1,1,255,184,0.4927007,"\int (d+e x)^4 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^4*(a + b*ArcTan[c*x]),x]","\frac{(d+e x)^5 \left(a+b \tan ^{-1}(c x)\right)-\frac{b \left(c^2 e^2 x \left(c^2 \left(120 d^3+60 d^2 e x+20 d e^2 x^2+3 e^3 x^3\right)-6 e^2 (10 d+e x)\right)+6 \left(-10 c^2 d^2 e^2 \left(\sqrt{-c^2} d+e\right)+e^4 \left(5 \sqrt{-c^2} d+e\right)+c^4 d^4 \left(\sqrt{-c^2} d+5 e\right)\right) \log \left(1-\sqrt{-c^2} x\right)-6 \left(-10 c^2 d^2 e^2 \left(\sqrt{-c^2} d-e\right)+e^4 \left(5 \sqrt{-c^2} d-e\right)+c^4 d^4 \left(\sqrt{-c^2} d-5 e\right)\right) \log \left(\sqrt{-c^2} x+1\right)\right)}{12 c^5}}{5 e}","\frac{(d+e x)^5 \left(a+b \tan ^{-1}(c x)\right)}{5 e}-\frac{b d \left(c^4 d^4-10 c^2 d^2 e^2+5 e^4\right) \tan ^{-1}(c x)}{5 c^4 e}-\frac{b e^2 x^2 \left(10 c^2 d^2-e^2\right)}{10 c^3}-\frac{b d e x \left(2 c^2 d^2-e^2\right)}{c^3}-\frac{b \left(5 c^4 d^4-10 c^2 d^2 e^2+e^4\right) \log \left(c^2 x^2+1\right)}{10 c^5}-\frac{b d e^3 x^3}{3 c}-\frac{b e^4 x^4}{20 c}",1,"((d + e*x)^5*(a + b*ArcTan[c*x]) - (b*(c^2*e^2*x*(-6*e^2*(10*d + e*x) + c^2*(120*d^3 + 60*d^2*e*x + 20*d*e^2*x^2 + 3*e^3*x^3)) + 6*(-10*c^2*d^2*e^2*(Sqrt[-c^2]*d + e) + e^4*(5*Sqrt[-c^2]*d + e) + c^4*d^4*(Sqrt[-c^2]*d + 5*e))*Log[1 - Sqrt[-c^2]*x] - 6*(c^4*d^4*(Sqrt[-c^2]*d - 5*e) - 10*c^2*d^2*(Sqrt[-c^2]*d - e)*e^2 + (5*Sqrt[-c^2]*d - e)*e^4)*Log[1 + Sqrt[-c^2]*x]))/(12*c^5))/(5*e)","A",1
2,1,218,144,0.4619813,"\int (d+e x)^3 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^3*(a + b*ArcTan[c*x]),x]","\frac{(d+e x)^4 \left(a+b \tan ^{-1}(c x)\right)-\frac{b c \left(2 \sqrt{-c^2} e^2 x \left(c^2 \left(18 d^2+6 d e x+e^2 x^2\right)-3 e^2\right)-3 \left(c^4 d^4-2 c^2 d^2 e \left(2 \sqrt{-c^2} d+3 e\right)+e^3 \left(4 \sqrt{-c^2} d+e\right)\right) \log \left(1-\sqrt{-c^2} x\right)+3 \left(c^4 d^4+2 c^2 d^2 e \left(2 \sqrt{-c^2} d-3 e\right)+e^3 \left(e-4 \sqrt{-c^2} d\right)\right) \log \left(\sqrt{-c^2} x+1\right)\right)}{6 \left(-c^2\right)^{5/2}}}{4 e}","\frac{(d+e x)^4 \left(a+b \tan ^{-1}(c x)\right)}{4 e}-\frac{b \left(c^4 d^4-6 c^2 d^2 e^2+e^4\right) \tan ^{-1}(c x)}{4 c^4 e}-\frac{b e x \left(6 c^2 d^2-e^2\right)}{4 c^3}-\frac{b d (c d-e) (c d+e) \log \left(c^2 x^2+1\right)}{2 c^3}-\frac{b d e^2 x^2}{2 c}-\frac{b e^3 x^3}{12 c}",1,"((d + e*x)^4*(a + b*ArcTan[c*x]) - (b*c*(2*Sqrt[-c^2]*e^2*x*(-3*e^2 + c^2*(18*d^2 + 6*d*e*x + e^2*x^2)) - 3*(c^4*d^4 + e^3*(4*Sqrt[-c^2]*d + e) - 2*c^2*d^2*e*(2*Sqrt[-c^2]*d + 3*e))*Log[1 - Sqrt[-c^2]*x] + 3*(c^4*d^4 + 2*c^2*d^2*(2*Sqrt[-c^2]*d - 3*e)*e + e^3*(-4*Sqrt[-c^2]*d + e))*Log[1 + Sqrt[-c^2]*x]))/(6*(-c^2)^(5/2)))/(4*e)","A",1
3,1,163,103,0.3523764,"\int (d+e x)^2 \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^2*(a + b*ArcTan[c*x]),x]","\frac{(d+e x)^3 \left(a+b \tan ^{-1}(c x)\right)-\frac{b \left(\left(c^2 d^2 \left(\sqrt{-c^2} d+3 e\right)-e^2 \left(3 \sqrt{-c^2} d+e\right)\right) \log \left(1-\sqrt{-c^2} x\right)-\left(c^2 d^2 \left(\sqrt{-c^2} d-3 e\right)+e^2 \left(e-3 \sqrt{-c^2} d\right)\right) \log \left(\sqrt{-c^2} x+1\right)+c^2 e^2 x (6 d+e x)\right)}{2 c^3}}{3 e}","\frac{(d+e x)^3 \left(a+b \tan ^{-1}(c x)\right)}{3 e}-\frac{b d \left(d^2-\frac{3 e^2}{c^2}\right) \tan ^{-1}(c x)}{3 e}-\frac{b \left(3 c^2 d^2-e^2\right) \log \left(c^2 x^2+1\right)}{6 c^3}-\frac{b d e x}{c}-\frac{b e^2 x^2}{6 c}",1,"((d + e*x)^3*(a + b*ArcTan[c*x]) - (b*(c^2*e^2*x*(6*d + e*x) + (-(e^2*(3*Sqrt[-c^2]*d + e)) + c^2*d^2*(Sqrt[-c^2]*d + 3*e))*Log[1 - Sqrt[-c^2]*x] - (c^2*d^2*(Sqrt[-c^2]*d - 3*e) + e^2*(-3*Sqrt[-c^2]*d + e))*Log[1 + Sqrt[-c^2]*x]))/(2*c^3))/(3*e)","A",1
4,1,77,76,0.006133,"\int (d+e x) \left(a+b \tan ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)*(a + b*ArcTan[c*x]),x]","a d x+\frac{1}{2} a e x^2-\frac{b d \log \left(c^2 x^2+1\right)}{2 c}+\frac{b e \tan ^{-1}(c x)}{2 c^2}+b d x \tan ^{-1}(c x)+\frac{1}{2} b e x^2 \tan ^{-1}(c x)-\frac{b e x}{2 c}","\frac{(d+e x)^2 \left(a+b \tan ^{-1}(c x)\right)}{2 e}-\frac{b \left(d^2-\frac{e^2}{c^2}\right) \tan ^{-1}(c x)}{2 e}-\frac{b d \log \left(c^2 x^2+1\right)}{2 c}-\frac{b e x}{2 c}",1,"a*d*x - (b*e*x)/(2*c) + (a*e*x^2)/2 + (b*e*ArcTan[c*x])/(2*c^2) + b*d*x*ArcTan[c*x] + (b*e*x^2*ArcTan[c*x])/2 - (b*d*Log[1 + c^2*x^2])/(2*c)","A",1
5,1,138,138,0.0652363,"\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
6,1,111,98,0.1965169,"\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{\frac{b c \left(\left(\sqrt{-c^2} d-e\right) \log \left(1-\sqrt{-c^2} x\right)-\left(\sqrt{-c^2} d+e\right) \log \left(\sqrt{-c^2} x+1\right)+2 e \log (d+e x)\right)}{2 \left(c^2 d^2+e^2\right)}-\frac{a+b \tan ^{-1}(c x)}{d+e x}}{e}","-\frac{a+b \tan ^{-1}(c x)}{e (d+e x)}-\frac{b c \log \left(c^2 x^2+1\right)}{2 \left(c^2 d^2+e^2\right)}+\frac{b c \log (d+e x)}{c^2 d^2+e^2}+\frac{b c^2 d \tan ^{-1}(c x)}{e \left(c^2 d^2+e^2\right)}",1,"(-((a + b*ArcTan[c*x])/(d + e*x)) + (b*c*((Sqrt[-c^2]*d - e)*Log[1 - Sqrt[-c^2]*x] - (Sqrt[-c^2]*d + e)*Log[1 + Sqrt[-c^2]*x] + 2*e*Log[d + e*x]))/(2*(c^2*d^2 + e^2)))/e","A",1
7,1,192,146,0.3318407,"\int \frac{a+b \tan ^{-1}(c x)}{(d+e x)^3} \, dx","Integrate[(a + b*ArcTan[c*x])/(d + e*x)^3,x]","-\frac{2 \left(a+b \tan ^{-1}(c x)\right)+\frac{b c (d+e x) \left(2 e \left(c^2 d^2+e^2\right)-\left(c^2 d \left(\sqrt{-c^2} d-2 e\right)-\sqrt{-c^2} e^2\right) \log \left(1-\sqrt{-c^2} x\right) (d+e x)-\left(\sqrt{-c^2} e^2-c^2 d \left(\sqrt{-c^2} d+2 e\right)\right) \log \left(\sqrt{-c^2} x+1\right) (d+e x)-4 c^2 d e (d+e x) \log (d+e x)\right)}{\left(c^2 d^2+e^2\right)^2}}{4 e (d+e x)^2}","-\frac{a+b \tan ^{-1}(c x)}{2 e (d+e x)^2}-\frac{b c}{2 \left(c^2 d^2+e^2\right) (d+e x)}+\frac{b c^2 (c d-e) (c d+e) \tan ^{-1}(c x)}{2 e \left(c^2 d^2+e^2\right)^2}-\frac{b c^3 d \log \left(c^2 x^2+1\right)}{2 \left(c^2 d^2+e^2\right)^2}+\frac{b c^3 d \log (d+e x)}{\left(c^2 d^2+e^2\right)^2}",1,"-1/4*(2*(a + b*ArcTan[c*x]) + (b*c*(d + e*x)*(2*e*(c^2*d^2 + e^2) - (c^2*d*(Sqrt[-c^2]*d - 2*e) - Sqrt[-c^2]*e^2)*(d + e*x)*Log[1 - Sqrt[-c^2]*x] - (Sqrt[-c^2]*e^2 - c^2*d*(Sqrt[-c^2]*d + 2*e))*(d + e*x)*Log[1 + Sqrt[-c^2]*x] - 4*c^2*d*e*(d + e*x)*Log[d + e*x]))/(c^2*d^2 + e^2)^2)/(e*(d + e*x)^2)","A",1
8,1,254,206,0.6505164,"\int \frac{a+b \tan ^{-1}(c x)}{(d+e x)^4} \, dx","Integrate[(a + b*ArcTan[c*x])/(d + e*x)^4,x]","-\frac{2 \left(a+b \tan ^{-1}(c x)\right)+\frac{b c (d+e x) \left(4 c^2 d e \left(c^2 d^2+e^2\right) (d+e x)-c^2 \left(c^2 d^2 \left(\sqrt{-c^2} d-3 e\right)+e^2 \left(e-3 \sqrt{-c^2} d\right)\right) \log \left(1-\sqrt{-c^2} x\right) (d+e x)^2-c^2 \left(e^2 \left(3 \sqrt{-c^2} d+e\right)-c^2 d^2 \left(\sqrt{-c^2} d+3 e\right)\right) \log \left(\sqrt{-c^2} x+1\right) (d+e x)^2-2 c^2 e \left(3 c^2 d^2-e^2\right) (d+e x)^2 \log (d+e x)+e \left(c^2 d^2+e^2\right)^2\right)}{\left(c^2 d^2+e^2\right)^3}}{6 e (d+e x)^3}","-\frac{a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}-\frac{b c}{6 \left(c^2 d^2+e^2\right) (d+e x)^2}+\frac{b c^4 d \left(c^2 d^2-3 e^2\right) \tan ^{-1}(c x)}{3 e \left(c^2 d^2+e^2\right)^3}-\frac{b c^3 \left(3 c^2 d^2-e^2\right) \log \left(c^2 x^2+1\right)}{6 \left(c^2 d^2+e^2\right)^3}-\frac{2 b c^3 d}{3 \left(c^2 d^2+e^2\right)^2 (d+e x)}+\frac{b c^3 \left(3 c^2 d^2-e^2\right) \log (d+e x)}{3 \left(c^2 d^2+e^2\right)^3}",1,"-1/6*(2*(a + b*ArcTan[c*x]) + (b*c*(d + e*x)*(e*(c^2*d^2 + e^2)^2 + 4*c^2*d*e*(c^2*d^2 + e^2)*(d + e*x) - c^2*(c^2*d^2*(Sqrt[-c^2]*d - 3*e) + e^2*(-3*Sqrt[-c^2]*d + e))*(d + e*x)^2*Log[1 - Sqrt[-c^2]*x] - c^2*(e^2*(3*Sqrt[-c^2]*d + e) - c^2*d^2*(Sqrt[-c^2]*d + 3*e))*(d + e*x)^2*Log[1 + Sqrt[-c^2]*x] - 2*c^2*e*(3*c^2*d^2 - e^2)*(d + e*x)^2*Log[d + e*x]))/(c^2*d^2 + e^2)^3)/(e*(d + e*x)^3)","A",1
