1,1,144,0,0.198872,"\int \frac{x^3 \left(a+b \cos ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Int[(x^3*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2),x]","\frac{i b \text{PolyLog}\left(2,e^{2 i \cos ^{-1}(c x)}\right)}{2 c^4 d}-\frac{x^2 \left(a+b \cos ^{-1}(c x)\right)}{2 c^2 d}+\frac{i \left(a+b \cos ^{-1}(c x)\right)^2}{2 b c^4 d}-\frac{\log \left(1-e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c^4 d}+\frac{b x \sqrt{1-c^2 x^2}}{4 c^3 d}-\frac{b \sin ^{-1}(c x)}{4 c^4 d}","\frac{i b \text{PolyLog}\left(2,e^{2 i \cos ^{-1}(c x)}\right)}{2 c^4 d}-\frac{x^2 \left(a+b \cos ^{-1}(c x)\right)}{2 c^2 d}+\frac{i \left(a+b \cos ^{-1}(c x)\right)^2}{2 b c^4 d}-\frac{\log \left(1-e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c^4 d}+\frac{b x \sqrt{1-c^2 x^2}}{4 c^3 d}-\frac{b \sin ^{-1}(c x)}{4 c^4 d}",1,"(b*x*Sqrt[1 - c^2*x^2])/(4*c^3*d) - (x^2*(a + b*ArcCos[c*x]))/(2*c^2*d) + ((I/2)*(a + b*ArcCos[c*x])^2)/(b*c^4*d) - (b*ArcSin[c*x])/(4*c^4*d) - ((a + b*ArcCos[c*x])*Log[1 - E^((2*I)*ArcCos[c*x])])/(c^4*d) + ((I/2)*b*PolyLog[2, E^((2*I)*ArcCos[c*x])])/(c^4*d)","A",8,8,25,0.3200,1,"{4716, 4676, 3717, 2190, 2279, 2391, 321, 216}"
2,1,115,0,0.1340216,"\int \frac{x^2 \left(a+b \cos ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Int[(x^2*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2),x]","-\frac{i b \text{PolyLog}\left(2,-e^{i \cos ^{-1}(c x)}\right)}{c^3 d}+\frac{i b \text{PolyLog}\left(2,e^{i \cos ^{-1}(c x)}\right)}{c^3 d}-\frac{x \left(a+b \cos ^{-1}(c x)\right)}{c^2 d}+\frac{2 \tanh ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c^3 d}+\frac{b \sqrt{1-c^2 x^2}}{c^3 d}","-\frac{i b \text{PolyLog}\left(2,-e^{i \cos ^{-1}(c x)}\right)}{c^3 d}+\frac{i b \text{PolyLog}\left(2,e^{i \cos ^{-1}(c x)}\right)}{c^3 d}-\frac{x \left(a+b \cos ^{-1}(c x)\right)}{c^2 d}+\frac{2 \tanh ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c^3 d}+\frac{b \sqrt{1-c^2 x^2}}{c^3 d}",1,"(b*Sqrt[1 - c^2*x^2])/(c^3*d) - (x*(a + b*ArcCos[c*x]))/(c^2*d) + (2*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/(c^3*d) - (I*b*PolyLog[2, -E^(I*ArcCos[c*x])])/(c^3*d) + (I*b*PolyLog[2, E^(I*ArcCos[c*x])])/(c^3*d)","A",8,6,25,0.2400,1,"{4716, 4658, 4183, 2279, 2391, 261}"
3,1,82,0,0.112598,"\int \frac{x \left(a+b \cos ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Int[(x*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2),x]","\frac{i b \text{PolyLog}\left(2,e^{2 i \cos ^{-1}(c x)}\right)}{2 c^2 d}+\frac{i \left(a+b \cos ^{-1}(c x)\right)^2}{2 b c^2 d}-\frac{\log \left(1-e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c^2 d}","\frac{i b \text{PolyLog}\left(2,e^{2 i \cos ^{-1}(c x)}\right)}{2 c^2 d}+\frac{i \left(a+b \cos ^{-1}(c x)\right)^2}{2 b c^2 d}-\frac{\log \left(1-e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c^2 d}",1,"((I/2)*(a + b*ArcCos[c*x])^2)/(b*c^2*d) - ((a + b*ArcCos[c*x])*Log[1 - E^((2*I)*ArcCos[c*x])])/(c^2*d) + ((I/2)*b*PolyLog[2, E^((2*I)*ArcCos[c*x])])/(c^2*d)","A",5,5,23,0.2174,1,"{4676, 3717, 2190, 2279, 2391}"
4,1,76,0,0.0619147,"\int \frac{a+b \cos ^{-1}(c x)}{d-c^2 d x^2} \, dx","Int[(a + b*ArcCos[c*x])/(d - c^2*d*x^2),x]","-\frac{i b \text{PolyLog}\left(2,-e^{i \cos ^{-1}(c x)}\right)}{c d}+\frac{i b \text{PolyLog}\left(2,e^{i \cos ^{-1}(c x)}\right)}{c d}+\frac{2 \tanh ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c d}","-\frac{i b \text{PolyLog}\left(2,-e^{i \cos ^{-1}(c x)}\right)}{c d}+\frac{i b \text{PolyLog}\left(2,e^{i \cos ^{-1}(c x)}\right)}{c d}+\frac{2 \tanh ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c d}",1,"(2*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/(c*d) - (I*b*PolyLog[2, -E^(I*ArcCos[c*x])])/(c*d) + (I*b*PolyLog[2, E^(I*ArcCos[c*x])])/(c*d)","A",6,4,22,0.1818,1,"{4658, 4183, 2279, 2391}"
5,1,71,0,0.11134,"\int \frac{a+b \cos ^{-1}(c x)}{x \left(d-c^2 d x^2\right)} \, dx","Int[(a + b*ArcCos[c*x])/(x*(d - c^2*d*x^2)),x]","-\frac{i b \text{PolyLog}\left(2,-e^{2 i \cos ^{-1}(c x)}\right)}{2 d}+\frac{i b \text{PolyLog}\left(2,e^{2 i \cos ^{-1}(c x)}\right)}{2 d}+\frac{2 \tanh ^{-1}\left(e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{d}","-\frac{i b \text{PolyLog}\left(2,-e^{2 i \cos ^{-1}(c x)}\right)}{2 d}+\frac{i b \text{PolyLog}\left(2,e^{2 i \cos ^{-1}(c x)}\right)}{2 d}+\frac{2 \tanh ^{-1}\left(e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{d}",1,"(2*(a + b*ArcCos[c*x])*ArcTanh[E^((2*I)*ArcCos[c*x])])/d - ((I/2)*b*PolyLog[2, -E^((2*I)*ArcCos[c*x])])/d + ((I/2)*b*PolyLog[2, E^((2*I)*ArcCos[c*x])])/d","A",7,5,25,0.2000,1,"{4680, 4419, 4183, 2279, 2391}"
