1,1,442,670,1.3285471,"\int (f+g x)^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]),x]","\frac{-3600 a c \sqrt{d} f \sqrt{1-c^2 x^2} \left(4 c^2 f^2+3 g^2\right) \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+240 a \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left(6 c^4 x \left(10 f^3+20 f^2 g x+15 f g^2 x^2+4 g^3 x^3\right)-c^2 g \left(120 f^2+45 f g x+8 g^2 x^2\right)-16 g^3\right)-2400 b c^2 f^2 g \sqrt{d-c^2 d x^2} \left(12 \left(1-c^2 x^2\right)^{3/2} \cos ^{-1}(c x)+9 c x-\cos \left(3 \cos ^{-1}(c x)\right)\right)+675 b c f g^2 \sqrt{d-c^2 d x^2} \left(-8 \cos ^{-1}(c x)^2+\cos \left(4 \cos ^{-1}(c x)\right)+4 \cos ^{-1}(c x) \sin \left(4 \cos ^{-1}(c x)\right)\right)-8 b g^3 \sqrt{d-c^2 d x^2} \left(15 \cos ^{-1}(c x) \left(30 \sqrt{1-c^2 x^2}-5 \sin \left(3 \cos ^{-1}(c x)\right)-3 \sin \left(5 \cos ^{-1}(c x)\right)\right)+16 c x \left(-9 c^4 x^4+5 c^2 x^2+30\right)\right)+3600 b c^3 f^3 \sqrt{d-c^2 d x^2} \left(\cos \left(2 \cos ^{-1}(c x)\right)+2 \cos ^{-1}(c x) \left(\sin \left(2 \cos ^{-1}(c x)\right)-\cos ^{-1}(c x)\right)\right)}{28800 c^4 \sqrt{1-c^2 x^2}}","\frac{1}{2} f^3 x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)-\frac{f^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{4 b c \sqrt{1-c^2 x^2}}-\frac{f^2 g \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{c^2}-\frac{3 f g^2 x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{8 c^2}+\frac{3}{4} f g^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)+\frac{g^3 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{5 c^4}-\frac{g^3 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{3 c^4}-\frac{3 f g^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{16 b c^3 \sqrt{1-c^2 x^2}}+\frac{b c f^3 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}-\frac{b f^2 g x \sqrt{d-c^2 d x^2}}{c \sqrt{1-c^2 x^2}}+\frac{b c f^2 g x^3 \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}-\frac{3 b f g^2 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}}+\frac{3 b c f g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{b c g^3 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}-\frac{b g^3 x^3 \sqrt{d-c^2 d x^2}}{45 c \sqrt{1-c^2 x^2}}-\frac{2 b g^3 x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{1-c^2 x^2}}",1,"(240*a*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(-16*g^3 - c^2*g*(120*f^2 + 45*f*g*x + 8*g^2*x^2) + 6*c^4*x*(10*f^3 + 20*f^2*g*x + 15*f*g^2*x^2 + 4*g^3*x^3)) - 3600*a*c*Sqrt[d]*f*(4*c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 2400*b*c^2*f^2*g*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*(1 - c^2*x^2)^(3/2)*ArcCos[c*x] - Cos[3*ArcCos[c*x]]) + 3600*b*c^3*f^3*Sqrt[d - c^2*d*x^2]*(Cos[2*ArcCos[c*x]] + 2*ArcCos[c*x]*(-ArcCos[c*x] + Sin[2*ArcCos[c*x]])) + 675*b*c*f*g^2*Sqrt[d - c^2*d*x^2]*(-8*ArcCos[c*x]^2 + Cos[4*ArcCos[c*x]] + 4*ArcCos[c*x]*Sin[4*ArcCos[c*x]]) - 8*b*g^3*Sqrt[d - c^2*d*x^2]*(16*c*x*(30 + 5*c^2*x^2 - 9*c^4*x^4) + 15*ArcCos[c*x]*(30*Sqrt[1 - c^2*x^2] - 5*Sin[3*ArcCos[c*x]] - 3*Sin[5*ArcCos[c*x]])))/(28800*c^4*Sqrt[1 - c^2*x^2])","A",1
2,1,320,450,1.4005159,"\int (f+g x)^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]),x]","\frac{48 a c \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left(12 c^2 f^2 x+16 f g \left(c^2 x^2-1\right)+3 g^2 x \left(2 c^2 x^2-1\right)\right)-144 a \sqrt{d} \sqrt{1-c^2 x^2} \left(4 c^2 f^2+g^2\right) \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+144 b c^2 f^2 \sqrt{d-c^2 d x^2} \left(\cos \left(2 \cos ^{-1}(c x)\right)+2 \cos ^{-1}(c x) \left(\sin \left(2 \cos ^{-1}(c x)\right)-\cos ^{-1}(c x)\right)\right)-64 b c f g \sqrt{d-c^2 d x^2} \left(12 \left(1-c^2 x^2\right)^{3/2} \cos ^{-1}(c x)+9 c x-\cos \left(3 \cos ^{-1}(c x)\right)\right)+9 b g^2 \sqrt{d-c^2 d x^2} \left(-8 \cos ^{-1}(c x)^2+\cos \left(4 \cos ^{-1}(c x)\right)+4 \cos ^{-1}(c x) \sin \left(4 \cos ^{-1}(c x)\right)\right)}{1152 c^3 \sqrt{1-c^2 x^2}}","\frac{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)-\frac{f^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{4 b c \sqrt{1-c^2 x^2}}-\frac{2 f g \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{3 c^2}-\frac{g^2 x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{8 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)-\frac{g^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{16 b c^3 \sqrt{1-c^2 x^2}}+\frac{b c f^2 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}-\frac{2 b f g x \sqrt{d-c^2 d x^2}}{3 c \sqrt{1-c^2 x^2}}+\frac{2 b c f g x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}-\frac{b g^2 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}}+\frac{b c g^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}",1,"(48*a*c*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(12*c^2*f^2*x + 16*f*g*(-1 + c^2*x^2) + 3*g^2*x*(-1 + 2*c^2*x^2)) - 144*a*Sqrt[d]*(4*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 64*b*c*f*g*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*(1 - c^2*x^2)^(3/2)*ArcCos[c*x] - Cos[3*ArcCos[c*x]]) + 144*b*c^2*f^2*Sqrt[d - c^2*d*x^2]*(Cos[2*ArcCos[c*x]] + 2*ArcCos[c*x]*(-ArcCos[c*x] + Sin[2*ArcCos[c*x]])) + 9*b*g^2*Sqrt[d - c^2*d*x^2]*(-8*ArcCos[c*x]^2 + Cos[4*ArcCos[c*x]] + 4*ArcCos[c*x]*Sin[4*ArcCos[c*x]]))/(1152*c^3*Sqrt[1 - c^2*x^2])","A",1
3,1,219,238,1.5938994,"\int (f+g x) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]),x]","\frac{12 a \sqrt{d-c^2 d x^2} \left(3 c^2 f x+2 g \left(c^2 x^2-1\right)\right)-36 a c \sqrt{d} f \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+\frac{9 b c f \sqrt{d-c^2 d x^2} \left(-2 \cos ^{-1}(c x)^2+\cos \left(2 \cos ^{-1}(c x)\right)+2 \cos ^{-1}(c x) \sin \left(2 \cos ^{-1}(c x)\right)\right)}{\sqrt{1-c^2 x^2}}+\frac{2 b g \sqrt{d-c^2 d x^2} \left(-12 \left(1-c^2 x^2\right)^{3/2} \cos ^{-1}(c x)-9 c x+\cos \left(3 \cos ^{-1}(c x)\right)\right)}{\sqrt{1-c^2 x^2}}}{72 c^2}","\frac{1}{2} f x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)-\frac{f \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{4 b c \sqrt{1-c^2 x^2}}-\frac{g \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{3 c^2}+\frac{b c f x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}-\frac{b g x \sqrt{d-c^2 d x^2}}{3 c \sqrt{1-c^2 x^2}}+\frac{b c g x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}",1,"(12*a*Sqrt[d - c^2*d*x^2]*(3*c^2*f*x + 2*g*(-1 + c^2*x^2)) - 36*a*c*Sqrt[d]*f*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (2*b*g*Sqrt[d - c^2*d*x^2]*(-9*c*x - 12*(1 - c^2*x^2)^(3/2)*ArcCos[c*x] + Cos[3*ArcCos[c*x]]))/Sqrt[1 - c^2*x^2] + (9*b*c*f*Sqrt[d - c^2*d*x^2]*(-2*ArcCos[c*x]^2 + Cos[2*ArcCos[c*x]] + 2*ArcCos[c*x]*Sin[2*ArcCos[c*x]]))/Sqrt[1 - c^2*x^2])/(72*c^2)","A",1
4,1,1095,725,3.6932188,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{f+g x} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(f + g*x),x]","-\frac{-2 a \sqrt{d-c^2 d x^2} g+2 a c \sqrt{d} f \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-2 a \sqrt{d} \sqrt{g^2-c^2 f^2} \log (f+g x)+2 a \sqrt{d} \sqrt{g^2-c^2 f^2} \log \left(d \left(f x c^2+g\right)+\sqrt{d} \sqrt{g^2-c^2 f^2} \sqrt{d-c^2 d x^2}\right)+b \sqrt{d-c^2 d x^2} \left(\frac{c f \cos ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}-2 g \cos ^{-1}(c x)+\frac{2 (g-c f) (c f+g) \left(2 \cos ^{-1}(c x) \tanh ^{-1}\left(\frac{(c f+g) \cot \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)-2 \cos ^{-1}\left(-\frac{c f}{g}\right) \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)+\left(\cos ^{-1}\left(-\frac{c f}{g}\right)-2 i \tanh ^{-1}\left(\frac{(c f+g) \cot \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)+2 i \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right) \log \left(\frac{e^{-\frac{1}{2} i \cos ^{-1}(c x)} \sqrt{g^2-c^2 f^2}}{\sqrt{2} \sqrt{g} \sqrt{c (f+g x)}}\right)+\left(\cos ^{-1}\left(-\frac{c f}{g}\right)+2 i \left(\tanh ^{-1}\left(\frac{(c f+g) \cot \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)-\tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right)\right) \log \left(\frac{e^{\frac{1}{2} i \cos ^{-1}(c x)} \sqrt{g^2-c^2 f^2}}{\sqrt{2} \sqrt{g} \sqrt{c (f+g x)}}\right)-\left(\cos ^{-1}\left(-\frac{c f}{g}\right)-2 i \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right) \log \left(\frac{(c f+g) \left(-i c f+i g+\sqrt{g^2-c^2 f^2}\right) \left(\tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)-i\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)-\left(\cos ^{-1}\left(-\frac{c f}{g}\right)+2 i \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right) \log \left(\frac{(c f+g) \left(i c f-i g+\sqrt{g^2-c^2 f^2}\right) \left(\tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)+i\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)+i \left(\text{Li}_2\left(\frac{\left(c f-i \sqrt{g^2-c^2 f^2}\right) \left(c f+g-\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)-\text{Li}_2\left(\frac{\left(c f+i \sqrt{g^2-c^2 f^2}\right) \left(c f+g-\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)\right)\right)}{\sqrt{g^2-c^2 f^2} \sqrt{1-c^2 x^2}}-\frac{2 c g x}{\sqrt{1-c^2 x^2}}\right)}{2 g^2}","\frac{\sqrt{d-c^2 d x^2} \left(1-\frac{c^2 f^2}{g^2}\right) \left(a+b \cos ^{-1}(c x)\right)^2}{2 b c \sqrt{1-c^2 x^2} (f+g x)}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{2 b c (f+g x)}-\frac{c x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{2 b g \sqrt{1-c^2 x^2}}-\frac{a \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \tan ^{-1}\left(\frac{c^2 f x+g}{\sqrt{1-c^2 x^2} \sqrt{c^2 f^2-g^2}}\right)}{g^2 \sqrt{1-c^2 x^2}}+\frac{a \sqrt{d-c^2 d x^2}}{g}-\frac{b \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^2 \sqrt{1-c^2 x^2}}+\frac{b \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^2 \sqrt{1-c^2 x^2}}-\frac{i b \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \cos ^{-1}(c x) \log \left(1+\frac{g e^{i \cos ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^2 \sqrt{1-c^2 x^2}}+\frac{i b \sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2} \cos ^{-1}(c x) \log \left(1+\frac{g e^{i \cos ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)}{g^2 \sqrt{1-c^2 x^2}}+\frac{b c x \sqrt{d-c^2 d x^2}}{g \sqrt{1-c^2 x^2}}+\frac{b \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g}",1,"-1/2*(-2*a*g*Sqrt[d - c^2*d*x^2] + 2*a*c*Sqrt[d]*f*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 2*a*Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Log[f + g*x] + 2*a*Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Log[d*(g + c^2*f*x) + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[d - c^2*d*x^2]] + b*Sqrt[d - c^2*d*x^2]*((-2*c*g*x)/Sqrt[1 - c^2*x^2] - 2*g*ArcCos[c*x] + (c*f*ArcCos[c*x]^2)/Sqrt[1 - c^2*x^2] + (2*(-(c*f) + g)*(c*f + g)*(2*ArcCos[c*x]*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - 2*ArcCos[-((c*f)/g)]*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^((I/2)*ArcCos[c*x])*Sqrt[g]*Sqrt[c*(f + g*x)])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^((I/2)*ArcCos[c*x])*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*(f + g*x)])] - (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*((-I)*c*f + I*g + Sqrt[-(c^2*f^2) + g^2])*(-I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - (ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(I*c*f - I*g + Sqrt[-(c^2*f^2) + g^2])*(I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] + I*(PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))])))/(Sqrt[-(c^2*f^2) + g^2]*Sqrt[1 - c^2*x^2])))/g^2","A",0
5,1,1130,851,9.7371853,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{(f+g x)^2} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCos[c*x]))/(f + g*x)^2,x]","\frac{a \sqrt{d} f \log (f+g x) c^2}{g^2 \sqrt{g^2-c^2 f^2}}-\frac{a \sqrt{d} f \log \left(d f x c^2+d g+\sqrt{d} \sqrt{g^2-c^2 f^2} \sqrt{-d \left(c^2 x^2-1\right)}\right) c^2}{g^2 \sqrt{g^2-c^2 f^2}}+\frac{a \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{-d \left(c^2 x^2-1\right)}}{\sqrt{d} \left(c^2 x^2-1\right)}\right) c}{g^2}-\frac{b \sqrt{d \left(1-c^2 x^2\right)} \left(-\frac{\cos ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}+\frac{2 g \cos ^{-1}(c x)}{c f+c g x}+\frac{2 \log \left(\frac{g x}{f}+1\right)}{\sqrt{1-c^2 x^2}}+\frac{2 c f \left(2 \cos ^{-1}(c x) \tanh ^{-1}\left(\frac{(c f+g) \cot \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)-2 \cos ^{-1}\left(-\frac{c f}{g}\right) \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)+\left(\cos ^{-1}\left(-\frac{c f}{g}\right)-2 i \tanh ^{-1}\left(\frac{(c f+g) \cot \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)+2 i \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right) \log \left(\frac{e^{-\frac{1}{2} i \cos ^{-1}(c x)} \sqrt{g^2-c^2 f^2}}{\sqrt{2} \sqrt{g} \sqrt{c f+c g x}}\right)+\left(\cos ^{-1}\left(-\frac{c f}{g}\right)+2 i \left(\tanh ^{-1}\left(\frac{(c f+g) \cot \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)-\tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right)\right) \log \left(\frac{e^{\frac{1}{2} i \cos ^{-1}(c x)} \sqrt{g^2-c^2 f^2}}{\sqrt{2} \sqrt{g} \sqrt{c f+c g x}}\right)-\left(\cos ^{-1}\left(-\frac{c f}{g}\right)-2 i \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right) \log \left(\frac{(c f+g) \left(-i c f+i g+\sqrt{g^2-c^2 f^2}\right) \left(\tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)-i\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)-\left(\cos ^{-1}\left(-\frac{c f}{g}\right)+2 i \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right) \log \left(\frac{(c f+g) \left(i c f-i g+\sqrt{g^2-c^2 f^2}\right) \left(\tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)+i\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)+i \left(\text{Li}_2\left(\frac{\left(c f-i \sqrt{g^2-c^2 f^2}\right) \left(c f+g-\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)-\text{Li}_2\left(\frac{\left(c f+i \sqrt{g^2-c^2 f^2}\right) \left(c f+g-\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)\right)\right)}{\sqrt{g^2-c^2 f^2} \sqrt{1-c^2 x^2}}\right) c}{2 g^2}-\frac{a \sqrt{-d \left(c^2 x^2-1\right)}}{g (f+g x)}","\frac{b f^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2 c^3}{2 g^2 \left(c^2 f^2-g^2\right) \sqrt{1-c^2 x^2}}-\frac{a f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) c^3}{g^2 \left(c^2 f^2-g^2\right) \sqrt{1-c^2 x^2}}+\frac{a f \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\frac{f x c^2+g}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left(\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}+1\right) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left(\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}+1\right) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{b f \sqrt{d-c^2 d x^2} \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{b f \sqrt{d-c^2 d x^2} \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{b \sqrt{d-c^2 d x^2} \log (f+g x) c}{g^2 \sqrt{1-c^2 x^2}}-\frac{b \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{2 b (f+g x)^2 c}-\frac{\left(f x c^2+g\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{2 b \left(c^2 f^2-g^2\right) (f+g x)^2 \sqrt{1-c^2 x^2} c}",1,"-((a*Sqrt[-(d*(-1 + c^2*x^2))])/(g*(f + g*x))) + (a*c*Sqrt[d]*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/g^2 + (a*c^2*Sqrt[d]*f*Log[f + g*x])/(g^2*Sqrt[-(c^2*f^2) + g^2]) - (a*c^2*Sqrt[d]*f*Log[d*g + c^2*d*f*x + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[-(d*(-1 + c^2*x^2))]])/(g^2*Sqrt[-(c^2*f^2) + g^2]) - (b*c*Sqrt[d*(1 - c^2*x^2)]*((2*g*ArcCos[c*x])/(c*f + c*g*x) - ArcCos[c*x]^2/Sqrt[1 - c^2*x^2] + (2*Log[1 + (g*x)/f])/Sqrt[1 - c^2*x^2] + (2*c*f*(2*ArcCos[c*x]*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - 2*ArcCos[-((c*f)/g)]*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^((I/2)*ArcCos[c*x])*Sqrt[g]*Sqrt[c*f + c*g*x])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^((I/2)*ArcCos[c*x])*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*f + c*g*x])] - (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*((-I)*c*f + I*g + Sqrt[-(c^2*f^2) + g^2])*(-I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - (ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(I*c*f - I*g + Sqrt[-(c^2*f^2) + g^2])*(I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] + I*(PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))])))/(Sqrt[-(c^2*f^2) + g^2]*Sqrt[1 - c^2*x^2])))/(2*g^2)","A",0
