1,1,165,179,0.1642456,"\int (d+e x)^3 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^3*(a + b*ArcSin[c*x]),x]","\frac{24 a c^4 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+b c \sqrt{1-c^2 x^2} \left(c^2 \left(96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right)+e^2 (64 d+9 e x)\right)+3 b \sin ^{-1}(c x) \left(8 c^4 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)-24 c^2 d^2 e-3 e^3\right)}{96 c^4}","\frac{(d+e x)^4 \left(a+b \sin ^{-1}(c x)\right)}{4 e}+\frac{b \sqrt{1-c^2 x^2} (d+e x)^3}{16 c}+\frac{7 b d \sqrt{1-c^2 x^2} (d+e x)^2}{48 c}-\frac{b \left(8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right) \sin ^{-1}(c x)}{32 c^4 e}+\frac{b \sqrt{1-c^2 x^2} \left(e x \left(26 c^2 d^2+9 e^2\right)+4 d \left(19 c^2 d^2+16 e^2\right)\right)}{96 c^3}",1,"(24*a*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + b*c*Sqrt[1 - c^2*x^2]*(e^2*(64*d + 9*e*x) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)) + 3*b*(-24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))*ArcSin[c*x])/(96*c^4)","A",1
2,1,121,124,0.1095764,"\int (d+e x)^2 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^2*(a + b*ArcSin[c*x]),x]","\frac{6 a c^3 x \left(3 d^2+3 d e x+e^2 x^2\right)+b \sqrt{1-c^2 x^2} \left(c^2 \left(18 d^2+9 d e x+2 e^2 x^2\right)+4 e^2\right)+3 b c \sin ^{-1}(c x) \left(6 c^2 d^2 x+3 d e \left(2 c^2 x^2-1\right)+2 c^2 e^2 x^3\right)}{18 c^3}","\frac{(d+e x)^3 \left(a+b \sin ^{-1}(c x)\right)}{3 e}-\frac{b d \left(\frac{3 e^2}{c^2}+2 d^2\right) \sin ^{-1}(c x)}{6 e}+\frac{b \sqrt{1-c^2 x^2} (d+e x)^2}{9 c}+\frac{b \sqrt{1-c^2 x^2} \left(4 \left(4 c^2 d^2+e^2\right)+5 c^2 d e x\right)}{18 c^3}",1,"(6*a*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) + b*Sqrt[1 - c^2*x^2]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + 3*b*c*(6*c^2*d^2*x + 2*c^2*e^2*x^3 + 3*d*e*(-1 + 2*c^2*x^2))*ArcSin[c*x])/(18*c^3)","A",1
3,1,92,98,0.0488955,"\int (d+e x) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)*(a + b*ArcSin[c*x]),x]","a d x+\frac{1}{2} a e x^2+\frac{b d \sqrt{1-c^2 x^2}}{c}+\frac{b e x \sqrt{1-c^2 x^2}}{4 c}-\frac{b e \sin ^{-1}(c x)}{4 c^2}+b d x \sin ^{-1}(c x)+\frac{1}{2} b e x^2 \sin ^{-1}(c x)","\frac{(d+e x)^2 \left(a+b \sin ^{-1}(c x)\right)}{2 e}-\frac{b \left(\frac{e^2}{c^2}+2 d^2\right) \sin ^{-1}(c x)}{4 e}+\frac{b \sqrt{1-c^2 x^2} (d+e x)}{4 c}+\frac{3 b d \sqrt{1-c^2 x^2}}{4 c}",1,"a*d*x + (a*e*x^2)/2 + (b*d*Sqrt[1 - c^2*x^2])/c + (b*e*x*Sqrt[1 - c^2*x^2])/(4*c) - (b*e*ArcSin[c*x])/(4*c^2) + b*d*x*ArcSin[c*x] + (b*e*x^2*ArcSin[c*x])/2","A",1
4,1,30,30,0.012388,"\int \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[a + b*ArcSin[c*x],x]","a x+\frac{b \sqrt{1-c^2 x^2}}{c}+b x \sin ^{-1}(c x)","a x+\frac{b \sqrt{1-c^2 x^2}}{c}+b x \sin ^{-1}(c x)",1,"a*x + (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcSin[c*x]","A",1
5,1,214,229,0.1801041,"\int \frac{a+b \sin ^{-1}(c x)}{d+e x} \, dx","Integrate[(a + b*ArcSin[c*x])/(d + e*x),x]","-\frac{i \left(\left(a+b \sin ^{-1}(c x)\right) \left(a+2 i b \log \left(1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)+2 i b \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)+b \sin ^{-1}(c x)\right)+2 b^2 \text{Li}_2\left(-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)+2 b^2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)\right)}{2 b e}","\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)}{e}-\frac{i \left(a+b \sin ^{-1}(c x)\right)^2}{2 b e}-\frac{i b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e}-\frac{i b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e}",1,"((-1/2*I)*((a + b*ArcSin[c*x])*(a + b*ArcSin[c*x] + (2*I)*b*Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + (2*I)*b*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) + 2*b^2*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + 2*b^2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/(b*e)","A",1
6,1,83,85,0.1618809,"\int \frac{a+b \sin ^{-1}(c x)}{(d+e x)^2} \, dx","Integrate[(a + b*ArcSin[c*x])/(d + e*x)^2,x]","\frac{\frac{b c \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right)}{\sqrt{c^2 d^2-e^2}}-\frac{a+b \sin ^{-1}(c x)}{d+e x}}{e}","\frac{b c \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right)}{e \sqrt{c^2 d^2-e^2}}-\frac{a+b \sin ^{-1}(c x)}{e (d+e x)}",1,"(-((a + b*ArcSin[c*x])/(d + e*x)) + (b*c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2])/e","A",1
7,1,207,135,0.4052232,"\int \frac{a+b \sin ^{-1}(c x)}{(d+e x)^3} \, dx","Integrate[(a + b*ArcSin[c*x])/(d + e*x)^3,x]","\frac{1}{2} \left(-\frac{a}{e (d+e x)^2}+\frac{b c \sqrt{1-c^2 x^2}}{\left(c^2 d^2-e^2\right) (d+e x)}-\frac{i b c^3 d \left(\log (4)+\log \left(\frac{e^2 \sqrt{c^2 d^2-e^2} \left(\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}+i c^2 d x+i e\right)}{b c^3 d (d+e x)}\right)\right)}{e (c d-e) (c d+e) \sqrt{c^2 d^2-e^2}}-\frac{b \sin ^{-1}(c x)}{e (d+e x)^2}\right)","-\frac{a+b \sin ^{-1}(c x)}{2 e (d+e x)^2}+\frac{b c \sqrt{1-c^2 x^2}}{2 \left(c^2 d^2-e^2\right) (d+e x)}+\frac{b c^3 d \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right)}{2 e \left(c^2 d^2-e^2\right)^{3/2}}",1,"(-(a/(e*(d + e*x)^2)) + (b*c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - (b*ArcSin[c*x])/(e*(d + e*x)^2) - (I*b*c^3*d*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))/(b*c^3*d*(d + e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2]))/2","C",1
8,1,241,191,0.5681645,"\int \frac{a+b \sin ^{-1}(c x)}{(d+e x)^4} \, dx","Integrate[(a + b*ArcSin[c*x])/(d + e*x)^4,x]","\frac{1}{6} \left(-\frac{2 a}{e (d+e x)^3}+\frac{b \sqrt{1-c^2 x^2} \left(c^3 d (4 d+3 e x)-c e^2\right)}{\left(e^2-c^2 d^2\right)^2 (d+e x)^2}-\frac{b c^3 \left(2 c^2 d^2+e^2\right) \log \left(\sqrt{1-c^2 x^2} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right)}{e (e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}+\frac{b c^3 \left(2 c^2 d^2+e^2\right) \log (d+e x)}{e (e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}-\frac{2 b \sin ^{-1}(c x)}{e (d+e x)^3}\right)","-\frac{a+b \sin ^{-1}(c x)}{3 e (d+e x)^3}+\frac{b c \sqrt{1-c^2 x^2}}{6 \left(c^2 d^2-e^2\right) (d+e x)^2}+\frac{b c^3 d \sqrt{1-c^2 x^2}}{2 \left(c^2 d^2-e^2\right)^2 (d+e x)}+\frac{b c^3 \left(2 c^2 d^2+e^2\right) \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right)}{6 e \left(c^2 d^2-e^2\right)^{5/2}}",1,"((-2*a)/(e*(d + e*x)^3) + (b*Sqrt[1 - c^2*x^2]*(-(c*e^2) + c^3*d*(4*d + 3*e*x)))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (2*b*ArcSin[c*x])/(e*(d + e*x)^3) + (b*c^3*(2*c^2*d^2 + e^2)*Log[d + e*x])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]) - (b*c^3*(2*c^2*d^2 + e^2)*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]))/6","A",1
9,1,355,374,0.6364453,"\int (d+e x)^3 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x)^3*(a + b*ArcSin[c*x])^2,x]","\frac{c \left(72 a^2 c^3 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+6 a b \sqrt{1-c^2 x^2} \left(c^2 \left(96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right)+e^2 (64 d+9 e x)\right)-b^2 c x \left(c^2 \left(576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right)+3 e^2 (128 d+9 e x)\right)\right)+6 b \sin ^{-1}(c x) \left(3 a \left(8 c^4 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)-24 c^2 d^2 e-3 e^3\right)+b c \sqrt{1-c^2 x^2} \left(c^2 \left(96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right)+e^2 (64 d+9 e x)\right)\right)+9 b^2 \sin ^{-1}(c x)^2 \left(8 c^4 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)-24 c^2 d^2 e-3 e^3\right)}{288 c^4}","-\frac{3 e^3 \left(a+b \sin ^{-1}(c x)\right)^2}{32 c^4}+\frac{2 b d^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{3 b d^2 e x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 c}-\frac{3 d^2 e \left(a+b \sin ^{-1}(c x)\right)^2}{4 c^2}+\frac{2 b d e^2 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 c}+\frac{b e^3 x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 c}+\frac{4 b d e^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 c^3}+\frac{3 b e^3 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{16 c^3}-\frac{d^4 \left(a+b \sin ^{-1}(c x)\right)^2}{4 e}+\frac{(d+e x)^4 \left(a+b \sin ^{-1}(c x)\right)^2}{4 e}-\frac{4 b^2 d e^2 x}{3 c^2}-\frac{3 b^2 e^3 x^2}{32 c^2}-2 b^2 d^3 x-\frac{3}{4} b^2 d^2 e x^2-\frac{2}{9} b^2 d e^2 x^3-\frac{1}{32} b^2 e^3 x^4",1,"(c*(72*a^2*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 6*a*b*Sqrt[1 - c^2*x^2]*(e^2*(64*d + 9*e*x) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)) - b^2*c*x*(3*e^2*(128*d + 9*e*x) + c^2*(576*d^3 + 216*d^2*e*x + 64*d*e^2*x^2 + 9*e^3*x^3))) + 6*b*(3*a*(-24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)) + b*c*Sqrt[1 - c^2*x^2]*(e^2*(64*d + 9*e*x) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)))*ArcSin[c*x] + 9*b^2*(-24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))*ArcSin[c*x]^2)/(288*c^4)","A",1
10,1,249,242,0.4072744,"\int (d+e x)^2 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x)^2*(a + b*ArcSin[c*x])^2,x]","\frac{18 a^2 c^3 x \left(3 d^2+3 d e x+e^2 x^2\right)+6 a b \sqrt{1-c^2 x^2} \left(c^2 \left(18 d^2+9 d e x+2 e^2 x^2\right)+4 e^2\right)+6 b \sin ^{-1}(c x) \left(6 a c^3 x \left(3 d^2+3 d e x+e^2 x^2\right)-9 a c d e+b \sqrt{1-c^2 x^2} \left(c^2 \left(18 d^2+9 d e x+2 e^2 x^2\right)+4 e^2\right)\right)-b^2 c x \left(c^2 \left(108 d^2+27 d e x+4 e^2 x^2\right)+24 e^2\right)+9 b^2 c \sin ^{-1}(c x)^2 \left(6 c^2 d^2 x+3 d e \left(2 c^2 x^2-1\right)+2 c^2 e^2 x^3\right)}{54 c^3}","\frac{2 b d^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{b d e x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}-\frac{d e \left(a+b \sin ^{-1}(c x)\right)^2}{2 c^2}+\frac{2 b e^2 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 c}+\frac{4 b e^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 c^3}-\frac{d^3 \left(a+b \sin ^{-1}(c x)\right)^2}{3 e}+\frac{(d+e x)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{3 e}-\frac{4 b^2 e^2 x}{9 c^2}-2 b^2 d^2 x-\frac{1}{2} b^2 d e x^2-\frac{2}{27} b^2 e^2 x^3",1,"(18*a^2*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) + 6*a*b*Sqrt[1 - c^2*x^2]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) - b^2*c*x*(24*e^2 + c^2*(108*d^2 + 27*d*e*x + 4*e^2*x^2)) + 6*b*(-9*a*c*d*e + 6*a*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) + b*Sqrt[1 - c^2*x^2]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)))*ArcSin[c*x] + 9*b^2*c*(6*c^2*d^2*x + 2*c^2*e^2*x^3 + 3*d*e*(-1 + 2*c^2*x^2))*ArcSin[c*x]^2)/(54*c^3)","A",1
11,1,147,142,0.3835012,"\int (d+e x) \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x)*(a + b*ArcSin[c*x])^2,x]","\frac{c \left(2 a^2 c x (2 d+e x)+2 a b \sqrt{1-c^2 x^2} (4 d+e x)-b^2 c x (8 d+e x)\right)+2 b \sin ^{-1}(c x) \left(4 a c^2 d x+a e \left(2 c^2 x^2-1\right)+b c \sqrt{1-c^2 x^2} (4 d+e x)\right)+b^2 \sin ^{-1}(c x)^2 \left(4 c^2 d x+e \left(2 c^2 x^2-1\right)\right)}{4 c^2}","\frac{2 b d \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{b e x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 c}-\frac{e \left(a+b \sin ^{-1}(c x)\right)^2}{4 c^2}-\frac{d^2 \left(a+b \sin ^{-1}(c x)\right)^2}{2 e}+\frac{(d+e x)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{2 e}-2 b^2 d x-\frac{1}{4} b^2 e x^2",1,"(c*(2*a^2*c*x*(2*d + e*x) - b^2*c*x*(8*d + e*x) + 2*a*b*(4*d + e*x)*Sqrt[1 - c^2*x^2]) + 2*b*(4*a*c^2*d*x + b*c*(4*d + e*x)*Sqrt[1 - c^2*x^2] + a*e*(-1 + 2*c^2*x^2))*ArcSin[c*x] + b^2*(4*c^2*d*x + e*(-1 + 2*c^2*x^2))*ArcSin[c*x]^2)/(4*c^2)","A",1
12,1,47,47,0.0464474,"\int \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(a + b*ArcSin[c*x])^2,x]","\frac{2 b \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+x \left(a+b \sin ^{-1}(c x)\right)^2-2 b^2 x","\frac{2 b \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+x \left(a+b \sin ^{-1}(c x)\right)^2-2 b^2 x",1,"-2*b^2*x + (2*b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + x*(a + b*ArcSin[c*x])^2","A",1
13,1,332,347,0.3359169,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{d+e x} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(d + e*x),x]","\frac{6 b \left(b \text{Li}_3\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)-i \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)\right)+6 b \left(b \text{Li}_3\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)-i \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)\right)+3 \left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)+3 \left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)-\frac{i \left(a+b \sin ^{-1}(c x)\right)^3}{b}}{3 e}","-\frac{2 i b \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e}-\frac{2 i b \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)}{e}-\frac{i \left(a+b \sin ^{-1}(c x)\right)^3}{3 b e}+\frac{2 b^2 \text{Li}_3\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e}+\frac{2 b^2 \text{Li}_3\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e}",1,"(((-I)*(a + b*ArcSin[c*x])^3)/b + 3*(a + b*ArcSin[c*x])^2*Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + 3*(a + b*ArcSin[c*x])^2*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] + 6*b*((-I)*(a + b*ArcSin[c*x])*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + b*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])]) + 6*b*((-I)*(a + b*ArcSin[c*x])*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] + b*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/(3*e)","A",1
14,1,231,309,0.3447418,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{(d+e x)^2} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(d + e*x)^2,x]","\frac{-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{d+e x}+\frac{2 b c \left(-i \left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)-\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)\right)-b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)+b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)\right)}{\sqrt{c^2 d^2-e^2}}}{e}","-\frac{2 i b c \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e \sqrt{c^2 d^2-e^2}}+\frac{2 i b c \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)}{e \sqrt{c^2 d^2-e^2}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{e (d+e x)}-\frac{2 b^2 c \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 c \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e \sqrt{c^2 d^2-e^2}}",1,"(-((a + b*ArcSin[c*x])^2/(d + e*x)) + (2*b*c*((-I)*(a + b*ArcSin[c*x])*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) - b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/Sqrt[c^2*d^2 - e^2])/e","A",1
15,1,315,401,1.07303,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{(d+e x)^3} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(d + e*x)^3,x]","\frac{\frac{2 b c e \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{\left(c^2 d^2-e^2\right) (d+e x)}+\frac{2 b c^3 d \left(-i \left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)-\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)\right)-b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)+b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)\right)}{\left(c^2 d^2-e^2\right)^{3/2}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{(d+e x)^2}-\frac{2 b^2 c^2 \log (d+e x)}{c^2 d^2-e^2}}{2 e}","\frac{b c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{\left(c^2 d^2-e^2\right) (d+e x)}-\frac{i b c^3 d \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e \left(c^2 d^2-e^2\right)^{3/2}}+\frac{i b c^3 d \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)}{e \left(c^2 d^2-e^2\right)^{3/2}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{2 e (d+e x)^2}-\frac{b^2 c^2 \log (d+e x)}{e \left(c^2 d^2-e^2\right)}-\frac{b^2 c^3 d \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e \left(c^2 d^2-e^2\right)^{3/2}}+\frac{b^2 c^3 d \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e \left(c^2 d^2-e^2\right)^{3/2}}",1,"((2*b*c*e*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/((c^2*d^2 - e^2)*(d + e*x)) - (a + b*ArcSin[c*x])^2/(d + e*x)^2 - (2*b^2*c^2*Log[d + e*x])/(c^2*d^2 - e^2) + (2*b*c^3*d*((-I)*(a + b*ArcSin[c*x])*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) - b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/(c^2*d^2 - e^2)^(3/2))/(2*e)","A",1
16,1,304,393,0.8326702,"\int \frac{(d+e x)^3}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(d + e*x)^3/(a + b*ArcSin[c*x]),x]","\frac{e^3 \left(-2 \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{4 a}{b}\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 \cos \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\cos \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)\right)}{8 b c^4}+\frac{3 d e^2 \left(\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-\cos \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-\sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)\right)}{4 b c^3}+\frac{3 d^2 e \left(\cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 a}{b}+2 \sin ^{-1}(c x)\right)-\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 a}{b}+2 \sin ^{-1}(c x)\right)\right)}{2 b c^2}+\frac{d^3 \left(\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{b c}","-\frac{e^3 \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 a}{b}+2 \sin ^{-1}(c x)\right)}{4 b c^4}+\frac{e^3 \sin \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 a}{b}+4 \sin ^{-1}(c x)\right)}{8 b c^4}+\frac{e^3 \cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 a}{b}+2 \sin ^{-1}(c x)\right)}{4 b c^4}-\frac{e^3 \cos \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 a}{b}+4 \sin ^{-1}(c x)\right)}{8 b c^4}+\frac{3 d e^2 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)}{4 b c^3}-\frac{3 d e^2 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 a}{b}+3 \sin ^{-1}(c x)\right)}{4 b c^3}+\frac{3 d e^2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)}{4 b c^3}-\frac{3 d e^2 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 a}{b}+3 \sin ^{-1}(c x)\right)}{4 b c^3}-\frac{3 d^2 e \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 a}{b}+2 \sin ^{-1}(c x)\right)}{2 b c^2}+\frac{3 d^2 e \cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 a}{b}+2 \sin ^{-1}(c x)\right)}{2 b c^2}+\frac{d^3 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)}{b c}+\frac{d^3 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)}{b c}",1,"(d^3*(Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] + Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]]))/(b*c) + (3*d*e^2*(Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] - Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] - Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])]))/(4*b*c^3) + (e^3*(-2*CosIntegral[2*(a/b + ArcSin[c*x])]*Sin[(2*a)/b] + CosIntegral[4*(a/b + ArcSin[c*x])]*Sin[(4*a)/b] + 2*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] - Cos[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])]))/(8*b*c^4) + (3*d^2*e*(-(CosIntegral[(2*a)/b + 2*ArcSin[c*x]]*Sin[(2*a)/b]) + Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSin[c*x]]))/(2*b*c^2)","A",1
17,1,187,244,0.4762298,"\int \frac{(d+e x)^2}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(d + e*x)^2/(a + b*ArcSin[c*x]),x]","\frac{\cos \left(\frac{a}{b}\right) \left(4 c^2 d^2+e^2\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+4 c^2 d^2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-4 c d e \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-e^2 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+4 c d e \cos \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+e^2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-e^2 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{4 b c^3}","\frac{e^2 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)}{4 b c^3}-\frac{e^2 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 a}{b}+3 \sin ^{-1}(c x)\right)}{4 b c^3}+\frac{e^2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)}{4 b c^3}-\frac{e^2 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 a}{b}+3 \sin ^{-1}(c x)\right)}{4 b c^3}-\frac{d e \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 a}{b}+2 \sin ^{-1}(c x)\right)}{b c^2}+\frac{d e \cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 a}{b}+2 \sin ^{-1}(c x)\right)}{b c^2}+\frac{d^2 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)}{b c}+\frac{d^2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)}{b c}",1,"((4*c^2*d^2 + e^2)*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] - e^2*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] - 4*c*d*e*CosIntegral[2*(a/b + ArcSin[c*x])]*Sin[(2*a)/b] + 4*c^2*d^2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + e^2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 4*c*d*e*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] - e^2*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])])/(4*b*c^3)","A",1
18,1,98,115,0.1991789,"\int \frac{d+e x}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(d + e*x)/(a + b*ArcSin[c*x]),x]","\frac{2 c d \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-e \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 c d \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+e \cos \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{2 b c^2}","-\frac{e \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 a}{b}+2 \sin ^{-1}(c x)\right)}{2 b c^2}+\frac{e \cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 a}{b}+2 \sin ^{-1}(c x)\right)}{2 b c^2}+\frac{d \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)}{b c}+\frac{d \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)}{b c}",1,"(2*c*d*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] - e*CosIntegral[2*(a/b + ArcSin[c*x])]*Sin[(2*a)/b] + 2*c*d*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + e*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])])/(2*b*c^2)","A",1
19,1,44,53,0.0263575,"\int \frac{1}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(a + b*ArcSin[c*x])^(-1),x]","\frac{\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)}{b c}","\frac{\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b c}+\frac{\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b c}",1,"(Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] + Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(b*c)","A",1
20,0,0,21,0.2114101,"\int \frac{1}{(d+e x) \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x)*(a + b*ArcSin[c*x])),x]","\int \frac{1}{(d+e x) \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{(d+e x) \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x)*(a + b*ArcSin[c*x])), x]","A",-1
21,0,0,21,0.433339,"\int \frac{1}{(d+e x)^2 \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x)^2*(a + b*ArcSin[c*x])),x]","\int \frac{1}{(d+e x)^2 \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{(d+e x)^2 \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x)^2*(a + b*ArcSin[c*x])), x]","A",-1
22,1,290,362,1.9203282,"\int \frac{(d+e x)^2}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(d + e*x)^2/(a + b*ArcSin[c*x])^2,x]","-\frac{-\sin \left(\frac{a}{b}\right) \left(4 c^2 d^2+e^2\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+4 c^2 d^2 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+\frac{4 b c^2 d^2 \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+\frac{8 b c^2 d e x \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+\frac{4 b c^2 e^2 x^2 \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}-8 c d e \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+3 e^2 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-8 c d e \sin \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+e^2 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-3 e^2 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{4 b^2 c^3}","\frac{e^2 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b^2 c^3}-\frac{3 e^2 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^3}-\frac{e^2 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b^2 c^3}+\frac{3 e^2 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^3}+\frac{2 d e \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{b^2 c^2}+\frac{2 d e \sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{b^2 c^2}+\frac{d^2 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{d^2 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{d^2 \sqrt{1-c^2 x^2}}{b c \left(a+b \sin ^{-1}(c x)\right)}-\frac{2 d e x \sqrt{1-c^2 x^2}}{b c \left(a+b \sin ^{-1}(c x)\right)}-\frac{e^2 x^2 \sqrt{1-c^2 x^2}}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"-1/4*((4*b*c^2*d^2*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + (8*b*c^2*d*e*x*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + (4*b*c^2*e^2*x^2*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) - 8*c*d*e*Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c*x])] - (4*c^2*d^2 + e^2)*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] + 3*e^2*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] + 4*c^2*d^2*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + e^2*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 8*c*d*e*Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] - 3*e^2*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])])/(b^2*c^3)","A",1
23,1,149,181,0.723212,"\int \frac{d+e x}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(d + e*x)/(a + b*ArcSin[c*x])^2,x]","\frac{-\frac{b c \sqrt{1-c^2 x^2} (d+e x)}{a+b \sin ^{-1}(c x)}+c d \left(\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+e \left(\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\log \left(a+b \sin ^{-1}(c x)\right)\right)+e \log \left(a+b \sin ^{-1}(c x)\right)}{b^2 c^2}","\frac{e \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{b^2 c^2}+\frac{e \sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{b^2 c^2}+\frac{d \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{d \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{d \sqrt{1-c^2 x^2}}{b c \left(a+b \sin ^{-1}(c x)\right)}-\frac{e x \sqrt{1-c^2 x^2}}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"(-((b*c*(d + e*x)*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])) + e*Log[a + b*ArcSin[c*x]] + c*d*(CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]]) + e*(Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c*x])] - Log[a + b*ArcSin[c*x]] + Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])]))/(b^2*c^2)","A",1
24,1,72,86,0.092327,"\int \frac{1}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])^(-2),x]","\frac{-\frac{b \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)}{b^2 c}","\frac{\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{\sqrt{1-c^2 x^2}}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"(-((b*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])) + CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(b^2*c)","A",1
25,0,0,21,6.495124,"\int \frac{1}{(d+e x) \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d + e*x)*(a + b*ArcSin[c*x])^2),x]","\int \frac{1}{(d+e x) \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{(d+e x) \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/((d + e*x)*(a + b*ArcSin[c*x])^2), x]","A",-1
26,0,0,21,12.6576221,"\int \frac{1}{(d+e x)^2 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d + e*x)^2*(a + b*ArcSin[c*x])^2),x]","\int \frac{1}{(d+e x)^2 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{(d+e x)^2 \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/((d + e*x)^2*(a + b*ArcSin[c*x])^2), x]","A",-1
