1,1,161,144,0.2530176,"\int \frac{x^3 \left(a+b \cos ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Integrate[(x^3*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2),x]","-\frac{2 a c^2 x^2+2 a \log \left(1-c^2 x^2\right)-b c x \sqrt{1-c^2 x^2}+2 b c^2 x^2 \cos ^{-1}(c x)-4 i b \text{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)-4 i b \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)+b \sin ^{-1}(c x)-2 i b \cos ^{-1}(c x)^2+4 b \cos ^{-1}(c x) \log \left(1-e^{i \cos ^{-1}(c x)}\right)+4 b \cos ^{-1}(c x) \log \left(1+e^{i \cos ^{-1}(c x)}\right)}{4 c^4 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{x^2 \left(a+b \cos ^{-1}(c x)\right)}{2 c^2 d}+\frac{i b \text{Li}_2\left(e^{2 i \cos ^{-1}(c x)}\right)}{2 c^4 d}-\frac{b \sin ^{-1}(c x)}{4 c^4 d}+\frac{b x \sqrt{1-c^2 x^2}}{4 c^3 d}",1,"-1/4*(2*a*c^2*x^2 - b*c*x*Sqrt[1 - c^2*x^2] + 2*b*c^2*x^2*ArcCos[c*x] - (2*I)*b*ArcCos[c*x]^2 + b*ArcSin[c*x] + 4*b*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] + 4*b*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])] + 2*a*Log[1 - c^2*x^2] - (4*I)*b*PolyLog[2, -E^(I*ArcCos[c*x])] - (4*I)*b*PolyLog[2, E^(I*ArcCos[c*x])])/(c^4*d)","A",0
2,1,138,115,0.17982,"\int \frac{x^2 \left(a+b \cos ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Integrate[(x^2*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2),x]","-\frac{2 a c x+a \log (1-c x)-a \log (c x+1)-2 b \sqrt{1-c^2 x^2}+2 i b \text{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)-2 i b \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)+2 b c x \cos ^{-1}(c x)+2 b \cos ^{-1}(c x) \log \left(1-e^{i \cos ^{-1}(c x)}\right)-2 b \cos ^{-1}(c x) \log \left(1+e^{i \cos ^{-1}(c x)}\right)}{2 c^3 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{x \left(a+b \cos ^{-1}(c x)\right)}{c^2 d}-\frac{i b \text{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)}{c^3 d}+\frac{i b \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)}{c^3 d}+\frac{b \sqrt{1-c^2 x^2}}{c^3 d}",1,"-1/2*(2*a*c*x - 2*b*Sqrt[1 - c^2*x^2] + 2*b*c*x*ArcCos[c*x] + 2*b*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] - 2*b*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])] + a*Log[1 - c*x] - a*Log[1 + c*x] + (2*I)*b*PolyLog[2, -E^(I*ArcCos[c*x])] - (2*I)*b*PolyLog[2, E^(I*ArcCos[c*x])])/(c^3*d)","A",0
3,1,115,82,0.1067954,"\int \frac{x \left(a+b \cos ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Integrate[(x*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2),x]","\frac{i \left(i a \log \left(1-c^2 x^2\right)+2 b \text{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)+2 b \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)+b \cos ^{-1}(c x)^2+2 i b \cos ^{-1}(c x) \log \left(1-e^{i \cos ^{-1}(c x)}\right)+2 i b \cos ^{-1}(c x) \log \left(1+e^{i \cos ^{-1}(c x)}\right)\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{Li}_2\left(e^{2 i \cos ^{-1}(c x)}\right)}{2 c^2 d}",1,"((I/2)*(b*ArcCos[c*x]^2 + (2*I)*b*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] + (2*I)*b*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])] + I*a*Log[1 - c^2*x^2] + 2*b*PolyLog[2, -E^(I*ArcCos[c*x])] + 2*b*PolyLog[2, E^(I*ArcCos[c*x])]))/(c^2*d)","A",0
4,1,107,76,0.0913319,"\int \frac{a+b \cos ^{-1}(c x)}{d-c^2 d x^2} \, dx","Integrate[(a + b*ArcCos[c*x])/(d - c^2*d*x^2),x]","\frac{-a \log (1-c x)+a \log (c x+1)-2 i b \text{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)+2 i b \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)-2 b \cos ^{-1}(c x) \log \left(1-e^{i \cos ^{-1}(c x)}\right)+2 b \cos ^{-1}(c x) \log \left(1+e^{i \cos ^{-1}(c x)}\right)}{2 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{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)}{c d}+\frac{i b \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)}{c d}",1,"(-2*b*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] + 2*b*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])] - a*Log[1 - c*x] + a*Log[1 + c*x] - (2*I)*b*PolyLog[2, -E^(I*ArcCos[c*x])] + (2*I)*b*PolyLog[2, E^(I*ArcCos[c*x])])/(2*c*d)","A",0
5,1,104,71,0.1775299,"\int \frac{a+b \cos ^{-1}(c x)}{x \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcCos[c*x])/(x*(d - c^2*d*x^2)),x]","-\frac{a \log \left(1-c^2 x^2\right)-2 a \log (x)+i b \text{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right)-i b \text{Li}_2\left(e^{2 i \cos ^{-1}(c x)}\right)+2 b \cos ^{-1}(c x) \log \left(1-e^{2 i \cos ^{-1}(c x)}\right)-2 b \cos ^{-1}(c x) \log \left(1+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{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right)}{2 d}+\frac{i b \text{Li}_2\left(e^{2 i \cos ^{-1}(c x)}\right)}{2 d}",1,"-1/2*(2*b*ArcCos[c*x]*Log[1 - E^((2*I)*ArcCos[c*x])] - 2*b*ArcCos[c*x]*Log[1 + E^((2*I)*ArcCos[c*x])] - 2*a*Log[x] + a*Log[1 - c^2*x^2] + I*b*PolyLog[2, -E^((2*I)*ArcCos[c*x])] - I*b*PolyLog[2, E^((2*I)*ArcCos[c*x])])/d","A",0