6,1,910,959,4.8400726,"\int (f+g x)^3 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]),x]","\frac{-88200 b c d f \left(2 c^2 f^2+g^2\right) \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2+140 b d \sqrt{d-c^2 d x^2} \left(6720 f^2 g x^2 \sqrt{1-c^2 x^2} c^4+1680 f^3 \sin \left(2 \cos ^{-1}(c x)\right) c^3-210 f^3 \sin \left(4 \cos ^{-1}(c x)\right) c^3-420 f^2 g \sin \left(3 \cos ^{-1}(c x)\right) c^2-252 f^2 g \sin \left(5 \cos ^{-1}(c x)\right) c^2-1256 g^3 x^2 \sqrt{1-c^2 x^2} c^2-4200 f^2 g \sqrt{1-c^2 x^2} c^2+315 f g^2 \sin \left(2 \cos ^{-1}(c x)\right) c+315 f g^2 \sin \left(4 \cos ^{-1}(c x)\right) c-105 f g^2 \sin \left(6 \cos ^{-1}(c x)\right) c+864 g^3 \left(1-c^2 x^2\right)^{3/2} \cos \left(2 \cos ^{-1}(c x)\right)+120 g^3 \left(1-c^2 x^2\right)^{3/2} \cos \left(4 \cos ^{-1}(c x)\right)+140 g^3 \sin \left(3 \cos ^{-1}(c x)\right)+84 g^3 \sin \left(5 \cos ^{-1}(c x)\right)+416 g^3 \sqrt{1-c^2 x^2}\right) \cos ^{-1}(c x)-176400 a c d^{3/2} f \left(2 c^2 f^2+g^2\right) \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-d \sqrt{d-c^2 d x^2} \left(134400 a g^3 x^6 \sqrt{1-c^2 x^2} c^6+470400 a f g^2 x^5 \sqrt{1-c^2 x^2} c^6+564480 a f^2 g x^4 \sqrt{1-c^2 x^2} c^6+235200 a f^3 x^3 \sqrt{1-c^2 x^2} c^6-215040 a g^3 x^4 \sqrt{1-c^2 x^2} c^4-823200 a f g^2 x^3 \sqrt{1-c^2 x^2} c^4-1128960 a f^2 g x^2 \sqrt{1-c^2 x^2} c^4-588000 a f^3 x \sqrt{1-c^2 x^2} c^4+352800 b f^2 g x c^3+7350 b f^3 \cos \left(4 \cos ^{-1}(c x)\right) c^3+7056 b f^2 g \cos \left(5 \cos ^{-1}(c x)\right) c^2+26880 a g^3 x^2 \sqrt{1-c^2 x^2} c^2+564480 a f^2 g \sqrt{1-c^2 x^2} c^2+176400 a f g^2 x \sqrt{1-c^2 x^2} c^2+44100 b g^3 x c-7350 b f \left(16 c^2 f^2+3 g^2\right) \cos \left(2 \cos ^{-1}(c x)\right) c-11025 b f g^2 \cos \left(4 \cos ^{-1}(c x)\right) c+2450 b f g^2 \cos \left(6 \cos ^{-1}(c x)\right) c-4900 b g \left(12 c^2 f^2+g^2\right) \cos \left(3 \cos ^{-1}(c x)\right)-588 b g^3 \cos \left(5 \cos ^{-1}(c x)\right)+300 b g^3 \cos \left(7 \cos ^{-1}(c x)\right)+53760 a g^3 \sqrt{1-c^2 x^2}\right)}{940800 c^4 \sqrt{1-c^2 x^2}}","-\frac{b c^3 d g^3 \sqrt{d-c^2 d x^2} x^7}{49 \sqrt{1-c^2 x^2}}-\frac{b c^3 d f g^2 \sqrt{d-c^2 d x^2} x^6}{12 \sqrt{1-c^2 x^2}}+\frac{8 b c d g^3 \sqrt{d-c^2 d x^2} x^5}{175 \sqrt{1-c^2 x^2}}-\frac{3 b c^3 d f^2 g \sqrt{d-c^2 d x^2} x^5}{25 \sqrt{1-c^2 x^2}}-\frac{b c^3 d f^3 \sqrt{d-c^2 d x^2} x^4}{16 \sqrt{1-c^2 x^2}}+\frac{7 b c d f g^2 \sqrt{d-c^2 d x^2} x^4}{32 \sqrt{1-c^2 x^2}}+\frac{3}{8} d f g^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x^3+\frac{1}{2} d f g^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x^3-\frac{b d g^3 \sqrt{d-c^2 d x^2} x^3}{105 c \sqrt{1-c^2 x^2}}+\frac{2 b c d f^2 g \sqrt{d-c^2 d x^2} x^3}{5 \sqrt{1-c^2 x^2}}+\frac{5 b c d f^3 \sqrt{d-c^2 d x^2} x^2}{16 \sqrt{1-c^2 x^2}}-\frac{3 b d f g^2 \sqrt{d-c^2 d x^2} x^2}{32 c \sqrt{1-c^2 x^2}}+\frac{3}{8} d f^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x-\frac{3 d f g^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x}{16 c^2}+\frac{1}{4} d f^3 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x-\frac{2 b d g^3 \sqrt{d-c^2 d x^2} x}{35 c^3 \sqrt{1-c^2 x^2}}-\frac{3 b d f^2 g \sqrt{d-c^2 d x^2} x}{5 c \sqrt{1-c^2 x^2}}-\frac{3 d f^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{16 b c \sqrt{1-c^2 x^2}}-\frac{3 d f g^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{32 b c^3 \sqrt{1-c^2 x^2}}+\frac{d g^3 \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{7 c^4}-\frac{d g^3 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{5 c^4}-\frac{3 d f^2 g \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{5 c^2}",1,"(-88200*b*c*d*f*(2*c^2*f^2 + g^2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 176400*a*c*d^(3/2)*f*(2*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - d*Sqrt[d - c^2*d*x^2]*(352800*b*c^3*f^2*g*x + 44100*b*c*g^3*x + 564480*a*c^2*f^2*g*Sqrt[1 - c^2*x^2] + 53760*a*g^3*Sqrt[1 - c^2*x^2] - 588000*a*c^4*f^3*x*Sqrt[1 - c^2*x^2] + 176400*a*c^2*f*g^2*x*Sqrt[1 - c^2*x^2] - 1128960*a*c^4*f^2*g*x^2*Sqrt[1 - c^2*x^2] + 26880*a*c^2*g^3*x^2*Sqrt[1 - c^2*x^2] + 235200*a*c^6*f^3*x^3*Sqrt[1 - c^2*x^2] - 823200*a*c^4*f*g^2*x^3*Sqrt[1 - c^2*x^2] + 564480*a*c^6*f^2*g*x^4*Sqrt[1 - c^2*x^2] - 215040*a*c^4*g^3*x^4*Sqrt[1 - c^2*x^2] + 470400*a*c^6*f*g^2*x^5*Sqrt[1 - c^2*x^2] + 134400*a*c^6*g^3*x^6*Sqrt[1 - c^2*x^2] - 7350*b*c*f*(16*c^2*f^2 + 3*g^2)*Cos[2*ArcCos[c*x]] - 4900*b*g*(12*c^2*f^2 + g^2)*Cos[3*ArcCos[c*x]] + 7350*b*c^3*f^3*Cos[4*ArcCos[c*x]] - 11025*b*c*f*g^2*Cos[4*ArcCos[c*x]] + 7056*b*c^2*f^2*g*Cos[5*ArcCos[c*x]] - 588*b*g^3*Cos[5*ArcCos[c*x]] + 2450*b*c*f*g^2*Cos[6*ArcCos[c*x]] + 300*b*g^3*Cos[7*ArcCos[c*x]]) + 140*b*d*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(-4200*c^2*f^2*g*Sqrt[1 - c^2*x^2] + 416*g^3*Sqrt[1 - c^2*x^2] + 6720*c^4*f^2*g*x^2*Sqrt[1 - c^2*x^2] - 1256*c^2*g^3*x^2*Sqrt[1 - c^2*x^2] + 864*g^3*(1 - c^2*x^2)^(3/2)*Cos[2*ArcCos[c*x]] + 120*g^3*(1 - c^2*x^2)^(3/2)*Cos[4*ArcCos[c*x]] + 1680*c^3*f^3*Sin[2*ArcCos[c*x]] + 315*c*f*g^2*Sin[2*ArcCos[c*x]] - 420*c^2*f^2*g*Sin[3*ArcCos[c*x]] + 140*g^3*Sin[3*ArcCos[c*x]] - 210*c^3*f^3*Sin[4*ArcCos[c*x]] + 315*c*f*g^2*Sin[4*ArcCos[c*x]] - 252*c^2*f^2*g*Sin[5*ArcCos[c*x]] + 84*g^3*Sin[5*ArcCos[c*x]] - 105*c*f*g^2*Sin[6*ArcCos[c*x]]))/(940800*c^4*Sqrt[1 - c^2*x^2])","A",1
7,1,591,680,2.2687717,"\int (f+g x)^2 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]),x]","\frac{-d \sqrt{d-c^2 d x^2} \left(23040 a c f g \sqrt{1-c^2 x^2}+3600 a c g^2 x \sqrt{1-c^2 x^2}+14400 a c^5 f^2 x^3 \sqrt{1-c^2 x^2}+23040 a c^5 f g x^4 \sqrt{1-c^2 x^2}+9600 a c^5 g^2 x^5 \sqrt{1-c^2 x^2}-36000 a c^3 f^2 x \sqrt{1-c^2 x^2}-46080 a c^3 f g x^2 \sqrt{1-c^2 x^2}-16800 a c^3 g^2 x^3 \sqrt{1-c^2 x^2}-450 b \left(16 c^2 f^2+g^2\right) \cos \left(2 \cos ^{-1}(c x)\right)+450 b c^2 f^2 \cos \left(4 \cos ^{-1}(c x)\right)+14400 b c^2 f g x-2400 b c f g \cos \left(3 \cos ^{-1}(c x)\right)+288 b c f g \cos \left(5 \cos ^{-1}(c x)\right)-225 b g^2 \cos \left(4 \cos ^{-1}(c x)\right)+50 b g^2 \cos \left(6 \cos ^{-1}(c x)\right)\right)-3600 a d^{3/2} \sqrt{1-c^2 x^2} \left(6 c^2 f^2+g^2\right) \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-1800 b d \sqrt{d-c^2 d x^2} \left(6 c^2 f^2+g^2\right) \cos ^{-1}(c x)^2+60 b d \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \left(15 \left(16 c^2 f^2+g^2\right) \sin \left(2 \cos ^{-1}(c x)\right)-30 c^2 f^2 \sin \left(4 \cos ^{-1}(c x)\right)-400 c f g \sqrt{1-c^2 x^2}+640 c^3 f g x^2 \sqrt{1-c^2 x^2}-40 c f g \sin \left(3 \cos ^{-1}(c x)\right)-24 c f g \sin \left(5 \cos ^{-1}(c x)\right)+15 g^2 \sin \left(4 \cos ^{-1}(c x)\right)-5 g^2 \sin \left(6 \cos ^{-1}(c x)\right)\right)}{57600 c^3 \sqrt{1-c^2 x^2}}","\frac{3}{8} d f^2 x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)+\frac{1}{4} d f^2 x \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)-\frac{3 d f^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{16 b c \sqrt{1-c^2 x^2}}-\frac{2 d f g \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{5 c^2}-\frac{d g^2 x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{16 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)+\frac{1}{6} d g^2 x^3 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)-\frac{d g^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{32 b c^3 \sqrt{1-c^2 x^2}}+\frac{5 b c d f^2 x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{2 b d f g x \sqrt{d-c^2 d x^2}}{5 c \sqrt{1-c^2 x^2}}+\frac{4 b c d f g x^3 \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}}-\frac{b d g^2 x^2 \sqrt{d-c^2 d x^2}}{32 c \sqrt{1-c^2 x^2}}+\frac{7 b c d g^2 x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}-\frac{b c^3 d f^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{2 b c^3 d f g x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}-\frac{b c^3 d g^2 x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{1-c^2 x^2}}",1,"(-1800*b*d*(6*c^2*f^2 + g^2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 3600*a*d^(3/2)*(6*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - d*Sqrt[d - c^2*d*x^2]*(14400*b*c^2*f*g*x + 23040*a*c*f*g*Sqrt[1 - c^2*x^2] - 36000*a*c^3*f^2*x*Sqrt[1 - c^2*x^2] + 3600*a*c*g^2*x*Sqrt[1 - c^2*x^2] - 46080*a*c^3*f*g*x^2*Sqrt[1 - c^2*x^2] + 14400*a*c^5*f^2*x^3*Sqrt[1 - c^2*x^2] - 16800*a*c^3*g^2*x^3*Sqrt[1 - c^2*x^2] + 23040*a*c^5*f*g*x^4*Sqrt[1 - c^2*x^2] + 9600*a*c^5*g^2*x^5*Sqrt[1 - c^2*x^2] - 450*b*(16*c^2*f^2 + g^2)*Cos[2*ArcCos[c*x]] - 2400*b*c*f*g*Cos[3*ArcCos[c*x]] + 450*b*c^2*f^2*Cos[4*ArcCos[c*x]] - 225*b*g^2*Cos[4*ArcCos[c*x]] + 288*b*c*f*g*Cos[5*ArcCos[c*x]] + 50*b*g^2*Cos[6*ArcCos[c*x]]) + 60*b*d*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(-400*c*f*g*Sqrt[1 - c^2*x^2] + 640*c^3*f*g*x^2*Sqrt[1 - c^2*x^2] + 15*(16*c^2*f^2 + g^2)*Sin[2*ArcCos[c*x]] - 40*c*f*g*Sin[3*ArcCos[c*x]] - 30*c^2*f^2*Sin[4*ArcCos[c*x]] + 15*g^2*Sin[4*ArcCos[c*x]] - 24*c*f*g*Sin[5*ArcCos[c*x]] - 5*g^2*Sin[6*ArcCos[c*x]]))/(57600*c^3*Sqrt[1 - c^2*x^2])","A",1
8,1,337,370,1.6173473,"\int (f+g x) \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]),x]","\frac{-d \sqrt{d-c^2 d x^2} \left(3 \left(80 a \sqrt{1-c^2 x^2} \left(5 c^2 f x \left(2 c^2 x^2-5\right)+8 g \left(c^2 x^2-1\right)^2\right)+25 b c f \cos \left(4 \cos ^{-1}(c x)\right)+400 b c g x+8 b g \cos \left(5 \cos ^{-1}(c x)\right)\right)-1200 b c f \cos \left(2 \cos ^{-1}(c x)\right)-200 b g \cos \left(3 \cos ^{-1}(c x)\right)\right)-3600 a c d^{3/2} f \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+20 b d \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \left(160 c^2 g x^2 \sqrt{1-c^2 x^2}-100 g \sqrt{1-c^2 x^2}+120 c f \sin \left(2 \cos ^{-1}(c x)\right)-15 c f \sin \left(4 \cos ^{-1}(c x)\right)-10 g \sin \left(3 \cos ^{-1}(c x)\right)-6 g \sin \left(5 \cos ^{-1}(c x)\right)\right)-1800 b c d f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2}{9600 c^2 \sqrt{1-c^2 x^2}}","\frac{3}{8} d f x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)+\frac{1}{4} d f x \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)-\frac{3 d f \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{16 b c \sqrt{1-c^2 x^2}}-\frac{d g \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{5 c^2}+\frac{5 b c d f x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{b d g x \sqrt{d-c^2 d x^2}}{5 c \sqrt{1-c^2 x^2}}+\frac{2 b c d g x^3 \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}}-\frac{b c^3 d f x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}-\frac{b c^3 d g x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}",1,"(-1800*b*c*d*f*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 3600*a*c*d^(3/2)*f*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - d*Sqrt[d - c^2*d*x^2]*(-1200*b*c*f*Cos[2*ArcCos[c*x]] - 200*b*g*Cos[3*ArcCos[c*x]] + 3*(400*b*c*g*x + 80*a*Sqrt[1 - c^2*x^2]*(8*g*(-1 + c^2*x^2)^2 + 5*c^2*f*x*(-5 + 2*c^2*x^2)) + 25*b*c*f*Cos[4*ArcCos[c*x]] + 8*b*g*Cos[5*ArcCos[c*x]])) + 20*b*d*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(-100*g*Sqrt[1 - c^2*x^2] + 160*c^2*g*x^2*Sqrt[1 - c^2*x^2] + 120*c*f*Sin[2*ArcCos[c*x]] - 10*g*Sin[3*ArcCos[c*x]] - 15*c*f*Sin[4*ArcCos[c*x]] - 6*g*Sin[5*ArcCos[c*x]]))/(9600*c^2*Sqrt[1 - c^2*x^2])","A",1
9,1,3034,1064,12.6589106,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cos ^{-1}(c x)\right)}{f+g x} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCos[c*x]))/(f + g*x),x]","\text{Result too large to show}","-\frac{b d x^3 \sqrt{d-c^2 d x^2} c^3}{9 g \sqrt{1-c^2 x^2}}+\frac{b d f x^2 \sqrt{d-c^2 d x^2} c^3}{4 g^2 \sqrt{1-c^2 x^2}}+\frac{d f x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) c^2}{2 g^2}+\frac{d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2 c}{2 b g^3 \sqrt{1-c^2 x^2}}-\frac{d f \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2 c}{4 b g^2 \sqrt{1-c^2 x^2}}-\frac{b d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} c}{g^3 \sqrt{1-c^2 x^2}}+\frac{b d x \sqrt{d-c^2 d x^2} c}{3 g \sqrt{1-c^2 x^2}}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^3}+\frac{d \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{3 g}+\frac{a d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\frac{f x c^2+g}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right)}{g^4 \sqrt{1-c^2 x^2}}+\frac{i b d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left(\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}+1\right)}{g^4 \sqrt{1-c^2 x^2}}-\frac{i b d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left(\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}+1\right)}{g^4 \sqrt{1-c^2 x^2}}+\frac{b d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^4 \sqrt{1-c^2 x^2}}-\frac{b d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^4 \sqrt{1-c^2 x^2}}-\frac{a d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}}{g^3}+\frac{d (c f-g) (c f+g) \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{2 b g^2 (f+g x) c}+\frac{d \left(c^2 f^2-g^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{2 b g^4 (f+g x) \sqrt{1-c^2 x^2} c}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*((a*d*(-3*c^2*f^2 + 4*g^2))/(3*g^3) + (a*c^2*d*f*x)/(2*g^2) - (a*c^2*d*x^2)/(3*g)) + (a*c*d^(3/2)*f*(2*c^2*f^2 - 3*g^2)*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(2*g^4) + (a*d^(3/2)*(-(c^2*f^2) + g^2)^(3/2)*Log[f + g*x])/g^4 - (a*d^(3/2)*(-(c^2*f^2) + g^2)^(3/2)*Log[d*g + c^2*d*f*x + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[-(d*(-1 + c^2*x^2))]])/g^4 - (b*d*Sqrt[d*(1 - c^2*x^2)]*((-2*c*g*x)/Sqrt[1 - c^2*x^2] - 2*g*ArcCos[c*x] + (c*f*ArcCos[c*x]^2)/Sqrt[1 - c^2*x^2] + (2*(-(c*f) + g)*(c*f + g)*(2*ArcCos[c*x]*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - 2*ArcCos[-((c*f)/g)]*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^((I/2)*ArcCos[c*x])*Sqrt[g]*Sqrt[c*f + c*g*x])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^((I/2)*ArcCos[c*x])*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*f + c*g*x])] - (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*((-I)*c*f + I*g + Sqrt[-(c^2*f^2) + g^2])*(-I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - (ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(I*c*f - I*g + Sqrt[-(c^2*f^2) + g^2])*(I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] + I*(PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))])))/(Sqrt[-(c^2*f^2) + g^2]*Sqrt[1 - c^2*x^2])))/(2*g^2) + (b*d*Sqrt[d*(1 - c^2*x^2)]*((9*(2*ArcCos[c*x]*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - 2*ArcCos[-((c*f)/g)]*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^((I/2)*ArcCos[c*x])*Sqrt[g]*Sqrt[c*f + c*g*x])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^((I/2)*ArcCos[c*x])*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*f + c*g*x])] - (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*((-I)*c*f + I*g + Sqrt[-(c^2*f^2) + g^2])*(-I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - (ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(I*c*f - I*g + Sqrt[-(c^2*f^2) + g^2])*(I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] + I*(PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))])))/Sqrt[-(c^2*f^2) + g^2] + (18*c*g*(-4*c^2*f^2 + g^2)*x + 18*g*(-4*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*ArcCos[c*x] + 18*c*f*(2*c^2*f^2 - g^2)*ArcCos[c*x]^2 + 9*c*f*g^2*Cos[2*ArcCos[c*x]] - 2*g^3*Cos[3*ArcCos[c*x]] - (9*(8*c^4*f^4 - 8*c^2*f^2*g^2 + g^4)*(2*ArcCos[c*x]*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - 2*ArcCos[-((c*f)/g)]*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^((I/2)*ArcCos[c*x])*Sqrt[g]*Sqrt[c*f + c*g*x])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^((I/2)*ArcCos[c*x])*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*f + c*g*x])] - (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*((-I)*c*f + I*g + Sqrt[-(c^2*f^2) + g^2])*(-I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - (ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(I*c*f - I*g + Sqrt[-(c^2*f^2) + g^2])*(I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] + I*(PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))])))/Sqrt[-(c^2*f^2) + g^2] + 18*c*f*g^2*ArcCos[c*x]*Sin[2*ArcCos[c*x]] - 6*g^3*ArcCos[c*x]*Sin[3*ArcCos[c*x]])/g^4))/(72*Sqrt[1 - c^2*x^2])","B",0
10,1,1144,1281,7.9859318,"\int (f+g x)^3 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]","\frac{d^2 \left(-3175200 b c f \left(8 c^2 f^2+3 g^2\right) \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2+504 b \sqrt{d-c^2 d x^2} \left(503424 f^2 g x^2 \sqrt{1-c^2 x^2} c^4+75600 f^3 \sin \left(2 \cos ^{-1}(c x)\right) c^3-15120 f^3 \sin \left(4 \cos ^{-1}(c x)\right) c^3+1680 f^3 \sin \left(6 \cos ^{-1}(c x)\right) c^3-40320 f^2 g \sin \left(3 \cos ^{-1}(c x)\right) c^2-24192 f^2 g \sin \left(5 \cos ^{-1}(c x)\right) c^2-120576 g^3 x^2 \sqrt{1-c^2 x^2} c^2-261504 f^2 g \sqrt{1-c^2 x^2} c^2+15120 f g^2 \sin \left(2 \cos ^{-1}(c x)\right) c+7560 f g^2 \sin \left(4 \cos ^{-1}(c x)\right) c-5040 f g^2 \sin \left(6 \cos ^{-1}(c x)\right) c+945 f g^2 \sin \left(8 \cos ^{-1}(c x)\right) c-41472 g \left(3 c^2 f^2-2 g^2\right) \left(1-c^2 x^2\right)^{3/2} \cos \left(2 \cos ^{-1}(c x)\right)-5760 g \left(3 c^2 f^2-2 g^2\right) \left(1-c^2 x^2\right)^{3/2} \cos \left(4 \cos ^{-1}(c x)\right)+6720 g^3 \sin \left(3 \cos ^{-1}(c x)\right)+6048 g^3 \sin \left(5 \cos ^{-1}(c x)\right)+900 g^3 \sin \left(7 \cos ^{-1}(c x)\right)+140 g^3 \sin \left(9 \cos ^{-1}(c x)\right)+62616 g^3 \sqrt{1-c^2 x^2}\right) \cos ^{-1}(c x)-6350400 a c \sqrt{d} f \left(8 c^2 f^2+3 g^2\right) \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+\sqrt{d-c^2 d x^2} \left(18063360 a g^3 x^8 \sqrt{1-c^2 x^2} c^8+60963840 a f g^2 x^7 \sqrt{1-c^2 x^2} c^8+69672960 a f^2 g x^6 \sqrt{1-c^2 x^2} c^8+27095040 a f^3 x^5 \sqrt{1-c^2 x^2} c^8-49029120 a g^3 x^6 \sqrt{1-c^2 x^2} c^6-172730880 a f g^2 x^5 \sqrt{1-c^2 x^2} c^6-209018880 a f^2 g x^4 \sqrt{1-c^2 x^2} c^6-88058880 a f^3 x^3 \sqrt{1-c^2 x^2} c^6+38707200 a g^3 x^4 \sqrt{1-c^2 x^2} c^4+149869440 a f g^2 x^3 \sqrt{1-c^2 x^2} c^4+209018880 a f^2 g x^2 \sqrt{1-c^2 x^2} c^4+111767040 a f^3 x \sqrt{1-c^2 x^2} c^4-38102400 b f^2 g x c^3-1905120 b f^3 \cos \left(4 \cos ^{-1}(c x)\right) c^3+141120 b f^3 \cos \left(6 \cos ^{-1}(c x)\right) c^3-1524096 b f^2 g \cos \left(5 \cos ^{-1}(c x)\right) c^2+155520 b f^2 g \cos \left(7 \cos ^{-1}(c x)\right) c^2-2580480 a g^3 x^2 \sqrt{1-c^2 x^2} c^2-69672960 a f^2 g \sqrt{1-c^2 x^2} c^2-19051200 a f g^2 x \sqrt{1-c^2 x^2} c^2-3810240 b g^3 x c+3810240 b f \left(5 c^2 f^2+g^2\right) \cos \left(2 \cos ^{-1}(c x)\right) c+952560 b f g^2 \cos \left(4 \cos ^{-1}(c x)\right) c-423360 b f g^2 \cos \left(6 \cos ^{-1}(c x)\right) c+59535 b f g^2 \cos \left(8 \cos ^{-1}(c x)\right) c+282240 b g \left(27 c^2 f^2+2 g^2\right) \cos \left(3 \cos ^{-1}(c x)\right)-38880 b g^3 \cos \left(7 \cos ^{-1}(c x)\right)+7840 b g^3 \cos \left(9 \cos ^{-1}(c x)\right)-5160960 a g^3 \sqrt{1-c^2 x^2}\right)\right)}{162570240 c^4 \sqrt{1-c^2 x^2}}","\frac{b c^5 d^2 g^3 \sqrt{d-c^2 d x^2} x^9}{81 \sqrt{1-c^2 x^2}}+\frac{3 b c^5 d^2 f g^2 \sqrt{d-c^2 d x^2} x^8}{64 \sqrt{1-c^2 x^2}}-\frac{19 b c^3 d^2 g^3 \sqrt{d-c^2 d x^2} x^7}{441 \sqrt{1-c^2 x^2}}+\frac{3 b c^5 d^2 f^2 g \sqrt{d-c^2 d x^2} x^7}{49 \sqrt{1-c^2 x^2}}-\frac{17 b c^3 d^2 f g^2 \sqrt{d-c^2 d x^2} x^6}{96 \sqrt{1-c^2 x^2}}+\frac{b c d^2 g^3 \sqrt{d-c^2 d x^2} x^5}{21 \sqrt{1-c^2 x^2}}-\frac{9 b c^3 d^2 f^2 g \sqrt{d-c^2 d x^2} x^5}{35 \sqrt{1-c^2 x^2}}-\frac{5 b c^3 d^2 f^3 \sqrt{d-c^2 d x^2} x^4}{96 \sqrt{1-c^2 x^2}}+\frac{59 b c d^2 f g^2 \sqrt{d-c^2 d x^2} x^4}{256 \sqrt{1-c^2 x^2}}+\frac{15}{64} d^2 f g^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x^3+\frac{3}{8} d^2 f g^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x^3+\frac{5}{16} d^2 f g^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x^3-\frac{b d^2 g^3 \sqrt{d-c^2 d x^2} x^3}{189 c \sqrt{1-c^2 x^2}}+\frac{3 b c d^2 f^2 g \sqrt{d-c^2 d x^2} x^3}{7 \sqrt{1-c^2 x^2}}+\frac{25 b c d^2 f^3 \sqrt{d-c^2 d x^2} x^2}{96 \sqrt{1-c^2 x^2}}-\frac{15 b d^2 f g^2 \sqrt{d-c^2 d x^2} x^2}{256 c \sqrt{1-c^2 x^2}}+\frac{5}{16} d^2 f^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x-\frac{15 d^2 f g^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x}{128 c^2}+\frac{1}{6} d^2 f^3 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x+\frac{5}{24} d^2 f^3 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x-\frac{2 b d^2 g^3 \sqrt{d-c^2 d x^2} x}{63 c^3 \sqrt{1-c^2 x^2}}-\frac{3 b d^2 f^2 g \sqrt{d-c^2 d x^2} x}{7 c \sqrt{1-c^2 x^2}}-\frac{5 d^2 f^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{32 b c \sqrt{1-c^2 x^2}}-\frac{15 d^2 f g^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{256 b c^3 \sqrt{1-c^2 x^2}}+\frac{d^2 g^3 \left(1-c^2 x^2\right)^4 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{9 c^4}-\frac{d^2 g^3 \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{7 c^4}-\frac{3 d^2 f^2 g \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{7 c^2}-\frac{b d^2 f^3 \left(1-c^2 x^2\right)^{5/2} \sqrt{d-c^2 d x^2}}{36 c}",1,"(d^2*(-3175200*b*c*f*(8*c^2*f^2 + 3*g^2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 6350400*a*c*Sqrt[d]*f*(8*c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d - c^2*d*x^2]*(-38102400*b*c^3*f^2*g*x - 3810240*b*c*g^3*x - 69672960*a*c^2*f^2*g*Sqrt[1 - c^2*x^2] - 5160960*a*g^3*Sqrt[1 - c^2*x^2] + 111767040*a*c^4*f^3*x*Sqrt[1 - c^2*x^2] - 19051200*a*c^2*f*g^2*x*Sqrt[1 - c^2*x^2] + 209018880*a*c^4*f^2*g*x^2*Sqrt[1 - c^2*x^2] - 2580480*a*c^2*g^3*x^2*Sqrt[1 - c^2*x^2] - 88058880*a*c^6*f^3*x^3*Sqrt[1 - c^2*x^2] + 149869440*a*c^4*f*g^2*x^3*Sqrt[1 - c^2*x^2] - 209018880*a*c^6*f^2*g*x^4*Sqrt[1 - c^2*x^2] + 38707200*a*c^4*g^3*x^4*Sqrt[1 - c^2*x^2] + 27095040*a*c^8*f^3*x^5*Sqrt[1 - c^2*x^2] - 172730880*a*c^6*f*g^2*x^5*Sqrt[1 - c^2*x^2] + 69672960*a*c^8*f^2*g*x^6*Sqrt[1 - c^2*x^2] - 49029120*a*c^6*g^3*x^6*Sqrt[1 - c^2*x^2] + 60963840*a*c^8*f*g^2*x^7*Sqrt[1 - c^2*x^2] + 18063360*a*c^8*g^3*x^8*Sqrt[1 - c^2*x^2] + 3810240*b*c*f*(5*c^2*f^2 + g^2)*Cos[2*ArcCos[c*x]] + 282240*b*g*(27*c^2*f^2 + 2*g^2)*Cos[3*ArcCos[c*x]] - 1905120*b*c^3*f^3*Cos[4*ArcCos[c*x]] + 952560*b*c*f*g^2*Cos[4*ArcCos[c*x]] - 1524096*b*c^2*f^2*g*Cos[5*ArcCos[c*x]] + 141120*b*c^3*f^3*Cos[6*ArcCos[c*x]] - 423360*b*c*f*g^2*Cos[6*ArcCos[c*x]] + 155520*b*c^2*f^2*g*Cos[7*ArcCos[c*x]] - 38880*b*g^3*Cos[7*ArcCos[c*x]] + 59535*b*c*f*g^2*Cos[8*ArcCos[c*x]] + 7840*b*g^3*Cos[9*ArcCos[c*x]]) + 504*b*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(-261504*c^2*f^2*g*Sqrt[1 - c^2*x^2] + 62616*g^3*Sqrt[1 - c^2*x^2] + 503424*c^4*f^2*g*x^2*Sqrt[1 - c^2*x^2] - 120576*c^2*g^3*x^2*Sqrt[1 - c^2*x^2] - 41472*g*(3*c^2*f^2 - 2*g^2)*(1 - c^2*x^2)^(3/2)*Cos[2*ArcCos[c*x]] - 5760*g*(3*c^2*f^2 - 2*g^2)*(1 - c^2*x^2)^(3/2)*Cos[4*ArcCos[c*x]] + 75600*c^3*f^3*Sin[2*ArcCos[c*x]] + 15120*c*f*g^2*Sin[2*ArcCos[c*x]] - 40320*c^2*f^2*g*Sin[3*ArcCos[c*x]] + 6720*g^3*Sin[3*ArcCos[c*x]] - 15120*c^3*f^3*Sin[4*ArcCos[c*x]] + 7560*c*f*g^2*Sin[4*ArcCos[c*x]] - 24192*c^2*f^2*g*Sin[5*ArcCos[c*x]] + 6048*g^3*Sin[5*ArcCos[c*x]] + 1680*c^3*f^3*Sin[6*ArcCos[c*x]] - 5040*c*f*g^2*Sin[6*ArcCos[c*x]] + 900*g^3*Sin[7*ArcCos[c*x]] + 945*c*f*g^2*Sin[8*ArcCos[c*x]] + 140*g^3*Sin[9*ArcCos[c*x]])))/(162570240*c^4*Sqrt[1 - c^2*x^2])","A",1