6,1,107,0,0.1411542,"\int \frac{a+b \cos ^{-1}(c x)}{x^2 \left(d-c^2 d x^2\right)} \, dx","Int[(a + b*ArcCos[c*x])/(x^2*(d - c^2*d*x^2)),x]","-\frac{i b c \text{PolyLog}\left(2,-e^{i \cos ^{-1}(c x)}\right)}{d}+\frac{i b c \text{PolyLog}\left(2,e^{i \cos ^{-1}(c x)}\right)}{d}-\frac{a+b \cos ^{-1}(c x)}{d x}+\frac{2 c \tanh ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{d}+\frac{b c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{d}","-\frac{i b c \text{PolyLog}\left(2,-e^{i \cos ^{-1}(c x)}\right)}{d}+\frac{i b c \text{PolyLog}\left(2,e^{i \cos ^{-1}(c x)}\right)}{d}-\frac{a+b \cos ^{-1}(c x)}{d x}+\frac{2 c \tanh ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{d}+\frac{b c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{d}",1,"-((a + b*ArcCos[c*x])/(d*x)) + (2*c*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/d + (b*c*ArcTanh[Sqrt[1 - c^2*x^2]])/d - (I*b*c*PolyLog[2, -E^(I*ArcCos[c*x])])/d + (I*b*c*PolyLog[2, E^(I*ArcCos[c*x])])/d","A",10,8,25,0.3200,1,"{4702, 4658, 4183, 2279, 2391, 266, 63, 208}"
7,1,124,0,0.1859643,"\int \frac{a+b \cos ^{-1}(c x)}{x^3 \left(d-c^2 d x^2\right)} \, dx","Int[(a + b*ArcCos[c*x])/(x^3*(d - c^2*d*x^2)),x]","-\frac{i b c^2 \text{PolyLog}\left(2,-e^{2 i \cos ^{-1}(c x)}\right)}{2 d}+\frac{i b c^2 \text{PolyLog}\left(2,e^{2 i \cos ^{-1}(c x)}\right)}{2 d}+\frac{2 c^2 \tanh ^{-1}\left(e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{d}-\frac{a+b \cos ^{-1}(c x)}{2 d x^2}+\frac{b c \sqrt{1-c^2 x^2}}{2 d x}","-\frac{i b c^2 \text{PolyLog}\left(2,-e^{2 i \cos ^{-1}(c x)}\right)}{2 d}+\frac{i b c^2 \text{PolyLog}\left(2,e^{2 i \cos ^{-1}(c x)}\right)}{2 d}+\frac{2 c^2 \tanh ^{-1}\left(e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{d}-\frac{a+b \cos ^{-1}(c x)}{2 d x^2}+\frac{b c \sqrt{1-c^2 x^2}}{2 d x}",1,"(b*c*Sqrt[1 - c^2*x^2])/(2*d*x) - (a + b*ArcCos[c*x])/(2*d*x^2) + (2*c^2*(a + b*ArcCos[c*x])*ArcTanh[E^((2*I)*ArcCos[c*x])])/d - ((I/2)*b*c^2*PolyLog[2, -E^((2*I)*ArcCos[c*x])])/d + ((I/2)*b*c^2*PolyLog[2, E^((2*I)*ArcCos[c*x])])/d","A",9,7,25,0.2800,1,"{4702, 4680, 4419, 4183, 2279, 2391, 264}"
8,1,180,0,0.2305066,"\int \frac{x^4 \left(a+b \cos ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Int[(x^4*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]","\frac{3 i b \text{PolyLog}\left(2,-e^{i \cos ^{-1}(c x)}\right)}{2 c^5 d^2}-\frac{3 i b \text{PolyLog}\left(2,e^{i \cos ^{-1}(c x)}\right)}{2 c^5 d^2}+\frac{x^3 \left(a+b \cos ^{-1}(c x)\right)}{2 c^2 d^2 \left(1-c^2 x^2\right)}+\frac{3 x \left(a+b \cos ^{-1}(c x)\right)}{2 c^4 d^2}-\frac{3 \tanh ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c^5 d^2}-\frac{b \sqrt{1-c^2 x^2}}{c^5 d^2}+\frac{b}{2 c^5 d^2 \sqrt{1-c^2 x^2}}","\frac{3 i b \text{PolyLog}\left(2,-e^{i \cos ^{-1}(c x)}\right)}{2 c^5 d^2}-\frac{3 i b \text{PolyLog}\left(2,e^{i \cos ^{-1}(c x)}\right)}{2 c^5 d^2}+\frac{x^3 \left(a+b \cos ^{-1}(c x)\right)}{2 c^2 d^2 \left(1-c^2 x^2\right)}+\frac{3 x \left(a+b \cos ^{-1}(c x)\right)}{2 c^4 d^2}-\frac{3 \tanh ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c^5 d^2}-\frac{b \sqrt{1-c^2 x^2}}{c^5 d^2}+\frac{b}{2 c^5 d^2 \sqrt{1-c^2 x^2}}",1,"b/(2*c^5*d^2*Sqrt[1 - c^2*x^2]) - (b*Sqrt[1 - c^2*x^2])/(c^5*d^2) + (3*x*(a + b*ArcCos[c*x]))/(2*c^4*d^2) + (x^3*(a + b*ArcCos[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) - (3*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/(c^5*d^2) + (((3*I)/2)*b*PolyLog[2, -E^(I*ArcCos[c*x])])/(c^5*d^2) - (((3*I)/2)*b*PolyLog[2, E^(I*ArcCos[c*x])])/(c^5*d^2)","A",12,9,25,0.3600,1,"{4704, 4716, 4658, 4183, 2279, 2391, 261, 266, 43}"
9,1,155,0,0.1867398,"\int \frac{x^3 \left(a+b \cos ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Int[(x^3*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]","-\frac{i b \text{PolyLog}\left(2,e^{2 i \cos ^{-1}(c x)}\right)}{2 c^4 d^2}+\frac{x^2 \left(a+b \cos ^{-1}(c x)\right)}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{i \left(a+b \cos ^{-1}(c x)\right)^2}{2 b c^4 d^2}+\frac{\log \left(1-e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c^4 d^2}+\frac{b x}{2 c^3 d^2 \sqrt{1-c^2 x^2}}-\frac{b \sin ^{-1}(c x)}{2 c^4 d^2}","-\frac{i b \text{PolyLog}\left(2,e^{2 i \cos ^{-1}(c x)}\right)}{2 c^4 d^2}+\frac{x^2 \left(a+b \cos ^{-1}(c x)\right)}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{i \left(a+b \cos ^{-1}(c x)\right)^2}{2 b c^4 d^2}+\frac{\log \left(1-e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c^4 d^2}+\frac{b x}{2 c^3 d^2 \sqrt{1-c^2 x^2}}-\frac{b \sin ^{-1}(c x)}{2 c^4 d^2}",1,"(b*x)/(2*c^3*d^2*Sqrt[1 - c^2*x^2]) + (x^2*(a + b*ArcCos[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) - ((I/2)*(a + b*ArcCos[c*x])^2)/(b*c^4*d^2) - (b*ArcSin[c*x])/(2*c^4*d^2) + ((a + b*ArcCos[c*x])*Log[1 - E^((2*I)*ArcCos[c*x])])/(c^4*d^2) - ((I/2)*b*PolyLog[2, E^((2*I)*ArcCos[c*x])])/(c^4*d^2)","A",8,8,25,0.3200,1,"{4704, 4676, 3717, 2190, 2279, 2391, 288, 216}"