6,1,158,107,0.2382058,"\int \frac{a+b \cos ^{-1}(c x)}{x^2 \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcCos[c*x])/(x^2*(d - c^2*d*x^2)),x]","-\frac{a c x \log (1-c x)-a c x \log (c x+1)+2 a-2 b c x \log \left(\sqrt{1-c^2 x^2}+1\right)+2 i b c x \text{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)-2 i b c x \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)+2 b c x \log (x)+2 b \cos ^{-1}(c x)+2 b c x \cos ^{-1}(c x) \log \left(1-e^{i \cos ^{-1}(c x)}\right)-2 b c x \cos ^{-1}(c x) \log \left(1+e^{i \cos ^{-1}(c x)}\right)}{2 d x}","-\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{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)}{d}+\frac{i b c \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)}{d}",1,"-1/2*(2*a + 2*b*ArcCos[c*x] + 2*b*c*x*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] - 2*b*c*x*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])] + 2*b*c*x*Log[x] + a*c*x*Log[1 - c*x] - a*c*x*Log[1 + c*x] - 2*b*c*x*Log[1 + Sqrt[1 - c^2*x^2]] + (2*I)*b*c*x*PolyLog[2, -E^(I*ArcCos[c*x])] - (2*I)*b*c*x*PolyLog[2, E^(I*ArcCos[c*x])])/(d*x)","A",0
7,1,150,124,0.4757613,"\int \frac{a+b \cos ^{-1}(c x)}{x^3 \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcCos[c*x])/(x^3*(d - c^2*d*x^2)),x]","-\frac{a c^2 \log \left(1-c^2 x^2\right)-2 a c^2 \log (x)+\frac{a}{x^2}+b c^2 \left(-\frac{\sqrt{1-c^2 x^2}}{c x}+\frac{\cos ^{-1}(c x)}{c^2 x^2}+i \text{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right)-i \text{Li}_2\left(e^{2 i \cos ^{-1}(c x)}\right)+2 \cos ^{-1}(c x) \log \left(1-e^{2 i \cos ^{-1}(c x)}\right)-2 \cos ^{-1}(c x) \log \left(1+e^{2 i \cos ^{-1}(c x)}\right)\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{i b c^2 \text{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right)}{2 d}+\frac{i b c^2 \text{Li}_2\left(e^{2 i \cos ^{-1}(c x)}\right)}{2 d}+\frac{b c \sqrt{1-c^2 x^2}}{2 d x}",1,"-1/2*(a/x^2 - 2*a*c^2*Log[x] + a*c^2*Log[1 - c^2*x^2] + b*c^2*(-(Sqrt[1 - c^2*x^2]/(c*x)) + ArcCos[c*x]/(c^2*x^2) + 2*ArcCos[c*x]*Log[1 - E^((2*I)*ArcCos[c*x])] - 2*ArcCos[c*x]*Log[1 + E^((2*I)*ArcCos[c*x])] + I*PolyLog[2, -E^((2*I)*ArcCos[c*x])] - I*PolyLog[2, E^((2*I)*ArcCos[c*x])]))/d","A",0
8,1,294,180,0.3668728,"\int \frac{x^4 \left(a+b \cos ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x^4*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]","\frac{3 a \log (1-c x)}{4 c^5 d^2}-\frac{3 a \log (c x+1)}{4 c^5 d^2}+\frac{a x}{c^4 d^2}-\frac{a x}{2 c^4 d^2 \left(c^2 x^2-1\right)}+\frac{b \left(-\frac{3 i \left(4 \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)+\cos ^{-1}(c x) \left(\cos ^{-1}(c x)+4 i \log \left(1-e^{i \cos ^{-1}(c x)}\right)\right)\right)}{8 c^5}-\frac{3 \left(-\frac{2 i \text{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)}{c}-\frac{i \cos ^{-1}(c x)^2}{2 c}+\frac{2 \cos ^{-1}(c x) \log \left(1+e^{i \cos ^{-1}(c x)}\right)}{c}\right)}{4 c^4}+\frac{c x \cos ^{-1}(c x)-\sqrt{1-c^2 x^2}}{c^5}+\frac{\sqrt{1-c^2 x^2}-\cos ^{-1}(c x)}{4 c^4 \left(c^2 x+c\right)}+\frac{\sqrt{1-c^2 x^2}+\cos ^{-1}(c x)}{4 c^4 \left(c-c^2 x\right)}\right)}{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{3 x \left(a+b \cos ^{-1}(c x)\right)}{2 c^4 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 i b \text{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)}{2 c^5 d^2}-\frac{3 i b \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)}{2 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,"(a*x)/(c^4*d^2) - (a*x)/(2*c^4*d^2*(-1 + c^2*x^2)) + (3*a*Log[1 - c*x])/(4*c^5*d^2) - (3*a*Log[1 + c*x])/(4*c^5*d^2) + (b*((Sqrt[1 - c^2*x^2] - ArcCos[c*x])/(4*c^4*(c + c^2*x)) + (Sqrt[1 - c^2*x^2] + ArcCos[c*x])/(4*c^4*(c - c^2*x)) + (-Sqrt[1 - c^2*x^2] + c*x*ArcCos[c*x])/c^5 - (3*(((-1/2*I)*ArcCos[c*x]^2)/c + (2*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])])/c - ((2*I)*PolyLog[2, -E^(I*ArcCos[c*x])])/c))/(4*c^4) - (((3*I)/8)*(ArcCos[c*x]*(ArcCos[c*x] + (4*I)*Log[1 - E^(I*ArcCos[c*x])]) + 4*PolyLog[2, E^(I*ArcCos[c*x])]))/c^5))/d^2","A",0