11,1,794,940,4.6935849,"\int (f+g x)^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]","\frac{d^2 \left(\sqrt{d-c^2 d x^2} \left(-5160960 a c f g \sqrt{1-c^2 x^2}-705600 a c g^2 x \sqrt{1-c^2 x^2}+3010560 a c^7 f^2 x^5 \sqrt{1-c^2 x^2}+5160960 a c^7 f g x^6 \sqrt{1-c^2 x^2}+2257920 a c^7 g^2 x^7 \sqrt{1-c^2 x^2}-9784320 a c^5 f^2 x^3 \sqrt{1-c^2 x^2}-15482880 a c^5 f g x^4 \sqrt{1-c^2 x^2}-6397440 a c^5 g^2 x^5 \sqrt{1-c^2 x^2}+12418560 a c^3 f^2 x \sqrt{1-c^2 x^2}+15482880 a c^3 f g x^2 \sqrt{1-c^2 x^2}+5550720 a c^3 g^2 x^3 \sqrt{1-c^2 x^2}+141120 b \left(15 c^2 f^2+g^2\right) \cos \left(2 \cos ^{-1}(c x)\right)-211680 b c^2 f^2 \cos \left(4 \cos ^{-1}(c x)\right)+15680 b c^2 f^2 \cos \left(6 \cos ^{-1}(c x)\right)-2822400 b c^2 f g x+564480 b c f g \cos \left(3 \cos ^{-1}(c x)\right)-112896 b c f g \cos \left(5 \cos ^{-1}(c x)\right)+11520 b c f g \cos \left(7 \cos ^{-1}(c x)\right)+35280 b g^2 \cos \left(4 \cos ^{-1}(c x)\right)-15680 b g^2 \cos \left(6 \cos ^{-1}(c x)\right)+2205 b g^2 \cos \left(8 \cos ^{-1}(c x)\right)\right)-705600 a \sqrt{d} \sqrt{1-c^2 x^2} \left(8 c^2 f^2+g^2\right) \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-352800 b \sqrt{d-c^2 d x^2} \left(8 c^2 f^2+g^2\right) \cos ^{-1}(c x)^2+168 b \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \left(25200 c^2 f^2 \sin \left(2 \cos ^{-1}(c x)\right)-5040 c^2 f^2 \sin \left(4 \cos ^{-1}(c x)\right)+560 c^2 f^2 \sin \left(6 \cos ^{-1}(c x)\right)-58112 c f g \sqrt{1-c^2 x^2}-27648 c f g \left(1-c^2 x^2\right)^{3/2} \cos \left(2 \cos ^{-1}(c x)\right)-3840 c f g \left(1-c^2 x^2\right)^{3/2} \cos \left(4 \cos ^{-1}(c x)\right)+111872 c^3 f g x^2 \sqrt{1-c^2 x^2}-8960 c f g \sin \left(3 \cos ^{-1}(c x)\right)-5376 c f g \sin \left(5 \cos ^{-1}(c x)\right)+1680 g^2 \sin \left(2 \cos ^{-1}(c x)\right)+840 g^2 \sin \left(4 \cos ^{-1}(c x)\right)-560 g^2 \sin \left(6 \cos ^{-1}(c x)\right)+105 g^2 \sin \left(8 \cos ^{-1}(c x)\right)\right)\right)}{18063360 c^3 \sqrt{1-c^2 x^2}}","\frac{b c^5 d^2 g^2 \sqrt{d-c^2 d x^2} x^8}{64 \sqrt{1-c^2 x^2}}+\frac{2 b c^5 d^2 f g \sqrt{d-c^2 d x^2} x^7}{49 \sqrt{1-c^2 x^2}}-\frac{17 b c^3 d^2 g^2 \sqrt{d-c^2 d x^2} x^6}{288 \sqrt{1-c^2 x^2}}-\frac{6 b c^3 d^2 f g \sqrt{d-c^2 d x^2} x^5}{35 \sqrt{1-c^2 x^2}}-\frac{5 b c^3 d^2 f^2 \sqrt{d-c^2 d x^2} x^4}{96 \sqrt{1-c^2 x^2}}+\frac{59 b c d^2 g^2 \sqrt{d-c^2 d x^2} x^4}{768 \sqrt{1-c^2 x^2}}+\frac{5}{64} d^2 g^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x^3+\frac{1}{8} d^2 g^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x^3+\frac{5}{48} d^2 g^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x^3+\frac{2 b c d^2 f g \sqrt{d-c^2 d x^2} x^3}{7 \sqrt{1-c^2 x^2}}+\frac{25 b c d^2 f^2 \sqrt{d-c^2 d x^2} x^2}{96 \sqrt{1-c^2 x^2}}-\frac{5 b d^2 g^2 \sqrt{d-c^2 d x^2} x^2}{256 c \sqrt{1-c^2 x^2}}+\frac{5}{16} d^2 f^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x}{128 c^2}+\frac{1}{6} d^2 f^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x+\frac{5}{24} d^2 f^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) x-\frac{2 b d^2 f g \sqrt{d-c^2 d x^2} x}{7 c \sqrt{1-c^2 x^2}}-\frac{5 d^2 f^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{32 b c \sqrt{1-c^2 x^2}}-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{256 b c^3 \sqrt{1-c^2 x^2}}-\frac{2 d^2 f g \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{7 c^2}-\frac{b d^2 f^2 \left(1-c^2 x^2\right)^{5/2} \sqrt{d-c^2 d x^2}}{36 c}",1,"(d^2*(-352800*b*(8*c^2*f^2 + g^2)*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 705600*a*Sqrt[d]*(8*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d - c^2*d*x^2]*(-2822400*b*c^2*f*g*x - 5160960*a*c*f*g*Sqrt[1 - c^2*x^2] + 12418560*a*c^3*f^2*x*Sqrt[1 - c^2*x^2] - 705600*a*c*g^2*x*Sqrt[1 - c^2*x^2] + 15482880*a*c^3*f*g*x^2*Sqrt[1 - c^2*x^2] - 9784320*a*c^5*f^2*x^3*Sqrt[1 - c^2*x^2] + 5550720*a*c^3*g^2*x^3*Sqrt[1 - c^2*x^2] - 15482880*a*c^5*f*g*x^4*Sqrt[1 - c^2*x^2] + 3010560*a*c^7*f^2*x^5*Sqrt[1 - c^2*x^2] - 6397440*a*c^5*g^2*x^5*Sqrt[1 - c^2*x^2] + 5160960*a*c^7*f*g*x^6*Sqrt[1 - c^2*x^2] + 2257920*a*c^7*g^2*x^7*Sqrt[1 - c^2*x^2] + 141120*b*(15*c^2*f^2 + g^2)*Cos[2*ArcCos[c*x]] + 564480*b*c*f*g*Cos[3*ArcCos[c*x]] - 211680*b*c^2*f^2*Cos[4*ArcCos[c*x]] + 35280*b*g^2*Cos[4*ArcCos[c*x]] - 112896*b*c*f*g*Cos[5*ArcCos[c*x]] + 15680*b*c^2*f^2*Cos[6*ArcCos[c*x]] - 15680*b*g^2*Cos[6*ArcCos[c*x]] + 11520*b*c*f*g*Cos[7*ArcCos[c*x]] + 2205*b*g^2*Cos[8*ArcCos[c*x]]) + 168*b*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(-58112*c*f*g*Sqrt[1 - c^2*x^2] + 111872*c^3*f*g*x^2*Sqrt[1 - c^2*x^2] - 27648*c*f*g*(1 - c^2*x^2)^(3/2)*Cos[2*ArcCos[c*x]] - 3840*c*f*g*(1 - c^2*x^2)^(3/2)*Cos[4*ArcCos[c*x]] + 25200*c^2*f^2*Sin[2*ArcCos[c*x]] + 1680*g^2*Sin[2*ArcCos[c*x]] - 8960*c*f*g*Sin[3*ArcCos[c*x]] - 5040*c^2*f^2*Sin[4*ArcCos[c*x]] + 840*g^2*Sin[4*ArcCos[c*x]] - 5376*c*f*g*Sin[5*ArcCos[c*x]] + 560*c^2*f^2*Sin[6*ArcCos[c*x]] - 560*g^2*Sin[6*ArcCos[c*x]] + 105*g^2*Sin[8*ArcCos[c*x]])))/(18063360*c^3*Sqrt[1 - c^2*x^2])","A",1
12,1,526,517,3.0279878,"\int (f+g x) \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cos ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]),x]","\frac{d^2 \left(\sqrt{d-c^2 d x^2} \left(388080 a c^2 f x \sqrt{1-c^2 x^2}+241920 a c^2 g x^2 \sqrt{1-c^2 x^2}-80640 a g \sqrt{1-c^2 x^2}+94080 a c^6 f x^5 \sqrt{1-c^2 x^2}+80640 a c^6 g x^6 \sqrt{1-c^2 x^2}-305760 a c^4 f x^3 \sqrt{1-c^2 x^2}-241920 a c^4 g x^4 \sqrt{1-c^2 x^2}+66150 b c f \cos \left(2 \cos ^{-1}(c x)\right)-6615 b c f \cos \left(4 \cos ^{-1}(c x)\right)+490 b c f \cos \left(6 \cos ^{-1}(c x)\right)-44100 b c g x+8820 b g \cos \left(3 \cos ^{-1}(c x)\right)-1764 b g \cos \left(5 \cos ^{-1}(c x)\right)+180 b g \cos \left(7 \cos ^{-1}(c x)\right)\right)-176400 a c \sqrt{d} f \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+84 b \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \left(3496 c^2 g x^2 \sqrt{1-c^2 x^2}-1816 g \sqrt{1-c^2 x^2}-864 g \left(1-c^2 x^2\right)^{3/2} \cos \left(2 \cos ^{-1}(c x)\right)-120 g \left(1-c^2 x^2\right)^{3/2} \cos \left(4 \cos ^{-1}(c x)\right)+1575 c f \sin \left(2 \cos ^{-1}(c x)\right)-315 c f \sin \left(4 \cos ^{-1}(c x)\right)+35 c f \sin \left(6 \cos ^{-1}(c x)\right)-280 g \sin \left(3 \cos ^{-1}(c x)\right)-168 g \sin \left(5 \cos ^{-1}(c x)\right)\right)-88200 b c f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2\right)}{564480 c^2 \sqrt{1-c^2 x^2}}","\frac{1}{6} d^2 f x \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)+\frac{5}{16} d^2 f x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)+\frac{5}{24} d^2 f x \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)-\frac{5 d^2 f \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{32 b c \sqrt{1-c^2 x^2}}-\frac{d^2 g \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{7 c^2}+\frac{25 b c d^2 f x^2 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}-\frac{b d^2 f \left(1-c^2 x^2\right)^{5/2} \sqrt{d-c^2 d x^2}}{36 c}-\frac{b d^2 g x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}}+\frac{b c d^2 g x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 g x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}-\frac{5 b c^3 d^2 f x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}-\frac{3 b c^3 d^2 g x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}",1,"(d^2*(-88200*b*c*f*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]^2 - 176400*a*c*Sqrt[d]*f*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d - c^2*d*x^2]*(-44100*b*c*g*x - 80640*a*g*Sqrt[1 - c^2*x^2] + 388080*a*c^2*f*x*Sqrt[1 - c^2*x^2] + 241920*a*c^2*g*x^2*Sqrt[1 - c^2*x^2] - 305760*a*c^4*f*x^3*Sqrt[1 - c^2*x^2] - 241920*a*c^4*g*x^4*Sqrt[1 - c^2*x^2] + 94080*a*c^6*f*x^5*Sqrt[1 - c^2*x^2] + 80640*a*c^6*g*x^6*Sqrt[1 - c^2*x^2] + 66150*b*c*f*Cos[2*ArcCos[c*x]] + 8820*b*g*Cos[3*ArcCos[c*x]] - 6615*b*c*f*Cos[4*ArcCos[c*x]] - 1764*b*g*Cos[5*ArcCos[c*x]] + 490*b*c*f*Cos[6*ArcCos[c*x]] + 180*b*g*Cos[7*ArcCos[c*x]]) + 84*b*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*(-1816*g*Sqrt[1 - c^2*x^2] + 3496*c^2*g*x^2*Sqrt[1 - c^2*x^2] - 864*g*(1 - c^2*x^2)^(3/2)*Cos[2*ArcCos[c*x]] - 120*g*(1 - c^2*x^2)^(3/2)*Cos[4*ArcCos[c*x]] + 1575*c*f*Sin[2*ArcCos[c*x]] - 280*g*Sin[3*ArcCos[c*x]] - 315*c*f*Sin[4*ArcCos[c*x]] - 168*g*Sin[5*ArcCos[c*x]] + 35*c*f*Sin[6*ArcCos[c*x]])))/(564480*c^2*Sqrt[1 - c^2*x^2])","A",1
13,1,6216,1637,20.2879379,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cos ^{-1}(c x)\right)}{f+g x} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCos[c*x]))/(f + g*x),x]","\text{Result too large to show}","\frac{b d^2 x^5 \sqrt{d-c^2 d x^2} c^5}{25 g \sqrt{1-c^2 x^2}}-\frac{b d^2 f x^4 \sqrt{d-c^2 d x^2} c^5}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{d^2 f x^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) c^4}{4 g^2}+\frac{b d^2 \left(c^2 f^2-2 g^2\right) x^3 \sqrt{d-c^2 d x^2} c^3}{9 g^3 \sqrt{1-c^2 x^2}}-\frac{b d^2 x^3 \sqrt{d-c^2 d x^2} c^3}{45 g \sqrt{1-c^2 x^2}}-\frac{b d^2 f \left(c^2 f^2-2 g^2\right) x^2 \sqrt{d-c^2 d x^2} c^3}{4 g^4 \sqrt{1-c^2 x^2}}+\frac{b d^2 f x^2 \sqrt{d-c^2 d x^2} c^3}{16 g^2 \sqrt{1-c^2 x^2}}-\frac{d^2 f \left(c^2 f^2-2 g^2\right) x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) c^2}{2 g^4}+\frac{d^2 f x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right) c^2}{8 g^2}+\frac{d^2 f \left(c^2 f^2-2 g^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2 c}{4 b g^4 \sqrt{1-c^2 x^2}}-\frac{d^2 \left(c^2 f^2-g^2\right)^2 x \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2 c}{2 b g^5 \sqrt{1-c^2 x^2}}+\frac{d^2 f \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2 c}{16 b g^2 \sqrt{1-c^2 x^2}}+\frac{b d^2 \left(c^2 f^2-g^2\right)^2 x \sqrt{d-c^2 d x^2} c}{g^5 \sqrt{1-c^2 x^2}}-\frac{b d^2 \left(c^2 f^2-2 g^2\right) x \sqrt{d-c^2 d x^2} c}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{2 b d^2 x \sqrt{d-c^2 d x^2} c}{15 g \sqrt{1-c^2 x^2}}+\frac{b d^2 \left(c^2 f^2-g^2\right)^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g^5}+\frac{d^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{5 g}-\frac{d^2 \left(c^2 f^2-2 g^2\right) \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{3 g^3}-\frac{d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)}{3 g}-\frac{a d^2 \left(c^2 f^2-g^2\right)^{5/2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\frac{f x c^2+g}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right)}{g^6 \sqrt{1-c^2 x^2}}-\frac{i b d^2 \left(c^2 f^2-g^2\right)^{5/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left(\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}+1\right)}{g^6 \sqrt{1-c^2 x^2}}+\frac{i b d^2 \left(c^2 f^2-g^2\right)^{5/2} \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left(\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}+1\right)}{g^6 \sqrt{1-c^2 x^2}}-\frac{b d^2 \left(c^2 f^2-g^2\right)^{5/2} \sqrt{d-c^2 d x^2} \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^6 \sqrt{1-c^2 x^2}}+\frac{b d^2 \left(c^2 f^2-g^2\right)^{5/2} \sqrt{d-c^2 d x^2} \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^6 \sqrt{1-c^2 x^2}}+\frac{a d^2 \left(c^2 f^2-g^2\right)^2 \sqrt{d-c^2 d x^2}}{g^5}-\frac{d^2 \left(c^2 f^2-g^2\right)^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{2 b g^4 (f+g x) c}-\frac{d^2 \left(c^2 f^2-g^2\right)^3 \sqrt{d-c^2 d x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{2 b g^6 (f+g x) \sqrt{1-c^2 x^2} c}",1,"Result too large to show","B",0
14,1,342,450,1.3465789,"\int \frac{(f+g x)^3 \left(a+b \cos ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[((f + g*x)^3*(a + b*ArcCos[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{\sqrt{d} g \left(c^2 x^2-1\right) \left(12 a \sqrt{1-c^2 x^2} \left(c^2 \left(18 f^2+9 f g x+2 g^2 x^2\right)+4 g^2\right)+8 b c x \left(c^2 \left(27 f^2+g^2 x^2\right)+6 g^2\right)+27 b c f g \cos \left(2 \cos ^{-1}(c x)\right)\right)-36 a c f \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left(2 c^2 f^2+3 g^2\right) \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+18 b c \sqrt{d} f \left(c^2 x^2-1\right) \left(2 c^2 f^2+3 g^2\right) \cos ^{-1}(c x)^2+6 b \sqrt{d} g \left(c^2 x^2-1\right) \cos ^{-1}(c x) \left(4 \sqrt{1-c^2 x^2} \left(c^2 \left(9 f^2+g^2 x^2\right)+2 g^2\right)+9 c f g \sin \left(2 \cos ^{-1}(c x)\right)\right)}{72 c^4 \sqrt{d} \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}","-\frac{f^3 \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{2 b c \sqrt{d-c^2 d x^2}}-\frac{3 f^2 g \left(1-c^2 x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{c^2 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x \left(1-c^2 x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 \left(1-c^2 x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{3 c^2 \sqrt{d-c^2 d x^2}}-\frac{2 g^3 \left(1-c^2 x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{3 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{4 b c^3 \sqrt{d-c^2 d x^2}}-\frac{3 b f^2 g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}-\frac{3 b f g^2 x^2 \sqrt{1-c^2 x^2}}{4 c \sqrt{d-c^2 d x^2}}-\frac{b g^3 x^3 \sqrt{1-c^2 x^2}}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 b g^3 x \sqrt{1-c^2 x^2}}{3 c^3 \sqrt{d-c^2 d x^2}}",1,"(18*b*c*Sqrt[d]*f*(2*c^2*f^2 + 3*g^2)*(-1 + c^2*x^2)*ArcCos[c*x]^2 - 36*a*c*f*(2*c^2*f^2 + 3*g^2)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d]*g*(-1 + c^2*x^2)*(8*b*c*x*(6*g^2 + c^2*(27*f^2 + g^2*x^2)) + 12*a*Sqrt[1 - c^2*x^2]*(4*g^2 + c^2*(18*f^2 + 9*f*g*x + 2*g^2*x^2)) + 27*b*c*f*g*Cos[2*ArcCos[c*x]]) + 6*b*Sqrt[d]*g*(-1 + c^2*x^2)*ArcCos[c*x]*(4*Sqrt[1 - c^2*x^2]*(2*g^2 + c^2*(9*f^2 + g^2*x^2)) + 9*c*f*g*Sin[2*ArcCos[c*x]]))/(72*c^4*Sqrt[d]*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])","A",1
15,1,266,270,0.8440296,"\int \frac{(f+g x)^2 \left(a+b \cos ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[((f + g*x)^2*(a + b*ArcCos[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{\sqrt{d} g \left(c^2 x^2-1\right) \left(4 c \left(a \sqrt{1-c^2 x^2} (4 f+g x)+4 b c f x\right)+b g \cos \left(2 \cos ^{-1}(c x)\right)\right)-4 a \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left(2 c^2 f^2+g^2\right) \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+2 b \sqrt{d} \left(c^2 x^2-1\right) \left(2 c^2 f^2+g^2\right) \cos ^{-1}(c x)^2+2 b \sqrt{d} g \left(c^2 x^2-1\right) \cos ^{-1}(c x) \left(8 c f \sqrt{1-c^2 x^2}+g \sin \left(2 \cos ^{-1}(c x)\right)\right)}{8 c^3 \sqrt{d} \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}","-\frac{f^2 \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{2 b c \sqrt{d-c^2 d x^2}}-\frac{2 f g \left(1-c^2 x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{c^2 \sqrt{d-c^2 d x^2}}-\frac{g^2 x \left(1-c^2 x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{g^2 \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{4 b c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b f g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}-\frac{b g^2 x^2 \sqrt{1-c^2 x^2}}{4 c \sqrt{d-c^2 d x^2}}",1,"(2*b*Sqrt[d]*(2*c^2*f^2 + g^2)*(-1 + c^2*x^2)*ArcCos[c*x]^2 - 4*a*(2*c^2*f^2 + g^2)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d]*g*(-1 + c^2*x^2)*(4*c*(4*b*c*f*x + a*(4*f + g*x)*Sqrt[1 - c^2*x^2]) + b*g*Cos[2*ArcCos[c*x]]) + 2*b*Sqrt[d]*g*(-1 + c^2*x^2)*ArcCos[c*x]*(8*c*f*Sqrt[1 - c^2*x^2] + g*Sin[2*ArcCos[c*x]]))/(8*c^3*Sqrt[d]*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])","A",1