27,0,0,76,5.2916542,"\int (d+e x)^m \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x)^m*(a + b*ArcSin[c*x])^2,x]","\int (d+e x)^m \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","\frac{(d+e x)^{m+1} \left(a+b \sin ^{-1}(c x)\right)^2}{e (m+1)}-\frac{2 b c \text{Int}\left(\frac{(d+e x)^{m+1} \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}},x\right)}{e (m+1)}",0,"Integrate[(d + e*x)^m*(a + b*ArcSin[c*x])^2, x]","A",-1
28,0,0,154,0.0462693,"\int (d+e x)^m \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^m*(a + b*ArcSin[c*x]),x]","\int (d+e x)^m \left(a+b \sin ^{-1}(c x)\right) \, dx","\frac{(d+e x)^{m+1} \left(a+b \sin ^{-1}(c x)\right)}{e (m+1)}-\frac{b c \sqrt{1-\frac{c (d+e x)}{c d-e}} \sqrt{1-\frac{c (d+e x)}{c d+e}} (d+e x)^{m+2} F_1\left(m+2;\frac{1}{2},\frac{1}{2};m+3;\frac{c (d+e x)}{c d-e},\frac{c (d+e x)}{c d+e}\right)}{e^2 (m+1) (m+2) \sqrt{1-c^2 x^2}}",1,"Integrate[(d + e*x)^m*(a + b*ArcSin[c*x]), x]","F",-1
29,0,0,21,0.4172876,"\int \frac{(d+e x)^m}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(d + e*x)^m/(a + b*ArcSin[c*x]),x]","\int \frac{(d+e x)^m}{a+b \sin ^{-1}(c x)} \, dx","\text{Int}\left(\frac{(d+e x)^m}{a+b \sin ^{-1}(c x)},x\right)",0,"Integrate[(d + e*x)^m/(a + b*ArcSin[c*x]), x]","A",-1
30,0,0,21,0.8794053,"\int \frac{(d+e x)^m}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(d + e*x)^m/(a + b*ArcSin[c*x])^2,x]","\int \frac{(d+e x)^m}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{(d+e x)^m}{\left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(d + e*x)^m/(a + b*ArcSin[c*x])^2, x]","A",-1
31,1,356,669,0.4694696,"\int (f+g x)^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]),x]","\frac{\sqrt{d-c^2 d x^2} \left(225 a^2 \left(4 c^3 f^3+3 c f g^2\right)+30 a b \sqrt{1-c^2 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)+30 b \sin ^{-1}(c x) \left(15 a \left(4 c^3 f^3+3 c f g^2\right)+b \sqrt{1-c^2 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)\right)+225 b^2 c f \left(4 c^2 f^2+3 g^2\right) \sin ^{-1}(c x)^2+b^2 c x \left(-3 c^4 x \left(300 f^3+400 f^2 g x+225 f g^2 x^2+48 g^3 x^3\right)+5 c^2 g \left(720 f^2+135 f g x+16 g^2 x^2\right)+480 g^3\right)\right)}{3600 b c^4 \sqrt{1-c^2 x^2}}","\frac{1}{2} f^3 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{f^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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 \sin ^{-1}(c x)\right)}{c^2}-\frac{3 f g^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right)}{3 c^4}+\frac{3 f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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,"(Sqrt[d - c^2*d*x^2]*(225*a^2*(4*c^3*f^3 + 3*c*f*g^2) + 30*a*b*Sqrt[1 - c^2*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)) + b^2*c*x*(480*g^3 + 5*c^2*g*(720*f^2 + 135*f*g*x + 16*g^2*x^2) - 3*c^4*x*(300*f^3 + 400*f^2*g*x + 225*f*g^2*x^2 + 48*g^3*x^3)) + 30*b*(15*a*(4*c^3*f^3 + 3*c*f*g^2) + b*Sqrt[1 - c^2*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)))*ArcSin[c*x] + 225*b^2*c*f*(4*c^2*f^2 + 3*g^2)*ArcSin[c*x]^2))/(3600*b*c^4*Sqrt[1 - c^2*x^2])","A",1
32,1,237,450,0.5568954,"\int (f+g x)^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]),x]","\frac{\sqrt{d-c^2 d x^2} \left(72 f^2 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{96 f g \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{c^2}+36 g^2 x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{9 g^2 \left(-2 c x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b}+b c^2 x^2\right)}{c^3}+\frac{36 f^2 \left(a+b \sin ^{-1}(c x)\right)^2}{b c}-\frac{32 b f g x \left(c^2 x^2-3\right)}{c}-36 b c f^2 x^2-9 b c g^2 x^4\right)}{144 \sqrt{1-c^2 x^2}}","\frac{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{f^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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 \sin ^{-1}(c x)\right)}{3 c^2}-\frac{g^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 c^2}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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,"(Sqrt[d - c^2*d*x^2]*(-36*b*c*f^2*x^2 - 9*b*c*g^2*x^4 - (32*b*f*g*x*(-3 + c^2*x^2))/c + 72*f^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) + 36*g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - (96*f*g*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/c^2 + (36*f^2*(a + b*ArcSin[c*x])^2)/(b*c) + (9*g^2*(b*c^2*x^2 - 2*c*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) + (a + b*ArcSin[c*x])^2/b))/c^3))/(144*Sqrt[1 - c^2*x^2])","A",1
33,1,132,238,0.3047652,"\int (f+g x) \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]),x]","\frac{\sqrt{d-c^2 d x^2} \left(18 f x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{12 g \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{c^2}+\frac{9 f \left(a+b \sin ^{-1}(c x)\right)^2}{b c}-\frac{4 b g x \left(c^2 x^2-3\right)}{c}-9 b c f x^2\right)}{36 \sqrt{1-c^2 x^2}}","\frac{1}{2} f x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{f \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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 \sin ^{-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,"(Sqrt[d - c^2*d*x^2]*(-9*b*c*f*x^2 - (4*b*g*x*(-3 + c^2*x^2))/c + 18*f*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - (12*g*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]))/c^2 + (9*f*(a + b*ArcSin[c*x])^2)/(b*c)))/(36*Sqrt[1 - c^2*x^2])","A",1
34,1,368,736,1.0892406,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{f+g x} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(f + g*x),x]","\frac{\sqrt{d-c^2 d x^2} \left(-2 b c (f+g x) \left(-i \sqrt{c^2 f^2-g^2} \left(\left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right)-\log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)\right)-i b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)+i b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)-g \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+b c g x\right)+\left(c^2 f^2-g^2\right) \left(a+b \sin ^{-1}(c x)\right)^2+c^2 g x (f+g x) \left(a+b \sin ^{-1}(c x)\right)^2+g^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2\right)}{2 b c g^2 \sqrt{1-c^2 x^2} (f+g x)}","-\frac{\sqrt{d-c^2 d x^2} \left(1-\frac{c^2 f^2}{g^2}\right) \left(a+b \sin ^{-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 \sin ^{-1}(c x)\right)^2}{2 b c (f+g x)}+\frac{c x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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{i e^{i \sin ^{-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{i e^{i \sin ^{-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} \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)}{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} \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)}{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} \sin ^{-1}(c x)}{g}",1,"(Sqrt[d - c^2*d*x^2]*((c^2*f^2 - g^2)*(a + b*ArcSin[c*x])^2 + c^2*g*x*(f + g*x)*(a + b*ArcSin[c*x])^2 + g^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2 - 2*b*c*(f + g*x)*(b*c*g*x - g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - I*Sqrt[c^2*f^2 - g^2]*((a + b*ArcSin[c*x])*(Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]) - I*b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + I*b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))))/(2*b*c*g^2*(f + g*x)*Sqrt[1 - c^2*x^2])","A",0
35,1,600,860,2.7377266,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{(f+g x)^2} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(f + g*x)^2,x]","\frac{\sqrt{d-c^2 d x^2} \left(\frac{2 b c^2 \left(\frac{c f \left(i \left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right)-\log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)\right)+b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)-b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)}{\sqrt{c^2 f^2-g^2}}-\frac{g \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c f+c g x}+b \log (f+g x)\right)}{g^2}+\frac{\left(c^2 f^2-g^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{g^2 (f+g x)^2}-\frac{2 c^2 f \left(a+b \sin ^{-1}(c x)\right)^2}{g^2 (f+g x)}+\frac{\left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{(f+g x)^2}+\frac{4 b c^3 f \left(-i \left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right)-\log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)\right)-b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)+b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)}{g^2 \sqrt{c^2 f^2-g^2}}\right)}{2 b c \sqrt{1-c^2 x^2}}","-\frac{b f^2 \sqrt{d-c^2 d x^2} \sin ^{-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} \sin ^{-1}(c x) \log \left(1-\frac{i e^{i \sin ^{-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{i b f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left(1-\frac{i e^{i \sin ^{-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{i e^{i \sin ^{-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{i e^{i \sin ^{-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} \sin ^{-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 \sin ^{-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 \sin ^{-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,"(Sqrt[d - c^2*d*x^2]*(((c^2*f^2 - g^2)*(a + b*ArcSin[c*x])^2)/(g^2*(f + g*x)^2) - (2*c^2*f*(a + b*ArcSin[c*x])^2)/(g^2*(f + g*x)) + ((1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(f + g*x)^2 + (4*b*c^3*f*((-I)*(a + b*ArcSin[c*x])*(Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]) - b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))/(g^2*Sqrt[c^2*f^2 - g^2]) + (2*b*c^2*(-((g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(c*f + c*g*x)) + b*Log[f + g*x] + (c*f*(I*(a + b*ArcSin[c*x])*(Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]) + b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] - b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))/Sqrt[c^2*f^2 - g^2]))/g^2))/(2*b*c*Sqrt[1 - c^2*x^2])","A",0
36,1,463,959,1.224141,"\int (f+g x)^3 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]","\frac{d \sqrt{d-c^2 d x^2} \left(11025 a^2 c f \left(2 c^2 f^2+g^2\right)-210 a b \sqrt{1-c^2 x^2} \left(4 c^6 x^3 \left(35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right)-2 c^4 x \left(175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right)+c^2 g \left(336 f^2+105 f g x+16 g^2 x^2\right)+32 g^3\right)-210 b \sin ^{-1}(c x) \left(b \sqrt{1-c^2 x^2} \left(4 c^6 x^3 \left(35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right)-2 c^4 x \left(175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right)+c^2 g \left(336 f^2+105 f g x+16 g^2 x^2\right)+32 g^3\right)-105 a c f \left(2 c^2 f^2+g^2\right)\right)+11025 b^2 c f \left(2 c^2 f^2+g^2\right) \sin ^{-1}(c x)^2+b^2 c x \left(2 c^6 x^3 \left(3675 f^3+7056 f^2 g x+4900 f g^2 x^2+1200 g^3 x^3\right)-21 c^4 x \left(1750 f^3+2240 f^2 g x+1225 f g^2 x^2+256 g^3 x^3\right)+35 c^2 g \left(2016 f^2+315 f g x+32 g^2 x^2\right)+6720 g^3\right)\right)}{117600 b 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 \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right) x-\frac{3 d f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right)}{5 c^2}",1,"(d*Sqrt[d - c^2*d*x^2]*(11025*a^2*c*f*(2*c^2*f^2 + g^2) - 210*a*b*Sqrt[1 - c^2*x^2]*(32*g^3 + c^2*g*(336*f^2 + 105*f*g*x + 16*g^2*x^2) + 4*c^6*x^3*(35*f^3 + 84*f^2*g*x + 70*f*g^2*x^2 + 20*g^3*x^3) - 2*c^4*x*(175*f^3 + 336*f^2*g*x + 245*f*g^2*x^2 + 64*g^3*x^3)) + b^2*c*x*(6720*g^3 + 35*c^2*g*(2016*f^2 + 315*f*g*x + 32*g^2*x^2) - 21*c^4*x*(1750*f^3 + 2240*f^2*g*x + 1225*f*g^2*x^2 + 256*g^3*x^3) + 2*c^6*x^3*(3675*f^3 + 7056*f^2*g*x + 4900*f*g^2*x^2 + 1200*g^3*x^3)) - 210*b*(-105*a*c*f*(2*c^2*f^2 + g^2) + b*Sqrt[1 - c^2*x^2]*(32*g^3 + c^2*g*(336*f^2 + 105*f*g*x + 16*g^2*x^2) + 4*c^6*x^3*(35*f^3 + 84*f^2*g*x + 70*f*g^2*x^2 + 20*g^3*x^3) - 2*c^4*x*(175*f^3 + 336*f^2*g*x + 245*f*g^2*x^2 + 64*g^3*x^3)))*ArcSin[c*x] + 11025*b^2*c*f*(2*c^2*f^2 + g^2)*ArcSin[c*x]^2))/(117600*b*c^4*Sqrt[1 - c^2*x^2])","A",1
37,1,332,680,0.5112027,"\int (f+g x)^2 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]","\frac{d \sqrt{d-c^2 d x^2} \left(225 a^2 \left(6 c^2 f^2+g^2\right)-30 a b c \sqrt{1-c^2 x^2} \left(30 c^2 f^2 x \left(2 c^2 x^2-5\right)+96 f g \left(c^2 x^2-1\right)^2+5 g^2 x \left(8 c^4 x^4-14 c^2 x^2+3\right)\right)+30 b \sin ^{-1}(c x) \left(15 a \left(6 c^2 f^2+g^2\right)-b c \sqrt{1-c^2 x^2} \left(30 c^2 f^2 x \left(2 c^2 x^2-5\right)+96 f g \left(c^2 x^2-1\right)^2+5 g^2 x \left(8 c^4 x^4-14 c^2 x^2+3\right)\right)\right)+225 b^2 \left(6 c^2 f^2+g^2\right) \sin ^{-1}(c x)^2+b^2 c^2 x \left(450 c^2 f^2 x \left(c^2 x^2-5\right)+192 f g \left(3 c^4 x^4-10 c^2 x^2+15\right)+25 g^2 x \left(8 c^4 x^4-21 c^2 x^2+9\right)\right)\right)}{7200 b 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 \sin ^{-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 \sin ^{-1}(c x)\right)+\frac{3 d f^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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 \sin ^{-1}(c x)\right)}{5 c^2}-\frac{d g^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right)+\frac{d g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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,"(d*Sqrt[d - c^2*d*x^2]*(225*a^2*(6*c^2*f^2 + g^2) + b^2*c^2*x*(450*c^2*f^2*x*(-5 + c^2*x^2) + 192*f*g*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 25*g^2*x*(9 - 21*c^2*x^2 + 8*c^4*x^4)) - 30*a*b*c*Sqrt[1 - c^2*x^2]*(96*f*g*(-1 + c^2*x^2)^2 + 30*c^2*f^2*x*(-5 + 2*c^2*x^2) + 5*g^2*x*(3 - 14*c^2*x^2 + 8*c^4*x^4)) + 30*b*(15*a*(6*c^2*f^2 + g^2) - b*c*Sqrt[1 - c^2*x^2]*(96*f*g*(-1 + c^2*x^2)^2 + 30*c^2*f^2*x*(-5 + 2*c^2*x^2) + 5*g^2*x*(3 - 14*c^2*x^2 + 8*c^4*x^4)))*ArcSin[c*x] + 225*b^2*(6*c^2*f^2 + g^2)*ArcSin[c*x]^2))/(7200*b*c^3*Sqrt[1 - c^2*x^2])","A",1
38,1,216,370,0.3089821,"\int (f+g x) \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]","\frac{d \sqrt{d-c^2 d x^2} \left(225 a^2 c f-30 a b \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)+30 b \sin ^{-1}(c x) \left(15 a c f+b \sqrt{1-c^2 x^2} \left(5 c^2 f x \left(5-2 c^2 x^2\right)-8 g \left(c^2 x^2-1\right)^2\right)\right)+b^2 c x \left(75 c^2 f x \left(c^2 x^2-5\right)+16 g \left(3 c^4 x^4-10 c^2 x^2+15\right)\right)+225 b^2 c f \sin ^{-1}(c x)^2\right)}{1200 b c^2 \sqrt{1-c^2 x^2}}","\frac{3}{8} d f x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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 \sin ^{-1}(c x)\right)+\frac{3 d f \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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 \sin ^{-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,"(d*Sqrt[d - c^2*d*x^2]*(225*a^2*c*f - 30*a*b*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)) + b^2*c*x*(75*c^2*f*x*(-5 + c^2*x^2) + 16*g*(15 - 10*c^2*x^2 + 3*c^4*x^4)) + 30*b*(15*a*c*f + b*Sqrt[1 - c^2*x^2]*(5*c^2*f*x*(5 - 2*c^2*x^2) - 8*g*(-1 + c^2*x^2)^2))*ArcSin[c*x] + 225*b^2*c*f*ArcSin[c*x]^2))/(1200*b*c^2*Sqrt[1 - c^2*x^2])","A",1
39,1,507,1073,1.4643054,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{f+g x} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(f + g*x),x]","\frac{d \sqrt{d-c^2 d x^2} \left(-\frac{18 \left(c^2 f^2-g^2\right) \left(-2 b c (f+g x) \left(-i \sqrt{c^2 f^2-g^2} \left(\left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right)-\log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)\right)-i b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)+i b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)-g \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+b c g x\right)+\left(c^2 f^2-g^2\right) \left(a+b \sin ^{-1}(c x)\right)^2+c^2 g x (f+g x) \left(a+b \sin ^{-1}(c x)\right)^2\right)}{b c g^2 (f+g x)}+\frac{18 \left(c^2 x^2-1\right) \left(c^2 f^2-g^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{b c (f+g x)}+18 c^2 f x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+12 g \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)+\frac{9 c f \left(a+b \sin ^{-1}(c x)\right)^2}{b}-9 b c^3 f x^2+4 b c g x \left(c^2 x^2-3\right)\right)}{36 g^2 \sqrt{1-c^2 x^2}}","\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 \sin ^{-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 \sin ^{-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 \sin ^{-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} \sin ^{-1}(c x)}{g^3}+\frac{d \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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} \sin ^{-1}(c x) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^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} \sin ^{-1}(c x) \log \left(1-\frac{i e^{i \sin ^{-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{i e^{i \sin ^{-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{i e^{i \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right)^2}{2 b g^4 (f+g x) \sqrt{1-c^2 x^2} c}",1,"(d*Sqrt[d - c^2*d*x^2]*(-9*b*c^3*f*x^2 + 4*b*c*g*x*(-3 + c^2*x^2) + 18*c^2*f*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) + 12*g*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]) + (9*c*f*(a + b*ArcSin[c*x])^2)/b + (18*(c^2*f^2 - g^2)*(-1 + c^2*x^2)*(a + b*ArcSin[c*x])^2)/(b*c*(f + g*x)) - (18*(c^2*f^2 - g^2)*((c^2*f^2 - g^2)*(a + b*ArcSin[c*x])^2 + c^2*g*x*(f + g*x)*(a + b*ArcSin[c*x])^2 - 2*b*c*(f + g*x)*(b*c*g*x - g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - I*Sqrt[c^2*f^2 - g^2]*((a + b*ArcSin[c*x])*(Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]) - I*b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + I*b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))))/(b*c*g^2*(f + g*x))))/(36*g^2*Sqrt[1 - c^2*x^2])","A",0
40,1,587,1281,1.0654784,"\int (f+g x)^3 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(99225 a^2 \left(8 c^3 f^3+3 c f g^2\right)+630 a b \sqrt{1-c^2 x^2} \left(16 c^8 x^5 \left(84 f^3+216 f^2 g x+189 f g^2 x^2+56 g^3 x^3\right)-8 c^6 x^3 \left(546 f^3+1296 f^2 g x+1071 f g^2 x^2+304 g^3 x^3\right)+6 c^4 x \left(924 f^3+1728 f^2 g x+1239 f g^2 x^2+320 g^3 x^3\right)-c^2 g \left(3456 f^2+945 f g x+128 g^2 x^2\right)-256 g^3\right)+630 b \sin ^{-1}(c x) \left(315 a \left(8 c^3 f^3+3 c f g^2\right)+b \sqrt{1-c^2 x^2} \left(16 c^8 x^5 \left(84 f^3+216 f^2 g x+189 f g^2 x^2+56 g^3 x^3\right)-8 c^6 x^3 \left(546 f^3+1296 f^2 g x+1071 f g^2 x^2+304 g^3 x^3\right)+6 c^4 x \left(924 f^3+1728 f^2 g x+1239 f g^2 x^2+320 g^3 x^3\right)-c^2 g \left(3456 f^2+945 f g x+128 g^2 x^2\right)-256 g^3\right)\right)+99225 b^2 c f \left(8 c^2 f^2+3 g^2\right) \sin ^{-1}(c x)^2+b^2 c x \left(-20 c^8 x^5 \left(7056 f^3+15552 f^2 g x+11907 f g^2 x^2+3136 g^3 x^3\right)+72 c^6 x^3 \left(9555 f^3+18144 f^2 g x+12495 f g^2 x^2+3040 g^3 x^3\right)-945 c^4 x \left(1848 f^3+2304 f^2 g x+1239 f g^2 x^2+256 g^3 x^3\right)+105 c^2 g \left(20736 f^2+2835 f g x+256 g^2 x^2\right)+161280 g^3\right)\right)}{5080320 b 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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right) x-\frac{15 d^2 f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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*Sqrt[d - c^2*d*x^2]*(99225*a^2*(8*c^3*f^3 + 3*c*f*g^2) + 630*a*b*Sqrt[1 - c^2*x^2]*(-256*g^3 - c^2*g*(3456*f^2 + 945*f*g*x + 128*g^2*x^2) + 16*c^8*x^5*(84*f^3 + 216*f^2*g*x + 189*f*g^2*x^2 + 56*g^3*x^3) - 8*c^6*x^3*(546*f^3 + 1296*f^2*g*x + 1071*f*g^2*x^2 + 304*g^3*x^3) + 6*c^4*x*(924*f^3 + 1728*f^2*g*x + 1239*f*g^2*x^2 + 320*g^3*x^3)) + b^2*c*x*(161280*g^3 + 105*c^2*g*(20736*f^2 + 2835*f*g*x + 256*g^2*x^2) - 945*c^4*x*(1848*f^3 + 2304*f^2*g*x + 1239*f*g^2*x^2 + 256*g^3*x^3) + 72*c^6*x^3*(9555*f^3 + 18144*f^2*g*x + 12495*f*g^2*x^2 + 3040*g^3*x^3) - 20*c^8*x^5*(7056*f^3 + 15552*f^2*g*x + 11907*f*g^2*x^2 + 3136*g^3*x^3)) + 630*b*(315*a*(8*c^3*f^3 + 3*c*f*g^2) + b*Sqrt[1 - c^2*x^2]*(-256*g^3 - c^2*g*(3456*f^2 + 945*f*g*x + 128*g^2*x^2) + 16*c^8*x^5*(84*f^3 + 216*f^2*g*x + 189*f*g^2*x^2 + 56*g^3*x^3) - 8*c^6*x^3*(546*f^3 + 1296*f^2*g*x + 1071*f*g^2*x^2 + 304*g^3*x^3) + 6*c^4*x*(924*f^3 + 1728*f^2*g*x + 1239*f*g^2*x^2 + 320*g^3*x^3)))*ArcSin[c*x] + 99225*b^2*c*f*(8*c^2*f^2 + 3*g^2)*ArcSin[c*x]^2))/(5080320*b*c^4*Sqrt[1 - c^2*x^2])","A",1
41,1,390,940,0.758829,"\int (f+g x)^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(11025 a^2 \left(8 c^2 f^2+g^2\right)+210 a b c \sqrt{1-c^2 x^2} \left(768 f g \left(c^2 x^2-1\right)^3+56 c^2 f^2 x \left(8 c^4 x^4-26 c^2 x^2+33\right)+7 g^2 x \left(48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right)\right)+210 b \sin ^{-1}(c x) \left(105 a \left(8 c^2 f^2+g^2\right)+b c \sqrt{1-c^2 x^2} \left(768 f g \left(c^2 x^2-1\right)^3+56 c^2 f^2 x \left(8 c^4 x^4-26 c^2 x^2+33\right)+7 g^2 x \left(48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right)\right)\right)+11025 b^2 \left(8 c^2 f^2+g^2\right) \sin ^{-1}(c x)^2+b^2 c^2 x \left(-1960 c^2 f^2 x \left(8 c^4 x^4-39 c^2 x^2+99\right)-4608 f g \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right)-245 g^2 x \left(36 c^6 x^6-136 c^4 x^4+177 c^2 x^2-45\right)\right)\right)}{564480 b 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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right) x-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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*Sqrt[d - c^2*d*x^2]*(11025*a^2*(8*c^2*f^2 + g^2) + b^2*c^2*x*(-1960*c^2*f^2*x*(99 - 39*c^2*x^2 + 8*c^4*x^4) - 4608*f*g*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) - 245*g^2*x*(-45 + 177*c^2*x^2 - 136*c^4*x^4 + 36*c^6*x^6)) + 210*a*b*c*Sqrt[1 - c^2*x^2]*(768*f*g*(-1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 7*g^2*x*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6)) + 210*b*(105*a*(8*c^2*f^2 + g^2) + b*c*Sqrt[1 - c^2*x^2]*(768*f*g*(-1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 7*g^2*x*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6)))*ArcSin[c*x] + 11025*b^2*(8*c^2*f^2 + g^2)*ArcSin[c*x]^2))/(564480*b*c^3*Sqrt[1 - c^2*x^2])","A",1
42,1,251,517,0.4517088,"\int (f+g x) \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(11025 a^2 c f+210 a b \sqrt{1-c^2 x^2} \left(48 g \left(c^2 x^2-1\right)^3+7 c^2 f x \left(8 c^4 x^4-26 c^2 x^2+33\right)\right)+210 b \sin ^{-1}(c x) \left(105 a c f+b \sqrt{1-c^2 x^2} \left(48 g \left(c^2 x^2-1\right)^3+7 c^2 f x \left(8 c^4 x^4-26 c^2 x^2+33\right)\right)\right)+b^2 c x \left(-245 c^2 f x \left(8 c^4 x^4-39 c^2 x^2+99\right)-288 g \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right)\right)+11025 b^2 c f \sin ^{-1}(c x)^2\right)}{70560 b 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 \sin ^{-1}(c x)\right)+\frac{5}{16} d^2 f x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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 \sin ^{-1}(c x)\right)+\frac{5 d^2 f \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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 \sin ^{-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*Sqrt[d - c^2*d*x^2]*(11025*a^2*c*f + 210*a*b*Sqrt[1 - c^2*x^2]*(48*g*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 - 26*c^2*x^2 + 8*c^4*x^4)) + b^2*c*x*(-245*c^2*f*x*(99 - 39*c^2*x^2 + 8*c^4*x^4) - 288*g*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)) + 210*b*(105*a*c*f + b*Sqrt[1 - c^2*x^2]*(48*g*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 - 26*c^2*x^2 + 8*c^4*x^4)))*ArcSin[c*x] + 11025*b^2*c*f*ArcSin[c*x]^2))/(70560*b*c^2*Sqrt[1 - c^2*x^2])","A",1
43,1,787,1648,2.7298036,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{f+g x} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(f + g*x),x]","-\frac{d^2 \sqrt{d-c^2 d x^2} \left(-\frac{1800 \left(g^2-c^2 f^2\right)^2 \left(-2 b c \left(-i \sqrt{c^2 f^2-g^2} \left(\left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right)-\log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)\right)-i b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)+i b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)-g \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+b c g x\right)+\frac{\left(c^2 f^2-g^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{f+g x}+c^2 g x \left(a+b \sin ^{-1}(c x)\right)^2\right)}{b c g^2}+1800 c^2 f x \sqrt{1-c^2 x^2} \left(c^2 f^2-2 g^2\right) \left(a+b \sin ^{-1}(c x)\right)+1200 g \left(1-c^2 x^2\right)^{3/2} \left(c^2 f^2-2 g^2\right) \left(a+b \sin ^{-1}(c x)\right)+\frac{1800 \left(c^2 x^2-1\right) \left(g^2-c^2 f^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{b c (f+g x)}+\frac{900 c f \left(c^2 f^2-2 g^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{b}+225 c f g^2 \left(-2 c x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b}+b c^2 x^2\right)+900 c^4 f g^2 x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)-720 c^4 g^3 x^4 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)-80 g^3 \left(-3 c^2 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)-6 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+b c^3 x^3+6 b c x\right)-225 b c^5 f g^2 x^4+144 b c^5 g^3 x^5+400 b c g x \left(c^2 x^2-3\right) \left(c^2 f^2-2 g^2\right)-900 b c^3 f x^2 \left(c^2 f^2-2 g^2\right)\right)}{3600 g^4 \sqrt{1-c^2 x^2}}","-\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 \sin ^{-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 \sin ^{-1}(c x)\right) c^2}{2 g^4}+\frac{d^2 f x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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} \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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} \sin ^{-1}(c x) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^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} \sin ^{-1}(c x) \log \left(1-\frac{i e^{i \sin ^{-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{i e^{i \sin ^{-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{i e^{i \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right)^2}{2 b g^6 (f+g x) \sqrt{1-c^2 x^2} c}",1,"-1/3600*(d^2*Sqrt[d - c^2*d*x^2]*(-900*b*c^3*f*(c^2*f^2 - 2*g^2)*x^2 - 225*b*c^5*f*g^2*x^4 + 144*b*c^5*g^3*x^5 + 400*b*c*g*(c^2*f^2 - 2*g^2)*x*(-3 + c^2*x^2) + 1800*c^2*f*(c^2*f^2 - 2*g^2)*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) + 900*c^4*f*g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - 720*c^4*g^3*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) + 1200*g*(c^2*f^2 - 2*g^2)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]) + (900*c*f*(c^2*f^2 - 2*g^2)*(a + b*ArcSin[c*x])^2)/b + (1800*(-(c^2*f^2) + g^2)^2*(-1 + c^2*x^2)*(a + b*ArcSin[c*x])^2)/(b*c*(f + g*x)) - 80*g^3*(6*b*c*x + b*c^3*x^3 - 6*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - 3*c^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])) + 225*c*f*g^2*(b*c^2*x^2 - 2*c*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) + (a + b*ArcSin[c*x])^2/b) - (1800*(-(c^2*f^2) + g^2)^2*(c^2*g*x*(a + b*ArcSin[c*x])^2 + ((c^2*f^2 - g^2)*(a + b*ArcSin[c*x])^2)/(f + g*x) - 2*b*c*(b*c*g*x - g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - I*Sqrt[c^2*f^2 - g^2]*((a + b*ArcSin[c*x])*(Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]) - I*b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + I*b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))))/(b*c*g^2)))/(g^4*Sqrt[1 - c^2*x^2])","A",0