9,1,203,155,0.482515,"\int \frac{x^3 \left(a+b \cos ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x^3*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]","\frac{-\frac{2 a}{c^2 x^2-1}+2 a \log \left(1-c^2 x^2\right)+\frac{b \sqrt{1-c^2 x^2}}{1-c x}-\frac{b \sqrt{1-c^2 x^2}}{c x+1}-4 i b \text{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)-4 i b \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)-2 i b \cos ^{-1}(c x)^2+\frac{b \cos ^{-1}(c x)}{1-c x}+\frac{b \cos ^{-1}(c x)}{c x+1}+4 b \cos ^{-1}(c x) \log \left(1-e^{i \cos ^{-1}(c x)}\right)+4 b \cos ^{-1}(c x) \log \left(1+e^{i \cos ^{-1}(c x)}\right)}{4 c^4 d^2}","-\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{x^2 \left(a+b \cos ^{-1}(c x)\right)}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{i b \text{Li}_2\left(e^{2 i \cos ^{-1}(c x)}\right)}{2 c^4 d^2}-\frac{b \sin ^{-1}(c x)}{2 c^4 d^2}+\frac{b x}{2 c^3 d^2 \sqrt{1-c^2 x^2}}",1,"((b*Sqrt[1 - c^2*x^2])/(1 - c*x) - (b*Sqrt[1 - c^2*x^2])/(1 + c*x) - (2*a)/(-1 + c^2*x^2) + (b*ArcCos[c*x])/(1 - c*x) + (b*ArcCos[c*x])/(1 + c*x) - (2*I)*b*ArcCos[c*x]^2 + 4*b*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] + 4*b*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])] + 2*a*Log[1 - c^2*x^2] - (4*I)*b*PolyLog[2, -E^(I*ArcCos[c*x])] - (4*I)*b*PolyLog[2, E^(I*ArcCos[c*x])])/(4*c^4*d^2)","A",0
10,1,251,136,0.3007054,"\int \frac{x^2 \left(a+b \cos ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x^2*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]","-\frac{-a c^2 x^2 \log (1-c x)+a c^2 x^2 \log (c x+1)+2 a c x+a \log (1-c x)-a \log (c x+1)-2 i b \left(c^2 x^2-1\right) \text{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)+2 i b \left(c^2 x^2-1\right) \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)+2 b \sqrt{1-c^2 x^2}-2 b c^2 x^2 \cos ^{-1}(c x) \log \left(1-e^{i \cos ^{-1}(c x)}\right)+2 b c^2 x^2 \cos ^{-1}(c x) \log \left(1+e^{i \cos ^{-1}(c x)}\right)+2 b c x \cos ^{-1}(c x)+2 b \cos ^{-1}(c x) \log \left(1-e^{i \cos ^{-1}(c x)}\right)-2 b \cos ^{-1}(c x) \log \left(1+e^{i \cos ^{-1}(c x)}\right)}{4 c^3 d^2 \left(c^2 x^2-1\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{x \left(a+b \cos ^{-1}(c x)\right)}{2 c^2 d^2 \left(1-c^2 x^2\right)}+\frac{i b \text{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)}{2 c^3 d^2}-\frac{i b \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)}{2 c^3 d^2}+\frac{b}{2 c^3 d^2 \sqrt{1-c^2 x^2}}",1,"-1/4*(2*a*c*x + 2*b*Sqrt[1 - c^2*x^2] + 2*b*c*x*ArcCos[c*x] + 2*b*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] - 2*b*c^2*x^2*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] - 2*b*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])] + 2*b*c^2*x^2*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])] + a*Log[1 - c*x] - a*c^2*x^2*Log[1 - c*x] - a*Log[1 + c*x] + a*c^2*x^2*Log[1 + c*x] - (2*I)*b*(-1 + c^2*x^2)*PolyLog[2, -E^(I*ArcCos[c*x])] + (2*I)*b*(-1 + c^2*x^2)*PolyLog[2, E^(I*ArcCos[c*x])])/(c^3*d^2*(-1 + c^2*x^2))","A",0
11,1,49,57,0.1134832,"\int \frac{x \left(a+b \cos ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x*(a + b*ArcCos[c*x]))/(d - c^2*d*x^2)^2,x]","\frac{a+b c x \sqrt{1-c^2 x^2}+b \cos ^{-1}(c x)}{2 c^2 d^2-2 c^4 d^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,"(a + b*c*x*Sqrt[1 - c^2*x^2] + b*ArcCos[c*x])/(2*c^2*d^2 - 2*c^4*d^2*x^2)","A",1
12,1,220,132,0.3002119,"\int \frac{a+b \cos ^{-1}(c x)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcCos[c*x])/(d - c^2*d*x^2)^2,x]","\frac{-\frac{2 a x}{c^2 x^2-1}-\frac{a \log (1-c x)}{c}+\frac{a \log (c x+1)}{c}+\frac{b \sqrt{1-c^2 x^2}}{c-c^2 x}+\frac{b \sqrt{1-c^2 x^2}}{c^2 x+c}+\frac{b \cos ^{-1}(c x)}{c-c^2 x}-\frac{b \cos ^{-1}(c x)}{c^2 x+c}-\frac{2 i b \text{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)}{c}+\frac{2 i b \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)}{c}-\frac{2 b \cos ^{-1}(c x) \log \left(1-e^{i \cos ^{-1}(c x)}\right)}{c}+\frac{2 b \cos ^{-1}(c x) \log \left(1+e^{i \cos ^{-1}(c x)}\right)}{c}}{4 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{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)}{2 c d^2}+\frac{i b \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)}{2 c d^2}",1,"((b*Sqrt[1 - c^2*x^2])/(c - c^2*x) + (b*Sqrt[1 - c^2*x^2])/(c + c^2*x) - (2*a*x)/(-1 + c^2*x^2) + (b*ArcCos[c*x])/(c - c^2*x) - (b*ArcCos[c*x])/(c + c^2*x) - (2*b*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])])/c + (2*b*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])])/c - (a*Log[1 - c*x])/c + (a*Log[1 + c*x])/c - ((2*I)*b*PolyLog[2, -E^(I*ArcCos[c*x])])/c + ((2*I)*b*PolyLog[2, E^(I*ArcCos[c*x])])/c)/(4*d^2)","A",0
13,1,152,122,0.5206019,"\int \frac{a+b \cos ^{-1}(c x)}{x \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcCos[c*x])/(x*(d - c^2*d*x^2)^2),x]","\frac{\frac{a}{1-c^2 x^2}-a \log \left(1-c^2 x^2\right)+2 a \log (x)+b \left(\frac{c x}{\sqrt{1-c^2 x^2}}+\frac{\cos ^{-1}(c x)}{1-c^2 x^2}-i \text{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right)+i \text{Li}_2\left(e^{2 i \cos ^{-1}(c x)}\right)-2 \cos ^{-1}(c x) \log \left(1-e^{2 i \cos ^{-1}(c x)}\right)+2 \cos ^{-1}(c x) \log \left(1+e^{2 i \cos ^{-1}(c x)}\right)\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{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right)}{2 d^2}+\frac{i b \text{Li}_2\left(e^{2 i \cos ^{-1}(c x)}\right)}{2 d^2}",1,"(a/(1 - c^2*x^2) + 2*a*Log[x] - a*Log[1 - c^2*x^2] + b*((c*x)/Sqrt[1 - c^2*x^2] + ArcCos[c*x]/(1 - c^2*x^2) - 2*ArcCos[c*x]*Log[1 - E^((2*I)*ArcCos[c*x])] + 2*ArcCos[c*x]*Log[1 + E^((2*I)*ArcCos[c*x])] - I*PolyLog[2, -E^((2*I)*ArcCos[c*x])] + I*PolyLog[2, E^((2*I)*ArcCos[c*x])]))/(2*d^2)","A",0