16,1,172,127,0.4515836,"\int \frac{(f+g x) \left(a+b \cos ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[((f + g*x)*(a + b*ArcCos[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{-2 \sqrt{d} g \left(-a c^2 x^2+a+b c x \sqrt{1-c^2 x^2}\right)-2 a c f \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-b c \sqrt{d} f \sqrt{1-c^2 x^2} \cos ^{-1}(c x)^2+2 b \sqrt{d} g \left(c^2 x^2-1\right) \cos ^{-1}(c x)}{2 c^2 \sqrt{d} \sqrt{d-c^2 d x^2}}","-\frac{f \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)^2}{2 b c \sqrt{d-c^2 d x^2}}-\frac{g \left(1-c^2 x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{c^2 \sqrt{d-c^2 d x^2}}-\frac{b g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}",1,"(-2*Sqrt[d]*g*(a - a*c^2*x^2 + b*c*x*Sqrt[1 - c^2*x^2]) + 2*b*Sqrt[d]*g*(-1 + c^2*x^2)*ArcCos[c*x] - b*c*Sqrt[d]*f*Sqrt[1 - c^2*x^2]*ArcCos[c*x]^2 - 2*a*c*f*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/(2*c^2*Sqrt[d]*Sqrt[d - c^2*d*x^2])","A",1
17,1,930,370,2.0985894,"\int \frac{a+b \cos ^{-1}(c x)}{(f+g x) \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCos[c*x])/((f + g*x)*Sqrt[d - c^2*d*x^2]),x]","\frac{\frac{a \log (f+g x)}{\sqrt{d}}-\frac{a \log \left(d \left(f x c^2+g\right)+\sqrt{d} \sqrt{g^2-c^2 f^2} \sqrt{d-c^2 d x^2}\right)}{\sqrt{d}}-\frac{b \sqrt{1-c^2 x^2} \left(2 \cos ^{-1}(c x) \tanh ^{-1}\left(\frac{(c f+g) \cot \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)-2 \cos ^{-1}\left(-\frac{c f}{g}\right) \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)+\left(\cos ^{-1}\left(-\frac{c f}{g}\right)-2 i \tanh ^{-1}\left(\frac{(c f+g) \cot \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)+2 i \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right) \log \left(\frac{e^{-\frac{1}{2} i \cos ^{-1}(c x)} \sqrt{g^2-c^2 f^2}}{\sqrt{2} \sqrt{g} \sqrt{c (f+g x)}}\right)+\left(\cos ^{-1}\left(-\frac{c f}{g}\right)+2 i \left(\tanh ^{-1}\left(\frac{(c f+g) \cot \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)-\tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right)\right) \log \left(\frac{e^{\frac{1}{2} i \cos ^{-1}(c x)} \sqrt{g^2-c^2 f^2}}{\sqrt{2} \sqrt{g} \sqrt{c (f+g x)}}\right)-\left(\cos ^{-1}\left(-\frac{c f}{g}\right)-2 i \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right) \log \left(\frac{(c f+g) \left(-i c f+i g+\sqrt{g^2-c^2 f^2}\right) \left(\tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)-i\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)-\left(\cos ^{-1}\left(-\frac{c f}{g}\right)+2 i \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right) \log \left(\frac{(c f+g) \left(i c f-i g+\sqrt{g^2-c^2 f^2}\right) \left(\tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)+i\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)+i \left(\text{Li}_2\left(\frac{\left(c f-i \sqrt{g^2-c^2 f^2}\right) \left(c f+g-\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)-\text{Li}_2\left(\frac{\left(c f+i \sqrt{g^2-c^2 f^2}\right) \left(c f+g-\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)\right)\right)}{\sqrt{d-c^2 d x^2}}}{\sqrt{g^2-c^2 f^2}}","\frac{i \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right) \log \left(1+\frac{g e^{i \cos ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right)}{\sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2}}-\frac{i \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right) \log \left(1+\frac{g e^{i \cos ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)}{\sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2}}+\frac{b \sqrt{1-c^2 x^2} \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{\sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2}}-\frac{b \sqrt{1-c^2 x^2} \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{\sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2}}",1,"((a*Log[f + g*x])/Sqrt[d] - (a*Log[d*(g + c^2*f*x) + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[d - c^2*d*x^2]])/Sqrt[d] - (b*Sqrt[1 - c^2*x^2]*(2*ArcCos[c*x]*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - 2*ArcCos[-((c*f)/g)]*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^((I/2)*ArcCos[c*x])*Sqrt[g]*Sqrt[c*(f + g*x)])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^((I/2)*ArcCos[c*x])*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*(f + g*x)])] - (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*((-I)*c*f + I*g + Sqrt[-(c^2*f^2) + g^2])*(-I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - (ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(I*c*f - I*g + Sqrt[-(c^2*f^2) + g^2])*(I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] + I*(PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))])))/Sqrt[d - c^2*d*x^2])/Sqrt[-(c^2*f^2) + g^2]","B",0
18,1,1108,496,5.5036266,"\int \frac{a+b \cos ^{-1}(c x)}{(f+g x)^2 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCos[c*x])/((f + g*x)^2*Sqrt[d - c^2*d*x^2]),x]","-\frac{a f \log (f+g x) c^2}{\sqrt{d} \left(g^2-c^2 f^2\right)^{3/2}}-\frac{a f \log \left(d \left(f x c^2+g\right)+\sqrt{d} \sqrt{g^2-c^2 f^2} \sqrt{d-c^2 d x^2}\right) c^2}{\sqrt{d} (c f-g) (c f+g) \sqrt{g^2-c^2 f^2}}-\frac{b \sqrt{1-c^2 x^2} \left(-\frac{g \sqrt{1-c^2 x^2} \cos ^{-1}(c x)}{(c f-g) (c f+g) (c f+c g x)}-\frac{\log \left(\frac{g x}{f}+1\right)}{c^2 f^2-g^2}-\frac{c f \left(2 \cos ^{-1}(c x) \tanh ^{-1}\left(\frac{(c f+g) \cot \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)-2 \cos ^{-1}\left(-\frac{c f}{g}\right) \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)+\left(\cos ^{-1}\left(-\frac{c f}{g}\right)-2 i \tanh ^{-1}\left(\frac{(c f+g) \cot \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)+2 i \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right) \log \left(\frac{e^{-\frac{1}{2} i \cos ^{-1}(c x)} \sqrt{g^2-c^2 f^2}}{\sqrt{2} \sqrt{g} \sqrt{c (f+g x)}}\right)+\left(\cos ^{-1}\left(-\frac{c f}{g}\right)+2 i \left(\tanh ^{-1}\left(\frac{(c f+g) \cot \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)-\tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right)\right) \log \left(\frac{e^{\frac{1}{2} i \cos ^{-1}(c x)} \sqrt{g^2-c^2 f^2}}{\sqrt{2} \sqrt{g} \sqrt{c (f+g x)}}\right)-\left(\cos ^{-1}\left(-\frac{c f}{g}\right)-2 i \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right) \log \left(\frac{(c f+g) \left(-i c f+i g+\sqrt{g^2-c^2 f^2}\right) \left(\tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)-i\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)-\left(\cos ^{-1}\left(-\frac{c f}{g}\right)+2 i \tanh ^{-1}\left(\frac{(g-c f) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{g^2-c^2 f^2}}\right)\right) \log \left(\frac{(c f+g) \left(i c f-i g+\sqrt{g^2-c^2 f^2}\right) \left(\tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)+i\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)+i \left(\text{Li}_2\left(\frac{\left(c f-i \sqrt{g^2-c^2 f^2}\right) \left(c f+g-\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)-\text{Li}_2\left(\frac{\left(c f+i \sqrt{g^2-c^2 f^2}\right) \left(c f+g-\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}{g \left(c f+g+\sqrt{g^2-c^2 f^2} \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)\right)}\right)\right)\right)}{\left(g^2-c^2 f^2\right)^{3/2}}\right) c}{\sqrt{d-c^2 d x^2}}-\frac{a g \sqrt{d-c^2 d x^2}}{d \left(g^2-c^2 f^2\right) (f+g x)}","\frac{g \left(1-c^2 x^2\right) \left(a+b \cos ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2} \left(c^2 f^2-g^2\right) (f+g x)}+\frac{i c^2 f \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right) \log \left(1+\frac{g e^{i \cos ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right)}{\sqrt{d-c^2 d x^2} \left(c^2 f^2-g^2\right)^{3/2}}-\frac{i c^2 f \sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right) \log \left(1+\frac{g e^{i \cos ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)}{\sqrt{d-c^2 d x^2} \left(c^2 f^2-g^2\right)^{3/2}}+\frac{b c^2 f \sqrt{1-c^2 x^2} \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{\sqrt{d-c^2 d x^2} \left(c^2 f^2-g^2\right)^{3/2}}-\frac{b c^2 f \sqrt{1-c^2 x^2} \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{\sqrt{d-c^2 d x^2} \left(c^2 f^2-g^2\right)^{3/2}}+\frac{b c \sqrt{1-c^2 x^2} \log (f+g x)}{\sqrt{d-c^2 d x^2} \left(c^2 f^2-g^2\right)}",1,"-((a*g*Sqrt[d - c^2*d*x^2])/(d*(-(c^2*f^2) + g^2)*(f + g*x))) - (a*c^2*f*Log[f + g*x])/(Sqrt[d]*(-(c^2*f^2) + g^2)^(3/2)) - (a*c^2*f*Log[d*(g + c^2*f*x) + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[d - c^2*d*x^2]])/(Sqrt[d]*(c*f - g)*(c*f + g)*Sqrt[-(c^2*f^2) + g^2]) - (b*c*Sqrt[1 - c^2*x^2]*(-((g*Sqrt[1 - c^2*x^2]*ArcCos[c*x])/((c*f - g)*(c*f + g)*(c*f + c*g*x))) - Log[1 + (g*x)/f]/(c^2*f^2 - g^2) - (c*f*(2*ArcCos[c*x]*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - 2*ArcCos[-((c*f)/g)]*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^((I/2)*ArcCos[c*x])*Sqrt[g]*Sqrt[c*(f + g*x)])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^((I/2)*ArcCos[c*x])*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*(f + g*x)])] - (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*((-I)*c*f + I*g + Sqrt[-(c^2*f^2) + g^2])*(-I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - (ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(I*c*f - I*g + Sqrt[-(c^2*f^2) + g^2])*(I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] + I*(PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))])))/(-(c^2*f^2) + g^2)^(3/2)))/Sqrt[d - c^2*d*x^2]","B",0
19,0,0,38,0.1638095,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^n \log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2}} \, dx","Integrate[((a + b*ArcCos[c*x])^n*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2],x]","\int \frac{\left(a+b \cos ^{-1}(c x)\right)^n \log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2}} \, dx","\text{Int}\left(\frac{\left(a+b \cos ^{-1}(c x)\right)^n \log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2}},x\right)",0,"Integrate[((a + b*ArcCos[c*x])^n*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2], x]","A",-1
20,0,0,496,27.4474018,"\int \frac{\left(a+b \cos ^{-1}(c x)\right)^2 \log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2}} \, dx","Integrate[((a + b*ArcCos[c*x])^2*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2],x]","\int \frac{\left(a+b \cos ^{-1}(c x)\right)^2 \log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2}} \, dx","-\frac{i m \left(a+b \cos ^{-1}(c x)\right)^4}{12 b^2 c}-\frac{i m \left(a+b \cos ^{-1}(c x)\right)^2 \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{c}-\frac{i m \left(a+b \cos ^{-1}(c x)\right)^2 \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{c}+\frac{2 b m \left(a+b \cos ^{-1}(c x)\right) \text{Li}_3\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{c}+\frac{2 b m \left(a+b \cos ^{-1}(c x)\right) \text{Li}_3\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{c}+\frac{m \left(a+b \cos ^{-1}(c x)\right)^3 \log \left(1+\frac{g e^{i \cos ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right)}{3 b c}+\frac{m \left(a+b \cos ^{-1}(c x)\right)^3 \log \left(1+\frac{g e^{i \cos ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)}{3 b c}-\frac{\left(a+b \cos ^{-1}(c x)\right)^3 \log \left(h (f+g x)^m\right)}{3 b c}+\frac{2 i b^2 m \text{Li}_4\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{c}+\frac{2 i b^2 m \text{Li}_4\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{c}",1,"Integrate[((a + b*ArcCos[c*x])^2*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2], x]","F",-1
21,1,1248,374,5.5885473,"\int \frac{\left(a+b \cos ^{-1}(c x)\right) \log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2}} \, dx","Integrate[((a + b*ArcCos[c*x])*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2],x]","\frac{-i b m \cos ^{-1}(c x)^3-3 i a m \cos ^{-1}(c x)^2+3 b m \log \left(\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}+1\right) \cos ^{-1}(c x)^2+3 b m \log \left(\frac{e^{i \cos ^{-1}(c x)} \left(c f-\sqrt{c^2 f^2-g^2}\right)}{g}+1\right) \cos ^{-1}(c x)^2+3 b m \log \left(\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}+1\right) \cos ^{-1}(c x)^2+3 b m \log \left(\frac{e^{i \cos ^{-1}(c x)} \left(c f+\sqrt{c^2 f^2-g^2}\right)}{g}+1\right) \cos ^{-1}(c x)^2-3 b \log \left(h (f+g x)^m\right) \cos ^{-1}(c x)^2-3 b m \log \left(\frac{\left(c f-\sqrt{c^2 f^2-g^2}\right) \left(c x+i \sqrt{1-c^2 x^2}\right)}{g}+1\right) \cos ^{-1}(c x)^2-3 b m \log \left(\frac{\left(c f+\sqrt{c^2 f^2-g^2}\right) \left(c x+i \sqrt{1-c^2 x^2}\right)}{g}+1\right) \cos ^{-1}(c x)^2+6 a m \log \left(\frac{e^{i \cos ^{-1}(c x)} \left(c f-\sqrt{c^2 f^2-g^2}\right)}{g}+1\right) \cos ^{-1}(c x)+12 b m \sin ^{-1}\left(\frac{\sqrt{\frac{c f}{g}+1}}{\sqrt{2}}\right) \log \left(\frac{e^{i \cos ^{-1}(c x)} \left(c f-\sqrt{c^2 f^2-g^2}\right)}{g}+1\right) \cos ^{-1}(c x)+6 a m \log \left(\frac{e^{i \cos ^{-1}(c x)} \left(c f+\sqrt{c^2 f^2-g^2}\right)}{g}+1\right) \cos ^{-1}(c x)-12 b m \sin ^{-1}\left(\frac{\sqrt{\frac{c f}{g}+1}}{\sqrt{2}}\right) \log \left(\frac{e^{i \cos ^{-1}(c x)} \left(c f+\sqrt{c^2 f^2-g^2}\right)}{g}+1\right) \cos ^{-1}(c x)-6 a m \log (f+g x) \cos ^{-1}(c x)-12 b m \sin ^{-1}\left(\frac{\sqrt{\frac{c f}{g}+1}}{\sqrt{2}}\right) \log \left(\frac{\left(c f-\sqrt{c^2 f^2-g^2}\right) \left(c x+i \sqrt{1-c^2 x^2}\right)}{g}+1\right) \cos ^{-1}(c x)+12 b m \sin ^{-1}\left(\frac{\sqrt{\frac{c f}{g}+1}}{\sqrt{2}}\right) \log \left(\frac{\left(c f+\sqrt{c^2 f^2-g^2}\right) \left(c x+i \sqrt{1-c^2 x^2}\right)}{g}+1\right) \cos ^{-1}(c x)-6 i b m \text{Li}_2\left(\frac{e^{i \cos ^{-1}(c x)} g}{\sqrt{c^2 f^2-g^2}-c f}\right) \cos ^{-1}(c x)-6 i b m \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right) \cos ^{-1}(c x)+24 i a m \sin ^{-1}\left(\frac{\sqrt{\frac{c f}{g}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{(c f-g) \tan \left(\frac{1}{2} \cos ^{-1}(c x)\right)}{\sqrt{c^2 f^2-g^2}}\right)+12 a m \sin ^{-1}\left(\frac{\sqrt{\frac{c f}{g}+1}}{\sqrt{2}}\right) \log \left(\frac{e^{i \cos ^{-1}(c x)} \left(c f-\sqrt{c^2 f^2-g^2}\right)}{g}+1\right)-12 a m \sin ^{-1}\left(\frac{\sqrt{\frac{c f}{g}+1}}{\sqrt{2}}\right) \log \left(\frac{e^{i \cos ^{-1}(c x)} \left(c f+\sqrt{c^2 f^2-g^2}\right)}{g}+1\right)-6 a m \sin ^{-1}(c x) \log (f+g x)+6 a \sin ^{-1}(c x) \log \left(h (f+g x)^m\right)-6 i a m \text{Li}_2\left(\frac{e^{i \cos ^{-1}(c x)} \left(\sqrt{c^2 f^2-g^2}-c f\right)}{g}\right)-6 i a m \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} \left(c f+\sqrt{c^2 f^2-g^2}\right)}{g}\right)+6 b m \text{Li}_3\left(\frac{e^{i \cos ^{-1}(c x)} g}{\sqrt{c^2 f^2-g^2}-c f}\right)+6 b m \text{Li}_3\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{6 c}","-\frac{i m \left(a+b \cos ^{-1}(c x)\right)^3}{6 b^2 c}-\frac{i m \left(a+b \cos ^{-1}(c x)\right) \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{c}-\frac{i m \left(a+b \cos ^{-1}(c x)\right) \text{Li}_2\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{c}+\frac{m \left(a+b \cos ^{-1}(c x)\right)^2 \log \left(1+\frac{g e^{i \cos ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right)}{2 b c}+\frac{m \left(a+b \cos ^{-1}(c x)\right)^2 \log \left(1+\frac{g e^{i \cos ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)}{2 b c}-\frac{\left(a+b \cos ^{-1}(c x)\right)^2 \log \left(h (f+g x)^m\right)}{2 b c}+\frac{b m \text{Li}_3\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{c}+\frac{b m \text{Li}_3\left(-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{c}",1,"((-3*I)*a*m*ArcCos[c*x]^2 - I*b*m*ArcCos[c*x]^3 + (24*I)*a*m*ArcSin[Sqrt[1 + (c*f)/g]/Sqrt[2]]*ArcTan[((c*f - g)*Tan[ArcCos[c*x]/2])/Sqrt[c^2*f^2 - g^2]] + 3*b*m*ArcCos[c*x]^2*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + 6*a*m*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*(c*f - Sqrt[c^2*f^2 - g^2]))/g] + 3*b*m*ArcCos[c*x]^2*Log[1 + (E^(I*ArcCos[c*x])*(c*f - Sqrt[c^2*f^2 - g^2]))/g] + 12*a*m*ArcSin[Sqrt[1 + (c*f)/g]/Sqrt[2]]*Log[1 + (E^(I*ArcCos[c*x])*(c*f - Sqrt[c^2*f^2 - g^2]))/g] + 12*b*m*ArcCos[c*x]*ArcSin[Sqrt[1 + (c*f)/g]/Sqrt[2]]*Log[1 + (E^(I*ArcCos[c*x])*(c*f - Sqrt[c^2*f^2 - g^2]))/g] + 3*b*m*ArcCos[c*x]^2*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + 6*a*m*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*(c*f + Sqrt[c^2*f^2 - g^2]))/g] + 3*b*m*ArcCos[c*x]^2*Log[1 + (E^(I*ArcCos[c*x])*(c*f + Sqrt[c^2*f^2 - g^2]))/g] - 12*a*m*ArcSin[Sqrt[1 + (c*f)/g]/Sqrt[2]]*Log[1 + (E^(I*ArcCos[c*x])*(c*f + Sqrt[c^2*f^2 - g^2]))/g] - 12*b*m*ArcCos[c*x]*ArcSin[Sqrt[1 + (c*f)/g]/Sqrt[2]]*Log[1 + (E^(I*ArcCos[c*x])*(c*f + Sqrt[c^2*f^2 - g^2]))/g] - 6*a*m*ArcCos[c*x]*Log[f + g*x] - 6*a*m*ArcSin[c*x]*Log[f + g*x] - 3*b*ArcCos[c*x]^2*Log[h*(f + g*x)^m] + 6*a*ArcSin[c*x]*Log[h*(f + g*x)^m] - 3*b*m*ArcCos[c*x]^2*Log[1 + ((c*f - Sqrt[c^2*f^2 - g^2])*(c*x + I*Sqrt[1 - c^2*x^2]))/g] - 12*b*m*ArcCos[c*x]*ArcSin[Sqrt[1 + (c*f)/g]/Sqrt[2]]*Log[1 + ((c*f - Sqrt[c^2*f^2 - g^2])*(c*x + I*Sqrt[1 - c^2*x^2]))/g] - 3*b*m*ArcCos[c*x]^2*Log[1 + ((c*f + Sqrt[c^2*f^2 - g^2])*(c*x + I*Sqrt[1 - c^2*x^2]))/g] + 12*b*m*ArcCos[c*x]*ArcSin[Sqrt[1 + (c*f)/g]/Sqrt[2]]*Log[1 + ((c*f + Sqrt[c^2*f^2 - g^2])*(c*x + I*Sqrt[1 - c^2*x^2]))/g] - (6*I)*b*m*ArcCos[c*x]*PolyLog[2, (E^(I*ArcCos[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - (6*I)*a*m*PolyLog[2, (E^(I*ArcCos[c*x])*(-(c*f) + Sqrt[c^2*f^2 - g^2]))/g] - (6*I)*b*m*ArcCos[c*x]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))] - (6*I)*a*m*PolyLog[2, -((E^(I*ArcCos[c*x])*(c*f + Sqrt[c^2*f^2 - g^2]))/g)] + 6*b*m*PolyLog[3, (E^(I*ArcCos[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] + 6*b*m*PolyLog[3, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(6*c)","B",0
22,1,246,237,0.0220539,"\int \frac{\log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2}} \, dx","Integrate[Log[h*(f + g*x)^m]/Sqrt[1 - c^2*x^2],x]","\frac{i m \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{c}+\frac{i m \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{c}-\frac{m \sin ^{-1}(c x) \log \left(1-\frac{i c g e^{i \sin ^{-1}(c x)}}{c^2 f-c \sqrt{c^2 f^2-g^2}}\right)}{c}-\frac{m \sin ^{-1}(c x) \log \left(1-\frac{i c g e^{i \sin ^{-1}(c x)}}{c \sqrt{c^2 f^2-g^2}+c^2 f}\right)}{c}+\frac{\sin ^{-1}(c x) \log \left(h (f+g x)^m\right)}{c}+\frac{i m \sin ^{-1}(c x)^2}{2 c}","\frac{i m \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{c}+\frac{i m \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{c}-\frac{m \sin ^{-1}(c x) \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right)}{c}-\frac{m \sin ^{-1}(c x) \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)}{c}+\frac{\sin ^{-1}(c x) \log \left(h (f+g x)^m\right)}{c}+\frac{i m \sin ^{-1}(c x)^2}{2 c}",1,"((I/2)*m*ArcSin[c*x]^2)/c - (m*ArcSin[c*x]*Log[1 - (I*c*E^(I*ArcSin[c*x])*g)/(c^2*f - c*Sqrt[c^2*f^2 - g^2])])/c - (m*ArcSin[c*x]*Log[1 - (I*c*E^(I*ArcSin[c*x])*g)/(c^2*f + c*Sqrt[c^2*f^2 - g^2])])/c + (ArcSin[c*x]*Log[h*(f + g*x)^m])/c + (I*m*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/c + (I*m*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/c","A",1