44,1,343,450,1.0924065,"\int \frac{(f+g x)^3 \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[((f + g*x)^3*(a + b*ArcSin[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 \sin ^{-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) \sin ^{-1}(c x)^2+6 b \sqrt{d} g \left(c^2 x^2-1\right) \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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)*ArcSin[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*ArcSin[c*x]]) + 6*b*Sqrt[d]*g*(-1 + c^2*x^2)*ArcSin[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*ArcSin[c*x]]))/(72*c^4*Sqrt[d]*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])","A",1
45,1,266,270,0.7181755,"\int \frac{(f+g x)^2 \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[((f + g*x)^2*(a + b*ArcSin[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 \sin ^{-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) \sin ^{-1}(c x)^2+2 b \sqrt{d} g \left(c^2 x^2-1\right) \sin ^{-1}(c x) \left(8 c f \sqrt{1-c^2 x^2}+g \sin \left(2 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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)*ArcSin[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*ArcSin[c*x]]) + 2*b*Sqrt[d]*g*(-1 + c^2*x^2)*ArcSin[c*x]*(8*c*f*Sqrt[1 - c^2*x^2] + g*Sin[2*ArcSin[c*x]]))/(8*c^3*Sqrt[d]*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])","A",1
46,1,172,126,0.3520986,"\int \frac{(f+g x) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[((f + g*x)*(a + b*ArcSin[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} \sin ^{-1}(c x)^2+2 b \sqrt{d} g \left(c^2 x^2-1\right) \sin ^{-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 \sin ^{-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 \sin ^{-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)*ArcSin[c*x] + b*c*Sqrt[d]*f*Sqrt[1 - c^2*x^2]*ArcSin[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
47,1,232,380,0.1858231,"\int \frac{a+b \sin ^{-1}(c x)}{(f+g x) \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])/((f + g*x)*Sqrt[d - c^2*d*x^2]),x]","\frac{\sqrt{1-c^2 x^2} \left(-i \left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right)-\log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)\right)-b \text{Li}_2\left(-\frac{i e^{i \sin ^{-1}(c x)} g}{\sqrt{c^2 f^2-g^2}-c f}\right)+b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\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 \sin ^{-1}(c x)\right) \log \left(1-\frac{i g e^{i \sin ^{-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 \sin ^{-1}(c x)\right) \log \left(1-\frac{i g e^{i \sin ^{-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{i e^{i \sin ^{-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{i e^{i \sin ^{-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,"(Sqrt[1 - c^2*x^2]*((-I)*(a + b*ArcSin[c*x])*(Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]) - b*PolyLog[2, ((-I)*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] + b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2])","A",1
48,1,295,507,0.4912631,"\int \frac{a+b \sin ^{-1}(c x)}{(f+g x)^2 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])/((f + g*x)^2*Sqrt[d - c^2*d*x^2]),x]","\frac{c \sqrt{1-c^2 x^2} \left(\frac{c f \left(-i \left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right)-\log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)\right)-b \text{Li}_2\left(-\frac{i e^{i \sin ^{-1}(c x)} g}{\sqrt{c^2 f^2-g^2}-c f}\right)+b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)}{\sqrt{c^2 f^2-g^2}}+\frac{g \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c f+c g x}-b \log (f+g x)\right)}{\sqrt{d-c^2 d x^2} \left(c^2 f^2-g^2\right)}","\frac{g \left(1-c^2 x^2\right) \left(a+b \sin ^{-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 \sin ^{-1}(c x)\right) \log \left(1-\frac{i g e^{i \sin ^{-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 \sin ^{-1}(c x)\right) \log \left(1-\frac{i g e^{i \sin ^{-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{i e^{i \sin ^{-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{i e^{i \sin ^{-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,"(c*Sqrt[1 - c^2*x^2]*((g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(c*f + c*g*x) - b*Log[f + g*x] + (c*f*((-I)*(a + b*ArcSin[c*x])*(Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]) - b*PolyLog[2, ((-I)*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] + b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))/Sqrt[c^2*f^2 - g^2]))/((c^2*f^2 - g^2)*Sqrt[d - c^2*d*x^2])","A",1
49,1,194,315,1.1000776,"\int \frac{(f+g x)^3 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[((f + g*x)^3*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(3/2),x]","\frac{\sqrt{1-c^2 x^2} \left(2 g^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{3 c f g^2 \left(a+b \sin ^{-1}(c x)\right)^2}{b}+(c f-g)^3 \left(2 b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-\cot \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(a+b \sin ^{-1}(c x)\right)\right)+(c f+g)^3 \left(\tan \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(a+b \sin ^{-1}(c x)\right)+2 b \log \left(\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)\right)-2 b c g^3 x\right)}{2 c^4 d \sqrt{d-c^2 d x^2}}","\frac{\left(c^2 f x \left(c^2 f^2+3 g^2\right)+g \left(3 c^2 f^2+g^2\right)\right) \left(a+b \sin ^{-1}(c x)\right)}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f-g)^3 \log (c x+1)}{2 c^4 d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f+g)^3 \log (1-c x)}{2 c^4 d \sqrt{d-c^2 d x^2}}-\frac{b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}",1,"(Sqrt[1 - c^2*x^2]*(-2*b*c*g^3*x + 2*g^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - (3*c*f*g^2*(a + b*ArcSin[c*x])^2)/b + (c*f - g)^3*(-((a + b*ArcSin[c*x])*Cot[(Pi + 2*ArcSin[c*x])/4]) + 2*b*Log[Sin[(Pi + 2*ArcSin[c*x])/4]]) + (c*f + g)^3*(2*b*Log[Cos[(Pi + 2*ArcSin[c*x])/4]] + (a + b*ArcSin[c*x])*Tan[(Pi + 2*ArcSin[c*x])/4])))/(2*c^4*d*Sqrt[d - c^2*d*x^2])","A",1
50,1,156,213,0.7696996,"\int \frac{(f+g x)^2 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[((f + g*x)^2*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(3/2),x]","\frac{\sqrt{1-c^2 x^2} \left((g-c f)^2 \left(2 b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-\cot \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(a+b \sin ^{-1}(c x)\right)\right)+(c f+g)^2 \left(\tan \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(a+b \sin ^{-1}(c x)\right)+2 b \log \left(\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)\right)-\frac{g^2 \left(a+b \sin ^{-1}(c x)\right)^2}{b}\right)}{2 c^3 d \sqrt{d-c^2 d x^2}}","\frac{\left(x \left(c^2 f^2+g^2\right)+2 f g\right) \left(a+b \sin ^{-1}(c x)\right)}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f-g)^2 \log (c x+1)}{2 c^3 d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f+g)^2 \log (1-c x)}{2 c^3 d \sqrt{d-c^2 d x^2}}",1,"(Sqrt[1 - c^2*x^2]*(-((g^2*(a + b*ArcSin[c*x])^2)/b) + (-(c*f) + g)^2*(-((a + b*ArcSin[c*x])*Cot[(Pi + 2*ArcSin[c*x])/4]) + 2*b*Log[Sin[(Pi + 2*ArcSin[c*x])/4]]) + (c*f + g)^2*(2*b*Log[Cos[(Pi + 2*ArcSin[c*x])/4]] + (a + b*ArcSin[c*x])*Tan[(Pi + 2*ArcSin[c*x])/4])))/(2*c^3*d*Sqrt[d - c^2*d*x^2])","A",1
51,1,135,144,0.5617522,"\int \frac{(f+g x) \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(3/2),x]","\frac{\sqrt{1-c^2 x^2} \left((c f-g) \left(2 b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-\cot \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(a+b \sin ^{-1}(c x)\right)\right)+(c f+g) \left(\tan \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(a+b \sin ^{-1}(c x)\right)+2 b \log \left(\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)\right)\right)}{2 c^2 d \sqrt{d-c^2 d x^2}}","\frac{\left(c^2 f x+g\right) \left(a+b \sin ^{-1}(c x)\right)}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f+g) \log (1-c x)}{2 c^2 d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f-g) \log (c x+1)}{2 c^2 d \sqrt{d-c^2 d x^2}}",1,"(Sqrt[1 - c^2*x^2]*((c*f - g)*(-((a + b*ArcSin[c*x])*Cot[(Pi + 2*ArcSin[c*x])/4]) + 2*b*Log[Sin[(Pi + 2*ArcSin[c*x])/4]]) + (c*f + g)*(2*b*Log[Cos[(Pi + 2*ArcSin[c*x])/4]] + (a + b*ArcSin[c*x])*Tan[(Pi + 2*ArcSin[c*x])/4])))/(2*c^2*d*Sqrt[d - c^2*d*x^2])","A",1
52,1,359,654,2.0370082,"\int \frac{a+b \sin ^{-1}(c x)}{(f+g x) \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/((f + g*x)*(d - c^2*d*x^2)^(3/2)),x]","\frac{\sqrt{1-c^2 x^2} \left(\frac{2 g^2 \left(i \left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right)-\log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)\right)+b \text{Li}_2\left(-\frac{i e^{i \sin ^{-1}(c x)} g}{\sqrt{c^2 f^2-g^2}-c f}\right)-b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)}{(c f-g) (c f+g) \sqrt{c^2 f^2-g^2}}+\frac{2 b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-\cot \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(a+b \sin ^{-1}(c x)\right)}{c f-g}+\frac{\tan \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(a+b \sin ^{-1}(c x)\right)+2 b \log \left(\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{c f+g}\right)}{2 d \sqrt{d-c^2 d x^2}}","\frac{i g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right)}{d \sqrt{d-c^2 d x^2} \left(c^2 f^2-g^2\right)^{3/2}}-\frac{i g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)}{d \sqrt{d-c^2 d x^2} \left(c^2 f^2-g^2\right)^{3/2}}+\frac{\sqrt{1-c^2 x^2} \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)}{2 d \sqrt{d-c^2 d x^2} (c f+g)}-\frac{\sqrt{1-c^2 x^2} \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)}{2 d \sqrt{d-c^2 d x^2} (c f-g)}+\frac{b g^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{d \sqrt{d-c^2 d x^2} \left(c^2 f^2-g^2\right)^{3/2}}-\frac{b g^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{d \sqrt{d-c^2 d x^2} \left(c^2 f^2-g^2\right)^{3/2}}+\frac{b \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)\right)}{d \sqrt{d-c^2 d x^2} (c f-g)}+\frac{b \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)\right)}{d \sqrt{d-c^2 d x^2} (c f+g)}",1,"(Sqrt[1 - c^2*x^2]*((-((a + b*ArcSin[c*x])*Cot[(Pi + 2*ArcSin[c*x])/4]) + 2*b*Log[Sin[(Pi + 2*ArcSin[c*x])/4]])/(c*f - g) + (2*g^2*(I*(a + b*ArcSin[c*x])*(Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]) + b*PolyLog[2, ((-I)*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))/((c*f - g)*(c*f + g)*Sqrt[c^2*f^2 - g^2]) + (2*b*Log[Cos[(Pi + 2*ArcSin[c*x])/4]] + (a + b*ArcSin[c*x])*Tan[(Pi + 2*ArcSin[c*x])/4])/(c*f + g)))/(2*d*Sqrt[d - c^2*d*x^2])","A",1
53,1,868,528,3.2086632,"\int \frac{(f+g x)^4 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[((f + g*x)^4*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]","\frac{b \left(4 c x \sin ^{-1}(c x)+\frac{\frac{2 c x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-1}{\sqrt{1-c^2 x^2}}+4 \sqrt{1-c^2 x^2} \log \left(\sqrt{1-c^2 x^2}\right)\right) f^4}{6 c d^2 \sqrt{d \left(1-c^2 x^2\right)}}+\frac{b g \left(8 \sin ^{-1}(c x)+\cos \left(3 \sin ^{-1}(c x)\right) \left(\log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-\log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)+3 \sqrt{1-c^2 x^2} \left(\log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-\log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-2 \sin \left(2 \sin ^{-1}(c x)\right)\right) f^3}{6 c^2 d \left(d \left(1-c^2 x^2\right)\right)^{3/2}}+\frac{b g^2 \left(-2 c x \sin ^{-1}(c x)+\frac{\frac{2 c x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-1}{\sqrt{1-c^2 x^2}}-2 \sqrt{1-c^2 x^2} \log \left(\sqrt{1-c^2 x^2}\right)\right) f^2}{c^3 d^2 \sqrt{d \left(1-c^2 x^2\right)}}-\frac{b g^3 \left(12 \cos \left(2 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+4 \sin ^{-1}(c x)+5 \cos \left(3 \sin ^{-1}(c x)\right) \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+15 \sqrt{1-c^2 x^2} \left(\log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-\log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-5 \cos \left(3 \sin ^{-1}(c x)\right) \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 \sin \left(2 \sin ^{-1}(c x)\right)\right) f}{6 c^4 d \left(d \left(1-c^2 x^2\right)\right)^{3/2}}+\sqrt{-d \left(c^2 x^2-1\right)} \left(\frac{a c^4 x f^4+4 a c^2 g f^3+6 a c^2 g^2 x f^2+4 a g^3 f+a g^4 x}{3 c^4 d^3 \left(c^2 x^2-1\right)^2}-\frac{2 a \left(c^4 x f^4-3 c^2 g^2 x f^2-6 g^3 f-2 g^4 x\right)}{3 c^4 d^3 \left(c^2 x^2-1\right)}\right)-\frac{a g^4 \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^5 d^{5/2}}+\frac{b g^4 \left(\sqrt{1-c^2 x^2} \left(3 \sin ^{-1}(c x)^2-8 \log \left(\sqrt{1-c^2 x^2}\right)\right)-\frac{\frac{2 \sin ^{-1}(c x) \sin \left(3 \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}+1}{\sqrt{1-c^2 x^2}}\right)}{6 c^5 d^2 \sqrt{d \left(1-c^2 x^2\right)}}","\frac{(f+g x)^3 \left(c^2 f x+g\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \left(a+b \sin ^{-1}(c x)\right)}{c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{f g \left(1-c^2 x^2\right) \left(2 c^2 f^2-5 g^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(f+g x) \left(2 c^2 f x \left(c^2 f^2-2 g^2\right)+g \left(c^2 f^2-3 g^2\right)\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f-2 g) (c f+g)^3 \log (1-c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f-g)^3 (c f+2 g) \log (c x+1)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b g^4 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (f+g x)^2 \left(c^2 f^2+2 c^2 f g x+g^2\right)}{6 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{b f g^3 x \sqrt{1-c^2 x^2}}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*((4*a*c^2*f^3*g + 4*a*f*g^3 + a*c^4*f^4*x + 6*a*c^2*f^2*g^2*x + a*g^4*x)/(3*c^4*d^3*(-1 + c^2*x^2)^2) - (2*a*(-6*f*g^3 + c^4*f^4*x - 3*c^2*f^2*g^2*x - 2*g^4*x))/(3*c^4*d^3*(-1 + c^2*x^2))) - (a*g^4*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(c^5*d^(5/2)) + (b*f^2*g^2*(-2*c*x*ArcSin[c*x] + (-1 + (2*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2])/Sqrt[1 - c^2*x^2] - 2*Sqrt[1 - c^2*x^2]*Log[Sqrt[1 - c^2*x^2]]))/(c^3*d^2*Sqrt[d*(1 - c^2*x^2)]) + (b*f^4*(4*c*x*ArcSin[c*x] + (-1 + (2*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2])/Sqrt[1 - c^2*x^2] + 4*Sqrt[1 - c^2*x^2]*Log[Sqrt[1 - c^2*x^2]]))/(6*c*d^2*Sqrt[d*(1 - c^2*x^2)]) + (b*f^3*g*(8*ArcSin[c*x] + 3*Sqrt[1 - c^2*x^2]*(Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + Cos[3*ArcSin[c*x]]*(Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) - 2*Sin[2*ArcSin[c*x]]))/(6*c^2*d*(d*(1 - c^2*x^2))^(3/2)) - (b*f*g^3*(4*ArcSin[c*x] + 12*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + 5*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 15*Sqrt[1 - c^2*x^2]*(Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) - 5*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + 2*Sin[2*ArcSin[c*x]]))/(6*c^4*d*(d*(1 - c^2*x^2))^(3/2)) + (b*g^4*(Sqrt[1 - c^2*x^2]*(3*ArcSin[c*x]^2 - 8*Log[Sqrt[1 - c^2*x^2]]) - (1 + (2*ArcSin[c*x]*Sin[3*ArcSin[c*x]])/Sqrt[1 - c^2*x^2])/Sqrt[1 - c^2*x^2]))/(6*c^5*d^2*Sqrt[d*(1 - c^2*x^2)])","A",1
54,1,366,410,1.3022644,"\int \frac{(f+g x)^3 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[((f + g*x)^3*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]","\frac{\sqrt{d-c^2 d x^2} \left(-\sqrt{-c^2} \left(4 a c^6 f^3 x^3-6 a c^4 f^3 x-6 a c^4 f g^2 x^3-6 a c^2 f^2 g-6 a c^2 g^3 x^2+4 a g^3-b c f \left(1-c^2 x^2\right)^{3/2} \left(2 c^2 f^2-3 g^2\right) \log \left(c^2 x^2-1\right)+3 b c f g^2 \sqrt{1-c^2 x^2}+b c g^3 x \sqrt{1-c^2 x^2}+b c^3 f^3 \sqrt{1-c^2 x^2}+3 b c^3 f^2 g x \sqrt{1-c^2 x^2}+2 b \sin ^{-1}(c x) \left(2 c^6 f^3 x^3-3 c^4 f x \left(f^2+g^2 x^2\right)-3 c^2 g \left(f^2+g^2 x^2\right)+2 g^3\right)\right)+i b c g \left(1-c^2 x^2\right)^{3/2} \left(3 c^2 f^2-5 g^2\right) F\left(\left.i \sinh ^{-1}\left(\sqrt{-c^2} x\right)\right|1\right)\right)}{6 c^4 \sqrt{-c^2} d^3 \left(c^2 x^2-1\right)^2}","\frac{(f+g x)^2 \left(c^2 f x+g\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{2 (c f+g) (c f-g) \left(c^2 f x+g\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b g \sqrt{1-c^2 x^2} (c f-g)^2 \log (c x+1)}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f+g) (c f-g)^2 \log (c x+1)}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f+g)^2 (c f-g) \log (1-c x)}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b g \sqrt{1-c^2 x^2} (c f+g)^2 \log (1-c x)}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (f+g x) \left(c^2 f^2+2 c^2 f g x+g^2\right)}{6 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(I*b*c*g*(3*c^2*f^2 - 5*g^2)*(1 - c^2*x^2)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], 1] - Sqrt[-c^2]*(-6*a*c^2*f^2*g + 4*a*g^3 - 6*a*c^4*f^3*x - 6*a*c^2*g^3*x^2 + 4*a*c^6*f^3*x^3 - 6*a*c^4*f*g^2*x^3 + b*c^3*f^3*Sqrt[1 - c^2*x^2] + 3*b*c*f*g^2*Sqrt[1 - c^2*x^2] + 3*b*c^3*f^2*g*x*Sqrt[1 - c^2*x^2] + b*c*g^3*x*Sqrt[1 - c^2*x^2] + 2*b*(2*g^3 + 2*c^6*f^3*x^3 - 3*c^2*g*(f^2 + g^2*x^2) - 3*c^4*f*x*(f^2 + g^2*x^2))*ArcSin[c*x] - b*c*f*(2*c^2*f^2 - 3*g^2)*(1 - c^2*x^2)^(3/2)*Log[-1 + c^2*x^2])))/(6*c^4*Sqrt[-c^2]*d^3*(-1 + c^2*x^2)^2)","C",1
55,1,285,271,1.0332862,"\int \frac{(f+g x)^2 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[((f + g*x)^2*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]","\frac{c \sqrt{d-c^2 d x^2} \left(-\sqrt{-c^2} \left(4 a c^5 f^2 x^3-6 a c^3 f^2 x-2 a c^3 g^2 x^3-4 a c f g-b \left(1-c^2 x^2\right)^{3/2} \left(2 c^2 f^2-g^2\right) \log \left(c^2 x^2-1\right)+2 b c \sin ^{-1}(c x) \left(c^2 f^2 x \left(2 c^2 x^2-3\right)-c^2 g^2 x^3-2 f g\right)+b c^2 f^2 \sqrt{1-c^2 x^2}+2 b c^2 f g x \sqrt{1-c^2 x^2}+b g^2 \sqrt{1-c^2 x^2}\right)+2 i b c^2 f g \left(1-c^2 x^2\right)^{3/2} F\left(\left.i \sinh ^{-1}\left(\sqrt{-c^2} x\right)\right|1\right)\right)}{6 \left(-c^2\right)^{5/2} d^3 \left(c^2 x^2-1\right)^2}","\frac{x (f+g x)^2 \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{2 f \left(c^2 f x+g\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{b x \left(x \left(c^2 f^2+g^2\right)+2 f g\right)}{6 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (2 c f-g) (c f+g) \log (1-c x)}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} (c f-g) (2 c f+g) \log (c x+1)}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}",1,"(c*Sqrt[d - c^2*d*x^2]*((2*I)*b*c^2*f*g*(1 - c^2*x^2)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], 1] - Sqrt[-c^2]*(-4*a*c*f*g - 6*a*c^3*f^2*x + 4*a*c^5*f^2*x^3 - 2*a*c^3*g^2*x^3 + b*c^2*f^2*Sqrt[1 - c^2*x^2] + b*g^2*Sqrt[1 - c^2*x^2] + 2*b*c^2*f*g*x*Sqrt[1 - c^2*x^2] + 2*b*c*(-2*f*g - c^2*g^2*x^3 + c^2*f^2*x*(-3 + 2*c^2*x^2))*ArcSin[c*x] - b*(2*c^2*f^2 - g^2)*(1 - c^2*x^2)^(3/2)*Log[-1 + c^2*x^2])))/(6*(-c^2)^(5/2)*d^3*(-1 + c^2*x^2)^2)","C",1
56,1,208,228,0.8305037,"\int \frac{(f+g x) \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]","-\frac{\sqrt{d-c^2 d x^2} \left(\sqrt{-c^2} \left(-4 a c^4 f x^3+6 a c^2 f x+2 a g+2 b \sin ^{-1}(c x) \left(c^2 f x \left(3-2 c^2 x^2\right)+g\right)-b c f \sqrt{1-c^2 x^2}+2 b c f \left(1-c^2 x^2\right)^{3/2} \log \left(c^2 x^2-1\right)-b c g x \sqrt{1-c^2 x^2}\right)+i b c g \left(1-c^2 x^2\right)^{3/2} F\left(\left.i \sinh ^{-1}\left(\sqrt{-c^2} x\right)\right|1\right)\right)}{6 \left(-c^2\right)^{3/2} d^3 \left(c^2 x^2-1\right)^2}","\frac{\left(c^2 f x+g\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{2 f x \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (f+g x)}{6 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{b f \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{b g \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{6 c^2 d^2 \sqrt{d-c^2 d x^2}}",1,"-1/6*(Sqrt[d - c^2*d*x^2]*(I*b*c*g*(1 - c^2*x^2)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], 1] + Sqrt[-c^2]*(2*a*g + 6*a*c^2*f*x - 4*a*c^4*f*x^3 - b*c*f*Sqrt[1 - c^2*x^2] - b*c*g*x*Sqrt[1 - c^2*x^2] + 2*b*(g + c^2*f*x*(3 - 2*c^2*x^2))*ArcSin[c*x] + 2*b*c*f*(1 - c^2*x^2)^(3/2)*Log[-1 + c^2*x^2])))/((-c^2)^(3/2)*d^3*(-1 + c^2*x^2)^2)","C",1
57,1,2078,1300,12.9056487,"\int \frac{a+b \sin ^{-1}(c x)}{(f+g x) \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/((f + g*x)*(d - c^2*d*x^2)^(5/2)),x]","\text{Result too large to show}","-\frac{i \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right) g^4}{d^2 (c f-g)^2 (c f+g)^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}+\frac{i \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right) g^4}{d^2 (c f-g)^2 (c f+g)^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{1-c^2 x^2} \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right) g^4}{d^2 (c f-g)^2 (c f+g)^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right) g^4}{d^2 (c f-g)^2 (c f+g)^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{24 d^2 (c f-g) \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{1-c^2 x^2} \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{24 d^2 (c f-g) \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{1-c^2 x^2} \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{24 d^2 (c f+g) \sqrt{d-c^2 d x^2}}-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{12 d^2 (c f-g) \sqrt{d-c^2 d x^2}}-\frac{(c f-2 g) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{4 d^2 (c f-g)^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f+2 g) \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)\right)}{2 d^2 (c f+g)^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)\right)}{6 d^2 (c f+g) \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)\right)}{6 d^2 (c f-g) \sqrt{d-c^2 d x^2}}+\frac{b (c f-2 g) \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)\right)}{2 d^2 (c f-g)^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{24 d^2 (c f+g) \sqrt{d-c^2 d x^2}}+\frac{(c f+2 g) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{4 d^2 (c f+g)^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{12 d^2 (c f+g) \sqrt{d-c^2 d x^2}}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*((a*g - a*c^2*f*x)/(3*d^3*(-(c^2*f^2) + g^2)*(-1 + c^2*x^2)^2) + (-3*a*g^3 - 2*a*c^4*f^3*x + 5*a*c^2*f*g^2*x)/(3*d^3*(-(c^2*f^2) + g^2)^2*(-1 + c^2*x^2))) + (a*g^4*Log[f + g*x])/(d^(5/2)*(-(c*f) + g)^2*(c*f + g)^2*Sqrt[-(c^2*f^2) + g^2]) - (a*g^4*Log[d*g + c^2*d*f*x + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[-(d*(-1 + c^2*x^2))]])/(d^(5/2)*(-(c*f) + g)^2*(c*f + g)^2*Sqrt[-(c^2*f^2) + g^2]) + (b*((g*(-(c^2*f^2) + 7*g^2)*(1 - c^2*x^2)^(3/2)*ArcSin[c*x])/(6*(-(c^2*f^2) + g^2)^2*(d*(1 - c^2*x^2))^(3/2)) + ((4*c*f + 7*g)*(1 - c^2*x^2)^(3/2)*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])/(6*(c*f + g)^2*(d*(1 - c^2*x^2))^(3/2)) + ((4*c*f - 7*g)*(1 - c^2*x^2)^(3/2)*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])/(6*(c*f - g)^2*(d*(1 - c^2*x^2))^(3/2)) + (g^4*(1 - c^2*x^2)^(3/2)*((Pi*ArcTan[(g + c*f*Tan[ArcSin[c*x]/2])/Sqrt[c^2*f^2 - g^2]])/Sqrt[c^2*f^2 - g^2] + (2*(Pi/2 - ArcSin[c*x])*ArcTanh[((c*f + g)*Cot[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]] - 2*ArcCos[-((c*f)/g)]*ArcTanh[((-(c*f) + g)*Tan[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] - (2*I)*(ArcTanh[((c*f + g)*Cot[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^((I/2)*(Pi/2 - ArcSin[c*x]))*Sqrt[g]*Sqrt[c*f + c*g*x])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^((I/2)*(Pi/2 - ArcSin[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[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[1 - ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))] + (-ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[1 - ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[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[(Pi/2 - ArcSin[c*x])/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[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[(Pi/2 - ArcSin[c*x])/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))]))/Sqrt[-(c^2*f^2) + g^2]))/((-(c*f) + g)^2*(c*f + g)^2*(d*(1 - c^2*x^2))^(3/2)) + ((1 - c^2*x^2)^(3/2)*(-1 + ArcSin[c*x]))/(12*(c*f + g)*(d*(1 - c^2*x^2))^(3/2)*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^2) + ((1 - c^2*x^2)^(3/2)*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(6*(c*f + g)*(d*(1 - c^2*x^2))^(3/2)*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3) + ((1 - c^2*x^2)^(3/2)*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(6*(c*f - g)*(d*(1 - c^2*x^2))^(3/2)*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3) + ((1 - c^2*x^2)^(3/2)*(-1 - ArcSin[c*x]))/(12*(c*f - g)*(d*(1 - c^2*x^2))^(3/2)*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + ((1 - c^2*x^2)^(3/2)*(4*c*f*ArcSin[c*x]*Sin[ArcSin[c*x]/2] - 7*g*ArcSin[c*x]*Sin[ArcSin[c*x]/2]))/(6*(c*f - g)^2*(d*(1 - c^2*x^2))^(3/2)*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) + ((1 - c^2*x^2)^(3/2)*(4*c*f*ArcSin[c*x]*Sin[ArcSin[c*x]/2] + 7*g*ArcSin[c*x]*Sin[ArcSin[c*x]/2]))/(6*(c*f + g)^2*(d*(1 - c^2*x^2))^(3/2)*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]))))/d","A",0
58,1,696,1154,1.1929854,"\int (f+g x)^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(f + g*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2,x]","\frac{\sqrt{d-c^2 d x^2} \left(-\frac{f g^2 \left(-3 b^2 \left(c x \left(2 a c x+b \sqrt{1-c^2 x^2}\right)+b \left(2 c^2 x^2-1\right) \sin ^{-1}(c x)\right)+6 b c x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-2 \left(a+b \sin ^{-1}(c x)\right)^3\right)}{16 b c^3}+\frac{1}{2} f^3 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{b f^3 \left(c x \left(2 a c x+b \sqrt{1-c^2 x^2}\right)+b \left(2 c^2 x^2-1\right) \sin ^{-1}(c x)\right)}{4 c}-\frac{f^2 g \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{c^2}-\frac{2 b f^2 g \left(3 a c x \left(c^2 x^2-3\right)+b \sqrt{1-c^2 x^2} \left(c^2 x^2-7\right)+3 b c x \left(c^2 x^2-3\right) \sin ^{-1}(c x)\right)}{9 c^2}+\frac{3}{4} f g^2 x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{5} g^3 x^4 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2 b g^3 \left(15 a c^5 x^5+15 b c^5 x^5 \sin ^{-1}(c x)+b \sqrt{1-c^2 x^2} \left(3 c^4 x^4+4 c^2 x^2+8\right)\right)}{375 c^4}-\frac{3 b f g^2 \left(8 a c^4 x^4+b \left(8 c^4 x^4-3\right) \sin ^{-1}(c x)+b c x \sqrt{1-c^2 x^2} \left(2 c^2 x^2+3\right)\right)}{64 c^3}-\frac{g^3 \left(9 c^2 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2+18 \left(\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-2 b \left(a c x+b \sqrt{1-c^2 x^2}+b c x \sin ^{-1}(c x)\right)\right)-2 b \left(3 c^3 x^3 \left(a+b \sin ^{-1}(c x)\right)+b \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right)\right)\right)}{135 c^4}+\frac{f^3 \left(a+b \sin ^{-1}(c x)\right)^3}{6 b c}\right)}{\sqrt{1-c^2 x^2}}","-\frac{2 b c g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^5}{25 \sqrt{1-c^2 x^2}}+\frac{1}{5} g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x^4-\frac{3 b c f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^4}{8 \sqrt{1-c^2 x^2}}+\frac{3}{4} f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x^3+\frac{2 b g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^3}{45 c \sqrt{1-c^2 x^2}}-\frac{2 b c f^2 g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^3}{3 \sqrt{1-c^2 x^2}}-\frac{3}{32} b^2 f g^2 \sqrt{d-c^2 d x^2} x^3-\frac{g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x^2}{15 c^2}-\frac{b c f^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^2}{2 \sqrt{1-c^2 x^2}}+\frac{3 b f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^2}{8 c \sqrt{1-c^2 x^2}}+\frac{1}{2} f^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x-\frac{3 f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x}{8 c^2}+\frac{4 b^2 g^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) x}{15 c^3 \sqrt{1-c^2 x^2}}+\frac{2 b f^2 g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x}{c \sqrt{1-c^2 x^2}}-\frac{1}{4} b^2 f^3 \sqrt{d-c^2 d x^2} x+\frac{3 b^2 f g^2 \sqrt{d-c^2 d x^2} x}{64 c^2}+\frac{4 a b g^3 \sqrt{d-c^2 d x^2} x}{15 c^3 \sqrt{1-c^2 x^2}}+\frac{f^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{6 b c \sqrt{1-c^2 x^2}}+\frac{f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{8 b c^3 \sqrt{1-c^2 x^2}}-\frac{2 g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{15 c^4}-\frac{f^2 g \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{c^2}+\frac{b^2 f^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}-\frac{3 b^2 f g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c^3 \sqrt{1-c^2 x^2}}+\frac{52 b^2 g^3 \sqrt{d-c^2 d x^2}}{225 c^4}-\frac{2 b^2 g^3 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{125 c^4}+\frac{4 b^2 f^2 g \sqrt{d-c^2 d x^2}}{3 c^2}+\frac{26 b^2 g^3 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{675 c^4}+\frac{2 b^2 f^2 g \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{9 c^2}",1,"(Sqrt[d - c^2*d*x^2]*((f^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/2 + (3*f*g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/4 + (g^3*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/5 - (f^2*g*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/c^2 + (f^3*(a + b*ArcSin[c*x])^3)/(6*b*c) - (2*b*g^3*(15*a*c^5*x^5 + b*Sqrt[1 - c^2*x^2]*(8 + 4*c^2*x^2 + 3*c^4*x^4) + 15*b*c^5*x^5*ArcSin[c*x]))/(375*c^4) - (2*b*f^2*g*(b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2) + 3*a*c*x*(-3 + c^2*x^2) + 3*b*c*x*(-3 + c^2*x^2)*ArcSin[c*x]))/(9*c^2) - (b*f^3*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x]))/(4*c) - (3*b*f*g^2*(8*a*c^4*x^4 + b*c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2) + b*(-3 + 8*c^4*x^4)*ArcSin[c*x]))/(64*c^3) - (f*g^2*(6*b*c*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*(a + b*ArcSin[c*x])^3 - 3*b^2*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x])))/(16*b*c^3) - (g^3*(9*c^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*(b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + 3*c^3*x^3*(a + b*ArcSin[c*x])) + 18*(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*(a*c*x + b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x]))))/(135*c^4)))/Sqrt[1 - c^2*x^2]","A",1