14,1,251,177,0.4923902,"\int \frac{a+b \cos ^{-1}(c x)}{x^2 \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcCos[c*x])/(x^2*(d - c^2*d*x^2)^2),x]","\frac{-\frac{2 a c^2 x}{c^2 x^2-1}-3 a c \log (1-c x)+3 a c \log (c x+1)-\frac{4 a}{x}+\frac{b c \sqrt{1-c^2 x^2}}{1-c x}+\frac{b c \sqrt{1-c^2 x^2}}{c x+1}+4 b c \log \left(\sqrt{1-c^2 x^2}+1\right)-6 i b c \text{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)+6 i b c \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)-4 b c \log (x)+\frac{b c \cos ^{-1}(c x)}{1-c x}-\frac{b c \cos ^{-1}(c x)}{c x+1}-\frac{4 b \cos ^{-1}(c x)}{x}-6 b c \cos ^{-1}(c x) \log \left(1-e^{i \cos ^{-1}(c x)}\right)+6 b c \cos ^{-1}(c x) \log \left(1+e^{i \cos ^{-1}(c x)}\right)}{4 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{Li}_2\left(-e^{i \cos ^{-1}(c x)}\right)}{2 d^2}+\frac{3 i b c \text{Li}_2\left(e^{i \cos ^{-1}(c x)}\right)}{2 d^2}",1,"((-4*a)/x + (b*c*Sqrt[1 - c^2*x^2])/(1 - c*x) + (b*c*Sqrt[1 - c^2*x^2])/(1 + c*x) - (2*a*c^2*x)/(-1 + c^2*x^2) - (4*b*ArcCos[c*x])/x + (b*c*ArcCos[c*x])/(1 - c*x) - (b*c*ArcCos[c*x])/(1 + c*x) - 6*b*c*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] + 6*b*c*ArcCos[c*x]*Log[1 + E^(I*ArcCos[c*x])] - 4*b*c*Log[x] - 3*a*c*Log[1 - c*x] + 3*a*c*Log[1 + c*x] + 4*b*c*Log[1 + Sqrt[1 - c^2*x^2]] - (6*I)*b*c*PolyLog[2, -E^(I*ArcCos[c*x])] + (6*I)*b*c*PolyLog[2, E^(I*ArcCos[c*x])])/(4*d^2)","A",0
15,1,217,159,0.7254335,"\int \frac{a+b \cos ^{-1}(c x)}{x^3 \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcCos[c*x])/(x^3*(d - c^2*d*x^2)^2),x]","\frac{\frac{a c^2}{1-c^2 x^2}-2 a c^2 \log \left(1-c^2 x^2\right)+4 a c^2 \log (x)-\frac{a}{x^2}-2 i b c^2 \text{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right)+2 i b c^2 \text{Li}_2\left(e^{2 i \cos ^{-1}(c x)}\right)+\frac{b c \sqrt{1-c^2 x^2}}{x}+\frac{b c^2 \cos ^{-1}(c x)}{1-c^2 x^2}-4 b c^2 \cos ^{-1}(c x) \log \left(1-e^{2 i \cos ^{-1}(c x)}\right)+4 b c^2 \cos ^{-1}(c x) \log \left(1+e^{2 i \cos ^{-1}(c x)}\right)+\frac{b c^3 x}{\sqrt{1-c^2 x^2}}-\frac{b \cos ^{-1}(c x)}{x^2}}{2 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{i b c^2 \text{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right)}{d^2}+\frac{i b c^2 \text{Li}_2\left(e^{2 i \cos ^{-1}(c x)}\right)}{d^2}+\frac{b c}{2 d^2 x \sqrt{1-c^2 x^2}}",1,"(-(a/x^2) + (a*c^2)/(1 - c^2*x^2) + (b*c^3*x)/Sqrt[1 - c^2*x^2] + (b*c*Sqrt[1 - c^2*x^2])/x - (b*ArcCos[c*x])/x^2 + (b*c^2*ArcCos[c*x])/(1 - c^2*x^2) - 4*b*c^2*ArcCos[c*x]*Log[1 - E^((2*I)*ArcCos[c*x])] + 4*b*c^2*ArcCos[c*x]*Log[1 + E^((2*I)*ArcCos[c*x])] + 4*a*c^2*Log[x] - 2*a*c^2*Log[1 - c^2*x^2] - (2*I)*b*c^2*PolyLog[2, -E^((2*I)*ArcCos[c*x])] + (2*I)*b*c^2*PolyLog[2, E^((2*I)*ArcCos[c*x])])/(2*d^2)","A",0
16,1,153,149,0.155606,"\int x^3 \left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)*(a + b*ArcCos[c*x]),x]","\frac{1}{4} a d x^4+\frac{1}{6} a e x^6+\frac{5 b e \sin ^{-1}(c x)}{96 c^6}+\frac{3 b d \sin ^{-1}(c x)}{32 c^4}+b d \sqrt{1-c^2 x^2} \left(-\frac{3 x}{32 c^3}-\frac{x^3}{16 c}\right)+b e \sqrt{1-c^2 x^2} \left(-\frac{5 x}{96 c^5}-\frac{5 x^3}{144 c^3}-\frac{x^5}{36 c}\right)+\frac{1}{4} b d x^4 \cos ^{-1}(c x)+\frac{1}{6} b e x^6 \cos ^{-1}(c 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 e x^5 \sqrt{1-c^2 x^2}}{36 c}+\frac{b \left(9 c^2 d+5 e\right) \sin ^{-1}(c x)}{96 c^6}-\frac{b x \sqrt{1-c^2 x^2} \left(9 c^2 d+5 e\right)}{96 c^5}-\frac{b x^3 \sqrt{1-c^2 x^2} \left(9 c^2 d+5 e\right)}{144 c^3}",1,"(a*d*x^4)/4 + (a*e*x^6)/6 + b*d*Sqrt[1 - c^2*x^2]*((-3*x)/(32*c^3) - x^3/(16*c)) + b*e*Sqrt[1 - c^2*x^2]*((-5*x)/(96*c^5) - (5*x^3)/(144*c^3) - x^5/(36*c)) + (b*d*x^4*ArcCos[c*x])/4 + (b*e*x^6*ArcCos[c*x])/6 + (3*b*d*ArcSin[c*x])/(32*c^4) + (5*b*e*ArcSin[c*x])/(96*c^6)","A",1
17,1,125,120,0.1176098,"\int x^2 \left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)*(a + b*ArcCos[c*x]),x]","\frac{1}{3} a d x^3+\frac{1}{5} a e x^5+b d \left(-\frac{2}{9 c^3}-\frac{x^2}{9 c}\right) \sqrt{1-c^2 x^2}+b e \sqrt{1-c^2 x^2} \left(-\frac{8}{75 c^5}-\frac{4 x^2}{75 c^3}-\frac{x^4}{25 c}\right)+\frac{1}{3} b d x^3 \cos ^{-1}(c x)+\frac{1}{5} b e x^5 \cos ^{-1}(c 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}",1,"(a*d*x^3)/3 + (a*e*x^5)/5 + b*d*(-2/(9*c^3) - x^2/(9*c))*Sqrt[1 - c^2*x^2] + b*e*Sqrt[1 - c^2*x^2]*(-8/(75*c^5) - (4*x^2)/(75*c^3) - x^4/(25*c)) + (b*d*x^3*ArcCos[c*x])/3 + (b*e*x^5*ArcCos[c*x])/5","A",1
