8 \sqrt{2 \pi } a \sqrt{a^2 x^2+1} C\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{2}{3 a c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c x^3 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[1/(x^2*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x]","A",-1
1100,0,0,202,16.2367288,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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{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{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}{3 c x^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}+\frac{4}{a^2 c x^4 \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}}",0,"Integrate[1/(x^3*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x]","A",-1
1101,0,0,206,28.4953796,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{3/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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)+\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}+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{16}{3 a^2 c x^5 \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{20}{3 c x^3 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[1/(x^4*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(5/2)), x]","A",-1
1102,0,0,29,1.1478014,"\int \frac{x^m}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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,"Integrate[x^m/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x]","A",-1
1103,1,255,190,0.8852993,"\int \frac{x^3}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[x^3/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]","\frac{\sqrt{6 \pi } \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^{3/2} \left(S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)-3 \sqrt{3} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)\right)-2 a^2 x^2 \left(a x+6 \tan ^{-1}(a x)\right)-\left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x) \left(3 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+3 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+\sqrt{3} \left(\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)+\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)\right)\right)}{3 a^4 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{2 x^3}{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} 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}}",1,"(-2*a^2*x^2*(a*x + 6*ArcTan[a*x]) + Sqrt[6*Pi]*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^(3/2)*(-3*Sqrt[3]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] + FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]]) - (1 + a^2*x^2)^(3/2)*ArcTan[a*x]*(3*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + 3*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/(3*a^4*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))","C",0
1104,1,311,224,0.8621157,"\int \frac{x^2}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[x^2/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]","\frac{\frac{-8 i a^3 x^3 \left(i \tan ^{-1}(a x)\right)^{3/2}-4 a^2 x^2 \sqrt{i \tan ^{-1}(a x)}-3 a^2 x^2 \sqrt{3 a^2 x^2+3} \tan ^{-1}(a x)^2 \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)+\left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^2 \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)-3 i \sqrt{3} \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x) \sqrt{\tan ^{-1}(a x)^2} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)-3 \sqrt{3 a^2 x^2+3} \tan ^{-1}(a x)^2 \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)+16 i a x \left(i \tan ^{-1}(a x)\right)^{3/2}}{\sqrt{i \tan ^{-1}(a x)}}-\left(a^2 x^2+1\right)^{3/2} \left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)}{6 a^3 c^2 \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}","-\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{4 x^3}{3 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} C\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} C\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)}{a^3 c^2 \sqrt{a^2 c x^2+c}}",1,"(-((1 + a^2*x^2)^(3/2)*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-I)*ArcTan[a*x]]) + (-4*a^2*x^2*Sqrt[I*ArcTan[a*x]] + (16*I)*a*x*(I*ArcTan[a*x])^(3/2) - (8*I)*a^3*x^3*(I*ArcTan[a*x])^(3/2) + (1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*Gamma[1/2, I*ArcTan[a*x]] - (3*I)*Sqrt[3]*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]*Sqrt[ArcTan[a*x]^2]*Gamma[1/2, (-3*I)*ArcTan[a*x]] - 3*Sqrt[3 + 3*a^2*x^2]*ArcTan[a*x]^2*Gamma[1/2, (3*I)*ArcTan[a*x]] - 3*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*ArcTan[a*x]^2*Gamma[1/2, (3*I)*ArcTan[a*x]])/Sqrt[I*ArcTan[a*x]])/(6*a^3*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))","C",0
1105,1,261,222,0.8714634,"\int \frac{x}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]","\frac{4 \sqrt{6 \pi } \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x)^{3/2} \left(3 \sqrt{3} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)-S\left(\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right)\right)-12 \left(\left(2-4 a^2 x^2\right) \tan ^{-1}(a x)+a x\right)+7 \left(a^2 x^2+1\right)^{3/2} \tan ^{-1}(a x) \left(3 \sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)+3 \sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)+\sqrt{3} \left(\sqrt{-i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)+\sqrt{i \tan ^{-1}(a x)} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)\right)\right)}{18 a^2 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} 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,"(-12*(a*x + (2 - 4*a^2*x^2)*ArcTan[a*x]) + 4*Sqrt[6*Pi]*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^(3/2)*(3*Sqrt[3]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]] - FresnelS[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]]) + 7*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]*(3*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-I)*ArcTan[a*x]] + 3*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*ArcTan[a*x]] + Sqrt[3]*(Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])))/(18*a^2*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^(3/2))","C",0
1106,1,300,183,0.5329886,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]","\frac{-3 a^2 x^2 \sqrt{3 a^2 x^2+3} \left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)-3 a^2 x^2 \sqrt{3 a^2 x^2+3} \left(i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)-3 \left(a^2 x^2+1\right)^{3/2} \left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-i \tan ^{-1}(a x)\right)-3 \left(a^2 x^2+1\right)^{3/2} \left(i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},i \tan ^{-1}(a x)\right)-3 \sqrt{3 a^2 x^2+3} \left(-i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-3 i \tan ^{-1}(a x)\right)-3 \sqrt{3 a^2 x^2+3} \left(i \tan ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},3 i \tan ^{-1}(a x)\right)+24 a x \tan ^{-1}(a x)-4}{6 c^2 \left(a^3 x^2+a\right) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^{3/2}}","-\frac{\sqrt{2 \pi } \sqrt{a^2 x^2+1} C\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} C\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,"(-4 + 24*a*x*ArcTan[a*x] - 3*(1 + a^2*x^2)^(3/2)*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-I)*ArcTan[a*x]] - 3*(1 + a^2*x^2)^(3/2)*(I*ArcTan[a*x])^(3/2)*Gamma[1/2, I*ArcTan[a*x]] - 3*Sqrt[3 + 3*a^2*x^2]*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-3*I)*ArcTan[a*x]] - 3*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*((-I)*ArcTan[a*x])^(3/2)*Gamma[1/2, (-3*I)*ArcTan[a*x]] - 3*Sqrt[3 + 3*a^2*x^2]*(I*ArcTan[a*x])^(3/2)*Gamma[1/2, (3*I)*ArcTan[a*x]] - 3*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*(I*ArcTan[a*x])^(3/2)*Gamma[1/2, (3*I)*ArcTan[a*x]])/(6*c^2*(a + a^3*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2))","C",0
1107,0,0,287,12.9824027,"\int \frac{1}{x \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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{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{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{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,"Integrate[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x]","A",-1
1108,0,0,292,14.8472098,"\int \frac{1}{x^2 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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{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{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{20 \sqrt{2 \pi } a \sqrt{a^2 x^2+1} C\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} C\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{2}{3 a c x^2 \left(a^2 c x^2+c\right)^{3/2} \tan ^{-1}(a x)^{3/2}}+\frac{8}{3 a^2 c x^3 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x]","A",-1
1109,0,0,198,17.6674761,"\int \frac{1}{x^3 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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)+\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}+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{8}{c x^2 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}+\frac{4}{a^2 c x^4 \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}}",0,"Integrate[1/(x^3*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x]","A",-1
1110,0,0,206,33.1561277,"\int \frac{1}{x^4 \left(c+a^2 c x^2\right)^{5/2} \tan ^{-1}(a x)^{5/2}} \, dx","Integrate[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{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{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{16}{3 a^2 c x^5 \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{28}{3 c x^3 \left(a^2 c x^2+c\right)^{3/2} \sqrt{\tan ^{-1}(a x)}}",0,"Integrate[1/(x^4*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x]","A",-1
1111,0,0,46,0.9451066,"\int \frac{x \tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx","Integrate[(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,"Integrate[(x*ArcTan[a*x]^n)/(c + a^2*c*x^2), x]","A",-1
1112,1,20,20,0.0036207,"\int \frac{\tan ^{-1}(a x)^n}{c+a^2 c x^2} \, dx","Integrate[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
1113,0,0,31,0.7177834,"\int (f x)^m \left(d+c^2 d x^2\right)^q \left(a+b \tan ^{-1}(c x)\right)^p \, dx","Integrate[(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,"Integrate[(f*x)^m*(d + c^2*d*x^2)^q*(a + b*ArcTan[c*x])^p, x]","A",-1
1114,1,127,107,0.0068628,"\int x^3 \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)*(a + b*ArcTan[c*x]),x]","\frac{1}{4} a d x^4+\frac{1}{6} a e x^6+\frac{b e \tan ^{-1}(c x)}{6 c^6}-\frac{b e x}{6 c^5}-\frac{b d \tan ^{-1}(c x)}{4 c^4}+\frac{b d x}{4 c^3}+\frac{b e x^3}{18 c^3}+\frac{1}{4} b d x^4 \tan ^{-1}(c x)-\frac{b d x^3}{12 c}+\frac{1}{6} b e x^6 \tan ^{-1}(c x)-\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 \left(3 c^2 d-2 e\right) \tan ^{-1}(c x)}{12 c^6}+\frac{b x \left(3 c^2 d-2 e\right)}{12 c^5}-\frac{b x^3 \left(3 c^2 d-2 e\right)}{36 c^3}-\frac{b e x^5}{30 c}",1,"(b*d*x)/(4*c^3) - (b*e*x)/(6*c^5) - (b*d*x^3)/(12*c) + (b*e*x^3)/(18*c^3) + (a*d*x^4)/4 - (b*e*x^5)/(30*c) + (a*e*x^6)/6 - (b*d*ArcTan[c*x])/(4*c^4) + (b*e*ArcTan[c*x])/(6*c^6) + (b*d*x^4*ArcTan[c*x])/4 + (b*e*x^6*ArcTan[c*x])/6","A",1
1115,1,119,94,0.0236422,"\int x^2 \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)*(a + b*ArcTan[c*x]),x]","\frac{1}{3} a d x^3+\frac{1}{5} a e x^5+\frac{b e x^2}{10 c^3}-\frac{b e \log \left(c^2 x^2+1\right)}{10 c^5}+\frac{b d \log \left(c^2 x^2+1\right)}{6 c^3}+\frac{1}{3} b d x^3 \tan ^{-1}(c x)-\frac{b d x^2}{6 c}+\frac{1}{5} b e x^5 \tan ^{-1}(c x)-\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 \left(5 c^2 d-3 e\right) \log \left(c^2 x^2+1\right)}{30 c^5}-\frac{b x^2 \left(5 c^2 d-3 e\right)}{30 c^3}-\frac{b e x^4}{20 c}",1,"-1/6*(b*d*x^2)/c + (b*e*x^2)/(10*c^3) + (a*d*x^3)/3 - (b*e*x^4)/(20*c) + (a*e*x^5)/5 + (b*d*x^3*ArcTan[c*x])/3 + (b*e*x^5*ArcTan[c*x])/5 + (b*d*Log[1 + c^2*x^2])/(6*c^3) - (b*e*Log[1 + c^2*x^2])/(10*c^5)","A",1
1116,1,103,82,0.0046899,"\int x \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)*(a + b*ArcTan[c*x]),x]","\frac{1}{2} a d x^2+\frac{1}{4} a e x^4-\frac{b e \tan ^{-1}(c x)}{4 c^4}+\frac{b e x}{4 c^3}+\frac{b d \tan ^{-1}(c x)}{2 c^2}+\frac{1}{2} b d x^2 \tan ^{-1}(c x)-\frac{b d x}{2 c}+\frac{1}{4} b e x^4 \tan ^{-1}(c x)-\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 \left(c^2 d-e\right)^2 \tan ^{-1}(c x)}{4 c^4 e}-\frac{b x \left(2 c^2 d-e\right)}{4 c^3}-\frac{b e x^3}{12 c}",1,"-1/2*(b*d*x)/c + (b*e*x)/(4*c^3) + (a*d*x^2)/2 - (b*e*x^3)/(12*c) + (a*e*x^4)/4 + (b*d*ArcTan[c*x])/(2*c^2) - (b*e*ArcTan[c*x])/(4*c^4) + (b*d*x^2*ArcTan[c*x])/2 + (b*e*x^4*ArcTan[c*x])/4","A",1
1117,1,85,68,0.010438,"\int \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)*(a + b*ArcTan[c*x]),x]","a d x+\frac{1}{3} a e x^3-\frac{b d \log \left(c^2 x^2+1\right)}{2 c}+\frac{b e \log \left(c^2 x^2+1\right)}{6 c^3}+b d x \tan ^{-1}(c x)+\frac{1}{3} b e x^3 \tan ^{-1}(c x)-\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,"a*d*x - (b*e*x^2)/(6*c) + (a*e*x^3)/3 + b*d*x*ArcTan[c*x] + (b*e*x^3*ArcTan[c*x])/3 - (b*d*Log[1 + c^2*x^2])/(2*c) + (b*e*Log[1 + c^2*x^2])/(6*c^3)","A",1
1118,1,83,77,0.0047162,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)*(a + b*ArcTan[c*x]))/x,x]","a d \log (x)+\frac{1}{2} a e x^2+\frac{b e \tan ^{-1}(c x)}{2 c^2}+\frac{1}{2} i b d \text{Li}_2(-i c x)-\frac{1}{2} i b d \text{Li}_2(i c x)+\frac{1}{2} b e x^2 \tan ^{-1}(c x)-\frac{b e x}{2 c}","\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{1}{2} i b d \text{Li}_2(-i c x)-\frac{1}{2} i b d \text{Li}_2(i c x)-\frac{b e x}{2 c}",1,"-1/2*(b*e*x)/c + (a*e*x^2)/2 + (b*e*ArcTan[c*x])/(2*c^2) + (b*e*x^2*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",1
1119,1,73,57,0.0049993,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)*(a + b*ArcTan[c*x]))/x^2,x]","-\frac{a d}{x}+a e x-\frac{1}{2} b c d \log \left(c^2 x^2+1\right)-\frac{b e \log \left(c^2 x^2+1\right)}{2 c}+b c d \log (x)-\frac{b d \tan ^{-1}(c x)}{x}+b e x \tan ^{-1}(c 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,"-((a*d)/x) + a*e*x - (b*d*ArcTan[c*x])/x + b*e*x*ArcTan[c*x] + b*c*d*Log[x] - (b*c*d*Log[1 + c^2*x^2])/2 - (b*e*Log[1 + c^2*x^2])/(2*c)","A",1
1120,1,86,77,0.0054346,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)*(a + b*ArcTan[c*x]))/x^3,x]","-\frac{a d}{2 x^2}+a e \log (x)-\frac{b c d \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)}{2 x}-\frac{b d \tan ^{-1}(c x)}{2 x^2}+\frac{1}{2} i b e \text{Li}_2(-i c x)-\frac{1}{2} i b e \text{Li}_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{Li}_2(-i c x)-\frac{1}{2} i b e \text{Li}_2(i c x)",1,"-1/2*(a*d)/x^2 - (b*d*ArcTan[c*x])/(2*x^2) - (b*c*d*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/(2*x) + a*e*Log[x] + (I/2)*b*e*PolyLog[2, (-I)*c*x] - (I/2)*b*e*PolyLog[2, I*c*x]","C",0
1121,1,98,83,0.0350011,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)*(a + b*ArcTan[c*x]))/x^4,x]","-\frac{a d}{3 x^3}-\frac{a e}{x}+\frac{1}{6} b c d \left(c^2 \log \left(c^2 x^2+1\right)-2 c^2 \log (x)-\frac{1}{x^2}\right)-\frac{1}{2} b c e \log \left(c^2 x^2+1\right)-\frac{b d \tan ^{-1}(c x)}{3 x^3}+b c e \log (x)-\frac{b e \tan ^{-1}(c x)}{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}",1,"-1/3*(a*d)/x^3 - (a*e)/x - (b*d*ArcTan[c*x])/(3*x^3) - (b*e*ArcTan[c*x])/x + b*c*e*Log[x] - (b*c*e*Log[1 + c^2*x^2])/2 + (b*c*d*(-x^(-2) - 2*c^2*Log[x] + c^2*Log[1 + c^2*x^2]))/6","A",1
1122,1,97,82,0.0054857,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d + e*x^2)*(a + b*ArcTan[c*x]))/x^5,x]","-\frac{a d}{4 x^4}-\frac{a e}{2 x^2}-\frac{b c d \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)}{12 x^3}-\frac{b c e \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)}{2 x}-\frac{b d \tan ^{-1}(c x)}{4 x^4}-\frac{b e \tan ^{-1}(c x)}{2 x^2}","-\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,"-1/4*(a*d)/x^4 - (a*e)/(2*x^2) - (b*d*ArcTan[c*x])/(4*x^4) - (b*e*ArcTan[c*x])/(2*x^2) - (b*c*d*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)])/(12*x^3) - (b*c*e*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/(2*x)","C",1
1123,1,123,110,0.0475152,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)*(a + b*ArcTan[c*x]))/x^6,x]","-\frac{a d}{5 x^5}-\frac{a e}{3 x^3}+\frac{1}{6} b c e \left(c^2 \log \left(c^2 x^2+1\right)-2 c^2 \log (x)-\frac{1}{x^2}\right)+\frac{1}{10} b c d \left(2 c^4 \log (x)+\frac{c^2}{x^2}-c^4 \log \left(c^2 x^2+1\right)-\frac{1}{2 x^4}\right)-\frac{b d \tan ^{-1}(c x)}{5 x^5}-\frac{b e \tan ^{-1}(c x)}{3 x^3}","-\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,"-1/5*(a*d)/x^5 - (a*e)/(3*x^3) - (b*d*ArcTan[c*x])/(5*x^5) - (b*e*ArcTan[c*x])/(3*x^3) + (b*c*e*(-x^(-2) - 2*c^2*Log[x] + c^2*Log[1 + c^2*x^2]))/6 + (b*c*d*(-1/2*1/x^4 + c^2/x^2 + 2*c^4*Log[x] - c^4*Log[1 + c^2*x^2]))/10","A",1
1124,1,97,105,0.0064136,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{x^7} \, dx","Integrate[((d + e*x^2)*(a + b*ArcTan[c*x]))/x^7,x]","-\frac{a d}{6 x^6}-\frac{a e}{4 x^4}-\frac{b c d \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-c^2 x^2\right)}{30 x^5}-\frac{b c e \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)}{12 x^3}-\frac{b d \tan ^{-1}(c x)}{6 x^6}-\frac{b e \tan ^{-1}(c x)}{4 x^4}","-\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{1}{12} b c^4 \left(2 c^2 d-3 e\right) \tan ^{-1}(c x)-\frac{b c^3 \left(2 c^2 d-3 e\right)}{12 x}-\frac{b c d}{30 x^5}",1,"-1/6*(a*d)/x^6 - (a*e)/(4*x^4) - (b*d*ArcTan[c*x])/(6*x^6) - (b*e*ArcTan[c*x])/(4*x^4) - (b*c*d*Hypergeometric2F1[-5/2, 1, -3/2, -(c^2*x^2)])/(30*x^5) - (b*c*e*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)])/(12*x^3)","C",1
1125,1,174,185,0.1522883,"\int x^3 \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^2*(a + b*ArcTan[c*x]),x]","\frac{105 a c^8 x^4 \left(6 d^2+8 d e x^2+3 e^2 x^4\right)+105 b \tan ^{-1}(c x) \left(c^8 \left(6 d^2 x^4+8 d e x^6+3 e^2 x^8\right)-6 c^4 d^2+8 c^2 d e-3 e^2\right)+b c x \left(-3 c^6 \left(70 d^2 x^2+56 d e x^4+15 e^2 x^6\right)+7 c^4 \left(90 d^2+40 d e x^2+9 e^2 x^4\right)-105 c^2 e \left(8 d+e x^2\right)+315 e^2\right)}{2520 c^8}","\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 e x^5 \left(8 c^2 d-3 e\right)}{120 c^3}-\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 x \left(6 c^4 d^2-8 c^2 d e+3 e^2\right)}{24 c^7}-\frac{b x^3 \left(6 c^4 d^2-8 c^2 d e+3 e^2\right)}{72 c^5}-\frac{b e^2 x^7}{56 c}",1,"(105*a*c^8*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + b*c*x*(315*e^2 - 105*c^2*e*(8*d + e*x^2) + 7*c^4*(90*d^2 + 40*d*e*x^2 + 9*e^2*x^4) - 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)) + 105*b*(-6*c^4*d^2 + 8*c^2*d*e - 3*e^2 + c^8*(6*d^2*x^4 + 8*d*e*x^6 + 3*e^2*x^8))*ArcTan[c*x])/(2520*c^8)","A",1
1126,1,162,161,0.1297995,"\int x^2 \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)^2*(a + b*ArcTan[c*x]),x]","\frac{c^2 x^2 \left(4 a c^5 x \left(35 d^2+42 d e x^2+15 e^2 x^4\right)-2 b c^4 \left(35 d^2+21 d e x^2+5 e^2 x^4\right)+3 b c^2 e \left(28 d+5 e x^2\right)-30 b e^2\right)+4 b c^7 x^3 \tan ^{-1}(c x) \left(35 d^2+42 d e x^2+15 e^2 x^4\right)+2 b \left(35 c^4 d^2-42 c^2 d e+15 e^2\right) \log \left(c^2 x^2+1\right)}{420 c^7}","\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 e x^4 \left(14 c^2 d-5 e\right)}{140 c^3}+\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 x^2 \left(35 c^4 d^2-42 c^2 d e+15 e^2\right)}{210 c^5}-\frac{b e^2 x^6}{42 c}",1,"(c^2*x^2*(-30*b*e^2 + 3*b*c^2*e*(28*d + 5*e*x^2) - 2*b*c^4*(35*d^2 + 21*d*e*x^2 + 5*e^2*x^4) + 4*a*c^5*x*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)) + 4*b*c^7*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcTan[c*x] + 2*b*(35*c^4*d^2 - 42*c^2*d*e + 15*e^2)*Log[1 + c^2*x^2])/(420*c^7)","A",1
1127,1,140,115,0.103467,"\int x \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^2*(a + b*ArcTan[c*x]),x]","\frac{c x \left(15 a c^5 x \left(3 d^2+3 d e x^2+e^2 x^4\right)-3 b c^4 \left(15 d^2+5 d e x^2+e^2 x^4\right)+5 b c^2 e \left(9 d+e x^2\right)-15 b e^2\right)+15 b \tan ^{-1}(c x) \left(c^6 \left(3 d^2 x^2+3 d e x^4+e^2 x^6\right)+3 c^4 d^2-3 c^2 d e+e^2\right)}{90 c^6}","\frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{6 e}-\frac{b \left(c^2 d-e\right)^3 \tan ^{-1}(c x)}{6 c^6 e}-\frac{b e x^3 \left(3 c^2 d-e\right)}{18 c^3}-\frac{b x \left(3 c^4 d^2-3 c^2 d e+e^2\right)}{6 c^5}-\frac{b e^2 x^5}{30 c}",1,"(c*x*(-15*b*e^2 + 5*b*c^2*e*(9*d + e*x^2) + 15*a*c^5*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) - 3*b*c^4*(15*d^2 + 5*d*e*x^2 + e^2*x^4)) + 15*b*(3*c^4*d^2 - 3*c^2*d*e + e^2 + c^6*(3*d^2*x^2 + 3*d*e*x^4 + e^2*x^6))*ArcTan[c*x])/(90*c^6)","A",1
1128,1,130,124,0.1026031,"\int \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^2*(a + b*ArcTan[c*x]),x]","\frac{c^2 x \left(4 a c^3 \left(15 d^2+10 d e x^2+3 e^2 x^4\right)+b e x \left(6 e-c^2 \left(20 d+3 e x^2\right)\right)\right)+4 b c^5 x \tan ^{-1}(c x) \left(15 d^2+10 d e x^2+3 e^2 x^4\right)-2 b \left(15 c^4 d^2-10 c^2 d e+3 e^2\right) \log \left(c^2 x^2+1\right)}{60 c^5}","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 e x^2 \left(10 c^2 d-3 e\right)}{30 c^3}-\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^2 x^4}{20 c}",1,"(c^2*x*(4*a*c^3*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + b*e*x*(6*e - c^2*(20*d + 3*e*x^2))) + 4*b*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcTan[c*x] - 2*b*(15*c^4*d^2 - 10*c^2*d*e + 3*e^2)*Log[1 + c^2*x^2])/(60*c^5)","A",1
1129,1,123,137,0.109631,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x,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 \left(c x-\tan ^{-1}(c x)\right)}{c^2}-\frac{b e^2 \left(c^3 x^3-3 c x+3 \tan ^{-1}(c x)\right)}{12 c^4}+\frac{1}{2} i b d^2 \text{Li}_2(-i c x)-\frac{1}{2} i b d^2 \text{Li}_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 e^2 \tan ^{-1}(c x)}{4 c^4}+\frac{b e^2 x}{4 c^3}+\frac{b d e \tan ^{-1}(c x)}{c^2}+\frac{1}{2} i b d^2 \text{Li}_2(-i c x)-\frac{1}{2} i b d^2 \text{Li}_2(i c x)-\frac{b d e x}{c}-\frac{b e^2 x^3}{12 c}",1,"-((b*d*e*(c*x - ArcTan[c*x]))/c^2) - (b*e^2*(-3*c*x + c^3*x^3 + 3*ArcTan[c*x]))/(12*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",1
1130,1,114,109,0.1139774,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^2,x]","\frac{1}{6} \left(-\frac{6 a d^2}{x}+12 a d e x+2 a e^2 x^3+\frac{b \left(-3 c^4 d^2-6 c^2 d e+e^2\right) \log \left(c^2 x^2+1\right)}{c^3}+\frac{2 b \tan ^{-1}(c x) \left(-3 d^2+6 d e x^2+e^2 x^4\right)}{x}+6 b c d^2 \log (x)-\frac{b e^2 x^2}{c}\right)","-\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,"((-6*a*d^2)/x + 12*a*d*e*x - (b*e^2*x^2)/c + 2*a*e^2*x^3 + (2*b*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)*ArcTan[c*x])/x + 6*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])/c^3)/6","A",1
1131,1,118,128,0.1151925,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^3,x]","\frac{1}{2} \left(-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{x^2}+e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)+4 a d e \log (x)-\frac{b c d^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)}{x}-\frac{b e^2 \left(c x-\tan ^{-1}(c x)\right)}{c^2}+2 i b d e \text{Li}_2(-i c x)-2 i b d e \text{Li}_2(i c x)\right)","-\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}+i b d e \text{Li}_2(-i c x)-i b d e \text{Li}_2(i c x)-\frac{b e^2 x}{2 c}",1,"(-((b*e^2*(c*x - ArcTan[c*x]))/c^2) - (d^2*(a + b*ArcTan[c*x]))/x^2 + e^2*x^2*(a + b*ArcTan[c*x]) - (b*c*d^2*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/x + 4*a*d*e*Log[x] + (2*I)*b*d*e*PolyLog[2, (-I)*c*x] - (2*I)*b*d*e*PolyLog[2, I*c*x])/2","C",0
1132,1,119,115,0.1248276,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^4,x]","\frac{1}{6} \left(-\frac{2 a d^2}{x^3}-\frac{12 a d e}{x}+6 a e^2 x-2 b c d \log (x) \left(c^2 d-6 e\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)}{c}-\frac{2 b \tan ^{-1}(c x) \left(d^2+6 d e x^2-3 e^2 x^4\right)}{x^3}-\frac{b c d^2}{x^2}\right)","-\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{1}{3} b c d \log (x) \left(c^2 d-6 e\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{b c d^2}{6 x^2}",1,"((-2*a*d^2)/x^3 - (b*c*d^2)/x^2 - (12*a*d*e)/x + 6*a*e^2*x - (2*b*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)*ArcTan[c*x])/x^3 - 2*b*c*d*(c^2*d - 6*e)*Log[x] + (b*(c^4*d^2 - 6*c^2*d*e - 3*e^2)*Log[1 + c^2*x^2])/c)/6","A",1
1133,1,130,139,0.1063641,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^5,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 d^2 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)}{12 x^3}-\frac{b c d e \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)}{x}+\frac{1}{2} i b e^2 \text{Li}_2(-i c x)-\frac{1}{2} i b e^2 \text{Li}_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{1}{4} b c^4 d^2 \tan ^{-1}(c x)+\frac{b c^3 d^2}{4 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{Li}_2(-i c x)-\frac{1}{2} i b e^2 \text{Li}_2(i c x)",1,"-1/4*(d^2*(a + b*ArcTan[c*x]))/x^4 - (d*e*(a + b*ArcTan[c*x]))/x^2 - (b*c*d^2*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)])/(12*x^3) - (b*c*d*e*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/x + 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]","C",0
1134,1,149,150,0.1588954,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^6,x]","\frac{1}{60} \left(-\frac{12 d^2 \left(a+b \tan ^{-1}(c x)\right)}{x^5}-\frac{40 d e \left(a+b \tan ^{-1}(c x)\right)}{x^3}-\frac{60 e^2 \left(a+b \tan ^{-1}(c x)\right)}{x}-20 b c d e \left(-c^2 \log \left(c^2 x^2+1\right)+2 c^2 \log (x)+\frac{1}{x^2}\right)+30 b c e^2 \left(2 \log (x)-\log \left(c^2 x^2+1\right)\right)-3 b c d^2 \left(-4 c^4 \log (x)-\frac{2 c^2}{x^2}+2 c^4 \log \left(c^2 x^2+1\right)+\frac{1}{x^4}\right)\right)","-\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{b c d \left(3 c^2 d-10 e\right)}{30 x^2}-\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^2}{20 x^4}",1,"((-12*d^2*(a + b*ArcTan[c*x]))/x^5 - (40*d*e*(a + b*ArcTan[c*x]))/x^3 - (60*e^2*(a + b*ArcTan[c*x]))/x + 30*b*c*e^2*(2*Log[x] - Log[1 + c^2*x^2]) - 20*b*c*d*e*(x^(-2) + 2*c^2*Log[x] - c^2*Log[1 + c^2*x^2]) - 3*b*c*d^2*(x^(-4) - (2*c^2)/x^2 - 4*c^4*Log[x] + 2*c^4*Log[1 + c^2*x^2]))/60","A",1
1135,1,112,111,0.10596,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^7} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^7,x]","-\frac{5 \left(\left(d^2+3 d e x^2+3 e^2 x^4\right) \left(a+b \tan ^{-1}(c x)\right)+b c d e x^3 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)+3 b c e^2 x^5 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)\right)+b c d^2 x \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-c^2 x^2\right)}{30 x^6}","-\frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{6 d x^6}+\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 \left(c^4 d^2-3 c^2 d e+3 e^2\right)}{6 x}-\frac{b c d^2}{30 x^5}",1,"-1/30*(b*c*d^2*x*Hypergeometric2F1[-5/2, 1, -3/2, -(c^2*x^2)] + 5*((d^2 + 3*d*e*x^2 + 3*e^2*x^4)*(a + b*ArcTan[c*x]) + b*c*d*e*x^3*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)] + 3*b*c*e^2*x^5*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)]))/x^6","C",1
1136,1,177,186,0.2290214,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^8,x]","\frac{1}{420} \left(-\frac{60 d^2 \left(a+b \tan ^{-1}(c x)\right)}{x^7}-\frac{168 d e \left(a+b \tan ^{-1}(c x)\right)}{x^5}-\frac{140 e^2 \left(a+b \tan ^{-1}(c x)\right)}{x^3}-70 b c e^2 \left(-c^2 \log \left(c^2 x^2+1\right)+2 c^2 \log (x)+\frac{1}{x^2}\right)-42 b c d e \left(-4 c^4 \log (x)-\frac{2 c^2}{x^2}+2 c^4 \log \left(c^2 x^2+1\right)+\frac{1}{x^4}\right)-5 b c d^2 \left(12 c^6 \log (x)-6 c^6 \log \left(c^2 x^2+1\right)+\frac{6 c^4 x^4-3 c^2 x^2+2}{x^6}\right)\right)","-\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 d \left(5 c^2 d-14 e\right)}{140 x^4}-\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^2}{42 x^6}",1,"((-60*d^2*(a + b*ArcTan[c*x]))/x^7 - (168*d*e*(a + b*ArcTan[c*x]))/x^5 - (140*e^2*(a + b*ArcTan[c*x]))/x^3 - 70*b*c*e^2*(x^(-2) + 2*c^2*Log[x] - c^2*Log[1 + c^2*x^2]) - 42*b*c*d*e*(x^(-4) - (2*c^2)/x^2 - 4*c^4*Log[x] + 2*c^4*Log[1 + c^2*x^2]) - 5*b*c*d^2*((2 - 3*c^2*x^2 + 6*c^4*x^4)/x^6 + 12*c^6*Log[x] - 6*c^6*Log[1 + c^2*x^2]))/420","A",1
1137,1,262,240,0.2445705,"\int x^3 \left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]","\frac{c x \left(315 a c^9 x^3 \left(10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right)-b \left(5 c^8 \left(210 d^3 x^2+252 d^2 e x^4+135 d e^2 x^6+28 e^3 x^8\right)-15 c^6 \left(210 d^3+140 d^2 e x^2+63 d e^2 x^4+12 e^3 x^6\right)+63 c^4 e \left(100 d^2+25 d e x^2+4 e^2 x^4\right)-105 c^2 e^2 \left(45 d+4 e x^2\right)+1260 e^3\right)\right)+315 b \tan ^{-1}(c x) \left(c^{10} x^4 \left(10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right)-10 c^6 d^3+20 c^4 d^2 e-15 c^2 d e^2+4 e^3\right)}{12600 c^{10}}","\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 \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^2 x^7 \left(15 c^2 d-4 e\right)}{280 c^3}-\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 \left(10 c^6 d^3-20 c^4 d^2 e+15 c^2 d e^2-4 e^3\right)}{40 c^9}-\frac{b x^3 \left(10 c^6 d^3-20 c^4 d^2 e+15 c^2 d e^2-4 e^3\right)}{120 c^7}-\frac{b e^3 x^9}{90 c}",1,"(c*x*(315*a*c^9*x^3*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6) - b*(1260*e^3 - 105*c^2*e^2*(45*d + 4*e*x^2) + 63*c^4*e*(100*d^2 + 25*d*e*x^2 + 4*e^2*x^4) - 15*c^6*(210*d^3 + 140*d^2*e*x^2 + 63*d*e^2*x^4 + 12*e^3*x^6) + 5*c^8*(210*d^3*x^2 + 252*d^2*e*x^4 + 135*d*e^2*x^6 + 28*e^3*x^8))) + 315*b*(-10*c^6*d^3 + 20*c^4*d^2*e - 15*c^2*d*e^2 + 4*e^3 + c^10*x^4*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6))*ArcTan[c*x])/(12600*c^10)","A",1
1138,1,252,239,0.2235821,"\int x^2 \left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]","\frac{1}{3} d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{3}{5} d^2 e x^5 \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{1}{6} b d^3 \left(\frac{x^2}{c}-\frac{\log \left(c^2 x^2+1\right)}{c^3}\right)+\frac{3}{20} b d^2 e \left(\frac{2 x^2}{c^3}-\frac{2 \log \left(c^2 x^2+1\right)}{c^5}-\frac{x^4}{c}\right)-\frac{1}{28} b d e^2 \left(\frac{6 x^2}{c^5}-\frac{3 x^4}{c^3}-\frac{6 \log \left(c^2 x^2+1\right)}{c^7}+\frac{2 x^6}{c}\right)+\frac{1}{216} b e^3 \left(\frac{12 x^2}{c^7}-\frac{6 x^4}{c^5}+\frac{4 x^6}{c^3}-\frac{12 \log \left(c^2 x^2+1\right)}{c^9}-\frac{3 x^8}{c}\right)","\frac{1}{3} d^3 x^3 \left(a+b \tan ^{-1}(c x)\right)+\frac{3}{5} d^2 e x^5 \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^2 x^6 \left(27 c^2 d-7 e\right)}{378 c^3}-\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 \left(105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right) \log \left(c^2 x^2+1\right)}{630 c^9}-\frac{b x^2 \left(105 c^6 d^3-189 c^4 d^2 e+135 c^2 d e^2-35 e^3\right)}{630 c^7}-\frac{b e^3 x^8}{72 c}",1,"(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*e^3*((12*x^2)/c^7 - (6*x^4)/c^5 + (4*x^6)/c^3 - (3*x^8)/c - (12*Log[1 + c^2*x^2])/c^9))/216 - (b*d*e^2*((6*x^2)/c^5 - (3*x^4)/c^3 + (2*x^6)/c - (6*Log[1 + c^2*x^2])/c^7))/28 + (3*b*d^2*e*((2*x^2)/c^3 - x^4/c - (2*Log[1 + c^2*x^2])/c^5))/20 - (b*d^3*(x^2/c - Log[1 + c^2*x^2]/c^3))/6","A",1
1139,1,217,158,0.1701959,"\int x \left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]","\frac{c x \left(105 a c^7 x \left(4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right)-3 b c^6 \left(140 d^3+70 d^2 e x^2+28 d e^2 x^4+5 e^3 x^6\right)+7 b c^4 e \left(90 d^2+20 d e x^2+3 e^2 x^4\right)-35 b c^2 e^2 \left(12 d+e x^2\right)+105 b e^3\right)+105 b \tan ^{-1}(c x) \left(c^8 \left(4 d^3 x^2+6 d^2 e x^4+4 d e^2 x^6+e^3 x^8\right)+4 c^6 d^3-6 c^4 d^2 e+4 c^2 d e^2-e^3\right)}{840 c^8}","\frac{\left(d+e x^2\right)^4 \left(a+b \tan ^{-1}(c x)\right)}{8 e}-\frac{b \left(c^2 d-e\right)^4 \tan ^{-1}(c x)}{8 c^8 e}-\frac{b e^2 x^5 \left(4 c^2 d-e\right)}{40 c^3}-\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 x^3 \left(6 c^4 d^2-4 c^2 d e+e^2\right)}{24 c^5}-\frac{b e^3 x^7}{56 c}",1,"(c*x*(105*b*e^3 - 35*b*c^2*e^2*(12*d + e*x^2) + 7*b*c^4*e*(90*d^2 + 20*d*e*x^2 + 3*e^2*x^4) + 105*a*c^7*x*(4*d^3 + 6*d^2*e*x^2 + 4*d*e^2*x^4 + e^3*x^6) - 3*b*c^6*(140*d^3 + 70*d^2*e*x^2 + 28*d*e^2*x^4 + 5*e^3*x^6)) + 105*b*(4*c^6*d^3 - 6*c^4*d^2*e + 4*c^2*d*e^2 - e^3 + c^8*(4*d^3*x^2 + 6*d^2*e*x^4 + 4*d*e^2*x^6 + e^3*x^8))*ArcTan[c*x])/(840*c^8)","A",1
1140,1,192,188,0.1547154,"\int \left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]","\frac{c^2 x \left(12 a c^5 \left(35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right)-b e x \left(c^4 \left(210 d^2+63 d e x^2+10 e^2 x^4\right)-3 c^2 e \left(42 d+5 e x^2\right)+30 e^2\right)\right)+12 b c^7 x \tan ^{-1}(c x) \left(35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right)-6 b \left(35 c^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3\right) \log \left(c^2 x^2+1\right)}{420 c^7}","d^3 x \left(a+b \tan ^{-1}(c x)\right)+d^2 e x^3 \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^2 x^4 \left(21 c^2 d-5 e\right)}{140 c^3}-\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^6 d^3-35 c^4 d^2 e+21 c^2 d e^2-5 e^3\right) \log \left(c^2 x^2+1\right)}{70 c^7}-\frac{b e^3 x^6}{42 c}",1,"(c^2*x*(12*a*c^5*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6) - b*e*x*(30*e^2 - 3*c^2*e*(42*d + 5*e*x^2) + c^4*(210*d^2 + 63*d*e*x^2 + 10*e^2*x^4))) + 12*b*c^7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6)*ArcTan[c*x] - 6*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])/(420*c^7)","A",1