23,0,0,38,0.2043489,"\int \frac{\log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)} \, dx","Integrate[Log[h*(f + g*x)^m]/(Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])),x]","\int \frac{\log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2} \left(a+b \cos ^{-1}(c x)\right)},x\right)",0,"Integrate[Log[h*(f + g*x)^m]/(Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])), x]","A",-1
24,1,104,137,0.1224812,"\int x^3 \cos ^{-1}(a+b x) \, dx","Integrate[x^3*ArcCos[a + b*x],x]","\frac{3 \left(8 a^4+24 a^2+3\right) \sin ^{-1}(a+b x)+\sqrt{-a^2-2 a b x-b^2 x^2+1} \left(50 a^3-26 a^2 b x+14 a b^2 x^2+55 a-6 b^3 x^3-9 b x\right)+24 b^4 x^4 \cos ^{-1}(a+b x)}{96 b^4}","\frac{\left(4 a \left(19 a^2+16\right)-\left(26 a^2+9\right) (a+b x)\right) \sqrt{1-(a+b x)^2}}{96 b^4}+\frac{\left(8 a^4+24 a^2+3\right) \sin ^{-1}(a+b x)}{32 b^4}+\frac{7 a x^2 \sqrt{1-(a+b x)^2}}{48 b^2}+\frac{1}{4} x^4 \cos ^{-1}(a+b x)-\frac{x^3 \sqrt{1-(a+b x)^2}}{16 b}",1,"(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(55*a + 50*a^3 - 9*b*x - 26*a^2*b*x + 14*a*b^2*x^2 - 6*b^3*x^3) + 24*b^4*x^4*ArcCos[a + b*x] + 3*(3 + 24*a^2 + 8*a^4)*ArcSin[a + b*x])/(96*b^4)","A",1
25,1,83,94,0.0889912,"\int x^2 \cos ^{-1}(a+b x) \, dx","Integrate[x^2*ArcCos[a + b*x],x]","-\frac{\sqrt{-a^2-2 a b x-b^2 x^2+1} \left(11 a^2-5 a b x+2 b^2 x^2+4\right)+3 a \left(2 a^2+3\right) \sin ^{-1}(a+b x)-6 b^3 x^3 \cos ^{-1}(a+b x)}{18 b^3}","-\frac{\left(11 a^2-5 a b x+4\right) \sqrt{1-(a+b x)^2}}{18 b^3}-\frac{a \left(2 a^2+3\right) \sin ^{-1}(a+b x)}{6 b^3}+\frac{1}{3} x^3 \cos ^{-1}(a+b x)-\frac{x^2 \sqrt{1-(a+b x)^2}}{9 b}",1,"-1/18*(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(4 + 11*a^2 - 5*a*b*x + 2*b^2*x^2) - 6*b^3*x^3*ArcCos[a + b*x] + 3*a*(3 + 2*a^2)*ArcSin[a + b*x])/b^3","A",1
26,1,69,80,0.0497368,"\int x \cos ^{-1}(a+b x) \, dx","Integrate[x*ArcCos[a + b*x],x]","\frac{(3 a-b x) \sqrt{-a^2-2 a b x-b^2 x^2+1}+\left(2 a^2+1\right) \sin ^{-1}(a+b x)+2 b^2 x^2 \cos ^{-1}(a+b x)}{4 b^2}","\frac{\left(2 a^2+1\right) \sin ^{-1}(a+b x)}{4 b^2}+\frac{3 a \sqrt{1-(a+b x)^2}}{4 b^2}+\frac{1}{2} x^2 \cos ^{-1}(a+b x)-\frac{x \sqrt{1-(a+b x)^2}}{4 b}",1,"((3*a - b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] + 2*b^2*x^2*ArcCos[a + b*x] + (1 + 2*a^2)*ArcSin[a + b*x])/(4*b^2)","A",1
27,1,47,36,0.0404712,"\int \cos ^{-1}(a+b x) \, dx","Integrate[ArcCos[a + b*x],x]","x \cos ^{-1}(a+b x)-\frac{\sqrt{-a^2-2 a b x-b^2 x^2+1}+a \sin ^{-1}(a+b x)}{b}","\frac{(a+b x) \cos ^{-1}(a+b x)}{b}-\frac{\sqrt{1-(a+b x)^2}}{b}",1,"x*ArcCos[a + b*x] - (Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] + a*ArcSin[a + b*x])/b","A",1
28,1,228,177,0.2059763,"\int \frac{\cos ^{-1}(a+b x)}{x} \, dx","Integrate[ArcCos[a + b*x]/x,x]","-i \left(\text{Li}_2\left(\left(a-\sqrt{a^2-1}\right) e^{i \cos ^{-1}(a+b x)}\right)+\text{Li}_2\left(\left(a+\sqrt{a^2-1}\right) e^{i \cos ^{-1}(a+b x)}\right)\right)+\log \left(1+\left(\sqrt{a^2-1}-a\right) e^{i \cos ^{-1}(a+b x)}\right) \left(\cos ^{-1}(a+b x)-2 \sin ^{-1}\left(\frac{\sqrt{1-a}}{\sqrt{2}}\right)\right)+\log \left(1-\left(\sqrt{a^2-1}+a\right) e^{i \cos ^{-1}(a+b x)}\right) \left(\cos ^{-1}(a+b x)+2 \sin ^{-1}\left(\frac{\sqrt{1-a}}{\sqrt{2}}\right)\right)-4 i \sin ^{-1}\left(\frac{\sqrt{1-a}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{(a+1) \tan \left(\frac{1}{2} \cos ^{-1}(a+b x)\right)}{\sqrt{a^2-1}}\right)-\frac{1}{2} i \cos ^{-1}(a+b x)^2","-i \text{Li}_2\left(\frac{e^{i \cos ^{-1}(a+b x)}}{a-i \sqrt{1-a^2}}\right)-i \text{Li}_2\left(\frac{e^{i \cos ^{-1}(a+b x)}}{a+i \sqrt{1-a^2}}\right)+\cos ^{-1}(a+b x) \log \left(1-\frac{e^{i \cos ^{-1}(a+b x)}}{a-i \sqrt{1-a^2}}\right)+\cos ^{-1}(a+b x) \log \left(1-\frac{e^{i \cos ^{-1}(a+b x)}}{a+i \sqrt{1-a^2}}\right)-\frac{1}{2} i \cos ^{-1}(a+b x)^2",1,"(-1/2*I)*ArcCos[a + b*x]^2 - (4*I)*ArcSin[Sqrt[1 - a]/Sqrt[2]]*ArcTan[((1 + a)*Tan[ArcCos[a + b*x]/2])/Sqrt[-1 + a^2]] + (ArcCos[a + b*x] - 2*ArcSin[Sqrt[1 - a]/Sqrt[2]])*Log[1 + (-a + Sqrt[-1 + a^2])*E^(I*ArcCos[a + b*x])] + (ArcCos[a + b*x] + 2*ArcSin[Sqrt[1 - a]/Sqrt[2]])*Log[1 - (a + Sqrt[-1 + a^2])*E^(I*ArcCos[a + b*x])] - I*(PolyLog[2, (a - Sqrt[-1 + a^2])*E^(I*ArcCos[a + b*x])] + PolyLog[2, (a + Sqrt[-1 + a^2])*E^(I*ArcCos[a + b*x])])","A",0
29,1,79,63,0.0600392,"\int \frac{\cos ^{-1}(a+b x)}{x^2} \, dx","Integrate[ArcCos[a + b*x]/x^2,x]","\frac{b \left(\log \left(\sqrt{1-a^2} \sqrt{-a^2-2 a b x-b^2 x^2+1}-a^2-a b x+1\right)-\log (x)\right)}{\sqrt{1-a^2}}-\frac{\cos ^{-1}(a+b x)}{x}","\frac{b \tanh ^{-1}\left(\frac{1-a (a+b x)}{\sqrt{1-a^2} \sqrt{1-(a+b x)^2}}\right)}{\sqrt{1-a^2}}-\frac{\cos ^{-1}(a+b x)}{x}",1,"-(ArcCos[a + b*x]/x) + (b*(-Log[x] + Log[1 - a^2 - a*b*x + Sqrt[1 - a^2]*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]]))/Sqrt[1 - a^2]","A",1
30,1,126,103,0.18477,"\int \frac{\cos ^{-1}(a+b x)}{x^3} \, dx","Integrate[ArcCos[a + b*x]/x^3,x]","-\frac{\cos ^{-1}(a+b x)-\frac{b x \left(\sqrt{1-a^2} \sqrt{-a^2-2 a b x-b^2 x^2+1}+a b x \log \left(\sqrt{1-a^2} \sqrt{-a^2-2 a b x-b^2 x^2+1}-a^2-a b x+1\right)-a b x \log (x)\right)}{\left(1-a^2\right)^{3/2}}}{2 x^2}","\frac{a b^2 \tanh ^{-1}\left(\frac{1-a (a+b x)}{\sqrt{1-a^2} \sqrt{1-(a+b x)^2}}\right)}{2 \left(1-a^2\right)^{3/2}}+\frac{b \sqrt{1-(a+b x)^2}}{2 \left(1-a^2\right) x}-\frac{\cos ^{-1}(a+b x)}{2 x^2}",1,"-1/2*(ArcCos[a + b*x] - (b*x*(Sqrt[1 - a^2]*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] - a*b*x*Log[x] + a*b*x*Log[1 - a^2 - a*b*x + Sqrt[1 - a^2]*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]]))/(1 - a^2)^(3/2))/x^2","A",1
31,1,168,144,0.1972551,"\int \frac{\cos ^{-1}(a+b x)}{x^4} \, dx","Integrate[ArcCos[a + b*x]/x^4,x]","\frac{-\left(\left(2 a^2+1\right) b^3 x^3 \log (x)\right)+\sqrt{1-a^2} b x \left(-a^2+3 a b x+1\right) \sqrt{-a^2-2 a b x-b^2 x^2+1}+\left(2 a^2+1\right) b^3 x^3 \log \left(\sqrt{1-a^2} \sqrt{-a^2-2 a b x-b^2 x^2+1}-a^2-a b x+1\right)-2 \left(1-a^2\right)^{5/2} \cos ^{-1}(a+b x)}{6 \left(1-a^2\right)^{5/2} x^3}","\frac{\left(2 a^2+1\right) b^3 \tanh ^{-1}\left(\frac{1-a (a+b x)}{\sqrt{1-a^2} \sqrt{1-(a+b x)^2}}\right)}{6 \left(1-a^2\right)^{5/2}}+\frac{a b^2 \sqrt{1-(a+b x)^2}}{2 \left(1-a^2\right)^2 x}+\frac{b \sqrt{1-(a+b x)^2}}{6 \left(1-a^2\right) x^2}-\frac{\cos ^{-1}(a+b x)}{3 x^3}",1,"(Sqrt[1 - a^2]*b*x*(1 - a^2 + 3*a*b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] - 2*(1 - a^2)^(5/2)*ArcCos[a + b*x] - (1 + 2*a^2)*b^3*x^3*Log[x] + (1 + 2*a^2)*b^3*x^3*Log[1 - a^2 - a*b*x + Sqrt[1 - a^2]*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]])/(6*(1 - a^2)^(5/2)*x^3)","A",1
32,1,74,82,0.038081,"\int \cos ^{-1}(a+b x)^3 \, dx","Integrate[ArcCos[a + b*x]^3,x]","\frac{6 \sqrt{1-(a+b x)^2}+(a+b x) \cos ^{-1}(a+b x)^3-3 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^2-6 (a+b x) \cos ^{-1}(a+b x)}{b}","\frac{6 \sqrt{1-(a+b x)^2}}{b}+\frac{(a+b x) \cos ^{-1}(a+b x)^3}{b}-\frac{3 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^2}{b}-\frac{6 (a+b x) \cos ^{-1}(a+b x)}{b}",1,"(6*Sqrt[1 - (a + b*x)^2] - 6*(a + b*x)*ArcCos[a + b*x] - 3*Sqrt[1 - (a + b*x)^2]*ArcCos[a + b*x]^2 + (a + b*x)*ArcCos[a + b*x]^3)/b","A",1
33,1,49,47,0.0233173,"\int \cos ^{-1}(a+b x)^2 \, dx","Integrate[ArcCos[a + b*x]^2,x]","\frac{-2 (a+b x)+(a+b x) \cos ^{-1}(a+b x)^2-2 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)}{b}","\frac{(a+b x) \cos ^{-1}(a+b x)^2}{b}-\frac{2 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)}{b}-2 x",1,"(-2*(a + b*x) - 2*Sqrt[1 - (a + b*x)^2]*ArcCos[a + b*x] + (a + b*x)*ArcCos[a + b*x]^2)/b","A",1
34,1,12,12,0.0267367,"\int \frac{1}{\cos ^{-1}(a+b x)} \, dx","Integrate[ArcCos[a + b*x]^(-1),x]","-\frac{\text{Si}\left(\cos ^{-1}(a+b x)\right)}{b}","-\frac{\text{Si}\left(\cos ^{-1}(a+b x)\right)}{b}",1,"-(SinIntegral[ArcCos[a + b*x]]/b)","A",1
35,1,40,40,0.053526,"\int \frac{1}{\cos ^{-1}(a+b x)^2} \, dx","Integrate[ArcCos[a + b*x]^(-2),x]","\frac{\sqrt{1-(a+b x)^2}}{b \cos ^{-1}(a+b x)}-\frac{\text{Ci}\left(\cos ^{-1}(a+b x)\right)}{b}","\frac{\sqrt{1-(a+b x)^2}}{b \cos ^{-1}(a+b x)}-\frac{\text{Ci}\left(\cos ^{-1}(a+b x)\right)}{b}",1,"Sqrt[1 - (a + b*x)^2]/(b*ArcCos[a + b*x]) - CosIntegral[ArcCos[a + b*x]]/b","A",1
36,1,65,65,0.0543628,"\int \frac{1}{\cos ^{-1}(a+b x)^3} \, dx","Integrate[ArcCos[a + b*x]^(-3),x]","\frac{\text{Si}\left(\cos ^{-1}(a+b x)\right)}{2 b}+\frac{a+b x}{2 b \cos ^{-1}(a+b x)}+\frac{\sqrt{1-(a+b x)^2}}{2 b \cos ^{-1}(a+b x)^2}","\frac{\text{Si}\left(\cos ^{-1}(a+b x)\right)}{2 b}+\frac{a+b x}{2 b \cos ^{-1}(a+b x)}+\frac{\sqrt{1-(a+b x)^2}}{2 b \cos ^{-1}(a+b x)^2}",1,"Sqrt[1 - (a + b*x)^2]/(2*b*ArcCos[a + b*x]^2) + (a + b*x)/(2*b*ArcCos[a + b*x]) + SinIntegral[ArcCos[a + b*x]]/(2*b)","A",1
37,1,90,111,0.0527443,"\int \cos ^{-1}(a+b x)^{5/2} \, dx","Integrate[ArcCos[a + b*x]^(5/2),x]","-\frac{\frac{\sqrt{\cos ^{-1}(a+b x)} \Gamma \left(\frac{7}{2},-i \cos ^{-1}(a+b x)\right)}{2 \sqrt{-i \cos ^{-1}(a+b x)}}+\frac{\sqrt{\cos ^{-1}(a+b x)} \Gamma \left(\frac{7}{2},i \cos ^{-1}(a+b x)\right)}{2 \sqrt{i \cos ^{-1}(a+b x)}}}{b}","\frac{15 \sqrt{\frac{\pi }{2}} C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right)}{4 b}+\frac{(a+b x) \cos ^{-1}(a+b x)^{5/2}}{b}-\frac{5 \sqrt{1-(a+b x)^2} \cos ^{-1}(a+b x)^{3/2}}{2 b}-\frac{15 (a+b x) \sqrt{\cos ^{-1}(a+b x)}}{4 b}",1,"-(((Sqrt[ArcCos[a + b*x]]*Gamma[7/2, (-I)*ArcCos[a + b*x]])/(2*Sqrt[(-I)*ArcCos[a + b*x]]) + (Sqrt[ArcCos[a + b*x]]*Gamma[7/2, I*ArcCos[a + b*x]])/(2*Sqrt[I*ArcCos[a + b*x]]))/b)","C",0
38,1,76,89,0.0385501,"\int \cos ^{-1}(a+b x)^{3/2} \, dx","Integrate[ArcCos[a + b*x]^(3/2),x]","-\frac{\sqrt{-i \cos ^{-1}(a+b x)} \Gamma \left(\frac{5}{2},-i \cos ^{-1}(a+b x)\right)+\sqrt{i \cos ^{-1}(a+b x)} \Gamma \left(\frac{5}{2},i \cos ^{-1}(a+b x)\right)}{2 b \sqrt{\cos ^{-1}(a+b x)}}","\frac{3 \sqrt{\frac{\pi }{2}} S\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right)}{2 b}+\frac{(a+b x) \cos ^{-1}(a+b x)^{3/2}}{b}-\frac{3 \sqrt{1-(a+b x)^2} \sqrt{\cos ^{-1}(a+b x)}}{2 b}",1,"-1/2*(Sqrt[(-I)*ArcCos[a + b*x]]*Gamma[5/2, (-I)*ArcCos[a + b*x]] + Sqrt[I*ArcCos[a + b*x]]*Gamma[5/2, I*ArcCos[a + b*x]])/(b*Sqrt[ArcCos[a + b*x]])","C",0
39,1,90,55,0.0410762,"\int \sqrt{\cos ^{-1}(a+b x)} \, dx","Integrate[Sqrt[ArcCos[a + b*x]],x]","-\frac{-\frac{\sqrt{\cos ^{-1}(a+b x)} \Gamma \left(\frac{3}{2},-i \cos ^{-1}(a+b x)\right)}{2 \sqrt{-i \cos ^{-1}(a+b x)}}-\frac{\sqrt{\cos ^{-1}(a+b x)} \Gamma \left(\frac{3}{2},i \cos ^{-1}(a+b x)\right)}{2 \sqrt{i \cos ^{-1}(a+b x)}}}{b}","\frac{(a+b x) \sqrt{\cos ^{-1}(a+b x)}}{b}-\frac{\sqrt{\frac{\pi }{2}} C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right)}{b}",1,"-((-1/2*(Sqrt[ArcCos[a + b*x]]*Gamma[3/2, (-I)*ArcCos[a + b*x]])/Sqrt[(-I)*ArcCos[a + b*x]] - (Sqrt[ArcCos[a + b*x]]*Gamma[3/2, I*ArcCos[a + b*x]])/(2*Sqrt[I*ArcCos[a + b*x]]))/b)","C",0
40,1,78,33,0.0338832,"\int \frac{1}{\sqrt{\cos ^{-1}(a+b x)}} \, dx","Integrate[1/Sqrt[ArcCos[a + b*x]],x]","-\frac{-\sqrt{-i \cos ^{-1}(a+b x)} \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a+b x)\right)-\sqrt{i \cos ^{-1}(a+b x)} \Gamma \left(\frac{1}{2},i \cos ^{-1}(a+b x)\right)}{2 b \sqrt{\cos ^{-1}(a+b x)}}","-\frac{\sqrt{2 \pi } S\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right)}{b}",1,"-1/2*(-(Sqrt[(-I)*ArcCos[a + b*x]]*Gamma[1/2, (-I)*ArcCos[a + b*x]]) - Sqrt[I*ArcCos[a + b*x]]*Gamma[1/2, I*ArcCos[a + b*x]])/(b*Sqrt[ArcCos[a + b*x]])","C",0
41,1,97,64,0.0533275,"\int \frac{1}{\cos ^{-1}(a+b x)^{3/2}} \, dx","Integrate[ArcCos[a + b*x]^(-3/2),x]","-\frac{-2 \sqrt{1-(a+b x)^2}-i \sqrt{-i \cos ^{-1}(a+b x)} \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a+b x)\right)+i \sqrt{i \cos ^{-1}(a+b x)} \Gamma \left(\frac{1}{2},i \cos ^{-1}(a+b x)\right)}{b \sqrt{\cos ^{-1}(a+b x)}}","\frac{2 \sqrt{1-(a+b x)^2}}{b \sqrt{\cos ^{-1}(a+b x)}}-\frac{2 \sqrt{2 \pi } C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right)}{b}",1,"-((-2*Sqrt[1 - (a + b*x)^2] - I*Sqrt[(-I)*ArcCos[a + b*x]]*Gamma[1/2, (-I)*ArcCos[a + b*x]] + I*Sqrt[I*ArcCos[a + b*x]]*Gamma[1/2, I*ArcCos[a + b*x]])/(b*Sqrt[ArcCos[a + b*x]]))","C",0
42,1,139,90,0.3069138,"\int \frac{1}{\cos ^{-1}(a+b x)^{5/2}} \, dx","Integrate[ArcCos[a + b*x]^(-5/2),x]","-\frac{2 \left(-\sqrt{1-(a+b x)^2}-e^{-i \cos ^{-1}(a+b x)} \cos ^{-1}(a+b x)-e^{i \cos ^{-1}(a+b x)} \cos ^{-1}(a+b x)+i \left(-i \cos ^{-1}(a+b x)\right)^{3/2} \Gamma \left(\frac{1}{2},-i \cos ^{-1}(a+b x)\right)-i \left(i \cos ^{-1}(a+b x)\right)^{3/2} \Gamma \left(\frac{1}{2},i \cos ^{-1}(a+b x)\right)\right)}{3 b \cos ^{-1}(a+b x)^{3/2}}","\frac{4 \sqrt{2 \pi } S\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right)}{3 b}+\frac{4 (a+b x)}{3 b \sqrt{\cos ^{-1}(a+b x)}}+\frac{2 \sqrt{1-(a+b x)^2}}{3 b \cos ^{-1}(a+b x)^{3/2}}",1,"(-2*(-Sqrt[1 - (a + b*x)^2] - ArcCos[a + b*x]/E^(I*ArcCos[a + b*x]) - E^(I*ArcCos[a + b*x])*ArcCos[a + b*x] + I*((-I)*ArcCos[a + b*x])^(3/2)*Gamma[1/2, (-I)*ArcCos[a + b*x]] - I*(I*ArcCos[a + b*x])^(3/2)*Gamma[1/2, I*ArcCos[a + b*x]]))/(3*b*ArcCos[a + b*x]^(3/2))","C",0
43,1,128,106,0.0517453,"\int \frac{1}{\sqrt{a+b \cos ^{-1}(c+d x)}} \, dx","Integrate[1/Sqrt[a + b*ArcCos[c + d*x]],x]","\frac{e^{-\frac{i a}{b}} \left(\sqrt{-\frac{i \left(a+b \cos ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \cos ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \cos ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \cos ^{-1}(c+d x)\right)}{b}\right)\right)}{2 d \sqrt{a+b \cos ^{-1}(c+d x)}}","\frac{\sqrt{2 \pi } \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d}-\frac{\sqrt{2 \pi } \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d}",1,"(Sqrt[((-I)*(a + b*ArcCos[c + d*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcCos[c + d*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcCos[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcCos[c + d*x]))/b])/(2*d*E^((I*a)/b)*Sqrt[a + b*ArcCos[c + d*x]])","C",0
44,1,133,108,0.1068234,"\int \frac{1}{\sqrt{a-b \cos ^{-1}(c+d x)}} \, dx","Integrate[1/Sqrt[a - b*ArcCos[c + d*x]],x]","\frac{e^{-\frac{i a}{b}} \left(\sqrt{-\frac{i \left(a-b \cos ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a-b \cos ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a-b \cos ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a-b \cos ^{-1}(c+d x)\right)}{b}\right)\right)}{2 d \sqrt{a-b \cos ^{-1}(c+d x)}}","\frac{\sqrt{2 \pi } \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a-b \cos ^{-1}(c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d}-\frac{\sqrt{2 \pi } \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a-b \cos ^{-1}(c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d}",1,"(Sqrt[((-I)*(a - b*ArcCos[c + d*x]))/b]*Gamma[1/2, ((-I)*(a - b*ArcCos[c + d*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a - b*ArcCos[c + d*x]))/b]*Gamma[1/2, (I*(a - b*ArcCos[c + d*x]))/b])/(2*d*E^((I*a)/b)*Sqrt[a - b*ArcCos[c + d*x]])","C",0
45,1,59,68,0.0437627,"\int \frac{\cos ^{-1}(a+b x)}{\frac{a d}{b}+d x} \, dx","Integrate[ArcCos[a + b*x]/((a*d)/b + d*x),x]","-\frac{i \left(\text{Li}_2\left(-e^{2 i \cos ^{-1}(a+b x)}\right)+\cos ^{-1}(a+b x) \left(\cos ^{-1}(a+b x)+2 i \log \left(1+e^{2 i \cos ^{-1}(a+b x)}\right)\right)\right)}{2 d}","-\frac{i \text{Li}_2\left(-e^{2 i \cos ^{-1}(a+b x)}\right)}{2 d}-\frac{i \cos ^{-1}(a+b x)^2}{2 d}+\frac{\cos ^{-1}(a+b x) \log \left(1+e^{2 i \cos ^{-1}(a+b x)}\right)}{d}",1,"((-1/2*I)*(ArcCos[a + b*x]*(ArcCos[a + b*x] + (2*I)*Log[1 + E^((2*I)*ArcCos[a + b*x])]) + PolyLog[2, -E^((2*I)*ArcCos[a + b*x])]))/d","A",1
46,1,30,34,0.0161023,"\int \sqrt{1-x^2} \cos ^{-1}(x) \, dx","Integrate[Sqrt[1 - x^2]*ArcCos[x],x]","\frac{1}{4} \left(x^2+2 \sqrt{1-x^2} x \cos ^{-1}(x)-\cos ^{-1}(x)^2\right)","\frac{x^2}{4}+\frac{1}{2} \sqrt{1-x^2} x \cos ^{-1}(x)-\frac{1}{4} \cos ^{-1}(x)^2",1,"(x^2 + 2*x*Sqrt[1 - x^2]*ArcCos[x] - ArcCos[x]^2)/4","A",1
47,1,48,51,0.0290243,"\int x^3 \cos ^{-1}\left(a x^2\right) \, dx","Integrate[x^3*ArcCos[a*x^2],x]","\frac{-a x^2 \sqrt{1-a^2 x^4}+2 a^2 x^4 \cos ^{-1}\left(a x^2\right)+\sin ^{-1}\left(a x^2\right)}{8 a^2}","\frac{\sin ^{-1}\left(a x^2\right)}{8 a^2}-\frac{x^2 \sqrt{1-a^2 x^4}}{8 a}+\frac{1}{4} x^4 \cos ^{-1}\left(a x^2\right)",1,"(-(a*x^2*Sqrt[1 - a^2*x^4]) + 2*a^2*x^4*ArcCos[a*x^2] + ArcSin[a*x^2])/(8*a^2)","A",1
48,1,63,55,0.1629904,"\int x^2 \cos ^{-1}\left(a x^2\right) \, dx","Integrate[x^2*ArcCos[a*x^2],x]","\frac{1}{9} \left(-\frac{2 x \sqrt{1-a^2 x^4}}{a}+3 x^3 \cos ^{-1}\left(a x^2\right)+\frac{2 i F\left(\left.i \sinh ^{-1}\left(\sqrt{-a} x\right)\right|-1\right)}{(-a)^{3/2}}\right)","\frac{2 F\left(\left.\sin ^{-1}\left(\sqrt{a} x\right)\right|-1\right)}{9 a^{3/2}}-\frac{2 x \sqrt{1-a^2 x^4}}{9 a}+\frac{1}{3} x^3 \cos ^{-1}\left(a x^2\right)",1,"((-2*x*Sqrt[1 - a^2*x^4])/a + 3*x^3*ArcCos[a*x^2] + ((2*I)*EllipticF[I*ArcSinh[Sqrt[-a]*x], -1])/(-a)^(3/2))/9","C",1
49,1,35,35,0.0150984,"\int x \cos ^{-1}\left(a x^2\right) \, dx","Integrate[x*ArcCos[a*x^2],x]","\frac{1}{2} x^2 \cos ^{-1}\left(a x^2\right)-\frac{\sqrt{1-a^2 x^4}}{2 a}","\frac{1}{2} x^2 \cos ^{-1}\left(a x^2\right)-\frac{\sqrt{1-a^2 x^4}}{2 a}",1,"-1/2*Sqrt[1 - a^2*x^4]/a + (x^2*ArcCos[a*x^2])/2","A",1
50,1,34,43,0.0052708,"\int \cos ^{-1}\left(a x^2\right) \, dx","Integrate[ArcCos[a*x^2],x]","\frac{2}{3} a x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};a^2 x^4\right)+x \cos ^{-1}\left(a x^2\right)","x \cos ^{-1}\left(a x^2\right)-\frac{2 F\left(\left.\sin ^{-1}\left(\sqrt{a} x\right)\right|-1\right)}{\sqrt{a}}+\frac{2 E\left(\left.\sin ^{-1}\left(\sqrt{a} x\right)\right|-1\right)}{\sqrt{a}}",1,"x*ArcCos[a*x^2] + (2*a*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, a^2*x^4])/3","C",1
51,1,56,62,0.0323499,"\int \frac{\cos ^{-1}\left(a x^2\right)}{x} \, dx","Integrate[ArcCos[a*x^2]/x,x]","-\frac{1}{4} i \left(\text{Li}_2\left(-e^{2 i \cos ^{-1}\left(a x^2\right)}\right)+\cos ^{-1}\left(a x^2\right) \left(\cos ^{-1}\left(a x^2\right)+2 i \log \left(1+e^{2 i \cos ^{-1}\left(a x^2\right)}\right)\right)\right)","-\frac{1}{4} i \text{Li}_2\left(-e^{2 i \cos ^{-1}\left(a x^2\right)}\right)-\frac{1}{4} i \cos ^{-1}\left(a x^2\right)^2+\frac{1}{2} \cos ^{-1}\left(a x^2\right) \log \left(1+e^{2 i \cos ^{-1}\left(a x^2\right)}\right)",1,"(-1/4*I)*(ArcCos[a*x^2]*(ArcCos[a*x^2] + (2*I)*Log[1 + E^((2*I)*ArcCos[a*x^2])]) + PolyLog[2, -E^((2*I)*ArcCos[a*x^2])])","A",1