10,1,136,0,0.129542,"\int \frac{x^2 \left(a+b \cos ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Int[(x^2*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]","\frac{i b \text{PolyLog}\left(2,-e^{i \cos ^{-1}(c x)}\right)}{2 c^3 d^2}-\frac{i b \text{PolyLog}\left(2,e^{i \cos ^{-1}(c x)}\right)}{2 c^3 d^2}+\frac{x \left(a+b \cos ^{-1}(c x)\right)}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{\tanh ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c^3 d^2}+\frac{b}{2 c^3 d^2 \sqrt{1-c^2 x^2}}","\frac{i b \text{PolyLog}\left(2,-e^{i \cos ^{-1}(c x)}\right)}{2 c^3 d^2}-\frac{i b \text{PolyLog}\left(2,e^{i \cos ^{-1}(c x)}\right)}{2 c^3 d^2}+\frac{x \left(a+b \cos ^{-1}(c x)\right)}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{\tanh ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c^3 d^2}+\frac{b}{2 c^3 d^2 \sqrt{1-c^2 x^2}}",1,"b/(2*c^3*d^2*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcCos[c*x]))/(2*c^2*d^2*(1 - c^2*x^2)) - ((a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/(c^3*d^2) + ((I/2)*b*PolyLog[2, -E^(I*ArcCos[c*x])])/(c^3*d^2) - ((I/2)*b*PolyLog[2, E^(I*ArcCos[c*x])])/(c^3*d^2)","A",8,6,25,0.2400,1,"{4704, 4658, 4183, 2279, 2391, 261}"
11,1,57,0,0.0461148,"\int \frac{x \left(a+b \cos ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Int[(x*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]","\frac{a+b \cos ^{-1}(c x)}{2 c^2 d^2 \left(1-c^2 x^2\right)}+\frac{b x}{2 c d^2 \sqrt{1-c^2 x^2}}","\frac{a+b \cos ^{-1}(c x)}{2 c^2 d^2 \left(1-c^2 x^2\right)}+\frac{b x}{2 c d^2 \sqrt{1-c^2 x^2}}",1,"(b*x)/(2*c*d^2*Sqrt[1 - c^2*x^2]) + (a + b*ArcCos[c*x])/(2*c^2*d^2*(1 - c^2*x^2))","A",2,2,23,0.08696,1,"{4678, 191}"
12,1,132,0,0.0915474,"\int \frac{a+b \cos ^{-1}(c x)}{\left(d-c^2 d x^2\right)^2} \, dx","Int[(a + b*ArcCos[c*x])/(d - c^2*d*x^2)^2,x]","-\frac{i b \text{PolyLog}\left(2,-e^{i \cos ^{-1}(c x)}\right)}{2 c d^2}+\frac{i b \text{PolyLog}\left(2,e^{i \cos ^{-1}(c x)}\right)}{2 c d^2}+\frac{x \left(a+b \cos ^{-1}(c x)\right)}{2 d^2 \left(1-c^2 x^2\right)}+\frac{\tanh ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c d^2}+\frac{b}{2 c d^2 \sqrt{1-c^2 x^2}}","-\frac{i b \text{PolyLog}\left(2,-e^{i \cos ^{-1}(c x)}\right)}{2 c d^2}+\frac{i b \text{PolyLog}\left(2,e^{i \cos ^{-1}(c x)}\right)}{2 c d^2}+\frac{x \left(a+b \cos ^{-1}(c x)\right)}{2 d^2 \left(1-c^2 x^2\right)}+\frac{\tanh ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{c d^2}+\frac{b}{2 c d^2 \sqrt{1-c^2 x^2}}",1,"b/(2*c*d^2*Sqrt[1 - c^2*x^2]) + (x*(a + b*ArcCos[c*x]))/(2*d^2*(1 - c^2*x^2)) + ((a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/(c*d^2) - ((I/2)*b*PolyLog[2, -E^(I*ArcCos[c*x])])/(c*d^2) + ((I/2)*b*PolyLog[2, E^(I*ArcCos[c*x])])/(c*d^2)","A",8,6,22,0.2727,1,"{4656, 4658, 4183, 2279, 2391, 261}"
13,1,122,0,0.1706443,"\int \frac{a+b \cos ^{-1}(c x)}{x \left(d-c^2 d x^2\right)^2} \, dx","Int[(a + b*ArcCos[c*x])/(x*(d - c^2*d*x^2)^2),x]","-\frac{i b \text{PolyLog}\left(2,-e^{2 i \cos ^{-1}(c x)}\right)}{2 d^2}+\frac{i b \text{PolyLog}\left(2,e^{2 i \cos ^{-1}(c x)}\right)}{2 d^2}+\frac{a+b \cos ^{-1}(c x)}{2 d^2 \left(1-c^2 x^2\right)}+\frac{2 \tanh ^{-1}\left(e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{d^2}+\frac{b c x}{2 d^2 \sqrt{1-c^2 x^2}}","-\frac{i b \text{PolyLog}\left(2,-e^{2 i \cos ^{-1}(c x)}\right)}{2 d^2}+\frac{i b \text{PolyLog}\left(2,e^{2 i \cos ^{-1}(c x)}\right)}{2 d^2}+\frac{a+b \cos ^{-1}(c x)}{2 d^2 \left(1-c^2 x^2\right)}+\frac{2 \tanh ^{-1}\left(e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{d^2}+\frac{b c x}{2 d^2 \sqrt{1-c^2 x^2}}",1,"(b*c*x)/(2*d^2*Sqrt[1 - c^2*x^2]) + (a + b*ArcCos[c*x])/(2*d^2*(1 - c^2*x^2)) + (2*(a + b*ArcCos[c*x])*ArcTanh[E^((2*I)*ArcCos[c*x])])/d^2 - ((I/2)*b*PolyLog[2, -E^((2*I)*ArcCos[c*x])])/d^2 + ((I/2)*b*PolyLog[2, E^((2*I)*ArcCos[c*x])])/d^2","A",9,7,25,0.2800,1,"{4706, 4680, 4419, 4183, 2279, 2391, 191}"