1141,1,190,228,0.1774351,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x,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 \left(c x-\tan ^{-1}(c x)\right)}{2 c^2}-\frac{b d e^2 \left(c^3 x^3-3 c x+3 \tan ^{-1}(c x)\right)}{4 c^4}-\frac{b e^3 \left(3 c^5 x^5-5 c^3 x^3+15 c x-15 \tan ^{-1}(c x)\right)}{90 c^6}+\frac{1}{2} i b d^3 \text{Li}_2(-i c x)-\frac{1}{2} i b d^3 \text{Li}_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{b e^3 \tan ^{-1}(c x)}{6 c^6}-\frac{b e^3 x}{6 c^5}-\frac{3 b d e^2 \tan ^{-1}(c x)}{4 c^4}+\frac{3 b d e^2 x}{4 c^3}+\frac{b e^3 x^3}{18 c^3}+\frac{3 b d^2 e \tan ^{-1}(c x)}{2 c^2}+\frac{1}{2} i b d^3 \text{Li}_2(-i c x)-\frac{1}{2} i b d^3 \text{Li}_2(i c x)-\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,"-1/90*(b*e^3*(15*c*x - 5*c^3*x^3 + 3*c^5*x^5 - 15*ArcTan[c*x]))/c^6 - (3*b*d^2*e*(c*x - ArcTan[c*x]))/(2*c^2) - (b*d*e^2*(-3*c*x + c^3*x^3 + 3*ArcTan[c*x]))/(4*c^4) + (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",1
1142,1,169,160,0.1598859,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^2,x]","\frac{1}{20} \left(-\frac{20 a d^3}{x}+60 a d^2 e x+20 a d e^2 x^3+4 a e^3 x^5+\frac{2 b e^2 x^2 \left(e-5 c^2 d\right)}{c^3}-\frac{2 b \left(5 c^6 d^3+15 c^4 d^2 e-5 c^2 d e^2+e^3\right) \log \left(c^2 x^2+1\right)}{c^5}+20 b c d^3 \log (x)+\frac{4 b \tan ^{-1}(c x) \left(-5 d^3+15 d^2 e x^2+5 d e^2 x^4+e^3 x^6\right)}{x}-\frac{b e^3 x^4}{c}\right)","-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{x}+3 d^2 e x \left(a+b \tan ^{-1}(c x)\right)+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 e^2 x^2 \left(5 c^2 d-e\right)}{10 c^3}-\frac{b \left(5 c^6 d^3+15 c^4 d^2 e-5 c^2 d e^2+e^3\right) \log \left(c^2 x^2+1\right)}{10 c^5}+b c d^3 \log (x)-\frac{b e^3 x^4}{20 c}",1,"((-20*a*d^3)/x + 60*a*d^2*e*x + (2*b*e^2*(-5*c^2*d + e)*x^2)/c^3 + 20*a*d*e^2*x^3 - (b*e^3*x^4)/c + 4*a*e^3*x^5 + (4*b*(-5*d^3 + 15*d^2*e*x^2 + 5*d*e^2*x^4 + e^3*x^6)*ArcTan[c*x])/x + 20*b*c*d^3*Log[x] - (2*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])/c^5)/20","A",1
1143,1,170,200,0.1635098,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^3,x]","\frac{1}{12} \left(-\frac{6 d^3 \left(a+b \tan ^{-1}(c x)\right)}{x^2}+18 d e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)+3 e^3 x^4 \left(a+b \tan ^{-1}(c x)\right)+36 a d^2 e \log (x)-\frac{6 b c d^3 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)}{x}-\frac{18 b d e^2 \left(c x-\tan ^{-1}(c x)\right)}{c^2}-\frac{b e^3 \left(c^3 x^3-3 c x+3 \tan ^{-1}(c x)\right)}{c^4}+18 i b d^2 e \text{Li}_2(-i c x)-18 i b d^2 e \text{Li}_2(i c x)\right)","-\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{b e^3 \tan ^{-1}(c x)}{4 c^4}+\frac{b e^3 x}{4 c^3}-\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 c d^3}{2 x}+\frac{3}{2} i b d^2 e \text{Li}_2(-i c x)-\frac{3}{2} i b d^2 e \text{Li}_2(i c x)-\frac{3 b d e^2 x}{2 c}-\frac{b e^3 x^3}{12 c}",1,"((-18*b*d*e^2*(c*x - ArcTan[c*x]))/c^2 - (b*e^3*(-3*c*x + c^3*x^3 + 3*ArcTan[c*x]))/c^4 - (6*d^3*(a + b*ArcTan[c*x]))/x^2 + 18*d*e^2*x^2*(a + b*ArcTan[c*x]) + 3*e^3*x^4*(a + b*ArcTan[c*x]) - (6*b*c*d^3*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/x + 36*a*d^2*e*Log[x] + (18*I)*b*d^2*e*PolyLog[2, (-I)*c*x] - (18*I)*b*d^2*e*PolyLog[2, I*c*x])/12","C",0
1144,1,166,158,0.1709797,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^4,x]","\frac{1}{6} \left(-\frac{2 a d^3}{x^3}-\frac{18 a d^2 e}{x}+18 a d e^2 x+2 a e^3 x^3-2 b c d^2 \log (x) \left(c^2 d-9 e\right)+\frac{b \left(c^6 d^3-9 c^4 d^2 e-9 c^2 d e^2+e^3\right) \log \left(c^2 x^2+1\right)}{c^3}-\frac{b c d^3}{x^2}+\frac{2 b \tan ^{-1}(c x) \left(-d^3-9 d^2 e x^2+9 d e^2 x^4+e^3 x^6\right)}{x^3}-\frac{b e^3 x^2}{c}\right)","-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{3 x^3}-\frac{3 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{x}+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{1}{3} b c d^2 \log (x) \left(c^2 d-9 e\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{b c d^3}{6 x^2}-\frac{b e^3 x^2}{6 c}",1,"((-2*a*d^3)/x^3 - (b*c*d^3)/x^2 - (18*a*d^2*e)/x + 18*a*d*e^2*x - (b*e^3*x^2)/c + 2*a*e^3*x^3 + (2*b*(-d^3 - 9*d^2*e*x^2 + 9*d*e^2*x^4 + e^3*x^6)*ArcTan[c*x])/x^3 - 2*b*c*d^2*(c^2*d - 9*e)*Log[x] + (b*(c^6*d^3 - 9*c^4*d^2*e - 9*c^2*d*e^2 + e^3)*Log[1 + c^2*x^2])/c^3)/6","A",1
1145,1,169,200,0.2203494,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^5,x]","\frac{1}{12} \left(-\frac{3 d^3 \left(a+b \tan ^{-1}(c x)\right)}{x^4}-\frac{18 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{x^2}+6 e^3 x^2 \left(a+b \tan ^{-1}(c x)\right)+36 a d e^2 \log (x)-\frac{b c d^3 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)}{x^3}-\frac{18 b c d^2 e \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)}{x}-\frac{6 b e^3 \left(c x-\tan ^{-1}(c x)\right)}{c^2}+18 i b d e^2 \text{Li}_2(-i c x)-18 i b d e^2 \text{Li}_2(i c x)\right)","-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{3 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}+\frac{1}{2} e^3 x^2 \left(a+b \tan ^{-1}(c x)\right)+3 a d e^2 \log (x)+\frac{1}{4} b c^4 d^3 \tan ^{-1}(c x)+\frac{b c^3 d^3}{4 x}-\frac{3}{2} b c^2 d^2 e \tan ^{-1}(c x)+\frac{b e^3 \tan ^{-1}(c x)}{2 c^2}-\frac{b c d^3}{12 x^3}-\frac{3 b c d^2 e}{2 x}+\frac{3}{2} i b d e^2 \text{Li}_2(-i c x)-\frac{3}{2} i b d e^2 \text{Li}_2(i c x)-\frac{b e^3 x}{2 c}",1,"((-6*b*e^3*(c*x - ArcTan[c*x]))/c^2 - (3*d^3*(a + b*ArcTan[c*x]))/x^4 - (18*d^2*e*(a + b*ArcTan[c*x]))/x^2 + 6*e^3*x^2*(a + b*ArcTan[c*x]) - (b*c*d^3*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)])/x^3 - (18*b*c*d^2*e*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/x + 36*a*d*e^2*Log[x] + (18*I)*b*d*e^2*PolyLog[2, (-I)*c*x] - (18*I)*b*d*e^2*PolyLog[2, I*c*x])/12","C",0
1146,1,184,177,0.1873245,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^6,x]","\frac{1}{20} \left(-\frac{4 a d^3}{x^5}-\frac{20 a d^2 e}{x^3}-\frac{60 a d e^2}{x}+20 a e^3 x+\frac{2 b c d^2 \left(c^2 d-5 e\right)}{x^2}+4 b c d \log (x) \left(c^4 d^2-5 c^2 d e+15 e^2\right)-\frac{2 b \left(c^6 d^3-5 c^4 d^2 e+15 c^2 d e^2+5 e^3\right) \log \left(c^2 x^2+1\right)}{c}-\frac{b c d^3}{x^4}-\frac{4 b \tan ^{-1}(c x) \left(d^3+5 d^2 e x^2+15 d e^2 x^4-5 e^3 x^6\right)}{x^5}\right)","-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\frac{d^2 e \left(a+b \tan ^{-1}(c x)\right)}{x^3}-\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 c d^2 \left(c^2 d-5 e\right)}{10 x^2}+\frac{1}{5} b c d \log (x) \left(c^4 d^2-5 c^2 d e+15 e^2\right)-\frac{b \left(c^6 d^3-5 c^4 d^2 e+15 c^2 d e^2+5 e^3\right) \log \left(c^2 x^2+1\right)}{10 c}-\frac{b c d^3}{20 x^4}",1,"((-4*a*d^3)/x^5 - (b*c*d^3)/x^4 - (20*a*d^2*e)/x^3 + (2*b*c*d^2*(c^2*d - 5*e))/x^2 - (60*a*d*e^2)/x + 20*a*e^3*x - (4*b*(d^3 + 5*d^2*e*x^2 + 15*d*e^2*x^4 - 5*e^3*x^6)*ArcTan[c*x])/x^5 + 4*b*c*d*(c^4*d^2 - 5*c^2*d*e + 15*e^2)*Log[x] - (2*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])/c)/20","A",1
1147,1,175,228,0.1453382,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^7} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^7,x]","\frac{1}{60} \left(-\frac{10 d^3 \left(a+b \tan ^{-1}(c x)\right)}{x^6}-\frac{45 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{x^4}-\frac{90 d e^2 \left(a+b \tan ^{-1}(c x)\right)}{x^2}+60 a e^3 \log (x)-\frac{2 b c d^3 \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-c^2 x^2\right)}{x^5}-\frac{15 b c d^2 e \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)}{x^3}-\frac{90 b c d e^2 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)}{x}+30 i b e^3 \text{Li}_2(-i c x)-30 i b e^3 \text{Li}_2(i c x)\right)","-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{6 x^6}-\frac{3 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{4 x^4}-\frac{3 d e^2 \left(a+b \tan ^{-1}(c x)\right)}{2 x^2}+a e^3 \log (x)-\frac{1}{6} b c^6 d^3 \tan ^{-1}(c x)-\frac{b c^5 d^3}{6 x}+\frac{3}{4} b c^4 d^2 e \tan ^{-1}(c x)+\frac{b c^3 d^3}{18 x^3}+\frac{3 b c^3 d^2 e}{4 x}-\frac{3}{2} b c^2 d e^2 \tan ^{-1}(c x)-\frac{b c d^3}{30 x^5}-\frac{b c d^2 e}{4 x^3}-\frac{3 b c d e^2}{2 x}+\frac{1}{2} i b e^3 \text{Li}_2(-i c x)-\frac{1}{2} i b e^3 \text{Li}_2(i c x)",1,"((-10*d^3*(a + b*ArcTan[c*x]))/x^6 - (45*d^2*e*(a + b*ArcTan[c*x]))/x^4 - (90*d*e^2*(a + b*ArcTan[c*x]))/x^2 - (2*b*c*d^3*Hypergeometric2F1[-5/2, 1, -3/2, -(c^2*x^2)])/x^5 - (15*b*c*d^2*e*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)])/x^3 - (90*b*c*d*e^2*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/x + 60*a*e^3*Log[x] + (30*I)*b*e^3*PolyLog[2, (-I)*c*x] - (30*I)*b*e^3*PolyLog[2, I*c*x])/60","C",0
1148,1,230,224,0.1842105,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^8,x]","-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{7 x^7}-\frac{3 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\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{1}{2} b c d e^2 \left(-c^2 \log \left(c^2 x^2+1\right)+2 c^2 \log (x)+\frac{1}{x^2}\right)+\frac{1}{2} b c e^3 \left(2 \log (x)-\log \left(c^2 x^2+1\right)\right)-\frac{3}{20} b c d^2 e \left(-4 c^4 \log (x)-\frac{2 c^2}{x^2}+2 c^4 \log \left(c^2 x^2+1\right)+\frac{1}{x^4}\right)-\frac{1}{84} b c d^3 \left(12 c^6 \log (x)+\frac{6 c^4}{x^2}-\frac{3 c^2}{x^4}-6 c^6 \log \left(c^2 x^2+1\right)+\frac{2}{x^6}\right)","-\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{7 x^7}-\frac{3 d^2 e \left(a+b \tan ^{-1}(c x)\right)}{5 x^5}-\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^2 \left(5 c^2 d-21 e\right)}{140 x^4}-\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(5 c^6 d^3-21 c^4 d^2 e+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(5 c^6 d^3-21 c^4 d^2 e+35 c^2 d e^2-35 e^3\right)-\frac{b c d^3}{42 x^6}",1,"-1/7*(d^3*(a + b*ArcTan[c*x]))/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*e^3*(2*Log[x] - Log[1 + c^2*x^2]))/2 - (b*c*d*e^2*(x^(-2) + 2*c^2*Log[x] - c^2*Log[1 + c^2*x^2]))/2 - (3*b*c*d^2*e*(x^(-4) - (2*c^2)/x^2 - 4*c^4*Log[x] + 2*c^4*Log[1 + c^2*x^2]))/20 - (b*c*d^3*(2/x^6 - (3*c^2)/x^4 + (6*c^4)/x^2 + 12*c^6*Log[x] - 6*c^6*Log[1 + c^2*x^2]))/84","A",1
1149,1,154,152,0.1871018,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right)}{x^9} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcTan[c*x]))/x^9,x]","-\frac{35 \left(\left(d^3+4 d^2 e x^2+6 d e^2 x^4+4 e^3 x^6\right) \left(a+b \tan ^{-1}(c x)\right)+2 b c d e^2 x^5 \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-c^2 x^2\right)+4 b c e^3 x^7 \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)\right)+5 b c d^3 x \, _2F_1\left(-\frac{7}{2},1;-\frac{5}{2};-c^2 x^2\right)+28 b c d^2 e x^3 \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};-c^2 x^2\right)}{280 x^8}","-\frac{\left(d+e x^2\right)^4 \left(a+b \tan ^{-1}(c x)\right)}{8 d x^8}+\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 \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^3}{56 x^7}",1,"-1/280*(5*b*c*d^3*x*Hypergeometric2F1[-7/2, 1, -5/2, -(c^2*x^2)] + 28*b*c*d^2*e*x^3*Hypergeometric2F1[-5/2, 1, -3/2, -(c^2*x^2)] + 35*((d^3 + 4*d^2*e*x^2 + 6*d*e^2*x^4 + 4*e^3*x^6)*(a + b*ArcTan[c*x]) + 2*b*c*d*e^2*x^5*Hypergeometric2F1[-3/2, 1, -1/2, -(c^2*x^2)] + 4*b*c*e^3*x^7*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)]))/x^8","C",1
1150,1,212,244,0.1870276,"\int \left(c+d x^2\right)^4 \tan ^{-1}(a x) \, dx","Integrate[(c + d*x^2)^4*ArcTan[a*x],x]","-\frac{-24 a^9 x \tan ^{-1}(a x) \left(315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right)+a^2 d x^2 \left(3 a^6 \left(1680 c^3+756 c^2 d x^2+240 c d^2 x^4+35 d^3 x^6\right)-4 a^4 d \left(1134 c^2+270 c d x^2+35 d^2 x^4\right)+30 a^2 d^2 \left(72 c+7 d x^2\right)-420 d^3\right)+12 \left(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\right) \log \left(a^2 x^2+1\right)}{7560 a^9}","-\frac{d^3 x^6 \left(36 a^2 c-7 d\right)}{378 a^3}-\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(420 a^6 c^3-378 a^4 c^2 d+180 a^2 c d^2-35 d^3\right)}{630 a^7}-\frac{\left(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\right) \log \left(a^2 x^2+1\right)}{630 a^9}+c^4 x \tan ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \tan ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \tan ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \tan ^{-1}(a x)+\frac{1}{9} d^4 x^9 \tan ^{-1}(a x)-\frac{d^4 x^8}{72 a}",1,"-1/7560*(a^2*d*x^2*(-420*d^3 + 30*a^2*d^2*(72*c + 7*d*x^2) - 4*a^4*d*(1134*c^2 + 270*c*d*x^2 + 35*d^2*x^4) + 3*a^6*(1680*c^3 + 756*c^2*d*x^2 + 240*c*d^2*x^4 + 35*d^3*x^6)) - 24*a^9*x*(315*c^4 + 420*c^3*d*x^2 + 378*c^2*d^2*x^4 + 180*c*d^3*x^6 + 35*d^4*x^8)*ArcTan[a*x] + 12*(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])/a^9","A",1
1151,1,503,361,0.2744517,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x]))/(d + e*x^2),x]","-\frac{a d \log \left(d+e x^2\right)}{2 e^2}+\frac{a x^2}{2 e}+\frac{b \tan ^{-1}(c x)}{2 c^2 e}-\frac{i b d \text{Li}_2\left(-\frac{\sqrt{e} (1-i c x)}{i c \sqrt{-d}-\sqrt{e}}\right)}{4 e^2}-\frac{i b d \text{Li}_2\left(\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 e^2}+\frac{i b d \text{Li}_2\left(-\frac{\sqrt{e} (i c x+1)}{i c \sqrt{-d}-\sqrt{e}}\right)}{4 e^2}+\frac{i b d \text{Li}_2\left(\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 e^2}+\frac{i b 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^2}-\frac{i b 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^2}-\frac{i b 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^2}+\frac{i b 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^2}+\frac{b x^2 \tan ^{-1}(c x)}{2 e}-\frac{b x}{2 c e}","\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{i b d \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 e^2}+\frac{i b d \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{4 e^2}+\frac{i b d \text{Li}_2\left(1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{4 e^2}-\frac{b x}{2 c e}",1,"-1/2*(b*x)/(c*e) + (a*x^2)/(2*e) + (b*ArcTan[c*x])/(2*c^2*e) + (b*x^2*ArcTan[c*x])/(2*e) + ((I/4)*b*d*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/e^2 - ((I/4)*b*d*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/e^2 - ((I/4)*b*d*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/e^2 + ((I/4)*b*d*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/e^2 - (a*d*Log[d + e*x^2])/(2*e^2) - ((I/4)*b*d*PolyLog[2, -((Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] - Sqrt[e]))])/e^2 - ((I/4)*b*d*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/e^2 + ((I/4)*b*d*PolyLog[2, -((Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] - Sqrt[e]))])/e^2 + ((I/4)*b*d*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/e^2","A",0
1152,1,441,311,0.1173349,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x*(a + b*ArcTan[c*x]))/(d + e*x^2),x]","\frac{a \log \left(d+e x^2\right)}{2 e}+\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} (1-i c x)}{i c \sqrt{-d}-\sqrt{e}}\right)}{4 e}+\frac{i b \text{Li}_2\left(\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 e}-\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} (i c x+1)}{i c \sqrt{-d}-\sqrt{e}}\right)}{4 e}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 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 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 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 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 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{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{4 e}-\frac{i b \text{Li}_2\left(1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{4 e}+\frac{i b \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 e}",1,"((-1/4*I)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/e + ((I/4)*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/e + ((I/4)*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/e - ((I/4)*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/e + (a*Log[d + e*x^2])/(2*e) + ((I/4)*b*PolyLog[2, -((Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] - Sqrt[e]))])/e + ((I/4)*b*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/e - ((I/4)*b*PolyLog[2, -((Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] - Sqrt[e]))])/e - ((I/4)*b*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/e","A",0
1153,1,429,353,0.2659364,"\int \frac{a+b \tan ^{-1}(c x)}{x \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcTan[c*x])/(x*(d + e*x^2)),x]","\frac{-2 a \log \left(d+e x^2\right)+4 a \log (x)+i b \text{Li}_2\left(\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right)-i b \text{Li}_2\left(\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)+i b \text{Li}_2\left(\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)-i b \text{Li}_2\left(\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+i \sqrt{e}}\right)+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)-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)-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)+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)+2 i b \text{Li}_2(-i c x)-2 i b \text{Li}_2(i c x)}{4 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{Li}_2\left(1-\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}+\frac{i b \text{Li}_2\left(1-\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}+\frac{i b \text{Li}_2(-i c x)}{2 d}-\frac{i b \text{Li}_2(i c x)}{2 d}-\frac{i b \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 d}",1,"(4*a*Log[x] + I*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])] - I*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])] - I*b*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])] + I*b*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])] - 2*a*Log[d + e*x^2] + (2*I)*b*PolyLog[2, (-I)*c*x] - (2*I)*b*PolyLog[2, I*c*x] + I*b*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])] - I*b*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])] + I*b*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])] - I*b*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*d)","A",0
1154,1,507,409,0.2973768,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)),x]","\frac{\frac{-a-b \tan ^{-1}(c x)}{2 x^2}-\frac{b c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-c^2 x^2\right)}{2 x}}{d}-\frac{e \left(-\frac{a \log \left(d+e x^2\right)}{2 d}+\frac{a \log (x)}{d}-\frac{i b \left(\text{Li}_2\left(-\frac{\sqrt{e} (1-i c x)}{i c \sqrt{-d}-\sqrt{e}}\right)+\log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)\right)}{4 d}-\frac{i b \left(\text{Li}_2\left(\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)+\log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)\right)}{4 d}+\frac{i b \left(\text{Li}_2\left(-\frac{\sqrt{e} (i c x+1)}{i c \sqrt{-d}-\sqrt{e}}\right)+\log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)\right)}{4 d}+\frac{i b \left(\text{Li}_2\left(\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)+\log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)\right)}{4 d}+\frac{i b \text{Li}_2(-i c x)}{2 d}-\frac{i b \text{Li}_2(i c x)}{2 d}\right)}{d}","-\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{i b e \text{Li}_2(-i c x)}{2 d^2}+\frac{i b e \text{Li}_2(i c x)}{2 d^2}+\frac{i b e \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 d^2}-\frac{i b e \text{Li}_2\left(1-\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^2}-\frac{i b e \text{Li}_2\left(1-\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^2}-\frac{b c}{2 d x}",1,"((-a - b*ArcTan[c*x])/(2*x^2) - (b*c*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/(2*x))/d - (e*((a*Log[x])/d - (a*Log[d + e*x^2])/(2*d) + ((I/2)*b*PolyLog[2, (-I)*c*x])/d - ((I/2)*b*PolyLog[2, I*c*x])/d - ((I/4)*b*(Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])] + PolyLog[2, -((Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] - Sqrt[e]))]))/d - ((I/4)*b*(Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])] + PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])]))/d + ((I/4)*b*(Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])] + PolyLog[2, -((Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] - Sqrt[e]))]))/d + ((I/4)*b*(Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])] + PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])]))/d))/d","C",0
1155,1,766,555,3.7123076,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x]))/(d + e*x^2),x]","-\frac{a \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{3/2}}+\frac{a x}{e}+\frac{b \left(\frac{c^2 d \left(i \left(\text{Li}_2\left(\frac{\left(d c^2+e+2 i \sqrt{-c^2 d e}\right) \left(c d+\sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(c d-\sqrt{-c^2 d e} x\right)}\right)-\text{Li}_2\left(\frac{\left(d c^2+e-2 i \sqrt{-c^2 d e}\right) \left(c d+\sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(c d-\sqrt{-c^2 d e} x\right)}\right)\right)-2 \cos ^{-1}\left(\frac{c^2 d+e}{e-c^2 d}\right) \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)-4 \tan ^{-1}(c x) \tanh ^{-1}\left(\frac{c d}{x \sqrt{-c^2 d e}}\right)-\log \left(\frac{2 c d (c x-i) \left(\sqrt{-c^2 d e}+i e\right)}{\left(c^2 d-e\right) \left(x \sqrt{-c^2 d e}-c d\right)}\right) \left(\cos ^{-1}\left(\frac{c^2 d+e}{e-c^2 d}\right)-2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)-\log \left(\frac{2 c d (c x+i) \left(\sqrt{-c^2 d e}-i e\right)}{\left(c^2 d-e\right) \left(x \sqrt{-c^2 d e}-c d\right)}\right) \left(\cos ^{-1}\left(\frac{c^2 d+e}{e-c^2 d}\right)+2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)+\left(2 i \tanh ^{-1}\left(\frac{c d}{x \sqrt{-c^2 d e}}\right)+2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)+\cos ^{-1}\left(\frac{c^2 d+e}{e-c^2 d}\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{-i \tan ^{-1}(c x)}}{\sqrt{e-c^2 d} \sqrt{\left(e-c^2 d\right) \cos \left(2 \tan ^{-1}(c x)\right)+c^2 (-d)-e}}\right)+\left(-2 i \tanh ^{-1}\left(\frac{c d}{x \sqrt{-c^2 d e}}\right)-2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)+\cos ^{-1}\left(\frac{c^2 d+e}{e-c^2 d}\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{i \tan ^{-1}(c x)}}{\sqrt{e-c^2 d} \sqrt{\left(e-c^2 d\right) \cos \left(2 \tan ^{-1}(c x)\right)+c^2 (-d)-e}}\right)\right)}{\sqrt{-c^2 d e}}-2 \log \left(c^2 x^2+1\right)+4 c x \tan ^{-1}(c x)\right)}{4 c e}","-\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} \text{Li}_2\left(\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 e^{3/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left(\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 e^{3/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left(\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 e^{3/2}}+\frac{i b \sqrt{-d} \text{Li}_2\left(\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+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{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 - (a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(3/2) + (b*(4*c*x*ArcTan[c*x] - 2*Log[1 + c^2*x^2] + (c^2*d*(-4*ArcTan[c*x]*ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] - 2*ArcCos[(c^2*d + e)/(-(c^2*d) + e)]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcCos[(c^2*d + e)/(-(c^2*d) + e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c*d*(I*e + Sqrt[-(c^2*d*e)])*(-I + c*x))/((c^2*d - e)*(-(c*d) + Sqrt[-(c^2*d*e)]*x))] - (ArcCos[(c^2*d + e)/(-(c^2*d) + e)] + (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c*d*((-I)*e + Sqrt[-(c^2*d*e)])*(I + c*x))/((c^2*d - e)*(-(c*d) + Sqrt[-(c^2*d*e)]*x))] + (ArcCos[(c^2*d + e)/(-(c^2*d) + e)] + (2*I)*ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[-(c^2*d) + e]*E^(I*ArcTan[c*x])*Sqrt[-(c^2*d) - e + (-(c^2*d) + e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[(c^2*d + e)/(-(c^2*d) + e)] - (2*I)*ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[-(c^2*d) + e]*Sqrt[-(c^2*d) - e + (-(c^2*d) + e)*Cos[2*ArcTan[c*x]]])] + I*(-PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(c*d + Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c*d - Sqrt[-(c^2*d*e)]*x))] + PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(c*d + Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c*d - Sqrt[-(c^2*d*e)]*x))])))/Sqrt[-(c^2*d*e)]))/(4*c*e)","A",0
1156,1,461,517,0.2498997,"\int \frac{a+b \tan ^{-1}(c x)}{d+e x^2} \, dx","Integrate[(a + b*ArcTan[c*x])/(d + e*x^2),x]","\frac{4 a \sqrt{-d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+i b \sqrt{d} \text{Li}_2\left(\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right)-i b \sqrt{d} \text{Li}_2\left(\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)-i b \sqrt{d} \text{Li}_2\left(\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)+i b \sqrt{d} \text{Li}_2\left(\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+i \sqrt{e}}\right)-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)+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)-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)+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 \sqrt{-d^2} \sqrt{e}}","\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} \sqrt{e}}+\frac{i b \text{Li}_2\left(\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 \sqrt{-d} \sqrt{e}}+\frac{i b \text{Li}_2\left(\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+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 \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,"(4*a*Sqrt[-d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - I*b*Sqrt[d]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])] + I*b*Sqrt[d]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])] - I*b*Sqrt[d]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])] + I*b*Sqrt[d]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])] + I*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])] - I*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])] - I*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])] + I*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(4*Sqrt[-d^2]*Sqrt[e])","A",1
1157,1,468,561,0.8204801,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)),x]","\frac{-\frac{\sqrt{e} \left(4 a \sqrt{-d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+i b \sqrt{d} \left(\text{Li}_2\left(\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right)+\log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)\right)-i b \sqrt{d} \left(\text{Li}_2\left(\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)+\log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}+\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)\right)-i b \sqrt{d} \left(\text{Li}_2\left(\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)+\log (1+i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}-i \sqrt{e}}\right)\right)+i b \sqrt{d} \left(\text{Li}_2\left(\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+i \sqrt{e}}\right)+\log (1-i c x) \log \left(\frac{c \left(\sqrt{-d}-\sqrt{e} x\right)}{c \sqrt{-d}+i \sqrt{e}}\right)\right)\right)}{4 \sqrt{-d^2}}-\frac{a+b \tan ^{-1}(c x)}{x}-\frac{1}{2} b c \log \left(c^2 x^2+1\right)+b c \log (x)}{d}","-\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} \text{Li}_2\left(\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 (-d)^{3/2}}+\frac{i b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+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{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])/x) + b*c*Log[x] - (b*c*Log[1 + c^2*x^2])/2 - (Sqrt[e]*(4*a*Sqrt[-d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + I*b*Sqrt[d]*(Log[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])] + PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])]) - I*b*Sqrt[d]*(Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])] + PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])]) - I*b*Sqrt[d]*(Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])] + PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])]) + I*b*Sqrt[d]*(Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])] + PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])))/(4*Sqrt[-d^2]))/d","A",0
1158,1,522,403,8.4691539,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x]))/(d + e*x^2)^2,x]","\frac{2 a \left(\frac{d}{d+e x^2}+\log \left(d+e x^2\right)\right)+b \left(-\frac{2 c^2 d \tan ^{-1}(c x)}{c^2 d-e}+\frac{2 c \sqrt{d} \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{c^2 d-e}-i \text{Li}_2\left(\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{c \sqrt{d}-\sqrt{e}}\right)+i \text{Li}_2\left(\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{d} c+\sqrt{e}}\right)+i \text{Li}_2\left(\frac{c \left(i \sqrt{e} x+\sqrt{d}\right)}{c \sqrt{d}-\sqrt{e}}\right)-i \text{Li}_2\left(\frac{c \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{d} c+\sqrt{e}}\right)+\frac{2 d \tan ^{-1}(c x)}{d+e x^2}+i \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (-1-i c x)}{c \sqrt{d}-\sqrt{e}}\right)-i \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (1-i c x)}{c \sqrt{d}+\sqrt{e}}\right)-i \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (-1+i c x)}{c \sqrt{d}-\sqrt{e}}\right)+i \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (1+i c x)}{c \sqrt{d}+\sqrt{e}}\right)+2 \tan ^{-1}(c x) \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right)+2 \tan ^{-1}(c x) \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)\right)}{4 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 \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 e^{3/2} \left(c^2 d-e\right)}-\frac{b c^2 d \tan ^{-1}(c x)}{2 e^2 \left(c^2 d-e\right)}-\frac{i b \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{4 e^2}-\frac{i b \text{Li}_2\left(1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{4 e^2}+\frac{i b \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 e^2}",1,"(2*a*(d/(d + e*x^2) + Log[d + e*x^2]) + b*((-2*c^2*d*ArcTan[c*x])/(c^2*d - e) + (2*d*ArcTan[c*x])/(d + e*x^2) + (2*c*Sqrt[d]*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(c^2*d - e) + 2*ArcTan[c*x]*Log[((-I)*Sqrt[d])/Sqrt[e] + x] + 2*ArcTan[c*x]*Log[(I*Sqrt[d])/Sqrt[e] + x] + I*Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(-1 - I*c*x))/(c*Sqrt[d] - Sqrt[e])] - I*Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(1 - I*c*x))/(c*Sqrt[d] + Sqrt[e])] - I*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(-1 + I*c*x))/(c*Sqrt[d] - Sqrt[e])] + I*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(1 + I*c*x))/(c*Sqrt[d] + Sqrt[e])] - I*PolyLog[2, (c*(Sqrt[d] - I*Sqrt[e]*x))/(c*Sqrt[d] - Sqrt[e])] + I*PolyLog[2, (c*(Sqrt[d] - I*Sqrt[e]*x))/(c*Sqrt[d] + Sqrt[e])] + I*PolyLog[2, (c*(Sqrt[d] + I*Sqrt[e]*x))/(c*Sqrt[d] - Sqrt[e])] - I*PolyLog[2, (c*(Sqrt[d] + I*Sqrt[e]*x))/(c*Sqrt[d] + Sqrt[e])]))/(4*e^2)","A",1
1159,1,98,91,0.15637,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x*(a + b*ArcTan[c*x]))/(d + e*x^2)^2,x]","\frac{a \sqrt{d} \left(c^2 d-e\right)-b \sqrt{d} e \left(c^2 x^2+1\right) \tan ^{-1}(c x)+b c \sqrt{e} \left(d+e x^2\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 \sqrt{d} e \left(e-c^2 d\right) \left(d+e x^2\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,"(a*Sqrt[d]*(c^2*d - e) - b*Sqrt[d]*e*(1 + c^2*x^2)*ArcTan[c*x] + b*c*Sqrt[e]*(d + e*x^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*Sqrt[d]*e*(-(c^2*d) + e)*(d + e*x^2))","A",1
1160,1,590,443,6.2881654,"\int \frac{a+b \tan ^{-1}(c x)}{x \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcTan[c*x])/(x*(d + e*x^2)^2),x]","\frac{2 a \left(\frac{d}{d+e x^2}-\log \left(d+e x^2\right)+2 \log (x)\right)+b \left(-\frac{2 c^2 d \tan ^{-1}(c x)}{c^2 d-e}+\frac{2 c \sqrt{d} \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{c^2 d-e}+i \text{Li}_2\left(\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{c \sqrt{d}-\sqrt{e}}\right)-i \text{Li}_2\left(\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{d} c+\sqrt{e}}\right)-i \text{Li}_2\left(\frac{c \left(i \sqrt{e} x+\sqrt{d}\right)}{c \sqrt{d}-\sqrt{e}}\right)+i \text{Li}_2\left(\frac{c \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{d} c+\sqrt{e}}\right)+\frac{2 d \tan ^{-1}(c x)}{d+e x^2}-i \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (-1-i c x)}{c \sqrt{d}-\sqrt{e}}\right)+i \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (1-i c x)}{c \sqrt{d}+\sqrt{e}}\right)+i \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (-1+i c x)}{c \sqrt{d}-\sqrt{e}}\right)-i \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (1+i c x)}{c \sqrt{d}+\sqrt{e}}\right)-2 \tan ^{-1}(c x) \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right)-2 \tan ^{-1}(c x) \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)+2 i \text{Li}_2(-i c x)-2 i \text{Li}_2(i c x)-2 i \log (x) \log (1-i c x)+2 i \log (x) \log (1+i c x)+4 \log (x) \tan ^{-1}(c x)\right)}{4 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{Li}_2\left(1-\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^2}+\frac{i b \text{Li}_2\left(1-\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^2}+\frac{i b \text{Li}_2(-i c x)}{2 d^2}-\frac{i b \text{Li}_2(i c x)}{2 d^2}-\frac{i b \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 d^2}",1,"(2*a*(d/(d + e*x^2) + 2*Log[x] - Log[d + e*x^2]) + b*((-2*c^2*d*ArcTan[c*x])/(c^2*d - e) + (2*d*ArcTan[c*x])/(d + e*x^2) + (2*c*Sqrt[d]*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(c^2*d - e) + 4*ArcTan[c*x]*Log[x] - 2*ArcTan[c*x]*Log[((-I)*Sqrt[d])/Sqrt[e] + x] - 2*ArcTan[c*x]*Log[(I*Sqrt[d])/Sqrt[e] + x] - I*Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(-1 - I*c*x))/(c*Sqrt[d] - Sqrt[e])] - (2*I)*Log[x]*Log[1 - I*c*x] + I*Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(1 - I*c*x))/(c*Sqrt[d] + Sqrt[e])] + I*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(-1 + I*c*x))/(c*Sqrt[d] - Sqrt[e])] + (2*I)*Log[x]*Log[1 + I*c*x] - I*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(1 + I*c*x))/(c*Sqrt[d] + Sqrt[e])] + (2*I)*PolyLog[2, (-I)*c*x] - (2*I)*PolyLog[2, I*c*x] + I*PolyLog[2, (c*(Sqrt[d] - I*Sqrt[e]*x))/(c*Sqrt[d] - Sqrt[e])] - I*PolyLog[2, (c*(Sqrt[d] - I*Sqrt[e]*x))/(c*Sqrt[d] + Sqrt[e])] - I*PolyLog[2, (c*(Sqrt[d] + I*Sqrt[e]*x))/(c*Sqrt[d] - Sqrt[e])] + I*PolyLog[2, (c*(Sqrt[d] + I*Sqrt[e]*x))/(c*Sqrt[d] + Sqrt[e])]))/(4*d^2)","A",1