9,1,472,376,1.0008453,"\int (d+e x)^3 \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x)^3*(a + b*ArcTan[c*x])^2,x]","\frac{12 a^2 c^4 d^3 x+18 a^2 c^4 d^2 e x^2+12 a^2 c^4 d e^2 x^3+3 a^2 c^4 e^3 x^4-36 a b c^3 d^2 e x-12 a b c^3 d e^2 x^2-2 a b c^3 e^3 x^3+12 a b c d e^2 \log \left(c^2 x^2+1\right)+2 b \tan ^{-1}(c x) \left(3 a \left(c^4 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+6 c^2 d^2 e-e^3\right)-b c e \left(18 c^2 d^2 x+6 d \left(c^2 e x^2+e\right)+e^2 x \left(c^2 x^2-3\right)\right)+12 b c d \left(c^2 d^2-e^2\right) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)-12 a b c^3 d^3 \log \left(c^2 x^2+1\right)+6 a b c e^3 x-12 i b^2 c d \left(c^2 d^2-e^2\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+18 b^2 c^2 d^2 e \log \left(c^2 x^2+1\right)+12 b^2 c^2 d e^2 x+b^2 c^2 e^3 x^2-4 b^2 e^3 \log \left(c^2 x^2+1\right)+3 b^2 \tan ^{-1}(c x)^2 \left(c^4 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)-4 i c^3 d^3+6 c^2 d^2 e+4 i c d e^2-e^3\right)+b^2 e^3}{12 c^4}","\frac{i d (c d-e) (c d+e) \left(a+b \tan ^{-1}(c x)\right)^2}{c^3}+\frac{2 b d (c d-e) (c d+e) \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3}-\frac{\left(c^4 d^4-6 c^2 d^2 e^2+e^4\right) \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4 e}-\frac{a b e x \left(6 c^2 d^2-e^2\right)}{2 c^3}-\frac{b d e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{c}+\frac{(d+e x)^4 \left(a+b \tan ^{-1}(c x)\right)^2}{4 e}-\frac{b e^3 x^3 \left(a+b \tan ^{-1}(c x)\right)}{6 c}-\frac{b^2 d e^2 \tan ^{-1}(c x)}{c^3}+\frac{i b^2 d (c d-e) (c d+e) \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c^3}+\frac{b^2 d e^2 x}{c^2}+\frac{b^2 e^3 x^2}{12 c^2}+\frac{b^2 e \left(6 c^2 d^2-e^2\right) \log \left(c^2 x^2+1\right)}{4 c^4}-\frac{b^2 e^3 \log \left(c^2 x^2+1\right)}{12 c^4}-\frac{b^2 e x \left(6 c^2 d^2-e^2\right) \tan ^{-1}(c x)}{2 c^3}",1,"(b^2*e^3 + 12*a^2*c^4*d^3*x - 36*a*b*c^3*d^2*e*x + 12*b^2*c^2*d*e^2*x + 6*a*b*c*e^3*x + 18*a^2*c^4*d^2*e*x^2 - 12*a*b*c^3*d*e^2*x^2 + b^2*c^2*e^3*x^2 + 12*a^2*c^4*d*e^2*x^3 - 2*a*b*c^3*e^3*x^3 + 3*a^2*c^4*e^3*x^4 + 3*b^2*((-4*I)*c^3*d^3 + 6*c^2*d^2*e + (4*I)*c*d*e^2 - e^3 + c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(-(b*c*e*(18*c^2*d^2*x + e^2*x*(-3 + c^2*x^2) + 6*d*(e + c^2*e*x^2))) + 3*a*(6*c^2*d^2*e - e^3 + c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)) + 12*b*c*d*(c^2*d^2 - e^2)*Log[1 + E^((2*I)*ArcTan[c*x])]) - 12*a*b*c^3*d^3*Log[1 + c^2*x^2] + 18*b^2*c^2*d^2*e*Log[1 + c^2*x^2] + 12*a*b*c*d*e^2*Log[1 + c^2*x^2] - 4*b^2*e^3*Log[1 + c^2*x^2] - (12*I)*b^2*c*d*(c^2*d^2 - e^2)*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(12*c^4)","A",0
10,1,312,270,0.594535,"\int (d+e x)^2 \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^2 x+3 a^2 c^3 d e x^2+a^2 c^3 e^2 x^3-3 a b c^2 d^2 \log \left(c^2 x^2+1\right)-6 a b c^2 d e x-a b c^2 e^2 x^2+a b e^2 \log \left(c^2 x^2+1\right)+b \tan ^{-1}(c x) \left(2 a c^3 x \left(3 d^2+3 d e x+e^2 x^2\right)+6 a c d e+2 b \left(3 c^2 d^2-e^2\right) \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-b e \left(6 c^2 d x+c^2 e x^2+e\right)\right)-i b^2 \left(3 c^2 d^2-e^2\right) \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)+3 b^2 c d e \log \left(c^2 x^2+1\right)+b^2 \tan ^{-1}(c x)^2 \left(c^3 x \left(3 d^2+3 d e x+e^2 x^2\right)-3 i c^2 d^2+3 c d e+i e^2\right)+b^2 c e^2 x}{3 c^3}","-\frac{d \left(d^2-\frac{3 e^2}{c^2}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{3 e}+\frac{i \left(3 c^2 d^2-e^2\right) \left(a+b \tan ^{-1}(c x)\right)^2}{3 c^3}+\frac{2 b \left(3 c^2 d^2-e^2\right) \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{3 c^3}-\frac{2 a b d e x}{c}+\frac{(d+e x)^3 \left(a+b \tan ^{-1}(c x)\right)^2}{3 e}-\frac{b e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)}{3 c}-\frac{b^2 e^2 \tan ^{-1}(c x)}{3 c^3}+\frac{b^2 d e \log \left(c^2 x^2+1\right)}{c^2}+\frac{b^2 e^2 x}{3 c^2}+\frac{i b^2 \left(3 c^2 d^2-e^2\right) \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{3 c^3}-\frac{2 b^2 d e x \tan ^{-1}(c x)}{c}",1,"(3*a^2*c^3*d^2*x - 6*a*b*c^2*d*e*x + b^2*c*e^2*x + 3*a^2*c^3*d*e*x^2 - a*b*c^2*e^2*x^2 + a^2*c^3*e^2*x^3 + b^2*((-3*I)*c^2*d^2 + 3*c*d*e + I*e^2 + c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2))*ArcTan[c*x]^2 + b*ArcTan[c*x]*(6*a*c*d*e - b*e*(e + 6*c^2*d*x + c^2*e*x^2) + 2*a*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) + 2*b*(3*c^2*d^2 - e^2)*Log[1 + E^((2*I)*ArcTan[c*x])]) - 3*a*b*c^2*d^2*Log[1 + c^2*x^2] + 3*b^2*c*d*e*Log[1 + c^2*x^2] + a*b*e^2*Log[1 + c^2*x^2] - I*b^2*(3*c^2*d^2 - e^2)*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(3*c^3)","A",0
11,1,172,171,0.2853483,"\int (d+e x) \left(a+b \tan ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x)*(a + b*ArcTan[c*x])^2,x]","\frac{2 a^2 c^2 d x+a^2 c^2 e x^2+2 b \tan ^{-1}(c x) \left(a \left(2 c^2 d x+c^2 e x^2+e\right)+2 b c d \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)-b c e x\right)-2 a b c d \log \left(c^2 x^2+1\right)-2 a b c e x+b^2 e \log \left(c^2 x^2+1\right)+b^2 (c x-i) \tan ^{-1}(c x)^2 (2 c d+c e x+i e)-2 i b^2 c d \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)}{2 c^2}","-\frac{\left(d^2-\frac{e^2}{c^2}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 e}+\frac{(d+e x)^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 e}+\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{a b e x}{c}+\frac{b^2 e \log \left(c^2 x^2+1\right)}{2 c^2}+\frac{i b^2 d \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c}-\frac{b^2 e x \tan ^{-1}(c x)}{c}",1,"(2*a^2*c^2*d*x - 2*a*b*c*e*x + a^2*c^2*e*x^2 + b^2*(-I + c*x)*(2*c*d + I*e + c*e*x)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*(-(b*c*e*x) + a*(e + 2*c^2*d*x + c^2*e*x^2) + 2*b*c*d*Log[1 + E^((2*I)*ArcTan[c*x])]) - 2*a*b*c*d*Log[1 + c^2*x^2] + b^2*e*Log[1 + c^2*x^2] - (2*I)*b^2*c*d*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(2*c^2)","A",0
12,1,741,223,15.1155576,"\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
13,1,300,341,3.3970848,"\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]","-\frac{a^2}{e (d+e x)}+\frac{a b \left(c (d+e x) \left(2 \log (c (d+e x))-\log \left(c^2 x^2+1\right)\right)-2 \tan ^{-1}(c x) \left(e-c^2 d x\right)\right)}{\left(c^2 d^2+e^2\right) (d+e x)}+\frac{b^2 \left(-\frac{c d \left(-\frac{1}{2} \pi  \log \left(c^2 x^2+1\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 \tan ^{-1}(c x) \left(\pi -2 \tan ^{-1}\left(\frac{c d}{e}\right)\right)-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 \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)-\pi  \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right)\right)}{c^2 d^2+e^2}-\frac{\tan ^{-1}(c x)^2 e^{i \tan ^{-1}\left(\frac{c d}{e}\right)}}{e \sqrt{\frac{c^2 d^2}{e^2}+1}}+\frac{x \tan ^{-1}(c x)^2}{d+e x}\right)}{d}","\frac{i c \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d^2+e^2}+\frac{c^2 d \left(a+b \tan ^{-1}(c x)\right)^2}{e \left(c^2 d^2+e^2\right)}-\frac{2 b c \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d^2+e^2}+\frac{2 b c \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d^2+e^2}+\frac{2 b c \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)}{c^2 d^2+e^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{e (d+e x)}+\frac{i b^2 c \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{c^2 d^2+e^2}+\frac{i b^2 c \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c^2 d^2+e^2}-\frac{i b^2 c \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{c^2 d^2+e^2}",1,"-(a^2/(e*(d + e*x))) + (a*b*(-2*(e - c^2*d*x)*ArcTan[c*x] + c*(d + e*x)*(2*Log[c*(d + e*x)] - Log[1 + c^2*x^2])))/((c^2*d^2 + e^2)*(d + e*x)) + (b^2*(-((E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^2)/(Sqrt[1 + (c^2*d^2)/e^2]*e)) + (x*ArcTan[c*x]^2)/(d + e*x) - (c*d*((-I)*(Pi - 2*ArcTan[(c*d)/e])*ArcTan[c*x] - Pi*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*Log[1 + c^2*x^2])/2 + 2*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + I*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/(c^2*d^2 + e^2)))/d","A",0