14,1,177,0,0.1818428,"\int \frac{a+b \cos ^{-1}(c x)}{x^2 \left(d-c^2 d x^2\right)^2} \, dx","Int[(a + b*ArcCos[c*x])/(x^2*(d - c^2*d*x^2)^2),x]","-\frac{3 i b c \text{PolyLog}\left(2,-e^{i \cos ^{-1}(c x)}\right)}{2 d^2}+\frac{3 i b c \text{PolyLog}\left(2,e^{i \cos ^{-1}(c x)}\right)}{2 d^2}+\frac{3 c^2 x \left(a+b \cos ^{-1}(c x)\right)}{2 d^2 \left(1-c^2 x^2\right)}-\frac{a+b \cos ^{-1}(c x)}{d^2 x \left(1-c^2 x^2\right)}+\frac{3 c \tanh ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{d^2}+\frac{b c}{2 d^2 \sqrt{1-c^2 x^2}}+\frac{b c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{d^2}","-\frac{3 i b c \text{PolyLog}\left(2,-e^{i \cos ^{-1}(c x)}\right)}{2 d^2}+\frac{3 i b c \text{PolyLog}\left(2,e^{i \cos ^{-1}(c x)}\right)}{2 d^2}+\frac{3 c^2 x \left(a+b \cos ^{-1}(c x)\right)}{2 d^2 \left(1-c^2 x^2\right)}-\frac{a+b \cos ^{-1}(c x)}{d^2 x \left(1-c^2 x^2\right)}+\frac{3 c \tanh ^{-1}\left(e^{i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{d^2}+\frac{b c}{2 d^2 \sqrt{1-c^2 x^2}}+\frac{b c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{d^2}",1,"(b*c)/(2*d^2*Sqrt[1 - c^2*x^2]) - (a + b*ArcCos[c*x])/(d^2*x*(1 - c^2*x^2)) + (3*c^2*x*(a + b*ArcCos[c*x]))/(2*d^2*(1 - c^2*x^2)) + (3*c*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/d^2 + (b*c*ArcTanh[Sqrt[1 - c^2*x^2]])/d^2 - (((3*I)/2)*b*c*PolyLog[2, -E^(I*ArcCos[c*x])])/d^2 + (((3*I)/2)*b*c*PolyLog[2, E^(I*ArcCos[c*x])])/d^2","A",13,11,25,0.4400,1,"{4702, 4656, 4658, 4183, 2279, 2391, 261, 266, 51, 63, 208}"
15,1,159,0,0.2569625,"\int \frac{a+b \cos ^{-1}(c x)}{x^3 \left(d-c^2 d x^2\right)^2} \, dx","Int[(a + b*ArcCos[c*x])/(x^3*(d - c^2*d*x^2)^2),x]","-\frac{i b c^2 \text{PolyLog}\left(2,-e^{2 i \cos ^{-1}(c x)}\right)}{d^2}+\frac{i b c^2 \text{PolyLog}\left(2,e^{2 i \cos ^{-1}(c x)}\right)}{d^2}+\frac{c^2 \left(a+b \cos ^{-1}(c x)\right)}{d^2 \left(1-c^2 x^2\right)}-\frac{a+b \cos ^{-1}(c x)}{2 d^2 x^2 \left(1-c^2 x^2\right)}+\frac{4 c^2 \tanh ^{-1}\left(e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{d^2}+\frac{b c}{2 d^2 x \sqrt{1-c^2 x^2}}","-\frac{i b c^2 \text{PolyLog}\left(2,-e^{2 i \cos ^{-1}(c x)}\right)}{d^2}+\frac{i b c^2 \text{PolyLog}\left(2,e^{2 i \cos ^{-1}(c x)}\right)}{d^2}+\frac{c^2 \left(a+b \cos ^{-1}(c x)\right)}{d^2 \left(1-c^2 x^2\right)}-\frac{a+b \cos ^{-1}(c x)}{2 d^2 x^2 \left(1-c^2 x^2\right)}+\frac{4 c^2 \tanh ^{-1}\left(e^{2 i \cos ^{-1}(c x)}\right) \left(a+b \cos ^{-1}(c x)\right)}{d^2}+\frac{b c}{2 d^2 x \sqrt{1-c^2 x^2}}",1,"(b*c)/(2*d^2*x*Sqrt[1 - c^2*x^2]) + (c^2*(a + b*ArcCos[c*x]))/(d^2*(1 - c^2*x^2)) - (a + b*ArcCos[c*x])/(2*d^2*x^2*(1 - c^2*x^2)) + (4*c^2*(a + b*ArcCos[c*x])*ArcTanh[E^((2*I)*ArcCos[c*x])])/d^2 - (I*b*c^2*PolyLog[2, -E^((2*I)*ArcCos[c*x])])/d^2 + (I*b*c^2*PolyLog[2, E^((2*I)*ArcCos[c*x])])/d^2","A",12,9,25,0.3600,1,"{4702, 4706, 4680, 4419, 4183, 2279, 2391, 191, 271}"
16,1,149,0,0.1187865,"\int x^3 \left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right) \, dx","Int[x^3*(d + e*x^2)*(a + b*ArcCos[c*x]),x]","\frac{1}{4} d x^4 \left(a+b \cos ^{-1}(c x)\right)+\frac{1}{6} e x^6 \left(a+b \cos ^{-1}(c x)\right)-\frac{b x^3 \sqrt{1-c^2 x^2} \left(9 c^2 d+5 e\right)}{144 c^3}-\frac{b x \sqrt{1-c^2 x^2} \left(9 c^2 d+5 e\right)}{96 c^5}+\frac{b \left(9 c^2 d+5 e\right) \sin ^{-1}(c x)}{96 c^6}-\frac{b e x^5 \sqrt{1-c^2 x^2}}{36 c}","\frac{1}{4} d x^4 \left(a+b \cos ^{-1}(c x)\right)+\frac{1}{6} e x^6 \left(a+b \cos ^{-1}(c x)\right)-\frac{b x^3 \sqrt{1-c^2 x^2} \left(9 c^2 d+5 e\right)}{144 c^3}-\frac{b x \sqrt{1-c^2 x^2} \left(9 c^2 d+5 e\right)}{96 c^5}+\frac{b \left(9 c^2 d+5 e\right) \sin ^{-1}(c x)}{96 c^6}-\frac{b e x^5 \sqrt{1-c^2 x^2}}{36 c}",1,"-(b*(9*c^2*d + 5*e)*x*Sqrt[1 - c^2*x^2])/(96*c^5) - (b*(9*c^2*d + 5*e)*x^3*Sqrt[1 - c^2*x^2])/(144*c^3) - (b*e*x^5*Sqrt[1 - c^2*x^2])/(36*c) + (d*x^4*(a + b*ArcCos[c*x]))/4 + (e*x^6*(a + b*ArcCos[c*x]))/6 + (b*(9*c^2*d + 5*e)*ArcSin[c*x])/(96*c^6)","A",6,6,19,0.3158,1,"{14, 4732, 12, 459, 321, 216}"
17,1,120,0,0.1206404,"\int x^2 \left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right) \, dx","Int[x^2*(d + e*x^2)*(a + b*ArcCos[c*x]),x]","\frac{1}{3} d x^3 \left(a+b \cos ^{-1}(c x)\right)+\frac{1}{5} e x^5 \left(a+b \cos ^{-1}(c x)\right)+\frac{b \left(1-c^2 x^2\right)^{3/2} \left(5 c^2 d+6 e\right)}{45 c^5}-\frac{b \sqrt{1-c^2 x^2} \left(5 c^2 d+3 e\right)}{15 c^5}-\frac{b e \left(1-c^2 x^2\right)^{5/2}}{25 c^5}","\frac{1}{3} d x^3 \left(a+b \cos ^{-1}(c x)\right)+\frac{1}{5} e x^5 \left(a+b \cos ^{-1}(c x)\right)+\frac{b \left(1-c^2 x^2\right)^{3/2} \left(5 c^2 d+6 e\right)}{45 c^5}-\frac{b \sqrt{1-c^2 x^2} \left(5 c^2 d+3 e\right)}{15 c^5}-\frac{b e \left(1-c^2 x^2\right)^{5/2}}{25 c^5}",1,"-(b*(5*c^2*d + 3*e)*Sqrt[1 - c^2*x^2])/(15*c^5) + (b*(5*c^2*d + 6*e)*(1 - c^2*x^2)^(3/2))/(45*c^5) - (b*e*(1 - c^2*x^2)^(5/2))/(25*c^5) + (d*x^3*(a + b*ArcCos[c*x]))/3 + (e*x^5*(a + b*ArcCos[c*x]))/5","A",5,5,19,0.2632,1,"{14, 4732, 12, 446, 77}"