1161,1,643,489,13.7040259,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)^2),x]","-\frac{a \left(d \left(\frac{e}{d+e x^2}+\frac{1}{x^2}\right)-2 e \log \left(d+e x^2\right)+4 e \log (x)\right)+b \left(\frac{c \sqrt{d} e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{c^2 d-e}+\frac{c^2 d \left(c^2 d-2 e\right) \tan ^{-1}(c x)}{c^2 d-e}-e \left(-i \text{Li}_2\left(\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{c \sqrt{d}-\sqrt{e}}\right)+i \text{Li}_2\left(\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{d} c+\sqrt{e}}\right)+i \text{Li}_2\left(\frac{c \left(i \sqrt{e} x+\sqrt{d}\right)}{c \sqrt{d}-\sqrt{e}}\right)-i \text{Li}_2\left(\frac{c \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{d} c+\sqrt{e}}\right)-2 \tan ^{-1}(c x) \log \left(d+e x^2\right)+i \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (-1-i c x)}{c \sqrt{d}-\sqrt{e}}\right)-i \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (1-i c x)}{c \sqrt{d}+\sqrt{e}}\right)-i \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (-1+i c x)}{c \sqrt{d}-\sqrt{e}}\right)+i \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (1+i c x)}{c \sqrt{d}+\sqrt{e}}\right)+2 \tan ^{-1}(c x) \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right)+2 \tan ^{-1}(c x) \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)\right)+d \tan ^{-1}(c x) \left(\frac{e}{d+e x^2}+\frac{1}{x^2}\right)-2 e \tan ^{-1}(c x) \log \left(d+e x^2\right)+\frac{c d}{x}-2 i e (-\text{Li}_2(-i c x)+\text{Li}_2(i c x)+\log (x) (\log (1-i c x)-\log (1+i c x)))+4 e \log (x) \tan ^{-1}(c x)\right)}{2 d^3}","-\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{e \left(a+b \tan ^{-1}(c x)\right)}{2 d^2 \left(d+e x^2\right)}-\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{i b e \text{Li}_2(-i c x)}{d^3}+\frac{i b e \text{Li}_2(i c x)}{d^3}+\frac{i b e \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{d^3}-\frac{i b e \text{Li}_2\left(1-\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^3}-\frac{i b e \text{Li}_2\left(1-\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^3}-\frac{b c}{2 d^2 x}",1,"-1/2*(a*(d*(x^(-2) + e/(d + e*x^2)) + 4*e*Log[x] - 2*e*Log[d + e*x^2]) + b*((c*d)/x + (c^2*d*(c^2*d - 2*e)*ArcTan[c*x])/(c^2*d - e) + d*(x^(-2) + e/(d + e*x^2))*ArcTan[c*x] + (c*Sqrt[d]*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(c^2*d - e) + 4*e*ArcTan[c*x]*Log[x] - 2*e*ArcTan[c*x]*Log[d + e*x^2] - (2*I)*e*(Log[x]*(Log[1 - I*c*x] - Log[1 + I*c*x]) - PolyLog[2, (-I)*c*x] + PolyLog[2, I*c*x]) - e*(2*ArcTan[c*x]*Log[((-I)*Sqrt[d])/Sqrt[e] + x] + 2*ArcTan[c*x]*Log[(I*Sqrt[d])/Sqrt[e] + x] + I*Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(-1 - I*c*x))/(c*Sqrt[d] - Sqrt[e])] - I*Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(1 - I*c*x))/(c*Sqrt[d] + Sqrt[e])] - I*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(-1 + I*c*x))/(c*Sqrt[d] - Sqrt[e])] + I*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(1 + I*c*x))/(c*Sqrt[d] + Sqrt[e])] - 2*ArcTan[c*x]*Log[d + e*x^2] - I*PolyLog[2, (c*(Sqrt[d] - I*Sqrt[e]*x))/(c*Sqrt[d] - Sqrt[e])] + I*PolyLog[2, (c*(Sqrt[d] - I*Sqrt[e]*x))/(c*Sqrt[d] + Sqrt[e])] + I*PolyLog[2, (c*(Sqrt[d] + I*Sqrt[e]*x))/(c*Sqrt[d] - Sqrt[e])] - I*PolyLog[2, (c*(Sqrt[d] + I*Sqrt[e]*x))/(c*Sqrt[d] + Sqrt[e])])))/d^3","A",1
1162,1,877,1335,10.5357386,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x]))/(d + e*x^2)^2,x]","-\frac{a x}{2 e \left(e x^2+d\right)}+\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 \sqrt{d} e^{3/2}}+\frac{b c \left(-\frac{2 \log \left(\frac{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}{d c^2+e}\right)}{c^2 d-e}+\frac{-4 \tan ^{-1}(c x) \tanh ^{-1}\left(\frac{\sqrt{-c^2 d e}}{c e x}\right)+2 \cos ^{-1}\left(\frac{d c^2+e}{e-c^2 d}\right) \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)+\left(\cos ^{-1}\left(\frac{d c^2+e}{e-c^2 d}\right)-2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(-\frac{2 c^2 d \left(i e+\sqrt{-c^2 d e}\right) (c x-i)}{\left(c^2 d-e\right) \left(c^2 d-c \sqrt{-c^2 d e} x\right)}\right)+\left(\cos ^{-1}\left(\frac{d c^2+e}{e-c^2 d}\right)+2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(\frac{2 i c^2 d \left(e+i \sqrt{-c^2 d e}\right) (c x+i)}{\left(c^2 d-e\right) \left(c^2 d-c \sqrt{-c^2 d e} x\right)}\right)-\left(\cos ^{-1}\left(\frac{d c^2+e}{e-c^2 d}\right)-2 i \tanh ^{-1}\left(\frac{\sqrt{-c^2 d e}}{c e x}\right)+2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{-i \tan ^{-1}(c x)}}{\sqrt{e-c^2 d} \sqrt{-d c^2-e+\left(e-c^2 d\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)-\left(\cos ^{-1}\left(\frac{d c^2+e}{e-c^2 d}\right)+2 i \tanh ^{-1}\left(\frac{\sqrt{-c^2 d e}}{c e x}\right)-2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{i \tan ^{-1}(c x)}}{\sqrt{e-c^2 d} \sqrt{-d c^2-e+\left(e-c^2 d\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(d c^2+e-2 i \sqrt{-c^2 d e}\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}{\left(c^2 d-e\right) \left(c^2 d-c \sqrt{-c^2 d e} x\right)}\right)-\text{Li}_2\left(\frac{\left(d c^2+e+2 i \sqrt{-c^2 d e}\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}{\left(c^2 d-e\right) \left(c^2 d-c \sqrt{-c^2 d e} x\right)}\right)\right)}{\sqrt{-c^2 d e}}-\frac{4 \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)}{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}\right)}{8 e}","-\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{Li}_2\left(\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{Li}_2\left(\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b c \text{Li}_2\left(\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{Li}_2\left(\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{Li}_2\left(\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{Li}_2\left(\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,"-1/2*(a*x)/(e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*Sqrt[d]*e^(3/2)) + (b*c*((-2*Log[(c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)])/(c^2*d - e) + (-4*ArcTan[c*x]*ArcTanh[Sqrt[-(c^2*d*e)]/(c*e*x)] + 2*ArcCos[(c^2*d + e)/(-(c^2*d) + e)]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] + (ArcCos[(c^2*d + e)/(-(c^2*d) + e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(-2*c^2*d*(I*e + Sqrt[-(c^2*d*e)])*(-I + c*x))/((c^2*d - e)*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[(c^2*d + e)/(-(c^2*d) + e)] + (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[((2*I)*c^2*d*(e + I*Sqrt[-(c^2*d*e)])*(I + c*x))/((c^2*d - e)*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))] - (ArcCos[(c^2*d + e)/(-(c^2*d) + e)] - (2*I)*ArcTanh[Sqrt[-(c^2*d*e)]/(c*e*x)] + (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[-(c^2*d) + e]*E^(I*ArcTan[c*x])*Sqrt[-(c^2*d) - e + (-(c^2*d) + e)*Cos[2*ArcTan[c*x]]])] - (ArcCos[(c^2*d + e)/(-(c^2*d) + e)] + (2*I)*ArcTanh[Sqrt[-(c^2*d*e)]/(c*e*x)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[-(c^2*d) + e]*Sqrt[-(c^2*d) - e + (-(c^2*d) + e)*Cos[2*ArcTan[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))]))/Sqrt[-(c^2*d*e)] - (4*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]])))/(8*e)","A",0
1163,1,861,819,10.5114759,"\int \frac{a+b \tan ^{-1}(c x)}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcTan[c*x])/(d + e*x^2)^2,x]","\frac{a x}{2 d \left(e x^2+d\right)}+\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{3/2} \sqrt{e}}+\frac{b c \left(\frac{2 \log \left(\frac{\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}{d c^2+e}+1\right)}{c^2 d-e}+\frac{-4 \tan ^{-1}(c x) \tanh ^{-1}\left(\frac{\sqrt{-c^2 d e}}{c e x}\right)+2 \cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right) \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)-\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)+2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(\frac{2 c^2 d \left(\sqrt{-c^2 d e}-i e\right) (c x-i)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)-\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)-2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(\frac{2 c^2 d \left(i e+\sqrt{-c^2 d e}\right) (c x+i)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)+\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)-2 i \left(\tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+\tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{-i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)+2 i \left(\tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+\tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(d c^2+e-2 i \sqrt{-c^2 d e}\right) \left(c^2 d-c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)-\text{Li}_2\left(\frac{\left(d c^2+e+2 i \sqrt{-c^2 d e}\right) \left(c^2 d-c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)\right)}{\sqrt{-c^2 d e}}+\frac{4 \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)}{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}\right)}{8 d}","\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{Li}_2\left(\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{Li}_2\left(\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{Li}_2\left(\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{Li}_2\left(\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,"(a*x)/(2*d*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*Sqrt[e]) + (b*c*((2*Log[1 + ((c^2*d - e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)])/(c^2*d - e) + (-4*ArcTan[c*x]*ArcTanh[Sqrt[-(c^2*d*e)]/(c*e*x)] + 2*ArcCos[-((c^2*d + e)/(c^2*d - e))]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*((-I)*e + Sqrt[-(c^2*d*e)])*(-I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*(I*e + Sqrt[-(c^2*d*e)])*(I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))]))/Sqrt[-(c^2*d*e)] + (4*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]])))/(8*d)","A",0
1164,1,992,1382,13.0137755,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^2),x]","b \left(-\frac{e \sin \left(2 \tan ^{-1}(c x)\right) \tan ^{-1}(c x)}{2 c^4 d^2 \left(d c^2+d \cos \left(2 \tan ^{-1}(c x)\right) c^2+e-e \cos \left(2 \tan ^{-1}(c x)\right)\right)}-\frac{\tan ^{-1}(c x)}{c^5 d^2 x}+\frac{\log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)}{c^4 d^2}-\frac{e \log \left(1-\frac{\left(e-c^2 d\right) \cos \left(2 \tan ^{-1}(c x)\right)}{d c^2+e}\right)}{4 c^4 d^2 \left(c^2 d-e\right)}-\frac{3 e \left(4 \tan ^{-1}(c x) \tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+2 \cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right) \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)-\left(\cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right)-2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(1-\frac{\left(d c^2+e-2 i \sqrt{-c^2 d e}\right) \left(2 c^2 d-2 c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(2 d c^2+2 \sqrt{-c^2 d e} x c\right)}\right)+\left(-\cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right)-2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(1-\frac{\left(d c^2+e+2 i \sqrt{-c^2 d e}\right) \left(2 c^2 d-2 c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(2 d c^2+2 \sqrt{-c^2 d e} x c\right)}\right)+\left(\cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right)-2 i \left(\tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+\tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{-i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+\left(\cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right)+2 i \left(\tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+\tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(d c^2+e-2 i \sqrt{-c^2 d e}\right) \left(2 c^2 d-2 c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(2 d c^2+2 \sqrt{-c^2 d e} x c\right)}\right)-\text{Li}_2\left(\frac{\left(d c^2+e+2 i \sqrt{-c^2 d e}\right) \left(2 c^2 d-2 c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(2 d c^2+2 \sqrt{-c^2 d e} x c\right)}\right)\right)\right)}{8 c^4 d^2 \sqrt{-c^2 d e}}\right) c^5-\frac{3 a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 d^{5/2}}-\frac{a}{d^2 x}-\frac{a e x}{2 d^2 \left(e x^2+d\right)}","-\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{Li}_2\left(\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 (-d)^{5/2}}+\frac{i b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right)}{4 (-d)^{5/2}}-\frac{i b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} (c x+i)}{\sqrt{-d} c+i \sqrt{e}}\right)}{4 (-d)^{5/2}}-\frac{i b c \sqrt{e} \text{Li}_2\left(\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{Li}_2\left(\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{Li}_2\left(\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{Li}_2\left(\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/(d^2*x)) - (a*e*x)/(2*d^2*(d + e*x^2)) - (3*a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(5/2)) + b*c^5*(-(ArcTan[c*x]/(c^5*d^2*x)) + Log[(c*x)/Sqrt[1 + c^2*x^2]]/(c^4*d^2) - (e*Log[1 - ((-(c^2*d) + e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)])/(4*c^4*d^2*(c^2*d - e)) - (3*e*(4*ArcTan[c*x]*ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + 2*ArcCos[(-(c^2*d) - e)/(c^2*d - e)]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (-ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))])))/(8*c^4*d^2*Sqrt[-(c^2*d*e)]) - (e*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(2*c^4*d^2*(c^2*d + e + c^2*d*Cos[2*ArcTan[c*x]] - e*Cos[2*ArcTan[c*x]])))","A",0
1165,1,589,532,13.4710399,"\int \frac{x^5 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^5*(a + b*ArcTan[c*x]))/(d + e*x^2)^3,x]","\frac{a \left(\frac{d \left(3 d+4 e x^2\right)}{\left(d+e x^2\right)^2}+2 \log \left(d+e x^2\right)\right)+b \left(-\frac{c d e x}{2 \left(c^2 d-e\right) \left(d+e x^2\right)}+\frac{c^2 d \left(4 e-3 c^2 d\right) \tan ^{-1}(c x)}{\left(e-c^2 d\right)^2}+\frac{c \sqrt{d} \sqrt{e} \left(5 c^2 d-7 e\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 \left(e-c^2 d\right)^2}-i \text{Li}_2\left(\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{c \sqrt{d}-\sqrt{e}}\right)+i \text{Li}_2\left(\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{d} c+\sqrt{e}}\right)+i \text{Li}_2\left(\frac{c \left(i \sqrt{e} x+\sqrt{d}\right)}{c \sqrt{d}-\sqrt{e}}\right)-i \text{Li}_2\left(\frac{c \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{d} c+\sqrt{e}}\right)+\frac{d \tan ^{-1}(c x) \left(3 d+4 e x^2\right)}{\left(d+e x^2\right)^2}+i \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (-1-i c x)}{c \sqrt{d}-\sqrt{e}}\right)-i \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (1-i c x)}{c \sqrt{d}+\sqrt{e}}\right)-i \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (-1+i c x)}{c \sqrt{d}-\sqrt{e}}\right)+i \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (1+i c x)}{c \sqrt{d}+\sqrt{e}}\right)+2 \tan ^{-1}(c x) \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right)+2 \tan ^{-1}(c x) \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)\right)}{4 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 \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{b c^2 d \tan ^{-1}(c x)}{e^3 \left(c^2 d-e\right)}-\frac{b c d x}{8 e^2 \left(c^2 d-e\right) \left(d+e x^2\right)}+\frac{b c^4 d^2 \tan ^{-1}(c x)}{4 e^3 \left(c^2 d-e\right)^2}-\frac{i b \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{4 e^3}-\frac{i b \text{Li}_2\left(1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{4 e^3}+\frac{i b \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 e^3}",1,"(a*((d*(3*d + 4*e*x^2))/(d + e*x^2)^2 + 2*Log[d + e*x^2]) + b*(-1/2*(c*d*e*x)/((c^2*d - e)*(d + e*x^2)) + (c^2*d*(-3*c^2*d + 4*e)*ArcTan[c*x])/(-(c^2*d) + e)^2 + (d*(3*d + 4*e*x^2)*ArcTan[c*x])/(d + e*x^2)^2 + (c*Sqrt[d]*(5*c^2*d - 7*e)*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*(-(c^2*d) + e)^2) + 2*ArcTan[c*x]*Log[((-I)*Sqrt[d])/Sqrt[e] + x] + 2*ArcTan[c*x]*Log[(I*Sqrt[d])/Sqrt[e] + x] + I*Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(-1 - I*c*x))/(c*Sqrt[d] - Sqrt[e])] - I*Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(1 - I*c*x))/(c*Sqrt[d] + Sqrt[e])] - I*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(-1 + I*c*x))/(c*Sqrt[d] - Sqrt[e])] + I*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(1 + I*c*x))/(c*Sqrt[d] + Sqrt[e])] - I*PolyLog[2, (c*(Sqrt[d] - I*Sqrt[e]*x))/(c*Sqrt[d] - Sqrt[e])] + I*PolyLog[2, (c*(Sqrt[d] - I*Sqrt[e]*x))/(c*Sqrt[d] + Sqrt[e])] + I*PolyLog[2, (c*(Sqrt[d] + I*Sqrt[e]*x))/(c*Sqrt[d] - Sqrt[e])] - I*PolyLog[2, (c*(Sqrt[d] + I*Sqrt[e]*x))/(c*Sqrt[d] + Sqrt[e])]))/(4*e^3)","A",1
1166,1,158,130,3.7635564,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x]))/(d + e*x^2)^3,x]","\frac{\frac{-4 a c^2 d+4 a e+b c e x}{\left(c^2 d-e\right) \left(d+e x^2\right)}+\frac{2 a d}{\left(d+e x^2\right)^2}+\frac{2 b c^2 \left(c^2 d-2 e\right) \tan ^{-1}(c x)}{\left(e-c^2 d\right)^2}-\frac{b c \sqrt{e} \left(c^2 d-3 e\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d} \left(e-c^2 d\right)^2}-\frac{2 b \tan ^{-1}(c x) \left(d+2 e x^2\right)}{\left(d+e x^2\right)^2}}{8 e^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,"((2*a*d)/(d + e*x^2)^2 + (-4*a*c^2*d + 4*a*e + b*c*e*x)/((c^2*d - e)*(d + e*x^2)) + (2*b*c^2*(c^2*d - 2*e)*ArcTan[c*x])/(-(c^2*d) + e)^2 - (2*b*(d + 2*e*x^2)*ArcTan[c*x])/(d + e*x^2)^2 - (b*c*(c^2*d - 3*e)*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*(-(c^2*d) + e)^2))/(8*e^2)","A",1
1167,1,131,131,1.1901888,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x*(a + b*ArcTan[c*x]))/(d + e*x^2)^3,x]","\frac{1}{8} \left(-\frac{\frac{2 a}{e}+\frac{b c x \left(d+e x^2\right)}{d \left(c^2 d-e\right)}}{\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)}{d^{3/2} \sqrt{e} \left(e-c^2 d\right)^2}+\frac{2 b \tan ^{-1}(c x) \left(\frac{c^4}{\left(e-c^2 d\right)^2}-\frac{1}{\left(d+e x^2\right)^2}\right)}{e}\right)","-\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,"(-(((2*a)/e + (b*c*x*(d + e*x^2))/(d*(c^2*d - e)))/(d + e*x^2)^2) + (2*b*(c^4/(-(c^2*d) + e)^2 - (d + e*x^2)^(-2))*ArcTan[c*x])/e - (b*c*(3*c^2*d - e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]*(-(c^2*d) + e)^2))/8","A",1
1168,1,645,574,14.4239484,"\int \frac{a+b \tan ^{-1}(c x)}{x \left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcTan[c*x])/(x*(d + e*x^2)^3),x]","\frac{2 a \left(\frac{d \left(3 d+2 e x^2\right)}{\left(d+e x^2\right)^2}-2 \log \left(d+e x^2\right)+4 \log (x)\right)+b \left(\frac{c d e x}{\left(c^2 d-e\right) \left(d+e x^2\right)}+\frac{2 c^2 d \left(2 e-3 c^2 d\right) \tan ^{-1}(c x)}{\left(e-c^2 d\right)^2}+\frac{c \sqrt{d} \sqrt{e} \left(7 c^2 d-5 e\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\left(e-c^2 d\right)^2}+2 i \text{Li}_2\left(\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{c \sqrt{d}-\sqrt{e}}\right)-2 i \text{Li}_2\left(\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{d} c+\sqrt{e}}\right)-2 i \text{Li}_2\left(\frac{c \left(i \sqrt{e} x+\sqrt{d}\right)}{c \sqrt{d}-\sqrt{e}}\right)+2 i \text{Li}_2\left(\frac{c \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{d} c+\sqrt{e}}\right)+\frac{2 d \tan ^{-1}(c x) \left(3 d+2 e x^2\right)}{\left(d+e x^2\right)^2}-2 i \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (-1-i c x)}{c \sqrt{d}-\sqrt{e}}\right)+2 i \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (1-i c x)}{c \sqrt{d}+\sqrt{e}}\right)+2 i \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (-1+i c x)}{c \sqrt{d}-\sqrt{e}}\right)-2 i \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (1+i c x)}{c \sqrt{d}+\sqrt{e}}\right)-4 \tan ^{-1}(c x) \log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right)-4 \tan ^{-1}(c x) \log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)-4 i (-\text{Li}_2(-i c x)+\text{Li}_2(i c x)+\log (x) (\log (1-i c x)-\log (1+i c x)))+8 \log (x) \tan ^{-1}(c x)\right)}{8 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{\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)}{2 d^2 \left(d+e x^2\right)}+\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 \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 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^4 \tan ^{-1}(c x)}{4 d \left(c^2 d-e\right)^2}+\frac{i b \text{Li}_2\left(1-\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}+\frac{i b \text{Li}_2\left(1-\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}+\frac{i b \text{Li}_2(-i c x)}{2 d^3}-\frac{i b \text{Li}_2(i c x)}{2 d^3}-\frac{i b \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 d^3}",1,"(2*a*((d*(3*d + 2*e*x^2))/(d + e*x^2)^2 + 4*Log[x] - 2*Log[d + e*x^2]) + b*((c*d*e*x)/((c^2*d - e)*(d + e*x^2)) + (2*c^2*d*(-3*c^2*d + 2*e)*ArcTan[c*x])/(-(c^2*d) + e)^2 + (2*d*(3*d + 2*e*x^2)*ArcTan[c*x])/(d + e*x^2)^2 + (c*Sqrt[d]*(7*c^2*d - 5*e)*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(-(c^2*d) + e)^2 + 8*ArcTan[c*x]*Log[x] - 4*ArcTan[c*x]*Log[((-I)*Sqrt[d])/Sqrt[e] + x] - 4*ArcTan[c*x]*Log[(I*Sqrt[d])/Sqrt[e] + x] - (2*I)*Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(-1 - I*c*x))/(c*Sqrt[d] - Sqrt[e])] + (2*I)*Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(1 - I*c*x))/(c*Sqrt[d] + Sqrt[e])] + (2*I)*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(-1 + I*c*x))/(c*Sqrt[d] - Sqrt[e])] - (2*I)*Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(1 + I*c*x))/(c*Sqrt[d] + Sqrt[e])] - (4*I)*(Log[x]*(Log[1 - I*c*x] - Log[1 + I*c*x]) - PolyLog[2, (-I)*c*x] + PolyLog[2, I*c*x]) + (2*I)*PolyLog[2, (c*(Sqrt[d] - I*Sqrt[e]*x))/(c*Sqrt[d] - Sqrt[e])] - (2*I)*PolyLog[2, (c*(Sqrt[d] - I*Sqrt[e]*x))/(c*Sqrt[d] + Sqrt[e])] - (2*I)*PolyLog[2, (c*(Sqrt[d] + I*Sqrt[e]*x))/(c*Sqrt[d] - Sqrt[e])] + (2*I)*PolyLog[2, (c*(Sqrt[d] + I*Sqrt[e]*x))/(c*Sqrt[d] + Sqrt[e])]))/(8*d^3)","A",1
1169,1,723,629,17.8276632,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 \left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)^3),x]","\frac{-a \left(\frac{d \left(2 d^2+9 d e x^2+6 e^2 x^4\right)}{x^2 \left(d+e x^2\right)^2}-6 e \log \left(d+e x^2\right)+12 e \log (x)\right)+b \left(\frac{c \sqrt{d} e^{3/2} \left(9 e-11 c^2 d\right) \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 \left(e-c^2 d\right)^2}-\frac{c d e^2 x}{2 \left(c^2 d-e\right) \left(d+e x^2\right)}+\frac{c^2 d \left(-2 c^4 d^2+9 c^2 d e-6 e^2\right) \tan ^{-1}(c x)}{\left(e-c^2 d\right)^2}-\frac{d \tan ^{-1}(c x) \left(2 d^2+9 d e x^2+6 e^2 x^4\right)}{x^2 \left(d+e x^2\right)^2}-3 i e \left(\text{Li}_2\left(\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{c \sqrt{d}-\sqrt{e}}\right)+\log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (-1+i c x)}{c \sqrt{d}-\sqrt{e}}\right)\right)+3 i e \left(\text{Li}_2\left(\frac{c \left(\sqrt{d}-i \sqrt{e} x\right)}{\sqrt{d} c+\sqrt{e}}\right)+\log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (1+i c x)}{c \sqrt{d}+\sqrt{e}}\right)\right)+3 i e \left(\text{Li}_2\left(\frac{c \left(i \sqrt{e} x+\sqrt{d}\right)}{c \sqrt{d}-\sqrt{e}}\right)+\log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (-1-i c x)}{c \sqrt{d}-\sqrt{e}}\right)\right)-3 i e \left(\text{Li}_2\left(\frac{c \left(i \sqrt{e} x+\sqrt{d}\right)}{\sqrt{d} c+\sqrt{e}}\right)+\log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right) \log \left(\frac{\sqrt{e} (1-i c x)}{c \sqrt{d}+\sqrt{e}}\right)\right)+6 e \tan ^{-1}(c x) \left(-\log \left(d+e x^2\right)+\log \left(x-\frac{i \sqrt{d}}{\sqrt{e}}\right)+\log \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)\right)+6 e \tan ^{-1}(c x) \log \left(d+e x^2\right)-\frac{2 c d}{x}-6 i e (\text{Li}_2(-i c x)+\log (x) \log (1+i c x))+6 i e (\text{Li}_2(i c x)+\log (x) \log (1-i c x))-12 e \log (x) \tan ^{-1}(c x)\right)}{4 d^4}","-\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{e \left(a+b \tan ^{-1}(c x)\right)}{d^3 \left(d+e x^2\right)}-\frac{a+b \tan ^{-1}(c x)}{2 d^3 x^2}-\frac{e \left(a+b \tan ^{-1}(c x)\right)}{4 d^2 \left(d+e x^2\right)^2}-\frac{3 a e \log (x)}{d^4}-\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 e^2 x}{8 d^3 \left(c^2 d-e\right) \left(d+e x^2\right)}+\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^4 e \tan ^{-1}(c x)}{4 d^2 \left(c^2 d-e\right)^2}-\frac{3 i b e \text{Li}_2(-i c x)}{2 d^4}+\frac{3 i b e \text{Li}_2(i c x)}{2 d^4}+\frac{3 i b e \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 d^4}-\frac{3 i b e \text{Li}_2\left(1-\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^4}-\frac{3 i b e \text{Li}_2\left(1-\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^4}-\frac{b c}{2 d^3 x}",1,"(-(a*((d*(2*d^2 + 9*d*e*x^2 + 6*e^2*x^4))/(x^2*(d + e*x^2)^2) + 12*e*Log[x] - 6*e*Log[d + e*x^2])) + b*((-2*c*d)/x - (c*d*e^2*x)/(2*(c^2*d - e)*(d + e*x^2)) + (c^2*d*(-2*c^4*d^2 + 9*c^2*d*e - 6*e^2)*ArcTan[c*x])/(-(c^2*d) + e)^2 - (d*(2*d^2 + 9*d*e*x^2 + 6*e^2*x^4)*ArcTan[c*x])/(x^2*(d + e*x^2)^2) + (c*Sqrt[d]*e^(3/2)*(-11*c^2*d + 9*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*(-(c^2*d) + e)^2) - 12*e*ArcTan[c*x]*Log[x] + 6*e*ArcTan[c*x]*(Log[((-I)*Sqrt[d])/Sqrt[e] + x] + Log[(I*Sqrt[d])/Sqrt[e] + x] - Log[d + e*x^2]) + 6*e*ArcTan[c*x]*Log[d + e*x^2] - (6*I)*e*(Log[x]*Log[1 + I*c*x] + PolyLog[2, (-I)*c*x]) + (6*I)*e*(Log[x]*Log[1 - I*c*x] + PolyLog[2, I*c*x]) - (3*I)*e*(Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(-1 + I*c*x))/(c*Sqrt[d] - Sqrt[e])] + PolyLog[2, (c*(Sqrt[d] - I*Sqrt[e]*x))/(c*Sqrt[d] - Sqrt[e])]) + (3*I)*e*(Log[(I*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(1 + I*c*x))/(c*Sqrt[d] + Sqrt[e])] + PolyLog[2, (c*(Sqrt[d] - I*Sqrt[e]*x))/(c*Sqrt[d] + Sqrt[e])]) + (3*I)*e*(Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(-1 - I*c*x))/(c*Sqrt[d] - Sqrt[e])] + PolyLog[2, (c*(Sqrt[d] + I*Sqrt[e]*x))/(c*Sqrt[d] - Sqrt[e])]) - (3*I)*e*(Log[((-I)*Sqrt[d])/Sqrt[e] + x]*Log[(Sqrt[e]*(1 - I*c*x))/(c*Sqrt[d] + Sqrt[e])] + PolyLog[2, (c*(Sqrt[d] + I*Sqrt[e]*x))/(c*Sqrt[d] + Sqrt[e])])))/(4*d^4)","A",1
1170,1,1914,966,13.0749567,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x]))/(d + e*x^2)^3,x]","b \left(-\frac{\log \left(1-\frac{\left(e-c^2 d\right) \cos \left(2 \tan ^{-1}(c x)\right)}{d c^2+e}\right)}{16 \left(c^2 d-e\right)^2 e}-\frac{\log \left(1-\frac{\left(e-c^2 d\right) \cos \left(2 \tan ^{-1}(c x)\right)}{d c^2+e}\right)}{16 c^2 d \left(c^2 d-e\right)^2}-\frac{4 \tan ^{-1}(c x) \tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+2 \cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right) \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)-\left(\cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right)-2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(1-\frac{\left(d c^2+e-2 i \sqrt{-c^2 d e}\right) \left(2 c^2 d-2 c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(2 d c^2+2 \sqrt{-c^2 d e} x c\right)}\right)+\left(-\cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right)-2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(1-\frac{\left(d c^2+e+2 i \sqrt{-c^2 d e}\right) \left(2 c^2 d-2 c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(2 d c^2+2 \sqrt{-c^2 d e} x c\right)}\right)+\left(\cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right)-2 i \left(\tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+\tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{-i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+\left(\cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right)+2 i \left(\tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+\tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(d c^2+e-2 i \sqrt{-c^2 d e}\right) \left(2 c^2 d-2 c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(2 d c^2+2 \sqrt{-c^2 d e} x c\right)}\right)-\text{Li}_2\left(\frac{\left(d c^2+e+2 i \sqrt{-c^2 d e}\right) \left(2 c^2 d-2 c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(2 d c^2+2 \sqrt{-c^2 d e} x c\right)}\right)\right)}{32 c^2 d \left(c^2 d-e\right) \sqrt{-c^2 d e}}+\frac{4 \tan ^{-1}(c x) \tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+2 \cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right) \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)-\left(\cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right)-2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(1-\frac{\left(d c^2+e-2 i \sqrt{-c^2 d e}\right) \left(2 c^2 d-2 c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(2 d c^2+2 \sqrt{-c^2 d e} x c\right)}\right)+\left(-\cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right)-2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(1-\frac{\left(d c^2+e+2 i \sqrt{-c^2 d e}\right) \left(2 c^2 d-2 c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(2 d c^2+2 \sqrt{-c^2 d e} x c\right)}\right)+\left(\cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right)-2 i \left(\tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+\tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{-i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+\left(\cos ^{-1}\left(\frac{-d c^2-e}{c^2 d-e}\right)+2 i \left(\tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+\tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(d c^2+e-2 i \sqrt{-c^2 d e}\right) \left(2 c^2 d-2 c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(2 d c^2+2 \sqrt{-c^2 d e} x c\right)}\right)-\text{Li}_2\left(\frac{\left(d c^2+e+2 i \sqrt{-c^2 d e}\right) \left(2 c^2 d-2 c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(2 d c^2+2 \sqrt{-c^2 d e} x c\right)}\right)\right)}{32 \left(c^2 d-e\right) e \sqrt{-c^2 d e}}+\frac{\tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)}{2 \left(c^2 d-e\right) \left(d c^2+d \cos \left(2 \tan ^{-1}(c x)\right) c^2+e-e \cos \left(2 \tan ^{-1}(c x)\right)\right)^2}+\frac{-d^2 \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right) c^4-2 d e c^2+e^2 \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)}{8 c^2 d \left(c^2 d-e\right)^2 e \left(d c^2+d \cos \left(2 \tan ^{-1}(c x)\right) c^2+e-e \cos \left(2 \tan ^{-1}(c x)\right)\right)}\right) c^3+\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{3/2} e^{3/2}}+\frac{a x}{8 d e \left(e x^2+d\right)}-\frac{a x}{4 e \left(e x^2+d\right)^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{Li}_2\left(\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{Li}_2\left(\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{Li}_2\left(\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{Li}_2\left(\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,"-1/4*(a*x)/(e*(d + e*x^2)^2) + (a*x)/(8*d*e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) + b*c^3*(-1/16*Log[1 - ((-(c^2*d) + e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)]/(c^2*d*(c^2*d - e)^2) - Log[1 - ((-(c^2*d) + e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)]/(16*(c^2*d - e)^2*e) - (4*ArcTan[c*x]*ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + 2*ArcCos[(-(c^2*d) - e)/(c^2*d - e)]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (-ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))]))/(32*c^2*d*(c^2*d - e)*Sqrt[-(c^2*d*e)]) + (4*ArcTan[c*x]*ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + 2*ArcCos[(-(c^2*d) - e)/(c^2*d - e)]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (-ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))]))/(32*(c^2*d - e)*e*Sqrt[-(c^2*d*e)]) + (ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(2*(c^2*d - e)*(c^2*d + e + c^2*d*Cos[2*ArcTan[c*x]] - e*Cos[2*ArcTan[c*x]])^2) + (-2*c^2*d*e - c^4*d^2*ArcTan[c*x]*Sin[2*ArcTan[c*x]] + e^2*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(8*c^2*d*(c^2*d - e)^2*e*(c^2*d + e + c^2*d*Cos[2*ArcTan[c*x]] - e*Cos[2*ArcTan[c*x]])))","A",0