59,1,441,737,0.9474415,"\int (f+g x)^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(f + g*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2,x]","\frac{\sqrt{d-c^2 d x^2} \left(-\frac{g^2 \left(-3 b^2 \left(c x \left(2 a c x+b \sqrt{1-c^2 x^2}\right)+b \left(2 c^2 x^2-1\right) \sin ^{-1}(c x)\right)+6 b c x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-2 \left(a+b \sin ^{-1}(c x)\right)^3\right)}{48 b c^3}+\frac{1}{2} f^2 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{b f^2 \left(c x \left(2 a c x+b \sqrt{1-c^2 x^2}\right)+b \left(2 c^2 x^2-1\right) \sin ^{-1}(c x)\right)}{4 c}-\frac{2 f g \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^2}-\frac{4 b f g \left(3 a c x \left(c^2 x^2-3\right)+b \sqrt{1-c^2 x^2} \left(c^2 x^2-7\right)+3 b c x \left(c^2 x^2-3\right) \sin ^{-1}(c x)\right)}{27 c^2}+\frac{1}{4} g^2 x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{b g^2 \left(8 a c^4 x^4+b \left(8 c^4 x^4-3\right) \sin ^{-1}(c x)+b c x \sqrt{1-c^2 x^2} \left(2 c^2 x^2+3\right)\right)}{64 c^3}+\frac{f^2 \left(a+b \sin ^{-1}(c x)\right)^3}{6 b c}\right)}{\sqrt{1-c^2 x^2}}","-\frac{b c f^2 x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{1-c^2 x^2}}+\frac{1}{2} f^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{f^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{6 b c \sqrt{1-c^2 x^2}}+\frac{4 b f g x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 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 \sin ^{-1}(c x)\right)^2}{3 c^2}-\frac{4 b c f g x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 \sqrt{1-c^2 x^2}}+\frac{b g^2 x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 c \sqrt{1-c^2 x^2}}-\frac{g^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{8 c^2}-\frac{b c g^2 x^4 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 \sqrt{1-c^2 x^2}}+\frac{1}{4} g^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{24 b c^3 \sqrt{1-c^2 x^2}}-\frac{1}{4} b^2 f^2 x \sqrt{d-c^2 d x^2}+\frac{b^2 f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}+\frac{8 b^2 f g \sqrt{d-c^2 d x^2}}{9 c^2}+\frac{4 b^2 f g \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{27 c^2}+\frac{b^2 g^2 x \sqrt{d-c^2 d x^2}}{64 c^2}-\frac{1}{32} b^2 g^2 x^3 \sqrt{d-c^2 d x^2}-\frac{b^2 g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c^3 \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*((f^2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/2 + (g^2*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/4 - (2*f*g*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*c^2) + (f^2*(a + b*ArcSin[c*x])^3)/(6*b*c) - (4*b*f*g*(b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2) + 3*a*c*x*(-3 + c^2*x^2) + 3*b*c*x*(-3 + c^2*x^2)*ArcSin[c*x]))/(27*c^2) - (b*f^2*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x]))/(4*c) - (b*g^2*(8*a*c^4*x^4 + b*c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2) + b*(-3 + 8*c^4*x^4)*ArcSin[c*x]))/(64*c^3) - (g^2*(6*b*c*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*(a + b*ArcSin[c*x])^3 - 3*b^2*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x])))/(48*b*c^3)))/Sqrt[1 - c^2*x^2]","A",1
60,1,224,396,0.2793135,"\int (f+g x) \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(f + g*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2,x]","\frac{\sqrt{d-c^2 d x^2} \left(-27 b^2 c f \left(c x \left(2 a c x+b \sqrt{1-c^2 x^2}\right)+b \left(2 c^2 x^2-1\right) \sin ^{-1}(c x)\right)+8 b^2 g \left(-3 c^3 x^3 \left(a+b \sin ^{-1}(c x)\right)+9 c x \left(a+b \sin ^{-1}(c x)\right)-b \sqrt{1-c^2 x^2} \left(c^2 x^2-7\right)\right)+54 b c^2 f x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-36 b g \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2+18 c f \left(a+b \sin ^{-1}(c x)\right)^3\right)}{108 b c^2 \sqrt{1-c^2 x^2}}","-\frac{b c f x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{1-c^2 x^2}}+\frac{1}{2} f x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{f \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{6 b c \sqrt{1-c^2 x^2}}+\frac{2 b g x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 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 \sin ^{-1}(c x)\right)^2}{3 c^2}-\frac{2 b c g x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 \sqrt{1-c^2 x^2}}-\frac{1}{4} b^2 f x \sqrt{d-c^2 d x^2}+\frac{b^2 f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}+\frac{4 b^2 g \sqrt{d-c^2 d x^2}}{9 c^2}+\frac{2 b^2 g \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{27 c^2}",1,"(Sqrt[d - c^2*d*x^2]*(54*b*c^2*f*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 36*b*g*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2 + 18*c*f*(a + b*ArcSin[c*x])^3 - 27*b^2*c*f*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x]) + 8*b^2*g*(-(b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2)) + 9*c*x*(a + b*ArcSin[c*x]) - 3*c^3*x^3*(a + b*ArcSin[c*x]))))/(108*b*c^2*Sqrt[1 - c^2*x^2])","A",1
61,1,516,1442,1.446866,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{f+g x} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(f + g*x),x]","\frac{\sqrt{d-c^2 d x^2} \left(3 b c (f+g x) \left(i \sqrt{c^2 f^2-g^2} \left(-2 i b \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)+2 i b \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)+\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right)-\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)+2 b^2 \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)-2 b^2 \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)+g \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-2 b g \left(a c x+b \sqrt{1-c^2 x^2}+b c x \sin ^{-1}(c x)\right)\right)+\left(c^2 f^2-g^2\right) \left(a+b \sin ^{-1}(c x)\right)^3+c^2 g x (f+g x) \left(a+b \sin ^{-1}(c x)\right)^3+g^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^3\right)}{3 b c g^2 \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 \sin ^{-1}(c x)\right)^3}{3 b c (f+g x)}+\frac{c x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b g \sqrt{1-c^2 x^2}}-\frac{\left(1-\frac{c^2 f^2}{g^2}\right) \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c (f+g x) \sqrt{1-c^2 x^2}}+\frac{b^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2}{g}-\frac{2 b^2 c x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g \sqrt{1-c^2 x^2}}+\frac{2 a b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g}-\frac{a^2 \sqrt{c^2 f^2-g^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^2 \sqrt{1-c^2 x^2}}+\frac{i b^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^2 \sqrt{1-c^2 x^2}}+\frac{2 i a b \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left(1-\frac{i e^{i \sin ^{-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^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^2 \sqrt{1-c^2 x^2}}-\frac{2 i a b \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^2 \sqrt{1-c^2 x^2}}+\frac{2 b^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^2 \sqrt{1-c^2 x^2}}+\frac{2 a b \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^2 \sqrt{1-c^2 x^2}}-\frac{2 b^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^2 \sqrt{1-c^2 x^2}}-\frac{2 a b \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^2 \sqrt{1-c^2 x^2}}+\frac{2 i b^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^2 \sqrt{1-c^2 x^2}}-\frac{2 i b^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2} \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^2 \sqrt{1-c^2 x^2}}-\frac{2 a b c x \sqrt{d-c^2 d x^2}}{g \sqrt{1-c^2 x^2}}+\frac{a^2 \sqrt{d-c^2 d x^2}}{g}-\frac{2 b^2 \sqrt{d-c^2 d x^2}}{g}",1,"(Sqrt[d - c^2*d*x^2]*((c^2*f^2 - g^2)*(a + b*ArcSin[c*x])^3 + c^2*g*x*(f + g*x)*(a + b*ArcSin[c*x])^3 + g^2*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^3 + 3*b*c*(f + g*x)*(g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*g*(a*c*x + b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x]) + I*Sqrt[c^2*f^2 - g^2]*((a + b*ArcSin[c*x])^2*Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - (a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] - (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] - 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))))/(3*b*c*g^2*(f + g*x)*Sqrt[1 - c^2*x^2])","A",0
62,1,872,1685,2.5895403,"\int (f+g x)^3 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(f + g*x)^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2,x]","\frac{d \sqrt{d-c^2 d x^2} \left(3087000 c f \left(2 c^2 f^2+g^2\right) a^3-88200 b \sqrt{1-c^2 x^2} \left(4 x^3 \left(35 f^3+84 g x f^2+70 g^2 x^2 f+20 g^3 x^3\right) c^6-2 x \left(175 f^3+336 g x f^2+245 g^2 x^2 f+64 g^3 x^3\right) c^4+g \left(336 f^2+105 g x f+16 g^2 x^2\right) c^2+32 g^3\right) a^2+840 b^2 c x \left(2 x^3 \left(3675 f^3+7056 g x f^2+4900 g^2 x^2 f+1200 g^3 x^3\right) c^6-21 x \left(1750 f^3+2240 g x f^2+1225 g^2 x^2 f+256 g^3 x^3\right) c^4+35 g \left(2016 f^2+315 g x f+32 g^2 x^2\right) c^2+6720 g^3\right) a+3087000 b^3 c f \left(2 c^2 f^2+g^2\right) \sin ^{-1}(c x)^3-88200 b^2 \left(b \sqrt{1-c^2 x^2} \left(4 x^3 \left(35 f^3+84 g x f^2+70 g^2 x^2 f+20 g^3 x^3\right) c^6-2 x \left(175 f^3+336 g x f^2+245 g^2 x^2 f+64 g^3 x^3\right) c^4+g \left(336 f^2+105 g x f+16 g^2 x^2\right) c^2+32 g^3\right)-105 a c f \left(2 c^2 f^2+g^2\right)\right) \sin ^{-1}(c x)^2+b^3 \sqrt{1-c^2 x^2} \left(4 x^3 \left(385875 f^3+592704 g x f^2+343000 g^2 x^2 f+72000 g^3 x^3\right) c^6-2 x \left(6559875 f^3+5005056 g x f^2+1843625 g^2 x^2 f+278784 g^3 x^3\right) c^4+g \left(39250176 f^2-900375 g x f-429824 g^2 x^2\right) c^2+4785152 g^3\right)+105 b \left(88200 c f \left(2 c^2 f^2+g^2\right) a^2-1680 b \sqrt{1-c^2 x^2} \left(4 x^3 \left(35 f^3+84 g x f^2+70 g^2 x^2 f+20 g^3 x^3\right) c^6-2 x \left(175 f^3+336 g x f^2+245 g^2 x^2 f+64 g^3 x^3\right) c^4+g \left(336 f^2+105 g x f+16 g^2 x^2\right) c^2+32 g^3\right) a+b^2 c \left(16 x^4 \left(3675 f^3+7056 g x f^2+4900 g^2 x^2 f+1200 g^3 x^3\right) c^6-168 x^2 \left(1750 f^3+2240 g x f^2+1225 g^2 x^2 f+256 g^3 x^3\right) c^4+70 \left(1785 f^3+8064 g x f^2+1260 g^2 x^2 f+128 g^3 x^3\right) c^2+35 g^2 (245 f+1536 g x)\right)\right) \sin ^{-1}(c x)\right)}{49392000 b c^4 \sqrt{1-c^2 x^2}}","\frac{2 b c^3 d g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^7}{49 \sqrt{1-c^2 x^2}}+\frac{b c^3 d f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^6}{6 \sqrt{1-c^2 x^2}}-\frac{16 b c d g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^5}{175 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d f^2 g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^5}{25 \sqrt{1-c^2 x^2}}+\frac{1}{36} b^2 c^2 d f g^2 \sqrt{d-c^2 d x^2} x^5+\frac{3}{35} d g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x^4+\frac{1}{7} d g^3 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x^4-\frac{7 b c d f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^4}{16 \sqrt{1-c^2 x^2}}+\frac{3}{8} d f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 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 \sin ^{-1}(c x)\right)^2 x^3+\frac{2 b d g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^3}{105 c \sqrt{1-c^2 x^2}}-\frac{4 b c d f^2 g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^3}{5 \sqrt{1-c^2 x^2}}-\frac{43}{576} b^2 d f g^2 \sqrt{d-c^2 d x^2} x^3-\frac{d g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x^2}{35 c^2}-\frac{3 b c d f^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^2}{8 \sqrt{1-c^2 x^2}}+\frac{3 b d f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^2}{16 c \sqrt{1-c^2 x^2}}+\frac{3}{8} d f^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x-\frac{3 d f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 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 \sin ^{-1}(c x)\right)^2 x+\frac{4 b^2 d g^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) x}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{6 b d f^2 g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x}{5 c \sqrt{1-c^2 x^2}}-\frac{15}{64} b^2 d f^3 \sqrt{d-c^2 d x^2} x-\frac{7 b^2 d f g^2 \sqrt{d-c^2 d x^2} x}{384 c^2}-\frac{1}{32} b^2 d f^3 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} x+\frac{4 a b d g^3 \sqrt{d-c^2 d x^2} x}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{d f^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{8 b c \sqrt{1-c^2 x^2}}+\frac{d f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{16 b c^3 \sqrt{1-c^2 x^2}}-\frac{2 d g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{35 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 \sin ^{-1}(c x)\right)^2}{5 c^2}+\frac{9 b^2 d f^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}+\frac{7 b^2 d f g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{384 c^3 \sqrt{1-c^2 x^2}}+\frac{b d f^3 \left(1-c^2 x^2\right)^{3/2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 c}+\frac{304 b^2 d g^3 \sqrt{d-c^2 d x^2}}{3675 c^4}-\frac{2 b^2 d g^3 \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2}}{343 c^4}+\frac{38 b^2 d g^3 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{6125 c^4}+\frac{6 b^2 d f^2 g \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{16 b^2 d f^2 g \sqrt{d-c^2 d x^2}}{25 c^2}+\frac{152 b^2 d g^3 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{11025 c^4}+\frac{8 b^2 d f^2 g \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{75 c^2}",1,"(d*Sqrt[d - c^2*d*x^2]*(3087000*a^3*c*f*(2*c^2*f^2 + g^2) - 88200*a^2*b*Sqrt[1 - c^2*x^2]*(32*g^3 + c^2*g*(336*f^2 + 105*f*g*x + 16*g^2*x^2) + 4*c^6*x^3*(35*f^3 + 84*f^2*g*x + 70*f*g^2*x^2 + 20*g^3*x^3) - 2*c^4*x*(175*f^3 + 336*f^2*g*x + 245*f*g^2*x^2 + 64*g^3*x^3)) + 840*a*b^2*c*x*(6720*g^3 + 35*c^2*g*(2016*f^2 + 315*f*g*x + 32*g^2*x^2) - 21*c^4*x*(1750*f^3 + 2240*f^2*g*x + 1225*f*g^2*x^2 + 256*g^3*x^3) + 2*c^6*x^3*(3675*f^3 + 7056*f^2*g*x + 4900*f*g^2*x^2 + 1200*g^3*x^3)) + b^3*Sqrt[1 - c^2*x^2]*(4785152*g^3 + c^2*g*(39250176*f^2 - 900375*f*g*x - 429824*g^2*x^2) + 4*c^6*x^3*(385875*f^3 + 592704*f^2*g*x + 343000*f*g^2*x^2 + 72000*g^3*x^3) - 2*c^4*x*(6559875*f^3 + 5005056*f^2*g*x + 1843625*f*g^2*x^2 + 278784*g^3*x^3)) + 105*b*(88200*a^2*c*f*(2*c^2*f^2 + g^2) - 1680*a*b*Sqrt[1 - c^2*x^2]*(32*g^3 + c^2*g*(336*f^2 + 105*f*g*x + 16*g^2*x^2) + 4*c^6*x^3*(35*f^3 + 84*f^2*g*x + 70*f*g^2*x^2 + 20*g^3*x^3) - 2*c^4*x*(175*f^3 + 336*f^2*g*x + 245*f*g^2*x^2 + 64*g^3*x^3)) + b^2*c*(35*g^2*(245*f + 1536*g*x) + 70*c^2*(1785*f^3 + 8064*f^2*g*x + 1260*f*g^2*x^2 + 128*g^3*x^3) - 168*c^4*x^2*(1750*f^3 + 2240*f^2*g*x + 1225*f*g^2*x^2 + 256*g^3*x^3) + 16*c^6*x^4*(3675*f^3 + 7056*f^2*g*x + 4900*f*g^2*x^2 + 1200*g^3*x^3)))*ArcSin[c*x] - 88200*b^2*(-105*a*c*f*(2*c^2*f^2 + g^2) + b*Sqrt[1 - c^2*x^2]*(32*g^3 + c^2*g*(336*f^2 + 105*f*g*x + 16*g^2*x^2) + 4*c^6*x^3*(35*f^3 + 84*f^2*g*x + 70*f*g^2*x^2 + 20*g^3*x^3) - 2*c^4*x*(175*f^3 + 336*f^2*g*x + 245*f*g^2*x^2 + 64*g^3*x^3)))*ArcSin[c*x]^2 + 3087000*b^3*c*f*(2*c^2*f^2 + g^2)*ArcSin[c*x]^3))/(49392000*b*c^4*Sqrt[1 - c^2*x^2])","A",1
63,1,616,1108,1.1146954,"\int (f+g x)^2 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(f + g*x)^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2,x]","\frac{d \sqrt{d-c^2 d x^2} \left(9000 a^3 \left(6 c^2 f^2+g^2\right)+15 b \sin ^{-1}(c x) \left(1800 a^2 \left(6 c^2 f^2+g^2\right)-240 a b c \sqrt{1-c^2 x^2} \left(30 c^2 f^2 x \left(2 c^2 x^2-5\right)+96 f g \left(c^2 x^2-1\right)^2+5 g^2 x \left(8 c^4 x^4-14 c^2 x^2+3\right)\right)+b^2 \left(16 c^6 x^4 \left(225 f^2+288 f g x+100 g^2 x^2\right)-120 c^4 x^2 \left(150 f^2+128 f g x+35 g^2 x^2\right)+90 c^2 \left(85 f^2+256 f g x+20 g^2 x^2\right)+175 g^2\right)\right)-1800 a^2 b c \sqrt{1-c^2 x^2} \left(30 c^2 f^2 x \left(2 c^2 x^2-5\right)+96 f g \left(c^2 x^2-1\right)^2+5 g^2 x \left(8 c^4 x^4-14 c^2 x^2+3\right)\right)+120 a b^2 c^2 x \left(450 c^2 f^2 x \left(c^2 x^2-5\right)+192 f g \left(3 c^4 x^4-10 c^2 x^2+15\right)+25 g^2 x \left(8 c^4 x^4-21 c^2 x^2+9\right)\right)+1800 b^2 \sin ^{-1}(c x)^2 \left(15 a \left(6 c^2 f^2+g^2\right)-b c \sqrt{1-c^2 x^2} \left(30 c^2 f^2 x \left(2 c^2 x^2-5\right)+96 f g \left(c^2 x^2-1\right)^2+5 g^2 x \left(8 c^4 x^4-14 c^2 x^2+3\right)\right)\right)+9000 b^3 \left(6 c^2 f^2+g^2\right) \sin ^{-1}(c x)^3+b^3 c \sqrt{1-c^2 x^2} \left(6750 c^2 f^2 x \left(2 c^2 x^2-17\right)+1536 f g \left(9 c^4 x^4-38 c^2 x^2+149\right)+125 g^2 x \left(32 c^4 x^4-86 c^2 x^2-21\right)\right)\right)}{432000 b c^3 \sqrt{1-c^2 x^2}}","\frac{b c^3 d g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^6}{18 \sqrt{1-c^2 x^2}}+\frac{4 b c^3 d f g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^5}{25 \sqrt{1-c^2 x^2}}+\frac{1}{108} b^2 c^2 d g^2 \sqrt{d-c^2 d x^2} x^5-\frac{7 b c d g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^4}{48 \sqrt{1-c^2 x^2}}+\frac{1}{8} d g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x^3+\frac{1}{6} d g^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x^3-\frac{8 b c d f g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^3}{15 \sqrt{1-c^2 x^2}}-\frac{43 b^2 d g^2 \sqrt{d-c^2 d x^2} x^3}{1728}-\frac{3 b c d f^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^2}{8 \sqrt{1-c^2 x^2}}+\frac{b d g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^2}{16 c \sqrt{1-c^2 x^2}}+\frac{3}{8} d f^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x-\frac{d g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x}{16 c^2}+\frac{1}{4} d f^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x+\frac{4 b d f g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x}{5 c \sqrt{1-c^2 x^2}}-\frac{15}{64} b^2 d f^2 \sqrt{d-c^2 d x^2} x-\frac{7 b^2 d g^2 \sqrt{d-c^2 d x^2} x}{1152 c^2}-\frac{1}{32} b^2 d f^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} x+\frac{d f^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{8 b c \sqrt{1-c^2 x^2}}+\frac{d g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{48 b c^3 \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 \sin ^{-1}(c x)\right)^2}{5 c^2}+\frac{9 b^2 d f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}+\frac{7 b^2 d g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c^3 \sqrt{1-c^2 x^2}}+\frac{b d f^2 \left(1-c^2 x^2\right)^{3/2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 c}+\frac{4 b^2 d f g \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{32 b^2 d f g \sqrt{d-c^2 d x^2}}{75 c^2}+\frac{16 b^2 d f g \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{225 c^2}",1,"(d*Sqrt[d - c^2*d*x^2]*(9000*a^3*(6*c^2*f^2 + g^2) + 120*a*b^2*c^2*x*(450*c^2*f^2*x*(-5 + c^2*x^2) + 192*f*g*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 25*g^2*x*(9 - 21*c^2*x^2 + 8*c^4*x^4)) - 1800*a^2*b*c*Sqrt[1 - c^2*x^2]*(96*f*g*(-1 + c^2*x^2)^2 + 30*c^2*f^2*x*(-5 + 2*c^2*x^2) + 5*g^2*x*(3 - 14*c^2*x^2 + 8*c^4*x^4)) + b^3*c*Sqrt[1 - c^2*x^2]*(6750*c^2*f^2*x*(-17 + 2*c^2*x^2) + 1536*f*g*(149 - 38*c^2*x^2 + 9*c^4*x^4) + 125*g^2*x*(-21 - 86*c^2*x^2 + 32*c^4*x^4)) + 15*b*(1800*a^2*(6*c^2*f^2 + g^2) + b^2*(175*g^2 + 90*c^2*(85*f^2 + 256*f*g*x + 20*g^2*x^2) - 120*c^4*x^2*(150*f^2 + 128*f*g*x + 35*g^2*x^2) + 16*c^6*x^4*(225*f^2 + 288*f*g*x + 100*g^2*x^2)) - 240*a*b*c*Sqrt[1 - c^2*x^2]*(96*f*g*(-1 + c^2*x^2)^2 + 30*c^2*f^2*x*(-5 + 2*c^2*x^2) + 5*g^2*x*(3 - 14*c^2*x^2 + 8*c^4*x^4)))*ArcSin[c*x] + 1800*b^2*(15*a*(6*c^2*f^2 + g^2) - b*c*Sqrt[1 - c^2*x^2]*(96*f*g*(-1 + c^2*x^2)^2 + 30*c^2*f^2*x*(-5 + 2*c^2*x^2) + 5*g^2*x*(3 - 14*c^2*x^2 + 8*c^4*x^4)))*ArcSin[c*x]^2 + 9000*b^3*(6*c^2*f^2 + g^2)*ArcSin[c*x]^3))/(432000*b*c^3*Sqrt[1 - c^2*x^2])","A",1
64,1,395,621,0.6900985,"\int (f+g x) \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(f + g*x)*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2,x]","\frac{d \sqrt{d-c^2 d x^2} \left(9000 a^3 c f+15 b \sin ^{-1}(c x) \left(1800 a^2 c f-240 a b \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)+b^2 c \left(75 f \left(8 c^4 x^4-40 c^2 x^2+17\right)+128 g x \left(3 c^4 x^4-10 c^2 x^2+15\right)\right)\right)-1800 a^2 b \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)+1800 b^2 \sin ^{-1}(c x)^2 \left(15 a c f+b \sqrt{1-c^2 x^2} \left(5 c^2 f x \left(5-2 c^2 x^2\right)-8 g \left(c^2 x^2-1\right)^2\right)\right)+120 a b^2 c x \left(75 c^2 f x \left(c^2 x^2-5\right)+16 g \left(3 c^4 x^4-10 c^2 x^2+15\right)\right)+b^3 \sqrt{1-c^2 x^2} \left(1125 c^2 f x \left(2 c^2 x^2-17\right)+128 g \left(9 c^4 x^4-38 c^2 x^2+149\right)\right)+9000 b^3 c f \sin ^{-1}(c x)^3\right)}{72000 b c^2 \sqrt{1-c^2 x^2}}","-\frac{3 b c d f x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 \sqrt{1-c^2 x^2}}+\frac{3}{8} d f x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{4} d f x \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{d f \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{8 b c \sqrt{1-c^2 x^2}}+\frac{b d f \left(1-c^2 x^2\right)^{3/2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 c}+\frac{2 b d g x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{5 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 \sin ^{-1}(c x)\right)^2}{5 c^2}-\frac{4 b c d g x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d g x^5 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{25 \sqrt{1-c^2 x^2}}-\frac{15}{64} b^2 d f x \sqrt{d-c^2 d x^2}-\frac{1}{32} b^2 d f x \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}+\frac{9 b^2 d f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}+\frac{2 b^2 d g \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{16 b^2 d g \sqrt{d-c^2 d x^2}}{75 c^2}+\frac{8 b^2 d g \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{225 c^2}",1,"(d*Sqrt[d - c^2*d*x^2]*(9000*a^3*c*f - 1800*a^2*b*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)) + 120*a*b^2*c*x*(75*c^2*f*x*(-5 + c^2*x^2) + 16*g*(15 - 10*c^2*x^2 + 3*c^4*x^4)) + b^3*Sqrt[1 - c^2*x^2]*(1125*c^2*f*x*(-17 + 2*c^2*x^2) + 128*g*(149 - 38*c^2*x^2 + 9*c^4*x^4)) + 15*b*(1800*a^2*c*f - 240*a*b*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)) + b^2*c*(128*g*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 75*f*(17 - 40*c^2*x^2 + 8*c^4*x^4)))*ArcSin[c*x] + 1800*b^2*(15*a*c*f + b*Sqrt[1 - c^2*x^2]*(5*c^2*f*x*(5 - 2*c^2*x^2) - 8*g*(-1 + c^2*x^2)^2))*ArcSin[c*x]^2 + 9000*b^3*c*f*ArcSin[c*x]^3))/(72000*b*c^2*Sqrt[1 - c^2*x^2])","A",1
65,1,740,1992,2.5573723,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{f+g x} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(f + g*x),x]","\frac{d \sqrt{d-c^2 d x^2} \left(-\frac{36 \left(c^2 f^2-g^2\right) \left(3 b c (f+g x) \left(i \sqrt{c^2 f^2-g^2} \left(-2 i b \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)+2 i b \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)+\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right)-\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)+2 b^2 \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)-2 b^2 \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)+g \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-2 b g \left(a c x+b \sqrt{1-c^2 x^2}+b c x \sin ^{-1}(c x)\right)\right)+\left(c^2 f^2-g^2\right) \left(a+b \sin ^{-1}(c x)\right)^3+c^2 g x (f+g x) \left(a+b \sin ^{-1}(c x)\right)^3\right)}{b c g^2 (f+g x)}+\frac{36 \left(c^2 x^2-1\right) \left(c^2 f^2-g^2\right) \left(a+b \sin ^{-1}(c x)\right)^3}{b c (f+g x)}+54 c^2 f x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-27 b c f \left(c x \left(2 a c x+b \sqrt{1-c^2 x^2}\right)+b \left(2 c^2 x^2-1\right) \sin ^{-1}(c x)\right)+36 g \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2-8 b g \left(-3 c^3 x^3 \left(a+b \sin ^{-1}(c x)\right)+9 c x \left(a+b \sin ^{-1}(c x)\right)-b \sqrt{1-c^2 x^2} \left(c^2 x^2-7\right)\right)+\frac{18 c f \left(a+b \sin ^{-1}(c x)\right)^3}{b}\right)}{108 g^2 \sqrt{1-c^2 x^2}}","\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) c^3}{9 g \sqrt{1-c^2 x^2}}-\frac{b d f x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) c^3}{2 g^2 \sqrt{1-c^2 x^2}}+\frac{d f x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 c^2}{2 g^2}-\frac{b^2 d f x \sqrt{d-c^2 d x^2} c^2}{4 g^2}-\frac{d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3 c}{3 b g^3 \sqrt{1-c^2 x^2}}+\frac{d f \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3 c}{6 b g^2 \sqrt{1-c^2 x^2}}+\frac{2 b^2 d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) c}{g^3 \sqrt{1-c^2 x^2}}+\frac{b^2 d f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) c}{4 g^2 \sqrt{1-c^2 x^2}}-\frac{2 b d x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) c}{3 g \sqrt{1-c^2 x^2}}+\frac{2 a 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^2 d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^3}+\frac{d \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 g}-\frac{2 a b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^3}+\frac{a^2 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^2 d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^4 \sqrt{1-c^2 x^2}}-\frac{2 i a b d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^4 \sqrt{1-c^2 x^2}}+\frac{i b^2 d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^4 \sqrt{1-c^2 x^2}}+\frac{2 i a b d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^4 \sqrt{1-c^2 x^2}}-\frac{2 b^2 d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^4 \sqrt{1-c^2 x^2}}-\frac{2 a b d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^4 \sqrt{1-c^2 x^2}}+\frac{2 b^2 d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^4 \sqrt{1-c^2 x^2}}+\frac{2 a b d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^4 \sqrt{1-c^2 x^2}}-\frac{2 i b^2 d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^4 \sqrt{1-c^2 x^2}}+\frac{2 i b^2 d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2} \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^4 \sqrt{1-c^2 x^2}}-\frac{a^2 d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}}{g^3}+\frac{2 b^2 d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}}{g^3}-\frac{2 b^2 d \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{27 g}-\frac{4 b^2 d \sqrt{d-c^2 d x^2}}{9 g}-\frac{d (c f-g) (c f+g) \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 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 \sin ^{-1}(c x)\right)^3}{3 b g^4 (f+g x) \sqrt{1-c^2 x^2} c}",1,"(d*Sqrt[d - c^2*d*x^2]*(54*c^2*f*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 + 36*g*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2 + (18*c*f*(a + b*ArcSin[c*x])^3)/b + (36*(c^2*f^2 - g^2)*(-1 + c^2*x^2)*(a + b*ArcSin[c*x])^3)/(b*c*(f + g*x)) - 27*b*c*f*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x]) - 8*b*g*(-(b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2)) + 9*c*x*(a + b*ArcSin[c*x]) - 3*c^3*x^3*(a + b*ArcSin[c*x])) - (36*(c^2*f^2 - g^2)*((c^2*f^2 - g^2)*(a + b*ArcSin[c*x])^3 + c^2*g*x*(f + g*x)*(a + b*ArcSin[c*x])^3 + 3*b*c*(f + g*x)*(g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*g*(a*c*x + b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x]) + I*Sqrt[c^2*f^2 - g^2]*((a + b*ArcSin[c*x])^2*Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - (a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] - (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] - 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))))/(b*c*g^2*(f + g*x))))/(108*g^2*Sqrt[1 - c^2*x^2])","A",0