18,1,122,0,0.0858386,"\int x \left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right) \, dx","Int[x*(d + e*x^2)*(a + b*ArcCos[c*x]),x]","\frac{\left(d+e x^2\right)^2 \left(a+b \cos ^{-1}(c x)\right)}{4 e}+\frac{b \left(8 c^4 d^2+8 c^2 d e+3 e^2\right) \sin ^{-1}(c x)}{32 c^4 e}-\frac{b x \sqrt{1-c^2 x^2} \left(d+e x^2\right)}{16 c}-\frac{3 b x \sqrt{1-c^2 x^2} \left(2 c^2 d+e\right)}{32 c^3}","\frac{\left(d+e x^2\right)^2 \left(a+b \cos ^{-1}(c x)\right)}{4 e}+\frac{b \left(8 c^4 d^2+8 c^2 d e+3 e^2\right) \sin ^{-1}(c x)}{32 c^4 e}-\frac{b x \sqrt{1-c^2 x^2} \left(d+e x^2\right)}{16 c}-\frac{3 b x \sqrt{1-c^2 x^2} \left(2 c^2 d+e\right)}{32 c^3}",1,"(-3*b*(2*c^2*d + e)*x*Sqrt[1 - c^2*x^2])/(32*c^3) - (b*x*Sqrt[1 - c^2*x^2]*(d + e*x^2))/(16*c) + ((d + e*x^2)^2*(a + b*ArcCos[c*x]))/(4*e) + (b*(8*c^4*d^2 + 8*c^2*d*e + 3*e^2)*ArcSin[c*x])/(32*c^4*e)","A",4,4,17,0.2353,1,"{4730, 416, 388, 216}"
19,1,81,0,0.0674421,"\int \left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right) \, dx","Int[(d + e*x^2)*(a + b*ArcCos[c*x]),x]","d x \left(a+b \cos ^{-1}(c x)\right)+\frac{1}{3} e x^3 \left(a+b \cos ^{-1}(c x)\right)-\frac{b \sqrt{1-c^2 x^2} \left(3 c^2 d+e\right)}{3 c^3}+\frac{b e \left(1-c^2 x^2\right)^{3/2}}{9 c^3}","d x \left(a+b \cos ^{-1}(c x)\right)+\frac{1}{3} e x^3 \left(a+b \cos ^{-1}(c x)\right)-\frac{b \sqrt{1-c^2 x^2} \left(3 c^2 d+e\right)}{3 c^3}+\frac{b e \left(1-c^2 x^2\right)^{3/2}}{9 c^3}",1,"-(b*(3*c^2*d + e)*Sqrt[1 - c^2*x^2])/(3*c^3) + (b*e*(1 - c^2*x^2)^(3/2))/(9*c^3) + d*x*(a + b*ArcCos[c*x]) + (e*x^3*(a + b*ArcCos[c*x]))/3","A",4,3,16,0.1875,1,"{4666, 444, 43}"
20,1,132,0,0.2390032,"\int \frac{\left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{x} \, dx","Int[((d + e*x^2)*(a + b*ArcCos[c*x]))/x,x]","\frac{1}{2} i b d \text{PolyLog}\left(2,e^{2 i \sin ^{-1}(c x)}\right)+d \log (x) \left(a+b \cos ^{-1}(c x)\right)+\frac{1}{2} e x^2 \left(a+b \cos ^{-1}(c x)\right)-\frac{b e x \sqrt{1-c^2 x^2}}{4 c}+\frac{b e \sin ^{-1}(c x)}{4 c^2}+\frac{1}{2} i b d \sin ^{-1}(c x)^2-b d \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+b d \log (x) \sin ^{-1}(c x)","\frac{1}{2} i b d \text{PolyLog}\left(2,e^{2 i \sin ^{-1}(c x)}\right)+d \log (x) \left(a+b \cos ^{-1}(c x)\right)+\frac{1}{2} e x^2 \left(a+b \cos ^{-1}(c x)\right)-\frac{b e x \sqrt{1-c^2 x^2}}{4 c}+\frac{b e \sin ^{-1}(c x)}{4 c^2}+\frac{1}{2} i b d \sin ^{-1}(c x)^2-b d \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+b d \log (x) \sin ^{-1}(c x)",1,"-(b*e*x*Sqrt[1 - c^2*x^2])/(4*c) + (e*x^2*(a + b*ArcCos[c*x]))/2 + (b*e*ArcSin[c*x])/(4*c^2) + (I/2)*b*d*ArcSin[c*x]^2 - b*d*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + d*(a + b*ArcCos[c*x])*Log[x] + b*d*ArcSin[c*x]*Log[x] + (I/2)*b*d*PolyLog[2, E^((2*I)*ArcSin[c*x])]","A",12,12,19,0.6316,1,"{14, 4732, 12, 6742, 321, 216, 2326, 4625, 3717, 2190, 2279, 2391}"
21,1,66,0,0.074847,"\int \frac{\left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{x^2} \, dx","Int[((d + e*x^2)*(a + b*ArcCos[c*x]))/x^2,x]","-\frac{d \left(a+b \cos ^{-1}(c x)\right)}{x}+e x \left(a+b \cos ^{-1}(c x)\right)+b c d \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-\frac{b e \sqrt{1-c^2 x^2}}{c}","-\frac{d \left(a+b \cos ^{-1}(c x)\right)}{x}+e x \left(a+b \cos ^{-1}(c x)\right)+b c d \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-\frac{b e \sqrt{1-c^2 x^2}}{c}",1,"-((b*e*Sqrt[1 - c^2*x^2])/c) - (d*(a + b*ArcCos[c*x]))/x + e*x*(a + b*ArcCos[c*x]) + b*c*d*ArcTanh[Sqrt[1 - c^2*x^2]]","A",5,6,19,0.3158,1,"{14, 4732, 446, 80, 63, 208}"
22,1,119,0,0.2207225,"\int \frac{\left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{x^3} \, dx","Int[((d + e*x^2)*(a + b*ArcCos[c*x]))/x^3,x]","\frac{1}{2} i b e \text{PolyLog}\left(2,e^{2 i \sin ^{-1}(c x)}\right)-\frac{d \left(a+b \cos ^{-1}(c x)\right)}{2 x^2}+e \log (x) \left(a+b \cos ^{-1}(c x)\right)+\frac{b c d \sqrt{1-c^2 x^2}}{2 x}+\frac{1}{2} i b e \sin ^{-1}(c x)^2-b e \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+b e \log (x) \sin ^{-1}(c x)","\frac{1}{2} i b e \text{PolyLog}\left(2,e^{2 i \sin ^{-1}(c x)}\right)-\frac{d \left(a+b \cos ^{-1}(c x)\right)}{2 x^2}+e \log (x) \left(a+b \cos ^{-1}(c x)\right)+\frac{b c d \sqrt{1-c^2 x^2}}{2 x}+\frac{1}{2} i b e \sin ^{-1}(c x)^2-b e \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+b e \log (x) \sin ^{-1}(c x)",1,"(b*c*d*Sqrt[1 - c^2*x^2])/(2*x) - (d*(a + b*ArcCos[c*x]))/(2*x^2) + (I/2)*b*e*ArcSin[c*x]^2 - b*e*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + e*(a + b*ArcCos[c*x])*Log[x] + b*e*ArcSin[c*x]*Log[x] + (I/2)*b*e*PolyLog[2, E^((2*I)*ArcSin[c*x])]","A",10,10,19,0.5263,1,"{14, 4732, 6742, 264, 2326, 4625, 3717, 2190, 2279, 2391}"