1171,1,1745,893,12.8205776,"\int \frac{a+b \tan ^{-1}(c x)}{\left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcTan[c*x])/(d + e*x^2)^3,x]","\frac{3 a x}{8 d^2 \left(e x^2+d\right)}+\frac{a x}{4 d \left(e x^2+d\right)^2}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{5/2} \sqrt{e}}+\frac{b c \left(10 d \log \left(\frac{\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}{d c^2+e}+1\right) c^2+\frac{3 d \left(c^2 d-e\right) \left(-4 \tan ^{-1}(c x) \tanh ^{-1}\left(\frac{\sqrt{-c^2 d e}}{c e x}\right)+2 \cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right) \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)-\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)+2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(\frac{2 c^2 d \left(\sqrt{-c^2 d e}-i e\right) (c x-i)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)-\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)-2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(\frac{2 c^2 d \left(i e+\sqrt{-c^2 d e}\right) (c x+i)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)+\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)-2 i \left(\tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+\tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{-i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)+2 i \left(\tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+\tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(d c^2+e-2 i \sqrt{-c^2 d e}\right) \left(c^2 d-c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)-\text{Li}_2\left(\frac{\left(d c^2+e+2 i \sqrt{-c^2 d e}\right) \left(c^2 d-c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)\right)\right) c^2}{\sqrt{-c^2 d e}}-\frac{16 d \left(c^2 d-e\right) e \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right) c^2}{\left(d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)\right)^2}-6 e \log \left(\frac{\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}{d c^2+e}+1\right)-\frac{3 \left(c^2 d-e\right) e \left(-4 \tan ^{-1}(c x) \tanh ^{-1}\left(\frac{\sqrt{-c^2 d e}}{c e x}\right)+2 \cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right) \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)-\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)+2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(\frac{2 c^2 d \left(\sqrt{-c^2 d e}-i e\right) (c x-i)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)-\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)-2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(\frac{2 c^2 d \left(i e+\sqrt{-c^2 d e}\right) (c x+i)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)+\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)-2 i \left(\tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+\tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{-i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)+2 i \left(\tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+\tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(d c^2+e-2 i \sqrt{-c^2 d e}\right) \left(c^2 d-c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)-\text{Li}_2\left(\frac{\left(d c^2+e+2 i \sqrt{-c^2 d e}\right) \left(c^2 d-c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)\right)\right)}{\sqrt{-c^2 d e}}+\frac{8 d e c^2+4 \left(5 d^2 c^4-8 d e c^2+3 e^2\right) \tan ^{-1}(c x) \sin \left(2 \tan ^{-1}(c x)\right)}{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}\right)}{32 d^2 \left(e-c^2 d\right)^2}","\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{Li}_2\left(\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{Li}_2\left(\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{Li}_2\left(\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{Li}_2\left(\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,"(a*x)/(4*d*(d + e*x^2)^2) + (3*a*x)/(8*d^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]) + (b*c*(10*c^2*d*Log[1 + ((c^2*d - e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)] - 6*e*Log[1 + ((c^2*d - e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)] + (3*c^2*d*(c^2*d - e)*(-4*ArcTan[c*x]*ArcTanh[Sqrt[-(c^2*d*e)]/(c*e*x)] + 2*ArcCos[-((c^2*d + e)/(c^2*d - e))]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*((-I)*e + Sqrt[-(c^2*d*e)])*(-I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*(I*e + Sqrt[-(c^2*d*e)])*(I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))])))/Sqrt[-(c^2*d*e)] - (3*(c^2*d - e)*e*(-4*ArcTan[c*x]*ArcTanh[Sqrt[-(c^2*d*e)]/(c*e*x)] + 2*ArcCos[-((c^2*d + e)/(c^2*d - e))]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*((-I)*e + Sqrt[-(c^2*d*e)])*(-I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*(I*e + Sqrt[-(c^2*d*e)])*(I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))])))/Sqrt[-(c^2*d*e)] - (16*c^2*d*(c^2*d - e)*e*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]])^2 + (8*c^2*d*e + 4*(5*c^4*d^2 - 8*c^2*d*e + 3*e^2)*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]])))/(32*d^2*(-(c^2*d) + e)^2)","A",0
1172,1,2005,1518,13.4252452,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 \left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^3),x]","\text{Result too large to show}","-\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{Li}_2\left(\frac{\sqrt{e} (i-c x)}{\sqrt{-d} c+i \sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} (1-i c x)}{i \sqrt{-d} c+\sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} (i c x+1)}{i \sqrt{-d} c+\sqrt{e}}\right) \sqrt{e}}{4 (-d)^{7/2}}+\frac{i b \text{Li}_2\left(\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{Li}_2\left(\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{Li}_2\left(\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{Li}_2\left(\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{Li}_2\left(\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,"-(a/(d^3*x)) - (a*e*x)/(4*d^2*(d + e*x^2)^2) - (7*a*e*x)/(8*d^3*(d + e*x^2)) - (15*a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(7/2)) + b*c^7*(-(ArcTan[c*x]/(c^7*d^3*x)) + Log[(c*x)/Sqrt[1 + c^2*x^2]]/(c^6*d^3) - (9*e*Log[1 - ((-(c^2*d) + e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)])/(16*c^4*d^2*(c^2*d - e)^2) + (7*e^2*Log[1 - ((-(c^2*d) + e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)])/(16*c^6*d^3*(c^2*d - e)^2) - (15*e*(4*ArcTan[c*x]*ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + 2*ArcCos[(-(c^2*d) - e)/(c^2*d - e)]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (-ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))])))/(32*c^4*d^2*(c^2*d - e)*Sqrt[-(c^2*d*e)]) + (15*e^2*(4*ArcTan[c*x]*ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + 2*ArcCos[(-(c^2*d) - e)/(c^2*d - e)]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (-ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))])))/(32*c^6*d^3*(c^2*d - e)*Sqrt[-(c^2*d*e)]) + (e^2*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(2*c^4*d^2*(c^2*d - e)*(c^2*d + e + c^2*d*Cos[2*ArcTan[c*x]] - e*Cos[2*ArcTan[c*x]])^2) + (-2*c^2*d*e^2 - 9*c^4*d^2*e*ArcTan[c*x]*Sin[2*ArcTan[c*x]] + 16*c^2*d*e^2*ArcTan[c*x]*Sin[2*ArcTan[c*x]] - 7*e^3*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(8*c^6*d^3*(c^2*d - e)^2*(c^2*d + e + c^2*d*Cos[2*ArcTan[c*x]] - e*Cos[2*ArcTan[c*x]])))","A",0
1173,1,391,223,0.5549489,"\int x^3 \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^3*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]),x]","\frac{-c^2 \sqrt{d+e x^2} \left(8 a c^3 \left(2 d^2-d e x^2-3 e^2 x^4\right)+b e x \left(c^2 \left(7 d+6 e x^2\right)-12 e\right)\right)-8 b c^5 \tan ^{-1}(c x) \sqrt{d+e x^2} \left(2 d^2-d e x^2-3 e^2 x^4\right)-4 i b \left(c^2 d-e\right)^{3/2} \left(2 c^2 d+3 e\right) \log \left(-\frac{60 i c^6 e^2 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \left(c^2 d-e\right)^{5/2} \left(2 c^2 d+3 e\right)}\right)+4 i b \left(c^2 d-e\right)^{3/2} \left(2 c^2 d+3 e\right) \log \left(\frac{60 i c^6 e^2 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \left(c^2 d-e\right)^{5/2} \left(2 c^2 d+3 e\right)}\right)+b \sqrt{e} \left(15 c^4 d^2+20 c^2 d e-24 e^2\right) \log \left(\sqrt{e} \sqrt{d+e x^2}+e x\right)}{120 c^5 e^2}","\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(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 \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 x \left(d+e x^2\right)^{3/2}}{20 c e}",1,"(-(c^2*Sqrt[d + e*x^2]*(8*a*c^3*(2*d^2 - d*e*x^2 - 3*e^2*x^4) + b*e*x*(-12*e + c^2*(7*d + 6*e*x^2)))) - 8*b*c^5*Sqrt[d + e*x^2]*(2*d^2 - d*e*x^2 - 3*e^2*x^4)*ArcTan[c*x] - (4*I)*b*(c^2*d - e)^(3/2)*(2*c^2*d + 3*e)*Log[((-60*I)*c^6*e^2*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(5/2)*(2*c^2*d + 3*e)*(I + c*x))] + (4*I)*b*(c^2*d - e)^(3/2)*(2*c^2*d + 3*e)*Log[((60*I)*c^6*e^2*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(5/2)*(2*c^2*d + 3*e)*(-I + c*x))] + b*Sqrt[e]*(15*c^4*d^2 + 20*c^2*d*e - 24*e^2)*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(120*c^5*e^2)","C",1
1174,0,0,97,11.2865654,"\int x^2 \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[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{a d x \sqrt{d+e x^2}}{8 e}+\frac{1}{4} a x^3 \sqrt{d+e x^2}",0,"Integrate[x^2*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]), x]","A",-1
1175,1,279,140,0.5611427,"\int x \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]),x]","\frac{c^2 \sqrt{d+e x^2} \left(2 a c \left(d+e x^2\right)-b e x\right)+2 b c^3 \tan ^{-1}(c x) \left(d+e x^2\right)^{3/2}+b \sqrt{e} \left(2 e-3 c^2 d\right) \log \left(\sqrt{e} \sqrt{d+e x^2}+e x\right)-i b \left(c^2 d-e\right)^{3/2} \log \left(\frac{12 c^4 e \left(-i \sqrt{c^2 d-e} \sqrt{d+e x^2}-i c d+e x\right)}{b (c x-i) \left(c^2 d-e\right)^{5/2}}\right)+i b \left(c^2 d-e\right)^{3/2} \log \left(\frac{12 c^4 e \left(i \sqrt{c^2 d-e} \sqrt{d+e x^2}+i c d+e x\right)}{b (c x+i) \left(c^2 d-e\right)^{5/2}}\right)}{6 c^3 e}","\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,"(c^2*Sqrt[d + e*x^2]*(-(b*e*x) + 2*a*c*(d + e*x^2)) + 2*b*c^3*(d + e*x^2)^(3/2)*ArcTan[c*x] - I*b*(c^2*d - e)^(3/2)*Log[(12*c^4*e*((-I)*c*d + e*x - I*Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(5/2)*(-I + c*x))] + I*b*(c^2*d - e)^(3/2)*Log[(12*c^4*e*(I*c*d + e*x + I*Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(5/2)*(I + c*x))] + b*Sqrt[e]*(-3*c^2*d + 2*e)*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(6*c^3*e)","C",1
1176,0,0,23,5.0999243,"\int \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[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,"Integrate[Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]), x]","A",-1
1177,0,0,64,82.6122089,"\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Integrate[(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,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x, x]","A",-1
1178,0,0,68,8.9497888,"\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Integrate[(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,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^2, x]","A",-1
1179,0,0,73,54.9252603,"\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Integrate[(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,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^3, x]","A",-1
1180,1,288,137,0.6766262,"\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^4,x]","-\frac{\sqrt{d+e x^2} \left(2 a \left(d+e x^2\right)+b c d x\right)+b c \sqrt{d} x^3 \log (x) \left(2 c^2 d-3 e\right)-b c \sqrt{d} x^3 \left(2 c^2 d-3 e\right) \log \left(\sqrt{d} \sqrt{d+e x^2}+d\right)+b x^3 \left(c^2 d-e\right)^{3/2} \log \left(\frac{12 c d \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \left(c^2 d-e\right)^{5/2}}\right)+b x^3 \left(c^2 d-e\right)^{3/2} \log \left(\frac{12 c d \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \left(c^2 d-e\right)^{5/2}}\right)+2 b \tan ^{-1}(c x) \left(d+e x^2\right)^{3/2}}{6 d x^3}","-\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,"-1/6*(Sqrt[d + e*x^2]*(b*c*d*x + 2*a*(d + e*x^2)) + 2*b*(d + e*x^2)^(3/2)*ArcTan[c*x] + b*c*Sqrt[d]*(2*c^2*d - 3*e)*x^3*Log[x] - b*c*Sqrt[d]*(2*c^2*d - 3*e)*x^3*Log[d + Sqrt[d]*Sqrt[d + e*x^2]] + b*(c^2*d - e)^(3/2)*x^3*Log[(12*c*d*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(5/2)*(I + c*x))] + b*(c^2*d - e)^(3/2)*x^3*Log[(12*c*d*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(5/2)*(-I + c*x))])/(d*x^3)","C",1
1181,0,0,98,57.9972218,"\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Integrate[(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,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^5, x]","A",-1
1182,1,413,224,0.5488736,"\int \frac{\sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]))/x^6,x]","\frac{-\sqrt{d+e x^2} \left(8 a \left(3 d^2+d e x^2-2 e^2 x^4\right)+b c d x \left(d \left(6-12 c^2 x^2\right)+7 e x^2\right)\right)+4 b x^5 \left(c^2 d-e\right)^{3/2} \left(3 c^2 d+2 e\right) \log \left(-\frac{60 c d^2 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \left(c^2 d-e\right)^{5/2} \left(3 c^2 d+2 e\right)}\right)+4 b x^5 \left(c^2 d-e\right)^{3/2} \left(3 c^2 d+2 e\right) \log \left(-\frac{60 c d^2 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \left(c^2 d-e\right)^{5/2} \left(3 c^2 d+2 e\right)}\right)+b c \sqrt{d} x^5 \log (x) \left(24 c^4 d^2-20 c^2 d e-15 e^2\right)-b c \sqrt{d} x^5 \left(24 c^4 d^2-20 c^2 d e-15 e^2\right) \log \left(\sqrt{d} \sqrt{d+e x^2}+d\right)-8 b \tan ^{-1}(c x) \sqrt{d+e x^2} \left(3 d^2+d e x^2-2 e^2 x^4\right)}{120 d^2 x^5}","\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 \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(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 c \left(d+e x^2\right)^{3/2}}{20 d x^4}",1,"(-(Sqrt[d + e*x^2]*(8*a*(3*d^2 + d*e*x^2 - 2*e^2*x^4) + b*c*d*x*(7*e*x^2 + d*(6 - 12*c^2*x^2)))) - 8*b*Sqrt[d + e*x^2]*(3*d^2 + d*e*x^2 - 2*e^2*x^4)*ArcTan[c*x] + b*c*Sqrt[d]*(24*c^4*d^2 - 20*c^2*d*e - 15*e^2)*x^5*Log[x] - b*c*Sqrt[d]*(24*c^4*d^2 - 20*c^2*d*e - 15*e^2)*x^5*Log[d + Sqrt[d]*Sqrt[d + e*x^2]] + 4*b*(c^2*d - e)^(3/2)*(3*c^2*d + 2*e)*x^5*Log[(-60*c*d^2*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(5/2)*(3*c^2*d + 2*e)*(I + c*x))] + 4*b*(c^2*d - e)^(3/2)*(3*c^2*d + 2*e)*x^5*Log[(-60*c*d^2*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(5/2)*(3*c^2*d + 2*e)*(-I + c*x))])/(120*d^2*x^5)","C",1
1183,1,418,279,0.7136085,"\int x^3 \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]),x]","-\frac{c^2 \sqrt{d+e x^2} \left(48 a c^5 \left(2 d-5 e x^2\right) \left(d+e x^2\right)^2+b e x \left(c^4 \left(57 d^2+106 d e x^2+40 e^2 x^4\right)-6 c^2 e \left(37 d+10 e x^2\right)+120 e^2\right)\right)+48 b c^7 \tan ^{-1}(c x) \left(2 d-5 e x^2\right) \left(d+e x^2\right)^{5/2}+24 i b \left(c^2 d-e\right)^{5/2} \left(2 c^2 d+5 e\right) \log \left(-\frac{140 i c^8 e^2 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \left(c^2 d-e\right)^{7/2} \left(2 c^2 d+5 e\right)}\right)-24 i b \left(c^2 d-e\right)^{5/2} \left(2 c^2 d+5 e\right) \log \left(\frac{140 i c^8 e^2 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \left(c^2 d-e\right)^{7/2} \left(2 c^2 d+5 e\right)}\right)-3 b \sqrt{e} \left(35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right) \log \left(\sqrt{e} \sqrt{d+e x^2}+e x\right)}{1680 c^7 e^2}","\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 \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(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(35 c^6 d^3+70 c^4 d^2 e-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 x \left(d+e x^2\right)^{5/2}}{42 c e}",1,"-1/1680*(c^2*Sqrt[d + e*x^2]*(48*a*c^5*(2*d - 5*e*x^2)*(d + e*x^2)^2 + b*e*x*(120*e^2 - 6*c^2*e*(37*d + 10*e*x^2) + c^4*(57*d^2 + 106*d*e*x^2 + 40*e^2*x^4))) + 48*b*c^7*(2*d - 5*e*x^2)*(d + e*x^2)^(5/2)*ArcTan[c*x] + (24*I)*b*(c^2*d - e)^(5/2)*(2*c^2*d + 5*e)*Log[((-140*I)*c^8*e^2*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(7/2)*(2*c^2*d + 5*e)*(I + c*x))] - (24*I)*b*(c^2*d - e)^(5/2)*(2*c^2*d + 5*e)*Log[((140*I)*c^8*e^2*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(7/2)*(2*c^2*d + 5*e)*(-I + c*x))] - 3*b*Sqrt[e]*(35*c^6*d^3 + 70*c^4*d^2*e - 168*c^2*d*e^2 + 80*e^3)*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(c^7*e^2)","C",1
1184,0,0,119,11.3087288,"\int x^2 \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[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,"Integrate[x^2*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]), x]","A",-1
1185,1,313,181,0.4626253,"\int x \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]),x]","\frac{c^2 \sqrt{d+e x^2} \left(8 a c^3 \left(d+e x^2\right)^2+b e x \left(4 e-c^2 \left(9 d+2 e x^2\right)\right)\right)+8 b c^5 \tan ^{-1}(c x) \left(d+e x^2\right)^{5/2}-4 i b \left(c^2 d-e\right)^{5/2} \log \left(\frac{20 c^6 e \left(-i \sqrt{c^2 d-e} \sqrt{d+e x^2}-i c d+e x\right)}{b (c x-i) \left(c^2 d-e\right)^{7/2}}\right)+4 i b \left(c^2 d-e\right)^{5/2} \log \left(\frac{20 c^6 e \left(i \sqrt{c^2 d-e} \sqrt{d+e x^2}+i c d+e x\right)}{b (c x+i) \left(c^2 d-e\right)^{7/2}}\right)-b \sqrt{e} \left(15 c^4 d^2-20 c^2 d e+8 e^2\right) \log \left(\sqrt{e} \sqrt{d+e x^2}+e x\right)}{40 c^5 e}","\frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{5 e}-\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(7 c^2 d-4 e\right) \sqrt{d+e x^2}}{40 c^3}-\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(d+e x^2\right)^{3/2}}{20 c}",1,"(c^2*Sqrt[d + e*x^2]*(8*a*c^3*(d + e*x^2)^2 + b*e*x*(4*e - c^2*(9*d + 2*e*x^2))) + 8*b*c^5*(d + e*x^2)^(5/2)*ArcTan[c*x] - (4*I)*b*(c^2*d - e)^(5/2)*Log[(20*c^6*e*((-I)*c*d + e*x - I*Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(7/2)*(-I + c*x))] + (4*I)*b*(c^2*d - e)^(5/2)*Log[(20*c^6*e*(I*c*d + e*x + I*Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(7/2)*(I + c*x))] - b*Sqrt[e]*(15*c^4*d^2 - 20*c^2*d*e + 8*e^2)*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(40*c^5*e)","C",1
1186,0,0,23,5.2523044,"\int \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[(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,"Integrate[(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]), x]","A",-1
1187,0,0,81,82.7765993,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Integrate[((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,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x, x]","A",-1
1188,0,0,90,9.2455108,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((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,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^2, x]","A",-1
1189,0,0,90,55.0929291,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((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,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^3, x]","A",-1
1190,0,0,88,31.6575701,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((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,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^4, x]","A",-1
1191,0,0,95,57.1667029,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((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,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^5, x]","A",-1
1192,1,334,178,0.5001105,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^6,x]","\frac{-\sqrt{d+e x^2} \left(8 a \left(d+e x^2\right)^2+b c d x \left(d \left(2-4 c^2 x^2\right)+9 e x^2\right)\right)+4 b x^5 \left(c^2 d-e\right)^{5/2} \log \left(-\frac{20 c d \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \left(c^2 d-e\right)^{7/2}}\right)+4 b x^5 \left(c^2 d-e\right)^{5/2} \log \left(-\frac{20 c d \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \left(c^2 d-e\right)^{7/2}}\right)+b c \sqrt{d} x^5 \log (x) \left(8 c^4 d^2-20 c^2 d e+15 e^2\right)-b c \sqrt{d} x^5 \left(8 c^4 d^2-20 c^2 d e+15 e^2\right) \log \left(\sqrt{d} \sqrt{d+e x^2}+d\right)-8 b \tan ^{-1}(c x) \left(d+e x^2\right)^{5/2}}{40 d x^5}","-\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(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(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(d+e x^2\right)^{3/2}}{20 x^4}",1,"(-(Sqrt[d + e*x^2]*(8*a*(d + e*x^2)^2 + b*c*d*x*(9*e*x^2 + d*(2 - 4*c^2*x^2)))) - 8*b*(d + e*x^2)^(5/2)*ArcTan[c*x] + b*c*Sqrt[d]*(8*c^4*d^2 - 20*c^2*d*e + 15*e^2)*x^5*Log[x] - b*c*Sqrt[d]*(8*c^4*d^2 - 20*c^2*d*e + 15*e^2)*x^5*Log[d + Sqrt[d]*Sqrt[d + e*x^2]] + 4*b*(c^2*d - e)^(5/2)*x^5*Log[(-20*c*d*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(7/2)*(I + c*x))] + 4*b*(c^2*d - e)^(5/2)*x^5*Log[(-20*c*d*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(7/2)*(-I + c*x))])/(40*d*x^5)","C",1
1193,1,470,345,0.9318211,"\int x^3 \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]),x]","-\frac{c^2 \sqrt{d+e x^2} \left(384 a c^7 \left(2 d-7 e x^2\right) \left(d+e x^2\right)^3+b e x \left(3 c^6 \left(187 d^3+558 d^2 e x^2+424 d e^2 x^4+112 e^3 x^6\right)-8 c^4 e \left(453 d^2+242 d e x^2+56 e^2 x^4\right)+48 c^2 e^2 \left(83 d+14 e x^2\right)-1344 e^3\right)\right)+384 b c^9 \tan ^{-1}(c x) \left(2 d-7 e x^2\right) \left(d+e x^2\right)^{7/2}+192 i b \left(c^2 d-e\right)^{7/2} \left(2 c^2 d+7 e\right) \log \left(-\frac{252 i c^{10} e^2 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \left(c^2 d-e\right)^{9/2} \left(2 c^2 d+7 e\right)}\right)-192 i b \left(c^2 d-e\right)^{7/2} \left(2 c^2 d+7 e\right) \log \left(\frac{252 i c^{10} e^2 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \left(c^2 d-e\right)^{9/2} \left(2 c^2 d+7 e\right)}\right)+3 b \sqrt{e} \left(-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\right) \log \left(\sqrt{e} \sqrt{d+e x^2}+e x\right)}{24192 c^9 e^2}","\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 \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(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(59 c^6 d^3+712 c^4 d^2 e-1104 c^2 d e^2+448 e^3\right) \sqrt{d+e x^2}}{8064 c^7 e}+\frac{b \left(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\right) \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{8064 c^9 e^{3/2}}-\frac{b x \left(d+e x^2\right)^{7/2}}{72 c e}",1,"-1/24192*(c^2*Sqrt[d + e*x^2]*(384*a*c^7*(2*d - 7*e*x^2)*(d + e*x^2)^3 + b*e*x*(-1344*e^3 + 48*c^2*e^2*(83*d + 14*e*x^2) - 8*c^4*e*(453*d^2 + 242*d*e*x^2 + 56*e^2*x^4) + 3*c^6*(187*d^3 + 558*d^2*e*x^2 + 424*d*e^2*x^4 + 112*e^3*x^6))) + 384*b*c^9*(2*d - 7*e*x^2)*(d + e*x^2)^(7/2)*ArcTan[c*x] + (192*I)*b*(c^2*d - e)^(7/2)*(2*c^2*d + 7*e)*Log[((-252*I)*c^10*e^2*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(9/2)*(2*c^2*d + 7*e)*(I + c*x))] - (192*I)*b*(c^2*d - e)^(7/2)*(2*c^2*d + 7*e)*Log[((252*I)*c^10*e^2*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(9/2)*(2*c^2*d + 7*e)*(-I + c*x))] + 3*b*Sqrt[e]*(-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)*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(c^9*e^2)","C",1
1194,0,0,141,11.803418,"\int x^2 \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[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,"Integrate[x^2*(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]), x]","A",-1
1195,1,353,233,0.5807159,"\int x \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]),x]","\frac{c^2 \sqrt{d+e x^2} \left(48 a c^5 \left(d+e x^2\right)^3-b e x \left(c^4 \left(87 d^2+38 d e x^2+8 e^2 x^4\right)-6 c^2 e \left(13 d+2 e x^2\right)+24 e^2\right)\right)+48 b c^7 \tan ^{-1}(c x) \left(d+e x^2\right)^{7/2}-24 i b \left(c^2 d-e\right)^{7/2} \log \left(\frac{28 c^8 e \left(-i \sqrt{c^2 d-e} \sqrt{d+e x^2}-i c d+e x\right)}{b (c x-i) \left(c^2 d-e\right)^{9/2}}\right)+24 i b \left(c^2 d-e\right)^{7/2} \log \left(\frac{28 c^8 e \left(i \sqrt{c^2 d-e} \sqrt{d+e x^2}+i c d+e x\right)}{b (c x+i) \left(c^2 d-e\right)^{9/2}}\right)+3 b \sqrt{e} \left(-35 c^6 d^3+70 c^4 d^2 e-56 c^2 d e^2+16 e^3\right) \log \left(\sqrt{e} \sqrt{d+e x^2}+e x\right)}{336 c^7 e}","\frac{\left(d+e x^2\right)^{7/2} \left(a+b \tan ^{-1}(c x)\right)}{7 e}-\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(11 c^2 d-6 e\right) \left(d+e x^2\right)^{3/2}}{168 c^3}-\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(35 c^6 d^3-70 c^4 d^2 e+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(d+e x^2\right)^{5/2}}{42 c}",1,"(c^2*Sqrt[d + e*x^2]*(48*a*c^5*(d + e*x^2)^3 - b*e*x*(24*e^2 - 6*c^2*e*(13*d + 2*e*x^2) + c^4*(87*d^2 + 38*d*e*x^2 + 8*e^2*x^4))) + 48*b*c^7*(d + e*x^2)^(7/2)*ArcTan[c*x] - (24*I)*b*(c^2*d - e)^(7/2)*Log[(28*c^8*e*((-I)*c*d + e*x - I*Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(9/2)*(-I + c*x))] + (24*I)*b*(c^2*d - e)^(7/2)*Log[(28*c^8*e*(I*c*d + e*x + I*Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(9/2)*(I + c*x))] + 3*b*Sqrt[e]*(-35*c^6*d^3 + 70*c^4*d^2*e - 56*c^2*d*e^2 + 16*e^3)*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(336*c^7*e)","C",1
1196,0,0,23,5.7505368,"\int \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[(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,"Integrate[(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]), x]","A",-1
1197,0,0,100,83.0726374,"\int \frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{x} \, dx","Integrate[((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^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)+a d^2 \sqrt{d+e x^2}+\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,"Integrate[((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/x, x]","A",-1
1198,0,0,111,9.199496,"\int \frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((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,"Integrate[((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/x^2, x]","A",-1
1199,0,0,108,55.4705737,"\int \frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((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,"Integrate[((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/x^3, x]","A",-1
1200,0,0,114,9.5460624,"\int \frac{\left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((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 d e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)+\frac{5}{2} a e^2 x \sqrt{d+e x^2}-\frac{5 a e \left(d+e x^2\right)^{3/2}}{3 x}-\frac{a \left(d+e x^2\right)^{5/2}}{3 x^3}",0,"Integrate[((d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]))/x^4, x]","A",-1
1201,1,377,176,0.5176508,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x]))/Sqrt[d + e*x^2],x]","\frac{-\frac{\sqrt{d+e x^2} \left(a c \left(4 d-2 e x^2\right)+b e x\right)}{c}+\frac{b \sqrt{e} \left(3 c^2 d+2 e\right) \log \left(\sqrt{e} \sqrt{d+e x^2}+e x\right)}{c^3}-\frac{i b \left(2 c^4 d^2-c^2 d e-e^2\right) \log \left(\frac{12 i c^4 e^2 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \sqrt{c^2 d-e} \left(-2 c^4 d^2+c^2 d e+e^2\right)}\right)}{c^3 \sqrt{c^2 d-e}}+\frac{i b \left(2 c^4 d^2-c^2 d e-e^2\right) \log \left(-\frac{12 i c^4 e^2 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \sqrt{c^2 d-e} \left(-2 c^4 d^2+c^2 d e+e^2\right)}\right)}{c^3 \sqrt{c^2 d-e}}+2 b \tan ^{-1}(c x) \left(e x^2-2 d\right) \sqrt{d+e x^2}}{6 e^2}","\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 \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 \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 x \sqrt{d+e x^2}}{6 c e}",1,"(-((Sqrt[d + e*x^2]*(b*e*x + a*c*(4*d - 2*e*x^2)))/c) + 2*b*(-2*d + e*x^2)*Sqrt[d + e*x^2]*ArcTan[c*x] - (I*b*(2*c^4*d^2 - c^2*d*e - e^2)*Log[((12*I)*c^4*e^2*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(-2*c^4*d^2 + c^2*d*e + e^2)*(I + c*x))])/(c^3*Sqrt[c^2*d - e]) + (I*b*(2*c^4*d^2 - c^2*d*e - e^2)*Log[((-12*I)*c^4*e^2*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(-2*c^4*d^2 + c^2*d*e + e^2)*(-I + c*x))])/(c^3*Sqrt[c^2*d - e]) + (b*Sqrt[e]*(3*c^2*d + 2*e)*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/c^3)/(6*e^2)","C",1
1202,0,0,75,9.3765413,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(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,"Integrate[(x^2*(a + b*ArcTan[c*x]))/Sqrt[d + e*x^2], x]","A",-1
1203,1,251,103,0.4061163,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x*(a + b*ArcTan[c*x]))/Sqrt[d + e*x^2],x]","\frac{2 a c \sqrt{d+e x^2}-i b \sqrt{c^2 d-e} \log \left(\frac{4 c^2 e \left(-i \sqrt{c^2 d-e} \sqrt{d+e x^2}-i c d+e x\right)}{b (c x-i) \left(c^2 d-e\right)^{3/2}}\right)+i b \sqrt{c^2 d-e} \log \left(\frac{4 c^2 e \left(i \sqrt{c^2 d-e} \sqrt{d+e x^2}+i c d+e x\right)}{b (c x+i) \left(c^2 d-e\right)^{3/2}}\right)+2 b c \tan ^{-1}(c x) \sqrt{d+e x^2}-2 b \sqrt{e} \log \left(\sqrt{e} \sqrt{d+e x^2}+e x\right)}{2 c 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,"(2*a*c*Sqrt[d + e*x^2] + 2*b*c*Sqrt[d + e*x^2]*ArcTan[c*x] - I*b*Sqrt[c^2*d - e]*Log[(4*c^2*e*((-I)*c*d + e*x - I*Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(3/2)*(-I + c*x))] + I*b*Sqrt[c^2*d - e]*Log[(4*c^2*e*(I*c*d + e*x + I*Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(3/2)*(I + c*x))] - 2*b*Sqrt[e]*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/(2*c*e)","C",1
1204,0,0,23,3.6609869,"\int \frac{a+b \tan ^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","Integrate[(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,"Integrate[(a + b*ArcTan[c*x])/Sqrt[d + e*x^2], x]","A",-1
1205,0,0,51,5.7794519,"\int \frac{a+b \tan ^{-1}(c x)}{x \sqrt{d+e x^2}} \, dx","Integrate[(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,"Integrate[(a + b*ArcTan[c*x])/(x*Sqrt[d + e*x^2]), x]","A",-1
1206,1,247,100,0.4479232,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^2*Sqrt[d + e*x^2]),x]","\frac{-2 a \sqrt{d+e x^2}+b x \sqrt{c^2 d-e} \log \left(-\frac{4 c d \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \left(c^2 d-e\right)^{3/2}}\right)+b x \sqrt{c^2 d-e} \log \left(-\frac{4 c d \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \left(c^2 d-e\right)^{3/2}}\right)-2 b c \sqrt{d} x \log \left(\sqrt{d} \sqrt{d+e x^2}+d\right)-2 b \tan ^{-1}(c x) \sqrt{d+e x^2}+2 b c \sqrt{d} x \log (x)}{2 d 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}}",1,"(-2*a*Sqrt[d + e*x^2] - 2*b*Sqrt[d + e*x^2]*ArcTan[c*x] + 2*b*c*Sqrt[d]*x*Log[x] - 2*b*c*Sqrt[d]*x*Log[d + Sqrt[d]*Sqrt[d + e*x^2]] + b*Sqrt[c^2*d - e]*x*Log[(-4*c*d*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(3/2)*(I + c*x))] + b*Sqrt[c^2*d - e]*x*Log[(-4*c*d*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - e)^(3/2)*(-I + c*x))])/(2*d*x)","C",1
1207,0,0,76,57.955414,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 \sqrt{d+e x^2}} \, dx","Integrate[(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,"Integrate[(a + b*ArcTan[c*x])/(x^3*Sqrt[d + e*x^2]), x]","A",-1
1208,1,372,179,0.5312886,"\int \frac{a+b \tan ^{-1}(c x)}{x^4 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^4*Sqrt[d + e*x^2]),x]","-\frac{\frac{\sqrt{d+e x^2} \left(2 a \left(d-2 e x^2\right)+b c d x\right)}{x^3}-b c \sqrt{d} \left(2 c^2 d+3 e\right) \log \left(\sqrt{d} \sqrt{d+e x^2}+d\right)+b c \sqrt{d} \log (x) \left(2 c^2 d+3 e\right)+\frac{b \left(c^4 d^2+c^2 d e-2 e^2\right) \log \left(\frac{12 c d^2 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \sqrt{c^2 d-e} \left(c^4 d^2+c^2 d e-2 e^2\right)}\right)}{\sqrt{c^2 d-e}}+\frac{b \left(c^4 d^2+c^2 d e-2 e^2\right) \log \left(\frac{12 c d^2 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \sqrt{c^2 d-e} \left(c^4 d^2+c^2 d e-2 e^2\right)}\right)}{\sqrt{c^2 d-e}}+\frac{2 b \tan ^{-1}(c x) \left(d-2 e x^2\right) \sqrt{d+e x^2}}{x^3}}{6 d^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,"-1/6*((Sqrt[d + e*x^2]*(b*c*d*x + 2*a*(d - 2*e*x^2)))/x^3 + (2*b*(d - 2*e*x^2)*Sqrt[d + e*x^2]*ArcTan[c*x])/x^3 + b*c*Sqrt[d]*(2*c^2*d + 3*e)*Log[x] - b*c*Sqrt[d]*(2*c^2*d + 3*e)*Log[d + Sqrt[d]*Sqrt[d + e*x^2]] + (b*(c^4*d^2 + c^2*d*e - 2*e^2)*Log[(12*c*d^2*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(c^4*d^2 + c^2*d*e - 2*e^2)*(I + c*x))])/Sqrt[c^2*d - e] + (b*(c^4*d^2 + c^2*d*e - 2*e^2)*Log[(12*c*d^2*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(c^4*d^2 + c^2*d*e - 2*e^2)*(-I + c*x))])/Sqrt[c^2*d - e])/d^2","C",1
1209,1,321,137,0.6577927,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x]))/(d + e*x^2)^(3/2),x]","\frac{\frac{2 a \left(2 d+e x^2\right)}{\sqrt{d+e x^2}}-\frac{i b \left(2 c^2 d-e\right) \log \left(\frac{4 c^2 e^2 \left(-i \sqrt{c^2 d-e} \sqrt{d+e x^2}-i c d+e x\right)}{b (c x-i) \sqrt{c^2 d-e} \left(2 c^2 d-e\right)}\right)}{c \sqrt{c^2 d-e}}+\frac{i b \left(2 c^2 d-e\right) \log \left(\frac{4 c^2 e^2 \left(i \sqrt{c^2 d-e} \sqrt{d+e x^2}+i c d+e x\right)}{b (c x+i) \sqrt{c^2 d-e} \left(2 c^2 d-e\right)}\right)}{c \sqrt{c^2 d-e}}-\frac{2 b \sqrt{e} \log \left(\sqrt{e} \sqrt{d+e x^2}+e x\right)}{c}+\frac{2 b \tan ^{-1}(c x) \left(2 d+e x^2\right)}{\sqrt{d+e x^2}}}{2 e^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,"((2*a*(2*d + e*x^2))/Sqrt[d + e*x^2] + (2*b*(2*d + e*x^2)*ArcTan[c*x])/Sqrt[d + e*x^2] - (I*b*(2*c^2*d - e)*Log[(4*c^2*e^2*((-I)*c*d + e*x - I*Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(2*c^2*d - e)*(-I + c*x))])/(c*Sqrt[c^2*d - e]) + (I*b*(2*c^2*d - e)*Log[(4*c^2*e^2*(I*c*d + e*x + I*Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(2*c^2*d - e)*(I + c*x))])/(c*Sqrt[c^2*d - e]) - (2*b*Sqrt[e]*Log[e*x + Sqrt[e]*Sqrt[d + e*x^2]])/c)/(2*e^2)","C",1
1210,0,0,69,20.1247224,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^2*(a + b*ArcTan[c*x]))/(d + e*x^2)^(3/2), x]","A",-1
1211,1,210,71,0.3978454,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x*(a + b*ArcTan[c*x]))/(d + e*x^2)^(3/2),x]","-\frac{\frac{2 a}{\sqrt{d+e x^2}}+\frac{i b c \log \left(-\frac{4 i e \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \sqrt{c^2 d-e}}\right)}{\sqrt{c^2 d-e}}-\frac{i b c \log \left(\frac{4 i e \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \sqrt{c^2 d-e}}\right)}{\sqrt{c^2 d-e}}+\frac{2 b \tan ^{-1}(c x)}{\sqrt{d+e x^2}}}{2 e}","\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,"-1/2*((2*a)/Sqrt[d + e*x^2] + (2*b*ArcTan[c*x])/Sqrt[d + e*x^2] + (I*b*c*Log[((-4*I)*e*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(I + c*x))])/Sqrt[c^2*d - e] - (I*b*c*Log[((4*I)*e*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(-I + c*x))])/Sqrt[c^2*d - e])/e","C",1
1212,1,202,70,0.28823,"\int \frac{a+b \tan ^{-1}(c x)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcTan[c*x])/(d + e*x^2)^(3/2),x]","\frac{\frac{2 a x}{\sqrt{d+e x^2}}+\frac{b \log \left(-\frac{4 c d \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \sqrt{c^2 d-e}}\right)}{\sqrt{c^2 d-e}}+\frac{b \log \left(-\frac{4 c d \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \sqrt{c^2 d-e}}\right)}{\sqrt{c^2 d-e}}+\frac{2 b x \tan ^{-1}(c x)}{\sqrt{d+e x^2}}}{2 d}","\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,"((2*a*x)/Sqrt[d + e*x^2] + (2*b*x*ArcTan[c*x])/Sqrt[d + e*x^2] + (b*Log[(-4*c*d*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(I + c*x))])/Sqrt[c^2*d - e] + (b*Log[(-4*c*d*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(-I + c*x))])/Sqrt[c^2*d - e])/(2*d)","C",1
1213,0,0,67,56.7305094,"\int \frac{a+b \tan ^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(a + b*ArcTan[c*x])/(x*(d + e*x^2)^(3/2)), x]","A",-1
1214,1,306,135,0.6975418,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^(3/2)),x]","\frac{-\frac{2 a \left(d+2 e x^2\right)}{x \sqrt{d+e x^2}}+\frac{b \left(c^2 d-2 e\right) \log \left(-\frac{4 c d^2 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \left(c^2 d-2 e\right) \sqrt{c^2 d-e}}\right)}{\sqrt{c^2 d-e}}+\frac{b \left(c^2 d-2 e\right) \log \left(-\frac{4 c d^2 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \left(c^2 d-2 e\right) \sqrt{c^2 d-e}}\right)}{\sqrt{c^2 d-e}}-2 b c \sqrt{d} \log \left(\sqrt{d} \sqrt{d+e x^2}+d\right)-\frac{2 b \tan ^{-1}(c x) \left(d+2 e x^2\right)}{x \sqrt{d+e x^2}}+2 b c \sqrt{d} \log (x)}{2 d^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,"((-2*a*(d + 2*e*x^2))/(x*Sqrt[d + e*x^2]) - (2*b*(d + 2*e*x^2)*ArcTan[c*x])/(x*Sqrt[d + e*x^2]) + 2*b*c*Sqrt[d]*Log[x] - 2*b*c*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d + e*x^2]] + (b*(c^2*d - 2*e)*Log[(-4*c*d^2*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - 2*e)*Sqrt[c^2*d - e]*(I + c*x))])/Sqrt[c^2*d - e] + (b*(c^2*d - 2*e)*Log[(-4*c*d^2*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^2*d - 2*e)*Sqrt[c^2*d - e]*(-I + c*x))])/Sqrt[c^2*d - e])/(2*d^2)","C",1