66,1,1114,2290,1.9922968,"\int (f+g x)^3 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(f + g*x)^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(333396000 \left(8 c^3 f^3+3 c g^2 f\right) a^3+3175200 b \sqrt{1-c^2 x^2} \left(16 x^5 \left(84 f^3+216 g x f^2+189 g^2 x^2 f+56 g^3 x^3\right) c^8-8 x^3 \left(546 f^3+1296 g x f^2+1071 g^2 x^2 f+304 g^3 x^3\right) c^6+6 x \left(924 f^3+1728 g x f^2+1239 g^2 x^2 f+320 g^3 x^3\right) c^4-g \left(3456 f^2+945 g x f+128 g^2 x^2\right) c^2-256 g^3\right) a^2-10080 b^2 c x \left(20 x^5 \left(7056 f^3+15552 g x f^2+11907 g^2 x^2 f+3136 g^3 x^3\right) c^8-72 x^3 \left(9555 f^3+18144 g x f^2+12495 g^2 x^2 f+3040 g^3 x^3\right) c^6+945 x \left(1848 f^3+2304 g x f^2+1239 g^2 x^2 f+256 g^3 x^3\right) c^4-105 g \left(20736 f^2+2835 g x f+256 g^2 x^2\right) c^2-161280 g^3\right) a+333396000 b^3 c f \left(8 c^2 f^2+3 g^2\right) \sin ^{-1}(c x)^3+3175200 b^2 \left(315 a \left(8 c^3 f^3+3 c g^2 f\right)+b \sqrt{1-c^2 x^2} \left(16 x^5 \left(84 f^3+216 g x f^2+189 g^2 x^2 f+56 g^3 x^3\right) c^8-8 x^3 \left(546 f^3+1296 g x f^2+1071 g^2 x^2 f+304 g^3 x^3\right) c^6+6 x \left(924 f^3+1728 g x f^2+1239 g^2 x^2 f+320 g^3 x^3\right) c^4-g \left(3456 f^2+945 g x f+128 g^2 x^2\right) c^2-256 g^3\right)\right) \sin ^{-1}(c x)^2-b^3 \sqrt{1-c^2 x^2} \left(400 x^5 \left(592704 f^3+1119744 g x f^2+750141 g^2 x^2 f+175616 g^3 x^3\right) c^8-8 x^3 \left(179663400 f^3+262020096 g x f^2+145166175 g^2 x^2 f+29363200 g^3 x^3\right) c^6+6 x \left(1107615600 f^3+753463296 g x f^2+249815475 g^2 x^2 f+34304000 g^3 x^3\right) c^4+g \left(-12905422848 f^2+748057275 g x f+184115200 g^2 x^2\right) c^2-1257472000 g^3\right)+315 b \left(3175200 \left(8 c^3 f^3+3 c g^2 f\right) a^2+20160 b \sqrt{1-c^2 x^2} \left(16 x^5 \left(84 f^3+216 g x f^2+189 g^2 x^2 f+56 g^3 x^3\right) c^8-8 x^3 \left(546 f^3+1296 g x f^2+1071 g^2 x^2 f+304 g^3 x^3\right) c^6+6 x \left(924 f^3+1728 g x f^2+1239 g^2 x^2 f+320 g^3 x^3\right) c^4-g \left(3456 f^2+945 g x f+128 g^2 x^2\right) c^2-256 g^3\right) a+b^2 c \left(-640 x^6 \left(7056 f^3+15552 g x f^2+11907 g^2 x^2 f+3136 g^3 x^3\right) c^8+2304 x^4 \left(9555 f^3+18144 g x f^2+12495 g^2 x^2 f+3040 g^3 x^3\right) c^6-30240 x^2 \left(1848 f^3+2304 g x f^2+1239 g^2 x^2 f+256 g^3 x^3\right) c^4+3360 \left(6279 f^3+20736 g x f^2+2835 g^2 x^2 f+256 g^3 x^3\right) c^2+315 g^2 (7539 f+16384 g x)\right)\right) \sin ^{-1}(c x)\right)}{25604812800 b c^4 \sqrt{1-c^2 x^2}}","-\frac{2 b c^5 d^2 g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) 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} \left(a+b \sin ^{-1}(c x)\right) x^8}{32 \sqrt{1-c^2 x^2}}+\frac{38 b c^3 d^2 g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^7}{441 \sqrt{1-c^2 x^2}}-\frac{6 b c^5 d^2 f^2 g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^7}{49 \sqrt{1-c^2 x^2}}-\frac{3}{256} b^2 c^4 d^2 f g^2 \sqrt{d-c^2 d x^2} x^7+\frac{17 b c^3 d^2 f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^6}{48 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^5}{21 \sqrt{1-c^2 x^2}}+\frac{18 b c^3 d^2 f^2 g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^5}{35 \sqrt{1-c^2 x^2}}+\frac{209 b^2 c^2 d^2 f g^2 \sqrt{d-c^2 d x^2} x^5}{4608}+\frac{1}{21} d^2 g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x^4+\frac{1}{9} d^2 g^3 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x^4+\frac{5}{63} d^2 g^3 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x^4-\frac{59 b c d^2 f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^4}{128 \sqrt{1-c^2 x^2}}+\frac{15}{64} d^2 f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 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 \sin ^{-1}(c x)\right)^2 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 \sin ^{-1}(c x)\right)^2 x^3+\frac{2 b d^2 g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^3}{189 c \sqrt{1-c^2 x^2}}-\frac{6 b c d^2 f^2 g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^3}{7 \sqrt{1-c^2 x^2}}-\frac{1079 b^2 d^2 f g^2 \sqrt{d-c^2 d x^2} x^3}{18432}-\frac{d^2 g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x^2}{63 c^2}-\frac{5 b c d^2 f^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^2}{16 \sqrt{1-c^2 x^2}}+\frac{15 b d^2 f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^2}{128 c \sqrt{1-c^2 x^2}}+\frac{5}{16} d^2 f^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x-\frac{15 d^2 f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 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 \sin ^{-1}(c x)\right)^2 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 \sin ^{-1}(c x)\right)^2 x+\frac{4 b^2 d^2 g^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) x}{63 c^3 \sqrt{1-c^2 x^2}}+\frac{6 b d^2 f^2 g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x}{7 c \sqrt{1-c^2 x^2}}-\frac{245 b^2 d^2 f^3 \sqrt{d-c^2 d x^2} x}{1152}-\frac{359 b^2 d^2 f g^2 \sqrt{d-c^2 d x^2} x}{12288 c^2}-\frac{1}{108} b^2 d^2 f^3 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} x-\frac{65 b^2 d^2 f^3 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} x}{1728}+\frac{4 a b d^2 g^3 \sqrt{d-c^2 d x^2} x}{63 c^3 \sqrt{1-c^2 x^2}}+\frac{5 d^2 f^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{48 b c \sqrt{1-c^2 x^2}}+\frac{5 d^2 f g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{128 b c^3 \sqrt{1-c^2 x^2}}-\frac{2 d^2 g^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{63 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 \sin ^{-1}(c x)\right)^2}{7 c^2}+\frac{115 b^2 d^2 f^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt{1-c^2 x^2}}+\frac{359 b^2 d^2 f g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{12288 c^3 \sqrt{1-c^2 x^2}}+\frac{b d^2 f^3 \left(1-c^2 x^2\right)^{5/2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{18 c}+\frac{5 b d^2 f^3 \left(1-c^2 x^2\right)^{3/2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{48 c}-\frac{2 b^2 d^2 g^3 \left(1-c^2 x^2\right)^4 \sqrt{d-c^2 d x^2}}{729 c^4}+\frac{160 b^2 d^2 g^3 \sqrt{d-c^2 d x^2}}{3969 c^4}+\frac{50 b^2 d^2 g^3 \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2}}{27783 c^4}+\frac{6 b^2 d^2 f^2 g \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2}}{343 c^2}+\frac{4 b^2 d^2 g^3 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{1323 c^4}+\frac{36 b^2 d^2 f^2 g \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{1225 c^2}+\frac{96 b^2 d^2 f^2 g \sqrt{d-c^2 d x^2}}{245 c^2}+\frac{80 b^2 d^2 g^3 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{11907 c^4}+\frac{16 b^2 d^2 f^2 g \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{245 c^2}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(333396000*a^3*(8*c^3*f^3 + 3*c*f*g^2) + 3175200*a^2*b*Sqrt[1 - c^2*x^2]*(-256*g^3 - c^2*g*(3456*f^2 + 945*f*g*x + 128*g^2*x^2) + 16*c^8*x^5*(84*f^3 + 216*f^2*g*x + 189*f*g^2*x^2 + 56*g^3*x^3) - 8*c^6*x^3*(546*f^3 + 1296*f^2*g*x + 1071*f*g^2*x^2 + 304*g^3*x^3) + 6*c^4*x*(924*f^3 + 1728*f^2*g*x + 1239*f*g^2*x^2 + 320*g^3*x^3)) - 10080*a*b^2*c*x*(-161280*g^3 - 105*c^2*g*(20736*f^2 + 2835*f*g*x + 256*g^2*x^2) + 945*c^4*x*(1848*f^3 + 2304*f^2*g*x + 1239*f*g^2*x^2 + 256*g^3*x^3) - 72*c^6*x^3*(9555*f^3 + 18144*f^2*g*x + 12495*f*g^2*x^2 + 3040*g^3*x^3) + 20*c^8*x^5*(7056*f^3 + 15552*f^2*g*x + 11907*f*g^2*x^2 + 3136*g^3*x^3)) - b^3*Sqrt[1 - c^2*x^2]*(-1257472000*g^3 + c^2*g*(-12905422848*f^2 + 748057275*f*g*x + 184115200*g^2*x^2) + 400*c^8*x^5*(592704*f^3 + 1119744*f^2*g*x + 750141*f*g^2*x^2 + 175616*g^3*x^3) - 8*c^6*x^3*(179663400*f^3 + 262020096*f^2*g*x + 145166175*f*g^2*x^2 + 29363200*g^3*x^3) + 6*c^4*x*(1107615600*f^3 + 753463296*f^2*g*x + 249815475*f*g^2*x^2 + 34304000*g^3*x^3)) + 315*b*(3175200*a^2*(8*c^3*f^3 + 3*c*f*g^2) + 20160*a*b*Sqrt[1 - c^2*x^2]*(-256*g^3 - c^2*g*(3456*f^2 + 945*f*g*x + 128*g^2*x^2) + 16*c^8*x^5*(84*f^3 + 216*f^2*g*x + 189*f*g^2*x^2 + 56*g^3*x^3) - 8*c^6*x^3*(546*f^3 + 1296*f^2*g*x + 1071*f*g^2*x^2 + 304*g^3*x^3) + 6*c^4*x*(924*f^3 + 1728*f^2*g*x + 1239*f*g^2*x^2 + 320*g^3*x^3)) + b^2*c*(315*g^2*(7539*f + 16384*g*x) - 30240*c^4*x^2*(1848*f^3 + 2304*f^2*g*x + 1239*f*g^2*x^2 + 256*g^3*x^3) + 3360*c^2*(6279*f^3 + 20736*f^2*g*x + 2835*f*g^2*x^2 + 256*g^3*x^3) + 2304*c^6*x^4*(9555*f^3 + 18144*f^2*g*x + 12495*f*g^2*x^2 + 3040*g^3*x^3) - 640*c^8*x^6*(7056*f^3 + 15552*f^2*g*x + 11907*f*g^2*x^2 + 3136*g^3*x^3)))*ArcSin[c*x] + 3175200*b^2*(315*a*(8*c^3*f^3 + 3*c*f*g^2) + b*Sqrt[1 - c^2*x^2]*(-256*g^3 - c^2*g*(3456*f^2 + 945*f*g*x + 128*g^2*x^2) + 16*c^8*x^5*(84*f^3 + 216*f^2*g*x + 189*f*g^2*x^2 + 56*g^3*x^3) - 8*c^6*x^3*(546*f^3 + 1296*f^2*g*x + 1071*f*g^2*x^2 + 304*g^3*x^3) + 6*c^4*x*(924*f^3 + 1728*f^2*g*x + 1239*f*g^2*x^2 + 320*g^3*x^3)))*ArcSin[c*x]^2 + 333396000*b^3*c*f*(8*c^2*f^2 + 3*g^2)*ArcSin[c*x]^3))/(25604812800*b*c^4*Sqrt[1 - c^2*x^2])","A",1
67,1,742,1533,1.4985707,"\int (f+g x)^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(f + g*x)^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(12348000 a^3 \left(8 c^2 f^2+g^2\right)+105 b \sin ^{-1}(c x) \left(352800 a^2 \left(8 c^2 f^2+g^2\right)+6720 a b c \sqrt{1-c^2 x^2} \left(768 f g \left(c^2 x^2-1\right)^3+56 c^2 f^2 x \left(8 c^4 x^4-26 c^2 x^2+33\right)+7 g^2 x \left(48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right)\right)+b^2 \left(-640 c^8 x^6 \left(784 f^2+1152 f g x+441 g^2 x^2\right)+1792 c^6 x^4 \left(1365 f^2+1728 f g x+595 g^2 x^2\right)-3360 c^4 x^2 \left(1848 f^2+1536 f g x+413 g^2 x^2\right)+1120 c^2 \left(2093 f^2+4608 f g x+315 g^2 x^2\right)+87955 g^2\right)\right)+352800 a^2 b c \sqrt{1-c^2 x^2} \left(768 f g \left(c^2 x^2-1\right)^3+56 c^2 f^2 x \left(8 c^4 x^4-26 c^2 x^2+33\right)+7 g^2 x \left(48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right)\right)-3360 a b^2 c^2 x \left(1960 c^2 f^2 x \left(8 c^4 x^4-39 c^2 x^2+99\right)+4608 f g \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right)+245 g^2 x \left(36 c^6 x^6-136 c^4 x^4+177 c^2 x^2-45\right)\right)+352800 b^2 \sin ^{-1}(c x)^2 \left(105 a \left(8 c^2 f^2+g^2\right)+b c \sqrt{1-c^2 x^2} \left(768 f g \left(c^2 x^2-1\right)^3+56 c^2 f^2 x \left(8 c^4 x^4-26 c^2 x^2+33\right)+7 g^2 x \left(48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right)\right)\right)+12348000 b^3 \left(8 c^2 f^2+g^2\right) \sin ^{-1}(c x)^3-b^3 c \sqrt{1-c^2 x^2} \left(274400 c^2 f^2 x \left(32 c^4 x^4-194 c^2 x^2+897\right)+147456 f g \left(75 c^6 x^6-351 c^4 x^4+757 c^2 x^2-2161\right)+8575 g^2 x \left(432 c^6 x^6-1672 c^4 x^4+2158 c^2 x^2+1077\right)\right)\right)}{948326400 b c^3 \sqrt{1-c^2 x^2}}","-\frac{b c^5 d^2 g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^8}{32 \sqrt{1-c^2 x^2}}-\frac{4 b c^5 d^2 f g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^7}{49 \sqrt{1-c^2 x^2}}-\frac{1}{256} b^2 c^4 d^2 g^2 \sqrt{d-c^2 d x^2} x^7+\frac{17 b c^3 d^2 g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^6}{144 \sqrt{1-c^2 x^2}}+\frac{12 b c^3 d^2 f g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^5}{35 \sqrt{1-c^2 x^2}}+\frac{209 b^2 c^2 d^2 g^2 \sqrt{d-c^2 d x^2} x^5}{13824}-\frac{59 b c d^2 g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^4}{384 \sqrt{1-c^2 x^2}}+\frac{5}{64} d^2 g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 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 \sin ^{-1}(c x)\right)^2 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 \sin ^{-1}(c x)\right)^2 x^3-\frac{4 b c d^2 f g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^3}{7 \sqrt{1-c^2 x^2}}-\frac{1079 b^2 d^2 g^2 \sqrt{d-c^2 d x^2} x^3}{55296}-\frac{5 b c d^2 f^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^2}{16 \sqrt{1-c^2 x^2}}+\frac{5 b d^2 g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^2}{128 c \sqrt{1-c^2 x^2}}+\frac{5}{16} d^2 f^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x-\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 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 \sin ^{-1}(c x)\right)^2 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 \sin ^{-1}(c x)\right)^2 x+\frac{4 b d^2 f g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x}{7 c \sqrt{1-c^2 x^2}}-\frac{245 b^2 d^2 f^2 \sqrt{d-c^2 d x^2} x}{1152}-\frac{359 b^2 d^2 g^2 \sqrt{d-c^2 d x^2} x}{36864 c^2}-\frac{1}{108} b^2 d^2 f^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} x-\frac{65 b^2 d^2 f^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} x}{1728}+\frac{5 d^2 f^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{48 b c \sqrt{1-c^2 x^2}}+\frac{5 d^2 g^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{384 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 \sin ^{-1}(c x)\right)^2}{7 c^2}+\frac{115 b^2 d^2 f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt{1-c^2 x^2}}+\frac{359 b^2 d^2 g^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{36864 c^3 \sqrt{1-c^2 x^2}}+\frac{b d^2 f^2 \left(1-c^2 x^2\right)^{5/2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{18 c}+\frac{5 b d^2 f^2 \left(1-c^2 x^2\right)^{3/2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{48 c}+\frac{4 b^2 d^2 f g \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2}}{343 c^2}+\frac{24 b^2 d^2 f g \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{1225 c^2}+\frac{64 b^2 d^2 f g \sqrt{d-c^2 d x^2}}{245 c^2}+\frac{32 b^2 d^2 f g \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{735 c^2}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(12348000*a^3*(8*c^2*f^2 + g^2) - 3360*a*b^2*c^2*x*(1960*c^2*f^2*x*(99 - 39*c^2*x^2 + 8*c^4*x^4) + 4608*f*g*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + 245*g^2*x*(-45 + 177*c^2*x^2 - 136*c^4*x^4 + 36*c^6*x^6)) + 352800*a^2*b*c*Sqrt[1 - c^2*x^2]*(768*f*g*(-1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 7*g^2*x*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6)) - b^3*c*Sqrt[1 - c^2*x^2]*(274400*c^2*f^2*x*(897 - 194*c^2*x^2 + 32*c^4*x^4) + 147456*f*g*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6) + 8575*g^2*x*(1077 + 2158*c^2*x^2 - 1672*c^4*x^4 + 432*c^6*x^6)) + 105*b*(352800*a^2*(8*c^2*f^2 + g^2) + b^2*(87955*g^2 + 1120*c^2*(2093*f^2 + 4608*f*g*x + 315*g^2*x^2) - 3360*c^4*x^2*(1848*f^2 + 1536*f*g*x + 413*g^2*x^2) - 640*c^8*x^6*(784*f^2 + 1152*f*g*x + 441*g^2*x^2) + 1792*c^6*x^4*(1365*f^2 + 1728*f*g*x + 595*g^2*x^2)) + 6720*a*b*c*Sqrt[1 - c^2*x^2]*(768*f*g*(-1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 7*g^2*x*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6)))*ArcSin[c*x] + 352800*b^2*(105*a*(8*c^2*f^2 + g^2) + b*c*Sqrt[1 - c^2*x^2]*(768*f*g*(-1 + c^2*x^2)^3 + 56*c^2*f^2*x*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 7*g^2*x*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6)))*ArcSin[c*x]^2 + 12348000*b^3*(8*c^2*f^2 + g^2)*ArcSin[c*x]^3))/(948326400*b*c^3*Sqrt[1 - c^2*x^2])","A",1
68,1,470,878,1.0110168,"\int (f+g x) \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(f + g*x)*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(3087000 a^3 c f+105 b \sin ^{-1}(c x) \left(88200 a^2 c f+1680 a b \sqrt{1-c^2 x^2} \left(48 g \left(c^2 x^2-1\right)^3+7 c^2 f x \left(8 c^4 x^4-26 c^2 x^2+33\right)\right)+b^2 c \left(-245 f \left(64 c^6 x^6-312 c^4 x^4+792 c^2 x^2-299\right)-2304 g x \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right)\right)\right)+88200 a^2 b \sqrt{1-c^2 x^2} \left(48 g \left(c^2 x^2-1\right)^3+7 c^2 f x \left(8 c^4 x^4-26 c^2 x^2+33\right)\right)+88200 b^2 \sin ^{-1}(c x)^2 \left(105 a c f+b \sqrt{1-c^2 x^2} \left(48 g \left(c^2 x^2-1\right)^3+7 c^2 f x \left(8 c^4 x^4-26 c^2 x^2+33\right)\right)\right)-840 a b^2 c x \left(245 c^2 f x \left(8 c^4 x^4-39 c^2 x^2+99\right)+288 g \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right)\right)+b^3 \sqrt{1-c^2 x^2} \left(-8575 c^2 f x \left(32 c^4 x^4-194 c^2 x^2+897\right)-2304 g \left(75 c^6 x^6-351 c^4 x^4+757 c^2 x^2-2161\right)\right)+3087000 b^3 c f \sin ^{-1}(c x)^3\right)}{29635200 b c^2 \sqrt{1-c^2 x^2}}","-\frac{2 b c^5 d^2 g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^7}{49 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^5}{35 \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^3}{7 \sqrt{1-c^2 x^2}}-\frac{5 b c d^2 f \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^2}{16 \sqrt{1-c^2 x^2}}+\frac{1}{6} d^2 f \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x+\frac{5}{16} d^2 f \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x+\frac{5}{24} d^2 f \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x+\frac{2 b d^2 g \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x}{7 c \sqrt{1-c^2 x^2}}-\frac{1}{108} b^2 d^2 f \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} x-\frac{245 b^2 d^2 f \sqrt{d-c^2 d x^2} x}{1152}-\frac{65 b^2 d^2 f \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} x}{1728}+\frac{5 d^2 f \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{48 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 \sin ^{-1}(c x)\right)^2}{7 c^2}+\frac{115 b^2 d^2 f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \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} \left(a+b \sin ^{-1}(c x)\right)}{18 c}+\frac{5 b d^2 f \left(1-c^2 x^2\right)^{3/2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{48 c}+\frac{2 b^2 d^2 g \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2}}{343 c^2}+\frac{12 b^2 d^2 g \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{1225 c^2}+\frac{32 b^2 d^2 g \sqrt{d-c^2 d x^2}}{245 c^2}+\frac{16 b^2 d^2 g \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{735 c^2}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(3087000*a^3*c*f + 88200*a^2*b*Sqrt[1 - c^2*x^2]*(48*g*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 - 26*c^2*x^2 + 8*c^4*x^4)) - 840*a*b^2*c*x*(245*c^2*f*x*(99 - 39*c^2*x^2 + 8*c^4*x^4) + 288*g*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)) + b^3*Sqrt[1 - c^2*x^2]*(-8575*c^2*f*x*(897 - 194*c^2*x^2 + 32*c^4*x^4) - 2304*g*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6)) + 105*b*(88200*a^2*c*f + 1680*a*b*Sqrt[1 - c^2*x^2]*(48*g*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 - 26*c^2*x^2 + 8*c^4*x^4)) + b^2*c*(-2304*g*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) - 245*f*(-299 + 792*c^2*x^2 - 312*c^4*x^4 + 64*c^6*x^6)))*ArcSin[c*x] + 88200*b^2*(105*a*c*f + b*Sqrt[1 - c^2*x^2]*(48*g*(-1 + c^2*x^2)^3 + 7*c^2*f*x*(33 - 26*c^2*x^2 + 8*c^4*x^4)))*ArcSin[c*x]^2 + 3087000*b^3*c*f*ArcSin[c*x]^3))/(29635200*b*c^2*Sqrt[1 - c^2*x^2])","A",1
69,1,1277,2989,4.9803121,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{f+g x} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(f + g*x),x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(\frac{x^4 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 c^4}{5 g}-\frac{f x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 c^4}{4 g^2}-\frac{f \left(c^2 f^2-2 g^2\right) x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 c^2}{2 g^4}-\frac{f \left(c^2 f^2-2 g^2\right) \left(a+b \sin ^{-1}(c x)\right)^3 c}{6 b g^4}+\frac{b f \left(c^2 f^2-2 g^2\right) \left(c x \left(\sqrt{1-c^2 x^2} b+2 a c x\right)+b \left(2 c^2 x^2-1\right) \sin ^{-1}(c x)\right) c}{4 g^4}+\frac{b f \left(8 a c^4 x^4+b c \sqrt{1-c^2 x^2} \left(2 c^2 x^2+3\right) x+b \left(8 c^4 x^4-3\right) \sin ^{-1}(c x)\right) c}{64 g^2}+\frac{f \left(-2 \left(a+b \sin ^{-1}(c x)\right)^3+6 b c x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-3 b^2 \left(c x \left(\sqrt{1-c^2 x^2} b+2 a c x\right)+b \left(2 c^2 x^2-1\right) \sin ^{-1}(c x)\right)\right) c}{48 b g^2}-\frac{\left(c^2 f^2-2 g^2\right) \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 g^3}+\frac{2 b \left(c^2 f^2-2 g^2\right) \left(-3 c^3 \left(a+b \sin ^{-1}(c x)\right) x^3+9 c \left(a+b \sin ^{-1}(c x)\right) x-b \sqrt{1-c^2 x^2} \left(c^2 x^2-7\right)\right)}{27 g^3}-\frac{2 b \left(15 c^5 \left(a+b \sin ^{-1}(c x)\right) x^5+b \sqrt{1-c^2 x^2} \left(3 c^4 x^4+4 c^2 x^2+8\right)\right)}{375 g}-\frac{9 c^2 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-2 b \left(3 c^3 \left(a+b \sin ^{-1}(c x)\right) x^3+b \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right)\right)+18 \left(\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-2 b \left(c x \sin ^{-1}(c x) b+\sqrt{1-c^2 x^2} b+a c x\right)\right)}{135 g}-\frac{\left(g^2-c^2 f^2\right)^2 \left(c^2 x^2-1\right) \left(a+b \sin ^{-1}(c x)\right)^3}{3 b g^4 (f+g x) c}+\frac{\left(g^2-c^2 f^2\right)^2 \left(\left(c^2 f^2-g^2\right) \left(a+b \sin ^{-1}(c x)\right)^3+c^2 g x (f+g x) \left(a+b \sin ^{-1}(c x)\right)^3+3 b c (f+g x) \left(g \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-2 b g \left(c x \sin ^{-1}(c x) b+\sqrt{1-c^2 x^2} b+a c x\right)+i \sqrt{c^2 f^2-g^2} \left(2 \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right) b^2-2 \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right) b^2-2 i \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right) b+2 i \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right) b+\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(\frac{i e^{i \sin ^{-1}(c x)} g}{\sqrt{c^2 f^2-g^2}-c f}+1\right)-\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)\right)\right)}{3 b g^6 (f+g x) c}\right)}{\sqrt{1-c^2 x^2}}","-\frac{2 b d^2 x^5 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) c^5}{25 g \sqrt{1-c^2 x^2}}+\frac{b d^2 f x^4 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) c^5}{8 g^2 \sqrt{1-c^2 x^2}}+\frac{d^2 x^4 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 c^4}{5 g}-\frac{d^2 f x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 c^4}{4 g^2}+\frac{b^2 d^2 f x^3 \sqrt{d-c^2 d x^2} c^4}{32 g^2}-\frac{2 b d^2 \left(c^2 f^2-2 g^2\right) x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) c^3}{9 g^3 \sqrt{1-c^2 x^2}}+\frac{2 b d^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) 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} \left(a+b \sin ^{-1}(c x)\right) c^3}{2 g^4 \sqrt{1-c^2 x^2}}-\frac{b d^2 f x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) c^3}{8 g^2 \sqrt{1-c^2 x^2}}-\frac{d^2 x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 c^2}{15 g}-\frac{d^2 f \left(c^2 f^2-2 g^2\right) x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 c^2}{2 g^4}+\frac{d^2 f x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 c^2}{8 g^2}+\frac{b^2 d^2 f \left(c^2 f^2-2 g^2\right) x \sqrt{d-c^2 d x^2} c^2}{4 g^4}-\frac{b^2 d^2 f x \sqrt{d-c^2 d x^2} c^2}{64 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 \sin ^{-1}(c x)\right)^3 c}{6 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 \sin ^{-1}(c x)\right)^3 c}{3 b g^5 \sqrt{1-c^2 x^2}}-\frac{d^2 f \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3 c}{24 b g^2 \sqrt{1-c^2 x^2}}-\frac{b^2 d^2 f \left(c^2 f^2-2 g^2\right) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) c}{4 g^4 \sqrt{1-c^2 x^2}}-\frac{2 b^2 d^2 \left(c^2 f^2-g^2\right)^2 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) c}{g^5 \sqrt{1-c^2 x^2}}+\frac{4 b^2 d^2 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) c}{15 g \sqrt{1-c^2 x^2}}+\frac{b^2 d^2 f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) c}{64 g^2 \sqrt{1-c^2 x^2}}+\frac{2 b d^2 \left(c^2 f^2-2 g^2\right) x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) c}{3 g^3 \sqrt{1-c^2 x^2}}-\frac{2 a 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{4 a b d^2 x \sqrt{d-c^2 d x^2} c}{15 g \sqrt{1-c^2 x^2}}+\frac{b^2 d^2 \left(c^2 f^2-g^2\right)^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2}{g^5}-\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 \sin ^{-1}(c x)\right)^2}{3 g^3}-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{15 g}+\frac{2 a b d^2 \left(c^2 f^2-g^2\right)^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^5}-\frac{a^2 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^2 d^2 \left(c^2 f^2-g^2\right)^{5/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^6 \sqrt{1-c^2 x^2}}+\frac{2 i a b d^2 \left(c^2 f^2-g^2\right)^{5/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^6 \sqrt{1-c^2 x^2}}-\frac{i b^2 d^2 \left(c^2 f^2-g^2\right)^{5/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^6 \sqrt{1-c^2 x^2}}-\frac{2 i a b d^2 \left(c^2 f^2-g^2\right)^{5/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^6 \sqrt{1-c^2 x^2}}+\frac{2 b^2 d^2 \left(c^2 f^2-g^2\right)^{5/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^6 \sqrt{1-c^2 x^2}}+\frac{2 a 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{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^6 \sqrt{1-c^2 x^2}}-\frac{2 b^2 d^2 \left(c^2 f^2-g^2\right)^{5/2} \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^6 \sqrt{1-c^2 x^2}}-\frac{2 a 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{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^6 \sqrt{1-c^2 x^2}}+\frac{2 i b^2 d^2 \left(c^2 f^2-g^2\right)^{5/2} \sqrt{d-c^2 d x^2} \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{g^6 \sqrt{1-c^2 x^2}}-\frac{2 i b^2 d^2 \left(c^2 f^2-g^2\right)^{5/2} \sqrt{d-c^2 d x^2} \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{g^6 \sqrt{1-c^2 x^2}}+\frac{a^2 d^2 \left(c^2 f^2-g^2\right)^2 \sqrt{d-c^2 d x^2}}{g^5}-\frac{2 b^2 d^2 \left(c^2 f^2-g^2\right)^2 \sqrt{d-c^2 d x^2}}{g^5}-\frac{2 b^2 d^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{125 g}+\frac{4 b^2 d^2 \left(c^2 f^2-2 g^2\right) \sqrt{d-c^2 d x^2}}{9 g^3}+\frac{2 b^2 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}}{27 g^3}+\frac{26 b^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{675 g}+\frac{52 b^2 d^2 \sqrt{d-c^2 d x^2}}{225 g}+\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 \sin ^{-1}(c x)\right)^3}{3 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 \sin ^{-1}(c x)\right)^3}{3 b g^6 (f+g x) \sqrt{1-c^2 x^2} c}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(-1/2*(c^2*f*(c^2*f^2 - 2*g^2)*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/g^4 - (c^4*f*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*g^2) + (c^4*x^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(5*g) - ((c^2*f^2 - 2*g^2)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*g^3) - (c*f*(c^2*f^2 - 2*g^2)*(a + b*ArcSin[c*x])^3)/(6*b*g^4) - ((-(c^2*f^2) + g^2)^2*(-1 + c^2*x^2)*(a + b*ArcSin[c*x])^3)/(3*b*c*g^4*(f + g*x)) + (b*c*f*(c^2*f^2 - 2*g^2)*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x]))/(4*g^4) + (b*c*f*(8*a*c^4*x^4 + b*c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2) + b*(-3 + 8*c^4*x^4)*ArcSin[c*x]))/(64*g^2) + (2*b*(c^2*f^2 - 2*g^2)*(-(b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2)) + 9*c*x*(a + b*ArcSin[c*x]) - 3*c^3*x^3*(a + b*ArcSin[c*x])))/(27*g^3) - (2*b*(b*Sqrt[1 - c^2*x^2]*(8 + 4*c^2*x^2 + 3*c^4*x^4) + 15*c^5*x^5*(a + b*ArcSin[c*x])))/(375*g) + (c*f*(6*b*c*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*(a + b*ArcSin[c*x])^3 - 3*b^2*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x])))/(48*b*g^2) - (9*c^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*(b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + 3*c^3*x^3*(a + b*ArcSin[c*x])) + 18*(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*(a*c*x + b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x])))/(135*g) + ((-(c^2*f^2) + g^2)^2*((c^2*f^2 - g^2)*(a + b*ArcSin[c*x])^3 + c^2*g*x*(f + g*x)*(a + b*ArcSin[c*x])^3 + 3*b*c*(f + g*x)*(g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - 2*b*g*(a*c*x + b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x]) + I*Sqrt[c^2*f^2 - g^2]*((a + b*ArcSin[c*x])^2*Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - (a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] - (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] - 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))))/(3*b*c*g^6*(f + g*x))))/Sqrt[1 - c^2*x^2]","A",0