52,1,40,29,0.0415398,"\int \frac{\cos ^{-1}\left(a x^2\right)}{x^2} \, dx","Integrate[ArcCos[a*x^2]/x^2,x]","-\frac{\cos ^{-1}\left(a x^2\right)+2 i \sqrt{-a} x F\left(\left.i \sinh ^{-1}\left(\sqrt{-a} x\right)\right|-1\right)}{x}","-\frac{\cos ^{-1}\left(a x^2\right)}{x}-2 \sqrt{a} F\left(\left.\sin ^{-1}\left(\sqrt{a} x\right)\right|-1\right)",1,"-((ArcCos[a*x^2] + (2*I)*Sqrt[-a]*x*EllipticF[I*ArcSinh[Sqrt[-a]*x], -1])/x)","C",1
53,1,61,58,0.0569321,"\int x^2 \cos ^{-1}\left(\frac{a}{x}\right) \, dx","Integrate[x^2*ArcCos[a/x],x]","\frac{1}{3} x^3 \cos ^{-1}\left(\frac{a}{x}\right)-\frac{1}{6} a \left(x^2 \sqrt{1-\frac{a^2}{x^2}}+a^2 \log \left(x \left(\sqrt{1-\frac{a^2}{x^2}}+1\right)\right)\right)","-\frac{1}{6} a x^2 \sqrt{1-\frac{a^2}{x^2}}-\frac{1}{6} a^3 \tanh ^{-1}\left(\sqrt{1-\frac{a^2}{x^2}}\right)+\frac{1}{3} x^3 \sec ^{-1}\left(\frac{x}{a}\right)",1,"(x^3*ArcCos[a/x])/3 - (a*(Sqrt[1 - a^2/x^2]*x^2 + a^2*Log[(1 + Sqrt[1 - a^2/x^2])*x]))/6","A",1
54,1,33,34,0.0230961,"\int x \cos ^{-1}\left(\frac{a}{x}\right) \, dx","Integrate[x*ArcCos[a/x],x]","\frac{1}{2} \left(x^2 \cos ^{-1}\left(\frac{a}{x}\right)-a x \sqrt{1-\frac{a^2}{x^2}}\right)","\frac{1}{2} x^2 \sec ^{-1}\left(\frac{x}{a}\right)-\frac{1}{2} a x \sqrt{1-\frac{a^2}{x^2}}",1,"(-(a*Sqrt[1 - a^2/x^2]*x) + x^2*ArcCos[a/x])/2","A",1
55,1,84,27,0.1093734,"\int \cos ^{-1}\left(\frac{a}{x}\right) \, dx","Integrate[ArcCos[a/x],x]","x \cos ^{-1}\left(\frac{a}{x}\right)-\frac{a \sqrt{x^2-a^2} \left(\log \left(\frac{x}{\sqrt{x^2-a^2}}+1\right)-\log \left(1-\frac{x}{\sqrt{x^2-a^2}}\right)\right)}{2 x \sqrt{1-\frac{a^2}{x^2}}}","x \sec ^{-1}\left(\frac{x}{a}\right)-a \tanh ^{-1}\left(\sqrt{1-\frac{a^2}{x^2}}\right)",1,"x*ArcCos[a/x] - (a*Sqrt[-a^2 + x^2]*(-Log[1 - x/Sqrt[-a^2 + x^2]] + Log[1 + x/Sqrt[-a^2 + x^2]]))/(2*Sqrt[1 - a^2/x^2]*x)","B",1
56,1,60,60,0.0213262,"\int \frac{\cos ^{-1}\left(\frac{a}{x}\right)}{x} \, dx","Integrate[ArcCos[a/x]/x,x]","\frac{1}{2} i \text{Li}_2\left(-e^{2 i \cos ^{-1}\left(\frac{a}{x}\right)}\right)+\frac{1}{2} i \cos ^{-1}\left(\frac{a}{x}\right)^2-\cos ^{-1}\left(\frac{a}{x}\right) \log \left(1+e^{2 i \cos ^{-1}\left(\frac{a}{x}\right)}\right)","\frac{1}{2} i \text{Li}_2\left(-e^{2 i \cos ^{-1}\left(\frac{a}{x}\right)}\right)+\frac{1}{2} i \cos ^{-1}\left(\frac{a}{x}\right)^2-\cos ^{-1}\left(\frac{a}{x}\right) \log \left(1+e^{2 i \cos ^{-1}\left(\frac{a}{x}\right)}\right)",1,"(I/2)*ArcCos[a/x]^2 - ArcCos[a/x]*Log[1 + E^((2*I)*ArcCos[a/x])] + (I/2)*PolyLog[2, -E^((2*I)*ArcCos[a/x])]","A",1
57,1,30,30,0.0175893,"\int \frac{\cos ^{-1}\left(\frac{a}{x}\right)}{x^2} \, dx","Integrate[ArcCos[a/x]/x^2,x]","\frac{\sqrt{1-\frac{a^2}{x^2}}}{a}-\frac{\cos ^{-1}\left(\frac{a}{x}\right)}{x}","\frac{\sqrt{1-\frac{a^2}{x^2}}}{a}-\frac{\sec ^{-1}\left(\frac{x}{a}\right)}{x}",1,"Sqrt[1 - a^2/x^2]/a - ArcCos[a/x]/x","A",1
58,1,50,51,0.026506,"\int \frac{\cos ^{-1}\left(\frac{a}{x}\right)}{x^3} \, dx","Integrate[ArcCos[a/x]/x^3,x]","\frac{a x \sqrt{1-\frac{a^2}{x^2}}-2 a^2 \cos ^{-1}\left(\frac{a}{x}\right)-x^2 \sin ^{-1}\left(\frac{a}{x}\right)}{4 a^2 x^2}","\frac{\sqrt{1-\frac{a^2}{x^2}}}{4 a x}-\frac{\csc ^{-1}\left(\frac{x}{a}\right)}{4 a^2}-\frac{\sec ^{-1}\left(\frac{x}{a}\right)}{2 x^2}",1,"(a*Sqrt[1 - a^2/x^2]*x - 2*a^2*ArcCos[a/x] - x^2*ArcSin[a/x])/(4*a^2*x^2)","A",1
59,1,47,56,0.0330396,"\int \frac{\cos ^{-1}\left(\frac{a}{x}\right)}{x^4} \, dx","Integrate[ArcCos[a/x]/x^4,x]","\frac{x \sqrt{1-\frac{a^2}{x^2}} \left(a^2+2 x^2\right)-3 a^3 \cos ^{-1}\left(\frac{a}{x}\right)}{9 a^3 x^3}","-\frac{\left(1-\frac{a^2}{x^2}\right)^{3/2}}{9 a^3}+\frac{\sqrt{1-\frac{a^2}{x^2}}}{3 a^3}-\frac{\sec ^{-1}\left(\frac{x}{a}\right)}{3 x^3}",1,"(Sqrt[1 - a^2/x^2]*x*(a^2 + 2*x^2) - 3*a^3*ArcCos[a/x])/(9*a^3*x^3)","A",1
60,1,46,78,0.0393252,"\int x^2 \cos ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x^2*ArcCos[Sqrt[x]],x]","\frac{1}{144} \left(48 x^3 \cos ^{-1}\left(\sqrt{x}\right)-\sqrt{-((x-1) x)} \left(8 x^2+10 x+15\right)+15 \sin ^{-1}\left(\sqrt{x}\right)\right)","-\frac{1}{18} \sqrt{1-x} x^{5/2}-\frac{5}{72} \sqrt{1-x} x^{3/2}+\frac{1}{3} x^3 \cos ^{-1}\left(\sqrt{x}\right)-\frac{5}{48} \sqrt{1-x} \sqrt{x}-\frac{5}{96} \sin ^{-1}(1-2 x)",1,"(-(Sqrt[-((-1 + x)*x)]*(15 + 10*x + 8*x^2)) + 48*x^3*ArcCos[Sqrt[x]] + 15*ArcSin[Sqrt[x]])/144","A",1
61,1,41,60,0.0309441,"\int x \cos ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x*ArcCos[Sqrt[x]],x]","\frac{1}{16} \left(8 x^2 \cos ^{-1}\left(\sqrt{x}\right)-\sqrt{-((x-1) x)} (2 x+3)+3 \sin ^{-1}\left(\sqrt{x}\right)\right)","-\frac{1}{8} \sqrt{1-x} x^{3/2}+\frac{1}{2} x^2 \cos ^{-1}\left(\sqrt{x}\right)-\frac{3}{16} \sqrt{1-x} \sqrt{x}-\frac{3}{32} \sin ^{-1}(1-2 x)",1,"(-(Sqrt[-((-1 + x)*x)]*(3 + 2*x)) + 8*x^2*ArcCos[Sqrt[x]] + 3*ArcSin[Sqrt[x]])/16","A",1
62,1,38,37,0.016917,"\int \cos ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[ArcCos[Sqrt[x]],x]","\frac{1}{2} \left(-\sqrt{-((x-1) x)}-\sin ^{-1}\left(\sqrt{1-x}\right)\right)+x \cos ^{-1}\left(\sqrt{x}\right)","-\frac{1}{2} \sqrt{1-x} \sqrt{x}-\frac{1}{4} \sin ^{-1}(1-2 x)+x \cos ^{-1}\left(\sqrt{x}\right)",1,"x*ArcCos[Sqrt[x]] + (-Sqrt[-((-1 + x)*x)] - ArcSin[Sqrt[1 - x]])/2","A",1
63,1,54,56,0.0275242,"\int \frac{\cos ^{-1}\left(\sqrt{x}\right)}{x} \, dx","Integrate[ArcCos[Sqrt[x]]/x,x]","-i \left(\text{Li}_2\left(-e^{2 i \cos ^{-1}\left(\sqrt{x}\right)}\right)+\cos ^{-1}\left(\sqrt{x}\right) \left(\cos ^{-1}\left(\sqrt{x}\right)+2 i \log \left(1+e^{2 i \cos ^{-1}\left(\sqrt{x}\right)}\right)\right)\right)","-i \text{Li}_2\left(-e^{2 i \cos ^{-1}\left(\sqrt{x}\right)}\right)-i \cos ^{-1}\left(\sqrt{x}\right)^2+2 \cos ^{-1}\left(\sqrt{x}\right) \log \left(1+e^{2 i \cos ^{-1}\left(\sqrt{x}\right)}\right)",1,"(-I)*(ArcCos[Sqrt[x]]*(ArcCos[Sqrt[x]] + (2*I)*Log[1 + E^((2*I)*ArcCos[Sqrt[x]])]) + PolyLog[2, -E^((2*I)*ArcCos[Sqrt[x]])])","A",1
64,1,24,27,0.017107,"\int \frac{\cos ^{-1}\left(\sqrt{x}\right)}{x^2} \, dx","Integrate[ArcCos[Sqrt[x]]/x^2,x]","\frac{\sqrt{x-x^2}-\cos ^{-1}\left(\sqrt{x}\right)}{x}","\frac{\sqrt{1-x}}{\sqrt{x}}-\frac{\cos ^{-1}\left(\sqrt{x}\right)}{x}",1,"(Sqrt[x - x^2] - ArcCos[Sqrt[x]])/x","A",1
65,1,43,50,0.0230167,"\int \frac{\cos ^{-1}\left(\sqrt{x}\right)}{x^3} \, dx","Integrate[ArcCos[Sqrt[x]]/x^3,x]","\left(\frac{1}{6 x^{3/2}}+\frac{1}{3 \sqrt{x}}\right) \sqrt{1-x}-\frac{\cos ^{-1}\left(\sqrt{x}\right)}{2 x^2}","\frac{\sqrt{1-x}}{6 x^{3/2}}-\frac{\cos ^{-1}\left(\sqrt{x}\right)}{2 x^2}+\frac{\sqrt{1-x}}{3 \sqrt{x}}",1,"(1/(6*x^(3/2)) + 1/(3*Sqrt[x]))*Sqrt[1 - x] - ArcCos[Sqrt[x]]/(2*x^2)","A",1
66,1,37,68,0.0408987,"\int \frac{\cos ^{-1}\left(\sqrt{x}\right)}{x^4} \, dx","Integrate[ArcCos[Sqrt[x]]/x^4,x]","\frac{\sqrt{-((x-1) x)} \left(8 x^2+4 x+3\right)-15 \cos ^{-1}\left(\sqrt{x}\right)}{45 x^3}","\frac{4 \sqrt{1-x}}{45 x^{3/2}}+\frac{\sqrt{1-x}}{15 x^{5/2}}-\frac{\cos ^{-1}\left(\sqrt{x}\right)}{3 x^3}+\frac{8 \sqrt{1-x}}{45 \sqrt{x}}",1,"(Sqrt[-((-1 + x)*x)]*(3 + 4*x + 8*x^2) - 15*ArcCos[Sqrt[x]])/(45*x^3)","A",1
67,1,42,86,0.047112,"\int \frac{\cos ^{-1}\left(\sqrt{x}\right)}{x^5} \, dx","Integrate[ArcCos[Sqrt[x]]/x^5,x]","\frac{\sqrt{-((x-1) x)} \left(16 x^3+8 x^2+6 x+5\right)-35 \cos ^{-1}\left(\sqrt{x}\right)}{140 x^4}","\frac{2 \sqrt{1-x}}{35 x^{3/2}}+\frac{3 \sqrt{1-x}}{70 x^{5/2}}+\frac{\sqrt{1-x}}{28 x^{7/2}}-\frac{\cos ^{-1}\left(\sqrt{x}\right)}{4 x^4}+\frac{4 \sqrt{1-x}}{35 \sqrt{x}}",1,"(Sqrt[-((-1 + x)*x)]*(5 + 6*x + 8*x^2 + 16*x^3) - 35*ArcCos[Sqrt[x]])/(140*x^4)","A",1
68,1,25,25,0.0083119,"\int \frac{\cos ^{-1}\left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Integrate[ArcCos[Sqrt[x]]/Sqrt[x],x]","2 \sqrt{x} \cos ^{-1}\left(\sqrt{x}\right)-2 \sqrt{1-x}","2 \sqrt{x} \cos ^{-1}\left(\sqrt{x}\right)-2 \sqrt{1-x}",1,"-2*Sqrt[1 - x] + 2*Sqrt[x]*ArcCos[Sqrt[x]]","A",1
69,1,141,68,0.1464548,"\int \frac{\cos ^{-1}\left(a x^n\right)}{x} \, dx","Integrate[ArcCos[a*x^n]/x,x]","\frac{a \left(\text{Li}_2\left(e^{-2 \sinh ^{-1}\left(\sqrt{-a^2} x^n\right)}\right)-\sinh ^{-1}\left(\sqrt{-a^2} x^n\right)^2-2 \sinh ^{-1}\left(\sqrt{-a^2} x^n\right) \log \left(1-e^{-2 \sinh ^{-1}\left(\sqrt{-a^2} x^n\right)}\right)+2 n \log (x) \log \left(\sqrt{1-a^2 x^{2 n}}+\sqrt{-a^2} x^n\right)\right)}{2 \sqrt{-a^2} n}+\log (x) \cos ^{-1}\left(a x^n\right)","-\frac{i \text{Li}_2\left(-e^{2 i \cos ^{-1}\left(a x^n\right)}\right)}{2 n}-\frac{i \cos ^{-1}\left(a x^n\right)^2}{2 n}+\frac{\cos ^{-1}\left(a x^n\right) \log \left(1+e^{2 i \cos ^{-1}\left(a x^n\right)}\right)}{n}",1,"ArcCos[a*x^n]*Log[x] + (a*(-ArcSinh[Sqrt[-a^2]*x^n]^2 - 2*ArcSinh[Sqrt[-a^2]*x^n]*Log[1 - E^(-2*ArcSinh[Sqrt[-a^2]*x^n])] + 2*n*Log[x]*Log[Sqrt[-a^2]*x^n + Sqrt[1 - a^2*x^(2*n)]] + PolyLog[2, E^(-2*ArcSinh[Sqrt[-a^2]*x^n])]))/(2*Sqrt[-a^2]*n)","B",0
70,1,56,62,0.0328478,"\int \frac{\cos ^{-1}\left(a x^5\right)}{x} \, dx","Integrate[ArcCos[a*x^5]/x,x]","-\frac{1}{10} i \left(\text{Li}_2\left(-e^{2 i \cos ^{-1}\left(a x^5\right)}\right)+\cos ^{-1}\left(a x^5\right) \left(\cos ^{-1}\left(a x^5\right)+2 i \log \left(1+e^{2 i \cos ^{-1}\left(a x^5\right)}\right)\right)\right)","-\frac{1}{10} i \text{Li}_2\left(-e^{2 i \cos ^{-1}\left(a x^5\right)}\right)-\frac{1}{10} i \cos ^{-1}\left(a x^5\right)^2+\frac{1}{5} \cos ^{-1}\left(a x^5\right) \log \left(1+e^{2 i \cos ^{-1}\left(a x^5\right)}\right)",1,"(-1/10*I)*(ArcCos[a*x^5]*(ArcCos[a*x^5] + (2*I)*Log[1 + E^((2*I)*ArcCos[a*x^5])]) + PolyLog[2, -E^((2*I)*ArcCos[a*x^5])])","A",1
71,1,43,47,0.0293961,"\int x^3 \cos ^{-1}\left(a+b x^4\right) \, dx","Integrate[x^3*ArcCos[a + b*x^4],x]","\frac{\left(a+b x^4\right) \cos ^{-1}\left(a+b x^4\right)-\sqrt{1-\left(a+b x^4\right)^2}}{4 b}","\frac{\left(a+b x^4\right) \cos ^{-1}\left(a+b x^4\right)}{4 b}-\frac{\sqrt{1-\left(a+b x^4\right)^2}}{4 b}",1,"(-Sqrt[1 - (a + b*x^4)^2] + (a + b*x^4)*ArcCos[a + b*x^4])/(4*b)","A",1
72,1,43,48,0.0431873,"\int x^{-1+n} \cos ^{-1}\left(a+b x^n\right) \, dx","Integrate[x^(-1 + n)*ArcCos[a + b*x^n],x]","\frac{\left(a+b x^n\right) \cos ^{-1}\left(a+b x^n\right)-\sqrt{1-\left(a+b x^n\right)^2}}{b n}","\frac{\left(a+b x^n\right) \cos ^{-1}\left(a+b x^n\right)}{b n}-\frac{\sqrt{1-\left(a+b x^n\right)^2}}{b n}",1,"(-Sqrt[1 - (a + b*x^n)^2] + (a + b*x^n)*ArcCos[a + b*x^n])/(b*n)","A",1
73,1,249,127,0.2454188,"\int \left(a+b \cos ^{-1}\left(1+d x^2\right)\right)^4 \, dx","Integrate[(a + b*ArcCos[1 + d*x^2])^4,x]","\frac{-8 a b \left(a^2-24 b^2\right) \sqrt{-d x^2 \left(d x^2+2\right)}+6 b^2 \cos ^{-1}\left(d x^2+1\right)^2 \left(a^2 d x^2-4 a b \sqrt{-d x^2 \left(d x^2+2\right)}-8 b^2 d x^2\right)+d x^2 \left(a^4-48 a^2 b^2+384 b^4\right)+4 b \cos ^{-1}\left(d x^2+1\right) \left(a^3 d x^2-6 a^2 b \sqrt{-d x^2 \left(d x^2+2\right)}-24 a b^2 d x^2+48 b^3 \sqrt{-d x^2 \left(d x^2+2\right)}\right)+4 b^3 \cos ^{-1}\left(d x^2+1\right)^3 \left(a d x^2-2 b \sqrt{-d x^2 \left(d x^2+2\right)}\right)+b^4 d x^2 \cos ^{-1}\left(d x^2+1\right)^4}{d x}","\frac{192 b^3 \sqrt{-d^2 x^4-2 d x^2} \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)}{d x}-48 b^2 x \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^2-\frac{8 b \sqrt{-d^2 x^4-2 d x^2} \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^3}{d x}+x \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^4+384 b^4 x",1,"((a^4 - 48*a^2*b^2 + 384*b^4)*d*x^2 - 8*a*b*(a^2 - 24*b^2)*Sqrt[-(d*x^2*(2 + d*x^2))] + 4*b*(a^3*d*x^2 - 24*a*b^2*d*x^2 - 6*a^2*b*Sqrt[-(d*x^2*(2 + d*x^2))] + 48*b^3*Sqrt[-(d*x^2*(2 + d*x^2))])*ArcCos[1 + d*x^2] + 6*b^2*(a^2*d*x^2 - 8*b^2*d*x^2 - 4*a*b*Sqrt[-(d*x^2*(2 + d*x^2))])*ArcCos[1 + d*x^2]^2 + 4*b^3*(a*d*x^2 - 2*b*Sqrt[-(d*x^2*(2 + d*x^2))])*ArcCos[1 + d*x^2]^3 + b^4*d*x^2*ArcCos[1 + d*x^2]^4)/(d*x)","A",1
74,1,162,110,0.1345711,"\int \left(a+b \cos ^{-1}\left(1+d x^2\right)\right)^3 \, dx","Integrate[(a + b*ArcCos[1 + d*x^2])^3,x]","\frac{a d x^2 \left(a^2-24 b^2\right)-6 b \left(a^2-8 b^2\right) \sqrt{-d x^2 \left(d x^2+2\right)}+3 b \cos ^{-1}\left(d x^2+1\right) \left(a^2 d x^2-4 a b \sqrt{-d x^2 \left(d x^2+2\right)}-8 b^2 d x^2\right)+3 b^2 \cos ^{-1}\left(d x^2+1\right)^2 \left(a d x^2-2 b \sqrt{-d x^2 \left(d x^2+2\right)}\right)+b^3 d x^2 \cos ^{-1}\left(d x^2+1\right)^3}{d x}","-24 a b^2 x-\frac{6 b \sqrt{-d^2 x^4-2 d x^2} \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^2}{d x}+x \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^3+\frac{48 b^3 \sqrt{-d^2 x^4-2 d x^2}}{d x}-24 b^3 x \cos ^{-1}\left(d x^2+1\right)",1,"(a*(a^2 - 24*b^2)*d*x^2 - 6*b*(a^2 - 8*b^2)*Sqrt[-(d*x^2*(2 + d*x^2))] + 3*b*(a^2*d*x^2 - 8*b^2*d*x^2 - 4*a*b*Sqrt[-(d*x^2*(2 + d*x^2))])*ArcCos[1 + d*x^2] + 3*b^2*(a*d*x^2 - 2*b*Sqrt[-(d*x^2*(2 + d*x^2))])*ArcCos[1 + d*x^2]^2 + b^3*d*x^2*ArcCos[1 + d*x^2]^3)/(d*x)","A",1
75,1,98,63,0.0698871,"\int \left(a+b \cos ^{-1}\left(1+d x^2\right)\right)^2 \, dx","Integrate[(a + b*ArcCos[1 + d*x^2])^2,x]","x \left(a^2-8 b^2\right)-\frac{4 a b \sqrt{-d x^2 \left(d x^2+2\right)}}{d x}+\frac{2 b \cos ^{-1}\left(d x^2+1\right) \left(a d x^2-2 b \sqrt{-d x^2 \left(d x^2+2\right)}\right)}{d x}+b^2 x \cos ^{-1}\left(d x^2+1\right)^2","-\frac{4 b \sqrt{-d^2 x^4-2 d x^2} \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)}{d x}+x \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^2-8 b^2 x",1,"(a^2 - 8*b^2)*x - (4*a*b*Sqrt[-(d*x^2*(2 + d*x^2))])/(d*x) + (2*b*(a*d*x^2 - 2*b*Sqrt[-(d*x^2*(2 + d*x^2))])*ArcCos[1 + d*x^2])/(d*x) + b^2*x*ArcCos[1 + d*x^2]^2","A",1
76,1,41,43,0.0305573,"\int \left(a+b \cos ^{-1}\left(1+d x^2\right)\right) \, dx","Integrate[a + b*ArcCos[1 + d*x^2],x]","a x-\frac{2 b \sqrt{-d x^2 \left(d x^2+2\right)}}{d x}+b x \cos ^{-1}\left(d x^2+1\right)","a x-\frac{2 b \sqrt{-d^2 x^4-2 d x^2}}{d x}+b x \cos ^{-1}\left(d x^2+1\right)",1,"a*x - (2*b*Sqrt[-(d*x^2*(2 + d*x^2))])/(d*x) + b*x*ArcCos[1 + d*x^2]","A",1
77,1,85,99,0.1131646,"\int \frac{1}{a+b \cos ^{-1}\left(1+d x^2\right)} \, dx","Integrate[(a + b*ArcCos[1 + d*x^2])^(-1),x]","-\frac{\sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) \left(\cos \left(\frac{a}{2 b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}\left(d x^2+1\right)}{2 b}\right)+\sin \left(\frac{a}{2 b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}\left(d x^2+1\right)}{2 b}\right)\right)}{b d x}","\frac{x \cos \left(\frac{a}{2 b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}\left(d x^2+1\right)}{2 b}\right)}{\sqrt{2} b \sqrt{-d x^2}}+\frac{x \sin \left(\frac{a}{2 b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}\left(d x^2+1\right)}{2 b}\right)}{\sqrt{2} b \sqrt{-d x^2}}",1,"-((Sin[ArcCos[1 + d*x^2]/2]*(Cos[a/(2*b)]*CosIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)] + Sin[a/(2*b)]*SinIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)]))/(b*d*x))","A",0
78,1,133,151,0.3501362,"\int \frac{1}{\left(a+b \cos ^{-1}\left(1+d x^2\right)\right)^2} \, dx","Integrate[(a + b*ArcCos[1 + d*x^2])^(-2),x]","\frac{\sqrt{-d x^2 \left(d x^2+2\right)} \left(\frac{b}{a+b \cos ^{-1}\left(d x^2+1\right)}-\frac{\cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) \left(\sin \left(\frac{a}{2 b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}\left(d x^2+1\right)}{2 b}\right)-\cos \left(\frac{a}{2 b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}\left(d x^2+1\right)}{2 b}\right)\right)}{d x^2+2}\right)}{2 b^2 d x}","\frac{x \sin \left(\frac{a}{2 b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}\left(d x^2+1\right)}{2 b}\right)}{2 \sqrt{2} b^2 \sqrt{-d x^2}}-\frac{x \cos \left(\frac{a}{2 b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}\left(d x^2+1\right)}{2 b}\right)}{2 \sqrt{2} b^2 \sqrt{-d x^2}}+\frac{\sqrt{-d^2 x^4-2 d x^2}}{2 b d x \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)}",1,"(Sqrt[-(d*x^2*(2 + d*x^2))]*(b/(a + b*ArcCos[1 + d*x^2]) - (Cos[ArcCos[1 + d*x^2]/2]*(CosIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)]*Sin[a/(2*b)] - Cos[a/(2*b)]*SinIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)]))/(2 + d*x^2)))/(2*b^2*d*x)","A",0
79,1,147,173,0.2593655,"\int \frac{1}{\left(a+b \cos ^{-1}\left(1+d x^2\right)\right)^3} \, dx","Integrate[(a + b*ArcCos[1 + d*x^2])^(-3),x]","\frac{\frac{2 b^2 \sqrt{-d x^2 \left(d x^2+2\right)}}{d \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^2}+\frac{\sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) \left(\cos \left(\frac{a}{2 b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}\left(d x^2+1\right)}{2 b}\right)+\sin \left(\frac{a}{2 b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}\left(d x^2+1\right)}{2 b}\right)\right)}{d}+\frac{b x^2}{a+b \cos ^{-1}\left(d x^2+1\right)}}{8 b^3 x}","-\frac{x \cos \left(\frac{a}{2 b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}\left(d x^2+1\right)}{2 b}\right)}{8 \sqrt{2} b^3 \sqrt{-d x^2}}-\frac{x \sin \left(\frac{a}{2 b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}\left(d x^2+1\right)}{2 b}\right)}{8 \sqrt{2} b^3 \sqrt{-d x^2}}+\frac{x}{8 b^2 \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)}+\frac{\sqrt{-d^2 x^4-2 d x^2}}{4 b d x \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^2}",1,"((2*b^2*Sqrt[-(d*x^2*(2 + d*x^2))])/(d*(a + b*ArcCos[1 + d*x^2])^2) + (b*x^2)/(a + b*ArcCos[1 + d*x^2]) + (Sin[ArcCos[1 + d*x^2]/2]*(Cos[a/(2*b)]*CosIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)] + Sin[a/(2*b)]*SinIntegral[(a + b*ArcCos[1 + d*x^2])/(2*b)]))/d)/(8*b^3*x)","A",0
80,1,249,127,0.3700767,"\int \left(a+b \cos ^{-1}\left(-1+d x^2\right)\right)^4 \, dx","Integrate[(a + b*ArcCos[-1 + d*x^2])^4,x]","\frac{-8 a b \left(a^2-24 b^2\right) \sqrt{d x^2 \left(2-d x^2\right)}+6 b^2 \cos ^{-1}\left(d x^2-1\right)^2 \left(a^2 d x^2-4 a b \sqrt{-d x^2 \left(d x^2-2\right)}-8 b^2 d x^2\right)+d x^2 \left(a^4-48 a^2 b^2+384 b^4\right)+4 b \cos ^{-1}\left(d x^2-1\right) \left(a^3 d x^2-6 a^2 b \sqrt{-d x^2 \left(d x^2-2\right)}-24 a b^2 d x^2+48 b^3 \sqrt{-d x^2 \left(d x^2-2\right)}\right)+4 b^3 \cos ^{-1}\left(d x^2-1\right)^3 \left(a d x^2-2 b \sqrt{-d x^2 \left(d x^2-2\right)}\right)+b^4 d x^2 \cos ^{-1}\left(d x^2-1\right)^4}{d x}","\frac{192 b^3 \sqrt{2 d x^2-d^2 x^4} \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)}{d x}-48 b^2 x \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^2-\frac{8 b \sqrt{2 d x^2-d^2 x^4} \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^3}{d x}+x \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^4+384 b^4 x",1,"((a^4 - 48*a^2*b^2 + 384*b^4)*d*x^2 - 8*a*b*(a^2 - 24*b^2)*Sqrt[d*x^2*(2 - d*x^2)] + 4*b*(a^3*d*x^2 - 24*a*b^2*d*x^2 - 6*a^2*b*Sqrt[-(d*x^2*(-2 + d*x^2))] + 48*b^3*Sqrt[-(d*x^2*(-2 + d*x^2))])*ArcCos[-1 + d*x^2] + 6*b^2*(a^2*d*x^2 - 8*b^2*d*x^2 - 4*a*b*Sqrt[-(d*x^2*(-2 + d*x^2))])*ArcCos[-1 + d*x^2]^2 + 4*b^3*(a*d*x^2 - 2*b*Sqrt[-(d*x^2*(-2 + d*x^2))])*ArcCos[-1 + d*x^2]^3 + b^4*d*x^2*ArcCos[-1 + d*x^2]^4)/(d*x)","A",1