18,1,131,122,0.112505,"\int x \left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)*(a + b*ArcCos[c*x]),x]","\frac{1}{2} a d x^2+\frac{1}{4} a e x^4+\frac{3 b e \sin ^{-1}(c x)}{32 c^4}-\frac{b d x \sqrt{1-c^2 x^2}}{4 c}+\frac{b d \sin ^{-1}(c x)}{4 c^2}+b e \sqrt{1-c^2 x^2} \left(-\frac{3 x}{32 c^3}-\frac{x^3}{16 c}\right)+\frac{1}{2} b d x^2 \cos ^{-1}(c x)+\frac{1}{4} b e x^4 \cos ^{-1}(c x)","\frac{\left(d+e x^2\right)^2 \left(a+b \cos ^{-1}(c x)\right)}{4 e}-\frac{b x \sqrt{1-c^2 x^2} \left(d+e x^2\right)}{16 c}+\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{3 b x \sqrt{1-c^2 x^2} \left(2 c^2 d+e\right)}{32 c^3}",1,"(a*d*x^2)/2 + (a*e*x^4)/4 - (b*d*x*Sqrt[1 - c^2*x^2])/(4*c) + b*e*Sqrt[1 - c^2*x^2]*((-3*x)/(32*c^3) - x^3/(16*c)) + (b*d*x^2*ArcCos[c*x])/2 + (b*e*x^4*ArcCos[c*x])/4 + (b*d*ArcSin[c*x])/(4*c^2) + (3*b*e*ArcSin[c*x])/(32*c^4)","A",1
19,1,91,81,0.0808129,"\int \left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)*(a + b*ArcCos[c*x]),x]","a d x+\frac{1}{3} a e x^3-\frac{b d \sqrt{1-c^2 x^2}}{c}+b e \left(-\frac{2}{9 c^3}-\frac{x^2}{9 c}\right) \sqrt{1-c^2 x^2}+b d x \cos ^{-1}(c x)+\frac{1}{3} b e x^3 \cos ^{-1}(c 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}",1,"a*d*x + (a*e*x^3)/3 - (b*d*Sqrt[1 - c^2*x^2])/c + b*e*(-2/(9*c^3) - x^2/(9*c))*Sqrt[1 - c^2*x^2] + b*d*x*ArcCos[c*x] + (b*e*x^3*ArcCos[c*x])/3","A",1
20,1,111,132,0.2110135,"\int \frac{\left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCos[c*x]))/x,x]","\frac{1}{4} \left(4 a d \log (x)+2 a e x^2-\frac{b e x \sqrt{1-c^2 x^2}}{c}+\frac{b e \sin ^{-1}(c x)}{c^2}+2 b \cos ^{-1}(c x) \left(e x^2+2 d \log \left(1+e^{2 i \cos ^{-1}(c x)}\right)\right)-2 i b d \text{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right)-2 i b d \cos ^{-1}(c x)^2\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 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+\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,"(2*a*e*x^2 - (b*e*x*Sqrt[1 - c^2*x^2])/c - (2*I)*b*d*ArcCos[c*x]^2 + (b*e*ArcSin[c*x])/c^2 + 2*b*ArcCos[c*x]*(e*x^2 + 2*d*Log[1 + E^((2*I)*ArcCos[c*x])]) + 4*a*d*Log[x] - (2*I)*b*d*PolyLog[2, -E^((2*I)*ArcCos[c*x])])/4","A",0
21,1,80,66,0.0769286,"\int \frac{\left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCos[c*x]))/x^2,x]","-\frac{a d}{x}+a e x+b c d \log \left(\sqrt{1-c^2 x^2}+1\right)-\frac{b e \sqrt{1-c^2 x^2}}{c}-b c d \log (x)-\frac{b d \cos ^{-1}(c x)}{x}+b e x \cos ^{-1}(c 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}",1,"-((a*d)/x) + a*e*x - (b*e*Sqrt[1 - c^2*x^2])/c - (b*d*ArcCos[c*x])/x + b*e*x*ArcCos[c*x] - b*c*d*Log[x] + b*c*d*Log[1 + Sqrt[1 - c^2*x^2]]","A",1
22,1,105,119,0.1673738,"\int \frac{\left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCos[c*x]))/x^3,x]","-\frac{a d-2 a e x^2 \log (x)-b c d x \sqrt{1-c^2 x^2}+b \cos ^{-1}(c x) \left(d-2 e x^2 \log \left(1+e^{2 i \cos ^{-1}(c x)}\right)\right)+i b e x^2 \text{Li}_2\left(-e^{2 i \cos ^{-1}(c x)}\right)+i b e x^2 \cos ^{-1}(c x)^2}{2 x^2}","-\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 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+\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,"-1/2*(a*d - b*c*d*x*Sqrt[1 - c^2*x^2] + I*b*e*x^2*ArcCos[c*x]^2 + b*ArcCos[c*x]*(d - 2*e*x^2*Log[1 + E^((2*I)*ArcCos[c*x])]) - 2*a*e*x^2*Log[x] + I*b*e*x^2*PolyLog[2, -E^((2*I)*ArcCos[c*x])])/x^2","A",0
23,1,130,85,0.067913,"\int \frac{\left(d+e x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCos[c*x]))/x^4,x]","-\frac{a d}{3 x^3}-\frac{a e}{x}-\frac{1}{6} b c^3 d \log (x)+\frac{b c d \sqrt{1-c^2 x^2}}{6 x^2}+b c e \log \left(\sqrt{1-c^2 x^2}+1\right)+\frac{1}{6} b c^3 d \log \left(\sqrt{1-c^2 x^2}+1\right)-\frac{b d \cos ^{-1}(c x)}{3 x^3}-b c e \log (x)-\frac{b e \cos ^{-1}(c x)}{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}",1,"-1/3*(a*d)/x^3 - (a*e)/x + (b*c*d*Sqrt[1 - c^2*x^2])/(6*x^2) - (b*d*ArcCos[c*x])/(3*x^3) - (b*e*ArcCos[c*x])/x - (b*c^3*d*Log[x])/6 - b*c*e*Log[x] + (b*c^3*d*Log[1 + Sqrt[1 - c^2*x^2]])/6 + b*c*e*Log[1 + Sqrt[1 - c^2*x^2]]","A",1
24,1,99,135,0.128212,"\int \left(c+d x^2\right)^2 \cos ^{-1}(a x) \, dx","Integrate[(c + d*x^2)^2*ArcCos[a*x],x]","\cos ^{-1}(a x) \left(c^2 x+\frac{2}{3} c d x^3+\frac{d^2 x^5}{5}\right)-\frac{\sqrt{1-a^2 x^2} \left(a^4 \left(225 c^2+50 c d x^2+9 d^2 x^4\right)+4 a^2 d \left(25 c+3 d x^2\right)+24 d^2\right)}{225 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}-\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}+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,"-1/225*(Sqrt[1 - a^2*x^2]*(24*d^2 + 4*a^2*d*(25*c + 3*d*x^2) + a^4*(225*c^2 + 50*c*d*x^2 + 9*d^2*x^4)))/a^5 + (c^2*x + (2*c*d*x^3)/3 + (d^2*x^5)/5)*ArcCos[a*x]","A",1