1215,0,0,96,60.608843,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(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 \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{2 d^{5/2}}-\frac{3 a e}{2 d^2 \sqrt{d+e x^2}}-\frac{a}{2 d x^2 \sqrt{d+e x^2}}",0,"Integrate[(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)^(3/2)), x]","A",-1
1216,1,405,249,0.7598414,"\int \frac{a+b \tan ^{-1}(c x)}{x^4 \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^4*(d + e*x^2)^(3/2)),x]","-\frac{\frac{2 a \left(d^2-4 d e x^2-8 e^2 x^4\right)+b c d x \left(d+e x^2\right)}{x^3 \sqrt{d+e x^2}}-b c \sqrt{d} \left(2 c^2 d+9 e\right) \log \left(\sqrt{d} \sqrt{d+e x^2}+d\right)+b c \sqrt{d} \log (x) \left(2 c^2 d+9 e\right)+\frac{b \left(c^4 d^2+4 c^2 d e-8 e^2\right) \log \left(\frac{12 c d^3 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \sqrt{c^2 d-e} \left(c^4 d^2+4 c^2 d e-8 e^2\right)}\right)}{\sqrt{c^2 d-e}}+\frac{b \left(c^4 d^2+4 c^2 d e-8 e^2\right) \log \left(\frac{12 c d^3 \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \sqrt{c^2 d-e} \left(c^4 d^2+4 c^2 d e-8 e^2\right)}\right)}{\sqrt{c^2 d-e}}+\frac{2 b \tan ^{-1}(c x) \left(d^2-4 d e x^2-8 e^2 x^4\right)}{x^3 \sqrt{d+e x^2}}}{6 d^3}","\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 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 \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 e \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{6 d^{5/2}}-\frac{b c \sqrt{d+e x^2}}{6 d^2 x^2}",1,"-1/6*((b*c*d*x*(d + e*x^2) + 2*a*(d^2 - 4*d*e*x^2 - 8*e^2*x^4))/(x^3*Sqrt[d + e*x^2]) + (2*b*(d^2 - 4*d*e*x^2 - 8*e^2*x^4)*ArcTan[c*x])/(x^3*Sqrt[d + e*x^2]) + b*c*Sqrt[d]*(2*c^2*d + 9*e)*Log[x] - b*c*Sqrt[d]*(2*c^2*d + 9*e)*Log[d + Sqrt[d]*Sqrt[d + e*x^2]] + (b*(c^4*d^2 + 4*c^2*d*e - 8*e^2)*Log[(12*c*d^3*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(c^4*d^2 + 4*c^2*d*e - 8*e^2)*(I + c*x))])/Sqrt[c^2*d - e] + (b*(c^4*d^2 + 4*c^2*d*e - 8*e^2)*Log[(12*c*d^3*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*Sqrt[c^2*d - e]*(c^4*d^2 + 4*c^2*d*e - 8*e^2)*(-I + c*x))])/Sqrt[c^2*d - e])/d^3","C",1
1217,0,0,91,12.0603017,"\int \frac{x^4 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(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 \tanh ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right)}{e^{5/2}}-\frac{a x}{e^2 \sqrt{d+e x^2}}-\frac{a x^3}{3 e \left(d+e x^2\right)^{3/2}}",0,"Integrate[(x^4*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2), x]","A",-1
1218,1,326,143,0.5985027,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2),x]","\frac{2 \sqrt{c^2 d-e} \left(b c e x \left(d+e x^2\right)-a \left(c^2 d-e\right) \left(2 d+3 e x^2\right)\right)-i b c \left(2 c^2 d-3 e\right) \left(d+e x^2\right)^{3/2} \log \left(-\frac{12 i e^2 \sqrt{c^2 d-e} \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \left(2 c^2 d-3 e\right)}\right)+i b c \left(2 c^2 d-3 e\right) \left(d+e x^2\right)^{3/2} \log \left(\frac{12 i e^2 \sqrt{c^2 d-e} \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \left(2 c^2 d-3 e\right)}\right)-2 b \left(c^2 d-e\right)^{3/2} \tan ^{-1}(c x) \left(2 d+3 e x^2\right)}{6 e^2 \left(c^2 d-e\right)^{3/2} \left(d+e x^2\right)^{3/2}}","-\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,"(2*Sqrt[c^2*d - e]*(b*c*e*x*(d + e*x^2) - a*(c^2*d - e)*(2*d + 3*e*x^2)) - 2*b*(c^2*d - e)^(3/2)*(2*d + 3*e*x^2)*ArcTan[c*x] - I*b*c*(2*c^2*d - 3*e)*(d + e*x^2)^(3/2)*Log[((-12*I)*Sqrt[c^2*d - e]*e^2*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(2*c^2*d - 3*e)*(I + c*x))] + I*b*c*(2*c^2*d - 3*e)*(d + e*x^2)^(3/2)*Log[((12*I)*Sqrt[c^2*d - e]*e^2*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(2*c^2*d - 3*e)*(-I + c*x))])/(6*(c^2*d - e)^(3/2)*e^2*(d + e*x^2)^(3/2))","C",1
1219,1,252,109,1.1109369,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2),x]","-\frac{-\frac{2 \left(a x \left(c^2 d-e\right)+b c d\right)}{e \left(c^2 d-e\right) \sqrt{d+e x^2}}+\frac{2 a d x}{e \left(d+e x^2\right)^{3/2}}+\frac{b \log \left(\frac{12 c d \sqrt{c^2 d-e} \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i)}\right)}{\left(c^2 d-e\right)^{3/2}}+\frac{b \log \left(\frac{12 c d \sqrt{c^2 d-e} \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i)}\right)}{\left(c^2 d-e\right)^{3/2}}-\frac{2 b x^3 \tan ^{-1}(c x)}{\left(d+e x^2\right)^{3/2}}}{6 d}","\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,"-1/6*((2*a*d*x)/(e*(d + e*x^2)^(3/2)) - (2*(b*c*d + a*(c^2*d - e)*x))/((c^2*d - e)*e*Sqrt[d + e*x^2]) - (2*b*x^3*ArcTan[c*x])/(d + e*x^2)^(3/2) + (b*Log[(12*c*d*Sqrt[c^2*d - e]*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(I + c*x))])/(c^2*d - e)^(3/2) + (b*Log[(12*c*d*Sqrt[c^2*d - e]*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(-I + c*x))])/(c^2*d - e)^(3/2))/d","C",1
1220,1,259,110,0.779211,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2),x]","\frac{1}{6} \left(-\frac{2 a}{e \left(d+e x^2\right)^{3/2}}-\frac{2 b c x}{\left(c^2 d^2-d e\right) \sqrt{d+e x^2}}-\frac{i b c^3 \log \left(-\frac{12 i e \sqrt{c^2 d-e} \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b c^2 (c x+i)}\right)}{e \left(c^2 d-e\right)^{3/2}}+\frac{i b c^3 \log \left(\frac{12 i e \sqrt{c^2 d-e} \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b c^2 (c x-i)}\right)}{e \left(c^2 d-e\right)^{3/2}}-\frac{2 b \tan ^{-1}(c x)}{e \left(d+e x^2\right)^{3/2}}\right)","-\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,"((-2*a)/(e*(d + e*x^2)^(3/2)) - (2*b*c*x)/((c^2*d^2 - d*e)*Sqrt[d + e*x^2]) - (2*b*ArcTan[c*x])/(e*(d + e*x^2)^(3/2)) - (I*b*c^3*Log[((-12*I)*Sqrt[c^2*d - e]*e*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*c^2*(I + c*x))])/((c^2*d - e)^(3/2)*e) + (I*b*c^3*Log[((12*I)*Sqrt[c^2*d - e]*e*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*c^2*(-I + c*x))])/((c^2*d - e)^(3/2)*e))/6","C",1
1221,1,317,144,0.5996509,"\int \frac{a+b \tan ^{-1}(c x)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcTan[c*x])/(d + e*x^2)^(5/2),x]","\frac{2 \sqrt{c^2 d-e} \left(a x \left(c^2 d-e\right) \left(3 d+2 e x^2\right)-b c d \left(d+e x^2\right)\right)+b \left(3 c^2 d-2 e\right) \left(d+e x^2\right)^{3/2} \log \left(-\frac{12 c d^2 \sqrt{c^2 d-e} \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \left(3 c^2 d-2 e\right)}\right)+b \left(3 c^2 d-2 e\right) \left(d+e x^2\right)^{3/2} \log \left(-\frac{12 c d^2 \sqrt{c^2 d-e} \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \left(3 c^2 d-2 e\right)}\right)+2 b x \left(c^2 d-e\right)^{3/2} \tan ^{-1}(c x) \left(3 d+2 e x^2\right)}{6 d^2 \left(c^2 d-e\right)^{3/2} \left(d+e x^2\right)^{3/2}}","\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,"(2*Sqrt[c^2*d - e]*(-(b*c*d*(d + e*x^2)) + a*(c^2*d - e)*x*(3*d + 2*e*x^2)) + 2*b*(c^2*d - e)^(3/2)*x*(3*d + 2*e*x^2)*ArcTan[c*x] + b*(3*c^2*d - 2*e)*(d + e*x^2)^(3/2)*Log[(-12*c*d^2*Sqrt[c^2*d - e]*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(3*c^2*d - 2*e)*(I + c*x))] + b*(3*c^2*d - 2*e)*(d + e*x^2)^(3/2)*Log[(-12*c*d^2*Sqrt[c^2*d - e]*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(3*c^2*d - 2*e)*(-I + c*x))])/(6*d^2*(c^2*d - e)^(3/2)*(d + e*x^2)^(3/2))","C",1
1222,0,0,86,57.9723686,"\int \frac{a+b \tan ^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}} \, dx","Integrate[(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 \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{d^{5/2}}+\frac{a}{d^2 \sqrt{d+e x^2}}+\frac{a}{3 d \left(d+e x^2\right)^{3/2}}",0,"Integrate[(a + b*ArcTan[c*x])/(x*(d + e*x^2)^(5/2)), x]","A",-1
1223,1,418,274,1.1847622,"\int \frac{a+b \tan ^{-1}(c x)}{x^2 \left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^(5/2)),x]","\frac{\frac{2 e \left(5 a x \left(e-c^2 d\right)+b c d\right)}{\left(c^2 d-e\right) \sqrt{d+e x^2}}-\frac{6 a \sqrt{d+e x^2}}{x}-\frac{2 a d e 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) \log \left(-\frac{12 c d^3 \sqrt{c^2 d-e} \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \left(3 c^4 d^2-12 c^2 d e+8 e^2\right)}\right)}{\left(c^2 d-e\right)^{3/2}}+\frac{b \left(3 c^4 d^2-12 c^2 d e+8 e^2\right) \log \left(-\frac{12 c d^3 \sqrt{c^2 d-e} \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \left(3 c^4 d^2-12 c^2 d e+8 e^2\right)}\right)}{\left(c^2 d-e\right)^{3/2}}-\frac{2 b \tan ^{-1}(c x) \left(3 d^2+12 d e x^2+8 e^2 x^4\right)}{x \left(d+e x^2\right)^{3/2}}-6 b c \sqrt{d} \log \left(\sqrt{d} \sqrt{d+e x^2}+d\right)+6 b c \sqrt{d} \log (x)}{6 d^3}","-\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 \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{d^{5/2}}-\frac{8 b e}{3 c d^3 \sqrt{d+e x^2}}+\frac{b c}{d^2 \sqrt{d+e x^2}}",1,"((-2*a*d*e*x)/(d + e*x^2)^(3/2) + (2*e*(b*c*d + 5*a*(-(c^2*d) + e)*x))/((c^2*d - e)*Sqrt[d + e*x^2]) - (6*a*Sqrt[d + e*x^2])/x - (2*b*(3*d^2 + 12*d*e*x^2 + 8*e^2*x^4)*ArcTan[c*x])/(x*(d + e*x^2)^(3/2)) + 6*b*c*Sqrt[d]*Log[x] - 6*b*c*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d + e*x^2]] + (b*(3*c^4*d^2 - 12*c^2*d*e + 8*e^2)*Log[(-12*c*d^3*Sqrt[c^2*d - e]*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(3*c^4*d^2 - 12*c^2*d*e + 8*e^2)*(I + c*x))])/(c^2*d - e)^(3/2) + (b*(3*c^4*d^2 - 12*c^2*d*e + 8*e^2)*Log[(-12*c*d^3*Sqrt[c^2*d - e]*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(3*c^4*d^2 - 12*c^2*d*e + 8*e^2)*(-I + c*x))])/(c^2*d - e)^(3/2))/(6*d^3)","C",1
1224,0,0,116,61.439934,"\int \frac{a+b \tan ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}} \, dx","Integrate[(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 \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{2 d^{7/2}}-\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{a}{2 d x^2 \left(d+e x^2\right)^{3/2}}",0,"Integrate[(a + b*ArcTan[c*x])/(x^3*(d + e*x^2)^(5/2)), x]","A",-1
1225,1,510,423,1.962606,"\int \frac{a+b \tan ^{-1}(c x)}{x^4 \left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcTan[c*x])/(x^4*(d + e*x^2)^(5/2)),x]","-\frac{\frac{2 a \left(d^3-6 d^2 e x^2-24 d e^2 x^4-16 e^3 x^6\right)}{x^3 \left(d+e x^2\right)^{3/2}}+\frac{b c d \left(c^2 d \left(d+e x^2\right)+e \left(e x^2-d\right)\right)}{x^2 \left(c^2 d-e\right) \sqrt{d+e x^2}}-b c \sqrt{d} \left(2 c^2 d+15 e\right) \log \left(\sqrt{d} \sqrt{d+e x^2}+d\right)+b c \sqrt{d} \log (x) \left(2 c^2 d+15 e\right)+\frac{b \left(c^6 d^3+6 c^4 d^2 e-24 c^2 d e^2+16 e^3\right) \log \left(\frac{12 c d^4 \sqrt{c^2 d-e} \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d-i e x\right)}{b (c x+i) \left(c^6 d^3+6 c^4 d^2 e-24 c^2 d e^2+16 e^3\right)}\right)}{\left(c^2 d-e\right)^{3/2}}+\frac{b \left(c^6 d^3+6 c^4 d^2 e-24 c^2 d e^2+16 e^3\right) \log \left(\frac{12 c d^4 \sqrt{c^2 d-e} \left(\sqrt{c^2 d-e} \sqrt{d+e x^2}+c d+i e x\right)}{b (c x-i) \left(c^6 d^3+6 c^4 d^2 e-24 c^2 d e^2+16 e^3\right)}\right)}{\left(c^2 d-e\right)^{3/2}}+\frac{2 b \tan ^{-1}(c x) \left(d^3-6 d^2 e x^2-24 d e^2 x^4-16 e^3 x^6\right)}{x^3 \left(d+e x^2\right)^{3/2}}}{6 d^4}","\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 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{b c \left(c^2 d+6 e\right)}{3 d^3 \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)}{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 e \tanh ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right)}{2 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}}",1,"-1/6*((2*a*(d^3 - 6*d^2*e*x^2 - 24*d*e^2*x^4 - 16*e^3*x^6))/(x^3*(d + e*x^2)^(3/2)) + (b*c*d*(e*(-d + e*x^2) + c^2*d*(d + e*x^2)))/((c^2*d - e)*x^2*Sqrt[d + e*x^2]) + (2*b*(d^3 - 6*d^2*e*x^2 - 24*d*e^2*x^4 - 16*e^3*x^6)*ArcTan[c*x])/(x^3*(d + e*x^2)^(3/2)) + b*c*Sqrt[d]*(2*c^2*d + 15*e)*Log[x] - b*c*Sqrt[d]*(2*c^2*d + 15*e)*Log[d + Sqrt[d]*Sqrt[d + e*x^2]] + (b*(c^6*d^3 + 6*c^4*d^2*e - 24*c^2*d*e^2 + 16*e^3)*Log[(12*c*d^4*Sqrt[c^2*d - e]*(c*d - I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^6*d^3 + 6*c^4*d^2*e - 24*c^2*d*e^2 + 16*e^3)*(I + c*x))])/(c^2*d - e)^(3/2) + (b*(c^6*d^3 + 6*c^4*d^2*e - 24*c^2*d*e^2 + 16*e^3)*Log[(12*c*d^4*Sqrt[c^2*d - e]*(c*d + I*e*x + Sqrt[c^2*d - e]*Sqrt[d + e*x^2]))/(b*(c^6*d^3 + 6*c^4*d^2*e - 24*c^2*d*e^2 + 16*e^3)*(-I + c*x))])/(c^2*d - e)^(3/2))/d^4","C",1
1226,1,345,208,0.9550834,"\int \frac{\tan ^{-1}(a x)}{\left(c+d x^2\right)^{7/2}} \, dx","Integrate[ArcTan[a*x]/(c + d*x^2)^(7/2),x]","\frac{-\frac{2 a c \left(a^2 c \left(8 c+7 d x^2\right)-d \left(5 c+4 d x^2\right)\right)}{\left(d-a^2 c\right)^2 \left(c+d x^2\right)^{3/2}}+\frac{\left(15 a^4 c^2-20 a^2 c d+8 d^2\right) \log \left(-\frac{60 a c^3 \left(a^2 c-d\right)^{3/2} \left(\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c-i d x\right)}{(a x+i) \left(15 a^4 c^2-20 a^2 c d+8 d^2\right)}\right)}{\left(a^2 c-d\right)^{5/2}}+\frac{\left(15 a^4 c^2-20 a^2 c d+8 d^2\right) \log \left(-\frac{60 a c^3 \left(a^2 c-d\right)^{3/2} \left(\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c+i d x\right)}{(a x-i) \left(15 a^4 c^2-20 a^2 c d+8 d^2\right)}\right)}{\left(a^2 c-d\right)^{5/2}}+\frac{2 x \tan ^{-1}(a x) \left(15 c^2+20 c d x^2+8 d^2 x^4\right)}{\left(c+d x^2\right)^{5/2}}}{30 c^3}","-\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{\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{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,"((-2*a*c*(-(d*(5*c + 4*d*x^2)) + a^2*c*(8*c + 7*d*x^2)))/((-(a^2*c) + d)^2*(c + d*x^2)^(3/2)) + (2*x*(15*c^2 + 20*c*d*x^2 + 8*d^2*x^4)*ArcTan[a*x])/(c + d*x^2)^(5/2) + ((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*Log[(-60*a*c^3*(a^2*c - d)^(3/2)*(a*c - I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*(I + a*x))])/(a^2*c - d)^(5/2) + ((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*Log[(-60*a*c^3*(a^2*c - d)^(3/2)*(a*c + I*d*x + Sqrt[a^2*c - d]*Sqrt[c + d*x^2]))/((15*a^4*c^2 - 20*a^2*c*d + 8*d^2)*(-I + a*x))])/(a^2*c - d)^(5/2))/(30*c^3)","C",1
1227,1,450,293,1.5361531,"\int \frac{\tan ^{-1}(a x)}{\left(c+d x^2\right)^{9/2}} \, dx","Integrate[ArcTan[a*x]/(c + d*x^2)^(9/2),x]","\frac{-\frac{2 a c \left(3 c^2 \left(d-a^2 c\right)^2+c \left(11 a^2 c-6 d\right) \left(a^2 c-d\right) \left(c+d x^2\right)+3 \left(19 a^4 c^2-22 a^2 c d+8 d^2\right) \left(c+d x^2\right)^2\right)}{\left(a^2 c-d\right)^3 \left(c+d x^2\right)^{5/2}}+\frac{3 \left(35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right) \log \left(-\frac{140 a c^4 \left(a^2 c-d\right)^{5/2} \left(\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c-i d x\right)}{(a x+i) \left(35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right)}\right)}{\left(a^2 c-d\right)^{7/2}}+\frac{3 \left(35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right) \log \left(-\frac{140 a c^4 \left(a^2 c-d\right)^{5/2} \left(\sqrt{a^2 c-d} \sqrt{c+d x^2}+a c+i d x\right)}{(a x-i) \left(35 a^6 c^3-70 a^4 c^2 d+56 a^2 c d^2-16 d^3\right)}\right)}{\left(a^2 c-d\right)^{7/2}}+\frac{6 x \tan ^{-1}(a x) \left(35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right)}{\left(c+d x^2\right)^{7/2}}}{210 c^4}","-\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{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(35 a^6 c^3-70 a^4 c^2 d+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{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,"((-2*a*c*(3*c^2*(-(a^2*c) + d)^2 + c*(11*a^2*c - 6*d)*(a^2*c - d)*(c + d*x^2) + 3*(19*a^4*c^2 - 22*a^2*c*d + 8*d^2)*(c + d*x^2)^2))/((a^2*c - d)^3*(c + d*x^2)^(5/2)) + (6*x*(35*c^3 + 70*c^2*d*x^2 + 56*c*d^2*x^4 + 16*d^3*x^6)*ArcTan[a*x])/(c + d*x^2)^(7/2) + (3*(35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*Log[(-140*a*c^4*(a^2*c - d)^(5/2)*(a*c - I*d*x + Sqrt[a^2*c - d]*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)*(I + a*x))])/(a^2*c - d)^(7/2) + (3*(35*a^6*c^3 - 70*a^4*c^2*d + 56*a^2*c*d^2 - 16*d^3)*Log[(-140*a*c^4*(a^2*c - d)^(5/2)*(a*c + I*d*x + Sqrt[a^2*c - d]*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)*(-I + a*x))])/(a^2*c - d)^(7/2))/(210*c^4)","C",1
1228,1,264,378,0.61065,"\int x^m \left(d+e x^2\right)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^m*(d + e*x^2)^3*(a + b*ArcTan[c*x]),x]","x^{m+1} \left(\frac{d^3 \left(a+b \tan ^{-1}(c x)\right)}{m+1}+\frac{3 d^2 e x^2 \left(a+b \tan ^{-1}(c x)\right)}{m+3}+\frac{3 d e^2 x^4 \left(a+b \tan ^{-1}(c x)\right)}{m+5}+\frac{e^3 x^6 \left(a+b \tan ^{-1}(c x)\right)}{m+7}-\frac{b c d^3 x \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{m^2+3 m+2}-\frac{3 b c d^2 e x^3 \, _2F_1\left(1,\frac{m+4}{2};\frac{m+6}{2};-c^2 x^2\right)}{m^2+7 m+12}-\frac{3 b c d e^2 x^5 \, _2F_1\left(1,\frac{m+6}{2};\frac{m+8}{2};-c^2 x^2\right)}{(m+5) (m+6)}-\frac{b c e^3 x^7 \, _2F_1\left(1,\frac{m}{2}+4;\frac{m}{2}+5;-c^2 x^2\right)}{(m+7) (m+8)}\right)","\frac{d^3 x^{m+1} \left(a+b \tan ^{-1}(c x)\right)}{m+1}+\frac{3 d^2 e x^{m+3} \left(a+b \tan ^{-1}(c x)\right)}{m+3}+\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 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 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 x^{m+2} \left(c^6 \left(-d^3\right) \left(m^3+15 m^2+71 m+105\right)+3 c^4 d^2 e \left(m^3+13 m^2+47 m+35\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^3 x^{m+6}}{c (m+6) (m+7)}",1,"x^(1 + m)*((d^3*(a + b*ArcTan[c*x]))/(1 + m) + (3*d^2*e*x^2*(a + b*ArcTan[c*x]))/(3 + m) + (3*d*e^2*x^4*(a + b*ArcTan[c*x]))/(5 + m) + (e^3*x^6*(a + b*ArcTan[c*x]))/(7 + m) - (b*c*e^3*x^7*Hypergeometric2F1[1, 4 + m/2, 5 + m/2, -(c^2*x^2)])/((7 + m)*(8 + m)) - (b*c*d^3*x*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/(2 + 3*m + m^2) - (3*b*c*d^2*e*x^3*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, -(c^2*x^2)])/(12 + 7*m + m^2) - (3*b*c*d*e^2*x^5*Hypergeometric2F1[1, (6 + m)/2, (8 + m)/2, -(c^2*x^2)])/((5 + m)*(6 + m)))","A",1
1229,1,193,230,0.2240222,"\int x^m \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^m*(d + e*x^2)^2*(a + b*ArcTan[c*x]),x]","x^{m+1} \left(\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)}{m+1}+\frac{2 d e x^2 \left(a+b \tan ^{-1}(c x)\right)}{m+3}+\frac{e^2 x^4 \left(a+b \tan ^{-1}(c x)\right)}{m+5}-\frac{b c d^2 x \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{m^2+3 m+2}-\frac{2 b c d e x^3 \, _2F_1\left(1,\frac{m+4}{2};\frac{m+6}{2};-c^2 x^2\right)}{m^2+7 m+12}-\frac{b c e^2 x^5 \, _2F_1\left(1,\frac{m+6}{2};\frac{m+8}{2};-c^2 x^2\right)}{(m+5) (m+6)}\right)","\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 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 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^2 x^{m+4}}{c (m+4) (m+5)}",1,"x^(1 + m)*((d^2*(a + b*ArcTan[c*x]))/(1 + m) + (2*d*e*x^2*(a + b*ArcTan[c*x]))/(3 + m) + (e^2*x^4*(a + b*ArcTan[c*x]))/(5 + m) - (b*c*d^2*x*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/(2 + 3*m + m^2) - (2*b*c*d*e*x^3*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, -(c^2*x^2)])/(12 + 7*m + m^2) - (b*c*e^2*x^5*Hypergeometric2F1[1, (6 + m)/2, (8 + m)/2, -(c^2*x^2)])/((5 + m)*(6 + m)))","A",1
1230,1,119,122,0.1895158,"\int x^m \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^m*(d + e*x^2)*(a + b*ArcTan[c*x]),x]","x^{m+1} \left(\frac{\frac{\left(d (m+3)+e (m+1) x^2\right) \left(a+b \tan ^{-1}(c x)\right)}{m+1}-\frac{b c e x^3 \, _2F_1\left(1,\frac{m+4}{2};\frac{m+6}{2};-c^2 x^2\right)}{m+4}}{m+3}-\frac{b c d x \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{m^2+3 m+2}\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,"x^(1 + m)*(-((b*c*d*x*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, -(c^2*x^2)])/(2 + 3*m + m^2)) + (((d*(3 + m) + e*(1 + m)*x^2)*(a + b*ArcTan[c*x]))/(1 + m) - (b*c*e*x^3*Hypergeometric2F1[1, (4 + m)/2, (6 + m)/2, -(c^2*x^2)])/(4 + m))/(3 + m))","A",1
1231,0,0,63,2.3932103,"\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(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,"Integrate[(x^m*(a + b*ArcTan[c*x]))/(d + e*x^2), x]","A",-1
1232,0,0,63,5.5284033,"\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(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,"Integrate[(x^m*(a + b*ArcTan[c*x]))/(d + e*x^2)^2, x]","A",-1
1233,0,0,76,4.1528497,"\int x^m \left(d+e x^2\right)^{5/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[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,"Integrate[x^m*(d + e*x^2)^(5/2)*(a + b*ArcTan[c*x]), x]","A",-1
1234,0,0,76,0.1283156,"\int x^m \left(d+e x^2\right)^{3/2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[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,"Integrate[x^m*(d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]), x]","A",-1
1235,0,0,76,0.1130759,"\int x^m \sqrt{d+e x^2} \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[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,"Integrate[x^m*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x]), x]","A",-1
1236,0,0,76,3.6610345,"\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(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,"Integrate[(x^m*(a + b*ArcTan[c*x]))/Sqrt[d + e*x^2], x]","A",-1
1237,0,0,74,4.6275838,"\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(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,"Integrate[(x^m*(a + b*ArcTan[c*x]))/(d + e*x^2)^(3/2), x]","A",-1
1238,0,0,76,6.1607167,"\int \frac{x^m \left(a+b \tan ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(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,"Integrate[(x^m*(a + b*ArcTan[c*x]))/(d + e*x^2)^(5/2), x]","A",-1
1239,0,0,77,3.3484873,"\int x^m \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[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,"Integrate[x^m*(d + e*x^2)^p*(a + b*ArcTan[c*x]), x]","A",-1
1240,0,0,81,3.2451732,"\int x^{-2-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[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,"Integrate[x^(-2 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]), x]","A",-1
1241,1,166,129,0.4516241,"\int x^{-3-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[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 \left(\frac{e x^2}{d}+1\right)^{-p} \left(c (2 p+1) \left(d+e x^2\right) \left(\frac{e x^2}{d}+1\right)^p \left(a+b \tan ^{-1}(c x)\right)+b x \left(c^2 d-e\right) F_1\left(-p-\frac{1}{2};-p,1;\frac{1}{2}-p;-\frac{e x^2}{d},-c^2 x^2\right)+b e x \, _2F_1\left(-p-\frac{1}{2},-p;\frac{1}{2}-p;-\frac{e x^2}{d}\right)\right)}{2 c d (p+1) (2 p+1)}","-\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,"-1/2*((d + e*x^2)^p*(b*(c^2*d - e)*x*AppellF1[-1/2 - p, -p, 1, 1/2 - p, -((e*x^2)/d), -(c^2*x^2)] + c*(1 + 2*p)*(d + e*x^2)*(1 + (e*x^2)/d)^p*(a + b*ArcTan[c*x]) + b*e*x*Hypergeometric2F1[-1/2 - p, -p, 1/2 - p, -((e*x^2)/d)]))/(c*d*(1 + p)*(1 + 2*p)*x^(2*(1 + p))*(1 + (e*x^2)/d)^p)","A",0
1242,0,0,81,3.5745899,"\int x^{-4-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[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,"Integrate[x^(-4 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]), x]","A",-1
1243,0,0,285,4.6306352,"\int x^{-5-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^(-5 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]","\int x^{-5-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","\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,"Integrate[x^(-5 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]), x]","F",-1
1244,0,0,81,4.1592542,"\int x^{-6-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[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,"Integrate[x^(-6 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]), x]","A",-1
1245,0,0,466,5.3216237,"\int x^{-7-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[x^(-7 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]),x]","\int x^{-7-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","-\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,"Integrate[x^(-7 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]), x]","F",-1
1246,0,0,81,3.2158263,"\int x^{-8-2 p} \left(d+e x^2\right)^p \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[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,"Integrate[x^(-8 - 2*p)*(d + e*x^2)^p*(a + b*ArcTan[c*x]), x]","A",-1
1247,1,240,271,0.2688192,"\int x^3 \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d + e*x^2)*(a + b*ArcTan[c*x])^2,x]","\frac{c x \left(15 a^2 c^5 x^3 \left(3 d+2 e x^2\right)-2 a b \left(3 c^4 \left(5 d x^2+2 e x^4\right)-5 c^2 \left(9 d+2 e x^2\right)+30 e\right)+b^2 c x \left(3 c^2 \left(5 d+e x^2\right)-16 e\right)\right)+2 b \tan ^{-1}(c x) \left(15 a \left(c^6 \left(3 d x^4+2 e x^6\right)-3 c^2 d+2 e\right)+b c x \left(-3 c^4 \left(5 d x^2+2 e x^4\right)+5 c^2 \left(9 d+2 e x^2\right)-30 e\right)\right)+2 b^2 \left(23 e-30 c^2 d\right) \log \left(c^2 x^2+1\right)+15 b^2 \tan ^{-1}(c x)^2 \left(c^6 \left(3 d x^4+2 e x^6\right)-3 c^2 d+2 e\right)}{180 c^6}","\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{6 c^6}-\frac{a b e x}{3 c^5}-\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}+\frac{a b d x}{2 c^3}+\frac{b e x^3 \left(a+b \tan ^{-1}(c x)\right)}{9 c^3}+\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 e x \tan ^{-1}(c x)}{3 c^5}-\frac{4 b^2 e x^2}{45 c^4}+\frac{b^2 d x \tan ^{-1}(c x)}{2 c^3}+\frac{b^2 d x^2}{12 c^2}+\frac{b^2 e x^4}{60 c^2}+\frac{23 b^2 e \log \left(c^2 x^2+1\right)}{90 c^6}-\frac{b^2 d \log \left(c^2 x^2+1\right)}{3 c^4}",1,"(c*x*(15*a^2*c^5*x^3*(3*d + 2*e*x^2) + b^2*c*x*(-16*e + 3*c^2*(5*d + e*x^2)) - 2*a*b*(30*e - 5*c^2*(9*d + 2*e*x^2) + 3*c^4*(5*d*x^2 + 2*e*x^4))) + 2*b*(b*c*x*(-30*e + 5*c^2*(9*d + 2*e*x^2) - 3*c^4*(5*d*x^2 + 2*e*x^4)) + 15*a*(-3*c^2*d + 2*e + c^6*(3*d*x^4 + 2*e*x^6)))*ArcTan[c*x] + 15*b^2*(-3*c^2*d + 2*e + c^6*(3*d*x^4 + 2*e*x^6))*ArcTan[c*x]^2 + 2*b^2*(-30*c^2*d + 23*e)*Log[1 + c^2*x^2])/(180*c^6)","A",1
1248,1,287,323,0.9320451,"\int x^2 \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d + e*x^2)*(a + b*ArcTan[c*x])^2,x]","\frac{10 a^2 c^5 d x^3+6 a^2 c^5 e x^5-10 a b c^4 d x^2-3 a b c^4 e x^4+10 a b c^2 d \log \left(c^2 x^2+1\right)+6 a b c^2 e x^2-6 a b e \log \left(c^2 x^2+1\right)-b \tan ^{-1}(c x) \left(-4 a c^5 x^3 \left(5 d+3 e x^2\right)+b \left(c^2 x^2+1\right) \left(c^2 \left(10 d+3 e x^2\right)-9 e\right)+4 b \left(5 c^2 d-3 e\right) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+9 a b e+10 b^2 c^3 d x+b^2 c^3 e x^3+2 i b^2 \left(5 c^2 d-3 e\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+2 b^2 \tan ^{-1}(c x)^2 \left(c^5 \left(5 d x^3+3 e x^5\right)+5 i c^2 d-3 i e\right)-9 b^2 c e x}{30 c^5}","\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{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{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{i b^2 e \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{5 c^5}+\frac{3 b^2 e \tan ^{-1}(c x)}{10 c^5}-\frac{3 b^2 e x}{10 c^4}-\frac{i b^2 d \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{3 c^3}-\frac{b^2 d \tan ^{-1}(c x)}{3 c^3}+\frac{b^2 d x}{3 c^2}+\frac{b^2 e x^3}{30 c^2}",1,"(9*a*b*e + 10*b^2*c^3*d*x - 9*b^2*c*e*x - 10*a*b*c^4*d*x^2 + 6*a*b*c^2*e*x^2 + 10*a^2*c^5*d*x^3 + b^2*c^3*e*x^3 - 3*a*b*c^4*e*x^4 + 6*a^2*c^5*e*x^5 + 2*b^2*((5*I)*c^2*d - (3*I)*e + c^5*(5*d*x^3 + 3*e*x^5))*ArcTan[c*x]^2 - b*ArcTan[c*x]*(-4*a*c^5*x^3*(5*d + 3*e*x^2) + b*(1 + c^2*x^2)*(-9*e + c^2*(10*d + 3*e*x^2)) + 4*b*(5*c^2*d - 3*e)*Log[1 + E^((2*I)*ArcTan[c*x])]) + 10*a*b*c^2*d*Log[1 + c^2*x^2] - 6*a*b*e*Log[1 + c^2*x^2] + (2*I)*b^2*(5*c^2*d - 3*e)*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(30*c^5)","A",0
1249,1,179,199,0.1882462,"\int x \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d + e*x^2)*(a + b*ArcTan[c*x])^2,x]","\frac{c x \left(3 a^2 c^3 x \left(2 d+e x^2\right)-2 a b c^2 \left(6 d+e x^2\right)+6 a b e+b^2 c e x\right)+2 b \tan ^{-1}(c x) \left(3 a c^4 \left(2 d x^2+e x^4\right)+6 a c^2 d-3 a e-b c^3 x \left(6 d+e x^2\right)+3 b c e x\right)+2 b^2 \left(3 c^2 d-2 e\right) \log \left(c^2 x^2+1\right)+3 b^2 \tan ^{-1}(c x)^2 \left(c^4 \left(2 d x^2+e x^4\right)+2 c^2 d-e\right)}{12 c^4}","-\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}+\frac{a b e x}{2 c^3}+\frac{d \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2}+\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 e x \tan ^{-1}(c x)}{2 c^3}+\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 d x \tan ^{-1}(c x)}{c}",1,"(c*x*(6*a*b*e + b^2*c*e*x + 3*a^2*c^3*x*(2*d + e*x^2) - 2*a*b*c^2*(6*d + e*x^2)) + 2*b*(6*a*c^2*d - 3*a*e + 3*b*c*e*x - b*c^3*x*(6*d + e*x^2) + 3*a*c^4*(2*d*x^2 + e*x^4))*ArcTan[c*x] + 3*b^2*(2*c^2*d - e + c^4*(2*d*x^2 + e*x^4))*ArcTan[c*x]^2 + 2*b^2*(3*c^2*d - 2*e)*Log[1 + c^2*x^2])/(12*c^4)","A",1
1250,1,208,231,0.4790965,"\int \left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x^2)*(a + b*ArcTan[c*x])^2,x]","\frac{3 a^2 c^3 d x+a^2 c^3 e x^3-3 a b c^2 d \log \left(c^2 x^2+1\right)-a b c^2 e x^2+a b e \log \left(c^2 x^2+1\right)-b \tan ^{-1}(c x) \left(-2 a c^3 x \left(3 d+e x^2\right)+2 b \left(e-3 c^2 d\right) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+b \left(c^2 e x^2+e\right)\right)-i b^2 \left(3 c^2 d-e\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+b^2 \tan ^{-1}(c x)^2 \left(c^3 \left(3 d x+e x^3\right)-3 i c^2 d+i e\right)+b^2 c e x}{3 c^3}","-\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{i b^2 e \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{3 c^3}-\frac{b^2 e \tan ^{-1}(c x)}{3 c^3}+\frac{b^2 e x}{3 c^2}+\frac{i b^2 d \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c}",1,"(3*a^2*c^3*d*x + b^2*c*e*x - a*b*c^2*e*x^2 + a^2*c^3*e*x^3 + b^2*((-3*I)*c^2*d + I*e + c^3*(3*d*x + e*x^3))*ArcTan[c*x]^2 - b*ArcTan[c*x]*(-2*a*c^3*x*(3*d + e*x^2) + b*(e + c^2*e*x^2) + 2*b*(-3*c^2*d + e)*Log[1 + E^((2*I)*ArcTan[c*x])]) - 3*a*b*c^2*d*Log[1 + c^2*x^2] + a*b*e*Log[1 + c^2*x^2] - I*b^2*(3*c^2*d - e)*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(3*c^3)","A",0
1251,1,263,217,0.3895372,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d + e*x^2)*(a + b*ArcTan[c*x])^2)/x,x]","a^2 d \log (x)+\frac{1}{2} a^2 e x^2+\frac{a b e \left(\left(c^2 x^2+1\right) \tan ^{-1}(c x)-c x\right)}{c^2}+i a b d (\text{Li}_2(-i c x)-\text{Li}_2(i c x))+\frac{b^2 e \left(\log \left(c^2 x^2+1\right)+\left(c^2 x^2+1\right) \tan ^{-1}(c x)^2-2 c x \tan ^{-1}(c x)\right)}{2 c^2}+b^2 d \left(i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+i \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+\frac{2}{3} i \tan ^{-1}(c x)^3+\tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-\tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-\frac{i \pi ^3}{24}\right)","\frac{e \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2}-i b d \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d \text{Li}_2\left(\frac{2}{i c x+1}-1\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+\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{1}{2} b^2 d \text{Li}_3\left(1-\frac{2}{i c x+1}\right)+\frac{1}{2} b^2 d \text{Li}_3\left(\frac{2}{i c x+1}-1\right)-\frac{b^2 e x \tan ^{-1}(c x)}{c}",1,"(a^2*e*x^2)/2 + (a*b*e*(-(c*x) + (1 + c^2*x^2)*ArcTan[c*x]))/c^2 + a^2*d*Log[x] + (b^2*e*(-2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 + Log[1 + c^2*x^2]))/(2*c^2) + I*a*b*d*(PolyLog[2, (-I)*c*x] - PolyLog[2, I*c*x]) + b^2*d*((-1/24*I)*Pi^3 + ((2*I)/3)*ArcTan[c*x]^3 + ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + PolyLog[3, E^((-2*I)*ArcTan[c*x])]/2 - PolyLog[3, -E^((2*I)*ArcTan[c*x])]/2)","A",0
1252,1,204,172,0.2990432,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d + e*x^2)*(a + b*ArcTan[c*x])^2)/x^2,x]","\frac{-a^2 c d+a^2 c e x^2+a b c d \left(c x \left(2 \log (c x)-\log \left(c^2 x^2+1\right)\right)-2 \tan ^{-1}(c x)\right)+a b e x \left(2 c x \tan ^{-1}(c x)-\log \left(c^2 x^2+1\right)\right)-b^2 c d \left(i c x \left(\tan ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)\right)+\tan ^{-1}(c x)^2-2 c x \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)\right)+b^2 e x \left(\tan ^{-1}(c x) \left((c x-i) \tan ^{-1}(c x)+2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)-i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)\right)}{c x}","-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{Li}_2\left(\frac{2}{1-i c x}-1\right)+\frac{i b^2 e \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c}",1,"(-(a^2*c*d) + a^2*c*e*x^2 + a*b*c*d*(-2*ArcTan[c*x] + c*x*(2*Log[c*x] - Log[1 + c^2*x^2])) + a*b*e*x*(2*c*x*ArcTan[c*x] - Log[1 + c^2*x^2]) + b^2*e*x*(ArcTan[c*x]*((-I + c*x)*ArcTan[c*x] + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) - b^2*c*d*(ArcTan[c*x]^2 - 2*c*x*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + I*c*x*(ArcTan[c*x]^2 + PolyLog[2, E^((2*I)*ArcTan[c*x])])))/(c*x)","A",0
1253,1,273,220,0.390524,"\int \frac{\left(d+e x^2\right) \left(a+b \tan ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d + e*x^2)*(a + b*ArcTan[c*x])^2)/x^3,x]","-\frac{a^2 d}{2 x^2}+a^2 e \log (x)-\frac{a b d \left(\tan ^{-1}(c x)+c x \left(c x \tan ^{-1}(c x)+1\right)\right)}{x^2}+i a b e (\text{Li}_2(-i c x)-\text{Li}_2(i c x))-\frac{b^2 d \left(-2 c^2 x^2 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+\left(c^2 x^2+1\right) \tan ^{-1}(c x)^2+2 c x \tan ^{-1}(c x)\right)}{2 x^2}+\frac{1}{24} b^2 e \left(24 i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+24 i \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+12 \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-12 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+16 i \tan ^{-1}(c x)^3+24 \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-24 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-i \pi ^3\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}-i b e \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)+i b e \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)+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)-\frac{1}{2} b^2 e \text{Li}_3\left(1-\frac{2}{i c x+1}\right)+\frac{1}{2} b^2 e \text{Li}_3\left(\frac{2}{i c x+1}-1\right)",1,"-1/2*(a^2*d)/x^2 - (a*b*d*(ArcTan[c*x] + c*x*(1 + c*x*ArcTan[c*x])))/x^2 + a^2*e*Log[x] - (b^2*d*(2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 - 2*c^2*x^2*Log[(c*x)/Sqrt[1 + c^2*x^2]]))/(2*x^2) + I*a*b*e*(PolyLog[2, (-I)*c*x] - PolyLog[2, I*c*x]) + (b^2*e*((-I)*Pi^3 + (16*I)*ArcTan[c*x]^3 + 24*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 24*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + (24*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - 12*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/24","A",0