70,1,582,692,1.5178049,"\int \frac{(f+g x)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[((f + g*x)^3*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\frac{-108 a^2 c \sqrt{d} 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)-36 a^2 d \left(1-c^2 x^2\right)^{3/2} \left(c^2 g \left(18 f^2+9 f g x+2 g^2 x^2\right)+4 g^3\right)-1296 a b c^2 d f^2 g \left(c^2 x^2-1\right) \left(c x-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)\right)+162 a b c d f g^2 \left(c^2 x^2-1\right) \left(-2 \sin ^{-1}(c x)^2+2 \sin \left(2 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+\cos \left(2 \sin ^{-1}(c x)\right)\right)-216 a b c^3 d f^3 \left(c^2 x^2-1\right) \sin ^{-1}(c x)^2-48 a b d g^3 \left(c^2 x^2-1\right) \left(c^3 x^3-3 \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right) \sin ^{-1}(c x)+6 c x\right)+648 b^2 c^2 d f^2 g \left(1-c^2 x^2\right) \left(2 c x \sin ^{-1}(c x)-\sqrt{1-c^2 x^2} \left(\sin ^{-1}(c x)^2-2\right)\right)+27 b^2 c d f g^2 \left(1-c^2 x^2\right) \left(4 \sin ^{-1}(c x)^3+\left(3-6 \sin ^{-1}(c x)^2\right) \sin \left(2 \sin ^{-1}(c x)\right)-6 \sin ^{-1}(c x) \cos \left(2 \sin ^{-1}(c x)\right)\right)-2 b^2 d g^3 \left(1-c^2 x^2\right) \left(81 \sqrt{1-c^2 x^2} \left(\sin ^{-1}(c x)^2-2\right)+6 \sin ^{-1}(c x) \left(\sin \left(3 \sin ^{-1}(c x)\right)-27 c x\right)-\left(9 \sin ^{-1}(c x)^2-2\right) \cos \left(3 \sin ^{-1}(c x)\right)\right)-72 b^2 c^3 d f^3 \left(c^2 x^2-1\right) \sin ^{-1}(c x)^3}{216 c^4 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 \sin ^{-1}(c x)\right)^3}{3 b c \sqrt{d-c^2 d x^2}}-\frac{3 f^2 g \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 \sqrt{d-c^2 d x^2}}+\frac{6 b f^2 g x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c \sqrt{d-c^2 d x^2}}-\frac{3 f g^2 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{3 b f g^2 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 c \sqrt{d-c^2 d x^2}}-\frac{g^3 x^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^2 \sqrt{d-c^2 d x^2}}+\frac{2 b g^3 x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 g^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^4 \sqrt{d-c^2 d x^2}}+\frac{f g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{2 b c^3 \sqrt{d-c^2 d x^2}}+\frac{4 b g^3 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 c^3 \sqrt{d-c^2 d x^2}}+\frac{6 b^2 f^2 g \left(1-c^2 x^2\right)}{c^2 \sqrt{d-c^2 d x^2}}+\frac{3 b^2 f g^2 x \left(1-c^2 x^2\right)}{4 c^2 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 \left(1-c^2 x^2\right)^2}{27 c^4 \sqrt{d-c^2 d x^2}}+\frac{14 b^2 g^3 \left(1-c^2 x^2\right)}{9 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 b^2 f g^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c^3 \sqrt{d-c^2 d x^2}}",1,"(-36*a^2*d*(1 - c^2*x^2)^(3/2)*(4*g^3 + c^2*g*(18*f^2 + 9*f*g*x + 2*g^2*x^2)) - 216*a*b*c^3*d*f^3*(-1 + c^2*x^2)*ArcSin[c*x]^2 - 72*b^2*c^3*d*f^3*(-1 + c^2*x^2)*ArcSin[c*x]^3 - 1296*a*b*c^2*d*f^2*g*(-1 + c^2*x^2)*(c*x - Sqrt[1 - c^2*x^2]*ArcSin[c*x]) - 48*a*b*d*g^3*(-1 + c^2*x^2)*(6*c*x + c^3*x^3 - 3*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2)*ArcSin[c*x]) + 648*b^2*c^2*d*f^2*g*(1 - c^2*x^2)*(2*c*x*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*(-2 + ArcSin[c*x]^2)) - 108*a^2*c*Sqrt[d]*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))] + 162*a*b*c*d*f*g^2*(-1 + c^2*x^2)*(-2*ArcSin[c*x]^2 + Cos[2*ArcSin[c*x]] + 2*ArcSin[c*x]*Sin[2*ArcSin[c*x]]) + 27*b^2*c*d*f*g^2*(1 - c^2*x^2)*(4*ArcSin[c*x]^3 - 6*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + (3 - 6*ArcSin[c*x]^2)*Sin[2*ArcSin[c*x]]) - 2*b^2*d*g^3*(1 - c^2*x^2)*(81*Sqrt[1 - c^2*x^2]*(-2 + ArcSin[c*x]^2) - (-2 + 9*ArcSin[c*x]^2)*Cos[3*ArcSin[c*x]] + 6*ArcSin[c*x]*(-27*c*x + Sin[3*ArcSin[c*x]])))/(216*c^4*d*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])","A",1
71,1,400,410,1.4018945,"\int \frac{(f+g x)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[((f + g*x)^2*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\frac{3 \sqrt{d} g \left(c^2 x^2-1\right) \left(4 c \left(a^2 \sqrt{1-c^2 x^2} (4 f+g x)-8 a b c f x-8 b^2 f \sqrt{1-c^2 x^2}\right)+2 a b g \cos \left(2 \sin ^{-1}(c x)\right)+b^2 (-g) \sin \left(2 \sin ^{-1}(c x)\right)\right)-12 a^2 \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)+6 b \sqrt{d} \left(c^2 x^2-1\right) \sin ^{-1}(c x)^2 \left(-2 a \left(2 c^2 f^2+g^2\right)+8 b c f g \sqrt{1-c^2 x^2}+b g^2 \sin \left(2 \sin ^{-1}(c x)\right)\right)+6 b \sqrt{d} g \left(c^2 x^2-1\right) \sin ^{-1}(c x) \left(16 c f \left(a \sqrt{1-c^2 x^2}-b c x\right)+2 a g \sin \left(2 \sin ^{-1}(c x)\right)+b g \cos \left(2 \sin ^{-1}(c x)\right)\right)-4 b^2 \sqrt{d} \left(c^2 x^2-1\right) \left(2 c^2 f^2+g^2\right) \sin ^{-1}(c x)^3}{24 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 \sin ^{-1}(c x)\right)^3}{3 b c \sqrt{d-c^2 d x^2}}-\frac{2 f g \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 \sqrt{d-c^2 d x^2}}+\frac{4 b f g x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c \sqrt{d-c^2 d x^2}}-\frac{g^2 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{b g^2 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 c \sqrt{d-c^2 d x^2}}+\frac{g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{6 b c^3 \sqrt{d-c^2 d x^2}}+\frac{4 b^2 f g \left(1-c^2 x^2\right)}{c^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 g^2 x \left(1-c^2 x^2\right)}{4 c^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 g^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c^3 \sqrt{d-c^2 d x^2}}",1,"(-4*b^2*Sqrt[d]*(2*c^2*f^2 + g^2)*(-1 + c^2*x^2)*ArcSin[c*x]^3 - 12*a^2*(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))] + 6*b*Sqrt[d]*g*(-1 + c^2*x^2)*ArcSin[c*x]*(16*c*f*(-(b*c*x) + a*Sqrt[1 - c^2*x^2]) + b*g*Cos[2*ArcSin[c*x]] + 2*a*g*Sin[2*ArcSin[c*x]]) + 3*Sqrt[d]*g*(-1 + c^2*x^2)*(4*c*(-8*a*b*c*f*x - 8*b^2*f*Sqrt[1 - c^2*x^2] + a^2*(4*f + g*x)*Sqrt[1 - c^2*x^2]) + 2*a*b*g*Cos[2*ArcSin[c*x]] - b^2*g*Sin[2*ArcSin[c*x]]) + 6*b*Sqrt[d]*(-1 + c^2*x^2)*ArcSin[c*x]^2*(-2*a*(2*c^2*f^2 + g^2) + 8*b*c*f*g*Sqrt[1 - c^2*x^2] + b*g^2*Sin[2*ArcSin[c*x]]))/(24*c^3*Sqrt[d]*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])","A",1
72,1,291,171,0.5987388,"\int \frac{(f+g x) \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[((f + g*x)*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\frac{3 \sqrt{d} g \left(c^2 x^2-1\right) \left(a^2 \sqrt{1-c^2 x^2}-2 a b c x-2 b^2 \sqrt{1-c^2 x^2}\right)-3 a^2 c f \sqrt{1-c^2 x^2} \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)+3 b \sqrt{d} \left(c^2 x^2-1\right) \sin ^{-1}(c x)^2 \left(b g \sqrt{1-c^2 x^2}-a c f\right)-6 b \sqrt{d} g \left(c^2 x^2-1\right) \sin ^{-1}(c x) \left(b c x-a \sqrt{1-c^2 x^2}\right)-b^2 c \sqrt{d} f \left(c^2 x^2-1\right) \sin ^{-1}(c x)^3}{3 c^2 \sqrt{d} \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}","\frac{f \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c \sqrt{d-c^2 d x^2}}-\frac{g \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 \sqrt{d-c^2 d x^2}}+\frac{2 b g x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c \sqrt{d-c^2 d x^2}}+\frac{2 b^2 g \left(1-c^2 x^2\right)}{c^2 \sqrt{d-c^2 d x^2}}",1,"(3*Sqrt[d]*g*(-1 + c^2*x^2)*(-2*a*b*c*x + a^2*Sqrt[1 - c^2*x^2] - 2*b^2*Sqrt[1 - c^2*x^2]) - 6*b*Sqrt[d]*g*(-1 + c^2*x^2)*(b*c*x - a*Sqrt[1 - c^2*x^2])*ArcSin[c*x] + 3*b*Sqrt[d]*(-1 + c^2*x^2)*(-(a*c*f) + b*g*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 - b^2*c*Sqrt[d]*f*(-1 + c^2*x^2)*ArcSin[c*x]^3 - 3*a^2*c*f*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))])/(3*c^2*Sqrt[d]*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])","A",1
73,1,357,589,0.3539771,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{(f+g x) \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/((f + g*x)*Sqrt[d - c^2*d*x^2]),x]","-\frac{i \sqrt{1-c^2 x^2} \left(-2 i b \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)+2 i b \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)+\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right)-\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)+2 b^2 \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)-2 b^2 \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)}{\sqrt{d-c^2 d x^2} \sqrt{c^2 f^2-g^2}}","-\frac{2 b \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-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{2 b \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-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{i \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i g e^{i \sin ^{-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 \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i g e^{i \sin ^{-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{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(\frac{i e^{i \sin ^{-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{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(\frac{i e^{i \sin ^{-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,"((-I)*Sqrt[1 - c^2*x^2]*((a + b*ArcSin[c*x])^2*Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - (a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] - (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] - 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))/(Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2])","A",1
74,1,651,1113,0.7457482,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{(f+g x)^2 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/((f + g*x)^2*Sqrt[d - c^2*d*x^2]),x]","\frac{c \sqrt{1-c^2 x^2} \left(-\frac{i c f \left(-2 i b \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)+\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right)+2 b^2 \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)\right)}{\sqrt{c^2 f^2-g^2}}+\frac{c f \left(i \left(\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)+2 b^2 \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)+2 b \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)}{\sqrt{c^2 f^2-g^2}}-2 b \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right)-2 b \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)+\frac{g \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{c f+c g x}+i \left(a+b \sin ^{-1}(c x)\right)^2+2 i b^2 \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)+2 i b^2 \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)}{\sqrt{d-c^2 d x^2} \left(c^2 f^2-g^2\right)}","\frac{2 i c \sqrt{1-c^2 x^2} \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right) b^2}{\left(c^2 f^2-g^2\right) \sqrt{d-c^2 d x^2}}+\frac{2 i c \sqrt{1-c^2 x^2} \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right) b^2}{\left(c^2 f^2-g^2\right) \sqrt{d-c^2 d x^2}}-\frac{2 i c^2 f \sqrt{1-c^2 x^2} \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right) b^2}{\left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2}}+\frac{2 i c^2 f \sqrt{1-c^2 x^2} \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right) b^2}{\left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2}}-\frac{2 c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right) b}{\left(c^2 f^2-g^2\right) \sqrt{d-c^2 d x^2}}-\frac{2 c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right) b}{\left(c^2 f^2-g^2\right) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 f \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right) b}{\left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2}}+\frac{2 c^2 f \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right) b}{\left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2}}+\frac{g \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{\left(c^2 f^2-g^2\right) (f+g x) \sqrt{d-c^2 d x^2}}+\frac{i c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{\left(c^2 f^2-g^2\right) \sqrt{d-c^2 d x^2}}-\frac{i c^2 f \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{\left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2}}+\frac{i c^2 f \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{\left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2}}",1,"(c*Sqrt[1 - c^2*x^2]*(I*(a + b*ArcSin[c*x])^2 + (g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c*f + c*g*x) - 2*b*(a + b*ArcSin[c*x])*Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - 2*b*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + (2*I)*b^2*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + (2*I)*b^2*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] - (I*c*f*((a + b*ArcSin[c*x])^2*Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])]))/Sqrt[c^2*f^2 - g^2] + (c*f*(2*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + I*((a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])))/Sqrt[c^2*f^2 - g^2]))/((c^2*f^2 - g^2)*Sqrt[d - c^2*d*x^2])","A",1
75,1,325,738,3.4359448,"\int \frac{(f+g x)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[((f + g*x)^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","\frac{\sqrt{1-c^2 x^2} \left(-(c f+g)^3 \left(-\tan \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(a+b \sin ^{-1}(c x)\right)^2+i \left(\left(a+b \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)+4 i b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)\right)+4 b^2 \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)\right)\right)+(c f-g)^3 \left(-\cot \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(a+b \sin ^{-1}(c x)\right)^2+i \left(\left(a+b \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)-4 i b \log \left(1+i e^{-i \sin ^{-1}(c x)}\right)\right)+4 b^2 \text{Li}_2\left(-i e^{-i \sin ^{-1}(c x)}\right)\right)\right)+2 g^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2-4 b g^3 \left(a c x+b \sqrt{1-c^2 x^2}+b c x \sin ^{-1}(c x)\right)-\frac{2 c f g^2 \left(a+b \sin ^{-1}(c x)\right)^3}{b}\right)}{2 c^4 d \sqrt{d-c^2 d x^2}}","\frac{f x \left(\frac{3 g^2}{c^2}+f^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d \sqrt{d-c^2 d x^2}}+\frac{g \left(3 c^2 f^2+g^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 i b g \sqrt{1-c^2 x^2} \left(3 c^2 f^2+g^2\right) \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{g^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{i f \sqrt{1-c^2 x^2} \left(c^2 f^2+3 g^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b f \sqrt{1-c^2 x^2} \left(c^2 f^2+3 g^2\right) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{f g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{b c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 a b g^3 x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 g \sqrt{1-c^2 x^2} \left(3 c^2 f^2+g^2\right) \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 g \sqrt{1-c^2 x^2} \left(3 c^2 f^2+g^2\right) \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 \left(1-c^2 x^2\right)}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{i b^2 f \sqrt{1-c^2 x^2} \left(c^2 f^2+3 g^2\right) \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 g^3 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}",1,"(Sqrt[1 - c^2*x^2]*(2*g^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2 - (2*c*f*g^2*(a + b*ArcSin[c*x])^3)/b - 4*b*g^3*(a*c*x + b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x]) + (c*f - g)^3*(-((a + b*ArcSin[c*x])^2*Cot[(Pi + 2*ArcSin[c*x])/4]) + I*((a + b*ArcSin[c*x])*(a + b*ArcSin[c*x] - (4*I)*b*Log[1 + I/E^(I*ArcSin[c*x])]) + 4*b^2*PolyLog[2, (-I)/E^(I*ArcSin[c*x])])) - (c*f + g)^3*(I*((a + b*ArcSin[c*x])*(a + b*ArcSin[c*x] + (4*I)*b*Log[1 + I*E^(I*ArcSin[c*x])]) + 4*b^2*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]) - (a + b*ArcSin[c*x])^2*Tan[(Pi + 2*ArcSin[c*x])/4])))/(2*c^4*d*Sqrt[d - c^2*d*x^2])","A",0
76,1,259,513,2.2573029,"\int \frac{(f+g x)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[((f + g*x)^2*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","\frac{\sqrt{1-c^2 x^2} \left(-3 (c f+g)^2 \left(-\tan \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(a+b \sin ^{-1}(c x)\right)^2+i \left(\left(a+b \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)+4 i b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)\right)+4 b^2 \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)\right)\right)+3 (g-c f)^2 \left(-\cot \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(a+b \sin ^{-1}(c x)\right)^2+i \left(\left(a+b \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)-4 i b \log \left(1+i e^{-i \sin ^{-1}(c x)}\right)\right)+4 b^2 \text{Li}_2\left(-i e^{-i \sin ^{-1}(c x)}\right)\right)\right)-\frac{2 g^2 \left(a+b \sin ^{-1}(c x)\right)^3}{b}\right)}{6 c^3 d \sqrt{d-c^2 d x^2}}","\frac{x \left(c^2 f^2+g^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 f g \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{8 i b f g \sqrt{1-c^2 x^2} \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{i \sqrt{1-c^2 x^2} \left(c^2 f^2+g^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b \sqrt{1-c^2 x^2} \left(c^2 f^2+g^2\right) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c^3 d \sqrt{d-c^2 d x^2}}-\frac{4 i b^2 f g \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{4 i b^2 f g \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{i b^2 \sqrt{1-c^2 x^2} \left(c^2 f^2+g^2\right) \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{c^3 d \sqrt{d-c^2 d x^2}}",1,"(Sqrt[1 - c^2*x^2]*((-2*g^2*(a + b*ArcSin[c*x])^3)/b + 3*(-(c*f) + g)^2*(-((a + b*ArcSin[c*x])^2*Cot[(Pi + 2*ArcSin[c*x])/4]) + I*((a + b*ArcSin[c*x])*(a + b*ArcSin[c*x] - (4*I)*b*Log[1 + I/E^(I*ArcSin[c*x])]) + 4*b^2*PolyLog[2, (-I)/E^(I*ArcSin[c*x])])) - 3*(c*f + g)^2*(I*((a + b*ArcSin[c*x])*(a + b*ArcSin[c*x] + (4*I)*b*Log[1 + I*E^(I*ArcSin[c*x])]) + 4*b^2*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]) - (a + b*ArcSin[c*x])^2*Tan[(Pi + 2*ArcSin[c*x])/4])))/(6*c^3*d*Sqrt[d - c^2*d*x^2])","A",1
77,1,237,410,1.4894138,"\int \frac{(f+g x) \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[((f + g*x)*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","\frac{\sqrt{1-c^2 x^2} \left((c f-g) \left(-\cot \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(a+b \sin ^{-1}(c x)\right)^2+i \left(\left(a+b \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)-4 i b \log \left(1+i e^{-i \sin ^{-1}(c x)}\right)\right)+4 b^2 \text{Li}_2\left(-i e^{-i \sin ^{-1}(c x)}\right)\right)\right)-(c f+g) \left(-\tan \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(a+b \sin ^{-1}(c x)\right)^2+i \left(\left(a+b \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)+4 i b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)\right)+4 b^2 \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)\right)\right)\right)}{2 c^2 d \sqrt{d-c^2 d x^2}}","\frac{f x \left(a+b \sin ^{-1}(c x)\right)^2}{d \sqrt{d-c^2 d x^2}}-\frac{i f \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{c d \sqrt{d-c^2 d x^2}}+\frac{2 b f \sqrt{1-c^2 x^2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c d \sqrt{d-c^2 d x^2}}+\frac{g \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{4 i b g \sqrt{1-c^2 x^2} \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{i b^2 f \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{c d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 g \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 g \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c^2 d \sqrt{d-c^2 d x^2}}",1,"(Sqrt[1 - c^2*x^2]*((c*f - g)*(-((a + b*ArcSin[c*x])^2*Cot[(Pi + 2*ArcSin[c*x])/4]) + I*((a + b*ArcSin[c*x])*(a + b*ArcSin[c*x] - (4*I)*b*Log[1 + I/E^(I*ArcSin[c*x])]) + 4*b^2*PolyLog[2, (-I)/E^(I*ArcSin[c*x])])) - (c*f + g)*(I*((a + b*ArcSin[c*x])*(a + b*ArcSin[c*x] + (4*I)*b*Log[1 + I*E^(I*ArcSin[c*x])]) + 4*b^2*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]) - (a + b*ArcSin[c*x])^2*Tan[(Pi + 2*ArcSin[c*x])/4])))/(2*c^2*d*Sqrt[d - c^2*d*x^2])","A",0
78,1,597,1137,5.228336,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{(f+g x) \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/((f + g*x)*(d - c^2*d*x^2)^(3/2)),x]","\frac{\sqrt{1-c^2 x^2} \left(\frac{2 i g^2 \left(-2 i b \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)+2 i b \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)+\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right)-\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)+2 b^2 \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)-2 b^2 \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)\right)}{(c f-g) (c f+g) \sqrt{c^2 f^2-g^2}}+\frac{\left(a+b \sin ^{-1}(c x)\right) \left(a \tan \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)-i a+4 b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+b \sin ^{-1}(c x) \left(\tan \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)-i\right)\right)-4 i b^2 \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c f+g}+\frac{4 i b^2 \text{Li}_2\left(-i e^{-i \sin ^{-1}(c x)}\right)-\left(a+b \sin ^{-1}(c x)\right) \left(a \cot \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)-i a-4 b \log \left(1+i e^{-i \sin ^{-1}(c x)}\right)+b \sin ^{-1}(c x) \left(\cot \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)-i\right)\right)}{c f-g}\right)}{2 d \sqrt{d-c^2 d x^2}}","\frac{2 i \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{-i \sin ^{-1}(c x)}\right) b^2}{d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{2 i \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) b^2}{d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{2 i g^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right) b^2}{d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2}}-\frac{2 i g^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right) b^2}{d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-i e^{-i \sin ^{-1}(c x)}\right) b}{d (c f+g) \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-i e^{i \sin ^{-1}(c x)}\right) b}{d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{2 g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right) b}{d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2}}-\frac{2 g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right) b}{d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2}}-\frac{i \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{i \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 d (c f+g) \sqrt{d-c^2 d x^2}}-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{2 d (c f-g) \sqrt{d-c^2 d x^2}}+\frac{i g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2}}-\frac{i g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{d \left(c^2 f^2-g^2\right)^{3/2} \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{2 d (c f+g) \sqrt{d-c^2 d x^2}}",1,"(Sqrt[1 - c^2*x^2]*((-((a + b*ArcSin[c*x])*((-I)*a + a*Cot[(Pi + 2*ArcSin[c*x])/4] + b*ArcSin[c*x]*(-I + Cot[(Pi + 2*ArcSin[c*x])/4]) - 4*b*Log[1 + I/E^(I*ArcSin[c*x])])) + (4*I)*b^2*PolyLog[2, (-I)/E^(I*ArcSin[c*x])])/(c*f - g) + ((2*I)*g^2*((a + b*ArcSin[c*x])^2*Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - (a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] - (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + (2*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])] + 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] - 2*b^2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))/((c*f - g)*(c*f + g)*Sqrt[c^2*f^2 - g^2]) + ((-4*I)*b^2*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (a + b*ArcSin[c*x])*((-I)*a + 4*b*Log[1 + I*E^(I*ArcSin[c*x])] + a*Tan[(Pi + 2*ArcSin[c*x])/4] + b*ArcSin[c*x]*(-I + Tan[(Pi + 2*ArcSin[c*x])/4])))/(c*f + g)))/(2*d*Sqrt[d - c^2*d*x^2])","A",0