25,1,149,205,0.1768864,"\int \left(c+d x^2\right)^3 \cos ^{-1}(a x) \, dx","Integrate[(c + d*x^2)^3*ArcCos[a*x],x]","\cos ^{-1}(a x) \left(c^3 x+c^2 d x^3+\frac{3}{5} c d^2 x^5+\frac{d^3 x^7}{7}\right)-\frac{\sqrt{1-a^2 x^2} \left(a^6 \left(3675 c^3+1225 c^2 d x^2+441 c d^2 x^4+75 d^3 x^6\right)+2 a^4 d \left(1225 c^2+294 c d x^2+45 d^2 x^4\right)+24 a^2 d^2 \left(49 c+5 d x^2\right)+240 d^3\right)}{3675 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}+\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^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right)}{35 a^7}+c^3 x \cos ^{-1}(a x)+c^2 d x^3 \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,"-1/3675*(Sqrt[1 - a^2*x^2]*(240*d^3 + 24*a^2*d^2*(49*c + 5*d*x^2) + 2*a^4*d*(1225*c^2 + 294*c*d*x^2 + 45*d^2*x^4) + a^6*(3675*c^3 + 1225*c^2*d*x^2 + 441*c*d^2*x^4 + 75*d^3*x^6)))/a^7 + (c^3*x + c^2*d*x^3 + (3*c*d^2*x^5)/5 + (d^3*x^7)/7)*ArcCos[a*x]","A",1
26,1,212,292,0.202279,"\int \left(c+d x^2\right)^4 \cos ^{-1}(a x) \, dx","Integrate[(c + d*x^2)^4*ArcCos[a*x],x]","\frac{1}{315} x \cos ^{-1}(a x) \left(315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right)-\frac{\sqrt{1-a^2 x^2} \left(a^8 \left(99225 c^4+44100 c^3 d x^2+23814 c^2 d^2 x^4+8100 c d^3 x^6+1225 d^4 x^8\right)+8 a^6 d \left(11025 c^3+3969 c^2 d x^2+1215 c d^2 x^4+175 d^3 x^6\right)+48 a^4 d^2 \left(1323 c^2+270 c d x^2+35 d^2 x^4\right)+320 a^2 d^3 \left(81 c+7 d x^2\right)+4480 d^4\right)}{99225 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{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(105 a^6 c^3+189 a^4 c^2 d+135 a^2 c d^2+35 d^3\right)}{945 a^9}-\frac{\sqrt{1-a^2 x^2} \left(315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right)}{315 a^9}+c^4 x \cos ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cos ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \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,"-1/99225*(Sqrt[1 - a^2*x^2]*(4480*d^4 + 320*a^2*d^3*(81*c + 7*d*x^2) + 48*a^4*d^2*(1323*c^2 + 270*c*d*x^2 + 35*d^2*x^4) + 8*a^6*d*(11025*c^3 + 3969*c^2*d*x^2 + 1215*c*d^2*x^4 + 175*d^3*x^6) + a^8*(99225*c^4 + 44100*c^3*d*x^2 + 23814*c^2*d^2*x^4 + 8100*c*d^3*x^6 + 1225*d^4*x^8)))/a^9 + (x*(315*c^4 + 420*c^3*d*x^2 + 378*c^2*d^2*x^4 + 180*c*d^3*x^6 + 35*d^4*x^8)*ArcCos[a*x])/315","A",1
27,1,811,521,1.2302912,"\int \frac{\cos ^{-1}(a x)}{c+d x^2} \, dx","Integrate[ArcCos[a*x]/(c + d*x^2),x]","\frac{4 \sin ^{-1}\left(\frac{\sqrt{1-\frac{i a \sqrt{c}}{\sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(a \sqrt{c}-i \sqrt{d}\right) \tan \left(\frac{1}{2} \cos ^{-1}(a x)\right)}{\sqrt{c a^2+d}}\right)-4 \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c} a}{\sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{c} a+i \sqrt{d}\right) \tan \left(\frac{1}{2} \cos ^{-1}(a x)\right)}{\sqrt{c a^2+d}}\right)+i \cos ^{-1}(a x) \log \left(1-\frac{i \left(\sqrt{c a^2+d}-a \sqrt{c}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)+2 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c} a}{\sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{c a^2+d}-a \sqrt{c}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)-i \cos ^{-1}(a x) \log \left(\frac{i e^{i \cos ^{-1}(a x)} \left(\sqrt{c a^2+d}-a \sqrt{c}\right)}{\sqrt{d}}+1\right)-2 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i a \sqrt{c}}{\sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \cos ^{-1}(a x)} \left(\sqrt{c a^2+d}-a \sqrt{c}\right)}{\sqrt{d}}+1\right)-i \cos ^{-1}(a x) \log \left(1-\frac{i \left(\sqrt{c} a+\sqrt{c a^2+d}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)+2 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i a \sqrt{c}}{\sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{c} a+\sqrt{c a^2+d}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)+i \cos ^{-1}(a x) \log \left(\frac{i e^{i \cos ^{-1}(a x)} \left(\sqrt{c} a+\sqrt{c a^2+d}\right)}{\sqrt{d}}+1\right)-2 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c} a}{\sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \cos ^{-1}(a x)} \left(\sqrt{c} a+\sqrt{c a^2+d}\right)}{\sqrt{d}}+1\right)-\text{Li}_2\left(-\frac{i \left(\sqrt{c a^2+d}-a \sqrt{c}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)+\text{Li}_2\left(\frac{i \left(\sqrt{c a^2+d}-a \sqrt{c}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)+\text{Li}_2\left(-\frac{i \left(\sqrt{c} a+\sqrt{c a^2+d}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)-\text{Li}_2\left(\frac{i \left(\sqrt{c} a+\sqrt{c a^2+d}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)}{2 \sqrt{c} \sqrt{d}}","\frac{i \text{Li}_2\left(-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{c a^2+d}}\right)}{2 \sqrt{-c} \sqrt{d}}-\frac{i \text{Li}_2\left(\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{c