1254,1,414,502,0.4984772,"\int x^3 \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d + e*x^2)^2*(a + b*ArcTan[c*x])^2,x]","\frac{c x \left(105 a^2 c^7 x^3 \left(6 d^2+8 d e x^2+3 e^2 x^4\right)-2 a b \left(3 c^6 \left(70 d^2 x^2+56 d e x^4+15 e^2 x^6\right)-7 c^4 \left(90 d^2+40 d e x^2+9 e^2 x^4\right)+105 c^2 e \left(8 d+e x^2\right)-315 e^2\right)+b^2 c x \left(3 c^4 \left(70 d^2+28 d e x^2+5 e^2 x^4\right)-2 c^2 e \left(224 d+27 e x^2\right)+213 e^2\right)\right)+2 b \tan ^{-1}(c x) \left(105 a \left(c^8 \left(6 d^2 x^4+8 d e x^6+3 e^2 x^8\right)-6 c^4 d^2+8 c^2 d e-3 e^2\right)+b c x \left(-3 c^6 \left(70 d^2 x^2+56 d e x^4+15 e^2 x^6\right)+7 c^4 \left(90 d^2+40 d e x^2+9 e^2 x^4\right)-105 c^2 e \left(8 d+e x^2\right)+315 e^2\right)\right)-8 b^2 \left(105 c^4 d^2-161 c^2 d e+66 e^2\right) \log \left(c^2 x^2+1\right)+105 b^2 \tan ^{-1}(c x)^2 \left(c^8 \left(6 d^2 x^4+8 d e x^6+3 e^2 x^8\right)-6 c^4 d^2+8 c^2 d e-3 e^2\right)}{2520 c^8}","-\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{8 c^8}+\frac{a b e^2 x}{4 c^7}+\frac{d e \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^6}-\frac{2 a b d e x}{3 c^5}-\frac{b e^2 x^3 \left(a+b \tan ^{-1}(c x)\right)}{12 c^5}-\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}+\frac{a b d^2 x}{2 c^3}+\frac{2 b d e x^3 \left(a+b \tan ^{-1}(c x)\right)}{9 c^3}+\frac{b e^2 x^5 \left(a+b \tan ^{-1}(c x)\right)}{20 c^3}+\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 e^2 x \tan ^{-1}(c x)}{4 c^7}+\frac{71 b^2 e^2 x^2}{840 c^6}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{3 c^5}-\frac{8 b^2 d e x^2}{45 c^4}-\frac{3 b^2 e^2 x^4}{140 c^4}+\frac{b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}+\frac{b^2 d^2 x^2}{12 c^2}+\frac{b^2 d e x^4}{30 c^2}+\frac{b^2 e^2 x^6}{168 c^2}-\frac{22 b^2 e^2 \log \left(c^2 x^2+1\right)}{105 c^8}+\frac{23 b^2 d e \log \left(c^2 x^2+1\right)}{45 c^6}-\frac{b^2 d^2 \log \left(c^2 x^2+1\right)}{3 c^4}",1,"(c*x*(105*a^2*c^7*x^3*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + b^2*c*x*(213*e^2 - 2*c^2*e*(224*d + 27*e*x^2) + 3*c^4*(70*d^2 + 28*d*e*x^2 + 5*e^2*x^4)) - 2*a*b*(-315*e^2 + 105*c^2*e*(8*d + e*x^2) - 7*c^4*(90*d^2 + 40*d*e*x^2 + 9*e^2*x^4) + 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6))) + 2*b*(b*c*x*(315*e^2 - 105*c^2*e*(8*d + e*x^2) + 7*c^4*(90*d^2 + 40*d*e*x^2 + 9*e^2*x^4) - 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)) + 105*a*(-6*c^4*d^2 + 8*c^2*d*e - 3*e^2 + c^8*(6*d^2*x^4 + 8*d*e*x^6 + 3*e^2*x^8)))*ArcTan[c*x] + 105*b^2*(-6*c^4*d^2 + 8*c^2*d*e - 3*e^2 + c^8*(6*d^2*x^4 + 8*d*e*x^6 + 3*e^2*x^8))*ArcTan[c*x]^2 - 8*b^2*(105*c^4*d^2 - 161*c^2*d*e + 66*e^2)*Log[1 + c^2*x^2])/(2520*c^8)","A",1
1255,1,513,580,1.7759644,"\int x^2 \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d + e*x^2)^2*(a + b*ArcTan[c*x])^2,x]","\frac{210 a^2 c^7 d^2 x^3+252 a^2 c^7 d e x^5+90 a^2 c^7 e^2 x^7-210 a b c^6 d^2 x^2-126 a b c^6 d e x^4-30 a b c^6 e^2 x^6+252 a b c^4 d e x^2+45 a b c^4 e^2 x^4-252 a b c^2 d e \log \left(c^2 x^2+1\right)+378 a b c^2 d e-90 a b c^2 e^2 x^2+90 a b e^2 \log \left(c^2 x^2+1\right)+210 a b c^4 d^2 \log \left(c^2 x^2+1\right)-3 b \tan ^{-1}(c x) \left(-4 a c^7 x^3 \left(35 d^2+42 d e x^2+15 e^2 x^4\right)+b \left(c^2 x^2+1\right) \left(2 c^4 \left(35 d^2+21 d e x^2+5 e^2 x^4\right)-c^2 e \left(126 d+25 e x^2\right)+55 e^2\right)+4 b \left(35 c^4 d^2-42 c^2 d e+15 e^2\right) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)-165 a b e^2+210 b^2 c^5 d^2 x+42 b^2 c^5 d e x^3+6 b^2 c^5 e^2 x^5-378 b^2 c^3 d e x-25 b^2 c^3 e^2 x^3+6 i b^2 \left(35 c^4 d^2-42 c^2 d e+15 e^2\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+6 b^2 \tan ^{-1}(c x)^2 \left(c^7 \left(35 d^2 x^3+42 d e x^5+15 e^2 x^7\right)+35 i c^4 d^2-42 i c^2 d e+15 i e^2\right)+165 b^2 c e^2 x}{630 c^7}","-\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{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^2 \left(a+b \tan ^{-1}(c x)\right)}{7 c^5}-\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{b e^2 x^4 \left(a+b \tan ^{-1}(c x)\right)}{14 c^3}+\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{i b^2 e^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{7 c^7}-\frac{11 b^2 e^2 \tan ^{-1}(c x)}{42 c^7}+\frac{11 b^2 e^2 x}{42 c^6}+\frac{2 i b^2 d e \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{5 c^5}+\frac{3 b^2 d e \tan ^{-1}(c x)}{5 c^5}-\frac{3 b^2 d e x}{5 c^4}-\frac{5 b^2 e^2 x^3}{126 c^4}-\frac{i b^2 d^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{3 c^3}-\frac{b^2 d^2 \tan ^{-1}(c x)}{3 c^3}+\frac{b^2 d^2 x}{3 c^2}+\frac{b^2 d e x^3}{15 c^2}+\frac{b^2 e^2 x^5}{105 c^2}",1,"(378*a*b*c^2*d*e - 165*a*b*e^2 + 210*b^2*c^5*d^2*x - 378*b^2*c^3*d*e*x + 165*b^2*c*e^2*x - 210*a*b*c^6*d^2*x^2 + 252*a*b*c^4*d*e*x^2 - 90*a*b*c^2*e^2*x^2 + 210*a^2*c^7*d^2*x^3 + 42*b^2*c^5*d*e*x^3 - 25*b^2*c^3*e^2*x^3 - 126*a*b*c^6*d*e*x^4 + 45*a*b*c^4*e^2*x^4 + 252*a^2*c^7*d*e*x^5 + 6*b^2*c^5*e^2*x^5 - 30*a*b*c^6*e^2*x^6 + 90*a^2*c^7*e^2*x^7 + 6*b^2*((35*I)*c^4*d^2 - (42*I)*c^2*d*e + (15*I)*e^2 + c^7*(35*d^2*x^3 + 42*d*e*x^5 + 15*e^2*x^7))*ArcTan[c*x]^2 - 3*b*ArcTan[c*x]*(-4*a*c^7*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) + b*(1 + c^2*x^2)*(55*e^2 - c^2*e*(126*d + 25*e*x^2) + 2*c^4*(35*d^2 + 21*d*e*x^2 + 5*e^2*x^4)) + 4*b*(35*c^4*d^2 - 42*c^2*d*e + 15*e^2)*Log[1 + E^((2*I)*ArcTan[c*x])]) + 210*a*b*c^4*d^2*Log[1 + c^2*x^2] - 252*a*b*c^2*d*e*Log[1 + c^2*x^2] + 90*a*b*e^2*Log[1 + c^2*x^2] + (6*I)*b^2*(35*c^4*d^2 - 42*c^2*d*e + 15*e^2)*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(630*c^7)","A",0
1256,1,317,380,0.3532742,"\int x \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d + e*x^2)^2*(a + b*ArcTan[c*x])^2,x]","\frac{c x \left(30 a^2 c^5 x \left(3 d^2+3 d e x^2+e^2 x^4\right)-4 a b \left(3 c^4 \left(15 d^2+5 d e x^2+e^2 x^4\right)-5 c^2 e \left(9 d+e x^2\right)+15 e^2\right)+b^2 c e x \left(3 c^2 \left(10 d+e x^2\right)-16 e\right)\right)+4 b \tan ^{-1}(c x) \left(15 a \left(c^6 \left(3 d^2 x^2+3 d e x^4+e^2 x^6\right)+3 c^4 d^2-3 c^2 d e+e^2\right)-b c x \left(3 c^4 \left(15 d^2+5 d e x^2+e^2 x^4\right)-5 c^2 e \left(9 d+e x^2\right)+15 e^2\right)\right)+2 b^2 \left(45 c^4 d^2-60 c^2 d e+23 e^2\right) \log \left(c^2 x^2+1\right)+30 b^2 \tan ^{-1}(c x)^2 \left(c^6 \left(3 d^2 x^2+3 d e x^4+e^2 x^6\right)+3 c^4 d^2-3 c^2 d e+e^2\right)}{180 c^6}","\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{6 c^6}-\frac{a b e^2 x}{3 c^5}-\frac{d e \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^4}+\frac{a b d e x}{c^3}+\frac{b e^2 x^3 \left(a+b \tan ^{-1}(c x)\right)}{9 c^3}+\frac{d^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2}+\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 e^2 x \tan ^{-1}(c x)}{3 c^5}-\frac{4 b^2 e^2 x^2}{45 c^4}+\frac{b^2 d e x \tan ^{-1}(c x)}{c^3}+\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{b^2 e^2 x^4}{60 c^2}+\frac{23 b^2 e^2 \log \left(c^2 x^2+1\right)}{90 c^6}-\frac{2 b^2 d e \log \left(c^2 x^2+1\right)}{3 c^4}-\frac{b^2 d^2 x \tan ^{-1}(c x)}{c}",1,"(c*x*(30*a^2*c^5*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) + b^2*c*e*x*(-16*e + 3*c^2*(10*d + e*x^2)) - 4*a*b*(15*e^2 - 5*c^2*e*(9*d + e*x^2) + 3*c^4*(15*d^2 + 5*d*e*x^2 + e^2*x^4))) + 4*b*(-(b*c*x*(15*e^2 - 5*c^2*e*(9*d + e*x^2) + 3*c^4*(15*d^2 + 5*d*e*x^2 + e^2*x^4))) + 15*a*(3*c^4*d^2 - 3*c^2*d*e + e^2 + c^6*(3*d^2*x^2 + 3*d*e*x^4 + e^2*x^6)))*ArcTan[c*x] + 30*b^2*(3*c^4*d^2 - 3*c^2*d*e + e^2 + c^6*(3*d^2*x^2 + 3*d*e*x^4 + e^2*x^6))*ArcTan[c*x]^2 + 2*b^2*(45*c^4*d^2 - 60*c^2*d*e + 23*e^2)*Log[1 + c^2*x^2])/(180*c^6)","A",1
1257,1,391,442,1.1148889,"\int \left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x^2)^2*(a + b*ArcTan[c*x])^2,x]","\frac{30 a^2 c^5 d^2 x+20 a^2 c^5 d e x^3+6 a^2 c^5 e^2 x^5-20 a b c^4 d e x^2-3 a b c^4 e^2 x^4+20 a b c^2 d e \log \left(c^2 x^2+1\right)+6 a b c^2 e^2 x^2-6 a b e^2 \log \left(c^2 x^2+1\right)-30 a b c^4 d^2 \log \left(c^2 x^2+1\right)+b \tan ^{-1}(c x) \left(4 a c^5 x \left(15 d^2+10 d e x^2+3 e^2 x^4\right)-b e \left(c^2 x^2+1\right) \left(c^2 \left(20 d+3 e x^2\right)-9 e\right)+4 b \left(15 c^4 d^2-10 c^2 d e+3 e^2\right) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+9 a b e^2+20 b^2 c^3 d e x+b^2 c^3 e^2 x^3-2 i b^2 \left(15 c^4 d^2-10 c^2 d e+3 e^2\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+2 b^2 \tan ^{-1}(c x)^2 \left(c^5 \left(15 d^2 x+10 d e x^3+3 e^2 x^5\right)-15 i c^4 d^2+10 i c^2 d e-3 i e^2\right)-9 b^2 c e^2 x}{30 c^5}","\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}-\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}+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{i b^2 e^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{5 c^5}+\frac{3 b^2 e^2 \tan ^{-1}(c x)}{10 c^5}-\frac{3 b^2 e^2 x}{10 c^4}-\frac{2 i b^2 d e \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{3 c^3}-\frac{2 b^2 d e \tan ^{-1}(c x)}{3 c^3}+\frac{2 b^2 d e x}{3 c^2}+\frac{b^2 e^2 x^3}{30 c^2}+\frac{i b^2 d^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c}",1,"(9*a*b*e^2 + 30*a^2*c^5*d^2*x + 20*b^2*c^3*d*e*x - 9*b^2*c*e^2*x - 20*a*b*c^4*d*e*x^2 + 6*a*b*c^2*e^2*x^2 + 20*a^2*c^5*d*e*x^3 + b^2*c^3*e^2*x^3 - 3*a*b*c^4*e^2*x^4 + 6*a^2*c^5*e^2*x^5 + 2*b^2*((-15*I)*c^4*d^2 + (10*I)*c^2*d*e - (3*I)*e^2 + c^5*(15*d^2*x + 10*d*e*x^3 + 3*e^2*x^5))*ArcTan[c*x]^2 + b*ArcTan[c*x]*(4*a*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - b*e*(1 + c^2*x^2)*(-9*e + c^2*(20*d + 3*e*x^2)) + 4*b*(15*c^4*d^2 - 10*c^2*d*e + 3*e^2)*Log[1 + E^((2*I)*ArcTan[c*x])]) - 30*a*b*c^4*d^2*Log[1 + c^2*x^2] + 20*a*b*c^2*d*e*Log[1 + c^2*x^2] - 6*a*b*e^2*Log[1 + c^2*x^2] - (2*I)*b^2*(15*c^4*d^2 - 10*c^2*d*e + 3*e^2)*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(30*c^5)","A",0
1258,1,389,355,0.7209917,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcTan[c*x])^2)/x,x]","a^2 d^2 \log (x)+a^2 d e x^2+\frac{1}{4} a^2 e^2 x^4+\frac{2 a b d e \left(\left(c^2 x^2+1\right) \tan ^{-1}(c x)-c x\right)}{c^2}+\frac{a b e^2 \left(3 \left(c^4 x^4-1\right) \tan ^{-1}(c x)-c^3 x^3+3 c x\right)}{6 c^4}+i a b d^2 (\text{Li}_2(-i c x)-\text{Li}_2(i c x))+\frac{b^2 d e \left(\log \left(c^2 x^2+1\right)+\left(c^2 x^2+1\right) \tan ^{-1}(c x)^2-2 c x \tan ^{-1}(c x)\right)}{c^2}+\frac{b^2 e^2 \left(3 \left(c^4 x^4-1\right) \tan ^{-1}(c x)^2+\left(6 c x-2 c^3 x^3\right) \tan ^{-1}(c x)+c^2 x^2-4 \log \left(c^2 x^2+1\right)+1\right)}{12 c^4}+b^2 d^2 \left(i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+i \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+\frac{2}{3} i \tan ^{-1}(c x)^3+\tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-\tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-\frac{i \pi ^3}{24}\right)","-\frac{e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}+\frac{a b e^2 x}{2 c^3}+\frac{d e \left(a+b \tan ^{-1}(c x)\right)^2}{c^2}-i b d^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)+i b d^2 \text{Li}_2\left(\frac{2}{i c x+1}-1\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+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 e^2 x \tan ^{-1}(c x)}{2 c^3}+\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{1}{2} b^2 d^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)+\frac{1}{2} b^2 d^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)-\frac{2 b^2 d e x \tan ^{-1}(c x)}{c}",1,"a^2*d*e*x^2 + (a^2*e^2*x^4)/4 + (2*a*b*d*e*(-(c*x) + (1 + c^2*x^2)*ArcTan[c*x]))/c^2 + (a*b*e^2*(3*c*x - c^3*x^3 + 3*(-1 + c^4*x^4)*ArcTan[c*x]))/(6*c^4) + a^2*d^2*Log[x] + (b^2*e^2*(1 + c^2*x^2 + (6*c*x - 2*c^3*x^3)*ArcTan[c*x] + 3*(-1 + c^4*x^4)*ArcTan[c*x]^2 - 4*Log[1 + c^2*x^2]))/(12*c^4) + (b^2*d*e*(-2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 + Log[1 + c^2*x^2]))/c^2 + I*a*b*d^2*(PolyLog[2, (-I)*c*x] - PolyLog[2, I*c*x]) + b^2*d^2*((-1/24*I)*Pi^3 + ((2*I)/3)*ArcTan[c*x]^3 + ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + I*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + PolyLog[3, E^((-2*I)*ArcTan[c*x])]/2 - PolyLog[3, -E^((2*I)*ArcTan[c*x])]/2)","A",0
1259,1,349,343,0.827428,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcTan[c*x])^2)/x^2,x]","\frac{1}{3} \left(-\frac{3 a^2 d^2}{x}+6 a^2 d e x+a^2 e^2 x^3-\frac{3 a b d^2 \left(c x \left(\log \left(c^2 x^2+1\right)-2 \log (c x)\right)+2 \tan ^{-1}(c x)\right)}{x}+\frac{6 a b d e \left(2 c x \tan ^{-1}(c x)-\log \left(c^2 x^2+1\right)\right)}{c}+\frac{a b e^2 \left(2 c^3 x^3 \tan ^{-1}(c x)-c^2 x^2+\log \left(c^2 x^2+1\right)\right)}{c^3}+\frac{b^2 e^2 \left(\left(c^3 x^3+i\right) \tan ^{-1}(c x)^2-\tan ^{-1}(c x) \left(c^2 x^2+2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+1\right)+i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+c x\right)}{c^3}+3 b^2 c d^2 \left(\tan ^{-1}(c x) \left(\left(-\frac{1}{c x}-i\right) \tan ^{-1}(c x)+2 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)\right)-i \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)\right)+\frac{6 b^2 d e \left(\tan ^{-1}(c x) \left((c x-i) \tan ^{-1}(c x)+2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)-i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)\right)}{c}\right)","-\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{i b^2 e^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{3 c^3}-\frac{b^2 e^2 \tan ^{-1}(c x)}{3 c^3}+\frac{b^2 e^2 x}{3 c^2}-i b^2 c d^2 \text{Li}_2\left(\frac{2}{1-i c x}-1\right)+\frac{2 i b^2 d e \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c}",1,"((-3*a^2*d^2)/x + 6*a^2*d*e*x + a^2*e^2*x^3 + (6*a*b*d*e*(2*c*x*ArcTan[c*x] - Log[1 + c^2*x^2]))/c + (a*b*e^2*(-(c^2*x^2) + 2*c^3*x^3*ArcTan[c*x] + Log[1 + c^2*x^2]))/c^3 - (3*a*b*d^2*(2*ArcTan[c*x] + c*x*(-2*Log[c*x] + Log[1 + c^2*x^2])))/x + (6*b^2*d*e*(ArcTan[c*x]*((-I + c*x)*ArcTan[c*x] + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/c + (b^2*e^2*(c*x + (I + c^3*x^3)*ArcTan[c*x]^2 - ArcTan[c*x]*(1 + c^2*x^2 + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) + I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/c^3 + 3*b^2*c*d^2*(ArcTan[c*x]*((-I - 1/(c*x))*ArcTan[c*x] + 2*Log[1 - E^((2*I)*ArcTan[c*x])]) - I*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/3","A",0
1260,1,367,320,0.6399114,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcTan[c*x])^2)/x^3,x]","\frac{1}{2} \left(-\frac{a^2 d^2}{x^2}+4 a^2 d e \log (x)+a^2 e^2 x^2+\frac{2 a b e^2 \left(\left(c^2 x^2+1\right) \tan ^{-1}(c x)-c x\right)}{c^2}-\frac{2 a b d^2 \left(\tan ^{-1}(c x)+c x \left(c x \tan ^{-1}(c x)+1\right)\right)}{x^2}+4 i a b d e (\text{Li}_2(-i c x)-\text{Li}_2(i c x))-\frac{b^2 d^2 \left(-2 c^2 x^2 \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+\left(c^2 x^2+1\right) \tan ^{-1}(c x)^2+2 c x \tan ^{-1}(c x)\right)}{x^2}+\frac{b^2 e^2 \left(\log \left(c^2 x^2+1\right)+\left(c^2 x^2+1\right) \tan ^{-1}(c x)^2-2 c x \tan ^{-1}(c x)\right)}{c^2}+\frac{1}{6} b^2 d e \left(24 i \tan ^{-1}(c x) \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right)+24 i \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+12 \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-12 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+16 i \tan ^{-1}(c x)^3+24 \tan ^{-1}(c x)^2 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right)-24 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-i \pi ^3\right)\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}-2 i b d e \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)+2 i b d e \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)+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}-b^2 d e \text{Li}_3\left(1-\frac{2}{i c x+1}\right)+b^2 d e \text{Li}_3\left(\frac{2}{i c x+1}-1\right)-\frac{b^2 e^2 x \tan ^{-1}(c x)}{c}",1,"(-((a^2*d^2)/x^2) + a^2*e^2*x^2 + (2*a*b*e^2*(-(c*x) + (1 + c^2*x^2)*ArcTan[c*x]))/c^2 - (2*a*b*d^2*(ArcTan[c*x] + c*x*(1 + c*x*ArcTan[c*x])))/x^2 + 4*a^2*d*e*Log[x] - (b^2*d^2*(2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 - 2*c^2*x^2*Log[(c*x)/Sqrt[1 + c^2*x^2]]))/x^2 + (b^2*e^2*(-2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 + Log[1 + c^2*x^2]))/c^2 + (4*I)*a*b*d*e*(PolyLog[2, (-I)*c*x] - PolyLog[2, I*c*x]) + (b^2*d*e*((-I)*Pi^3 + (16*I)*ArcTan[c*x]^3 + 24*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 24*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + (24*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + (24*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - 12*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/6)/2","A",0
1261,1,1569,590,10.8852367,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{d+e x^2} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x])^2)/(d + e*x^2),x]","\frac{2 e x^2 a^2-2 d \log \left(e x^2+d\right) a^2+4 b \left(-i d \tan ^{-1}(c x)^2+\left(e \left(x^2+\frac{1}{c^2}\right)+2 d \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right) \tan ^{-1}(c x)-\frac{e x}{c}-i d \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+\frac{2 d \left(e-c^2 d\right) \left(-i \tan ^{-1}(c x)^2+2 i \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \tan ^{-1}\left(\frac{c e x}{\sqrt{c^2 d e}}\right)+\left(\tan ^{-1}(c x)-\sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right)\right) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e+2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right)+\left(\sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right)+\tan ^{-1}(c x)\right) \log \left(\frac{d \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 \sqrt{c^2 d e} e^{2 i \tan ^{-1}(c x)}+e \left(-1+e^{2 i \tan ^{-1}(c x)}\right)}{c^2 d-e}\right)-\frac{1}{2} i \left(\text{Li}_2\left(\frac{\left(-d c^2-e+2 \sqrt{c^2 d e}\right) e^{2 i \tan ^{-1}(c x)}}{c^2 d-e}\right)+\text{Li}_2\left(-\frac{\left(d c^2+e+2 \sqrt{c^2 d e}\right) e^{2 i \tan ^{-1}(c x)}}{c^2 d-e}\right)\right)\right)}{2 c^2 d-2 e}\right) a+\frac{b^2 \left(2 e x^2 \tan ^{-1}(c x)^2 c^2+4 d \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 d \tan ^{-1}(c x)^2 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(c \sqrt{d}-\sqrt{e}\right)}{\sqrt{d} c+\sqrt{e}}+1\right) c^2-2 d \tan ^{-1}(c x)^2 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(\sqrt{d} c+\sqrt{e}\right)}{c \sqrt{d}-\sqrt{e}}+1\right) c^2+2 d \tan ^{-1}(c x)^2 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e-2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right) c^2-2 d \tan ^{-1}(c x)^2 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e+2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right) c^2+4 d \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \tan ^{-1}(c x) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e+2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right) c^2-4 d \tan ^{-1}(c x)^2 \log \left(\frac{d \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 \sqrt{c^2 d e} e^{2 i \tan ^{-1}(c x)}+e \left(-1+e^{2 i \tan ^{-1}(c x)}\right)}{c^2 d-e}\right) c^2-4 d \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \tan ^{-1}(c x) \log \left(\frac{d \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 \sqrt{c^2 d e} e^{2 i \tan ^{-1}(c x)}+e \left(-1+e^{2 i \tan ^{-1}(c x)}\right)}{c^2 d-e}\right) c^2+2 d \tan ^{-1}(c x)^2 \log \left(\frac{2 i d c^2+2 \left(\sqrt{c^2 d e}-e\right) x c-2 i \sqrt{c^2 d e}}{\left(c^2 d-e\right) (c x+i)}\right) c^2+4 d \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \tan ^{-1}(c x) \log \left(\frac{2 i d c^2+2 \left(\sqrt{c^2 d e}-e\right) x c-2 i \sqrt{c^2 d e}}{\left(c^2 d-e\right) (c x+i)}\right) c^2+2 d \tan ^{-1}(c x)^2 \log \left(\frac{\left(d c^2+e+2 \sqrt{c^2 d e}\right) \left(\cos \left(2 \tan ^{-1}(c x)\right)+i \sin \left(2 \tan ^{-1}(c x)\right)\right)}{c^2 d-e}+1\right) c^2-4 d \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \tan ^{-1}(c x) \log \left(\frac{\left(d c^2+e+2 \sqrt{c^2 d e}\right) \left(\cos \left(2 \tan ^{-1}(c x)\right)+i \sin \left(2 \tan ^{-1}(c x)\right)\right)}{c^2 d-e}+1\right) c^2-4 i d \tan ^{-1}(c x) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right) c^2+2 i d \tan ^{-1}(c x) \text{Li}_2\left(\frac{\left(\sqrt{e}-c \sqrt{d}\right) e^{2 i \tan ^{-1}(c x)}}{\sqrt{d} c+\sqrt{e}}\right) c^2+2 i d \tan ^{-1}(c x) \text{Li}_2\left(-\frac{\left(\sqrt{d} c+\sqrt{e}\right) e^{2 i \tan ^{-1}(c x)}}{c \sqrt{d}-\sqrt{e}}\right) c^2+2 d \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right) c^2-d \text{Li}_3\left(\frac{\left(\sqrt{e}-c \sqrt{d}\right) e^{2 i \tan ^{-1}(c x)}}{\sqrt{d} c+\sqrt{e}}\right) c^2-d \text{Li}_3\left(-\frac{\left(\sqrt{d} c+\sqrt{e}\right) e^{2 i \tan ^{-1}(c x)}}{c \sqrt{d}-\sqrt{e}}\right) c^2-4 e x \tan ^{-1}(c x) c+2 e \tan ^{-1}(c x)^2+2 e \log \left(c^2 x^2+1\right)\right)}{c^2}}{4 e^2}","\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2 e}-\frac{i b d \text{Li}_2\left(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{Li}_2\left(1-\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{i b d \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\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{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 d \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{2 e^2}-\frac{b^2 d \text{Li}_3\left(1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{4 e^2}-\frac{b^2 d \text{Li}_3\left(1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{4 e^2}-\frac{b^2 x \tan ^{-1}(c x)}{c e}",1,"(2*a^2*e*x^2 - 2*a^2*d*Log[d + e*x^2] + 4*a*b*(-((e*x)/c) - I*d*ArcTan[c*x]^2 + ArcTan[c*x]*(e*(c^(-2) + x^2) + 2*d*Log[1 + E^((2*I)*ArcTan[c*x])]) - I*d*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (2*d*(-(c^2*d) + e)*((-I)*ArcTan[c*x]^2 + (2*I)*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[(c*e*x)/Sqrt[c^2*d*e]] + (-ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]] + ArcTan[c*x])*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] + (ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]] + ArcTan[c*x])*Log[(-2*Sqrt[c^2*d*e]*E^((2*I)*ArcTan[c*x]) + e*(-1 + E^((2*I)*ArcTan[c*x])) + c^2*d*(1 + E^((2*I)*ArcTan[c*x])))/(c^2*d - e)] - (I/2)*(PolyLog[2, ((-(c^2*d) - e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] + PolyLog[2, -(((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e))])))/(2*c^2*d - 2*e)) + (b^2*(-4*c*e*x*ArcTan[c*x] + 2*e*ArcTan[c*x]^2 + 2*c^2*e*x^2*ArcTan[c*x]^2 + 4*c^2*d*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - 2*c^2*d*ArcTan[c*x]^2*Log[1 + ((c*Sqrt[d] - Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] + Sqrt[e])] - 2*c^2*d*ArcTan[c*x]^2*Log[1 + ((c*Sqrt[d] + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] - Sqrt[e])] + 2*c^2*d*ArcTan[c*x]^2*Log[1 + ((c^2*d + e - 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] + 4*c^2*d*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] - 2*c^2*d*ArcTan[c*x]^2*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] - 4*c^2*d*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[(-2*Sqrt[c^2*d*e]*E^((2*I)*ArcTan[c*x]) + e*(-1 + E^((2*I)*ArcTan[c*x])) + c^2*d*(1 + E^((2*I)*ArcTan[c*x])))/(c^2*d - e)] - 4*c^2*d*ArcTan[c*x]^2*Log[(-2*Sqrt[c^2*d*e]*E^((2*I)*ArcTan[c*x]) + e*(-1 + E^((2*I)*ArcTan[c*x])) + c^2*d*(1 + E^((2*I)*ArcTan[c*x])))/(c^2*d - e)] + 4*c^2*d*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[((2*I)*c^2*d - (2*I)*Sqrt[c^2*d*e] + 2*c*(-e + Sqrt[c^2*d*e])*x)/((c^2*d - e)*(I + c*x))] + 2*c^2*d*ArcTan[c*x]^2*Log[((2*I)*c^2*d - (2*I)*Sqrt[c^2*d*e] + 2*c*(-e + Sqrt[c^2*d*e])*x)/((c^2*d - e)*(I + c*x))] + 2*e*Log[1 + c^2*x^2] - 4*c^2*d*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*(Cos[2*ArcTan[c*x]] + I*Sin[2*ArcTan[c*x]]))/(c^2*d - e)] + 2*c^2*d*ArcTan[c*x]^2*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*(Cos[2*ArcTan[c*x]] + I*Sin[2*ArcTan[c*x]]))/(c^2*d - e)] - (4*I)*c^2*d*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (2*I)*c^2*d*ArcTan[c*x]*PolyLog[2, ((-(c*Sqrt[d]) + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] + Sqrt[e])] + (2*I)*c^2*d*ArcTan[c*x]*PolyLog[2, -(((c*Sqrt[d] + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] - Sqrt[e]))] + 2*c^2*d*PolyLog[3, -E^((2*I)*ArcTan[c*x])] - c^2*d*PolyLog[3, ((-(c*Sqrt[d]) + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] + Sqrt[e])] - c^2*d*PolyLog[3, -(((c*Sqrt[d] + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] - Sqrt[e]))]))/c^2)/(4*e^2)","B",0
1262,-1,0,554,180.0033387,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{d+e x^2} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x])^2)/(d + e*x^2),x]","\text{\$Aborted}","-\frac{i b \sqrt{-d} \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\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^{3/2}}+\frac{i b \sqrt{-d} \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\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^{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{\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{b^2 \sqrt{-d} \text{Li}_3\left(1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{4 e^{3/2}}-\frac{b^2 \sqrt{-d} \text{Li}_3\left(1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{4 e^{3/2}}+\frac{i b^2 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c e}",1,"$Aborted","F",-1
1263,1,1529,492,9.276029,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{d+e x^2} \, dx","Integrate[(x*(a + b*ArcTan[c*x])^2)/(d + e*x^2),x]","\frac{2 \log \left(e x^2+d\right) a^2+8 i b \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \tan ^{-1}\left(\frac{c e x}{\sqrt{c^2 d e}}\right) a-8 b \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right) a-4 b \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e+2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right) a+4 b \tan ^{-1}(c x) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e+2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right) a+4 b \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \log \left(\frac{d \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 \sqrt{c^2 d e} e^{2 i \tan ^{-1}(c x)}+e \left(-1+e^{2 i \tan ^{-1}(c x)}\right)}{c^2 d-e}\right) a+4 b \tan ^{-1}(c x) \log \left(\frac{d \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 \sqrt{c^2 d e} e^{2 i \tan ^{-1}(c x)}+e \left(-1+e^{2 i \tan ^{-1}(c x)}\right)}{c^2 d-e}\right) a-2 i b \text{Li}_2\left(\frac{\left(-d c^2-e+2 \sqrt{c^2 d e}\right) e^{2 i \tan ^{-1}(c x)}}{c^2 d-e}\right) a-2 i b \text{Li}_2\left(-\frac{\left(d c^2+e+2 \sqrt{c^2 d e}\right) e^{2 i \tan ^{-1}(c x)}}{c^2 d-e}\right) a-4 b^2 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+2 b^2 \tan ^{-1}(c x)^2 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(c \sqrt{d}-\sqrt{e}\right)}{\sqrt{d} c+\sqrt{e}}+1\right)+2 b^2 \tan ^{-1}(c x)^2 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(\sqrt{d} c+\sqrt{e}\right)}{c \sqrt{d}-\sqrt{e}}+1\right)-2 b^2 \tan ^{-1}(c x)^2 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e-2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right)+2 b^2 \tan ^{-1}(c x)^2 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e+2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right)-4 b^2 \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \tan ^{-1}(c x) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e+2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right)+4 b^2 \tan ^{-1}(c x)^2 \log \left(\frac{d \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 \sqrt{c^2 d e} e^{2 i \tan ^{-1}(c x)}+e \left(-1+e^{2 i \tan ^{-1}(c x)}\right)}{c^2 d-e}\right)+4 b^2 \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \tan ^{-1}(c x) \log \left(\frac{d \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 \sqrt{c^2 d e} e^{2 i \tan ^{-1}(c x)}+e \left(-1+e^{2 i \tan ^{-1}(c x)}\right)}{c^2 d-e}\right)-2 b^2 \tan ^{-1}(c x)^2 \log \left(\frac{2 i d c^2+2 \left(\sqrt{c^2 d e}-e\right) x c-2 i \sqrt{c^2 d e}}{\left(c^2 d-e\right) (c x+i)}\right)-4 b^2 \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \tan ^{-1}(c x) \log \left(\frac{2 i d c^2+2 \left(\sqrt{c^2 d e}-e\right) x c-2 i \sqrt{c^2 d e}}{\left(c^2 d-e\right) (c x+i)}\right)-2 b^2 \tan ^{-1}(c x)^2 \log \left(\frac{\left(d c^2+e+2 \sqrt{c^2 d e}\right) \left(\cos \left(2 \tan ^{-1}(c x)\right)+i \sin \left(2 \tan ^{-1}(c x)\right)\right)}{c^2 d-e}+1\right)+4 b^2 \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \tan ^{-1}(c x) \log \left(\frac{\left(d c^2+e+2 \sqrt{c^2 d e}\right) \left(\cos \left(2 \tan ^{-1}(c x)\right)+i \sin \left(2 \tan ^{-1}(c x)\right)\right)}{c^2 d-e}+1\right)+4 i b \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-2 i b^2 \tan ^{-1}(c x) \text{Li}_2\left(\frac{\left(\sqrt{e}-c \sqrt{d}\right) e^{2 i \tan ^{-1}(c x)}}{\sqrt{d} c+\sqrt{e}}\right)-2 i b^2 \tan ^{-1}(c x) \text{Li}_2\left(-\frac{\left(\sqrt{d} c+\sqrt{e}\right) e^{2 i \tan ^{-1}(c x)}}{c \sqrt{d}-\sqrt{e}}\right)-2 b^2 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)+b^2 \text{Li}_3\left(\frac{\left(\sqrt{e}-c \sqrt{d}\right) e^{2 i \tan ^{-1}(c x)}}{\sqrt{d} c+\sqrt{e}}\right)+b^2 \text{Li}_3\left(-\frac{\left(\sqrt{d} c+\sqrt{e}\right) e^{2 i \tan ^{-1}(c x)}}{c \sqrt{d}-\sqrt{e}}\right)}{4 e}","-\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\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}-\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\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}+\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{i b \text{Li}_2\left(1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{e}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{e}+\frac{b^2 \text{Li}_3\left(1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{4 e}+\frac{b^2 \text{Li}_3\left(1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\right)}{4 e}-\frac{b^2 \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{2 e}",1,"((8*I)*a*b*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[(c*e*x)/Sqrt[c^2*d*e]] - 8*a*b*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - 4*b^2*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] + 2*b^2*ArcTan[c*x]^2*Log[1 + ((c*Sqrt[d] - Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] + Sqrt[e])] + 2*b^2*ArcTan[c*x]^2*Log[1 + ((c*Sqrt[d] + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] - Sqrt[e])] - 2*b^2*ArcTan[c*x]^2*Log[1 + ((c^2*d + e - 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] - 4*a*b*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] + 4*a*b*ArcTan[c*x]*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] - 4*b^2*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] + 2*b^2*ArcTan[c*x]^2*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] + 4*a*b*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*Log[(-2*Sqrt[c^2*d*e]*E^((2*I)*ArcTan[c*x]) + e*(-1 + E^((2*I)*ArcTan[c*x])) + c^2*d*(1 + E^((2*I)*ArcTan[c*x])))/(c^2*d - e)] + 4*a*b*ArcTan[c*x]*Log[(-2*Sqrt[c^2*d*e]*E^((2*I)*ArcTan[c*x]) + e*(-1 + E^((2*I)*ArcTan[c*x])) + c^2*d*(1 + E^((2*I)*ArcTan[c*x])))/(c^2*d - e)] + 4*b^2*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[(-2*Sqrt[c^2*d*e]*E^((2*I)*ArcTan[c*x]) + e*(-1 + E^((2*I)*ArcTan[c*x])) + c^2*d*(1 + E^((2*I)*ArcTan[c*x])))/(c^2*d - e)] + 4*b^2*ArcTan[c*x]^2*Log[(-2*Sqrt[c^2*d*e]*E^((2*I)*ArcTan[c*x]) + e*(-1 + E^((2*I)*ArcTan[c*x])) + c^2*d*(1 + E^((2*I)*ArcTan[c*x])))/(c^2*d - e)] - 4*b^2*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[((2*I)*c^2*d - (2*I)*Sqrt[c^2*d*e] + 2*c*(-e + Sqrt[c^2*d*e])*x)/((c^2*d - e)*(I + c*x))] - 2*b^2*ArcTan[c*x]^2*Log[((2*I)*c^2*d - (2*I)*Sqrt[c^2*d*e] + 2*c*(-e + Sqrt[c^2*d*e])*x)/((c^2*d - e)*(I + c*x))] + 2*a^2*Log[d + e*x^2] + 4*b^2*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*(Cos[2*ArcTan[c*x]] + I*Sin[2*ArcTan[c*x]]))/(c^2*d - e)] - 2*b^2*ArcTan[c*x]^2*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*(Cos[2*ArcTan[c*x]] + I*Sin[2*ArcTan[c*x]]))/(c^2*d - e)] + (4*I)*b*(a + b*ArcTan[c*x])*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - (2*I)*b^2*ArcTan[c*x]*PolyLog[2, ((-(c*Sqrt[d]) + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] + Sqrt[e])] - (2*I)*b^2*ArcTan[c*x]*PolyLog[2, -(((c*Sqrt[d] + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] - Sqrt[e]))] - (2*I)*a*b*PolyLog[2, ((-(c^2*d) - e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] - (2*I)*a*b*PolyLog[2, -(((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e))] - 2*b^2*PolyLog[3, -E^((2*I)*ArcTan[c*x])] + b^2*PolyLog[3, ((-(c*Sqrt[d]) + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] + Sqrt[e])] + b^2*PolyLog[3, -(((c*Sqrt[d] + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] - Sqrt[e]))])/(4*e)","B",0