79,1,715,1589,6.2760481,"\int \frac{(f+g x)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[((f + g*x)^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","\frac{\sqrt{1-c^2 x^2} \left(-\frac{(c f-g)^3 \left(-2 \left(-\tan \left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)\right)^2+i b \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b}-4 \left(i \log \left(1+e^{\frac{1}{2} i \left(\pi -2 \sin ^{-1}(c x)\right)}\right) \left(a+b \sin ^{-1}(c x)\right)-b \text{Li}_2\left(-e^{\frac{1}{2} i \left(\pi -2 \sin ^{-1}(c x)\right)}\right)\right)\right)\right)+2 b \sec ^2\left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)\right)+\tan \left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \sec ^2\left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)\right)^2+4 b^2 \tan \left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{24 c^4}-\frac{(c f+g)^3 \left(2 \left(-\tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2+i b \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b}+4 \left(i \log \left(1+e^{\frac{1}{2} i \left(2 \sin ^{-1}(c x)+\pi \right)}\right) \left(a+b \sin ^{-1}(c x)\right)+b \text{Li}_2\left(-e^{\frac{1}{2} i \left(2 \sin ^{-1}(c x)+\pi \right)}\right)\right)\right)\right)+2 b \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)-\tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2-4 b^2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)\right)}{24 c^4}+\frac{(c f+2 g) (c f-g)^2 \left(-\tan \left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)\right)^2+i b \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b}-4 \left(i \log \left(1+e^{\frac{1}{2} i \left(\pi -2 \sin ^{-1}(c x)\right)}\right) \left(a+b \sin ^{-1}(c x)\right)-b \text{Li}_2\left(-e^{\frac{1}{2} i \left(\pi -2 \sin ^{-1}(c x)\right)}\right)\right)\right)\right)}{4 c^4}-\frac{(c f-2 g) (c f+g)^2 \left(-\tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2+i b \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b}+4 \left(i \log \left(1+e^{\frac{1}{2} i \left(2 \sin ^{-1}(c x)+\pi \right)}\right) \left(a+b \sin ^{-1}(c x)\right)+b \text{Li}_2\left(-e^{\frac{1}{2} i \left(2 \sin ^{-1}(c x)+\pi \right)}\right)\right)\right)\right)}{4 c^4}\right)}{d^2 \sqrt{d-c^2 d x^2}}","-\frac{i \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 (c f-g)^3}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) (c f-g)^3}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) (c f-g)^3}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) (c f-g)^3}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{1-c^2 x^2} \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) (c f-g)^3}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-i e^{i \sin ^{-1}(c x)}\right) (c f-g)^3}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) (c f-g)^3}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{i (c f+2 g) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 (c f-g)^2}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f+2 g) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) (c f-g)^2}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f+2 g) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-i e^{i \sin ^{-1}(c x)}\right) (c f-g)^2}{c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{i b^2 (c f+2 g) \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) (c f-g)^2}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i (c f+g)^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (c f+g)^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f+g)^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-i e^{-i \sin ^{-1}(c x)}\right)}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-i e^{-i \sin ^{-1}(c x)}\right)}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i b^2 (c f+g)^3 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{-i \sin ^{-1}(c x)}\right)}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i b^2 (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{-i \sin ^{-1}(c x)}\right)}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 (c f+g)^3 \sqrt{1-c^2 x^2} \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}",1,"(Sqrt[1 - c^2*x^2]*(((c*f - g)^2*(c*f + 2*g)*(I*b*((a + b*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi - 2*ArcSin[c*x]))] - b*PolyLog[2, -E^((I/2)*(Pi - 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 - ArcSin[c*x]/2]))/(4*c^4) - ((c*f - g)^3*(2*b*(a + b*ArcSin[c*x])*Sec[Pi/4 - ArcSin[c*x]/2]^2 + 4*b^2*Tan[Pi/4 - ArcSin[c*x]/2] + (a + b*ArcSin[c*x])^2*Sec[Pi/4 - ArcSin[c*x]/2]^2*Tan[Pi/4 - ArcSin[c*x]/2] - 2*(I*b*((a + b*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi - 2*ArcSin[c*x]))] - b*PolyLog[2, -E^((I/2)*(Pi - 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 - ArcSin[c*x]/2])))/(24*c^4) - ((c*f - 2*g)*(c*f + g)^2*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[c*x]))] + b*PolyLog[2, -E^((I/2)*(Pi + 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2]))/(4*c^4) - ((c*f + g)^3*(2*b*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcSin[c*x]/2]^2 - 4*b^2*Tan[Pi/4 + ArcSin[c*x]/2] - (a + b*ArcSin[c*x])^2*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2] + 2*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[c*x]))] + b*PolyLog[2, -E^((I/2)*(Pi + 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2])))/(24*c^4)))/(d^2*Sqrt[d - c^2*d*x^2])","A",0
80,1,711,1025,6.2605911,"\int \frac{(f+g x)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[((f + g*x)^2*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","\frac{\sqrt{1-c^2 x^2} \left(-\frac{(c f-g)^2 \left(-2 \left(-\tan \left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)\right)^2+i b \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b}-4 \left(i \log \left(1+e^{\frac{1}{2} i \left(\pi -2 \sin ^{-1}(c x)\right)}\right) \left(a+b \sin ^{-1}(c x)\right)-b \text{Li}_2\left(-e^{\frac{1}{2} i \left(\pi -2 \sin ^{-1}(c x)\right)}\right)\right)\right)\right)+2 b \sec ^2\left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)\right)+\tan \left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \sec ^2\left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)\right)^2+4 b^2 \tan \left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{24 c^3}-\frac{(c f+g)^2 \left(2 \left(-\tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2+i b \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b}+4 \left(i \log \left(1+e^{\frac{1}{2} i \left(2 \sin ^{-1}(c x)+\pi \right)}\right) \left(a+b \sin ^{-1}(c x)\right)+b \text{Li}_2\left(-e^{\frac{1}{2} i \left(2 \sin ^{-1}(c x)+\pi \right)}\right)\right)\right)\right)+2 b \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)-\tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2-4 b^2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)\right)}{24 c^3}+\frac{\left(c^2 f^2-g^2\right) \left(-\tan \left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)\right)^2+i b \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b}-4 \left(i \log \left(1+e^{\frac{1}{2} i \left(\pi -2 \sin ^{-1}(c x)\right)}\right) \left(a+b \sin ^{-1}(c x)\right)-b \text{Li}_2\left(-e^{\frac{1}{2} i \left(\pi -2 \sin ^{-1}(c x)\right)}\right)\right)\right)\right)}{4 c^3}-\frac{\left(c^2 f^2-g^2\right) \left(-\tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2+i b \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b}+4 \left(i \log \left(1+e^{\frac{1}{2} i \left(2 \sin ^{-1}(c x)+\pi \right)}\right) \left(a+b \sin ^{-1}(c x)\right)+b \text{Li}_2\left(-e^{\frac{1}{2} i \left(2 \sin ^{-1}(c x)+\pi \right)}\right)\right)\right)\right)}{4 c^3}\right)}{d^2 \sqrt{d-c^2 d x^2}}","\frac{g^2 \left(a+b \sin ^{-1}(c x)\right)^2 x^3}{3 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}-\frac{b g^2 \left(a+b \sin ^{-1}(c x)\right) x^2}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 f^2 \left(a+b \sin ^{-1}(c x)\right)^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{f^2 \left(a+b \sin ^{-1}(c x)\right)^2 x}{3 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}-\frac{2 b f g \left(a+b \sin ^{-1}(c x)\right) x}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{b^2 f^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 g^2 x}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i f^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{i g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 f g \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}-\frac{b^2 g^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b f^2 \left(a+b \sin ^{-1}(c x)\right)}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{4 i b f g \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right)}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 b f^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{2 b g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 f g \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 f g \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 f^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{i b^2 g^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 f g}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}",1,"(Sqrt[1 - c^2*x^2]*(((c^2*f^2 - g^2)*(I*b*((a + b*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi - 2*ArcSin[c*x]))] - b*PolyLog[2, -E^((I/2)*(Pi - 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 - ArcSin[c*x]/2]))/(4*c^3) - ((c*f - g)^2*(2*b*(a + b*ArcSin[c*x])*Sec[Pi/4 - ArcSin[c*x]/2]^2 + 4*b^2*Tan[Pi/4 - ArcSin[c*x]/2] + (a + b*ArcSin[c*x])^2*Sec[Pi/4 - ArcSin[c*x]/2]^2*Tan[Pi/4 - ArcSin[c*x]/2] - 2*(I*b*((a + b*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi - 2*ArcSin[c*x]))] - b*PolyLog[2, -E^((I/2)*(Pi - 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 - ArcSin[c*x]/2])))/(24*c^3) - ((c^2*f^2 - g^2)*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[c*x]))] + b*PolyLog[2, -E^((I/2)*(Pi + 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2]))/(4*c^3) - ((c*f + g)^2*(2*b*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcSin[c*x]/2]^2 - 4*b^2*Tan[Pi/4 + ArcSin[c*x]/2] - (a + b*ArcSin[c*x])^2*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2] + 2*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[c*x]))] + b*PolyLog[2, -E^((I/2)*(Pi + 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2])))/(24*c^3)))/(d^2*Sqrt[d - c^2*d*x^2])","A",0
81,1,683,641,6.2395707,"\int \frac{(f+g x) \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[((f + g*x)*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","\frac{\sqrt{1-c^2 x^2} \left(-\frac{(c f-g) \left(-2 \left(-\tan \left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)\right)^2+i b \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b}-4 \left(i \log \left(1+e^{\frac{1}{2} i \left(\pi -2 \sin ^{-1}(c x)\right)}\right) \left(a+b \sin ^{-1}(c x)\right)-b \text{Li}_2\left(-e^{\frac{1}{2} i \left(\pi -2 \sin ^{-1}(c x)\right)}\right)\right)\right)\right)+2 b \sec ^2\left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)\right)+\tan \left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \sec ^2\left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)\right)^2+4 b^2 \tan \left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{24 c^2}-\frac{(c f+g) \left(2 \left(-\tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2+i b \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b}+4 \left(i \log \left(1+e^{\frac{1}{2} i \left(2 \sin ^{-1}(c x)+\pi \right)}\right) \left(a+b \sin ^{-1}(c x)\right)+b \text{Li}_2\left(-e^{\frac{1}{2} i \left(2 \sin ^{-1}(c x)+\pi \right)}\right)\right)\right)\right)+2 b \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)-\tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2-4 b^2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)\right)}{24 c^2}+\frac{f \left(-\tan \left(\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right) \left(a+b \sin ^{-1}(c x)\right)^2+i b \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b}-4 \left(i \log \left(1+e^{\frac{1}{2} i \left(\pi -2 \sin ^{-1}(c x)\right)}\right) \left(a+b \sin ^{-1}(c x)\right)-b \text{Li}_2\left(-e^{\frac{1}{2} i \left(\pi -2 \sin ^{-1}(c x)\right)}\right)\right)\right)\right)}{4 c}-\frac{f \left(-\tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2+i b \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b}+4 \left(i \log \left(1+e^{\frac{1}{2} i \left(2 \sin ^{-1}(c x)+\pi \right)}\right) \left(a+b \sin ^{-1}(c x)\right)+b \text{Li}_2\left(-e^{\frac{1}{2} i \left(2 \sin ^{-1}(c x)+\pi \right)}\right)\right)\right)\right)}{4 c}\right)}{d^2 \sqrt{d-c^2 d x^2}}","-\frac{b f \left(a+b \sin ^{-1}(c x)\right)}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{2 f x \left(a+b \sin ^{-1}(c x)\right)^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i f \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{f x \left(a+b \sin ^{-1}(c x)\right)^2}{3 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{4 b f \sqrt{1-c^2 x^2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{b g x \left(a+b \sin ^{-1}(c x)\right)}{3 c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{g \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{2 i b g \sqrt{1-c^2 x^2} \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 f \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 f x}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{i b^2 g \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{i b^2 g \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 g}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}",1,"(Sqrt[1 - c^2*x^2]*((f*(I*b*((a + b*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi - 2*ArcSin[c*x]))] - b*PolyLog[2, -E^((I/2)*(Pi - 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 - ArcSin[c*x]/2]))/(4*c) - ((c*f - g)*(2*b*(a + b*ArcSin[c*x])*Sec[Pi/4 - ArcSin[c*x]/2]^2 + 4*b^2*Tan[Pi/4 - ArcSin[c*x]/2] + (a + b*ArcSin[c*x])^2*Sec[Pi/4 - ArcSin[c*x]/2]^2*Tan[Pi/4 - ArcSin[c*x]/2] - 2*(I*b*((a + b*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi - 2*ArcSin[c*x]))] - b*PolyLog[2, -E^((I/2)*(Pi - 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 - ArcSin[c*x]/2])))/(24*c^2) - (f*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[c*x]))] + b*PolyLog[2, -E^((I/2)*(Pi + 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2]))/(4*c) - ((c*f + g)*(2*b*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcSin[c*x]/2]^2 - 4*b^2*Tan[Pi/4 + ArcSin[c*x]/2] - (a + b*ArcSin[c*x])^2*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2] + 2*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[c*x]))] + b*PolyLog[2, -E^((I/2)*(Pi + 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2])))/(24*c^2)))/(d^2*Sqrt[d - c^2*d*x^2])","A",0
82,0,0,38,0.1608743,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^n \log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2}} \, dx","Integrate[((a + b*ArcSin[c*x])^n*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2],x]","\int \frac{\left(a+b \sin ^{-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 \sin ^{-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*ArcSin[c*x])^n*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2], x]","A",-1
83,0,0,634,157.0833963,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^3 \log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2}} \, dx","Integrate[((a + b*ArcSin[c*x])^3*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2],x]","\int \frac{\left(a+b \sin ^{-1}(c x)\right)^3 \log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2}} \, dx","-\frac{6 i b^2 m \left(a+b \sin ^{-1}(c x)\right) \text{Li}_4\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{c}-\frac{6 i b^2 m \left(a+b \sin ^{-1}(c x)\right) \text{Li}_4\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{c}+\frac{i m \left(a+b \sin ^{-1}(c x)\right)^5}{20 b^2 c}+\frac{i m \left(a+b \sin ^{-1}(c x)\right)^3 \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 \left(a+b \sin ^{-1}(c x)\right)^3 \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{3 b m \left(a+b \sin ^{-1}(c x)\right)^2 \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{c}-\frac{3 b m \left(a+b \sin ^{-1}(c x)\right)^2 \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{c}-\frac{m \left(a+b \sin ^{-1}(c x)\right)^4 \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right)}{4 b c}-\frac{m \left(a+b \sin ^{-1}(c x)\right)^4 \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)}{4 b c}+\frac{\left(a+b \sin ^{-1}(c x)\right)^4 \log \left(h (f+g x)^m\right)}{4 b c}+\frac{6 b^3 m \text{Li}_5\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{c}+\frac{6 b^3 m \text{Li}_5\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{c}",1,"Integrate[((a + b*ArcSin[c*x])^3*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2], x]","F",-1
84,0,0,514,78.457824,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2}} \, dx","Integrate[((a + b*ArcSin[c*x])^2*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2],x]","\int \frac{\left(a+b \sin ^{-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 \sin ^{-1}(c x)\right)^4}{12 b^2 c}+\frac{i m \left(a+b \sin ^{-1}(c x)\right)^2 \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 \left(a+b \sin ^{-1}(c x)\right)^2 \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{2 b m \left(a+b \sin ^{-1}(c x)\right) \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{c}-\frac{2 b m \left(a+b \sin ^{-1}(c x)\right) \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{c}-\frac{m \left(a+b \sin ^{-1}(c x)\right)^3 \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right)}{3 b c}-\frac{m \left(a+b \sin ^{-1}(c x)\right)^3 \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)}{3 b c}+\frac{\left(a+b \sin ^{-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{i e^{i \sin ^{-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{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{c}",1,"Integrate[((a + b*ArcSin[c*x])^2*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2], x]","F",-1
85,1,2724,390,9.7865216,"\int \frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2}} \, dx","Integrate[((a + b*ArcSin[c*x])*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2],x]","\text{Result too large to show}","\frac{i m \left(a+b \sin ^{-1}(c x)\right)^3}{6 b^2 c}+\frac{i m \left(a+b \sin ^{-1}(c x)\right) \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 \left(a+b \sin ^{-1}(c x)\right) \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 \left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right)}{2 b c}-\frac{m \left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right)}{2 b c}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(h (f+g x)^m\right)}{2 b c}-\frac{b m \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right)}{c}-\frac{b m \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right)}{c}",1,"(m*ArcSin[c*x]*(2*a + b*ArcSin[c*x])*Log[f + g*x])/(2*c) + (a*ArcSin[c*x]*(-(m*Log[f + g*x]) + Log[h*(f + g*x)^m]))/c + (b*f*(-(m*Log[f + g*x]) + Log[h*(f + g*x)^m])*((-I)*ArcSin[c*x]*(Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]) - PolyLog[2, ((-I)*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] + PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))/Sqrt[c^2*f^2 - g^2] + (a*g*m*(-1/2*(((3*I)/2)*Pi*ArcSin[c*x] - (I/2)*ArcSin[c*x]^2 + 2*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - 2*Pi*Log[Cos[ArcSin[c*x]/2]] + Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(-c^(-1) - f/g)*g) + ((I/2)*Pi*ArcSin[c*x] - (I/2)*ArcSin[c*x]^2 + 2*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + Pi*Log[1 - I*E^(I*ArcSin[c*x])] + 2*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 2*Pi*Log[Cos[ArcSin[c*x]/2]] - Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(2*c*(c^(-1) - f/g)*g) + (((-1/2*I)*ArcSin[c*x]^2)/g + (ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g + (ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g - (I*PolyLog[2, ((-I)*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])])/g - (I*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g)/(c^2*(-c^(-1) - f/g)*(c^(-1) - f/g))))/c - a*c*g*m*(-1/2*(((3*I)/2)*Pi*ArcSin[c*x] - (I/2)*ArcSin[c*x]^2 + 2*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - 2*Pi*Log[Cos[ArcSin[c*x]/2]] + Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c^3*(-c^(-1) - f/g)*g) + ((I/2)*Pi*ArcSin[c*x] - (I/2)*ArcSin[c*x]^2 + 2*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + Pi*Log[1 - I*E^(I*ArcSin[c*x])] + 2*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 2*Pi*Log[Cos[ArcSin[c*x]/2]] - Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*PolyLog[2, I*E^(I*ArcSin[c*x])])/(2*c^3*(c^(-1) - f/g)*g) + (f^2*(((-1/2*I)*ArcSin[c*x]^2)/g + (ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g + (ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g - (I*PolyLog[2, ((-I)*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])])/g - (I*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g))/(c^2*(-c^(-1) - f/g)*(c^(-1) - f/g)*g^2)) + (b*(-(m*Log[f + g*x]) + Log[h*(f + g*x)^m])*(ArcSin[c*x]^2 - 2*c*f*((Pi*ArcTan[(g + c*f*Tan[ArcSin[c*x]/2])/Sqrt[c^2*f^2 - g^2]])/Sqrt[c^2*f^2 - g^2] + (2*ArcCos[-((c*f)/g)]*ArcTanh[((c*f - g)*Cot[(Pi + 2*ArcSin[c*x])/4])/Sqrt[-(c^2*f^2) + g^2]] + (Pi - 2*ArcSin[c*x])*ArcTanh[((c*f + g)*Tan[(Pi + 2*ArcSin[c*x])/4])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f - g)*Cot[(Pi + 2*ArcSin[c*x])/4])/Sqrt[-(c^2*f^2) + g^2]] + ArcTanh[((c*f + g)*Tan[(Pi + 2*ArcSin[c*x])/4])/Sqrt[-(c^2*f^2) + g^2]]))*Log[((1/2 + I/2)*Sqrt[-(c^2*f^2) + g^2])/(E^((I/2)*ArcSin[c*x])*Sqrt[g]*Sqrt[c*f + c*g*x])] + (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((c*f - g)*Cot[(Pi + 2*ArcSin[c*x])/4])/Sqrt[-(c^2*f^2) + g^2]] - (2*I)*ArcTanh[((c*f + g)*Tan[(Pi + 2*ArcSin[c*x])/4])/Sqrt[-(c^2*f^2) + g^2]])*Log[((1/2 - I/2)*E^((I/2)*ArcSin[c*x])*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[g]*Sqrt[c*f + c*g*x])] - (ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((c*f - g)*Cot[(Pi + 2*ArcSin[c*x])/4])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(-(c*f) + g - I*Sqrt[-(c^2*f^2) + g^2])*(1 + I*Cot[(Pi + 2*ArcSin[c*x])/4]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Cot[(Pi + 2*ArcSin[c*x])/4]))] - (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((c*f - g)*Cot[(Pi + 2*ArcSin[c*x])/4])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(I*c*f - I*g + Sqrt[-(c^2*f^2) + g^2])*(I + Cot[(Pi + 2*ArcSin[c*x])/4]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Cot[(Pi + 2*ArcSin[c*x])/4]))] + I*(PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Cot[(Pi + 2*ArcSin[c*x])/4]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Cot[(Pi + 2*ArcSin[c*x])/4]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Cot[(Pi + 2*ArcSin[c*x])/4]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Cot[(Pi + 2*ArcSin[c*x])/4]))]))/Sqrt[-(c^2*f^2) + g^2])))/(2*c) - (b*g*m*(((-1/3*I)*ArcSin[c*x]^3)/g + (ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g + (ArcSin[c*x]^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g - ((2*I)*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g - ((2*I)*ArcSin[c*x]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g + (2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g + (2*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g))/(2*c)","B",0
86,1,246,237,0.0242964,"\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
87,0,0,38,0.2112765,"\int \frac{\log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[Log[h*(f + g*x)^m]/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])),x]","\int \frac{\log \left(h (f+g x)^m\right)}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-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 \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[Log[h*(f + g*x)^m]/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]","A",-1
88,1,305,351,0.4293593,"\int (d+e x)^3 (f+g x) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^3*(f + g*x)*(a + b*ArcSin[c*x]),x]","\frac{120 a c^5 x \left(10 d^3 (2 f+g x)+10 d^2 e x (3 f+2 g x)+5 d e^2 x^2 (4 f+3 g x)+e^3 x^3 (5 f+4 g x)\right)+b \sqrt{1-c^2 x^2} \left(2 c^4 \left(300 d^3 (4 f+g x)+100 d^2 e x (9 f+4 g x)+25 d e^2 x^2 (16 f+9 g x)+3 e^3 x^3 (25 f+16 g x)\right)+c^2 e \left(1600 d^2 g+25 d e (64 f+27 g x)+e^2 x (225 f+128 g x)\right)+256 e^3 g\right)+15 b c \sin ^{-1}(c x) \left(8 c^4 x \left(10 d^3 (2 f+g x)+10 d^2 e x (3 f+2 g x)+5 d e^2 x^2 (4 f+3 g x)+e^3 x^3 (5 f+4 g x)\right)-40 c^2 d^2 (d g+3 e f)-15 e^2 (3 d g+e f)\right)}{2400 c^5}","d^3 f x \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{2} d^2 x^2 (d g+3 e f) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{4} e^2 x^4 (3 d g+e f) \left(a+b \sin ^{-1}(c x)\right)+d e x^3 (d g+e f) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{5} e^3 g x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{b e^2 x^3 \sqrt{1-c^2 x^2} (3 d g+e f)}{16 c}+\frac{b e^3 g x^4 \sqrt{1-c^2 x^2}}{25 c}-\frac{b \sin ^{-1}(c x) \left(8 c^2 d^2 (d g+3 e f)+3 e^2 (3 d g+e f)\right)}{32 c^4}+\frac{b e x^2 \sqrt{1-c^2 x^2} \left(25 c^2 d (d g+e f)+4 e^2 g\right)}{75 c^3}+\frac{b \sqrt{1-c^2 x^2} \left(75 c^2 x \left(8 c^2 d^2 (d g+3 e f)+3 e^2 (3 d g+e f)\right)+32 \left(75 c^4 d^3 f+50 c^2 d e (d g+e f)+8 e^3 g\right)\right)}{2400 c^5}",1,"(120*a*c^5*x*(10*d^3*(2*f + g*x) + 10*d^2*e*x*(3*f + 2*g*x) + 5*d*e^2*x^2*(4*f + 3*g*x) + e^3*x^3*(5*f + 4*g*x)) + b*Sqrt[1 - c^2*x^2]*(256*e^3*g + 2*c^4*(300*d^3*(4*f + g*x) + 100*d^2*e*x*(9*f + 4*g*x) + 25*d*e^2*x^2*(16*f + 9*g*x) + 3*e^3*x^3*(25*f + 16*g*x)) + c^2*e*(1600*d^2*g + 25*d*e*(64*f + 27*g*x) + e^2*x*(225*f + 128*g*x))) + 15*b*c*(-40*c^2*d^2*(3*e*f + d*g) - 15*e^2*(e*f + 3*d*g) + 8*c^4*x*(10*d^3*(2*f + g*x) + 10*d^2*e*x*(3*f + 2*g*x) + 5*d*e^2*x^2*(4*f + 3*g*x) + e^3*x^3*(5*f + 4*g*x)))*ArcSin[c*x])/(2400*c^5)","A",1
89,1,211,248,0.3146826,"\int (d+e x)^2 (f+g x) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^2*(f + g*x)*(a + b*ArcSin[c*x]),x]","\frac{24 a c^4 x \left(6 d^2 (2 f+g x)+4 d e x (3 f+2 g x)+e^2 x^2 (4 f+3 g x)\right)+b c \sqrt{1-c^2 x^2} \left(2 c^2 \left(36 d^2 (4 f+g x)+8 d e x (9 f+4 g x)+e^2 x^2 (16 f+9 g x)\right)+e (128 d g+64 e f+27 e g x)\right)+3 b \sin ^{-1}(c x) \left(8 c^4 x \left(6 d^2 (2 f+g x)+4 d e x (3 f+2 g x)+e^2 x^2 (4 f+3 g x)\right)-24 c^2 d (d g+2 e f)-9 e^2 g\right)}{288 c^4}","d^2 f x \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} e x^3 (2 d g+e f) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{2} d x^2 (d g+2 e f) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{4} e^2 g x^4 \left(a+b \sin ^{-1}(c x)\right)+\frac{b e x^2 \sqrt{1-c^2 x^2} (2 d g+e f)}{9 c}+\frac{b e^2 g x^3 \sqrt{1-c^2 x^2}}{16 c}-\frac{b \sin ^{-1}(c x) \left(8 c^2 d (d g+2 e f)+3 e^2 g\right)}{32 c^4}+\frac{b \sqrt{1-c^2 x^2} \left(32 \left(9 c^2 d^2 f+2 e (2 d g+e f)\right)+9 x \left(8 c^2 d (d g+2 e f)+3 e^2 g\right)\right)}{288 c^3}",1,"(24*a*c^4*x*(6*d^2*(2*f + g*x) + 4*d*e*x*(3*f + 2*g*x) + e^2*x^2*(4*f + 3*g*x)) + b*c*Sqrt[1 - c^2*x^2]*(e*(64*e*f + 128*d*g + 27*e*g*x) + 2*c^2*(36*d^2*(4*f + g*x) + 8*d*e*x*(9*f + 4*g*x) + e^2*x^2*(16*f + 9*g*x))) + 3*b*(-9*e^2*g - 24*c^2*d*(2*e*f + d*g) + 8*c^4*x*(6*d^2*(2*f + g*x) + 4*d*e*x*(3*f + 2*g*x) + e^2*x^2*(4*f + 3*g*x)))*ArcSin[c*x])/(288*c^4)","A",1
90,1,138,148,0.1858915,"\int (d+e x) (f+g x) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)*(f + g*x)*(a + b*ArcSin[c*x]),x]","\frac{6 a c^3 x (3 d (2 f+g x)+e x (3 f+2 g x))+b \sqrt{1-c^2 x^2} \left(c^2 (9 d (4 f+g x)+e x (9 f+4 g x))+8 e g\right)+3 b c \sin ^{-1}(c x) \left(12 c^2 d f x+3 d g \left(2 c^2 x^2-1\right)+e f \left(6 c^2 x^2-3\right)+4 c^2 e g x^3\right)}{36 c^3}","\frac{1}{2} x^2 (d g+e f) \left(a+b \sin ^{-1}(c x)\right)+d f x \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} e g x^3 \left(a+b \sin ^{-1}(c x)\right)-\frac{b \sin ^{-1}(c x) (d g+e f)}{4 c^2}+\frac{b e g x^2 \sqrt{1-c^2 x^2}}{9 c}+\frac{b \sqrt{1-c^2 x^2} \left(9 c^2 x (d g+e f)+4 \left(9 c^2 d f+2 e g\right)\right)}{36 c^3}",1,"(6*a*c^3*x*(3*d*(2*f + g*x) + e*x*(3*f + 2*g*x)) + b*Sqrt[1 - c^2*x^2]*(8*e*g + c^2*(9*d*(4*f + g*x) + e*x*(9*f + 4*g*x))) + 3*b*c*(12*c^2*d*f*x + 4*c^2*e*g*x^3 + 3*d*g*(-1 + 2*c^2*x^2) + e*f*(-3 + 6*c^2*x^2))*ArcSin[c*x])/(36*c^3)","A",1
91,1,282,344,0.565178,"\int \frac{(f+g x) \left(a+b \sin ^{-1}(c x)\right)}{d+e x} \, dx","Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x),x]","\frac{(e f-d g) \log (d+e x) \left(a+b \sin ^{-1}(c x)\right)+e g x \left(a+b \sin ^{-1}(c x)\right)-\frac{1}{2} i b (e f-d g) \left(2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)+2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)+\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)\right)\right)\right)+\frac{b e g \sqrt{1-c^2 x^2}}{c}-b \sin ^{-1}(c x) (e f-d g) \log (d+e x)}{e^2}","\frac{(e f-d g) \log (d+e x) \left(a+b \sin ^{-1}(c x)\right)}{e^2}+\frac{g x \left(a+b \sin ^{-1}(c x)\right)}{e}-\frac{i b (e f-d g) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^2}-\frac{i b (e f-d g) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^2}+\frac{b \sin ^{-1}(c x) (e f-d g) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^2}+\frac{b \sin ^{-1}(c x) (e f-d g) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)}{e^2}+\frac{b g \sqrt{1-c^2 x^2}}{c e}-\frac{i b \sin ^{-1}(c x)^2 (e f-d g)}{2 e^2}-\frac{b \sin ^{-1}(c x) (e f-d g) \log (d+e x)}{e^2}",1,"((b*e*g*Sqrt[1 - c^2*x^2])/c + e*g*x*(a + b*ArcSin[c*x]) - b*(e*f - d*g)*ArcSin[c*x]*Log[d + e*x] + (e*f - d*g)*(a + b*ArcSin[c*x])*Log[d + e*x] - (I/2)*b*(e*f - d*g)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])) + 2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + 2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/e^2","A",1