a^2+d}}\right)}{2 \sqrt{-c} \sqrt{d}}+\frac{i \text{Li}_2\left(-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{\sqrt{-c} a+i \sqrt{c a^2+d}}\right)}{2 \sqrt{-c} \sqrt{d}}-\frac{i \text{Li}_2\left(\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{\sqrt{-c} a+i \sqrt{c a^2+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,"(4*ArcSin[Sqrt[1 - (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*ArcTan[((a*Sqrt[c] - I*Sqrt[d])*Tan[ArcCos[a*x]/2])/Sqrt[a^2*c + d]] - 4*ArcSin[Sqrt[1 + (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*ArcTan[((a*Sqrt[c] + I*Sqrt[d])*Tan[ArcCos[a*x]/2])/Sqrt[a^2*c + d]] + I*ArcCos[a*x]*Log[1 - (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + (2*I)*ArcSin[Sqrt[1 + (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 - (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - I*ArcCos[a*x]*Log[1 + (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - (2*I)*ArcSin[Sqrt[1 - (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 + (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - I*ArcCos[a*x]*Log[1 - (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + (2*I)*ArcSin[Sqrt[1 - (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 - (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + I*ArcCos[a*x]*Log[1 + (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - (2*I)*ArcSin[Sqrt[1 + (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 + (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - PolyLog[2, ((-I)*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + PolyLog[2, (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + PolyLog[2, ((-I)*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - PolyLog[2, (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]])/(2*Sqrt[c]*Sqrt[d])","A",0
28,1,1065,727,2.4692592,"\int \frac{\cos ^{-1}(a x)}{\left(c+d x^2\right)^2} \, dx","Integrate[ArcCos[a*x]/(c + d*x^2)^2,x]","\frac{4 \sin ^{-1}\left(\frac{\sqrt{1-\frac{i a \sqrt{c}}{\sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(a \sqrt{c}-i \sqrt{d}\right) \tan \left(\frac{1}{2} \cos ^{-1}(a x)\right)}{\sqrt{c a^2+d}}\right)-4 \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c} a}{\sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{c} a+i \sqrt{d}\right) \tan \left(\frac{1}{2} \cos ^{-1}(a x)\right)}{\sqrt{c a^2+d}}\right)+i \cos ^{-1}(a x) \log \left(1-\frac{i \left(\sqrt{c a^2+d}-a \sqrt{c}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)+2 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c} a}{\sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{c a^2+d}-a \sqrt{c}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)-i \cos ^{-1}(a x) \log \left(\frac{i e^{i \cos ^{-1}(a x)} \left(\sqrt{c a^2+d}-a \sqrt{c}\right)}{\sqrt{d}}+1\right)-2 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i a \sqrt{c}}{\sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \cos ^{-1}(a x)} \left(\sqrt{c a^2+d}-a \sqrt{c}\right)}{\sqrt{d}}+1\right)-i \cos ^{-1}(a x) \log \left(1-\frac{i \left(\sqrt{c} a+\sqrt{c a^2+d}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)+2 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i a \sqrt{c}}{\sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{c} a+\sqrt{c a^2+d}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)+i \cos ^{-1}(a x) \log \left(\frac{i e^{i \cos ^{-1}(a x)} \left(\sqrt{c} a+\sqrt{c a^2+d}\right)}{\sqrt{d}}+1\right)-2 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{c} a}{\sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \cos ^{-1}(a x)} \left(\sqrt{c} a+\sqrt{c a^2+d}\right)}{\sqrt{d}}+1\right)+\sqrt{c} \left(\frac{\cos ^{-1}(a x)}{\sqrt{d} x-i \sqrt{c}}-\frac{a \log \left(\frac{2 d \left(-i \sqrt{c} x a^2+\sqrt{d}+\sqrt{c a^2+d} \sqrt{1-a^2 x^2}\right)}{a \sqrt{c a^2+d} \left(\sqrt{d} x-i \sqrt{c}\right)}\right)}{\sqrt{c a^2+d}}\right)+\sqrt{c} \left(\frac{\cos ^{-1}(a x)}{\sqrt{d} x+i \sqrt{c}}-\frac{a \log \left(-\frac{2 d \left(i \sqrt{c} x a^2+\sqrt{d}+\sqrt{c a^2+d} \sqrt{1-a^2 x^2}\right)}{a \sqrt{c a^2+d} \left(\sqrt{d} x+i \sqrt{c}\right)}\right)}{\sqrt{c a^2+d}}\right)-\text{Li}_2\left(-\frac{i \left(\sqrt{c a^2+d}-a \sqrt{c}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)+\text{Li}_2\left(\frac{i \left(\sqrt{c a^2+d}-a \sqrt{c}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)+\text{Li}_2\left(-\frac{i \left(\sqrt{c} a+\sqrt{c a^2+d}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)-\text{Li}_2\left(\frac{i \left(\sqrt{c} a+\sqrt{c a^2+d}\right) e^{i \cos ^{-1}(a x)}}{\sqrt{d}}\right)}{4 c^{3/2} \sqrt{d}}","-\frac{i \text{Li}_2\left(-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{c a^2+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}+\frac{i \text{Li}_2\left(\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{a \sqrt{-c}-i \sqrt{c a^2+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}-\frac{i \text{Li}_2\left(-\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{\sqrt{-c} a+i \sqrt{c a^2+d}}\right)}{4 (-c)^{3/2} \sqrt{d}}+\frac{i \text{Li}_2\left(\frac{\sqrt{d} e^{i \cos ^{-1}(a x)}}{\sqrt{-c} a+i \sqrt{c a^2+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,"(4*ArcSin[Sqrt[1 - (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*ArcTan[((a*Sqrt[c] - I*Sqrt[d])*Tan[ArcCos[a*x]/2])/Sqrt[a^2*c + d]] - 4*ArcSin[Sqrt[1 + (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*ArcTan[((a*Sqrt[c] + I*Sqrt[d])*Tan[ArcCos[a*x]/2])/Sqrt[a^2*c + d]] + I*ArcCos[a*x]*Log[1 - (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + (2*I)*ArcSin[Sqrt[1 + (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 - (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - I*ArcCos[a*x]*Log[1 + (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - (2*I)*ArcSin[Sqrt[1 - (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 + (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - I*ArcCos[a*x]*Log[1 - (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + (2*I)*ArcSin[Sqrt[1 - (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 - (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + I*ArcCos[a*x]*Log[1 + (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - (2*I)*ArcSin[Sqrt[1 + (I*a*Sqrt[c])/Sqrt[d]]/Sqrt[2]]*Log[1 + (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + Sqrt[c]*(ArcCos[a*x]/((-I)*Sqrt[c] + Sqrt[d]*x) - (a*Log[(2*d*(Sqrt[d] - I*a^2*Sqrt[c]*x + Sqrt[a^2*c + d]*Sqrt[1 - a^2*x^2]))/(a*Sqrt[a^2*c + d]*((-I)*Sqrt[c] + Sqrt[d]*x))])/Sqrt[a^2*c + d]) + Sqrt[c]*(ArcCos[a*x]/(I*Sqrt[c] + Sqrt[d]*x) - (a*Log[(-2*d*(Sqrt[d] + I*a^2*Sqrt[c]*x + Sqrt[a^2*c + d]*Sqrt[1 - a^2*x^2]))/(a*Sqrt[a^2*c + d]*(I*Sqrt[c] + Sqrt[d]*x))])/Sqrt[a^2*c + d]) - PolyLog[2, ((-I)*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + PolyLog[2, (I*(-(a*Sqrt[c]) + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] + PolyLog[2, ((-I)*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]] - PolyLog[2, (I*(a*Sqrt[c] + Sqrt[a^2*c + d])*E^(I*ArcCos[a*x]))/Sqrt[d]])/(4*c^(3/2)*Sqrt[d])","A",0
29,0,0,19,3.7897111,"\int \sqrt{c+d x^2} \cos ^{-1}(a x) \, dx","Integrate[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,"Integrate[Sqrt[c + d*x^2]*ArcCos[a*x], x]","A",-1
30,0,0,19,2.3721776,"\int \frac{\cos ^{-1}(a x)}{\sqrt{c+d x^2}} \, dx","Integrate[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,"Integrate[ArcCos[a*x]/Sqrt[c + d*x^2], x]","A",-1
31,1,68,66,0.0822292,"\int \frac{\cos ^{-1}(a x)}{\left(c+d x^2\right)^{3/2}} \, dx","Integrate[ArcCos[a*x]/(c + d*x^2)^(3/2),x]","\frac{x \left(a x \sqrt{\frac{d x^2}{c}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;a^2 x^2,-\frac{d x^2}{c}\right)+2 \cos ^{-1}(a x)\right)}{2 c \sqrt{c+d x^2}}","\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*(a*x*Sqrt[1 + (d*x^2)/c]*AppellF1[1, 1/2, 1/2, 2, a^2*x^2, -((d*x^2)/c)] + 2*ArcCos[a*x]))/(2*c*Sqrt[c + d*x^2])","C",0
32,1,120,136,0.2008027,"\int \frac{\cos ^{-1}(a x)}{\left(c+d x^2\right)^{5/2}} \, dx","Integrate[ArcCos[a*x]/(c + d*x^2)^(5/2),x]","\frac{a x^2 \left(c+d x^2\right) \sqrt{\frac{d x^2}{c}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;a^2 x^2,-\frac{d x^2}{c}\right)-\frac{a c \sqrt{1-a^2 x^2} \left(c+d x^2\right)}{a^2 c+d}+\cos ^{-1}(a x) \left(3 c x+2 d x^3\right)}{3 c^2 \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*c*Sqrt[1 - a^2*x^2]*(c + d*x^2))/(a^2*c + d)) + a*x^2*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*AppellF1[1, 1/2, 1/2, 2, a^2*x^2, -((d*x^2)/c)] + (3*c*x + 2*d*x^3)*ArcCos[a*x])/(3*c^2*(c + d*x^2)^(3/2))","C",0
33,1,162,211,0.3456225,"\int \frac{\cos ^{-1}(a x)}{\left(c+d x^2\right)^{7/2}} \, dx","Integrate[ArcCos[a*x]/(c + d*x^2)^(7/2),x]","\frac{4 a x^2 \sqrt{\frac{d x^2}{c}+1} \left(c+d x^2\right)^2 F_1\left(1;\frac{1}{2},\frac{1}{2};2;a^2 x^2,-\frac{d x^2}{c}\right)-\frac{a c \sqrt{1-a^2 x^2} \left(c+d x^2\right) \left(a^2 c \left(7 c+6 d x^2\right)+d \left(5 c+4 d x^2\right)\right)}{\left(a^2 c+d\right)^2}+x \cos ^{-1}(a x) \left(15 c^2+20 c d x^2+8 d^2 x^4\right)}{15 c^3 \left(c+d x^2\right)^{5/2}}","-\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{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{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*c*Sqrt[1 - a^2*x^2]*(c + d*x^2)*(d*(5*c + 4*d*x^2) + a^2*c*(7*c + 6*d*x^2)))/(a^2*c + d)^2) + 4*a*x^2*(c + d*x^2)^2*Sqrt[1 + (d*x^2)/c]*AppellF1[1, 1/2, 1/2, 2, a^2*x^2, -((d*x^2)/c)] + x*(15*c^2 + 20*c*d*x^2 + 8*d^2*x^4)*ArcCos[a*x])/(15*c^3*(c + d*x^2)^(5/2))","C",0