14,1,479,496,7.083335,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^2}{(d+e x)^3} \, dx","Integrate[(a + b*ArcTan[c*x])^2/(d + e*x)^3,x]","-\frac{a^2}{2 e (d+e x)^2}+\frac{a b \left(c (d+e x) \left(c^2 \left(-d^2\right)-c^2 d \log \left(c^2 x^2+1\right) (d+e x)+2 c^2 d (d+e x) \log (c (d+e x))-e^2\right)+\tan ^{-1}(c x) \left(c^4 d^2 x (2 d+e x)-c^2 e \left(3 d^2+2 d e x+e^2 x^2\right)-e^3\right)\right)}{\left(c^2 d^2+e^2\right)^2 (d+e x)^2}+\frac{b^2 c^2 \left(-\frac{2 c d \left(-\frac{1}{2} \pi  \log \left(c^2 x^2+1\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 \tan ^{-1}(c x) \left(\pi -2 \tan ^{-1}\left(\frac{c d}{e}\right)\right)-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 \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)-\pi  \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right)\right)}{c^2 d^2+e^2}-\frac{2 \tan ^{-1}(c x)^2 e^{i \tan ^{-1}\left(\frac{c d}{e}\right)}}{e \sqrt{\frac{c^2 d^2}{e^2}+1}}-\frac{e \left(c^2 x^2+1\right) \tan ^{-1}(c x)^2}{c^2 (d+e x)^2}+\frac{2 c d e \log \left(\frac{c (d+e x)}{\sqrt{c^2 x^2+1}}\right)-2 e^2 \tan ^{-1}(c x)}{c^3 d^3+c d e^2}+\frac{2 x \tan ^{-1}(c x) \left(c d \tan ^{-1}(c x)+e\right)}{c d (d+e x)}\right)}{2 \left(c^2 d^2+e^2\right)}","\frac{c^2 (c d-e) (c d+e) \left(a+b \tan ^{-1}(c x)\right)^2}{2 e \left(c^2 d^2+e^2\right)^2}-\frac{b c \left(a+b \tan ^{-1}(c x)\right)}{\left(c^2 d^2+e^2\right) (d+e x)}+\frac{i c^3 d \left(a+b \tan ^{-1}(c x)\right)^2}{\left(c^2 d^2+e^2\right)^2}-\frac{2 b c^3 d \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{\left(c^2 d^2+e^2\right)^2}+\frac{2 b c^3 d \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{\left(c^2 d^2+e^2\right)^2}+\frac{2 b c^3 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)}{\left(c^2 d^2+e^2\right)^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^2}{2 e (d+e x)^2}-\frac{b^2 c^2 e \log \left(c^2 x^2+1\right)}{2 \left(c^2 d^2+e^2\right)^2}+\frac{b^2 c^2 e \log (d+e x)}{\left(c^2 d^2+e^2\right)^2}+\frac{i b^2 c^3 d \text{Li}_2\left(1-\frac{2}{1-i c x}\right)}{\left(c^2 d^2+e^2\right)^2}+\frac{i b^2 c^3 d \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{\left(c^2 d^2+e^2\right)^2}-\frac{i b^2 c^3 d \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{\left(c^2 d^2+e^2\right)^2}+\frac{b^2 c^3 d \tan ^{-1}(c x)}{\left(c^2 d^2+e^2\right)^2}",1,"-1/2*a^2/(e*(d + e*x)^2) + (a*b*((-e^3 + c^4*d^2*x*(2*d + e*x) - c^2*e*(3*d^2 + 2*d*e*x + e^2*x^2))*ArcTan[c*x] + c*(d + e*x)*(-(c^2*d^2) - e^2 + 2*c^2*d*(d + e*x)*Log[c*(d + e*x)] - c^2*d*(d + e*x)*Log[1 + c^2*x^2])))/((c^2*d^2 + e^2)^2*(d + e*x)^2) + (b^2*c^2*((-2*E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^2)/(Sqrt[1 + (c^2*d^2)/e^2]*e) - (e*(1 + c^2*x^2)*ArcTan[c*x]^2)/(c^2*(d + e*x)^2) + (2*x*ArcTan[c*x]*(e + c*d*ArcTan[c*x]))/(c*d*(d + e*x)) + (-2*e^2*ArcTan[c*x] + 2*c*d*e*Log[(c*(d + e*x))/Sqrt[1 + c^2*x^2]])/(c^3*d^3 + c*d*e^2) - (2*c*d*((-I)*(Pi - 2*ArcTan[(c*d)/e])*ArcTan[c*x] - Pi*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*Log[1 + c^2*x^2])/2 + 2*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + I*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/(c^2*d^2 + e^2)))/(2*(c^2*d^2 + e^2))","A",0
15,1,855,652,2.233286,"\int (d+e x)^3 \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Integrate[(d + e*x)^3*(a + b*ArcTan[c*x])^3,x]","\frac{a^3 e^3 x^4 c^4+3 a^2 b x \left(4 d^3+6 e x d^2+4 e^2 x^2 d+e^3 x^3\right) \tan ^{-1}(c x) c^4+a^2 e^2 (4 a c d-b e) x^3 c^3+6 a^2 d e (a c d-b e) x^2 c^3+12 a b^2 d^3 \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^3+2 b^3 d^3 \left(2 \left((c x-i) \tan ^{-1}(c x)+3 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right) \tan ^{-1}(c x)^2-6 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)+3 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)\right) c^3+18 a b^2 d^2 e \left(\left(c^2 x^2+1\right) \tan ^{-1}(c x)^2-2 c x \tan ^{-1}(c x)+\log \left(c^2 x^2+1\right)\right) c^2+6 b^3 d^2 e \left(\tan ^{-1}(c x) \left(\left(c^2 x^2+1\right) \tan ^{-1}(c x)^2+(3 i-3 c x) \tan ^{-1}(c x)-6 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+3 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)\right) c^2+a^2 \left(4 a c^3 d^3+3 b e \left(e^2-6 c^2 d^2\right)\right) x c-6 a^2 b d \left(c^2 d^2-e^2\right) \log \left(c^2 x^2+1\right) c+12 a b^2 d e^2 \left(\left(c^3 x^3+i\right) \tan ^{-1}(c x)^2-\left(c^2 x^2+2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+1\right) \tan ^{-1}(c x)+c x+i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)\right) c+2 b^3 d e^2 \left(2 c^3 x^3 \tan ^{-1}(c x)^3+2 i \tan ^{-1}(c x)^3-3 c^2 x^2 \tan ^{-1}(c x)^2-6 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)^2-3 \tan ^{-1}(c x)^2+6 c x \tan ^{-1}(c x)+6 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right) \tan ^{-1}(c x)-3 \log \left(c^2 x^2+1\right)-3 \text{Li}_3\left(-e^{2 i \tan ^{-1}(c x)}\right)\right) c+3 a^2 b \left(6 c^2 d^2 e-e^3\right) \tan ^{-1}(c x)+a b^2 e^3 \left(c^2 x^2+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)-4 \log \left(c^2 x^2+1\right)+1\right)+b^3 e^3 \left(\left(c^4 x^4-1\right) \tan ^{-1}(c x)^3-\left(c^3 x^3-3 c x+4 i\right) \tan ^{-1}(c x)^2+\left(c^2 x^2+8 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)+1\right) \tan ^{-1}(c x)-c x-4 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)\right)}{4 c^4}","\frac{b^2 e^3 \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c^4}+\frac{3 i b^2 d (c d-e) (c d+e) \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3}+\frac{3 a b^2 d e^2 x}{c^2}+\frac{b^2 e^3 x^2 \left(a+b \tan ^{-1}(c x)\right)}{4 c^2}-\frac{3 b^2 e \left(6 c^2 d^2-e^2\right) \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 c^4}+\frac{i b e^3 \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}-\frac{3 b d e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3}+\frac{i d (c d-e) (c d+e) \left(a+b \tan ^{-1}(c x)\right)^3}{c^3}+\frac{3 b d (c d-e) (c d+e) \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^3}-\frac{3 i b e \left(6 c^2 d^2-e^2\right) \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^4}-\frac{\left(c^4 d^4-6 c^2 d^2 e^2+e^4\right) \left(a+b \tan ^{-1}(c x)\right)^3}{4 c^4 e}-\frac{3 b e x \left(6 c^2 d^2-e^2\right) \left(a+b \tan ^{-1}(c x)\right)^2}{4 c^3}-\frac{3 b d e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c}+\frac{(d+e x)^4 \left(a+b \tan ^{-1}(c x)\right)^3}{4 e}-\frac{b e^3 x^3 \left(a+b \tan ^{-1}(c x)\right)^2}{4 c}+\frac{i b^3 e^3 \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{4 c^4}+\frac{b^3 e^3 \tan ^{-1}(c x)}{4 c^4}+\frac{3 b^3 d (c d-e) (c d+e) \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 