81,1,162,110,0.1967488,"\int \left(a+b \cos ^{-1}\left(-1+d x^2\right)\right)^3 \, dx","Integrate[(a + b*ArcCos[-1 + d*x^2])^3,x]","\frac{a d x^2 \left(a^2-24 b^2\right)-6 b \left(a^2-8 b^2\right) \sqrt{d x^2 \left(2-d x^2\right)}+3 b \cos ^{-1}\left(d x^2-1\right) \left(a^2 d x^2-4 a b \sqrt{-d x^2 \left(d x^2-2\right)}-8 b^2 d x^2\right)+3 b^2 \cos ^{-1}\left(d x^2-1\right)^2 \left(a d x^2-2 b \sqrt{-d x^2 \left(d x^2-2\right)}\right)+b^3 d x^2 \cos ^{-1}\left(d x^2-1\right)^3}{d x}","-24 a b^2 x-\frac{6 b \sqrt{2 d x^2-d^2 x^4} \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^2}{d x}+x \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^3+\frac{48 b^3 \sqrt{2 d x^2-d^2 x^4}}{d x}-24 b^3 x \cos ^{-1}\left(d x^2-1\right)",1,"(a*(a^2 - 24*b^2)*d*x^2 - 6*b*(a^2 - 8*b^2)*Sqrt[d*x^2*(2 - d*x^2)] + 3*b*(a^2*d*x^2 - 8*b^2*d*x^2 - 4*a*b*Sqrt[-(d*x^2*(-2 + d*x^2))])*ArcCos[-1 + d*x^2] + 3*b^2*(a*d*x^2 - 2*b*Sqrt[-(d*x^2*(-2 + d*x^2))])*ArcCos[-1 + d*x^2]^2 + b^3*d*x^2*ArcCos[-1 + d*x^2]^3)/(d*x)","A",1
82,1,98,63,0.0760705,"\int \left(a+b \cos ^{-1}\left(-1+d x^2\right)\right)^2 \, dx","Integrate[(a + b*ArcCos[-1 + d*x^2])^2,x]","x \left(a^2-8 b^2\right)-\frac{4 a b \sqrt{-d x^2 \left(d x^2-2\right)}}{d x}+\frac{2 b \cos ^{-1}\left(d x^2-1\right) \left(a d x^2-2 b \sqrt{-d x^2 \left(d x^2-2\right)}\right)}{d x}+b^2 x \cos ^{-1}\left(d x^2-1\right)^2","-\frac{4 b \sqrt{2 d x^2-d^2 x^4} \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)}{d x}+x \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^2-8 b^2 x",1,"(a^2 - 8*b^2)*x - (4*a*b*Sqrt[-(d*x^2*(-2 + d*x^2))])/(d*x) + (2*b*(a*d*x^2 - 2*b*Sqrt[-(d*x^2*(-2 + d*x^2))])*ArcCos[-1 + d*x^2])/(d*x) + b^2*x*ArcCos[-1 + d*x^2]^2","A",1
83,1,41,43,0.0341238,"\int \left(a+b \cos ^{-1}\left(-1+d x^2\right)\right) \, dx","Integrate[a + b*ArcCos[-1 + d*x^2],x]","a x-\frac{2 b \sqrt{-d x^2 \left(d x^2-2\right)}}{d x}+b x \cos ^{-1}\left(d x^2-1\right)","a x-\frac{2 b \sqrt{2 d x^2-d^2 x^4}}{d x}+b x \cos ^{-1}\left(d x^2-1\right)",1,"a*x - (2*b*Sqrt[-(d*x^2*(-2 + d*x^2))])/(d*x) + b*x*ArcCos[-1 + d*x^2]","A",1
84,1,85,98,0.1117489,"\int \frac{1}{a+b \cos ^{-1}\left(-1+d x^2\right)} \, dx","Integrate[(a + b*ArcCos[-1 + d*x^2])^(-1),x]","\frac{\cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) \left(\sin \left(\frac{a}{2 b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}\left(d x^2-1\right)}{2 b}\right)-\cos \left(\frac{a}{2 b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}\left(d x^2-1\right)}{2 b}\right)\right)}{b d x}","\frac{x \sin \left(\frac{a}{2 b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}\left(d x^2-1\right)}{2 b}\right)}{\sqrt{2} b \sqrt{d x^2}}-\frac{x \cos \left(\frac{a}{2 b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}\left(d x^2-1\right)}{2 b}\right)}{\sqrt{2} b \sqrt{d x^2}}",1,"(Cos[ArcCos[-1 + d*x^2]/2]*(CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]*Sin[a/(2*b)] - Cos[a/(2*b)]*SinIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]))/(b*d*x)","A",0
85,1,131,149,0.4004669,"\int \frac{1}{\left(a+b \cos ^{-1}\left(-1+d x^2\right)\right)^2} \, dx","Integrate[(a + b*ArcCos[-1 + d*x^2])^(-2),x]","\frac{\sqrt{-d x^2 \left(d x^2-2\right)} \left(\frac{\sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) \left(\cos \left(\frac{a}{2 b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}\left(d x^2-1\right)}{2 b}\right)+\sin \left(\frac{a}{2 b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}\left(d x^2-1\right)}{2 b}\right)\right)}{d x^2-2}+\frac{b}{a+b \cos ^{-1}\left(d x^2-1\right)}\right)}{2 b^2 d x}","-\frac{x \cos \left(\frac{a}{2 b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}\left(d x^2-1\right)}{2 b}\right)}{2 \sqrt{2} b^2 \sqrt{d x^2}}-\frac{x \sin \left(\frac{a}{2 b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}\left(d x^2-1\right)}{2 b}\right)}{2 \sqrt{2} b^2 \sqrt{d x^2}}+\frac{\sqrt{2 d x^2-d^2 x^4}}{2 b d x \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)}",1,"(Sqrt[-(d*x^2*(-2 + d*x^2))]*(b/(a + b*ArcCos[-1 + d*x^2]) + (Sin[ArcCos[-1 + d*x^2]/2]*(Cos[a/(2*b)]*CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)] + Sin[a/(2*b)]*SinIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]))/(-2 + d*x^2)))/(2*b^2*d*x)","A",0
86,1,149,171,0.2186264,"\int \frac{1}{\left(a+b \cos ^{-1}\left(-1+d x^2\right)\right)^3} \, dx","Integrate[(a + b*ArcCos[-1 + d*x^2])^(-3),x]","\frac{\frac{2 b^2 \sqrt{-d x^2 \left(d x^2-2\right)}}{d \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^2}-\frac{\cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) \left(\sin \left(\frac{a}{2 b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}\left(d x^2-1\right)}{2 b}\right)-\cos \left(\frac{a}{2 b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}\left(d x^2-1\right)}{2 b}\right)\right)}{d}+\frac{b x^2}{a+b \cos ^{-1}\left(d x^2-1\right)}}{8 b^3 x}","-\frac{x \sin \left(\frac{a}{2 b}\right) \text{Ci}\left(\frac{a+b \cos ^{-1}\left(d x^2-1\right)}{2 b}\right)}{8 \sqrt{2} b^3 \sqrt{d x^2}}+\frac{x \cos \left(\frac{a}{2 b}\right) \text{Si}\left(\frac{a+b \cos ^{-1}\left(d x^2-1\right)}{2 b}\right)}{8 \sqrt{2} b^3 \sqrt{d x^2}}+\frac{x}{8 b^2 \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)}+\frac{\sqrt{2 d x^2-d^2 x^4}}{4 b d x \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^2}",1,"((2*b^2*Sqrt[-(d*x^2*(-2 + d*x^2))])/(d*(a + b*ArcCos[-1 + d*x^2])^2) + (b*x^2)/(a + b*ArcCos[-1 + d*x^2]) - (Cos[ArcCos[-1 + d*x^2]/2]*(CosIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]*Sin[a/(2*b)] - Cos[a/(2*b)]*SinIntegral[(a + b*ArcCos[-1 + d*x^2])/(2*b)]))/d)/(8*b^3*x)","A",0
87,1,256,249,2.7514967,"\int \left(a+b \cos ^{-1}\left(1+d x^2\right)\right)^{5/2} \, dx","Integrate[(a + b*ArcCos[1 + d*x^2])^(5/2),x]","-\frac{2 \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) \left(\sqrt{a+b \cos ^{-1}\left(d x^2+1\right)} \left(\left(a^2-15 b^2\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)+5 a b \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)+b \cos ^{-1}\left(d x^2+1\right) \left(2 a \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)+5 b \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)\right)+b^2 \cos ^{-1}\left(d x^2+1\right)^2 \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)\right)-\frac{15 \sqrt{\pi } \sin \left(\frac{a}{2 b}\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\left(\frac{1}{b}\right)^{5/2}}+\frac{15 \sqrt{\pi } \cos \left(\frac{a}{2 b}\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\left(\frac{1}{b}\right)^{5/2}}\right)}{d x}","\frac{30 b^2 \sin ^2\left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{d x}-\frac{5 b \sqrt{-d^2 x^4-2 d x^2} \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^{3/2}}{d x}+\frac{30 \sqrt{\pi } \sin \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\left(\frac{1}{b}\right)^{5/2} d x}-\frac{30 \sqrt{\pi } \cos \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\left(\frac{1}{b}\right)^{5/2} d x}+x \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^{5/2}",1,"(-2*Sin[ArcCos[1 + d*x^2]/2]*((15*Sqrt[Pi]*Cos[a/(2*b)]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]])/(b^(-1))^(5/2) - (15*Sqrt[Pi]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)])/(b^(-1))^(5/2) + Sqrt[a + b*ArcCos[1 + d*x^2]]*(5*a*b*Cos[ArcCos[1 + d*x^2]/2] + (a^2 - 15*b^2)*Sin[ArcCos[1 + d*x^2]/2] + b^2*ArcCos[1 + d*x^2]^2*Sin[ArcCos[1 + d*x^2]/2] + b*ArcCos[1 + d*x^2]*(5*b*Cos[ArcCos[1 + d*x^2]/2] + 2*a*Sin[ArcCos[1 + d*x^2]/2]))))/(d*x)","A",1
88,1,200,207,0.5680581,"\int \left(a+b \cos ^{-1}\left(1+d x^2\right)\right)^{3/2} \, dx","Integrate[(a + b*ArcCos[1 + d*x^2])^(3/2),x]","-\frac{2 \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) \left(-3 \sqrt{\pi } \cos \left(\frac{a}{2 b}\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)-3 \sqrt{\pi } \sin \left(\frac{a}{2 b}\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)+\left(\frac{1}{b}\right)^{3/2} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)} \left(a \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)+3 b \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)+b \cos ^{-1}\left(d x^2+1\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)\right)\right)}{\left(\frac{1}{b}\right)^{3/2} d x}","-\frac{3 b \sqrt{-d^2 x^4-2 d x^2} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{d x}+\frac{6 \sqrt{\pi } \cos \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\left(\frac{1}{b}\right)^{3/2} d x}+\frac{6 \sqrt{\pi } \sin \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\left(\frac{1}{b}\right)^{3/2} d x}+x \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^{3/2}",1,"(-2*Sin[ArcCos[1 + d*x^2]/2]*(-3*Sqrt[Pi]*Cos[a/(2*b)]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]] - 3*Sqrt[Pi]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)] + (b^(-1))^(3/2)*Sqrt[a + b*ArcCos[1 + d*x^2]]*(3*b*Cos[ArcCos[1 + d*x^2]/2] + a*Sin[ArcCos[1 + d*x^2]/2] + b*ArcCos[1 + d*x^2]*Sin[ArcCos[1 + d*x^2]/2])))/((b^(-1))^(3/2)*d*x)","A",1
89,1,157,184,0.0804933,"\int \sqrt{a+b \cos ^{-1}\left(1+d x^2\right)} \, dx","Integrate[Sqrt[a + b*ArcCos[1 + d*x^2]],x]","-\frac{2 \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) \left(\sqrt{\pi } \sin \left(\frac{a}{2 b}\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)-\sqrt{\pi } \cos \left(\frac{a}{2 b}\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)+\sqrt{\frac{1}{b}} \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}\right)}{\sqrt{\frac{1}{b}} d x}","-\frac{2 \sqrt{\pi } \sin \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\sqrt{\frac{1}{b}} d x}+\frac{2 \sqrt{\pi } \cos \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\sqrt{\frac{1}{b}} d x}-\frac{2 \sin ^2\left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{d x}",1,"(-2*Sin[ArcCos[1 + d*x^2]/2]*(-(Sqrt[Pi]*Cos[a/(2*b)]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]) + Sqrt[Pi]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)] + Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]]*Sin[ArcCos[1 + d*x^2]/2]))/(Sqrt[b^(-1)]*d*x)","A",1
90,1,114,145,0.17114,"\int \frac{1}{\sqrt{a+b \cos ^{-1}\left(1+d x^2\right)}} \, dx","Integrate[1/Sqrt[a + b*ArcCos[1 + d*x^2]],x]","-\frac{2 \sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) \left(\cos \left(\frac{a}{2 b}\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)+\sin \left(\frac{a}{2 b}\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)\right)}{d x}","-\frac{2 \sqrt{\pi } \sqrt{\frac{1}{b}} \cos \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{d x}-\frac{2 \sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{d x}",1,"(-2*Sqrt[b^(-1)]*Sqrt[Pi]*(Cos[a/(2*b)]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]] + FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)])*Sin[ArcCos[1 + d*x^2]/2])/(d*x)","A",1
91,1,177,190,0.4166156,"\int \frac{1}{\left(a+b \cos ^{-1}\left(1+d x^2\right)\right)^{3/2}} \, dx","Integrate[(a + b*ArcCos[1 + d*x^2])^(-3/2),x]","\frac{-2 \sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)+2 \sqrt{\pi } \sqrt{\frac{1}{b}} \cos \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)+\frac{\sqrt{-d x^2 \left(d x^2+2\right)}}{\sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}}{b d x}","\frac{\sqrt{-d^2 x^4-2 d x^2}}{b d x \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}-\frac{2 \sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} \sin \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{d x}+\frac{2 \sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} \cos \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{d x}",1,"(Sqrt[-(d*x^2*(2 + d*x^2))]/Sqrt[a + b*ArcCos[1 + d*x^2]] + 2*Sqrt[b^(-1)]*Sqrt[Pi]*Cos[a/(2*b)]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[ArcCos[1 + d*x^2]/2] - 2*Sqrt[b^(-1)]*Sqrt[Pi]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)]*Sin[ArcCos[1 + d*x^2]/2])/(b*d*x)","A",1
92,1,234,221,0.7634475,"\int \frac{1}{\left(a+b \cos ^{-1}\left(1+d x^2\right)\right)^{5/2}} \, dx","Integrate[(a + b*ArcCos[1 + d*x^2])^(-5/2),x]","\frac{2 \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) \left(\sqrt{\pi } \sqrt{\frac{1}{b}} \cos \left(\frac{a}{2 b}\right) \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^{3/2} C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left(\frac{a}{2 b}\right) \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^{3/2} S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)-a \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)+b \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)-b \cos ^{-1}\left(d x^2+1\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)\right)}{3 b^2 d x \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^{3/2}}","\frac{x}{3 b^2 \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}+\frac{\sqrt{-d^2 x^4-2 d x^2}}{3 b d x \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^{3/2}}+\frac{2 \sqrt{\pi } \left(\frac{1}{b}\right)^{5/2} \cos \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{3 d x}+\frac{2 \sqrt{\pi } \left(\frac{1}{b}\right)^{5/2} \sin \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{3 d x}",1,"(2*Sin[ArcCos[1 + d*x^2]/2]*(b*Cos[ArcCos[1 + d*x^2]/2] + Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcCos[1 + d*x^2])^(3/2)*Cos[a/(2*b)]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]] + Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcCos[1 + d*x^2])^(3/2)*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)] - a*Sin[ArcCos[1 + d*x^2]/2] - b*ArcCos[1 + d*x^2]*Sin[ArcCos[1 + d*x^2]/2]))/(3*b^2*d*x*(a + b*ArcCos[1 + d*x^2])^(3/2))","A",1
93,1,308,269,0.5683262,"\int \frac{1}{\left(a+b \cos ^{-1}\left(1+d x^2\right)\right)^{7/2}} \, dx","Integrate[(a + b*ArcCos[1 + d*x^2])^(-7/2),x]","-\frac{2 \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) \left(a^2 \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)-\sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left(\frac{a}{2 b}\right) \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^{5/2} C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \sqrt{\frac{1}{b}} \cos \left(\frac{a}{2 b}\right) \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^{5/2} S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)+2 a b \cos ^{-1}\left(d x^2+1\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)+a b \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)-3 b^2 \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)+b^2 \cos ^{-1}\left(d x^2+1\right)^2 \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)+b^2 \cos ^{-1}\left(d x^2+1\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right)\right)}{15 b^3 d x \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^{5/2}}","-\frac{\sqrt{-d^2 x^4-2 d x^2}}{15 b^3 d x \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}+\frac{x}{15 b^2 \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^{3/2}}+\frac{\sqrt{-d^2 x^4-2 d x^2}}{5 b d x \left(a+b \cos ^{-1}\left(d x^2+1\right)\right)^{5/2}}+\frac{2 \sqrt{\pi } \left(\frac{1}{b}\right)^{7/2} \sin \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{15 d x}-\frac{2 \sqrt{\pi } \left(\frac{1}{b}\right)^{7/2} \cos \left(\frac{a}{2 b}\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2+1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{15 d x}",1,"(-2*Sin[ArcCos[1 + d*x^2]/2]*(a^2*Cos[ArcCos[1 + d*x^2]/2] - 3*b^2*Cos[ArcCos[1 + d*x^2]/2] + 2*a*b*ArcCos[1 + d*x^2]*Cos[ArcCos[1 + d*x^2]/2] + b^2*ArcCos[1 + d*x^2]^2*Cos[ArcCos[1 + d*x^2]/2] + Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcCos[1 + d*x^2])^(5/2)*Cos[a/(2*b)]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]] - Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcCos[1 + d*x^2])^(5/2)*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)] + a*b*Sin[ArcCos[1 + d*x^2]/2] + b^2*ArcCos[1 + d*x^2]*Sin[ArcCos[1 + d*x^2]/2]))/(15*b^3*d*x*(a + b*ArcCos[1 + d*x^2])^(5/2))","A",1
94,1,256,249,2.2443217,"\int \left(a+b \cos ^{-1}\left(-1+d x^2\right)\right)^{5/2} \, dx","Integrate[(a + b*ArcCos[-1 + d*x^2])^(5/2),x]","\frac{2 \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) \left(\sqrt{a+b \cos ^{-1}\left(d x^2-1\right)} \left(\left(a^2-15 b^2\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)+b \cos ^{-1}\left(d x^2-1\right) \left(2 a \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)-5 b \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)\right)-5 a b \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)+b^2 \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) \cos ^{-1}\left(d x^2-1\right)^2\right)+\frac{15 \sqrt{\pi } \cos \left(\frac{a}{2 b}\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{\left(\frac{1}{b}\right)^{5/2}}+\frac{15 \sqrt{\pi } \sin \left(\frac{a}{2 b}\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{\left(\frac{1}{b}\right)^{5/2}}\right)}{d x}","-\frac{30 b^2 \cos ^2\left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{d x}-\frac{5 b \sqrt{2 d x^2-d^2 x^4} \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^{3/2}}{d x}+\frac{30 \sqrt{\pi } \cos \left(\frac{a}{2 b}\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{\left(\frac{1}{b}\right)^{5/2} d x}+\frac{30 \sqrt{\pi } \sin \left(\frac{a}{2 b}\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{\left(\frac{1}{b}\right)^{5/2} d x}+x \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^{5/2}",1,"(2*Cos[ArcCos[-1 + d*x^2]/2]*((15*Sqrt[Pi]*Cos[a/(2*b)]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]])/(b^(-1))^(5/2) + (15*Sqrt[Pi]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)])/(b^(-1))^(5/2) + Sqrt[a + b*ArcCos[-1 + d*x^2]]*((a^2 - 15*b^2)*Cos[ArcCos[-1 + d*x^2]/2] + b^2*ArcCos[-1 + d*x^2]^2*Cos[ArcCos[-1 + d*x^2]/2] - 5*a*b*Sin[ArcCos[-1 + d*x^2]/2] + b*ArcCos[-1 + d*x^2]*(2*a*Cos[ArcCos[-1 + d*x^2]/2] - 5*b*Sin[ArcCos[-1 + d*x^2]/2]))))/(d*x)","A",1