23,1,85,0,0.0909705,"\int \frac{\left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{x^4} \, dx","Int[((d + e*x^2)*(a + b*ArcCos[c*x]))/x^4,x]","-\frac{d \left(a+b \cos ^{-1}(c x)\right)}{3 x^3}-\frac{e \left(a+b \cos ^{-1}(c x)\right)}{x}+\frac{1}{6} b c \left(c^2 d+6 e\right) \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+\frac{b c d \sqrt{1-c^2 x^2}}{6 x^2}","-\frac{d \left(a+b \cos ^{-1}(c x)\right)}{3 x^3}-\frac{e \left(a+b \cos ^{-1}(c x)\right)}{x}+\frac{1}{6} b c \left(c^2 d+6 e\right) \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+\frac{b c d \sqrt{1-c^2 x^2}}{6 x^2}",1,"(b*c*d*Sqrt[1 - c^2*x^2])/(6*x^2) - (d*(a + b*ArcCos[c*x]))/(3*x^3) - (e*(a + b*ArcCos[c*x]))/x + (b*c*(c^2*d + 6*e)*ArcTanh[Sqrt[1 - c^2*x^2]])/6","A",6,7,19,0.3684,1,"{14, 4732, 12, 446, 78, 63, 208}"
24,1,135,0,0.1272575,"\int \left(c+d x^2\right)^2 \cos ^{-1}(a x) \, dx","Int[(c + d*x^2)^2*ArcCos[a*x],x]","-\frac{\sqrt{1-a^2 x^2} \left(15 a^4 c^2+10 a^2 c d+3 d^2\right)}{15 a^5}+\frac{2 d \left(1-a^2 x^2\right)^{3/2} \left(5 a^2 c+3 d\right)}{45 a^5}-\frac{d^2 \left(1-a^2 x^2\right)^{5/2}}{25 a^5}+c^2 x \cos ^{-1}(a x)+\frac{2}{3} c d x^3 \cos ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cos ^{-1}(a x)","-\frac{\sqrt{1-a^2 x^2} \left(15 a^4 c^2+10 a^2 c d+3 d^2\right)}{15 a^5}+\frac{2 d \left(1-a^2 x^2\right)^{3/2} \left(5 a^2 c+3 d\right)}{45 a^5}-\frac{d^2 \left(1-a^2 x^2\right)^{5/2}}{25 a^5}+c^2 x \cos ^{-1}(a x)+\frac{2}{3} c d x^3 \cos ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cos ^{-1}(a x)",1,"-((15*a^4*c^2 + 10*a^2*c*d + 3*d^2)*Sqrt[1 - a^2*x^2])/(15*a^5) + (2*d*(5*a^2*c + 3*d)*(1 - a^2*x^2)^(3/2))/(45*a^5) - (d^2*(1 - a^2*x^2)^(5/2))/(25*a^5) + c^2*x*ArcCos[a*x] + (2*c*d*x^3*ArcCos[a*x])/3 + (d^2*x^5*ArcCos[a*x])/5","A",5,5,14,0.3571,1,"{194, 4666, 12, 1247, 698}"
25,1,205,0,0.2437082,"\int \left(c+d x^2\right)^3 \cos ^{-1}(a x) \, dx","Int[(c + d*x^2)^3*ArcCos[a*x],x]","\frac{d \left(1-a^2 x^2\right)^{3/2} \left(35 a^4 c^2+42 a^2 c d+15 d^2\right)}{105 a^7}-\frac{\sqrt{1-a^2 x^2} \left(35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2+5 d^3\right)}{35 a^7}-\frac{3 d^2 \left(1-a^2 x^2\right)^{5/2} \left(7 a^2 c+5 d\right)}{175 a^7}+\frac{d^3 \left(1-a^2 x^2\right)^{7/2}}{49 a^7}+c^2 d x^3 \cos ^{-1}(a x)+c^3 x \cos ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cos ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cos ^{-1}(a x)","\frac{d \left(1-a^2 x^2\right)^{3/2} \left(35 a^4 c^2+42 a^2 c d+15 d^2\right)}{105 a^7}-\frac{\sqrt{1-a^2 x^2} \left(35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2+5 d^3\right)}{35 a^7}-\frac{3 d^2 \left(1-a^2 x^2\right)^{5/2} \left(7 a^2 c+5 d\right)}{175 a^7}+\frac{d^3 \left(1-a^2 x^2\right)^{7/2}}{49 a^7}+c^2 d x^3 \cos ^{-1}(a x)+c^3 x \cos ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cos ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cos ^{-1}(a x)",1,"-((35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*Sqrt[1 - a^2*x^2])/(35*a^7) + (d*(35*a^4*c^2 + 42*a^2*c*d + 15*d^2)*(1 - a^2*x^2)^(3/2))/(105*a^7) - (3*d^2*(7*a^2*c + 5*d)*(1 - a^2*x^2)^(5/2))/(175*a^7) + (d^3*(1 - a^2*x^2)^(7/2))/(49*a^7) + c^3*x*ArcCos[a*x] + c^2*d*x^3*ArcCos[a*x] + (3*c*d^2*x^5*ArcCos[a*x])/5 + (d^3*x^7*ArcCos[a*x])/7","A",5,5,14,0.3571,1,"{194, 4666, 12, 1799, 1850}"
26,1,292,0,0.3260992,"\int \left(c+d x^2\right)^4 \cos ^{-1}(a x) \, dx","Int[(c + d*x^2)^4*ArcCos[a*x],x]","-\frac{2 d^2 \left(1-a^2 x^2\right)^{5/2} \left(63 a^4 c^2+90 a^2 c d+35 d^2\right)}{525 a^9}+\frac{4 d \left(1-a^2 x^2\right)^{3/2} \left(189 a^4 c^2 d+105 a^6 c^3+135 a^2 c d^2+35 d^3\right)}{945 a^9}-\frac{\sqrt{1-a^2 x^2} \left(378 a^4 c^2 d^2+420 a^6 c^3 d+315 a^8 c^4+180 a^2 c d^3+35 d^4\right)}{315 a^9}+\frac{4 d^3 \left(1-a^2 x^2\right)^{7/2} \left(9 a^2 c+7 d\right)}{441 a^9}-\frac{d^4 \left(1-a^2 x^2\right)^{9/2}}{81 a^9}+\frac{6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cos ^{-1}(a x)+c^4 x \cos ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cos ^{-1}(a x)","-\frac{2 d^2 \left(1-a^2 x^2\right)^{5/2} \left(63 a^4 c^2+90 a^2 c d+35 d^2\right)}{525 a^9}+\frac{4 d \left(1-a^2 x^2\right)^{3/2} \left(189 a^4 c^2 d+105 a^6 c^3+135 a^2 c d^2+35 d^3\right)}{945 a^9}-\frac{\sqrt{1-a^2 x^2} \left(378 a^4 c^2 d^2+420 a^6 c^3 d+315 a^8 c^4+180 a^2 c d^3+35 d^4\right)}{315 a^9}+\frac{4 d^3 \left(1-a^2 x^2\right)^{7/2} \left(9 a^2 c+7 d\right)}{441 a^9}-\frac{d^4 \left(1-a^2 x^2\right)^{9/2}}{81 a^9}+\frac{6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cos ^{-1}(a x)+c^4 x \cos ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cos ^{-1}(a x)",1,"-((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)*Sqrt[1 - a^2*x^2])/(315*a^9) + (4*d*(105*a^6*c^3 + 189*a^4*c^2*d + 135*a^2*c*d^2 + 35*d^3)*(1 - a^2*x^2)^(3/2))/(945*a^9) - (2*d^2*(63*a^4*c^2 + 90*a^2*c*d + 35*d^2)*(1 - a^2*x^2)^(5/2))/(525*a^9) + (4*d^3*(9*a^2*c + 7*d)*(1 - a^2*x^2)^(7/2))/(441*a^9) - (d^4*(1 - a^2*x^2)^(9/2))/(81*a^9) + c^4*x*ArcCos[a*x] + (4*c^3*d*x^3*ArcCos[a*x])/3 + (6*c^2*d^2*x^5*ArcCos[a*x])/5 + (4*c*d^3*x^7*ArcCos[a*x])/7 + (d^4*x^9*ArcCos[a*x])/9","A",5,5,14,0.3571,1,"{194, 4666, 12, 1799, 1850}"