c^3}-\frac{b^3 e^3 x}{4 c^3}+\frac{3 b^3 d e^2 x \tan ^{-1}(c x)}{c^2}-\frac{3 i b^3 e \left(6 c^2 d^2-e^2\right) \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{4 c^4}-\frac{3 b^3 d e^2 \log \left(c^2 x^2+1\right)}{2 c^3}",1,"(a^2*c*(4*a*c^3*d^3 + 3*b*e*(-6*c^2*d^2 + e^2))*x + 6*a^2*c^3*d*e*(a*c*d - b*e)*x^2 + a^2*c^3*e^2*(4*a*c*d - b*e)*x^3 + a^3*c^4*e^3*x^4 + 3*a^2*b*(6*c^2*d^2*e - e^3)*ArcTan[c*x] + 3*a^2*b*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcTan[c*x] + a*b^2*e^3*(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]) - 6*a^2*b*c*d*(c^2*d^2 - e^2)*Log[1 + c^2*x^2] + 18*a*b^2*c^2*d^2*e*(-2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 + Log[1 + c^2*x^2]) + 12*a*b^2*c^3*d^3*(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])]) + 12*a*b^2*c*d*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])]) + 6*b^3*c^2*d^2*e*(ArcTan[c*x]*((3*I - 3*c*x)*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 - 6*Log[1 + E^((2*I)*ArcTan[c*x])]) + (3*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + b^3*e^3*(-(c*x) - (4*I - 3*c*x + c^3*x^3)*ArcTan[c*x]^2 + (-1 + c^4*x^4)*ArcTan[c*x]^3 + ArcTan[c*x]*(1 + c^2*x^2 + 8*Log[1 + E^((2*I)*ArcTan[c*x])]) - (4*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 2*b^3*c*d*e^2*(6*c*x*ArcTan[c*x] - 3*ArcTan[c*x]^2 - 3*c^2*x^2*ArcTan[c*x]^2 + (2*I)*ArcTan[c*x]^3 + 2*c^3*x^3*ArcTan[c*x]^3 - 6*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - 3*Log[1 + c^2*x^2] + (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]) + 2*b^3*c^3*d^3*(2*ArcTan[c*x]^2*((-I + c*x)*ArcTan[c*x] + 3*Log[1 + E^((2*I)*ArcTan[c*x])]) - (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/(4*c^4)","A",0
16,1,621,411,1.3180264,"\int (d+e x)^2 \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Integrate[(d + e*x)^2*(a + b*ArcTan[c*x])^3,x]","\frac{2 a^3 c^3 e^2 x^3+6 a^2 b c^3 x \tan ^{-1}(c x) \left(3 d^2+3 d e x+e^2 x^2\right)-3 a^2 b \left(3 c^2 d^2-e^2\right) \log \left(c^2 x^2+1\right)+3 a^2 c^2 e x^2 (2 a c d-b e)+6 a^2 c^2 d x (a c d-3 b e)+18 a^2 b c d e \tan ^{-1}(c x)+18 a b^2 c^2 d^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)+18 a b^2 c 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)+6 a 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)+3 b^3 c^2 d^2 \left(-6 i \tan ^{-1}(c x) \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)+2 \tan ^{-1}(c x)^2 \left((c x-i) \tan ^{-1}(c x)+3 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)\right)+6 b^3 c d e \left(\tan ^{-1}(c x) \left(\left(c^2 x^2+1\right) \tan ^{-1}(c x)^2+(-3 c x+3 i) \tan ^{-1}(c x)-6 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+3 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)\right)+b^3 e^2 \left(2 c^3 x^3 \tan ^{-1}(c x)^3-3 \log \left(c^2 x^2+1\right)-3 c^2 x^2 \tan ^{-1}(c x)^2+6 i \tan ^{-1}(c x) \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)+2 i \tan ^{-1}(c x)^3-3 \tan ^{-1}(c x)^2+6 c x \tan ^{-1}(c x)-6 \tan ^{-1}(c x)^2 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)}{6 c^3}","-\frac{6 b^2 d e \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2}+\frac{a b^2 e^2 x}{c^2}+\frac{i b^2 \left(3 c^2 d^2-e^2\right) \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^3}-\frac{b e^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^3}-\frac{d \left(d^2-\frac{3 e^2}{c^2}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{3 e}-\frac{3 i b d e \left(a+b \tan ^{-1}(c x)\right)^2}{c^2}+\frac{i \left(3 c^2 d^2-e^2\right) \left(a+b \tan ^{-1}(c x)\right)^3}{3 c^3}+\frac{b \left(3 c^2 d^2-e^2\right) \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^3}-\frac{3 b d e x \left(a+b \tan ^{-1}(c x)\right)^2}{c}+\frac{(d+e x)^3 \left(a+b \tan ^{-1}(c x)\right)^3}{3 e}-\frac{b e^2 x^2 \left(a+b \tan ^{-1}(c x)\right)^2}{2 c}-\frac{3 i b^3 d e \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{c^2}+\frac{b^3 e^2 x \tan ^{-1}(c x)}{c^2}+\frac{b^3 \left(3 c^2 d^2-e^2\right) \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 c^3}-\frac{b^3 e^2 \log \left(c^2 x^2+1\right)}{2 c^3}",1,"(6*a^2*c^2*d*(a*c*d - 3*b*e)*x + 3*a^2*c^2*e*(2*a*c*d - b*e)*x^2 + 2*a^3*c^3*e^2*x^3 + 18*a^2*b*c*d*e*ArcTan[c*x] + 6*a^2*b*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTan[c*x] - 3*a^2*b*(3*c^2*d^2 - e^2)*Log[1 + c^2*x^2] + 18*a*b^2*c*d*e*(-2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 + Log[1 + c^2*x^2]) + 18*a*b^2*c^2*d^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*a*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])]) + 6*b^3*c*d*e*(ArcTan[c*x]*((3*I - 3*c*x)*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 - 6*Log[1 + E^((2*I)*ArcTan[c*x])]) + (3*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + b^3*e^2*(6*c*x*ArcTan[c*x] - 3*ArcTan[c*x]^2 - 3*c^2*x^2*ArcTan[c*x]^2 + (2*I)*ArcTan[c*x]^3 + 2*c^3*x^3*ArcTan[c*x]^3 - 6*ArcTan[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - 3*Log[1 + c^2*x^2] + (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]) + 3*b^3*c^2*d^2*(2*ArcTan[c*x]^2*((-I + c*x)*ArcTan[c*x] + 3*Log[1 + E^((2*I)*ArcTan[c*x])]) - (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/(6*c^3)","A",0
17,1,342,264,0.7092216,"\int (d+e x) \left(a+b \tan ^{-1}(c x)\right)^3 \, dx","Integrate[(d + e*x)*(a + b*ArcTan[c*x])^3,x]","\frac{a^3 c^2 e x^2+3 a^2 b c^2 x \tan ^{-1}(c x) (2 d+e x)-3 a^2 b c d \log \left(c^2 x^2+1\right)+a^2 c x (2 a c d-3 b e)+3 a^2 b e \tan ^{-1}(c x)+3 a 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)+6 a b^2 c d \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)+b^3 e \left(\tan ^{-1}(c x) \left(\left(c^2 x^2+1\right) \tan ^{-1}(c x)^2+(-3 c x+3 i) \tan ^{-1}(c x)-6 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)+3 i \text{Li}_2\left(-e^{2 i \tan ^{-1}(c x)}\right)\right)+b^3 c d \left(-6 i \tan ^{-1}(c x) \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)+2 \tan ^{-1}(c x)^2 \left((c x-i) \tan ^{-1}(c x)+3 \log \left(1+e^{2 i \tan ^{-1}(c x)}\right)\right)\right)}{2 c^2}","-\frac{3 b^2 e \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2}+\frac{3 i b^2 d \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{c}-\frac{\left(d^2-\frac{e^2}{c^2}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{2 e}-\frac{3 i b e \left(a+b \tan ^{-1}(c x)\right)^2}{2 c^2}+\frac{(d+e x)^2 \left(a+b \tan ^{-1}(c x)\right)^3}{2 e}+\frac{i d \left(a+b \tan ^{-1}(c x)\right)^3}{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 e x \left(a+b \tan ^{-1}(c x)\right)^2}{2 c}-\frac{3 i b^3 e \text{Li}_2\left(1-\frac{2}{i c x+1}\right)}{2 c^2}+\frac{3 b^3 d \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 c}",1,"(a^2*c*(2*a*c*d - 3*b*e)*x + a^3*c^2*e*x^2 + 3*a^2*b*e*ArcTan[c*x] + 3*a^2*b*c^2*x*(2*d + e*x)*ArcTan[c*x] - 3*a^2*b*c*d*Log[1 + c^2*x^2] + 3*a*b^2*e*(-2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 + Log[1 + c^2*x^2]) + 6*a*b^2*c*d*(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^3*e*(ArcTan[c*x]*((3*I - 3*c*x)*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 - 6*Log[1 + E^((2*I)*ArcTan[c*x])]) + (3*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + b^3*c*d*(2*ArcTan[c*x]^2*((-I + c*x)*ArcTan[c*x] + 3*Log[1 + E^((2*I)*ArcTan[c*x])]) - (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]))/(2*c^2)","A",0