1264,0,0,460,160.2981272,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d+e x^2} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(d + e*x^2),x]","\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{d+e x^2} \, dx","-\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-d}-\sqrt{e} x\right)}{\left(c \sqrt{-d}-i \sqrt{e}\right) (1-i c x)}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2 c \left(\sqrt{e} x+\sqrt{-d}\right)}{\left(\sqrt{-d} c+i \sqrt{e}\right) (1-i c x)}\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{\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{b^2 \text{Li}_3\left(1-\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} \sqrt{e}}-\frac{b^2 \text{Li}_3\left(1-\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} \sqrt{e}}",1,"Integrate[(a + b*ArcTan[c*x])^2/(d + e*x^2), x]","F",-1
1265,1,1412,637,7.5914866,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x*(d + e*x^2)),x]","\frac{24 \log (x) a^2-12 \log \left(e x^2+d\right) a^2-24 b \left(-i \tan ^{-1}(c x)^2-2 \log \left(1-e^{2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)+2 i \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \tan ^{-1}\left(\frac{c e x}{\sqrt{c^2 d e}}\right)+\left(\tan ^{-1}(c x)-\sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right)\right) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e+2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right)+\left(\sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right)+\tan ^{-1}(c x)\right) \log \left(\frac{d \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 \sqrt{c^2 d e} e^{2 i \tan ^{-1}(c x)}+e \left(-1+e^{2 i \tan ^{-1}(c x)}\right)}{c^2 d-e}\right)+i \left(\tan ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)\right)-\frac{1}{2} i \left(\text{Li}_2\left(\frac{\left(-d c^2-e+2 \sqrt{c^2 d e}\right) e^{2 i \tan ^{-1}(c x)}}{c^2 d-e}\right)+\text{Li}_2\left(-\frac{\left(d c^2+e+2 \sqrt{c^2 d e}\right) e^{2 i \tan ^{-1}(c x)}}{c^2 d-e}\right)\right)\right) a+b^2 \left(16 i \tan ^{-1}(c x)^3+24 \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)^2-12 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(c \sqrt{d}-\sqrt{e}\right)}{\sqrt{d} c+\sqrt{e}}+1\right) \tan ^{-1}(c x)^2-12 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(\sqrt{d} c+\sqrt{e}\right)}{c \sqrt{d}-\sqrt{e}}+1\right) \tan ^{-1}(c x)^2+12 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e-2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right) \tan ^{-1}(c x)^2-12 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e+2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right) \tan ^{-1}(c x)^2-24 \log \left(\frac{d \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 \sqrt{c^2 d e} e^{2 i \tan ^{-1}(c x)}+e \left(-1+e^{2 i \tan ^{-1}(c x)}\right)}{c^2 d-e}\right) \tan ^{-1}(c x)^2+12 \log \left(\frac{2 i d c^2+2 \left(\sqrt{c^2 d e}-e\right) x c-2 i \sqrt{c^2 d e}}{\left(c^2 d-e\right) (c x+i)}\right) \tan ^{-1}(c x)^2+12 \log \left(\frac{\left(d c^2+e+2 \sqrt{c^2 d e}\right) \left(\cos \left(2 \tan ^{-1}(c x)\right)+i \sin \left(2 \tan ^{-1}(c x)\right)\right)}{c^2 d-e}+1\right) \tan ^{-1}(c x)^2+24 \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e+2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right) \tan ^{-1}(c x)-24 \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \log \left(\frac{d \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 \sqrt{c^2 d e} e^{2 i \tan ^{-1}(c x)}+e \left(-1+e^{2 i \tan ^{-1}(c x)}\right)}{c^2 d-e}\right) \tan ^{-1}(c x)+24 \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \log \left(\frac{2 i d c^2+2 \left(\sqrt{c^2 d e}-e\right) x c-2 i \sqrt{c^2 d e}}{\left(c^2 d-e\right) (c x+i)}\right) \tan ^{-1}(c x)-24 \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \log \left(\frac{\left(d c^2+e+2 \sqrt{c^2 d e}\right) \left(\cos \left(2 \tan ^{-1}(c x)\right)+i \sin \left(2 \tan ^{-1}(c x)\right)\right)}{c^2 d-e}+1\right) \tan ^{-1}(c x)+24 i \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)+12 i \text{Li}_2\left(\frac{\left(\sqrt{e}-c \sqrt{d}\right) e^{2 i \tan ^{-1}(c x)}}{\sqrt{d} c+\sqrt{e}}\right) \tan ^{-1}(c x)+12 i \text{Li}_2\left(-\frac{\left(\sqrt{d} c+\sqrt{e}\right) e^{2 i \tan ^{-1}(c x)}}{c \sqrt{d}-\sqrt{e}}\right) \tan ^{-1}(c x)+12 \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-6 \text{Li}_3\left(\frac{\left(\sqrt{e}-c \sqrt{d}\right) e^{2 i \tan ^{-1}(c x)}}{\sqrt{d} c+\sqrt{e}}\right)-6 \text{Li}_3\left(-\frac{\left(\sqrt{d} c+\sqrt{e}\right) e^{2 i \tan ^{-1}(c x)}}{c \sqrt{d}-\sqrt{e}}\right)-i \pi ^3\right)}{24 d}","\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\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}+\frac{i b \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\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}-\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{i b \text{Li}_2\left(1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}-\frac{i b \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{d}+\frac{i b \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\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{b^2 \text{Li}_3\left(1-\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}-\frac{b^2 \text{Li}_3\left(1-\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}+\frac{b^2 \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{2 d}-\frac{b^2 \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 d}+\frac{b^2 \text{Li}_3\left(\frac{2}{i c x+1}-1\right)}{2 d}",1,"(24*a^2*Log[x] - 12*a^2*Log[d + e*x^2] - 24*a*b*((-I)*ArcTan[c*x]^2 + (2*I)*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[(c*e*x)/Sqrt[c^2*d*e]] - 2*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + (-ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]] + ArcTan[c*x])*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] + (ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]] + ArcTan[c*x])*Log[(-2*Sqrt[c^2*d*e]*E^((2*I)*ArcTan[c*x]) + e*(-1 + E^((2*I)*ArcTan[c*x])) + c^2*d*(1 + E^((2*I)*ArcTan[c*x])))/(c^2*d - e)] + I*(ArcTan[c*x]^2 + PolyLog[2, E^((2*I)*ArcTan[c*x])]) - (I/2)*(PolyLog[2, ((-(c^2*d) - e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] + PolyLog[2, -(((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e))])) + b^2*((-I)*Pi^3 + (16*I)*ArcTan[c*x]^3 + 24*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 12*ArcTan[c*x]^2*Log[1 + ((c*Sqrt[d] - Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] + Sqrt[e])] - 12*ArcTan[c*x]^2*Log[1 + ((c*Sqrt[d] + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] - Sqrt[e])] + 12*ArcTan[c*x]^2*Log[1 + ((c^2*d + e - 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] + 24*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] - 12*ArcTan[c*x]^2*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] - 24*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[(-2*Sqrt[c^2*d*e]*E^((2*I)*ArcTan[c*x]) + e*(-1 + E^((2*I)*ArcTan[c*x])) + c^2*d*(1 + E^((2*I)*ArcTan[c*x])))/(c^2*d - e)] - 24*ArcTan[c*x]^2*Log[(-2*Sqrt[c^2*d*e]*E^((2*I)*ArcTan[c*x]) + e*(-1 + E^((2*I)*ArcTan[c*x])) + c^2*d*(1 + E^((2*I)*ArcTan[c*x])))/(c^2*d - e)] + 24*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[((2*I)*c^2*d - (2*I)*Sqrt[c^2*d*e] + 2*c*(-e + Sqrt[c^2*d*e])*x)/((c^2*d - e)*(I + c*x))] + 12*ArcTan[c*x]^2*Log[((2*I)*c^2*d - (2*I)*Sqrt[c^2*d*e] + 2*c*(-e + Sqrt[c^2*d*e])*x)/((c^2*d - e)*(I + c*x))] - 24*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*(Cos[2*ArcTan[c*x]] + I*Sin[2*ArcTan[c*x]]))/(c^2*d - e)] + 12*ArcTan[c*x]^2*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*(Cos[2*ArcTan[c*x]] + I*Sin[2*ArcTan[c*x]]))/(c^2*d - e)] + (24*I)*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + (12*I)*ArcTan[c*x]*PolyLog[2, ((-(c*Sqrt[d]) + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] + Sqrt[e])] + (12*I)*ArcTan[c*x]*PolyLog[2, -(((c*Sqrt[d] + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] - Sqrt[e]))] + 12*PolyLog[3, E^((-2*I)*ArcTan[c*x])] - 6*PolyLog[3, ((-(c*Sqrt[d]) + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] + Sqrt[e])] - 6*PolyLog[3, -(((c*Sqrt[d] + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] - Sqrt[e]))]))/(24*d)","B",0
1266,-1,0,553,180.0015954,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^2 \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x^2*(d + e*x^2)),x]","\text{\$Aborted}","-\frac{i b \sqrt{e} \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\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)^{3/2}}+\frac{i b \sqrt{e} \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\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)^{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{\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{b^2 \sqrt{e} \text{Li}_3\left(1-\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}}-\frac{b^2 \sqrt{e} \text{Li}_3\left(1-\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}}-\frac{i b^2 c \text{Li}_2\left(\frac{2}{1-i c x}-1\right)}{d}",1,"$Aborted","F",-1
1267,1,1557,745,11.7283291,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^3 \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x^3*(d + e*x^2)),x]","-\frac{24 e \log (x) a^2-12 e \log \left(e x^2+d\right) a^2+\frac{12 d a^2}{x^2}+\frac{24 b d \left(c^2 x^2+1\right) \tan ^{-1}(c x) a}{x^2}-24 i b e \left(\tan ^{-1}(c x) \left(\tan ^{-1}(c x)+2 i \log \left(1-e^{2 i \tan ^{-1}(c x)}\right)\right)+\text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right)\right) a-\frac{48 b \left(c^2 d-e\right) e \left(-i \tan ^{-1}(c x)^2+2 i \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \tan ^{-1}\left(\frac{c e x}{\sqrt{c^2 d e}}\right)+\left(\tan ^{-1}(c x)-\sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right)\right) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e+2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right)+\left(\sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right)+\tan ^{-1}(c x)\right) \log \left(\frac{d \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 \sqrt{c^2 d e} e^{2 i \tan ^{-1}(c x)}+e \left(-1+e^{2 i \tan ^{-1}(c x)}\right)}{c^2 d-e}\right)-\frac{1}{2} i \left(\text{Li}_2\left(\frac{\left(-d c^2-e+2 \sqrt{c^2 d e}\right) e^{2 i \tan ^{-1}(c x)}}{c^2 d-e}\right)+\text{Li}_2\left(-\frac{\left(d c^2+e+2 \sqrt{c^2 d e}\right) e^{2 i \tan ^{-1}(c x)}}{c^2 d-e}\right)\right)\right) a}{2 c^2 d-2 e}+\frac{24 b c d a}{x}+b^2 \left(8 i e \tan ^{-1}(c x)^3+\frac{12 d \left(c^2 x^2+1\right) \tan ^{-1}(c x)^2}{x^2}+24 e \log \left(1-e^{-2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)^2+24 i e \text{Li}_2\left(e^{-2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)+\frac{24 c d \tan ^{-1}(c x)}{x}-24 c^2 d \log \left(\frac{c x}{\sqrt{c^2 x^2+1}}\right)+12 e \text{Li}_3\left(e^{-2 i \tan ^{-1}(c x)}\right)-i e \pi ^3\right)+2 b^2 e \left(4 i \tan ^{-1}(c x)^3-6 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(c \sqrt{d}-\sqrt{e}\right)}{\sqrt{d} c+\sqrt{e}}+1\right) \tan ^{-1}(c x)^2-6 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(\sqrt{d} c+\sqrt{e}\right)}{c \sqrt{d}-\sqrt{e}}+1\right) \tan ^{-1}(c x)^2+6 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e-2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right) \tan ^{-1}(c x)^2-6 \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e+2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right) \tan ^{-1}(c x)^2-12 \log \left(\frac{d \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 \sqrt{c^2 d e} e^{2 i \tan ^{-1}(c x)}+e \left(-1+e^{2 i \tan ^{-1}(c x)}\right)}{c^2 d-e}\right) \tan ^{-1}(c x)^2+6 \log \left(\frac{2 i d c^2+2 \left(\sqrt{c^2 d e}-e\right) x c-2 i \sqrt{c^2 d e}}{\left(c^2 d-e\right) (c x+i)}\right) \tan ^{-1}(c x)^2+6 \log \left(\frac{\left(d c^2+e+2 \sqrt{c^2 d e}\right) \left(\cos \left(2 \tan ^{-1}(c x)\right)+i \sin \left(2 \tan ^{-1}(c x)\right)\right)}{c^2 d-e}+1\right) \tan ^{-1}(c x)^2+12 \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(d c^2+e+2 \sqrt{c^2 d e}\right)}{c^2 d-e}+1\right) \tan ^{-1}(c x)-12 \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \log \left(\frac{d \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2-2 \sqrt{c^2 d e} e^{2 i \tan ^{-1}(c x)}+e \left(-1+e^{2 i \tan ^{-1}(c x)}\right)}{c^2 d-e}\right) \tan ^{-1}(c x)+12 \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \log \left(\frac{2 i d c^2+2 \left(\sqrt{c^2 d e}-e\right) x c-2 i \sqrt{c^2 d e}}{\left(c^2 d-e\right) (c x+i)}\right) \tan ^{-1}(c x)-12 \sin ^{-1}\left(\sqrt{\frac{c^2 d}{c^2 d-e}}\right) \log \left(\frac{\left(d c^2+e+2 \sqrt{c^2 d e}\right) \left(\cos \left(2 \tan ^{-1}(c x)\right)+i \sin \left(2 \tan ^{-1}(c x)\right)\right)}{c^2 d-e}+1\right) \tan ^{-1}(c x)+6 i \text{Li}_2\left(\frac{\left(\sqrt{e}-c \sqrt{d}\right) e^{2 i \tan ^{-1}(c x)}}{\sqrt{d} c+\sqrt{e}}\right) \tan ^{-1}(c x)+6 i \text{Li}_2\left(-\frac{\left(\sqrt{d} c+\sqrt{e}\right) e^{2 i \tan ^{-1}(c x)}}{c \sqrt{d}-\sqrt{e}}\right) \tan ^{-1}(c x)-3 \text{Li}_3\left(\frac{\left(\sqrt{e}-c \sqrt{d}\right) e^{2 i \tan ^{-1}(c x)}}{\sqrt{d} c+\sqrt{e}}\right)-3 \text{Li}_3\left(-\frac{\left(\sqrt{d} c+\sqrt{e}\right) e^{2 i \tan ^{-1}(c x)}}{c \sqrt{d}-\sqrt{e}}\right)\right)}{24 d^2}","-\frac{c^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 d}+\frac{i b e \text{Li}_2\left(1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}+\frac{i b e \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{i b e \text{Li}_2\left(\frac{2}{i c x+1}-1\right) \left(a+b \tan ^{-1}(c x)\right)}{d^2}-\frac{i b e \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\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{i b e \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\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{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{b^2 e \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{2 d^2}+\frac{b^2 e \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 d^2}-\frac{b^2 e \text{Li}_3\left(\frac{2}{i c x+1}-1\right)}{2 d^2}+\frac{b^2 e \text{Li}_3\left(1-\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^2}+\frac{b^2 e \text{Li}_3\left(1-\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^2}",1,"-1/24*((12*a^2*d)/x^2 + (24*a*b*c*d)/x + (24*a*b*d*(1 + c^2*x^2)*ArcTan[c*x])/x^2 + 24*a^2*e*Log[x] - 12*a^2*e*Log[d + e*x^2] - (24*I)*a*b*e*(ArcTan[c*x]*(ArcTan[c*x] + (2*I)*Log[1 - E^((2*I)*ArcTan[c*x])]) + PolyLog[2, E^((2*I)*ArcTan[c*x])]) - (48*a*b*(c^2*d - e)*e*((-I)*ArcTan[c*x]^2 + (2*I)*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[(c*e*x)/Sqrt[c^2*d*e]] + (-ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]] + ArcTan[c*x])*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] + (ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]] + ArcTan[c*x])*Log[(-2*Sqrt[c^2*d*e]*E^((2*I)*ArcTan[c*x]) + e*(-1 + E^((2*I)*ArcTan[c*x])) + c^2*d*(1 + E^((2*I)*ArcTan[c*x])))/(c^2*d - e)] - (I/2)*(PolyLog[2, ((-(c^2*d) - e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] + PolyLog[2, -(((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e))])))/(2*c^2*d - 2*e) + b^2*((-I)*e*Pi^3 + (24*c*d*ArcTan[c*x])/x + (12*d*(1 + c^2*x^2)*ArcTan[c*x]^2)/x^2 + (8*I)*e*ArcTan[c*x]^3 + 24*e*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] - 24*c^2*d*Log[(c*x)/Sqrt[1 + c^2*x^2]] + (24*I)*e*ArcTan[c*x]*PolyLog[2, E^((-2*I)*ArcTan[c*x])] + 12*e*PolyLog[3, E^((-2*I)*ArcTan[c*x])]) + 2*b^2*e*((4*I)*ArcTan[c*x]^3 - 6*ArcTan[c*x]^2*Log[1 + ((c*Sqrt[d] - Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] + Sqrt[e])] - 6*ArcTan[c*x]^2*Log[1 + ((c*Sqrt[d] + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] - Sqrt[e])] + 6*ArcTan[c*x]^2*Log[1 + ((c^2*d + e - 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] + 12*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] - 6*ArcTan[c*x]^2*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*E^((2*I)*ArcTan[c*x]))/(c^2*d - e)] - 12*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[(-2*Sqrt[c^2*d*e]*E^((2*I)*ArcTan[c*x]) + e*(-1 + E^((2*I)*ArcTan[c*x])) + c^2*d*(1 + E^((2*I)*ArcTan[c*x])))/(c^2*d - e)] - 12*ArcTan[c*x]^2*Log[(-2*Sqrt[c^2*d*e]*E^((2*I)*ArcTan[c*x]) + e*(-1 + E^((2*I)*ArcTan[c*x])) + c^2*d*(1 + E^((2*I)*ArcTan[c*x])))/(c^2*d - e)] + 12*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[((2*I)*c^2*d - (2*I)*Sqrt[c^2*d*e] + 2*c*(-e + Sqrt[c^2*d*e])*x)/((c^2*d - e)*(I + c*x))] + 6*ArcTan[c*x]^2*Log[((2*I)*c^2*d - (2*I)*Sqrt[c^2*d*e] + 2*c*(-e + Sqrt[c^2*d*e])*x)/((c^2*d - e)*(I + c*x))] - 12*ArcSin[Sqrt[(c^2*d)/(c^2*d - e)]]*ArcTan[c*x]*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*(Cos[2*ArcTan[c*x]] + I*Sin[2*ArcTan[c*x]]))/(c^2*d - e)] + 6*ArcTan[c*x]^2*Log[1 + ((c^2*d + e + 2*Sqrt[c^2*d*e])*(Cos[2*ArcTan[c*x]] + I*Sin[2*ArcTan[c*x]]))/(c^2*d - e)] + (6*I)*ArcTan[c*x]*PolyLog[2, ((-(c*Sqrt[d]) + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] + Sqrt[e])] + (6*I)*ArcTan[c*x]*PolyLog[2, -(((c*Sqrt[d] + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] - Sqrt[e]))] - 3*PolyLog[3, ((-(c*Sqrt[d]) + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] + Sqrt[e])] - 3*PolyLog[3, -(((c*Sqrt[d] + Sqrt[e])*E^((2*I)*ArcTan[c*x]))/(c*Sqrt[d] - Sqrt[e]))]))/d^2","B",0
1268,0,0,943,19.7538424,"\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^3*(a + b*ArcTan[c*x])^2)/(d + e*x^2)^2,x]","\int \frac{x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^2} \, dx","\frac{i c \sqrt{-d} \text{Li}_2\left(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{Li}_2\left(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{Li}_3\left(1-\frac{2}{1-i c x}\right) b^2}{2 e^2}+\frac{\text{Li}_3\left(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{Li}_3\left(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{Li}_2\left(1-\frac{2}{1-i c x}\right) b}{e^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(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{Li}_2\left(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,"Integrate[(x^3*(a + b*ArcTan[c*x])^2)/(d + e*x^2)^2, x]","F",-1
1269,0,0,1033,45.3451395,"\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^2*(a + b*ArcTan[c*x])^2)/(d + e*x^2)^2,x]","\int \frac{x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^2} \, dx","-\frac{i c \text{Li}_2\left(1-\frac{2}{1-i c x}\right) b^2}{2 \left(c^2 d-e\right) e}-\frac{i c \text{Li}_2\left(1-\frac{2}{i c x+1}\right) b^2}{2 \left(c^2 d-e\right) e}+\frac{i c \text{Li}_2\left(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{Li}_2\left(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{Li}_3\left(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{Li}_3\left(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{Li}_2\left(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{Li}_2\left(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,"Integrate[(x^2*(a + b*ArcTan[c*x])^2)/(d + e*x^2)^2, x]","F",-1
1270,1,885,457,8.8751192,"\int \frac{x \left(a+b \tan ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^2} \, dx","Integrate[(x*(a + b*ArcTan[c*x])^2)/(d + e*x^2)^2,x]","-\frac{a^2}{2 e \left(e x^2+d\right)}+2 b c^2 \left(\frac{c \tan ^{-1}(c x)-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d}}}{2 e \left(c^3 d-c e\right)}-\frac{\tan ^{-1}(c x)}{2 e \left(e x^2 c^2+d c^2\right)}\right) a+\frac{b^2 c^2 \left(\frac{4 \tan ^{-1}(c x)^2}{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}+\frac{4 \tan ^{-1}(c x) \tanh ^{-1}\left(\frac{\sqrt{-c^2 d e}}{c e x}\right)-2 \cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right) \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)+\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)+2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(\frac{2 c^2 d \left(\sqrt{-c^2 d e}-i e\right) (c x-i)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)+\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)-2 i \tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right) \log \left(\frac{2 c^2 d \left(i e+\sqrt{-c^2 d e}\right) (c x+i)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)-\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)-2 i \left(\tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+\tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{-i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)-\left(\cos ^{-1}\left(-\frac{d c^2+e}{c^2 d-e}\right)+2 i \left(\tanh ^{-1}\left(\frac{c d}{\sqrt{-c^2 d e} x}\right)+\tanh ^{-1}\left(\frac{c e x}{\sqrt{-c^2 d e}}\right)\right)\right) \log \left(\frac{\sqrt{2} \sqrt{-c^2 d e} e^{i \tan ^{-1}(c x)}}{\sqrt{c^2 d-e} \sqrt{d c^2+e+\left(c^2 d-e\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)-i \left(\text{Li}_2\left(\frac{\left(d c^2+e-2 i \sqrt{-c^2 d e}\right) \left(c^2 d-c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)-\text{Li}_2\left(\frac{\left(d c^2+e+2 i \sqrt{-c^2 d e}\right) \left(c^2 d-c \sqrt{-c^2 d e} x\right)}{\left(c^2 d-e\right) \left(d c^2+\sqrt{-c^2 d e} x c\right)}\right)\right)}{\sqrt{-c^2 d e}}\right)}{4 \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{Li}_2\left(1-\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} \sqrt{e} \left(c^2 d-e\right)}-\frac{i b^2 c \text{Li}_2\left(1-\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} \sqrt{e} \left(c^2 d-e\right)}",1,"-1/2*a^2/(e*(d + e*x^2)) + 2*a*b*c^2*(-1/2*ArcTan[c*x]/(e*(c^2*d + c^2*e*x^2)) + (c*ArcTan[c*x] - (Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d])/(2*e*(c^3*d - c*e))) + (b^2*c^2*((4*ArcTan[c*x]^2)/(c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]) + (4*ArcTan[c*x]*ArcTanh[Sqrt[-(c^2*d*e)]/(c*e*x)] - 2*ArcCos[-((c^2*d + e)/(c^2*d - e))]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*((-I)*e + Sqrt[-(c^2*d*e)])*(-I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[(2*c^2*d*(I*e + Sqrt[-(c^2*d*e)])*(I + c*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] - (ArcCos[-((c^2*d + e)/(c^2*d - e))] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] - I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(c^2*d - c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(c^2*d + c*Sqrt[-(c^2*d*e)]*x))]))/Sqrt[-(c^2*d*e)]))/(4*(c^2*d - e))","A",0
1271,0,0,1039,24.4019162,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(d + e*x^2)^2,x]","\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^2} \, dx","\frac{i c \text{Li}_2\left(1-\frac{2}{1-i c x}\right) b^2}{2 d \left(c^2 d-e\right)}+\frac{i c \text{Li}_2\left(1-\frac{2}{i c x+1}\right) b^2}{2 d \left(c^2 d-e\right)}-\frac{i c \text{Li}_2\left(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{Li}_2\left(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{Li}_3\left(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{Li}_3\left(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{Li}_2\left(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{Li}_2\left(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,"Integrate[(a + b*ArcTan[c*x])^2/(d + e*x^2)^2, x]","F",-1
1272,0,0,1087,17.4104823,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x*(d + e*x^2)^2),x]","\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x \left(d+e x^2\right)^2} \, dx","\frac{i c \sqrt{e} \text{Li}_2\left(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{Li}_2\left(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{Li}_3\left(1-\frac{2}{1-i c x}\right) b^2}{2 d^2}-\frac{\text{Li}_3\left(1-\frac{2}{i c x+1}\right) b^2}{2 d^2}+\frac{\text{Li}_3\left(\frac{2}{i c x+1}-1\right) b^2}{2 d^2}-\frac{\text{Li}_3\left(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{Li}_3\left(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{Li}_2\left(1-\frac{2}{1-i c x}\right) b}{d^2}-\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2}{i c x+1}\right) b}{d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(\frac{2}{i c x+1}-1\right) b}{d^2}+\frac{i \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(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{Li}_2\left(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,"Integrate[(a + b*ArcTan[c*x])^2/(x*(d + e*x^2)^2), x]","F",-1
1273,-1,0,1141,180.0027445,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^2 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x^2*(d + e*x^2)^2),x]","\text{\$Aborted}","-\frac{i c e \text{Li}_2\left(1-\frac{2}{1-i c x}\right) b^2}{2 d^2 \left(c^2 d-e\right)}-\frac{i c \text{Li}_2\left(\frac{2}{1-i c x}-1\right) b^2}{d^2}-\frac{i c e \text{Li}_2\left(1-\frac{2}{i c x+1}\right) b^2}{2 d^2 \left(c^2 d-e\right)}+\frac{i c e \text{Li}_2\left(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{Li}_2\left(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{Li}_3\left(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{Li}_3\left(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{Li}_2\left(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{Li}_2\left(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,"$Aborted","F",-1
1274,0,0,1181,32.7642055,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^3 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(x^3*(d + e*x^2)^2),x]","\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{x^3 \left(d+e x^2\right)^2} \, dx","\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{Li}_2\left(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{Li}_2\left(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{Li}_3\left(1-\frac{2}{1-i c x}\right) b^2}{d^3}+\frac{e \text{Li}_3\left(1-\frac{2}{i c x+1}\right) b^2}{d^3}-\frac{e \text{Li}_3\left(\frac{2}{i c x+1}-1\right) b^2}{d^3}+\frac{e \text{Li}_3\left(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{Li}_3\left(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{Li}_2\left(1-\frac{2}{1-i c x}\right) b}{d^3}+\frac{2 i e \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2}{i c x+1}\right) b}{d^3}-\frac{2 i e \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(\frac{2}{i c x+1}-1\right) b}{d^3}-\frac{i e \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(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{Li}_2\left(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,"Integrate[(a + b*ArcTan[c*x])^2/(x^3*(d + e*x^2)^2), x]","F",-1
1275,1,79,111,0.0292007,"\int x^4 \tan ^{-1}(x) \log \left(1+x^2\right) \, dx","Integrate[x^4*ArcTan[x]*Log[1 + x^2],x]","\frac{1}{600} \left(\left(27 x^2-154\right) x^2-30 \log ^2\left(x^2+1\right)+\left(-30 x^4+60 x^2+274\right) \log \left(x^2+1\right)+8 x \left(-6 x^4+10 x^2+15 x^4 \log \left(x^2+1\right)-30\right) \tan ^{-1}(x)+120 \tan ^{-1}(x)^2\right)","-\frac{2}{25} x^5 \tan ^{-1}(x)+\frac{9 x^4}{200}+\frac{2}{15} x^3 \tan ^{-1}(x)-\frac{77 x^2}{300}-\frac{1}{20} \log ^2\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{1}{5} x^5 \log \left(x^2+1\right) \tan ^{-1}(x)-\frac{1}{20} x^4 \log \left(x^2+1\right)-\frac{2}{5} x \tan ^{-1}(x)+\frac{1}{5} \tan ^{-1}(x)^2",1,"(x^2*(-154 + 27*x^2) + 120*ArcTan[x]^2 + (274 + 60*x^2 - 30*x^4)*Log[1 + x^2] - 30*Log[1 + x^2]^2 + 8*x*ArcTan[x]*(-30 + 10*x^2 - 6*x^4 + 15*x^4*Log[1 + x^2]))/600","A",1
1276,1,56,88,0.0268296,"\int x^3 \tan ^{-1}(x) \log \left(1+x^2\right) \, dx","Integrate[x^3*ArcTan[x]*Log[1 + x^2],x]","\frac{1}{72} \left(x \left(7 x^2-6 \left(x^2-3\right) \log \left(x^2+1\right)-75\right)+3 \left(-3 x^4+6 x^2+6 \left(x^4-1\right) \log \left(x^2+1\right)+25\right) \tan ^{-1}(x)\right)","-\frac{1}{8} x^4 \tan ^{-1}(x)+\frac{7 x^3}{72}+\frac{1}{4} x \log \left(x^2+1\right)+\frac{1}{4} x^2 \tan ^{-1}(x)-\frac{1}{4} \log \left(x^2+1\right) \tan ^{-1}(x)+\frac{1}{4} x^4 \log \left(x^2+1\right) \tan ^{-1}(x)-\frac{1}{12} x^3 \log \left(x^2+1\right)-\frac{25 x}{24}+\frac{25}{24} \tan ^{-1}(x)",1,"(x*(-75 + 7*x^2 - 6*(-3 + x^2)*Log[1 + x^2]) + 3*ArcTan[x]*(25 + 6*x^2 - 3*x^4 + 6*(-1 + x^4)*Log[1 + x^2]))/72","A",1
1277,1,64,82,0.023526,"\int x^2 \tan ^{-1}(x) \log \left(1+x^2\right) \, dx","Integrate[x^2*ArcTan[x]*Log[1 + x^2],x]","\frac{1}{36} \left(10 x^2+3 \log ^2\left(x^2+1\right)-2 \left(3 x^2+11\right) \log \left(x^2+1\right)+4 x \left(-2 x^2+3 x^2 \log \left(x^2+1\right)+6\right) \tan ^{-1}(x)-12 \tan ^{-1}(x)^2\right)","-\frac{2}{9} x^3 \tan ^{-1}(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{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,"(10*x^2 - 12*ArcTan[x]^2 - 2*(11 + 3*x^2)*Log[1 + x^2] + 3*Log[1 + x^2]^2 + 4*x*ArcTan[x]*(6 - 2*x^2 + 3*x^2*Log[1 + x^2]))/36","A",1
1278,1,38,49,0.0203891,"\int x \tan ^{-1}(x) \log \left(1+x^2\right) \, dx","Integrate[x*ArcTan[x]*Log[1 + x^2],x]","\frac{1}{2} \left(x^2 \left(-\tan ^{-1}(x)\right)+\log \left(x^2+1\right) \left(\left(x^2+1\right) \tan ^{-1}(x)-x\right)+3 x-3 \tan ^{-1}(x)\right)","-\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 - 3*ArcTan[x] - x^2*ArcTan[x] + (-x + (1 + x^2)*ArcTan[x])*Log[1 + x^2])/2","A",1
1279,1,38,38,0.0100564,"\int \tan ^{-1}(x) \log \left(1+x^2\right) \, dx","Integrate[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",1
1280,0,0,189,0.7813687,"\int \frac{\tan ^{-1}(x) \log \left(1+x^2\right)}{x} \, dx","Integrate[(ArcTan[x]*Log[1 + x^2])/x,x]","\int \frac{\tan ^{-1}(x) \log \left(1+x^2\right)}{x} \, dx","-\frac{1}{2} i \text{Li}_2(-i x) \left(-\log \left(x^2+1\right)+\log (1-i x)+\log (1+i x)\right)+\frac{1}{2} i \text{Li}_2(i x) \left(-\log \left(x^2+1\right)+\log (1-i x)+\log (1+i x)\right)-i \text{Li}_3(1-i x)+i \text{Li}_3(i x+1)+i \text{Li}_2(1-i x) \log (1-i x)-i \text{Li}_2(i x+1) \log (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,"Integrate[(ArcTan[x]*Log[1 + x^2])/x, x]","F",-1
1281,1,41,41,0.0107659,"\int \frac{\tan ^{-1}(x) \log \left(1+x^2\right)}{x^2} \, dx","Integrate[(ArcTan[x]*Log[1 + x^2])/x^2,x]","-\frac{\text{Li}_2\left(-x^2\right)}{2}-\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{\text{Li}_2\left(-x^2\right)}{2}-\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",1
1282,1,49,69,0.0316941,"\int \frac{\tan ^{-1}(x) \log \left(1+x^2\right)}{x^3} \, dx","Integrate[(ArcTan[x]*Log[1 + x^2])/x^3,x]","\frac{1}{2} i (\text{Li}_2(-i x)-\text{Li}_2(i x))-\frac{\log \left(x^2+1\right) \left(x^2 \tan ^{-1}(x)+x+\tan ^{-1}(x)\right)}{2 x^2}+\tan ^{-1}(x)","\frac{1}{2} i \text{Li}_2(-i x)-\frac{1}{2} i \text{Li}_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] - ((x + ArcTan[x] + x^2*ArcTan[x])*Log[1 + x^2])/(2*x^2) + (I/2)*(PolyLog[2, (-I)*x] - PolyLog[2, I*x])","A",1
1283,1,81,81,0.0146927,"\int \frac{\tan ^{-1}(x) \log \left(1+x^2\right)}{x^4} \, dx","Integrate[(ArcTan[x]*Log[1 + x^2])/x^4,x]","\frac{\text{Li}_2\left(-x^2\right)}{6}+\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{\text{Li}_2\left(-x^2\right)}{6}+\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",1
1284,1,98,102,0.0388522,"\int \frac{\tan ^{-1}(x) \log \left(1+x^2\right)}{x^5} \, dx","Integrate[(ArcTan[x]*Log[1 + x^2])/x^5,x]","-\frac{1}{4} i (\text{Li}_2(-i x)-\text{Li}_2(i x))+\frac{1}{2} \left(\frac{1}{2} \left(-\frac{1}{x}-\tan ^{-1}(x)\right)-\frac{\tan ^{-1}(x)}{2 x^2}\right)+\frac{\log \left(x^2+1\right) \left(3 x^4 \tan ^{-1}(x)+3 x^3-x-3 \tan ^{-1}(x)\right)}{12 x^4}-\frac{1}{6 x}-\frac{2}{3} \tan ^{-1}(x)","-\frac{1}{4} i \text{Li}_2(-i x)+\frac{1}{4} i \text{Li}_2(i x)+\frac{\log \left(x^2+1\right)}{4 x}-\frac{\tan ^{-1}(x)}{4 x^2}+\frac{1}{4} \log \left(x^2+1\right) \tan ^{-1}(x)-\frac{\log \left(x^2+1\right) \tan ^{-1}(x)}{4 x^4}-\frac{\log \left(x^2+1\right)}{12 x^3}-\frac{5}{12 x}-\frac{11}{12} \tan ^{-1}(x)",1,"-1/6*1/x - (2*ArcTan[x])/3 + ((-x^(-1) - ArcTan[x])/2 - ArcTan[x]/(2*x^2))/2 + ((-x + 3*x^3 - 3*ArcTan[x] + 3*x^4*ArcTan[x])*Log[1 + x^2])/(12*x^4) - (I/4)*(PolyLog[2, (-I)*x] - PolyLog[2, I*x])","A",1
1285,1,114,114,0.0276377,"\int \frac{\tan ^{-1}(x) \log \left(1+x^2\right)}{x^6} \, dx","Integrate[(ArcTan[x]*Log[1 + x^2])/x^6,x]","-\frac{\text{Li}_2\left(-x^2\right)}{10}-\frac{2 \tan ^{-1}(x)}{15 x^3}-\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{5}{12} \log \left(x^2+1\right)-\frac{\log \left(x^2+1\right) \tan ^{-1}(x)}{5 x^5}-\frac{\log \left(x^2+1\right)}{20 x^4}-\frac{5 \log (x)}{6}+\frac{1}{5} \tan ^{-1}(x)^2+\frac{2 \tan ^{-1}(x)}{5 x}","-\frac{\text{Li}_2\left(-x^2\right)}{10}-\frac{2 \tan ^{-1}(x)}{15 x^3}-\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{5}{12} \log \left(x^2+1\right)-\frac{\log \left(x^2+1\right) \tan ^{-1}(x)}{5 x^5}-\frac{\log \left(x^2+1\right)}{20 x^4}-\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",1
1286,1,214,278,0.192311,"\int x^4 \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right) \, dx","Integrate[x^4*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]),x]","\frac{c x \left(8 a \left(15 c^4 d x^4-2 e \left(3 c^4 x^4-5 c^2 x^2+15\right)\right)+b c x \left(e \left(27 c^2 x^2-154\right)-30 d \left(c^2 x^2-2\right)\right)\right)+\log \left(c^2 x^2+1\right) \left(120 a c^5 e x^5+2 b e \left(-15 c^4 x^4+30 c^2 x^2+137\right)-60 b d\right)+8 \tan ^{-1}(c x) \left(30 a e+15 b c^5 d x^5+15 b c^5 e x^5 \log \left(c^2 x^2+1\right)-2 b c e x \left(3 c^4 x^4-5 c^2 x^2+15\right)\right)-30 b e \log ^2\left(c^2 x^2+1\right)+120 b e \tan ^{-1}(c x)^2}{600 c^5}","\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 \tan ^{-1}(c x)}{5 c^5}-\frac{2 a e x}{5 c^4}+\frac{2 a e x^3}{15 c^2}-\frac{2}{25} a e x^5+\frac{b e \tan ^{-1}(c x)^2}{5 c^5}-\frac{2 b e x \tan ^{-1}(c x)}{5 c^4}-\frac{77 b e x^2}{300 c^3}-\frac{b x^4 \left(e \log \left(c^2 x^2+1\right)+d\right)}{20 c}+\frac{2 b e x^3 \tan ^{-1}(c x)}{15 c^2}-\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{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{b x^2 \left(e \log \left(c^2 x^2+1\right)+d\right)}{10 c^3}-\frac{2}{25} b e x^5 \tan ^{-1}(c x)+\frac{9 b e x^4}{200 c}",1,"(c*x*(b*c*x*(-30*d*(-2 + c^2*x^2) + e*(-154 + 27*c^2*x^2)) + 8*a*(15*c^4*d*x^4 - 2*e*(15 - 5*c^2*x^2 + 3*c^4*x^4))) + 120*b*e*ArcTan[c*x]^2 + (-60*b*d + 120*a*c^5*e*x^5 + 2*b*e*(137 + 30*c^2*x^2 - 15*c^4*x^4))*Log[1 + c^2*x^2] - 30*b*e*Log[1 + c^2*x^2]^2 + 8*ArcTan[c*x]*(30*a*e + 15*b*c^5*d*x^5 - 2*b*c*e*x*(15 - 5*c^2*x^2 + 3*c^4*x^4) + 15*b*c^5*e*x^5*Log[1 + c^2*x^2]))/(600*c^5)","A",1