27,1,521,0,0.8095977,"\int \frac{\cos ^{-1}(a x)}{c+d x^2} \, dx","Int[ArcCos[a*x]/(c + d*x^2),x]","\frac{i \text{PolyLog}\left(2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}-\frac{i \text{PolyLog}\left(2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}+\frac{i \text{PolyLog}\left(2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}-\frac{i \text{PolyLog}\left(2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left(1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left(1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left(1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left(1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}","\frac{i \text{PolyLog}\left(2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}-\frac{i \text{PolyLog}\left(2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}+\frac{i \text{PolyLog}\left(2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}-\frac{i \text{PolyLog}\left(2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left(1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left(1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left(1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left(1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{2 \sqrt{-c} \sqrt{d}}",1,"(ArcCos[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) - (ArcCos[a*x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) + (ArcCos[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) - (ArcCos[a*x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(2*Sqrt[-c]*Sqrt[d]) + ((I/2)*PolyLog[2, -((Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d]))])/(Sqrt[-c]*Sqrt[d]) - ((I/2)*PolyLog[2, (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(Sqrt[-c]*Sqrt[d]) + ((I/2)*PolyLog[2, -((Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d]))])/(Sqrt[-c]*Sqrt[d]) - ((I/2)*PolyLog[2, (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(Sqrt[-c]*Sqrt[d])","A",18,6,14,0.4286,1,"{4668, 4742, 4522, 2190, 2279, 2391}"
28,1,727,0,1.0690591,"\int \frac{\cos ^{-1}(a x)}{\left(c+d x^2\right)^2} \, dx","Int[ArcCos[a*x]/(c + d*x^2)^2,x]","-\frac{i \text{PolyLog}\left(2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}+\frac{i \text{PolyLog}\left(2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}-\frac{i \text{PolyLog}\left(2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}+\frac{i \text{PolyLog}\left(2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{d}-a^2 \sqrt{-c} x}{\sqrt{1-a^2 x^2} \sqrt{a^2 c+d}}\right)}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{a \tanh ^{-1}\left(\frac{a^2 \sqrt{-c} x+\sqrt{d}}{\sqrt{1-a^2 x^2} \sqrt{a^2 c+d}}\right)}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{\cos ^{-1}(a x) \log \left(1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left(1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left(1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left(1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left(\sqrt{-c}-\sqrt{d} x\right)}+\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left(\sqrt{-c}+\sqrt{d} x\right)}","-\frac{i \text{PolyLog}\left(2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}+\frac{i \text{PolyLog}\left(2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}-\frac{i \text{PolyLog}\left(2,-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}+\frac{i \text{PolyLog}\left(2,\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{d}-a^2 \sqrt{-c} x}{\sqrt{1-a^2 x^2} \sqrt{a^2 c+d}}\right)}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{a \tanh ^{-1}\left(\frac{a^2 \sqrt{-c} x+\sqrt{d}}{\sqrt{1-a^2 x^2} \sqrt{a^2 c+d}}\right)}{4 c \sqrt{d} \sqrt{a^2 c+d}}-\frac{\cos ^{-1}(a x) \log \left(1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left(1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cos ^{-1}(a x) \log \left(1-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}+\frac{\cos ^{-1}(a x) \log \left(1+\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}+i \sqrt{a^2 c+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}-\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left(\sqrt{-c}-\sqrt{d} x\right)}+\frac{\cos ^{-1}(a x)}{4 c \sqrt{d} \left(\sqrt{-c}+\sqrt{d} x\right)}",1,"-ArcCos[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] - Sqrt[d]*x)) + ArcCos[a*x]/(4*c*Sqrt[d]*(Sqrt[-c] + Sqrt[d]*x)) - (a*ArcTanh[(Sqrt[d] - a^2*Sqrt[-c]*x)/(Sqrt[a^2*c + d]*Sqrt[1 - a^2*x^2])])/(4*c*Sqrt[d]*Sqrt[a^2*c + d]) - (a*ArcTanh[(Sqrt[d] + a^2*Sqrt[-c]*x)/(Sqrt[a^2*c + d]*Sqrt[1 - a^2*x^2])])/(4*c*Sqrt[d]*Sqrt[a^2*c + d]) - (ArcCos[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) + (ArcCos[a*x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) - (ArcCos[a*x]*Log[1 - (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) + (ArcCos[a*x]*Log[1 + (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/(4*(-c)^(3/2)*Sqrt[d]) - ((I/4)*PolyLog[2, -((Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d]))])/((-c)^(3/2)*Sqrt[d]) + ((I/4)*PolyLog[2, (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] - I*Sqrt[a^2*c + d])])/((-c)^(3/2)*Sqrt[d]) - ((I/4)*PolyLog[2, -((Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d]))])/((-c)^(3/2)*Sqrt[d]) + ((I/4)*PolyLog[2, (Sqrt[d]*E^(I*ArcCos[a*x]))/(a*Sqrt[-c] + I*Sqrt[a^2*c + d])])/((-c)^(3/2)*Sqrt[d])","A",26,9,14,0.6429,1,"{4668, 4744, 725, 206, 4742, 4522, 2190, 2279, 2391}"
29,0,0,0,0.017767,"\int \sqrt{c+d x^2} \cos ^{-1}(a x) \, dx","Int[Sqrt[c + d*x^2]*ArcCos[a*x],x]","\int \sqrt{c+d x^2} \cos ^{-1}(a x) \, dx","\text{Int}\left(\cos ^{-1}(a x) \sqrt{c+d x^2},x\right)",0,"Defer[Int][Sqrt[c + d*x^2]*ArcCos[a*x], x]","A",0,0,0,0,-1,"{}"
30,0,0,0,0.018808,"\int \frac{\cos ^{-1}(a x)}{\sqrt{c+d x^2}} \, dx","Int[ArcCos[a*x]/Sqrt[c + d*x^2],x]","\int \frac{\cos ^{-1}(a x)}{\sqrt{c+d x^2}} \, dx","\text{Int}\left(\frac{\cos ^{-1}(a x)}{\sqrt{c+d x^2}},x\right)",0,"Defer[Int][ArcCos[a*x]/Sqrt[c + d*x^2], x]","A",0,0,0,0,-1,"{}"
31,1,66,0,0.0964348,"\int \frac{\cos ^{-1}(a x)}{\left(c+d x^2\right)^{3/2}} \, dx","Int[ArcCos[a*x]/(c + d*x^2)^(3/2),x]","\frac{x \cos ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-a^2 x^2}}{a \sqrt{c+d x^2}}\right)}{c \sqrt{d}}","\frac{x \cos ^{-1}(a x)}{c \sqrt{c+d x^2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-a^2 x^2}}{a \sqrt{c+d x^2}}\right)}{c \sqrt{d}}",1,"(x*ArcCos[a*x])/(c*Sqrt[c + d*x^2]) - ArcTan[(Sqrt[d]*Sqrt[1 - a^2*x^2])/(a*Sqrt[c + d*x^2])]/(c*Sqrt[d])","A",6,7,16,0.4375,1,"{191, 4666, 12, 444, 63, 217, 203}"
32,1,136,0,0.1533414,"\int \frac{\cos ^{-1}(a x)}{\left(c+d x^2\right)^{5/2}} \, dx","Int[ArcCos[a*x]/(c + d*x^2)^(5/2),x]","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-a^2 x^2}}{a \sqrt{c+d x^2}}\right)}{3 c^2 \sqrt{d}}-\frac{a \sqrt{1-a^2 x^2}}{3 c \left(a^2 c+d\right) \sqrt{c+d x^2}}+\frac{2 x \cos ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{x \cos ^{-1}(a x)}{3 c \left(c+d x^2\right)^{3/2}}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-a^2 x^2}}{a \sqrt{c+d x^2}}\right)}{3 c^2 \sqrt{d}}-\frac{a \sqrt{1-a^2 x^2}}{3 c \left(a^2 c+d\right) \sqrt{c+d x^2}}+\frac{2 x \cos ^{-1}(a x)}{3 c^2 \sqrt{c+d x^2}}+\frac{x \cos ^{-1}(a x)}{3 c \left(c+d x^2\right)^{3/2}}",1,"-(a*Sqrt[1 - a^2*x^2])/(3*c*(a^2*c + d)*Sqrt[c + d*x^2]) + (x*ArcCos[a*x])/(3*c*(c + d*x^2)^(3/2)) + (2*x*ArcCos[a*x])/(3*c^2*Sqrt[c + d*x^2]) - (2*ArcTan[(Sqrt[d]*Sqrt[1 - a^2*x^2])/(a*Sqrt[c + d*x^2])])/(3*c^2*Sqrt[d])","A",7,9,16,0.5625,1,"{192, 191, 4666, 12, 571, 78, 63, 217, 203}"
33,1,211,0,0.8484325,"\int \frac{\cos ^{-1}(a x)}{\left(c+d x^2\right)^{7/2}} \, dx","Int[ArcCos[a*x]/(c + d*x^2)^(7/2),x]","-\frac{2 a \sqrt{1-a^2 x^2} \left(3 a^2 c+2 d\right)}{15 c^2 \left(a^2 c+d\right)^2 \sqrt{c+d x^2}}-\frac{8 \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-a^2 x^2}}{a \sqrt{c+d x^2}}\right)}{15 c^3 \sqrt{d}}-\frac{a \sqrt{1-a^2 x^2}}{15 c \left(a^2 c+d\right) \left(c+d x^2\right)^{3/2}}+\frac{8 x \cos ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}+\frac{4 x \cos ^{-1}(a x)}{15 c^2 \left(c+d x^2\right)^{3/2}}+\frac{x \cos ^{-1}(a x)}{5 c \left(c+d x^2\right)^{5/2}}","-\frac{2 a \sqrt{1-a^2 x^2} \left(3 a^2 c+2 d\right)}{15 c^2 \left(a^2 c+d\right)^2 \sqrt{c+d x^2}}-\frac{8 \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{1-a^2 x^2}}{a \sqrt{c+d x^2}}\right)}{15 c^3 \sqrt{d}}-\frac{a \sqrt{1-a^2 x^2}}{15 c \left(a^2 c+d\right) \left(c+d x^2\right)^{3/2}}+\frac{8 x \cos ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}+\frac{4 x \cos ^{-1}(a x)}{15 c^2 \left(c+d x^2\right)^{3/2}}+\frac{x \cos ^{-1}(a x)}{5 c \left(c+d x^2\right)^{5/2}}",1,"-(a*Sqrt[1 - a^2*x^2])/(15*c*(a^2*c + d)*(c + d*x^2)^(3/2)) - (2*a*(3*a^2*c + 2*d)*Sqrt[1 - a^2*x^2])/(15*c^2*(a^2*c + d)^2*Sqrt[c + d*x^2]) + (x*ArcCos[a*x])/(5*c*(c + d*x^2)^(5/2)) + (4*x*ArcCos[a*x])/(15*c^2*(c + d*x^2)^(3/2)) + (8*x*ArcCos[a*x])/(15*c^3*Sqrt[c + d*x^2]) - (8*ArcTan[(Sqrt[d]*Sqrt[1 - a^2*x^2])/(a*Sqrt[c + d*x^2])])/(15*c^3*Sqrt[d])","A",8,10,16,0.6250,1,"{192, 191, 4666, 12, 6715, 949, 78, 63, 217, 203}"