18,0,0,320,102.4589619,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{d+e x} \, dx","Integrate[(a + b*ArcTan[c*x])^3/(d + e*x),x]","\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{d+e x} \, dx","\frac{3 b^2 \left(a+b \tan ^{-1}(c x)\right) \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 e}-\frac{3 b^2 \text{Li}_3\left(1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{2 e}-\frac{3 i b \left(a+b \tan ^{-1}(c x)\right)^2 \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 e}+\frac{\left(a+b \tan ^{-1}(c x)\right)^3 \log \left(\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right)}{e}+\frac{3 i b \text{Li}_2\left(1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{2 e}-\frac{\log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^3}{e}+\frac{3 i b^3 \text{Li}_4\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{4 e}-\frac{3 i b^3 \text{Li}_4\left(1-\frac{2}{1-i c x}\right)}{4 e}",1,"Integrate[(a + b*ArcTan[c*x])^3/(d + e*x), x]","F",-1
19,1,770,499,21.652675,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{(d+e x)^2} \, dx","Integrate[(a + b*ArcTan[c*x])^3/(d + e*x)^2,x]","-\frac{a^3}{e (d+e x)}+\frac{3 a^2 b c^2 d \tan ^{-1}(c x)}{c^2 d^2 e+e^3}-\frac{3 a^2 b c \log \left(c^2 x^2+1\right)}{2 \left(c^2 d^2+e^2\right)}+\frac{3 a^2 b c \log (d+e x)}{c^2 d^2+e^2}-\frac{3 a^2 b \tan ^{-1}(c x)}{e (d+e x)}+\frac{3 a b^2 \left(-\frac{c d \left(-\frac{1}{2} \pi  \log \left(c^2 x^2+1\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 \tan ^{-1}(c x) \left(\pi -2 \tan ^{-1}\left(\frac{c d}{e}\right)\right)-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 \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)-\pi  \log \left(1+e^{-2 i \tan ^{-1}(c x)}\right)\right)}{c^2 d^2+e^2}-\frac{\tan ^{-1}(c x)^2 e^{i \tan ^{-1}\left(\frac{c d}{e}\right)}}{e \sqrt{\frac{c^2 d^2}{e^2}+1}}+\frac{x \tan ^{-1}(c x)^2}{d+e x}\right)}{d}+\frac{b^3 \left(\frac{2 x \tan ^{-1}(c x)^3}{d+e x}+\frac{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(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)+\pi  \left(\log \left(1+e^{-2 i \tan ^{-1}(c x)}\right)-\log \left(-\frac{2 i}{c x-i}\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)}{c^2 d^2+e^2}\right)}{2 d}","\frac{3 i b^2 c \text{Li}_2\left(1-\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d^2+e^2}+\frac{3 i b^2 c \text{Li}_2\left(1-\frac{2}{i c x+1}\right) \left(a+b \tan ^{-1}(c x)\right)}{c^2 d^2+e^2}-\frac{3 i b^2 c \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)}{c^2 d^2+e^2}+\frac{i c \left(a+b \tan ^{-1}(c x)\right)^3}{c^2 d^2+e^2}+\frac{c^2 d \left(a+b \tan ^{-1}(c x)\right)^3}{e \left(c^2 d^2+e^2\right)}-\frac{3 b c \log \left(\frac{2}{1-i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d^2+e^2}+\frac{3 b c \log \left(\frac{2}{1+i c x}\right) \left(a+b \tan ^{-1}(c x)\right)^2}{c^2 d^2+e^2}+\frac{3 b c \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)}{c^2 d^2+e^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{e (d+e x)}-\frac{3 b^3 c \text{Li}_3\left(1-\frac{2}{1-i c x}\right)}{2 \left(c^2 d^2+e^2\right)}+\frac{3 b^3 c \text{Li}_3\left(1-\frac{2}{i c x+1}\right)}{2 \left(c^2 d^2+e^2\right)}+\frac{3 b^3 c \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right)}{2 \left(c^2 d^2+e^2\right)}",1,"-(a^3/(e*(d + e*x))) + (3*a^2*b*c^2*d*ArcTan[c*x])/(c^2*d^2*e + e^3) - (3*a^2*b*ArcTan[c*x])/(e*(d + e*x)) + (3*a^2*b*c*Log[d + e*x])/(c^2*d^2 + e^2) - (3*a^2*b*c*Log[1 + c^2*x^2])/(2*(c^2*d^2 + e^2)) + (3*a*b^2*(-((E^(I*ArcTan[(c*d)/e])*ArcTan[c*x]^2)/(Sqrt[1 + (c^2*d^2)/e^2]*e)) + (x*ArcTan[c*x]^2)/(d + e*x) - (c*d*((-I)*(Pi - 2*ArcTan[(c*d)/e])*ArcTan[c*x] - Pi*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*Log[1 + c^2*x^2])/2 + 2*ArcTan[(c*d)/e]*Log[Sin[ArcTan[(c*d)/e] + ArcTan[c*x]]] + I*PolyLog[2, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))]))/(c^2*d^2 + e^2)))/d + (b^3*((2*x*ArcTan[c*x]^3)/(d + e*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*d)/e] + ArcTan[c*x]))] + 3*c*d*PolyLog[3, E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))])/(c^2*d^2 + e^2)))/(2*d)","A",0
20,0,0,936,69.395746,"\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{(d+e x)^3} \, dx","Integrate[(a + b*ArcTan[c*x])^3/(d + e*x)^3,x]","\int \frac{\left(a+b \tan ^{-1}(c x)\right)^3}{(d+e x)^3} \, dx","\frac{3 i c^2 e \text{Li}_2\left(1-\frac{2}{1-i c x}\right) b^3}{2 \left(c^2 d^2+e^2\right)^2}+\frac{3 i c^2 e \text{Li}_2\left(1-\frac{2}{i c x+1}\right) b^3}{2 \left(c^2 d^2+e^2\right)^2}-\frac{3 i c^2 e \text{Li}_2\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right) b^3}{2 \left(c^2 d^2+e^2\right)^2}-\frac{3 c^3 d \text{Li}_3\left(1-\frac{2}{1-i c x}\right) b^3}{2 \left(c^2 d^2+e^2\right)^2}+\frac{3 c^3 d \text{Li}_3\left(1-\frac{2}{i c x+1}\right) b^3}{2 \left(c^2 d^2+e^2\right)^2}+\frac{3 c^3 d \text{Li}_3\left(1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right) b^3}{2 \left(c^2 d^2+e^2\right)^2}-\frac{3 c^2 e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2}{1-i c x}\right) b^2}{\left(c^2 d^2+e^2\right)^2}+\frac{3 c^2 e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2}{i c x+1}\right) b^2}{\left(c^2 d^2+e^2\right)^2}+\frac{3 c^2 e \left(a+b \tan ^{-1}(c x)\right) \log \left(\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right) b^2}{\left(c^2 d^2+e^2\right)^2}+\frac{3 i c^3 d \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2}{1-i c x}\right) b^2}{\left(c^2 d^2+e^2\right)^2}+\frac{3 i c^3 d \left(a+b \tan ^{-1}(c x)\right) \text{Li}_2\left(1-\frac{2}{i c x+1}\right) b^2}{\left(c^2 d^2+e^2\right)^2}-\frac{3 i c^3 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) b^2}{\left(c^2 d^2+e^2\right)^2}-\frac{3 c \left(a+b \tan ^{-1}(c x)\right)^2 b}{2 \left(c^2 d^2+e^2\right) (d+e x)}+\frac{3 c^3 d \left(a+b \tan ^{-1}(c x)\right)^2 b}{2 \left(c^2 d^2+e^2\right)^2}+\frac{3 i c^2 e \left(a+b \tan ^{-1}(c x)\right)^2 b}{2 \left(c^2 d^2+e^2\right)^2}-\frac{3 c^3 d \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2}{1-i c x}\right) b}{\left(c^2 d^2+e^2\right)^2}+\frac{3 c^3 d \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2}{i c x+1}\right) b}{\left(c^2 d^2+e^2\right)^2}+\frac{3 c^3 d \left(a+b \tan ^{-1}(c x)\right)^2 \log \left(\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right) b}{\left(c^2 d^2+e^2\right)^2}+\frac{i c^3 d \left(a+b \tan ^{-1}(c x)\right)^3}{\left(c^2 d^2+e^2\right)^2}+\frac{c^2 (c d-e) (c d+e) \left(a+b \tan ^{-1}(c x)\right)^3}{2 e \left(c^2 d^2+e^2\right)^2}-\frac{\left(a+b \tan ^{-1}(c x)\right)^3}{2 e (d+e x)^2}",1,"Integrate[(a + b*ArcTan[c*x])^3/(d + e*x)^3, x]","F",-1
21,1,252,250,3.4802333,"\int (d+e x)^2 \left(a+b \tan ^{-1}\left(c x^2\right)\right) \, dx","Integrate[(d + e*x)^2*(a + b*ArcTan[c*x^2]),x]","\frac{1}{12} \left(12 a d^2 x+12 a d e x^2+4 a e^2 x^3-\frac{\sqrt{2} b \left(3 c d^2+e^2\right) \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{c^{3/2}}+\frac{\sqrt{2} b \left(3 c d^2+e^2\right) \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{c^{3/2}}+\frac{2 \sqrt{2} b \left(3 c d^2-e^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{c} x\right)}{c^{3/2}}-\frac{2 \sqrt{2} b \left(3 c d^2-e^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{c} x+1\right)}{c^{3/2}}-\frac{6 b d e \log \left(c^2 x^4+1\right)}{c}+4 b x \tan ^{-1}\left(c x^2\right) \left(3 d^2+3 d e x+e^2 x^2\right)-\frac{8 b e^2 x}{c}\right)","\frac{(d+e x)^3 \left(a+b \tan ^{-1}\left(c x^2\right)\right)}{3 e}-\frac{b \left(3 c d^2+e^2\right) \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{6 \sqrt{2} c^{3/2}}+\frac{b \left(3 c d^2+e^2\right) \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{6 \sqrt{2} c^{3/2}}+\frac{b \left(3 c d^2-e^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{c} x\right)}{3 \sqrt{2} c^{3/2}}-\frac{b \left(3 c d^2-e^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{c} x+1\right)}{3 \sqrt{2} c^{3/2}}-\frac{b d e \log \left(c^2 x^4+1\right)}{2 c}-\frac{b d^3 \tan ^{-1}\left(c x^2\right)}{3 e}-\frac{2 b e^2 x}{3 c}",1,"(12*a*d^2*x - (8*b*e^2*x)/c + 12*a*d*e*x^2 + 4*a*e^2*x^3 + 4*b*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTan[c*x^2] + (2*Sqrt[2]*b*(3*c*d^2 - e^2)*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/c^(3/2) - (2*Sqrt[2]*b*(3*c*d^2 - e^2)*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/c^(3/2) - (Sqrt[2]*b*(3*c*d^2 + e^2)*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/c^(3/2) + (Sqrt[2]*b*(3*c*d^2 + e^2)*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/c^(3/2) - (6*b*d*e*Log[1 + c^2*x^4])/c)/12","A",1