1287,1,164,221,0.1780988,"\int x^3 \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right) \, dx","Integrate[x^3*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]),x]","\frac{-6 e \log \left(c^2 x^2+1\right) \left(a \left(3-3 c^4 x^4\right)+b c x \left(c^2 x^2-3\right)\right)+c x \left(18 a c^3 d x^3-9 a c e x \left(c^2 x^2-2\right)-6 b d \left(c^2 x^2-3\right)+b e \left(7 c^2 x^2-75\right)\right)+3 b \tan ^{-1}(c x) \left(6 d \left(c^4 x^4-1\right)+e \left(-3 c^4 x^4+6 c^2 x^2+25\right)+6 e \left(c^4 x^4-1\right) \log \left(c^2 x^2+1\right)\right)}{72 c^4}","\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 (2 d-3 e) \tan ^{-1}(c x)}{8 c^4}+\frac{2 b e \tan ^{-1}(c x)}{3 c^4}+\frac{b x (2 d-3 e)}{8 c^3}-\frac{2 b e x}{3 c^3}-\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{b x^3 (2 d-e)}{24 c}+\frac{b e x^3}{18 c}",1,"(c*x*(18*a*c^3*d*x^3 - 6*b*d*(-3 + c^2*x^2) - 9*a*c*e*x*(-2 + c^2*x^2) + b*e*(-75 + 7*c^2*x^2)) - 6*e*(b*c*x*(-3 + c^2*x^2) + a*(3 - 3*c^4*x^4))*Log[1 + c^2*x^2] + 3*b*ArcTan[c*x]*(e*(25 + 6*c^2*x^2 - 3*c^4*x^4) + 6*d*(-1 + c^4*x^4) + 6*e*(-1 + c^4*x^4)*Log[1 + c^2*x^2]))/(72*c^4)","A",1
1288,1,171,213,0.1414943,"\int x^2 \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right) \, dx","Integrate[x^2*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]),x]","\frac{2 c x \left(6 a c^2 d x^2-4 a e \left(c^2 x^2-3\right)+b c x (5 e-3 d)\right)+2 \log \left(c^2 x^2+1\right) \left(6 a c^3 e x^3-b e \left(3 c^2 x^2+11\right)+3 b d\right)-4 \tan ^{-1}(c x) \left(6 a e+b c x \left(-3 c^2 d x^2+2 c^2 e x^2-6 e\right)-3 b c^3 e x^3 \log \left(c^2 x^2+1\right)\right)+3 b e \log ^2\left(c^2 x^2+1\right)-12 b e \tan ^{-1}(c x)^2}{36 c^3}","\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 \tan ^{-1}(c x)}{3 c^3}+\frac{2 a e x}{3 c^2}-\frac{2}{9} a e x^3-\frac{b e \tan ^{-1}(c x)^2}{3 c^3}-\frac{b x^2 \left(e \log \left(c^2 x^2+1\right)+d\right)}{6 c}+\frac{2 b e x \tan ^{-1}(c x)}{3 c^2}+\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}{9} b e x^3 \tan ^{-1}(c x)+\frac{5 b e x^2}{18 c}",1,"(2*c*x*(b*c*(-3*d + 5*e)*x + 6*a*c^2*d*x^2 - 4*a*e*(-3 + c^2*x^2)) - 12*b*e*ArcTan[c*x]^2 + 2*(3*b*d + 6*a*c^3*e*x^3 - b*e*(11 + 3*c^2*x^2))*Log[1 + c^2*x^2] + 3*b*e*Log[1 + c^2*x^2]^2 - 4*ArcTan[c*x]*(6*a*e + b*c*x*(-6*e - 3*c^2*d*x^2 + 2*c^2*e*x^2) - 3*b*c^3*e*x^3*Log[1 + c^2*x^2]))/(36*c^3)","A",1
1289,1,105,137,0.0986746,"\int x \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right) \, dx","Integrate[x*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]),x]","\frac{e \log \left(c^2 x^2+1\right) \left(a c^2 x^2+a-b c x\right)+c x (a c x (d-e)-b (d-3 e))+b \tan ^{-1}(c x) \left(c^2 d x^2-e \left(c^2 x^2+3\right)+\left(c^2 e x^2+e\right) \log \left(c^2 x^2+1\right)+d\right)}{2 c^2}","\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,"(c*x*(-(b*(d - 3*e)) + a*c*(d - e)*x) + e*(a - b*c*x + a*c^2*x^2)*Log[1 + c^2*x^2] + b*ArcTan[c*x]*(d + c^2*d*x^2 - e*(3 + c^2*x^2) + (e + c^2*e*x^2)*Log[1 + c^2*x^2]))/(2*c^2)","A",1
1290,1,138,100,0.0171834,"\int \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right) \, dx","Integrate[(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]),x]","a e x \log \left(c^2 x^2+1\right)+\frac{2 a e \tan ^{-1}(c x)}{c}+a d x-2 a e x-\frac{b d \log \left(c^2 x^2+1\right)}{2 c}-\frac{b e \log ^2\left(c^2 x^2+1\right)}{4 c}+\frac{b e \log \left(c^2 x^2+1\right)}{c}+b e x \log \left(c^2 x^2+1\right) \tan ^{-1}(c x)+b d x \tan ^{-1}(c x)+\frac{b e \tan ^{-1}(c x)^2}{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,"a*d*x - 2*a*e*x + (2*a*e*ArcTan[c*x])/c + b*d*x*ArcTan[c*x] - 2*b*e*x*ArcTan[c*x] + (b*e*ArcTan[c*x]^2)/c - (b*d*Log[1 + c^2*x^2])/(2*c) + (b*e*Log[1 + c^2*x^2])/c + a*e*x*Log[1 + c^2*x^2] + b*e*x*ArcTan[c*x]*Log[1 + c^2*x^2] - (b*e*Log[1 + c^2*x^2]^2)/(4*c)","A",1
1291,0,0,282,0.2109479,"\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right)}{x} \, dx","Integrate[((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x,x]","\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(1+c^2 x^2\right)\right)}{x} \, dx","-\frac{1}{2} a e \text{Li}_2\left(-c^2 x^2\right)+a d \log (x)-\frac{1}{2} i b e \text{Li}_2(-i c x) \left(-\log \left(c^2 x^2+1\right)+\log (1-i c x)+\log (1+i c x)\right)+\frac{1}{2} i b e \text{Li}_2(i c x) \left(-\log \left(c^2 x^2+1\right)+\log (1-i c x)+\log (1+i c x)\right)+\frac{1}{2} i b d \text{Li}_2(-i c x)-\frac{1}{2} i b d \text{Li}_2(i c x)-i b e \text{Li}_3(1-i c x)+i b e \text{Li}_3(i c x+1)+i b e \text{Li}_2(1-i c x) \log (1-i c x)-i b e \text{Li}_2(i c x+1) \log (1+i c 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,"Integrate[((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x, x]","F",-1
1292,1,111,100,0.1062257,"\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","Integrate[((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x^2,x]","-\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}+b c \left(\frac{1}{2} e \text{Li}_2\left(c^2 x^2+1\right)-\frac{\left(e \log \left(c^2 x^2+1\right)+d\right) \left(-2 e \log \left(-c^2 x^2\right)+e \log \left(c^2 x^2+1\right)+d\right)}{4 e}\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)-\frac{1}{2} b c e \text{Li}_2\left(\frac{1}{c^2 x^2+1}\right)",1,"(c*e*(a + b*ArcTan[c*x])^2)/b - ((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x + b*c*(-1/4*((d + e*Log[1 + c^2*x^2])*(d - 2*e*Log[-(c^2*x^2)] + e*Log[1 + c^2*x^2]))/e + (e*PolyLog[2, 1 + c^2*x^2])/2)","A",1
1293,1,189,154,0.1280723,"\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","Integrate[((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x^3,x]","-\frac{-2 a c^2 e x^2 \log (x)+a c^2 e x^2 \log \left(c^2 x^2+1\right)+a e \log \left(c^2 x^2+1\right)+a d+b c^2 d x^2 \tan ^{-1}(c x)-i b c^2 e x^2 \text{Li}_2(-i c x)+i b c^2 e x^2 \text{Li}_2(i c x)+b c e x \log \left(c^2 x^2+1\right)-2 b c^2 e x^2 \tan ^{-1}(c x)+b c^2 e x^2 \log \left(c^2 x^2+1\right) \tan ^{-1}(c x)+b e \log \left(c^2 x^2+1\right) \tan ^{-1}(c x)+b c d x+b d \tan ^{-1}(c x)}{2 x^2}","-\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)+\frac{1}{2} i b c^2 e \text{Li}_2(-i c x)-\frac{1}{2} i b c^2 e \text{Li}_2(i c x)+b c^2 e \tan ^{-1}(c x)",1,"-1/2*(a*d + b*c*d*x + b*d*ArcTan[c*x] + b*c^2*d*x^2*ArcTan[c*x] - 2*b*c^2*e*x^2*ArcTan[c*x] - 2*a*c^2*e*x^2*Log[x] + a*e*Log[1 + c^2*x^2] + b*c*e*x*Log[1 + c^2*x^2] + a*c^2*e*x^2*Log[1 + c^2*x^2] + b*e*ArcTan[c*x]*Log[1 + c^2*x^2] + b*c^2*e*x^2*ArcTan[c*x]*Log[1 + c^2*x^2] - I*b*c^2*e*x^2*PolyLog[2, (-I)*c*x] + I*b*c^2*e*x^2*PolyLog[2, I*c*x])/x^2","A",1
1294,1,181,189,0.1817559,"\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","Integrate[((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x^4,x]","\frac{1}{12} \left(-\frac{4 c^3 e \left(a+b \tan ^{-1}(c x)\right)^2}{b}-\frac{4 \left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{x^3}-\frac{8 c^2 e \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{2 b c \left(e \log \left(c^2 x^2+1\right)+d\right)}{x^2}-2 b c^3 \left(\log \left(-c^2 x^2\right) \left(e \log \left(c^2 x^2+1\right)+d\right)+e \text{Li}_2\left(c^2 x^2+1\right)\right)+\frac{b c^3 \left(e \log \left(c^2 x^2+1\right)+d\right)^2}{e}+6 b c^3 e \left(2 \log (x)-\log \left(c^2 x^2+1\right)\right)\right)","-\frac{c^3 e \left(a+b \tan ^{-1}(c x)\right)^2}{3 b}-\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{2 c^2 e \left(a+b \tan ^{-1}(c x)\right)}{3 x}+b c^3 e \log (x)-\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}{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{1}{6} b c^3 e \text{Li}_2\left(\frac{1}{c^2 x^2+1}\right)-\frac{1}{3} b c^3 e \log \left(c^2 x^2+1\right)",1,"((-8*c^2*e*(a + b*ArcTan[c*x]))/x - (4*c^3*e*(a + b*ArcTan[c*x])^2)/b + 6*b*c^3*e*(2*Log[x] - Log[1 + c^2*x^2]) - (2*b*c*(d + e*Log[1 + c^2*x^2]))/x^2 - (4*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x^3 + (b*c^3*(d + e*Log[1 + c^2*x^2])^2)/e - 2*b*c^3*(Log[-(c^2*x^2)]*(d + e*Log[1 + c^2*x^2]) + e*PolyLog[2, 1 + c^2*x^2]))/12","A",1
1295,1,260,225,0.190916,"\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","Integrate[((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x^5,x]","-\frac{6 a c^4 e x^4 \log (x)+3 a c^2 e x^2+3 a e \log \left(c^2 x^2+1\right)-3 a c^4 e x^4 \log \left(c^2 x^2+1\right)+3 a d-3 b c^4 d x^4 \tan ^{-1}(c x)+3 i b c^4 e x^4 \text{Li}_2(-i c x)-3 i b c^4 e x^4 \text{Li}_2(i c x)+11 b c^4 e x^4 \tan ^{-1}(c x)-3 b c^3 d x^3+5 b c^3 e x^3+b c e x \log \left(c^2 x^2+1\right)+3 b c^2 e x^2 \tan ^{-1}(c x)+3 b e \log \left(c^2 x^2+1\right) \tan ^{-1}(c x)-3 b c^4 e x^4 \log \left(c^2 x^2+1\right) \tan ^{-1}(c x)-3 b c^3 e x^3 \log \left(c^2 x^2+1\right)+b c d x+3 b d \tan ^{-1}(c x)}{12 x^4}","-\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{1}{2} a c^4 e \log (x)-\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}{4} i b c^4 e \text{Li}_2(-i c x)+\frac{1}{4} i b c^4 e \text{Li}_2(i c x)-\frac{11}{12} b c^4 e \tan ^{-1}(c x)-\frac{5 b c^3 e}{12 x}-\frac{b c \left(e \log \left(c^2 x^2+1\right)+d\right)}{12 x^3}-\frac{b c^2 e \tan ^{-1}(c x)}{4 x^2}+\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^3 \left(e \log \left(c^2 x^2+1\right)+d\right)}{4 x}",1,"-1/12*(3*a*d + b*c*d*x + 3*a*c^2*e*x^2 - 3*b*c^3*d*x^3 + 5*b*c^3*e*x^3 + 3*b*d*ArcTan[c*x] + 3*b*c^2*e*x^2*ArcTan[c*x] - 3*b*c^4*d*x^4*ArcTan[c*x] + 11*b*c^4*e*x^4*ArcTan[c*x] + 6*a*c^4*e*x^4*Log[x] + 3*a*e*Log[1 + c^2*x^2] + b*c*e*x*Log[1 + c^2*x^2] - 3*b*c^3*e*x^3*Log[1 + c^2*x^2] - 3*a*c^4*e*x^4*Log[1 + c^2*x^2] + 3*b*e*ArcTan[c*x]*Log[1 + c^2*x^2] - 3*b*c^4*e*x^4*ArcTan[c*x]*Log[1 + c^2*x^2] + (3*I)*b*c^4*e*x^4*PolyLog[2, (-I)*c*x] - (3*I)*b*c^4*e*x^4*PolyLog[2, I*c*x])/x^4","A",1
1296,1,259,248,0.3407489,"\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","Integrate[((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x^6,x]","\frac{1}{60} \left(\frac{12 c^5 e \left(a+b \tan ^{-1}(c x)\right)^2}{b}+\frac{24 c^4 e \left(a+b \tan ^{-1}(c x)\right)}{x}-\frac{12 \left(a+b \tan ^{-1}(c x)\right) \left(e \log \left(c^2 x^2+1\right)+d\right)}{x^5}-\frac{8 c^2 e \left(a+b \tan ^{-1}(c x)\right)}{x^3}-\frac{3 b c \left(e \log \left(c^2 x^2+1\right)+d\right)}{x^4}-\frac{3 b c^5 \left(e \log \left(c^2 x^2+1\right)+d\right)^2}{e}+6 b c^5 \log \left(-c^2 x^2\right) \left(e \log \left(c^2 x^2+1\right)+d\right)+6 b c^5 e \text{Li}_2\left(c^2 x^2+1\right)-18 b c^5 e \left(2 \log (x)-\log \left(c^2 x^2+1\right)\right)+\frac{6 b c^3 \left(e \log \left(c^2 x^2+1\right)+d\right)}{x^2}+7 b c^3 e \left(c^2 \log \left(c^2 x^2+1\right)-2 c^2 \log (x)-\frac{1}{x^2}\right)\right)","\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{\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{5}{6} b c^5 e \log (x)-\frac{7 b c^3 e}{60 x^2}-\frac{b c \left(e \log \left(c^2 x^2+1\right)+d\right)}{20 x^4}+\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{1}{10} b c^5 e \text{Li}_2\left(\frac{1}{c^2 x^2+1}\right)+\frac{19}{60} b c^5 e \log \left(c^2 x^2+1\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}",1,"((-8*c^2*e*(a + b*ArcTan[c*x]))/x^3 + (24*c^4*e*(a + b*ArcTan[c*x]))/x + (12*c^5*e*(a + b*ArcTan[c*x])^2)/b - 18*b*c^5*e*(2*Log[x] - Log[1 + c^2*x^2]) + 7*b*c^3*e*(-x^(-2) - 2*c^2*Log[x] + c^2*Log[1 + c^2*x^2]) - (3*b*c*(d + e*Log[1 + c^2*x^2]))/x^4 + (6*b*c^3*(d + e*Log[1 + c^2*x^2]))/x^2 - (12*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x^5 + 6*b*c^5*Log[-(c^2*x^2)]*(d + e*Log[1 + c^2*x^2]) - (3*b*c^5*(d + e*Log[1 + c^2*x^2])^2)/e + 6*b*c^5*e*PolyLog[2, 1 + c^2*x^2])/60","A",1
1297,1,1140,562,8.9728461,"\int x \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right) \, dx","Integrate[x*(a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]),x]","\frac{2 a d g x^2 c^2-2 a e g x^2 c^2+2 b d g x^2 \tan ^{-1}(c x) c^2-2 b e g x^2 \tan ^{-1}(c x) c^2+4 i b e f \sin ^{-1}\left(\sqrt{\frac{c^2 f}{c^2 f-g}}\right) \tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right) c^2-4 b e f \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right) c^2+2 b e f \sin ^{-1}\left(\sqrt{\frac{c^2 f}{c^2 f-g}}\right) \log \left(\frac{\left(1+e^{2 i \tan ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 i \tan ^{-1}(c x)}\right) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right) c^2+2 b e f \tan ^{-1}(c x) \log \left(\frac{\left(1+e^{2 i \tan ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 i \tan ^{-1}(c x)}\right) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right) c^2-2 b e f \sin ^{-1}\left(\sqrt{\frac{c^2 f}{c^2 f-g}}\right) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(f c^2+g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}+1\right) c^2+2 b e f \tan ^{-1}(c x) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(f c^2+g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}+1\right) c^2+2 a e g x^2 \log \left(g x^2+f\right) c^2+2 a e f \log \left(g x^2+f\right) c^2+2 b e g x^2 \tan ^{-1}(c x) \log \left(g x^2+f\right) c^2-i b e f \text{Li}_2\left(-\frac{e^{2 i \tan ^{-1}(c x)} \left(f c^2+g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}\right) c^2-2 b d g x c+6 b e g x c-4 b e \sqrt{f} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right) c-2 b e g x \log \left(g x^2+f\right) c+2 b d g \tan ^{-1}(c x)-2 b e g \tan ^{-1}(c x)-4 i b e g \sin ^{-1}\left(\sqrt{\frac{c^2 f}{c^2 f-g}}\right) \tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right)+4 b e g \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-2 b e g \sin ^{-1}\left(\sqrt{\frac{c^2 f}{c^2 f-g}}\right) \log \left(\frac{\left(1+e^{2 i \tan ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 i \tan ^{-1}(c x)}\right) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right)-2 b e g \tan ^{-1}(c x) \log \left(\frac{\left(1+e^{2 i \tan ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 i \tan ^{-1}(c x)}\right) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right)+2 b e g \sin ^{-1}\left(\sqrt{\frac{c^2 f}{c^2 f-g}}\right) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(f c^2+g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}+1\right)-2 b e g \tan ^{-1}(c x) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(f c^2+g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}+1\right)+2 b e g \tan ^{-1}(c x) \log \left(g x^2+f\right)+2 i b e \left(c^2 f-g\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)-i b e \left(c^2 f-g\right) \text{Li}_2\left(\frac{e^{2 i \tan ^{-1}(c x)} \left(-f c^2-g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}\right)+i b e g \text{Li}_2\left(-\frac{e^{2 i \tan ^{-1}(c x)} \left(f c^2+g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}\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{i b e \left(c^2 f-g\right) \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 c^2 g}-\frac{i b e \left(c^2 f-g\right) \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{\left(c \sqrt{-f}-i \sqrt{g}\right) (1-i c x)}\right)}{4 c^2 g}-\frac{i b e \left(c^2 f-g\right) \text{Li}_2\left(1-\frac{2 c \left(\sqrt{g} x+\sqrt{-f}\right)}{\left(\sqrt{-f} c+i \sqrt{g}\right) (1-i c x)}\right)}{4 c^2 g}-\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,"(-2*b*c*d*g*x + 6*b*c*e*g*x + 2*a*c^2*d*g*x^2 - 2*a*c^2*e*g*x^2 + 2*b*d*g*ArcTan[c*x] - 2*b*e*g*ArcTan[c*x] + 2*b*c^2*d*g*x^2*ArcTan[c*x] - 2*b*c^2*e*g*x^2*ArcTan[c*x] - 4*b*c*e*Sqrt[f]*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]] + (4*I)*b*c^2*e*f*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] - (4*I)*b*e*g*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] - 4*b*c^2*e*f*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 4*b*e*g*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 2*b*c^2*e*f*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[(c^2*(1 + E^((2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcTan[c*x]))*g - 2*E^((2*I)*ArcTan[c*x])*Sqrt[c^2*f*g])/(c^2*f - g)] - 2*b*e*g*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[(c^2*(1 + E^((2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcTan[c*x]))*g - 2*E^((2*I)*ArcTan[c*x])*Sqrt[c^2*f*g])/(c^2*f - g)] + 2*b*c^2*e*f*ArcTan[c*x]*Log[(c^2*(1 + E^((2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcTan[c*x]))*g - 2*E^((2*I)*ArcTan[c*x])*Sqrt[c^2*f*g])/(c^2*f - g)] - 2*b*e*g*ArcTan[c*x]*Log[(c^2*(1 + E^((2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcTan[c*x]))*g - 2*E^((2*I)*ArcTan[c*x])*Sqrt[c^2*f*g])/(c^2*f - g)] - 2*b*c^2*e*f*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[1 + (E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] + 2*b*e*g*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[1 + (E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] + 2*b*c^2*e*f*ArcTan[c*x]*Log[1 + (E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] - 2*b*e*g*ArcTan[c*x]*Log[1 + (E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] + 2*a*c^2*e*f*Log[f + g*x^2] - 2*b*c*e*g*x*Log[f + g*x^2] + 2*a*c^2*e*g*x^2*Log[f + g*x^2] + 2*b*e*g*ArcTan[c*x]*Log[f + g*x^2] + 2*b*c^2*e*g*x^2*ArcTan[c*x]*Log[f + g*x^2] + (2*I)*b*e*(c^2*f - g)*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - I*b*e*(c^2*f - g)*PolyLog[2, (E^((2*I)*ArcTan[c*x])*(-(c^2*f) - g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] - I*b*c^2*e*f*PolyLog[2, -((E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g))] + I*b*e*g*PolyLog[2, -((E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g))])/(4*c^2*g)","B",0
1298,1,1352,656,4.5545468,"\int \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right) \, dx","Integrate[(a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]),x]","a d x-2 a e x+b d \tan ^{-1}(c x) x+a e \log \left(g x^2+f\right) x+\frac{2 a e \sqrt{f} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{g}}-\frac{b d \log \left(c^2 x^2+1\right)}{2 c}+b e \left(x \tan ^{-1}(c x)-\frac{\log \left(c^2 x^2+1\right)}{2 c}\right) \log \left(g x^2+f\right)+\frac{b e g \left(\frac{\left(-\log \left(x-\frac{i}{c}\right)-\log \left(x+\frac{i}{c}\right)+\log \left(c^2 x^2+1\right)\right) \log \left(g x^2+f\right)}{2 g}+\frac{\log \left(x-\frac{i}{c}\right) \log \left(1-\frac{\sqrt{g} \left(x-\frac{i}{c}\right)}{-i \sqrt{f}-\frac{i \sqrt{g}}{c}}\right)+\text{Li}_2\left(\frac{\sqrt{g} \left(x-\frac{i}{c}\right)}{-i \sqrt{f}-\frac{i \sqrt{g}}{c}}\right)}{2 g}+\frac{\log \left(x-\frac{i}{c}\right) \log \left(1-\frac{\sqrt{g} \left(x-\frac{i}{c}\right)}{i \sqrt{f}-\frac{i \sqrt{g}}{c}}\right)+\text{Li}_2\left(\frac{\sqrt{g} \left(x-\frac{i}{c}\right)}{i \sqrt{f}-\frac{i \sqrt{g}}{c}}\right)}{2 g}+\frac{\log \left(x+\frac{i}{c}\right) \log \left(1-\frac{\sqrt{g} \left(x+\frac{i}{c}\right)}{\frac{i \sqrt{g}}{c}-i \sqrt{f}}\right)+\text{Li}_2\left(\frac{\sqrt{g} \left(x+\frac{i}{c}\right)}{\frac{i \sqrt{g}}{c}-i \sqrt{f}}\right)}{2 g}+\frac{\log \left(x+\frac{i}{c}\right) \log \left(1-\frac{\sqrt{g} \left(x+\frac{i}{c}\right)}{i \sqrt{f}+\frac{i \sqrt{g}}{c}}\right)+\text{Li}_2\left(\frac{\sqrt{g} \left(x+\frac{i}{c}\right)}{i \sqrt{f}+\frac{i \sqrt{g}}{c}}\right)}{2 g}\right)}{c}-\frac{b e \left(\frac{f \left(4 \tan ^{-1}(c x) \tanh ^{-1}\left(\frac{\sqrt{-c^2 f g}}{c g x}\right)-2 \cos ^{-1}\left(\frac{f c^2+g}{g-c^2 f}\right) \tanh ^{-1}\left(\frac{c g x}{\sqrt{-c^2 f g}}\right)-\left(\cos ^{-1}\left(\frac{f c^2+g}{g-c^2 f}\right)-2 i \tanh ^{-1}\left(\frac{c g x}{\sqrt{-c^2 f g}}\right)\right) \log \left(-\frac{2 c^2 f \left(i g+\sqrt{-c^2 f g}\right) (c x-i)}{\left(c^2 f-g\right) \left(c^2 f-c \sqrt{-c^2 f g} x\right)}\right)-\left(\cos ^{-1}\left(\frac{f c^2+g}{g-c^2 f}\right)+2 i \tanh ^{-1}\left(\frac{c g x}{\sqrt{-c^2 f g}}\right)\right) \log \left(\frac{2 i c^2 f \left(g+i \sqrt{-c^2 f g}\right) (c x+i)}{\left(c^2 f-g\right) \left(c^2 f-c \sqrt{-c^2 f g} x\right)}\right)+\left(\cos ^{-1}\left(\frac{f c^2+g}{g-c^2 f}\right)-2 i \tanh ^{-1}\left(\frac{\sqrt{-c^2 f g}}{c g x}\right)+2 i \tanh ^{-1}\left(\frac{c g x}{\sqrt{-c^2 f g}}\right)\right) \log \left(\frac{\sqrt{2} e^{-i \tan ^{-1}(c x)} \sqrt{-c^2 f g}}{\sqrt{g-c^2 f} \sqrt{-f c^2-g+\left(g-c^2 f\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+\left(\cos ^{-1}\left(\frac{f c^2+g}{g-c^2 f}\right)+2 i \tanh ^{-1}\left(\frac{\sqrt{-c^2 f g}}{c g x}\right)-2 i \tanh ^{-1}\left(\frac{c g x}{\sqrt{-c^2 f g}}\right)\right) \log \left(\frac{\sqrt{2} e^{i \tan ^{-1}(c x)} \sqrt{-c^2 f g}}{\sqrt{g-c^2 f} \sqrt{-f c^2-g+\left(g-c^2 f\right) \cos \left(2 \tan ^{-1}(c x)\right)}}\right)+i \left(\text{Li}_2\left(\frac{\left(f c^2+g+2 i \sqrt{-c^2 f g}\right) \left(f c^2+\sqrt{-c^2 f g} x c\right)}{\left(c^2 f-g\right) \left(c^2 f-c \sqrt{-c^2 f g} x\right)}\right)-\text{Li}_2\left(\frac{\left(f c^2+g-2 i \sqrt{-c^2 f g}\right) \left(f c^2+\sqrt{-c^2 f g} x c\right)}{\left(c^2 f-g\right) \left(c^2 f-c \sqrt{-c^2 f g} x\right)}\right)\right)\right) c^2}{\sqrt{-c^2 f g}}+4 x \tan ^{-1}(c x) c+4 \log \left(\frac{1}{\sqrt{c^2 x^2+1}}\right)\right)}{2 c}","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 \text{Li}_2\left(\frac{c^2 \left(g x^2+f\right)}{c^2 f-g}\right)}{2 c}+\frac{b e \log \left(c^2 x^2+1\right)}{c}-\frac{i b e \sqrt{-f} \text{Li}_2\left(\frac{\sqrt{g} (i-c x)}{\sqrt{-f} c+i \sqrt{g}}\right)}{2 \sqrt{g}}+\frac{i b e \sqrt{-f} \text{Li}_2\left(\frac{\sqrt{g} (1-i c x)}{i \sqrt{-f} c+\sqrt{g}}\right)}{2 \sqrt{g}}+\frac{i b e \sqrt{-f} \text{Li}_2\left(\frac{\sqrt{g} (i c x+1)}{i \sqrt{-f} c+\sqrt{g}}\right)}{2 \sqrt{g}}-\frac{i b e \sqrt{-f} \text{Li}_2\left(\frac{\sqrt{g} (c x+i)}{\sqrt{-f} c+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}}-\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,"a*d*x - 2*a*e*x + b*d*x*ArcTan[c*x] + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] - (b*d*Log[1 + c^2*x^2])/(2*c) + a*e*x*Log[f + g*x^2] + b*e*(x*ArcTan[c*x] - Log[1 + c^2*x^2]/(2*c))*Log[f + g*x^2] + (b*e*g*(((-Log[(-I)/c + x] - Log[I/c + x] + Log[1 + c^2*x^2])*Log[f + g*x^2])/(2*g) + (Log[(-I)/c + x]*Log[1 - (Sqrt[g]*((-I)/c + x))/((-I)*Sqrt[f] - (I*Sqrt[g])/c)] + PolyLog[2, (Sqrt[g]*((-I)/c + x))/((-I)*Sqrt[f] - (I*Sqrt[g])/c)])/(2*g) + (Log[(-I)/c + x]*Log[1 - (Sqrt[g]*((-I)/c + x))/(I*Sqrt[f] - (I*Sqrt[g])/c)] + PolyLog[2, (Sqrt[g]*((-I)/c + x))/(I*Sqrt[f] - (I*Sqrt[g])/c)])/(2*g) + (Log[I/c + x]*Log[1 - (Sqrt[g]*(I/c + x))/((-I)*Sqrt[f] + (I*Sqrt[g])/c)] + PolyLog[2, (Sqrt[g]*(I/c + x))/((-I)*Sqrt[f] + (I*Sqrt[g])/c)])/(2*g) + (Log[I/c + x]*Log[1 - (Sqrt[g]*(I/c + x))/(I*Sqrt[f] + (I*Sqrt[g])/c)] + PolyLog[2, (Sqrt[g]*(I/c + x))/(I*Sqrt[f] + (I*Sqrt[g])/c)])/(2*g)))/c - (b*e*(4*c*x*ArcTan[c*x] + 4*Log[1/Sqrt[1 + c^2*x^2]] + (c^2*f*(4*ArcTan[c*x]*ArcTanh[Sqrt[-(c^2*f*g)]/(c*g*x)] - 2*ArcCos[(c^2*f + g)/(-(c^2*f) + g)]*ArcTanh[(c*g*x)/Sqrt[-(c^2*f*g)]] - (ArcCos[(c^2*f + g)/(-(c^2*f) + g)] - (2*I)*ArcTanh[(c*g*x)/Sqrt[-(c^2*f*g)]])*Log[(-2*c^2*f*(I*g + Sqrt[-(c^2*f*g)])*(-I + c*x))/((c^2*f - g)*(c^2*f - c*Sqrt[-(c^2*f*g)]*x))] - (ArcCos[(c^2*f + g)/(-(c^2*f) + g)] + (2*I)*ArcTanh[(c*g*x)/Sqrt[-(c^2*f*g)]])*Log[((2*I)*c^2*f*(g + I*Sqrt[-(c^2*f*g)])*(I + c*x))/((c^2*f - g)*(c^2*f - c*Sqrt[-(c^2*f*g)]*x))] + (ArcCos[(c^2*f + g)/(-(c^2*f) + g)] - (2*I)*ArcTanh[Sqrt[-(c^2*f*g)]/(c*g*x)] + (2*I)*ArcTanh[(c*g*x)/Sqrt[-(c^2*f*g)]])*Log[(Sqrt[2]*Sqrt[-(c^2*f*g)])/(E^(I*ArcTan[c*x])*Sqrt[-(c^2*f) + g]*Sqrt[-(c^2*f) - g + (-(c^2*f) + g)*Cos[2*ArcTan[c*x]]])] + (ArcCos[(c^2*f + g)/(-(c^2*f) + g)] + (2*I)*ArcTanh[Sqrt[-(c^2*f*g)]/(c*g*x)] - (2*I)*ArcTanh[(c*g*x)/Sqrt[-(c^2*f*g)]])*Log[(Sqrt[2]*E^(I*ArcTan[c*x])*Sqrt[-(c^2*f*g)])/(Sqrt[-(c^2*f) + g]*Sqrt[-(c^2*f) - g + (-(c^2*f) + g)*Cos[2*ArcTan[c*x]]])] + I*(-PolyLog[2, ((c^2*f + g - (2*I)*Sqrt[-(c^2*f*g)])*(c^2*f + c*Sqrt[-(c^2*f*g)]*x))/((c^2*f - g)*(c^2*f - c*Sqrt[-(c^2*f*g)]*x))] + PolyLog[2, ((c^2*f + g + (2*I)*Sqrt[-(c^2*f*g)])*(c^2*f + c*Sqrt[-(c^2*f*g)]*x))/((c^2*f - g)*(c^2*f - c*Sqrt[-(c^2*f*g)]*x))])))/Sqrt[-(c^2*f*g)]))/(2*c)","B",1
1299,0,0,102,0.2715416,"\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x} \, dx","Integrate[((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","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 \text{Li}_2\left(\frac{g x^2}{f}+1\right)+\frac{1}{2} a e \log \left(-\frac{g x^2}{f}\right) \log \left(f+g x^2\right)+\frac{1}{2} i b d \text{Li}_2(-i c x)-\frac{1}{2} i b d \text{Li}_2(i c x)",0,"Integrate[((a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]))/x, x]","A",-1
1300,1,552,672,1.1226901,"\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x^2} \, dx","Integrate[((a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]))/x^2,x]","\frac{1}{2} \left(-\frac{2 \left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x}+\frac{e \sqrt{g} \left(4 a \sqrt{-f} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)+i b \sqrt{f} \left(\text{Li}_2\left(\frac{\sqrt{g} (i-c x)}{\sqrt{-f} c+i \sqrt{g}}\right)+\log (1+i c x) \log \left(\frac{c \left(\sqrt{-f}+\sqrt{g} x\right)}{c \sqrt{-f}+i \sqrt{g}}\right)\right)-i b \sqrt{f} \left(\text{Li}_2\left(\frac{\sqrt{g} (1-i c x)}{i \sqrt{-f} c+\sqrt{g}}\right)+\log (1-i c x) \log \left(\frac{c \left(\sqrt{-f}+\sqrt{g} x\right)}{c \sqrt{-f}-i \sqrt{g}}\right)\right)-i b \sqrt{f} \left(\text{Li}_2\left(\frac{\sqrt{g} (i c x+1)}{i \sqrt{-f} c+\sqrt{g}}\right)+\log (1+i c x) \log \left(\frac{c \left(\sqrt{-f}-\sqrt{g} x\right)}{c \sqrt{-f}-i \sqrt{g}}\right)\right)+i b \sqrt{f} \left(\text{Li}_2\left(\frac{\sqrt{g} (c x+i)}{\sqrt{-f} c+i \sqrt{g}}\right)+\log (1-i c x) \log \left(\frac{c \left(\sqrt{-f}-\sqrt{g} x\right)}{c \sqrt{-f}+i \sqrt{g}}\right)\right)\right)}{\sqrt{-f^2}}+b c \left(\left(\log \left(-\frac{g x^2}{f}\right)-\log \left(-\frac{g \left(c^2 x^2+1\right)}{c^2 f-g}\right)\right) \left(d+e \log \left(f+g x^2\right)\right)-e \text{Li}_2\left(\frac{c^2 \left(g x^2+f\right)}{c^2 f-g}\right)+e \text{Li}_2\left(\frac{g x^2}{f}+1\right)\right)\right)","-\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 e \text{Li}_2\left(\frac{c^2 \left(g x^2+f\right)}{c^2 f-g}\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{1}{2} b c e \text{Li}_2\left(\frac{g x^2}{f}+1\right)+\frac{i b e \sqrt{g} \text{Li}_2\left(\frac{\sqrt{g} (i-c x)}{\sqrt{-f} c+i \sqrt{g}}\right)}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \text{Li}_2\left(\frac{\sqrt{g} (1-i c x)}{i \sqrt{-f} c+\sqrt{g}}\right)}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \text{Li}_2\left(\frac{\sqrt{g} (i c x+1)}{i \sqrt{-f} c+\sqrt{g}}\right)}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \text{Li}_2\left(\frac{\sqrt{g} (c x+i)}{\sqrt{-f} c+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{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 + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]))/x + (e*Sqrt[g]*(4*a*Sqrt[-f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]] + I*b*Sqrt[f]*(Log[1 + I*c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] + I*Sqrt[g])] + PolyLog[2, (Sqrt[g]*(I - c*x))/(c*Sqrt[-f] + I*Sqrt[g])]) - I*b*Sqrt[f]*(Log[1 - I*c*x]*Log[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])] + PolyLog[2, (Sqrt[g]*(1 - I*c*x))/(I*c*Sqrt[-f] + Sqrt[g])]) - I*b*Sqrt[f]*(Log[1 + I*c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])] + PolyLog[2, (Sqrt[g]*(1 + I*c*x))/(I*c*Sqrt[-f] + Sqrt[g])]) + I*b*Sqrt[f]*(Log[1 - I*c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] + I*Sqrt[g])] + PolyLog[2, (Sqrt[g]*(I + c*x))/(c*Sqrt[-f] + I*Sqrt[g])])))/Sqrt[-f^2] + b*c*((Log[-((g*x^2)/f)] - Log[-((g*(1 + c^2*x^2))/(c^2*f - g))])*(d + e*Log[f + g*x^2]) - e*PolyLog[2, (c^2*(f + g*x^2))/(c^2*f - g)] + e*PolyLog[2, 1 + (g*x^2)/f]))/2","A",0
1301,1,1217,528,7.2059011,"\int \frac{\left(a+b \tan ^{-1}(c x)\right) \left(d+e \log \left(f+g x^2\right)\right)}{x^3} \, dx","Integrate[((a + b*ArcTan[c*x])*(d + e*Log[f + g*x^2]))/x^3,x]","-\frac{2 b c^2 d f \tan ^{-1}(c x) x^2-4 b c e \sqrt{f} \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right) x^2-4 i b c^2 e f \sin ^{-1}\left(\sqrt{\frac{c^2 f}{c^2 f-g}}\right) \tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right) x^2+4 i b e g \sin ^{-1}\left(\sqrt{\frac{c^2 f}{c^2 f-g}}\right) \tan ^{-1}\left(\frac{c g x}{\sqrt{c^2 f g}}\right) x^2-4 b e g \tan ^{-1}(c x) \log \left(1-e^{2 i \tan ^{-1}(c x)}\right) x^2+4 b c^2 e f \tan ^{-1}(c x) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right) x^2-2 b c^2 e f \sin ^{-1}\left(\sqrt{\frac{c^2 f}{c^2 f-g}}\right) \log \left(\frac{\left(1+e^{2 i \tan ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 i \tan ^{-1}(c x)}\right) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right) x^2+2 b e g \sin ^{-1}\left(\sqrt{\frac{c^2 f}{c^2 f-g}}\right) \log \left(\frac{\left(1+e^{2 i \tan ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 i \tan ^{-1}(c x)}\right) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right) x^2-2 b c^2 e f \tan ^{-1}(c x) \log \left(\frac{\left(1+e^{2 i \tan ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 i \tan ^{-1}(c x)}\right) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right) x^2+2 b e g \tan ^{-1}(c x) \log \left(\frac{\left(1+e^{2 i \tan ^{-1}(c x)}\right) f c^2+\left(-1+e^{2 i \tan ^{-1}(c x)}\right) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right) x^2+2 b c^2 e f \sin ^{-1}\left(\sqrt{\frac{c^2 f}{c^2 f-g}}\right) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(f c^2+g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}+1\right) x^2-2 b e g \sin ^{-1}\left(\sqrt{\frac{c^2 f}{c^2 f-g}}\right) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(f c^2+g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}+1\right) x^2-2 b c^2 e f \tan ^{-1}(c x) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(f c^2+g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}+1\right) x^2+2 b e g \tan ^{-1}(c x) \log \left(\frac{e^{2 i \tan ^{-1}(c x)} \left(f c^2+g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}+1\right) x^2-4 a e g \log (x) x^2+2 a e g \log \left(g x^2+f\right) x^2+2 b c^2 e f \tan ^{-1}(c x) \log \left(g x^2+f\right) x^2-2 i b c^2 e f \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right) x^2+2 i b e g \text{Li}_2\left(e^{2 i \tan ^{-1}(c x)}\right) x^2+i b c^2 e f \text{Li}_2\left(\frac{e^{2 i \tan ^{-1}(c x)} \left(-f c^2-g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}\right) x^2-i b e g \text{Li}_2\left(\frac{e^{2 i \tan ^{-1}(c x)} \left(-f c^2-g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}\right) x^2+i b c^2 e f \text{Li}_2\left(-\frac{e^{2 i \tan ^{-1}(c x)} \left(f c^2+g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}\right) x^2-i b e g \text{Li}_2\left(-\frac{e^{2 i \tan ^{-1}(c x)} \left(f c^2+g+2 \sqrt{c^2 f g}\right)}{c^2 f-g}\right) x^2+2 b c d f x+2 b c e f \log \left(g x^2+f\right) x+2 a d f+2 b d f \tan ^{-1}(c x)+2 a e f \log \left(g x^2+f\right)+2 b e f \tan ^{-1}(c x) \log \left(g x^2+f\right)}{4 f x^2}","-\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{i b e \left(c^2 f-g\right) \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{2 f}-\frac{i b e \left(c^2 f-g\right) \text{Li}_2\left(1-\frac{2 c \left(\sqrt{-f}-\sqrt{g} x\right)}{\left(c \sqrt{-f}-i \sqrt{g}\right) (1-i c x)}\right)}{4 f}-\frac{i b e \left(c^2 f-g\right) \text{Li}_2\left(1-\frac{2 c \left(\sqrt{g} x+\sqrt{-f}\right)}{\left(\sqrt{-f} c+i \sqrt{g}\right) (1-i c x)}\right)}{4 f}-\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{i b e g \text{Li}_2(-i c x)}{2 f}-\frac{i b e g \text{Li}_2(i c x)}{2 f}+\frac{b c e \sqrt{g} \tan ^{-1}\left(\frac{\sqrt{g} x}{\sqrt{f}}\right)}{\sqrt{f}}",1,"-1/4*(2*a*d*f + 2*b*c*d*f*x + 2*b*d*f*ArcTan[c*x] + 2*b*c^2*d*f*x^2*ArcTan[c*x] - 4*b*c*e*Sqrt[f]*Sqrt[g]*x^2*ArcTan[(Sqrt[g]*x)/Sqrt[f]] - (4*I)*b*c^2*e*f*x^2*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] + (4*I)*b*e*g*x^2*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] - 4*b*e*g*x^2*ArcTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + 4*b*c^2*e*f*x^2*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - 2*b*c^2*e*f*x^2*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[(c^2*(1 + E^((2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcTan[c*x]))*g - 2*E^((2*I)*ArcTan[c*x])*Sqrt[c^2*f*g])/(c^2*f - g)] + 2*b*e*g*x^2*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[(c^2*(1 + E^((2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcTan[c*x]))*g - 2*E^((2*I)*ArcTan[c*x])*Sqrt[c^2*f*g])/(c^2*f - g)] - 2*b*c^2*e*f*x^2*ArcTan[c*x]*Log[(c^2*(1 + E^((2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcTan[c*x]))*g - 2*E^((2*I)*ArcTan[c*x])*Sqrt[c^2*f*g])/(c^2*f - g)] + 2*b*e*g*x^2*ArcTan[c*x]*Log[(c^2*(1 + E^((2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcTan[c*x]))*g - 2*E^((2*I)*ArcTan[c*x])*Sqrt[c^2*f*g])/(c^2*f - g)] + 2*b*c^2*e*f*x^2*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[1 + (E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] - 2*b*e*g*x^2*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[1 + (E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] - 2*b*c^2*e*f*x^2*ArcTan[c*x]*Log[1 + (E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] + 2*b*e*g*x^2*ArcTan[c*x]*Log[1 + (E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] - 4*a*e*g*x^2*Log[x] + 2*a*e*f*Log[f + g*x^2] + 2*b*c*e*f*x*Log[f + g*x^2] + 2*a*e*g*x^2*Log[f + g*x^2] + 2*b*e*f*ArcTan[c*x]*Log[f + g*x^2] + 2*b*c^2*e*f*x^2*ArcTan[c*x]*Log[f + g*x^2] - (2*I)*b*c^2*e*f*x^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + (2*I)*b*e*g*x^2*PolyLog[2, E^((2*I)*ArcTan[c*x])] + I*b*c^2*e*f*x^2*PolyLog[2, (E^((2*I)*ArcTan[c*x])*(-(c^2*f) - g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] - I*b*e*g*x^2*PolyLog[2, (E^((2*I)*ArcTan[c*x])*(-(c^2*f) - g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] + I*b*c^2*e*f*x^2*PolyLog[2, -((E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g))] - I*b*e*g*x^2*PolyLog[2, -((E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g))])/(f*x^2)","B",0