95,1,200,207,0.557534,"\int \left(a+b \cos ^{-1}\left(-1+d x^2\right)\right)^{3/2} \, dx","Integrate[(a + b*ArcCos[-1 + d*x^2])^(3/2),x]","\frac{2 \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) \left(-3 \sqrt{\pi } \sin \left(\frac{a}{2 b}\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)+3 \sqrt{\pi } \cos \left(\frac{a}{2 b}\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)+\left(\frac{1}{b}\right)^{3/2} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)} \left(a \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)+b \cos ^{-1}\left(d x^2-1\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)-3 b \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)\right)\right)}{\left(\frac{1}{b}\right)^{3/2} d x}","-\frac{3 b \sqrt{2 d x^2-d^2 x^4} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{d x}-\frac{6 \sqrt{\pi } \sin \left(\frac{a}{2 b}\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{\left(\frac{1}{b}\right)^{3/2} d x}+\frac{6 \sqrt{\pi } \cos \left(\frac{a}{2 b}\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{\left(\frac{1}{b}\right)^{3/2} d x}+x \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^{3/2}",1,"(2*Cos[ArcCos[-1 + d*x^2]/2]*(3*Sqrt[Pi]*Cos[a/(2*b)]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]] - 3*Sqrt[Pi]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)] + (b^(-1))^(3/2)*Sqrt[a + b*ArcCos[-1 + d*x^2]]*(a*Cos[ArcCos[-1 + d*x^2]/2] + b*ArcCos[-1 + d*x^2]*Cos[ArcCos[-1 + d*x^2]/2] - 3*b*Sin[ArcCos[-1 + d*x^2]/2])))/((b^(-1))^(3/2)*d*x)","A",1
96,1,157,184,0.1129945,"\int \sqrt{a+b \cos ^{-1}\left(-1+d x^2\right)} \, dx","Integrate[Sqrt[a + b*ArcCos[-1 + d*x^2]],x]","-\frac{2 \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) \left(\sqrt{\pi } \cos \left(\frac{a}{2 b}\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \sin \left(\frac{a}{2 b}\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)-\sqrt{\frac{1}{b}} \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}\right)}{\sqrt{\frac{1}{b}} d x}","-\frac{2 \sqrt{\pi } \cos \left(\frac{a}{2 b}\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{\sqrt{\frac{1}{b}} d x}-\frac{2 \sqrt{\pi } \sin \left(\frac{a}{2 b}\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{\sqrt{\frac{1}{b}} d x}+\frac{2 \cos ^2\left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{d x}",1,"(-2*Cos[ArcCos[-1 + d*x^2]/2]*(-(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]]*Cos[ArcCos[-1 + d*x^2]/2]) + Sqrt[Pi]*Cos[a/(2*b)]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]] + Sqrt[Pi]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)]))/(Sqrt[b^(-1)]*d*x)","A",1
97,1,115,145,0.1638942,"\int \frac{1}{\sqrt{a+b \cos ^{-1}\left(-1+d x^2\right)}} \, dx","Integrate[1/Sqrt[a + b*ArcCos[-1 + d*x^2]],x]","-\frac{2 \sqrt{\pi } \sqrt{\frac{1}{b}} \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) \left(\cos \left(\frac{a}{2 b}\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)-\sin \left(\frac{a}{2 b}\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)\right)}{d x}","\frac{2 \sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left(\frac{a}{2 b}\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{d x}-\frac{2 \sqrt{\pi } \sqrt{\frac{1}{b}} \cos \left(\frac{a}{2 b}\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{d x}",1,"(-2*Sqrt[b^(-1)]*Sqrt[Pi]*Cos[ArcCos[-1 + d*x^2]/2]*(Cos[a/(2*b)]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]] - FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)]))/(d*x)","A",1
98,1,161,190,0.3425621,"\int \frac{1}{\left(a+b \cos ^{-1}\left(-1+d x^2\right)\right)^{3/2}} \, dx","Integrate[(a + b*ArcCos[-1 + d*x^2])^(-3/2),x]","\frac{2 \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) \left(\sqrt{\pi } \left(-\sqrt{\frac{1}{b}}\right) \cos \left(\frac{a}{2 b}\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)-\sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left(\frac{a}{2 b}\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)+\frac{\sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)}{\sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}\right)}{b d x}","\frac{\sqrt{2 d x^2-d^2 x^4}}{b d x \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}-\frac{2 \sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} \cos \left(\frac{a}{2 b}\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{d x}-\frac{2 \sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} \sin \left(\frac{a}{2 b}\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{d x}",1,"(2*Cos[ArcCos[-1 + d*x^2]/2]*(-(Sqrt[b^(-1)]*Sqrt[Pi]*Cos[a/(2*b)]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]) - Sqrt[b^(-1)]*Sqrt[Pi]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)] + Sin[ArcCos[-1 + d*x^2]/2]/Sqrt[a + b*ArcCos[-1 + d*x^2]]))/(b*d*x)","A",1
99,1,233,221,0.6232739,"\int \frac{1}{\left(a+b \cos ^{-1}\left(-1+d x^2\right)\right)^{5/2}} \, dx","Integrate[(a + b*ArcCos[-1 + d*x^2])^(-5/2),x]","\frac{2 \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) \left(-\sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left(\frac{a}{2 b}\right) \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^{3/2} C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \sqrt{\frac{1}{b}} \cos \left(\frac{a}{2 b}\right) \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^{3/2} S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)+a \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)+b \cos ^{-1}\left(d x^2-1\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)+b \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)\right)}{3 b^2 d x \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^{3/2}}","\frac{x}{3 b^2 \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}+\frac{\sqrt{2 d x^2-d^2 x^4}}{3 b d x \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^{3/2}}-\frac{2 \sqrt{\pi } \left(\frac{1}{b}\right)^{5/2} \sin \left(\frac{a}{2 b}\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{3 d x}+\frac{2 \sqrt{\pi } \left(\frac{1}{b}\right)^{5/2} \cos \left(\frac{a}{2 b}\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{3 d x}",1,"(2*Cos[ArcCos[-1 + d*x^2]/2]*(a*Cos[ArcCos[-1 + d*x^2]/2] + b*ArcCos[-1 + d*x^2]*Cos[ArcCos[-1 + d*x^2]/2] + Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcCos[-1 + d*x^2])^(3/2)*Cos[a/(2*b)]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]] - Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcCos[-1 + d*x^2])^(3/2)*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)] + b*Sin[ArcCos[-1 + d*x^2]/2]))/(3*b^2*d*x*(a + b*ArcCos[-1 + d*x^2])^(3/2))","A",1
100,1,309,269,0.5719739,"\int \frac{1}{\left(a+b \cos ^{-1}\left(-1+d x^2\right)\right)^{7/2}} \, dx","Integrate[(a + b*ArcCos[-1 + d*x^2])^(-7/2),x]","\frac{2 \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) \left(-a^2 \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)+\sqrt{\pi } \sqrt{\frac{1}{b}} \cos \left(\frac{a}{2 b}\right) \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^{5/2} C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left(\frac{a}{2 b}\right) \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^{5/2} S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)+a b \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)-2 a b \cos ^{-1}\left(d x^2-1\right) \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)+b^2 \cos ^{-1}\left(d x^2-1\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)+3 b^2 \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)-b^2 \cos ^{-1}\left(d x^2-1\right)^2 \sin \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right)\right)}{15 b^3 d x \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^{5/2}}","-\frac{\sqrt{2 d x^2-d^2 x^4}}{15 b^3 d x \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}+\frac{x}{15 b^2 \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^{3/2}}+\frac{\sqrt{2 d x^2-d^2 x^4}}{5 b d x \left(a+b \cos ^{-1}\left(d x^2-1\right)\right)^{5/2}}+\frac{2 \sqrt{\pi } \left(\frac{1}{b}\right)^{7/2} \cos \left(\frac{a}{2 b}\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{15 d x}+\frac{2 \sqrt{\pi } \left(\frac{1}{b}\right)^{7/2} \sin \left(\frac{a}{2 b}\right) \cos \left(\frac{1}{2} \cos ^{-1}\left(d x^2-1\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}\left(d x^2-1\right)}}{\sqrt{\pi }}\right)}{15 d x}",1,"(2*Cos[ArcCos[-1 + d*x^2]/2]*(a*b*Cos[ArcCos[-1 + d*x^2]/2] + b^2*ArcCos[-1 + d*x^2]*Cos[ArcCos[-1 + d*x^2]/2] + Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcCos[-1 + d*x^2])^(5/2)*Cos[a/(2*b)]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]] + Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcCos[-1 + d*x^2])^(5/2)*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[-1 + d*x^2]])/Sqrt[Pi]]*Sin[a/(2*b)] - a^2*Sin[ArcCos[-1 + d*x^2]/2] + 3*b^2*Sin[ArcCos[-1 + d*x^2]/2] - 2*a*b*ArcCos[-1 + d*x^2]*Sin[ArcCos[-1 + d*x^2]/2] - b^2*ArcCos[-1 + d*x^2]^2*Sin[ArcCos[-1 + d*x^2]/2]))/(15*b^3*d*x*(a + b*ArcCos[-1 + d*x^2])^(5/2))","A",1
101,0,0,43,0.107077,"\int \frac{\left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^n}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2),x]","\int \frac{\left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^n}{1-c^2 x^2} \, dx","\text{Int}\left(\frac{\left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^n}{1-c^2 x^2},x\right)",0,"Integrate[(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]","A",-1
102,0,0,279,0.3073435,"\int \frac{\left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^3}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2),x]","\int \frac{\left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^3}{1-c^2 x^2} \, dx","-\frac{3 b^2 \text{Li}_3\left(-e^{2 i \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{2 c}+\frac{3 i b \text{Li}_2\left(-e^{2 i \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 c}+\frac{i \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^4}{4 b c}-\frac{\log \left(1+e^{2 i \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^3}{c}-\frac{3 i b^3 \text{Li}_4\left(-e^{2 i \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right)}{4 c}",1,"Integrate[(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2), x]","F",-1
103,0,0,207,0.7301564,"\int \frac{\left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2),x]","\int \frac{\left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2}{1-c^2 x^2} \, dx","\frac{i b \text{Li}_2\left(-e^{2 i \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}+\frac{i \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^3}{3 b c}-\frac{\log \left(1+e^{2 i \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{c}-\frac{b^2 \text{Li}_3\left(-e^{2 i \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right)}{2 c}",1,"Integrate[(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2), x]","F",-1
104,0,0,141,0.3926945,"\int \frac{a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(1 - c^2*x^2),x]","\int \frac{a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)}{1-c^2 x^2} \, dx","\frac{i \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 b c}-\frac{\log \left(1+e^{2 i \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}+\frac{i b \text{Li}_2\left(-e^{2 i \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right)}{2 c}",1,"Integrate[(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(1 - c^2*x^2), x]","F",-1
105,0,0,43,0.1067424,"\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)} \, dx","Integrate[1/((1 - c^2*x^2)*(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])),x]","\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)} \, dx","\text{Int}\left(\frac{1}{\left(1-c^2 x^2\right) \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)},x\right)",0,"Integrate[1/((1 - c^2*x^2)*(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]","A",-1
106,0,0,43,1.7881102,"\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2} \, dx","Integrate[1/((1 - c^2*x^2)*(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2),x]","\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(1-c^2 x^2\right) \left(a+b \cos ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2},x\right)",0,"Integrate[1/((1 - c^2*x^2)*(a + b*ArcCos[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]","A",-1
107,0,0,84,0.9907059,"\int \cos ^{-1}\left(c e^{a+b x}\right) \, dx","Integrate[ArcCos[c*E^(a + b*x)],x]","\int \cos ^{-1}\left(c e^{a+b x}\right) \, dx","-\frac{i \text{Li}_2\left(-e^{2 i \cos ^{-1}\left(c e^{a+b x}\right)}\right)}{2 b}-\frac{i \cos ^{-1}\left(c e^{a+b x}\right)^2}{2 b}+\frac{\cos ^{-1}\left(c e^{a+b x}\right) \log \left(1+e^{2 i \cos ^{-1}\left(c e^{a+b x}\right)}\right)}{b}",1,"Integrate[ArcCos[c*E^(a + b*x)], x]","F",-1
108,1,50,81,0.1450977,"\int e^{\cos ^{-1}(a x)} x^3 \, dx","Integrate[E^ArcCos[a*x]*x^3,x]","-\frac{e^{\cos ^{-1}(a x)} \left(-68 \cos \left(2 \cos ^{-1}(a x)\right)-20 \cos \left(4 \cos ^{-1}(a x)\right)+34 \sin \left(2 \cos ^{-1}(a x)\right)+5 \sin \left(4 \cos ^{-1}(a x)\right)\right)}{680 a^4}","\frac{e^{\cos ^{-1}(a x)} \cos \left(2 \cos ^{-1}(a x)\right)}{10 a^4}+\frac{e^{\cos ^{-1}(a x)} \cos \left(4 \cos ^{-1}(a x)\right)}{34 a^4}-\frac{e^{\cos ^{-1}(a x)} \sin \left(2 \cos ^{-1}(a x)\right)}{20 a^4}-\frac{e^{\cos ^{-1}(a x)} \sin \left(4 \cos ^{-1}(a x)\right)}{136 a^4}",1,"-1/680*(E^ArcCos[a*x]*(-68*Cos[2*ArcCos[a*x]] - 20*Cos[4*ArcCos[a*x]] + 34*Sin[2*ArcCos[a*x]] + 5*Sin[4*ArcCos[a*x]]))/a^4","A",1
109,1,50,82,0.1126025,"\int e^{\cos ^{-1}(a x)} x^2 \, dx","Integrate[E^ArcCos[a*x]*x^2,x]","-\frac{e^{\cos ^{-1}(a x)} \left(5 \sqrt{1-a^2 x^2}-5 a x-3 \cos \left(3 \cos ^{-1}(a x)\right)+\sin \left(3 \cos ^{-1}(a x)\right)\right)}{40 a^3}","\frac{3 e^{\cos ^{-1}(a x)} \cos \left(3 \cos ^{-1}(a x)\right)}{40 a^3}-\frac{e^{\cos ^{-1}(a x)} \sin \left(3 \cos ^{-1}(a x)\right)}{40 a^3}+\frac{x e^{\cos ^{-1}(a x)}}{8 a^2}-\frac{\sqrt{1-a^2 x^2} e^{\cos ^{-1}(a x)}}{8 a^3}",1,"-1/40*(E^ArcCos[a*x]*(-5*a*x + 5*Sqrt[1 - a^2*x^2] - 3*Cos[3*ArcCos[a*x]] + Sin[3*ArcCos[a*x]]))/a^3","A",1
110,1,30,41,0.0399394,"\int e^{\cos ^{-1}(a x)} x \, dx","Integrate[E^ArcCos[a*x]*x,x]","-\frac{e^{\cos ^{-1}(a x)} \left(\sin \left(2 \cos ^{-1}(a x)\right)-2 \cos \left(2 \cos ^{-1}(a x)\right)\right)}{10 a^2}","\frac{e^{\cos ^{-1}(a x)} \cos \left(2 \cos ^{-1}(a x)\right)}{5 a^2}-\frac{e^{\cos ^{-1}(a x)} \sin \left(2 \cos ^{-1}(a x)\right)}{10 a^2}",1,"-1/10*(E^ArcCos[a*x]*(-2*Cos[2*ArcCos[a*x]] + Sin[2*ArcCos[a*x]]))/a^2","A",1
111,1,32,39,0.0344273,"\int e^{\cos ^{-1}(a x)} \, dx","Integrate[E^ArcCos[a*x],x]","-\frac{\left(\sqrt{1-a^2 x^2}-a x\right) e^{\cos ^{-1}(a x)}}{2 a}","\frac{1}{2} x e^{\cos ^{-1}(a x)}-\frac{\sqrt{1-a^2 x^2} e^{\cos ^{-1}(a x)}}{2 a}",1,"-1/2*(E^ArcCos[a*x]*(-(a*x) + Sqrt[1 - a^2*x^2]))/a","A",1
112,1,79,45,0.0595196,"\int \frac{e^{\cos ^{-1}(a x)}}{x} \, dx","Integrate[E^ArcCos[a*x]/x,x]","i \left(\left(\frac{1}{5}-\frac{2 i}{5}\right) e^{(1+2 i) \cos ^{-1}(a x)} \, _2F_1\left(1,1-\frac{i}{2};2-\frac{i}{2};-e^{2 i \cos ^{-1}(a x)}\right)-e^{\cos ^{-1}(a x)} \, _2F_1\left(-\frac{i}{2},1;1-\frac{i}{2};-e^{2 i \cos ^{-1}(a x)}\right)\right)","i e^{\cos ^{-1}(a x)}-2 i e^{\cos ^{-1}(a x)} \, _2F_1\left(-\frac{i}{2},1;1-\frac{i}{2};-e^{2 i \cos ^{-1}(a x)}\right)",1,"I*(-(E^ArcCos[a*x]*Hypergeometric2F1[-1/2*I, 1, 1 - I/2, -E^((2*I)*ArcCos[a*x])]) + (1/5 - (2*I)/5)*E^((1 + 2*I)*ArcCos[a*x])*Hypergeometric2F1[1, 1 - I/2, 2 - I/2, -E^((2*I)*ArcCos[a*x])])","A",0
113,1,55,87,0.0658166,"\int \frac{e^{\cos ^{-1}(a x)}}{x^2} \, dx","Integrate[E^ArcCos[a*x]/x^2,x]","-\frac{e^{\cos ^{-1}(a x)}}{x}+(1-i) a e^{(1+i) \cos ^{-1}(a x)} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};-e^{2 i \cos ^{-1}(a x)}\right)","(1+i) a e^{(1+i) \cos ^{-1}(a x)} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};-e^{2 i \cos ^{-1}(a x)}\right)-(2+2 i) a e^{(1+i) \cos ^{-1}(a x)} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},2;\frac{3}{2}-\frac{i}{2};-e^{2 i \cos ^{-1}(a x)}\right)",1,"-(E^ArcCos[a*x]/x) + (1 - I)*a*E^((1 + I)*ArcCos[a*x])*Hypergeometric2F1[1/2 - I/2, 1, 3/2 - I/2, -E^((2*I)*ArcCos[a*x])]","A",0
114,1,141,48,0.1707098,"\int \cos ^{-1}\left(\frac{c}{a+b x}\right) \, dx","Integrate[ArcCos[c/(a + b*x)],x]","x \cos ^{-1}\left(\frac{c}{a+b x}\right)-\frac{(a+b x) \sqrt{\frac{a^2+2 a b x+b^2 x^2-c^2}{(a+b x)^2}} \left(c \tanh ^{-1}\left(\frac{a+b x}{\sqrt{a^2+2 a b x+b^2 x^2-c^2}}\right)-a \tan ^{-1}\left(\frac{\sqrt{(a+b x)^2-c^2}}{c}\right)\right)}{b \sqrt{a^2+2 a b x+b^2 x^2-c^2}}","\frac{(a+b x) \sec ^{-1}\left(\frac{a}{c}+\frac{b x}{c}\right)}{b}-\frac{c \tanh ^{-1}\left(\sqrt{1-\frac{c^2}{(a+b x)^2}}\right)}{b}",1,"x*ArcCos[c/(a + b*x)] - ((a + b*x)*Sqrt[(a^2 - c^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*(-(a*ArcTan[Sqrt[-c^2 + (a + b*x)^2]/c]) + c*ArcTanh[(a + b*x)/Sqrt[a^2 - c^2 + 2*a*b*x + b^2*x^2]]))/(b*Sqrt[a^2 - c^2 + 2*a*b*x + b^2*x^2])","B",1
115,1,56,26,0.0937049,"\int \frac{x}{\sqrt{1-x^2} \sqrt{\cos ^{-1}(x)}} \, dx","Integrate[x/(Sqrt[1 - x^2]*Sqrt[ArcCos[x]]),x]","\frac{i \left(\sqrt{-i \cos ^{-1}(x)} \Gamma \left(\frac{1}{2},-i \cos ^{-1}(x)\right)-\sqrt{i \cos ^{-1}(x)} \Gamma \left(\frac{1}{2},i \cos ^{-1}(x)\right)\right)}{2 \sqrt{\cos ^{-1}(x)}}","-\sqrt{2 \pi } C\left(\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(x)}\right)",1,"((I/2)*(Sqrt[(-I)*ArcCos[x]]*Gamma[1/2, (-I)*ArcCos[x]] - Sqrt[I*ArcCos[x]]*Gamma[1/2, I*ArcCos[x]]))/Sqrt[ArcCos[x]]","C",0
116,1,5,5,0.0419415,"\int \frac{x}{\sqrt{1-x^2} \cos ^{-1}(x)} \, dx","Integrate[x/(Sqrt[1 - x^2]*ArcCos[x]),x]","-\text{Ci}\left(\cos ^{-1}(x)\right)","-\text{Ci}\left(\cos ^{-1}(x)\right)",1,"-CosIntegral[ArcCos[x]]","A",1
117,1,39,39,0.0512546,"\int \frac{\cos ^{-1}\left(\sqrt{1+b x^2}\right)^n}{\sqrt{1+b x^2}} \, dx","Integrate[ArcCos[Sqrt[1 + b*x^2]]^n/Sqrt[1 + b*x^2],x]","-\frac{\sqrt{-b x^2} \cos ^{-1}\left(\sqrt{b x^2+1}\right)^{n+1}}{b (n+1) x}","-\frac{\sqrt{-b x^2} \cos ^{-1}\left(\sqrt{b x^2+1}\right)^{n+1}}{b (n+1) x}",1,"-((Sqrt[-(b*x^2)]*ArcCos[Sqrt[1 + b*x^2]]^(1 + n))/(b*(1 + n)*x))","A",1
118,1,25,31,0.0263497,"\int \frac{1}{\sqrt{1+b x^2} \cos ^{-1}\left(\sqrt{1+b x^2}\right)} \, dx","Integrate[1/(Sqrt[1 + b*x^2]*ArcCos[Sqrt[1 + b*x^2]]),x]","\frac{x \log \left(\cos ^{-1}\left(\sqrt{b x^2+1}\right)\right)}{\sqrt{-b x^2}}","-\frac{\sqrt{-b x^2} \log \left(\cos ^{-1}\left(\sqrt{b x^2+1}\right)\right)}{b x}",1,"(x*Log[ArcCos[Sqrt[1 + b*x^2]]])/Sqrt[-(b*x^2)]","A",1