22,1,153,192,0.0992147,"\int (d+e x) \left(a+b \tan ^{-1}\left(c x^2\right)\right) \, dx","Integrate[(d + e*x)*(a + b*ArcTan[c*x^2]),x]","a d x+\frac{1}{2} a e x^2-\frac{b e \log \left(c^2 x^4+1\right)}{4 c}+b d x \tan ^{-1}\left(c x^2\right)-\frac{b d \left(\log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)-\log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)-2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{c} x\right)+2 \tan ^{-1}\left(\sqrt{2} \sqrt{c} x+1\right)\right)}{2 \sqrt{2} \sqrt{c}}+\frac{1}{2} b e x^2 \tan ^{-1}\left(c x^2\right)","\frac{(d+e x)^2 \left(a+b \tan ^{-1}\left(c x^2\right)\right)}{2 e}-\frac{b e \log \left(c^2 x^4+1\right)}{4 c}-\frac{b d^2 \tan ^{-1}\left(c x^2\right)}{2 e}-\frac{b d \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{2 \sqrt{2} \sqrt{c}}+\frac{b d \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{2 \sqrt{2} \sqrt{c}}+\frac{b d \tan ^{-1}\left(1-\sqrt{2} \sqrt{c} x\right)}{\sqrt{2} \sqrt{c}}-\frac{b d \tan ^{-1}\left(\sqrt{2} \sqrt{c} x+1\right)}{\sqrt{2} \sqrt{c}}",1,"a*d*x + (a*e*x^2)/2 + b*d*x*ArcTan[c*x^2] + (b*e*x^2*ArcTan[c*x^2])/2 - (b*d*(-2*ArcTan[1 - Sqrt[2]*Sqrt[c]*x] + 2*ArcTan[1 + Sqrt[2]*Sqrt[c]*x] + Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2] - Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2]))/(2*Sqrt[2]*Sqrt[c]) - (b*e*Log[1 + c^2*x^4])/(4*c)","A",1
23,1,326,501,34.978856,"\int \frac{a+b \tan ^{-1}\left(c x^2\right)}{d+e x} \, dx","Integrate[(a + b*ArcTan[c*x^2])/(d + e*x),x]","\frac{a \log (d+e x)}{e}+\frac{b \left(2 \tan ^{-1}\left(c x^2\right) \log (d+e x)+i \left(\text{Li}_2\left(\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt[4]{-1} e}\right)+\text{Li}_2\left(\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt[4]{-1} e}\right)-\text{Li}_2\left(\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-(-1)^{3/4} e}\right)-\text{Li}_2\left(\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+(-1)^{3/4} e}\right)+\log (d+e x) \log \left(1-\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt[4]{-1} e}\right)+\log (d+e x) \log \left(1-\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt[4]{-1} e}\right)-\log (d+e x) \log \left(1-\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-(-1)^{3/4} e}\right)-\log (d+e x) \log \left(1-\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+(-1)^{3/4} e}\right)\right)\right)}{2 e}","\frac{\log (d+e x) \left(a+b \tan ^{-1}\left(c x^2\right)\right)}{e}+\frac{b c \text{Li}_2\left(\frac{\sqrt[4]{-c^2} (d+e x)}{\sqrt[4]{-c^2} d-e}\right)}{2 \sqrt{-c^2} e}-\frac{b c \text{Li}_2\left(\frac{\sqrt{-\sqrt{-c^2}} (d+e x)}{\sqrt{-\sqrt{-c^2}} d-e}\right)}{2 \sqrt{-c^2} e}+\frac{b c \text{Li}_2\left(\frac{\sqrt[4]{-c^2} (d+e x)}{\sqrt[4]{-c^2} d+e}\right)}{2 \sqrt{-c^2} e}-\frac{b c \text{Li}_2\left(\frac{\sqrt{-\sqrt{-c^2}} (d+e x)}{\sqrt{-\sqrt{-c^2}} d+e}\right)}{2 \sqrt{-c^2} e}+\frac{b c \log (d+e x) \log \left(\frac{e \left(1-\sqrt[4]{-c^2} x\right)}{\sqrt[4]{-c^2} d+e}\right)}{2 \sqrt{-c^2} e}+\frac{b c \log (d+e x) \log \left(-\frac{e \left(\sqrt[4]{-c^2} x+1\right)}{\sqrt[4]{-c^2} d-e}\right)}{2 \sqrt{-c^2} e}-\frac{b c \log (d+e x) \log \left(\frac{e \left(1-\sqrt{-\sqrt{-c^2}} x\right)}{\sqrt{-\sqrt{-c^2}} d+e}\right)}{2 \sqrt{-c^2} e}-\frac{b c \log (d+e x) \log \left(-\frac{e \left(\sqrt{-\sqrt{-c^2}} x+1\right)}{\sqrt{-\sqrt{-c^2}} d-e}\right)}{2 \sqrt{-c^2} e}",1,"(a*Log[d + e*x])/e + (b*(2*ArcTan[c*x^2]*Log[d + e*x] + I*(Log[d + e*x]*Log[1 - (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - (-1)^(1/4)*e)] + Log[d + e*x]*Log[1 - (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + (-1)^(1/4)*e)] - Log[d + e*x]*Log[1 - (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - (-1)^(3/4)*e)] - Log[d + e*x]*Log[1 - (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + (-1)^(3/4)*e)] + PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - (-1)^(1/4)*e)] + PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + (-1)^(1/4)*e)] - PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - (-1)^(3/4)*e)] - PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + (-1)^(3/4)*e)])))/(2*e)","C",1
24,1,321,328,0.8027789,"\int \frac{a+b \tan ^{-1}\left(c x^2\right)}{(d+e x)^2} \, dx","Integrate[(a + b*ArcTan[c*x^2])/(d + e*x)^2,x]","-\frac{4 a \left(c^2 d^4+e^4\right)+2 b \sqrt{c} \left(2 c^{3/2} d^3-\sqrt{2} c d^2 e+\sqrt{2} e^3\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{c} x\right) (d+e x)+2 b \sqrt{c} \left(2 c^{3/2} d^3+\sqrt{2} c d^2 e-\sqrt{2} e^3\right) \tan ^{-1}\left(\sqrt{2} \sqrt{c} x+1\right) (d+e x)+4 b \left(c^2 d^4+e^4\right) \tan ^{-1}\left(c x^2\right)-2 b c d e^2 \log \left(c^2 x^4+1\right) (d+e x)+\sqrt{2} b \sqrt{c} e \left(c d^2+e^2\right) \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right) (d+e x)-\sqrt{2} b \sqrt{c} e \left(c d^2+e^2\right) \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right) (d+e x)+8 b c d e^2 (d+e x) \log (d+e x)}{4 e \left(c^2 d^4+e^4\right) (d+e x)}","-\frac{a+b \tan ^{-1}\left(c x^2\right)}{e (d+e x)}+\frac{b c d e \log \left(c^2 x^4+1\right)}{2 \left(c^2 d^4+e^4\right)}-\frac{2 b c d e \log (d+e x)}{c^2 d^4+e^4}+\frac{b c^2 d^3 \tan ^{-1}\left(c x^2\right)}{e \left(c^2 d^4+e^4\right)}-\frac{b \sqrt{c} \left(c d^2+e^2\right) \log \left(c x^2-\sqrt{2} \sqrt{c} x+1\right)}{2 \sqrt{2} \left(c^2 d^4+e^4\right)}+\frac{b \sqrt{c} \left(c d^2+e^2\right) \log \left(c x^2+\sqrt{2} \sqrt{c} x+1\right)}{2 \sqrt{2} \left(c^2 d^4+e^4\right)}+\frac{b \sqrt{c} \left(c d^2-e^2\right) \tan ^{-1}\left(1-\sqrt{2} \sqrt{c} x\right)}{\sqrt{2} \left(c^2 d^4+e^4\right)}-\frac{b \sqrt{c} \left(c d^2-e^2\right) \tan ^{-1}\left(\sqrt{2} \sqrt{c} x+1\right)}{\sqrt{2} \left(c^2 d^4+e^4\right)}",1,"-1/4*(4*a*(c^2*d^4 + e^4) + 4*b*(c^2*d^4 + e^4)*ArcTan[c*x^2] + 2*b*Sqrt[c]*(2*c^(3/2)*d^3 - Sqrt[2]*c*d^2*e + Sqrt[2]*e^3)*(d + e*x)*ArcTan[1 - Sqrt[2]*Sqrt[c]*x] + 2*b*Sqrt[c]*(2*c^(3/2)*d^3 + Sqrt[2]*c*d^2*e - Sqrt[2]*e^3)*(d + e*x)*ArcTan[1 + Sqrt[2]*Sqrt[c]*x] + 8*b*c*d*e^2*(d + e*x)*Log[d + e*x] + Sqrt[2]*b*Sqrt[c]*e*(c*d^2 + e^2)*(d + e*x)*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2] - Sqrt[2]*b*Sqrt[c]*e*(c*d^2 + e^2)*(d + e*x)*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2] - 2*b*c*d*e^2*(d + e*x)*Log[1 + c^2*x^4])/(e*(c^2*d^4 + e^4)*(d + e*x))","A",1
25,1,5591,1325,31.4032555,"\int (d+e x) \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2 \, dx","Integrate[(d + e*x)*(a + b*ArcTan[c*x^2])^2,x]","\text{Result too large to show}","d x a^2-\frac{2 (-1)^{3/4} b d \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) a}{\sqrt{c}}+\frac{2 (-1)^{3/4} b d \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) a}{\sqrt{c}}+i b d x \log \left(1-i c x^2\right) a-i b d x \log \left(i c x^2+1\right) a+\frac{(-1)^{3/4} b^2 d \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)^2}{\sqrt{c}}+\frac{1}{2} e x^2 \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2+\frac{i e \left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{2 c}-\frac{\sqrt[4]{-1} b^2 d \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right)^2}{\sqrt{c}}-\frac{1}{4} b^2 d x \log ^2\left(1-i c x^2\right)-\frac{1}{4} b^2 d x \log ^2\left(i c x^2+1\right)+\frac{2 \sqrt[4]{-1} b^2 d \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{1-\sqrt[4]{-1} \sqrt{c} x}\right)}{\sqrt{c}}-\frac{2 \sqrt[4]{-1} b^2 d \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{\sqrt[4]{-1} b^2 d \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{\sqrt{2} \left(\sqrt{c} x+\sqrt[4]{-1}\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{2 \sqrt[4]{-1} b^2 d \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right)}{\sqrt{c}}-\frac{2 \sqrt[4]{-1} b^2 d \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{2}{(-1)^{3/4} \sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{\sqrt[4]{-1} b^2 d \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(-\frac{\sqrt{2} \left(\sqrt{c} x+(-1)^{3/4}\right)}{(-1)^{3/4} \sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{\sqrt[4]{-1} b^2 d \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{(1+i) \left(\sqrt[4]{-1} \sqrt{c} x+1\right)}{(-1)^{3/4} \sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{\sqrt[4]{-1} b^2 d \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(\frac{(1-i) \left((-1)^{3/4} \sqrt{c} x+1\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{\sqrt{c}}+\frac{\sqrt[4]{-1} b^2 d \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(1-i c x^2\right)}{\sqrt{c}}-\frac{\sqrt[4]{-1} b^2 d \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(1-i c x^2\right)}{\sqrt{c}}+\frac{b e \left(a+b \tan ^{-1}\left(c x^2\right)\right) \log \left(\frac{2}{i c x^2+1}\right)}{c}-\frac{\sqrt[4]{-1} b^2 d \tan ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(i c x^2+1\right)}{\sqrt{c}}+\frac{\sqrt[4]{-1} b^2 d \tanh ^{-1}\left((-1)^{3/4} \sqrt{c} x\right) \log \left(i c x^2+1\right)}{\sqrt{c}}+\frac{1}{2} b^2 d x \log \left(1-i c x^2\right) \log \left(i c x^2+1\right)+\frac{(-1)^{3/4} b^2 d \text{Li}_2\left(1-\frac{2}{1-\sqrt[4]{-1} \sqrt{c} x}\right)}{\sqrt{c}}+\frac{(-1)^{3/4} b^2 d \text{Li}_2\left(1-\frac{2}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{\sqrt{c}}-\frac{(-1)^{3/4} b^2 d \text{Li}_2\left(1-\frac{\sqrt{2} \left(\sqrt{c} x+\sqrt[4]{-1}\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{2 \sqrt{c}}+\frac{\sqrt[4]{-1} b^2 d \text{Li}_2\left(1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right)}{\sqrt{c}}+\frac{\sqrt[4]{-1} b^2 d \text{Li}_2\left(1-\frac{2}{(-1)^{3/4} \sqrt{c} x+1}\right)}{\sqrt{c}}-\frac{\sqrt[4]{-1} b^2 d \text{Li}_2\left(\frac{\sqrt{2} \left(\sqrt{c} x+(-1)^{3/4}\right)}{(-1)^{3/4} \sqrt{c} x+1}+1\right)}{2 \sqrt{c}}-\frac{\sqrt[4]{-1} b^2 d \text{Li}_2\left(1-\frac{(1+i) \left(\sqrt[4]{-1} \sqrt{c} x+1\right)}{(-1)^{3/4} \sqrt{c} x+1}\right)}{2 \sqrt{c}}-\frac{(-1)^{3/4} b^2 d \text{Li}_2\left(1-\frac{(1-i) \left((-1)^{3/4} \sqrt{c} x+1\right)}{\sqrt[4]{-1} \sqrt{c} x+1}\right)}{2 \sqrt{c}}+\frac{i b^2 e \text{Li}_2\left(1-\frac{2}{i c x^2+1}\right)}{2 c}",1,"Result too large to show","C",0
26,0,0,23,75.9780362,"\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{d+e x} \, dx","Integrate[(a + b*ArcTan[c*x^2])^2/(d + e*x),x]","\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{d+e x} \, dx","\text{Int}\left(\frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{d+e x},x\right)",0,"Integrate[(a + b*ArcTan[c*x^2])^2/(d + e*x), x]","A",-1
27,0,0,23,66.471575,"\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{(d+e x)^2} \, dx","Integrate[(a + b*ArcTan[c*x^2])^2/(d + e*x)^2,x]","\int \frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{(d+e x)^2} \, dx","\text{Int}\left(\frac{\left(a+b \tan ^{-1}\left(c x^2\right)\right)^2}{(d+e x)^2},x\right)",0,"Integrate[(a + b*ArcTan[c*x^2])^2/(d + e*x)^2, x]","A",-1
28,1,297,315,168.7074027,"\int (d+e x)^2 \left(a+b \tan ^{-1}\left(c x^3\right)\right) \, dx","Integrate[(d + e*x)^2*(a + b*ArcTan[c*x^3]),x]","\frac{12 a c d^2 x+12 a c d e x^2+4 a c e^2 x^3+6 b c^{2/3} d^2 \log \left(c^{2/3} x^2+1\right)-3 b \sqrt[3]{c} d \left(\sqrt[3]{c} d+\sqrt{3} e\right) \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)-3 b \sqrt[3]{c} d \left(\sqrt[3]{c} d-\sqrt{3} e\right) \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)-2 b e^2 \log \left(c^2 x^6+1\right)+4 b c x \tan ^{-1}\left(c x^3\right) \left(3 d^2+3 d e x+e^2 x^2\right)-12 b \sqrt[3]{c} d e \tan ^{-1}\left(\sqrt[3]{c} x\right)+6 b \sqrt[3]{c} d \left(\sqrt{3} \sqrt[3]{c} d+e\right) \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)+6 b \sqrt[3]{c} d \left(\sqrt{3} \sqrt[3]{c} d-e\right) \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)}{12 c}","\frac{(d+e x)^3 \left(a+b \tan ^{-1}\left(c x^3\right)\right)}{3 e}+\frac{b d^2 \log \left(c^{2/3} x^2+1\right)}{2 \sqrt[3]{c}}+\frac{\sqrt{3} b d^2 \tan ^{-1}\left(\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right)}{2 \sqrt[3]{c}}-\frac{b d^2 \log \left(c^{4/3} x^4-c^{2/3} x^2+1\right)}{4 \sqrt[3]{c}}-\frac{\sqrt{3} b d e \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)}{4 c^{2/3}}+\frac{\sqrt{3} b d e \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)}{4 c^{2/3}}-\frac{b d e \tan ^{-1}\left(\sqrt[3]{c} x\right)}{c^{2/3}}+\frac{b d e \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)}{2 c^{2/3}}-\frac{b d e \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)}{2 c^{2/3}}-\frac{b e^2 \log \left(c^2 x^6+1\right)}{6 c}-\frac{b d^3 \tan ^{-1}\left(c x^3\right)}{3 e}",1,"(12*a*c*d^2*x + 12*a*c*d*e*x^2 + 4*a*c*e^2*x^3 - 12*b*c^(1/3)*d*e*ArcTan[c^(1/3)*x] + 4*b*c*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTan[c*x^3] + 6*b*c^(1/3)*d*(Sqrt[3]*c^(1/3)*d + e)*ArcTan[Sqrt[3] - 2*c^(1/3)*x] + 6*b*c^(1/3)*d*(Sqrt[3]*c^(1/3)*d - e)*ArcTan[Sqrt[3] + 2*c^(1/3)*x] + 6*b*c^(2/3)*d^2*Log[1 + c^(2/3)*x^2] - 3*b*c^(1/3)*d*(c^(1/3)*d + Sqrt[3]*e)*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2] - 3*b*c^(1/3)*d*(c^(1/3)*d - Sqrt[3]*e)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2] - 2*b*e^2*Log[1 + c^2*x^6])/(12*c)","A",1
29,1,310,285,0.0956151,"\int (d+e x) \left(a+b \tan ^{-1}\left(c x^3\right)\right) \, dx","Integrate[(d + e*x)*(a + b*ArcTan[c*x^3]),x]","a d x+\frac{1}{2} a e x^2-\frac{b d \left(-2 \log \left(c^{2/3} x^2+1\right)+\log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)+\log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)-2 \sqrt{3} \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)-2 \sqrt{3} \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)\right)}{4 \sqrt[3]{c}}-\frac{\sqrt{3} b e \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)}{8 c^{2/3}}+\frac{\sqrt{3} b e \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)}{8 c^{2/3}}-\frac{b e \tan ^{-1}\left(\sqrt[3]{c} x\right)}{2 c^{2/3}}+\frac{b e \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)}{4 c^{2/3}}-\frac{b e \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)}{4 c^{2/3}}+b d x \tan ^{-1}\left(c x^3\right)+\frac{1}{2} b e x^2 \tan ^{-1}\left(c x^3\right)","\frac{(d+e x)^2 \left(a+b \tan ^{-1}\left(c x^3\right)\right)}{2 e}+\frac{b d \log \left(c^{2/3} x^2+1\right)}{2 \sqrt[3]{c}}+\frac{\sqrt{3} b d \tan ^{-1}\left(\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right)}{2 \sqrt[3]{c}}-\frac{b d \log \left(c^{4/3} x^4-c^{2/3} x^2+1\right)}{4 \sqrt[3]{c}}-\frac{\sqrt{3} b e \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right)}{8 c^{2/3}}+\frac{\sqrt{3} b e \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right)}{8 c^{2/3}}-\frac{b e \tan ^{-1}\left(\sqrt[3]{c} x\right)}{2 c^{2/3}}+\frac{b e \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right)}{4 c^{2/3}}-\frac{b e \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right)}{4 c^{2/3}}-\frac{b d^2 \tan ^{-1}\left(c x^3\right)}{2 e}",1,"a*d*x + (a*e*x^2)/2 - (b*e*ArcTan[c^(1/3)*x])/(2*c^(2/3)) + b*d*x*ArcTan[c*x^3] + (b*e*x^2*ArcTan[c*x^3])/2 + (b*e*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(4*c^(2/3)) - (b*e*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(4*c^(2/3)) - (Sqrt[3]*b*e*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) + (Sqrt[3]*b*e*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) - (b*d*(-2*Sqrt[3]*ArcTan[Sqrt[3] - 2*c^(1/3)*x] - 2*Sqrt[3]*ArcTan[Sqrt[3] + 2*c^(1/3)*x] - 2*Log[1 + c^(2/3)*x^2] + Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2] + Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2]))/(4*c^(1/3))","A",1
30,-1,0,739,180.002523,"\int \frac{a+b \tan ^{-1}\left(c x^3\right)}{d+e x} \, dx","Integrate[(a + b*ArcTan[c*x^3])/(d + e*x),x]","\text{\$Aborted}","\frac{\log (d+e x) \left(a+b \tan ^{-1}\left(c x^3\right)\right)}{e}-\frac{b c \text{Li}_2\left(\frac{\sqrt[6]{-c^2} (d+e x)}{\sqrt[6]{-c^2} d-e}\right)}{2 \sqrt{-c^2} e}+\frac{b c \text{Li}_2\left(\frac{\sqrt[6]{-c^2} (d+e x)}{\sqrt[6]{-c^2} d+e}\right)}{2 \sqrt{-c^2} e}+\frac{b c \text{Li}_2\left(\frac{\sqrt[6]{-c^2} (d+e x)}{\sqrt[6]{-c^2} d-\sqrt[3]{-1} e}\right)}{2 \sqrt{-c^2} e}-\frac{b c \text{Li}_2\left(\frac{\sqrt[6]{-c^2} (d+e x)}{\sqrt[6]{-c^2} d+\sqrt[3]{-1} e}\right)}{2 \sqrt{-c^2} e}-\frac{b c \text{Li}_2\left(\frac{\sqrt[6]{-c^2} (d+e x)}{\sqrt[6]{-c^2} d-(-1)^{2/3} e}\right)}{2 \sqrt{-c^2} e}+\frac{b c \text{Li}_2\left(\frac{\sqrt[6]{-c^2} (d+e x)}{\sqrt[6]{-c^2} d+(-1)^{2/3} e}\right)}{2 \sqrt{-c^2} e}+\frac{b c \log (d+e x) \log \left(\frac{e \left(1-\sqrt[6]{-c^2} x\right)}{\sqrt[6]{-c^2} d+e}\right)}{2 \sqrt{-c^2} e}-\frac{b c \log (d+e x) \log \left(-\frac{e \left(\sqrt[6]{-c^2} x+1\right)}{\sqrt[6]{-c^2} d-e}\right)}{2 \sqrt{-c^2} e}+\frac{b c \log (d+e x) \log \left(-\frac{e \left(\sqrt[6]{-c^2} x+\sqrt[3]{-1}\right)}{\sqrt[6]{-c^2} d-\sqrt[3]{-1} e}\right)}{2 \sqrt{-c^2} e}-\frac{b c \log (d+e x) \log \left(-\frac{e \left(\sqrt[6]{-c^2} x+(-1)^{2/3}\right)}{\sqrt[6]{-c^2} d-(-1)^{2/3} e}\right)}{2 \sqrt{-c^2} e}+\frac{b c \log (d+e x) \log \left(\frac{(-1)^{2/3} e \left(\sqrt[3]{-1} \sqrt[6]{-c^2} x+1\right)}{\sqrt[6]{-c^2} d+(-1)^{2/3} e}\right)}{2 \sqrt{-c^2} e}-\frac{b c \log (d+e x) \log \left(\frac{\sqrt[3]{-1} e \left((-1)^{2/3} \sqrt[6]{-c^2} x+1\right)}{\sqrt[6]{-c^2} d+\sqrt[3]{-1} e}\right)}{2 \sqrt{-c^2} e}",1,"$Aborted","F",-1
31,1,536,906,16.8092791,"\int \frac{a+b \tan ^{-1}\left(c x^3\right)}{(d+e x)^2} \, dx","Integrate[(a + b*ArcTan[c*x^3])/(d + e*x)^2,x]","\frac{-4 a \sqrt[3]{c} \left(c^2 d^6+e^6\right)+12 b c^{4/3} d^2 e^3 (d+e x) \log (d+e x)-b c^{2/3} e \left(c^{4/3} d^4-\sqrt{3} c d^3 e-\sqrt{3} \sqrt[3]{c} d e^3+e^4\right) \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right) (d+e x)-b c^{2/3} e \left(c^{4/3} d^4+\sqrt{3} c d^3 e+\sqrt{3} \sqrt[3]{c} d e^3+e^4\right) \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right) (d+e x)-4 b c d \left(c^{4/3} d^4-c^{2/3} d^2 e^2+e^4\right) \tan ^{-1}\left(\sqrt[3]{c} x\right) (d+e x)-2 b c^{2/3} \left(2 c^{5/3} d^5-\sqrt{3} c^{4/3} d^4 e+c d^3 e^2-\sqrt[3]{c} d e^4+\sqrt{3} e^5\right) \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right) (d+e x)+2 b c^{2/3} \left(2 c^{5/3} d^5+\sqrt{3} c^{4/3} d^4 e+c d^3 e^2-\sqrt[3]{c} d e^4-\sqrt{3} e^5\right) \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right) (d+e x)-4 b \sqrt[3]{c} \left(c^2 d^6+e^6\right) \tan ^{-1}\left(c x^3\right)+2 b e \left(c^{2/3} e^4+c^2 d^4\right) \log \left(c^{2/3} x^2+1\right) (d+e x)-2 b c^{4/3} d^2 e^3 \log \left(c^2 x^6+1\right) (d+e x)}{4 \sqrt[3]{c} e \left(c^2 d^6+e^6\right) (d+e x)}","\frac{b c^2 \tan ^{-1}\left(c x^3\right) d^5}{e \left(c^2 d^6+e^6\right)}+\frac{b c^{5/3} \log \left(c^{2/3} x^2+1\right) d^4}{2 \left(c^2 d^6+e^6\right)}+\frac{3 b c e^2 \log (d+e x) d^2}{c^2 d^6+e^6}-\frac{b c e^2 \log \left(c^2 x^6+1\right) d^2}{2 \left(c^2 d^6+e^6\right)}-\frac{b c^{2/3} e^3 \tan ^{-1}\left(\sqrt[3]{c} x\right) d}{c^2 d^6+e^6}+\frac{b c^{2/3} \left(\sqrt{3} c d^3+e^3\right) \tan ^{-1}\left(\sqrt{3}-2 \sqrt[3]{c} x\right) d}{2 \left(c^2 d^6+e^6\right)}+\frac{b c^{2/3} \left(\sqrt{3} c d^3-e^3\right) \tan ^{-1}\left(2 \sqrt[3]{c} x+\sqrt{3}\right) d}{2 \left(c^2 d^6+e^6\right)}-\frac{b c^{2/3} \left(c d^3-\sqrt{3} e^3\right) \log \left(c^{2/3} x^2-\sqrt{3} \sqrt[3]{c} x+1\right) d}{4 \left(c^2 d^6+e^6\right)}-\frac{b c^{2/3} \left(c d^3+\sqrt{3} e^3\right) \log \left(c^{2/3} x^2+\sqrt{3} \sqrt[3]{c} x+1\right) d}{4 \left(c^2 d^6+e^6\right)}-\frac{a+b \tan ^{-1}\left(c x^3\right)}{e (d+e x)}+\frac{\sqrt{3} b c^{5/3} e \left(\sqrt{-c^2} d^3+e^3\right) \tan ^{-1}\left(\frac{\frac{2 c^{2/3} x}{\sqrt[6]{-c^2}}+1}{\sqrt{3}}\right)}{2 \left(-c^2\right)^{2/3} \left(c^2 d^6+e^6\right)}-\frac{\sqrt{3} b c^{5/3} e \left(\sqrt{-c^2} d^3-e^3\right) \tan ^{-1}\left(\frac{c^{4/3}+2 \left(-c^2\right)^{5/6} x}{\sqrt{3} c^{4/3}}\right)}{2 \left(-c^2\right)^{2/3} \left(c^2 d^6+e^6\right)}+\frac{b c^{5/3} e \left(\sqrt{-c^2} d^3+e^3\right) \log \left(\sqrt[6]{-c^2}-c^{2/3} x\right)}{2 \left(-c^2\right)^{2/3} \left(c^2 d^6+e^6\right)}-\frac{b c^{5/3} e \left(\sqrt{-c^2} d^3-e^3\right) \log \left(c^{2/3} x+\sqrt[6]{-c^2}\right)}{2 \left(-c^2\right)^{2/3} \left(c^2 d^6+e^6\right)}+\frac{b c^{5/3} e \left(\sqrt{-c^2} d^3-e^3\right) \log \left(c^{4/3} x^2-c^{2/3} \sqrt[6]{-c^2} x+\sqrt[3]{-c^2}\right)}{4 \left(-c^2\right)^{2/3} \left(c^2 d^6+e^6\right)}-\frac{b c^{5/3} e \left(\sqrt{-c^2} d^3+e^3\right) \log \left(c^{4/3} x^2+c^{2/3} \sqrt[6]{-c^2} x+\sqrt[3]{-c^2}\right)}{4 \left(-c^2\right)^{2/3} \left(c^2 d^6+e^6\right)}",1,"(-4*a*c^(1/3)*(c^2*d^6 + e^6) - 4*b*c*d*(c^(4/3)*d^4 - c^(2/3)*d^2*e^2 + e^4)*(d + e*x)*ArcTan[c^(1/3)*x] - 4*b*c^(1/3)*(c^2*d^6 + e^6)*ArcTan[c*x^3] - 2*b*c^(2/3)*(2*c^(5/3)*d^5 - Sqrt[3]*c^(4/3)*d^4*e + c*d^3*e^2 - c^(1/3)*d*e^4 + Sqrt[3]*e^5)*(d + e*x)*ArcTan[Sqrt[3] - 2*c^(1/3)*x] + 2*b*c^(2/3)*(2*c^(5/3)*d^5 + Sqrt[3]*c^(4/3)*d^4*e + c*d^3*e^2 - c^(1/3)*d*e^4 - Sqrt[3]*e^5)*(d + e*x)*ArcTan[Sqrt[3] + 2*c^(1/3)*x] + 12*b*c^(4/3)*d^2*e^3*(d + e*x)*Log[d + e*x] + 2*b*e*(c^2*d^4 + c^(2/3)*e^4)*(d + e*x)*Log[1 + c^(2/3)*x^2] - b*c^(2/3)*e*(c^(4/3)*d^4 - Sqrt[3]*c*d^3*e - Sqrt[3]*c^(1/3)*d*e^3 + e^4)*(d + e*x)*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2] - b*c^(2/3)*e*(c^(4/3)*d^4 + Sqrt[3]*c*d^3*e + Sqrt[3]*c^(1/3)*d*e^3 + e^4)*(d + e*x)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2] - 2*b*c^(4/3)*d^2*e^3*(d + e*x)*Log[1 + c^2*x^6])/(4*c^(1/3)*e*(c^2*d^6 + e^6)*(d + e*x))","A",1