92,1,322,358,0.4884117,"\int \frac{(f+g x) \left(a+b \sin ^{-1}(c x)\right)}{(d+e x)^2} \, dx","Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^2,x]","\frac{-\frac{(e f-d g) \left(a+b \sin ^{-1}(c x)\right)}{d+e x}+g \log (d+e x) \left(a+b \sin ^{-1}(c x)\right)+\frac{b c (e f-d g) \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right)}{\sqrt{c^2 d^2-e^2}}-\frac{1}{2} i b g \left(2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)+2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)+\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)\right)\right)\right)-b g \sin ^{-1}(c x) \log (d+e x)}{e^2}","-\frac{(e f-d g) \left(a+b \sin ^{-1}(c x)\right)}{e^2 (d+e x)}+\frac{g \log (d+e x) \left(a+b \sin ^{-1}(c x)\right)}{e^2}+\frac{b c (e f-d g) \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right)}{e^2 \sqrt{c^2 d^2-e^2}}-\frac{i b g \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^2}-\frac{i b g \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^2}+\frac{b g \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^2}+\frac{b g \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)}{e^2}-\frac{b g \sin ^{-1}(c x) \log (d+e x)}{e^2}-\frac{i b g \sin ^{-1}(c x)^2}{2 e^2}",1,"(-(((e*f - d*g)*(a + b*ArcSin[c*x]))/(d + e*x)) + (b*c*(e*f - d*g)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2] - b*g*ArcSin[c*x]*Log[d + e*x] + g*(a + b*ArcSin[c*x])*Log[d + e*x] - (I/2)*b*g*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])) + 2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + 2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/e^2","A",1
93,1,263,202,0.5398598,"\int \frac{(f+g x) \left(a+b \sin ^{-1}(c x)\right)}{(d+e x)^3} \, dx","Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^3,x]","\frac{\frac{a (d g-e f)}{(d+e x)^2}-\frac{2 a g}{d+e x}-\frac{b c e \sqrt{1-c^2 x^2} (e f-d g)}{\left(e^2-c^2 d^2\right) (d+e x)}+\frac{b c \left(c^2 d (d g+e f)-2 e^2 g\right) \log \left(\sqrt{1-c^2 x^2} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right)}{(e-c d) (c d+e) \sqrt{e^2-c^2 d^2}}+\frac{b c \log (d+e x) \left(c^2 d (d g+e f)-2 e^2 g\right)}{(c d-e) (c d+e) \sqrt{e^2-c^2 d^2}}-\frac{b \sin ^{-1}(c x) (d g+e (f+2 g x))}{(d+e x)^2}}{2 e^2}","-\frac{(f+g x)^2 \left(a+b \sin ^{-1}(c x)\right)}{2 (d+e x)^2 (e f-d g)}+\frac{b c \sqrt{1-c^2 x^2} (e f-d g)}{2 e \left(c^2 d^2-e^2\right) (d+e x)}-\frac{b c \left(2 e^2 g-c^2 d (d g+e f)\right) \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right)}{2 e^2 \left(c^2 d^2-e^2\right)^{3/2}}+\frac{b g^2 \sin ^{-1}(c x)}{2 e^2 (e f-d g)}",1,"((a*(-(e*f) + d*g))/(d + e*x)^2 - (2*a*g)/(d + e*x) - (b*c*e*(e*f - d*g)*Sqrt[1 - c^2*x^2])/((-(c^2*d^2) + e^2)*(d + e*x)) - (b*(d*g + e*(f + 2*g*x))*ArcSin[c*x])/(d + e*x)^2 + (b*c*(-2*e^2*g + c^2*d*(e*f + d*g))*Log[d + e*x])/((c*d - e)*(c*d + e)*Sqrt[-(c^2*d^2) + e^2]) + (b*c*(-2*e^2*g + c^2*d*(e*f + d*g))*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/((-(c*d) + e)*(c*d + e)*Sqrt[-(c^2*d^2) + e^2]))/(2*e^2)","A",1
94,1,321,257,0.7729415,"\int \frac{(f+g x) \left(a+b \sin ^{-1}(c x)\right)}{(d+e x)^4} \, dx","Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^4,x]","\frac{\frac{a (2 d g-2 e f)}{(d+e x)^3}-\frac{3 a g}{(d+e x)^2}+\frac{b c e \sqrt{1-c^2 x^2} \left(c^2 d \left(d^2 (-g)+4 d e f+3 e^2 f x\right)-e^2 (2 d g+e (f+3 g x))\right)}{\left(e^2-c^2 d^2\right)^2 (d+e x)^2}-\frac{b c^3 \left(c^2 d^2 (d g+2 e f)+e^2 (e f-4 d g)\right) \log \left(\sqrt{1-c^2 x^2} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right)}{(e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}+\frac{b c^3 \log (d+e x) \left(c^2 d^2 (d g+2 e f)+e^2 (e f-4 d g)\right)}{(e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}-\frac{b \sin ^{-1}(c x) (d g+2 e f+3 e g x)}{(d+e x)^3}}{6 e^2}","-\frac{(e f-d g) \left(a+b \sin ^{-1}(c x)\right)}{3 e^2 (d+e x)^3}-\frac{g \left(a+b \sin ^{-1}(c x)\right)}{2 e^2 (d+e x)^2}+\frac{b c \sqrt{1-c^2 x^2} \left(c^2 d f-e g\right)}{2 \left(c^2 d^2-e^2\right)^2 (d+e x)}+\frac{b c \sqrt{1-c^2 x^2} (e f-d g)}{6 e \left(c^2 d^2-e^2\right) (d+e x)^2}+\frac{b c^3 \left(c^2 d^2 (d g+2 e f)+e^2 (e f-4 d g)\right) \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right)}{6 e^2 \left(c^2 d^2-e^2\right)^{5/2}}",1,"((a*(-2*e*f + 2*d*g))/(d + e*x)^3 - (3*a*g)/(d + e*x)^2 + (b*c*e*Sqrt[1 - c^2*x^2]*(c^2*d*(4*d*e*f - d^2*g + 3*e^2*f*x) - e^2*(2*d*g + e*(f + 3*g*x))))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (b*(2*e*f + d*g + 3*e*g*x)*ArcSin[c*x])/(d + e*x)^3 + (b*c^3*(e^2*(e*f - 4*d*g) + c^2*d^2*(2*e*f + d*g))*Log[d + e*x])/((-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]) - (b*c^3*(e^2*(e*f - 4*d*g) + c^2*d^2*(2*e*f + d*g))*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/((-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]))/(6*e^2)","A",1
95,1,418,360,1.2756297,"\int \frac{(f+g x) \left(a+b \sin ^{-1}(c x)\right)}{(d+e x)^5} \, dx","Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^5,x]","\frac{\frac{a (6 d g-6 e f)}{(d+e x)^4}-\frac{8 a g}{(d+e x)^3}-\frac{b e \sqrt{1-c^2 x^2} \left(c^5 d^2 \left(-2 d^3 g+d^2 e (18 f+g x)+d e^2 x (27 f+g x)+11 e^3 f x^2\right)-c^3 e^2 \left(15 d^3 g+5 d^2 e (f+7 g x)+d e^2 x (16 g x-3 f)-4 e^3 f x^2\right)+2 c e^4 (d g+e (f+2 g x))\right)}{\left(e^2-c^2 d^2\right)^3 (d+e x)^3}+\frac{b c^3 \left(2 c^4 d^3 (d g+3 e f)+c^2 d e^2 (9 e f-13 d g)-4 e^4 g\right) \log \left(\sqrt{1-c^2 x^2} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right)}{(e-c d)^3 (c d+e)^3 \sqrt{e^2-c^2 d^2}}+\frac{b c^3 \log (d+e x) \left(-2 c^4 d^3 (d g+3 e f)+c^2 d e^2 (13 d g-9 e f)+4 e^4 g\right)}{(e-c d)^3 (c d+e)^3 \sqrt{e^2-c^2 d^2}}-\frac{2 b \sin ^{-1}(c x) (d g+3 e f+4 e g x)}{(d+e x)^4}}{24 e^2}","-\frac{(e f-d g) \left(a+b \sin ^{-1}(c x)\right)}{4 e^2 (d+e x)^4}-\frac{g \left(a+b \sin ^{-1}(c x)\right)}{3 e^2 (d+e x)^3}-\frac{b c \sqrt{1-c^2 x^2} \left(4 e^2 g-c^2 d (5 e f-d g)\right)}{24 e \left(c^2 d^2-e^2\right)^2 (d+e x)^2}+\frac{b c \sqrt{1-c^2 x^2} (e f-d g)}{12 e \left(c^2 d^2-e^2\right) (d+e x)^3}+\frac{b c^3 \sqrt{1-c^2 x^2} \left(c^2 d^2 (d g+11 e f)+4 e^2 (e f-4 d g)\right)}{24 e \left(c^2 d^2-e^2\right)^3 (d+e x)}-\frac{b c^3 \left(-2 c^4 d^3 (d g+3 e f)-c^2 d e^2 (9 e f-13 d g)+4 e^4 g\right) \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right)}{24 e^2 \left(c^2 d^2-e^2\right)^{7/2}}",1,"((a*(-6*e*f + 6*d*g))/(d + e*x)^4 - (8*a*g)/(d + e*x)^3 - (b*e*Sqrt[1 - c^2*x^2]*(c^5*d^2*(-2*d^3*g + 11*e^3*f*x^2 + d^2*e*(18*f + g*x) + d*e^2*x*(27*f + g*x)) + 2*c*e^4*(d*g + e*(f + 2*g*x)) - c^3*e^2*(15*d^3*g - 4*e^3*f*x^2 + 5*d^2*e*(f + 7*g*x) + d*e^2*x*(-3*f + 16*g*x))))/((-(c^2*d^2) + e^2)^3*(d + e*x)^3) - (2*b*(3*e*f + d*g + 4*e*g*x)*ArcSin[c*x])/(d + e*x)^4 + (b*c^3*(4*e^4*g - 2*c^4*d^3*(3*e*f + d*g) + c^2*d*e^2*(-9*e*f + 13*d*g))*Log[d + e*x])/((-(c*d) + e)^3*(c*d + e)^3*Sqrt[-(c^2*d^2) + e^2]) + (b*c^3*(-4*e^4*g + c^2*d*e^2*(9*e*f - 13*d*g) + 2*c^4*d^3*(3*e*f + d*g))*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/((-(c*d) + e)^3*(c*d + e)^3*Sqrt[-(c^2*d^2) + e^2]))/(24*e^2)","A",1
96,1,494,457,1.3829666,"\int \frac{(f+g x) \left(a+b \sin ^{-1}(c x)\right)}{(d+e x)^6} \, dx","Integrate[((f + g*x)*(a + b*ArcSin[c*x]))/(d + e*x)^6,x]","\frac{\frac{3 a (8 d g-8 e f)}{(d+e x)^5}-\frac{30 a g}{(d+e x)^4}+\frac{b c e \sqrt{1-c^2 x^2} \left(-2 \left(e^2-c^2 d^2\right)^2 (d+e x) \left(c^2 d (2 d g-7 e f)+5 e^2 g\right)-c^2 \left(c^2 d^2-e^2\right) (d+e x)^2 \left(c^2 d^2 (d g-26 e f)+e^2 (34 d g-9 e f)\right)-6 \left(e^2-c^2 d^2\right)^3 (e f-d g)+5 c^2 (d+e x)^3 \left(c^4 d^3 (d g+10 e f)+c^2 d e^2 (11 e f-18 d g)-4 e^4 g\right)\right)}{\left(e^2-c^2 d^2\right)^4 (d+e x)^4}-\frac{3 b c^5 \left(2 c^4 d^4 (d g+4 e f)+c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)\right) \log \left(\sqrt{1-c^2 x^2} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right)}{(e-c d)^4 (c d+e)^4 \sqrt{e^2-c^2 d^2}}+\frac{3 b c^5 \log (d+e x) \left(2 c^4 d^4 (d g+4 e f)+c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)\right)}{(e-c d)^4 (c d+e)^4 \sqrt{e^2-c^2 d^2}}-\frac{6 b \sin ^{-1}(c x) (d g+4 e f+5 e g x)}{(d+e x)^5}}{120 e^2}","-\frac{(e f-d g) \left(a+b \sin ^{-1}(c x)\right)}{5 e^2 (d+e x)^5}-\frac{g \left(a+b \sin ^{-1}(c x)\right)}{4 e^2 (d+e x)^4}-\frac{b c \sqrt{1-c^2 x^2} \left(5 e^2 g-c^2 d (7 e f-2 d g)\right)}{60 e \left(c^2 d^2-e^2\right)^2 (d+e x)^3}+\frac{b c \sqrt{1-c^2 x^2} (e f-d g)}{20 e \left(c^2 d^2-e^2\right) (d+e x)^4}+\frac{b c^3 \sqrt{1-c^2 x^2} \left(c^2 d^2 (26 e f-d g)+e^2 (9 e f-34 d g)\right)}{120 e \left(c^2 d^2-e^2\right)^3 (d+e x)^2}+\frac{b c^5 \left(2 c^4 d^4 (d g+4 e f)+c^2 d^2 e^2 (24 e f-19 d g)+3 e^4 (e f-6 d g)\right) \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right)}{40 e^2 \left(c^2 d^2-e^2\right)^{9/2}}-\frac{b c^3 \sqrt{1-c^2 x^2} \left(c^4 \left(-d^3\right) (d g+10 e f)-c^2 d e^2 (11 e f-18 d g)+4 e^4 g\right)}{24 e \left(c^2 d^2-e^2\right)^4 (d+e x)}",1,"((3*a*(-8*e*f + 8*d*g))/(d + e*x)^5 - (30*a*g)/(d + e*x)^4 + (b*c*e*Sqrt[1 - c^2*x^2]*(-6*(-(c^2*d^2) + e^2)^3*(e*f - d*g) - 2*(-(c^2*d^2) + e^2)^2*(5*e^2*g + c^2*d*(-7*e*f + 2*d*g))*(d + e*x) - c^2*(c^2*d^2 - e^2)*(c^2*d^2*(-26*e*f + d*g) + e^2*(-9*e*f + 34*d*g))*(d + e*x)^2 + 5*c^2*(-4*e^4*g + c^2*d*e^2*(11*e*f - 18*d*g) + c^4*d^3*(10*e*f + d*g))*(d + e*x)^3))/((-(c^2*d^2) + e^2)^4*(d + e*x)^4) - (6*b*(4*e*f + d*g + 5*e*g*x)*ArcSin[c*x])/(d + e*x)^5 + (3*b*c^5*(c^2*d^2*e^2*(24*e*f - 19*d*g) + 3*e^4*(e*f - 6*d*g) + 2*c^4*d^4*(4*e*f + d*g))*Log[d + e*x])/((-(c*d) + e)^4*(c*d + e)^4*Sqrt[-(c^2*d^2) + e^2]) - (3*b*c^5*(c^2*d^2*e^2*(24*e*f - 19*d*g) + 3*e^4*(e*f - 6*d*g) + 2*c^4*d^4*(4*e*f + d*g))*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/((-(c*d) + e)^4*(c*d + e)^4*Sqrt[-(c^2*d^2) + e^2]))/(120*e^2)","A",1
97,1,463,512,0.5557489,"\int (d+e x)^3 \left(f+g x+h x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^3*(f + g*x + h*x^2)*(a + b*ArcSin[c*x]),x]","a d^3 f x+\frac{1}{4} a e x^4 \left(3 d^2 h+3 d e g+e^2 f\right)+\frac{1}{3} a d x^3 \left(d^2 h+3 d e g+3 e^2 f\right)+\frac{1}{2} a d^2 x^2 (d g+3 e f)+\frac{1}{5} a e^2 x^5 (3 d h+e g)+\frac{1}{6} a e^3 h x^6-\frac{b \sin ^{-1}(c x) \left(24 c^4 d^2 (d g+3 e f)+9 c^2 e \left(3 d^2 h+3 d e g+e^2 f\right)+5 e^3 h\right)}{96 c^6}+\frac{b \sqrt{1-c^2 x^2} \left(2 c^4 \left(100 d^3 (36 f+x (9 g+4 h x))+75 d^2 e x (36 f+x (16 g+9 h x))+3 d e^2 x^2 (400 f+9 x (25 g+16 h x))+e^3 x^3 (225 f+4 x (36 g+25 h x))\right)+c^2 \left(1600 d^3 h+75 d^2 e (64 g+27 h x)+3 d e^2 \left(1600 f+675 g x+384 h x^2\right)+e^3 x \left(675 f+384 g x+250 h x^2\right)\right)+3 e^2 (768 d h+256 e g+125 e h x)\right)}{7200 c^5}+\frac{1}{60} b x \sin ^{-1}(c x) \left(10 d^3 (6 f+x (3 g+2 h x))+15 d^2 e x (6 f+x (4 g+3 h x))+3 d e^2 x^2 (20 f+3 x (5 g+4 h x))+e^3 x^3 (15 f+2 x (6 g+5 h x))\right)","d^3 f x \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{4} e x^4 \left(a+b \sin ^{-1}(c x)\right) \left(3 d^2 h+3 d e g+e^2 f\right)+\frac{1}{3} d x^3 \left(a+b \sin ^{-1}(c x)\right) \left(d^2 h+3 d e g+3 e^2 f\right)+\frac{1}{2} d^2 x^2 (d g+3 e f) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{5} e^2 x^5 (3 d h+e g) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{6} e^3 h x^6 \left(a+b \sin ^{-1}(c x)\right)+\frac{b e^2 x^4 \sqrt{1-c^2 x^2} (3 d h+e g)}{25 c}+\frac{b e^3 h x^5 \sqrt{1-c^2 x^2}}{36 c}+\frac{b x^2 \sqrt{1-c^2 x^2} \left(25 c^2 d \left(d^2 h+3 d e g+3 e^2 f\right)+12 e^2 (3 d h+e g)\right)}{225 c^3}+\frac{b e x^3 \sqrt{1-c^2 x^2} \left(9 c^2 \left(3 d^2 h+3 d e g+e^2 f\right)+5 e^2 h\right)}{144 c^3}-\frac{b \sin ^{-1}(c x) \left(24 c^4 d^2 (d g+3 e f)+9 c^2 e \left(3 d^2 h+3 d e g+e^2 f\right)+5 e^3 h\right)}{96 c^6}+\frac{b \sqrt{1-c^2 x^2} \left(75 x \left(24 c^4 d^2 (d g+3 e f)+9 c^2 e \left(3 d^2 h+3 d e g+e^2 f\right)+5 e^3 h\right)+32 \left(225 c^4 d^3 f+50 c^2 d \left(d^2 h+3 d e g+3 e^2 f\right)+24 e^2 (3 d h+e g)\right)\right)}{7200 c^5}",1,"a*d^3*f*x + (a*d^2*(3*e*f + d*g)*x^2)/2 + (a*d*(3*e^2*f + 3*d*e*g + d^2*h)*x^3)/3 + (a*e*(e^2*f + 3*d*e*g + 3*d^2*h)*x^4)/4 + (a*e^2*(e*g + 3*d*h)*x^5)/5 + (a*e^3*h*x^6)/6 + (b*Sqrt[1 - c^2*x^2]*(3*e^2*(256*e*g + 768*d*h + 125*e*h*x) + c^2*(1600*d^3*h + 75*d^2*e*(64*g + 27*h*x) + e^3*x*(675*f + 384*g*x + 250*h*x^2) + 3*d*e^2*(1600*f + 675*g*x + 384*h*x^2)) + 2*c^4*(100*d^3*(36*f + x*(9*g + 4*h*x)) + 75*d^2*e*x*(36*f + x*(16*g + 9*h*x)) + 3*d*e^2*x^2*(400*f + 9*x*(25*g + 16*h*x)) + e^3*x^3*(225*f + 4*x*(36*g + 25*h*x)))))/(7200*c^5) - (b*(24*c^4*d^2*(3*e*f + d*g) + 5*e^3*h + 9*c^2*e*(e^2*f + 3*d*e*g + 3*d^2*h))*ArcSin[c*x])/(96*c^6) + (b*x*(10*d^3*(6*f + x*(3*g + 2*h*x)) + 15*d^2*e*x*(6*f + x*(4*g + 3*h*x)) + 3*d*e^2*x^2*(20*f + 3*x*(5*g + 4*h*x)) + e^3*x^3*(15*f + 2*x*(6*g + 5*h*x)))*ArcSin[c*x])/60","A",1
98,1,307,361,0.534807,"\int (d+e x)^2 \left(f+g x+h x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^2*(f + g*x + h*x^2)*(a + b*ArcSin[c*x]),x]","\frac{120 a c^5 x \left(10 d^2 (6 f+x (3 g+2 h x))+10 d e x (6 f+x (4 g+3 h x))+e^2 x^2 (20 f+3 x (5 g+4 h x))\right)+b \sqrt{1-c^2 x^2} \left(2 c^4 \left(100 d^2 (36 f+x (9 g+4 h x))+50 d e x (36 f+x (16 g+9 h x))+e^2 x^2 (400 f+9 x (25 g+16 h x))\right)+c^2 \left(1600 d^2 h+50 d e (64 g+27 h x)+e^2 \left(1600 f+675 g x+384 h x^2\right)\right)+768 e^2 h\right)+15 b c \sin ^{-1}(c x) \left(8 c^4 x \left(10 d^2 (6 f+x (3 g+2 h x))+10 d e x (6 f+x (4 g+3 h x))+e^2 x^2 (20 f+3 x (5 g+4 h x))\right)-120 c^2 d (d g+2 e f)-45 e (2 d h+e g)\right)}{7200 c^5}","\frac{1}{3} x^3 \left(a+b \sin ^{-1}(c x)\right) \left(d^2 h+2 d e g+e^2 f\right)+d^2 f x \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{2} d x^2 (d g+2 e f) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{4} e x^4 (2 d h+e g) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{5} e^2 h x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{b e x^3 \sqrt{1-c^2 x^2} (2 d h+e g)}{16 c}+\frac{b e^2 h x^4 \sqrt{1-c^2 x^2}}{25 c}-\frac{b \sin ^{-1}(c x) \left(8 c^2 d (d g+2 e f)+3 e (2 d h+e g)\right)}{32 c^4}+\frac{b x^2 \sqrt{1-c^2 x^2} \left(25 c^2 \left(d^2 h+2 d e g+e^2 f\right)+12 e^2 h\right)}{225 c^3}+\frac{b \sqrt{1-c^2 x^2} \left(225 c^2 x \left(8 c^2 d (d g+2 e f)+3 e (2 d h+e g)\right)+32 \left(225 c^4 d^2 f+50 c^2 \left(d^2 h+2 d e g+e^2 f\right)+24 e^2 h\right)\right)}{7200 c^5}",1,"(120*a*c^5*x*(10*d^2*(6*f + x*(3*g + 2*h*x)) + 10*d*e*x*(6*f + x*(4*g + 3*h*x)) + e^2*x^2*(20*f + 3*x*(5*g + 4*h*x))) + b*Sqrt[1 - c^2*x^2]*(768*e^2*h + c^2*(1600*d^2*h + 50*d*e*(64*g + 27*h*x) + e^2*(1600*f + 675*g*x + 384*h*x^2)) + 2*c^4*(100*d^2*(36*f + x*(9*g + 4*h*x)) + 50*d*e*x*(36*f + x*(16*g + 9*h*x)) + e^2*x^2*(400*f + 9*x*(25*g + 16*h*x)))) + 15*b*c*(-120*c^2*d*(2*e*f + d*g) - 45*e*(e*g + 2*d*h) + 8*c^4*x*(10*d^2*(6*f + x*(3*g + 2*h*x)) + 10*d*e*x*(6*f + x*(4*g + 3*h*x)) + e^2*x^2*(20*f + 3*x*(5*g + 4*h*x))))*ArcSin[c*x])/(7200*c^5)","A",1
99,1,186,223,0.320344,"\int (d+e x) \left(f+g x+h x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)*(f + g*x + h*x^2)*(a + b*ArcSin[c*x]),x]","\frac{24 a c^4 x (2 d (6 f+x (3 g+2 h x))+e x (6 f+x (4 g+3 h x)))+b c \sqrt{1-c^2 x^2} \left(2 c^2 \left(4 d \left(36 f+9 g x+4 h x^2\right)+e x \left(36 f+16 g x+9 h x^2\right)\right)+64 d h+64 e g+27 e h x\right)+3 b \sin ^{-1}(c x) \left(8 c^4 x \left(2 d \left(6 f+3 g x+2 h x^2\right)+e x \left(6 f+4 g x+3 h x^2\right)\right)-24 c^2 (d g+e f)-9 e h\right)}{288 c^4}","\frac{1}{2} x^2 (d g+e f) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} x^3 (d h+e g) \left(a+b \sin ^{-1}(c x)\right)+d f x \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{4} e h x^4 \left(a+b \sin ^{-1}(c x)\right)+\frac{b x^2 \sqrt{1-c^2 x^2} (d h+e g)}{9 c}+\frac{b e h x^3 \sqrt{1-c^2 x^2}}{16 c}-\frac{b \sin ^{-1}(c x) \left(8 c^2 (d g+e f)+3 e h\right)}{32 c^4}+\frac{b \sqrt{1-c^2 x^2} \left(9 x \left(8 c^2 (d g+e f)+3 e h\right)+32 \left(9 c^2 d f+2 d h+2 e g\right)\right)}{288 c^3}",1,"(24*a*c^4*x*(2*d*(6*f + x*(3*g + 2*h*x)) + e*x*(6*f + x*(4*g + 3*h*x))) + b*c*Sqrt[1 - c^2*x^2]*(64*e*g + 64*d*h + 27*e*h*x + 2*c^2*(4*d*(36*f + 9*g*x + 4*h*x^2) + e*x*(36*f + 16*g*x + 9*h*x^2))) + 3*b*(-24*c^2*(e*f + d*g) - 9*e*h + 8*c^4*x*(2*d*(6*f + 3*g*x + 2*h*x^2) + e*x*(6*f + 4*g*x + 3*h*x^2)))*ArcSin[c*x])/(288*c^4)","A",1
100,1,381,459,0.6535727,"\int \frac{\left(f+g x+h x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{d+e x} \, dx","Integrate[((f + g*x + h*x^2)*(a + b*ArcSin[c*x]))/(d + e*x),x]","\frac{2 \log (d+e x) \left(a+b \sin ^{-1}(c x)\right) \left(d^2 h-d e g+e^2 f\right)+2 e x (e g-d h) \left(a+b \sin ^{-1}(c x)\right)+e^2 h x^2 \left(a+b \sin ^{-1}(c x)\right)+\frac{b \left(-2 i c^2 \left(d^2 h-d e g+e^2 f\right) \left(2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)+2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)+\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)\right)\right)\right)-4 c^2 \sin ^{-1}(c x) \log (d+e x) \left(d^2 h-d e g+e^2 f\right)+4 c e \sqrt{1-c^2 x^2} (e g-d h)+c e^2 h x \sqrt{1-c^2 x^2}-e^2 h \sin ^{-1}(c x)\right)}{2 c^2}}{2 e^3}","\frac{\log (d+e x) \left(a+b \sin ^{-1}(c x)\right) \left(d^2 h-d e g+e^2 f\right)}{e^3}+\frac{x (e g-d h) \left(a+b \sin ^{-1}(c x)\right)}{e^2}+\frac{h x^2 \left(a+b \sin ^{-1}(c x)\right)}{2 e}-\frac{i b \left(d^2 h-d e g+e^2 f\right) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^3}-\frac{i b \left(d^2 h-d e g+e^2 f\right) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^3}+\frac{b \sin ^{-1}(c x) \left(d^2 h-d e g+e^2 f\right) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^3}+\frac{b \sin ^{-1}(c x) \left(d^2 h-d e g+e^2 f\right) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)}{e^3}+\frac{b \sqrt{1-c^2 x^2} (4 (e g-d h)+e h x)}{4 c e^2}-\frac{b h \sin ^{-1}(c x)}{4 c^2 e}-\frac{i b \sin ^{-1}(c x)^2 \left(d^2 h-d e g+e^2 f\right)}{2 e^3}-\frac{b \sin ^{-1}(c x) \log (d+e x) \left(d^2 h-d e g+e^2 f\right)}{e^3}",1,"(2*e*(e*g - d*h)*x*(a + b*ArcSin[c*x]) + e^2*h*x^2*(a + b*ArcSin[c*x]) + 2*(e^2*f - d*e*g + d^2*h)*(a + b*ArcSin[c*x])*Log[d + e*x] + (b*(4*c*e*(e*g - d*h)*Sqrt[1 - c^2*x^2] + c*e^2*h*x*Sqrt[1 - c^2*x^2] - e^2*h*ArcSin[c*x] - 4*c^2*(e^2*f - d*e*g + d^2*h)*ArcSin[c*x]*Log[d + e*x] - (2*I)*c^2*(e^2*f - d*e*g + d^2*h)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])) + 2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + 2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])))/(2*c^2))/(2*e^3)","A",1
101,1,392,460,0.9739044,"\int \frac{\left(f+g x+h x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{(d+e x)^2} \, dx","Integrate[((f + g*x + h*x^2)*(a + b*ArcSin[c*x]))/(d + e*x)^2,x]","\frac{-\frac{\left(a+b \sin ^{-1}(c x)\right) \left(d^2 h-d e g+e^2 f\right)}{d+e x}+(e g-2 d h) \log (d+e x) \left(a+b \sin ^{-1}(c x)\right)+e h x \left(a+b \sin ^{-1}(c x)\right)+\frac{b c \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right) \left(d^2 h-d e g+e^2 f\right)}{\sqrt{c^2 d^2-e^2}}-\frac{1}{2} i b (e g-2 d h) \left(2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)+2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)+\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)\right)\right)\right)+\frac{b e h \sqrt{1-c^2 x^2}}{c}-b \sin ^{-1}(c x) (e g-2 d h) \log (d+e x)}{e^3}","-\frac{\left(a+b \sin ^{-1}(c x)\right) \left(d^2 h-d e g+e^2 f\right)}{e^3 (d+e x)}+\frac{(e g-2 d h) \log (d+e x) \left(a+b \sin ^{-1}(c x)\right)}{e^3}+\frac{h x \left(a+b \sin ^{-1}(c x)\right)}{e^2}+\frac{b c \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right) \left(d^2 h-d e g+e^2 f\right)}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{i b (e g-2 d h) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^3}-\frac{i b (e g-2 d h) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^3}+\frac{b \sin ^{-1}(c x) (e g-2 d h) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^3}+\frac{b \sin ^{-1}(c x) (e g-2 d h) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)}{e^3}+\frac{b h \sqrt{1-c^2 x^2}}{c e^2}-\frac{i b \sin ^{-1}(c x)^2 (e g-2 d h)}{2 e^3}-\frac{b \sin ^{-1}(c x) (e g-2 d h) \log (d+e x)}{e^3}",1,"((b*e*h*Sqrt[1 - c^2*x^2])/c + e*h*x*(a + b*ArcSin[c*x]) - ((e^2*f - d*e*g + d^2*h)*(a + b*ArcSin[c*x]))/(d + e*x) + (b*c*(e^2*f - d*e*g + d^2*h)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2] - b*(e*g - 2*d*h)*ArcSin[c*x]*Log[d + e*x] + (e*g - 2*d*h)*(a + b*ArcSin[c*x])*Log[d + e*x] - (I/2)*b*(e*g - 2*d*h)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])) + 2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + 2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/e^3","A",1
102,1,996,488,6.6906536,"\int \frac{\left(f+g x+h x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{(d+e x)^3} \, dx","Integrate[((f + g*x + h*x^2)*(a + b*ArcSin[c*x]))/(d + e*x)^3,x]","\frac{2 a d h-a e g}{e^3 (d+e x)}+b f \left(-\frac{c \sqrt{\frac{-d-\sqrt{\frac{1}{c^2}} e}{d+e x}+1} \sqrt{\frac{\sqrt{\frac{1}{c^2}} e-d}{d+e x}+1} F_1\left(2;\frac{1}{2},\frac{1}{2};3;-\frac{\sqrt{\frac{1}{c^2}} e-d}{d+e x},-\frac{-d-\sqrt{\frac{1}{c^2}} e}{d+e x}\right)}{4 e^2 (d+e x) \sqrt{1-c^2 x^2}}-\frac{\sin ^{-1}(c x)}{2 e (d+e x)^2}\right)+\frac{a h \log (d+e x)}{e^3}+b g \left(\frac{\frac{c \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right)}{\sqrt{c^2 d^2-e^2}}-\frac{\sin ^{-1}(c x)}{d+e x}}{e^2}-\frac{d \left(-\frac{i d \left(\log \left(\frac{e^2 \sqrt{c^2 d^2-e^2} \left(i d x c^2+i e+\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}\right)}{c^3 d (d+e x)}\right)+\log (4)\right) c^3}{(c d-e) e (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{\sqrt{1-c^2 x^2} c}{\left(c^2 d^2-e^2\right) (d+e x)}-\frac{\sin ^{-1}(c x)}{e (d+e x)^2}\right)}{2 e}\right)+b h \left(\frac{\left(-\frac{i d \left(\log \left(\frac{e^2 \sqrt{c^2 d^2-e^2} \left(i d x c^2+i e+\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}\right)}{c^3 d (d+e x)}\right)+\log (4)\right) c^3}{(c d-e) e (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{\sqrt{1-c^2 x^2} c}{\left(c^2 d^2-e^2\right) (d+e x)}-\frac{\sin ^{-1}(c x)}{e (d+e x)^2}\right) d^2}{2 e^2}-\frac{2 \left(\frac{c \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right)}{\sqrt{c^2 d^2-e^2}}-\frac{\sin ^{-1}(c x)}{d+e x}\right) d}{e^3}+\frac{-\frac{i \sin ^{-1}(c x)^2}{2 e}+\frac{\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right) \sin ^{-1}(c x)}{e}+\frac{\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right) \sin ^{-1}(c x)}{e}-\frac{i \text{Li}_2\left(-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)}{e}-\frac{i \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e}}{e^2}\right)+\frac{-a h d^2+a e g d-a e^2 f}{2 e^3 (d+e x)^2}","-\frac{\left(a+b \sin ^{-1}(c x)\right) \left(d^2 h-d e g+e^2 f\right)}{2 e^3 (d+e x)^2}-\frac{(e g-2 d h) \left(a+b \sin ^{-1}(c x)\right)}{e^3 (d+e x)}+\frac{h \log (d+e x) \left(a+b \sin ^{-1}(c x)\right)}{e^3}+\frac{b c \sqrt{1-c^2 x^2} \left(d^2 h-d e g+e^2 f\right)}{2 e^2 \left(c^2 d^2-e^2\right) (d+e x)}-\frac{b c \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right) \left(2 e^2 (e g-2 d h)-c^2 d \left(-3 d^2 h+d e g+e^2 f\right)\right)}{2 e^3 \left(c^2 d^2-e^2\right)^{3/2}}-\frac{i b h \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^3}-\frac{i b h \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^3}+\frac{b h \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^3}+\frac{b h \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)}{e^3}-\frac{b h \sin ^{-1}(c x) \log (d+e x)}{e^3}-\frac{i b h \sin ^{-1}(c x)^2}{2 e^3}",1,"(-(a*e^2*f) + a*d*e*g - a*d^2*h)/(2*e^3*(d + e*x)^2) + (-(a*e*g) + 2*a*d*h)/(e^3*(d + e*x)) + b*f*(-1/4*(c*Sqrt[1 + (-d - Sqrt[c^(-2)]*e)/(d + e*x)]*Sqrt[1 + (-d + Sqrt[c^(-2)]*e)/(d + e*x)]*AppellF1[2, 1/2, 1/2, 3, -((-d + Sqrt[c^(-2)]*e)/(d + e*x)), -((-d - Sqrt[c^(-2)]*e)/(d + e*x))])/(e^2*(d + e*x)*Sqrt[1 - c^2*x^2]) - ArcSin[c*x]/(2*e*(d + e*x)^2)) + (a*h*Log[d + e*x])/e^3 + b*g*((-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2])/e^2 - (d*((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2) - (I*c^3*d*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*(d + e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2])))/(2*e)) + b*h*((-2*d*(-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2]))/e^3 + (d^2*((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2) - (I*c^3*d*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*(d + e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2])))/(2*e^2) + (((-1/2*I)*ArcSin[c*x]^2)/e + (ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e + (ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e - (I*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])])/e - (I*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e)/e^2)","C",0
103,1,442,349,1.831416,"\int \frac{\left(f+g x+h x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{(d+e x)^4} \, dx","Integrate[((f + g*x + h*x^2)*(a + b*ArcSin[c*x]))/(d + e*x)^4,x]","-\frac{\frac{2 a \left(d^2 h-d e g+e^2 f\right)}{(d+e x)^3}+\frac{3 a (e g-2 d h)}{(d+e x)^2}+\frac{6 a h}{d+e x}+\frac{b c e \sqrt{1-c^2 x^2} \left(c^2 d \left(2 d^3 h+d^2 e (g+3 h x)-4 d e^2 f-3 e^3 f x\right)+e^2 \left(-5 d^2 h+2 d e (g-3 h x)+e^2 (f+3 g x)\right)\right)}{\left(e^2-c^2 d^2\right)^2 (d+e x)^2}+\frac{b c \log \left(\sqrt{1-c^2 x^2} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right) \left(c^4 d^2 \left(2 d^2 h+d e g+2 e^2 f\right)+c^2 e^2 \left(-5 d^2 h-4 d e g+e^2 f\right)+6 e^4 h\right)}{(e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}-\frac{b c \log (d+e x) \left(c^4 d^2 \left(2 d^2 h+d e g+2 e^2 f\right)+c^2 e^2 \left(-5 d^2 h-4 d e g+e^2 f\right)+6 e^4 h\right)}{(e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}+\frac{b \sin ^{-1}(c x) \left(2 d^2 h+d e (g+6 h x)+e^2 (2 f+3 x (g+2 h x))\right)}{(d+e x)^3}}{6 e^3}","-\frac{\left(a+b \sin ^{-1}(c x)\right) \left(d^2 h-d e g+e^2 f\right)}{3 e^3 (d+e x)^3}-\frac{(e g-2 d h) \left(a+b \sin ^{-1}(c x)\right)}{2 e^3 (d+e x)^2}-\frac{h \left(a+b \sin ^{-1}(c x)\right)}{e^3 (d+e x)}+\frac{b c \sqrt{1-c^2 x^2} \left(d^2 h-d e g+e^2 f\right)}{6 e^2 \left(c^2 d^2-e^2\right) (d+e x)^2}-\frac{b c \sqrt{1-c^2 x^2} \left(e^2 (e g-2 d h)-c^2 \left(d e^2 f-d^3 h\right)\right)}{2 e^2 \left(c^2 d^2-e^2\right)^2 (d+e x)}+\frac{b c \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right) \left(c^4 d^2 \left(2 d^2 h+d e g+2 e^2 f\right)+c^2 e^2 \left(-5 d^2 h-4 d e g+e^2 f\right)+6 e^4 h\right)}{6 e^3 \left(c^2 d^2-e^2\right)^{5/2}}",1,"-1/6*((2*a*(e^2*f - d*e*g + d^2*h))/(d + e*x)^3 + (3*a*(e*g - 2*d*h))/(d + e*x)^2 + (6*a*h)/(d + e*x) + (b*c*e*Sqrt[1 - c^2*x^2]*(e^2*(-5*d^2*h + e^2*(f + 3*g*x) + 2*d*e*(g - 3*h*x)) + c^2*d*(-4*d*e^2*f + 2*d^3*h - 3*e^3*f*x + d^2*e*(g + 3*h*x))))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) + (b*(2*d^2*h + d*e*(g + 6*h*x) + e^2*(2*f + 3*x*(g + 2*h*x)))*ArcSin[c*x])/(d + e*x)^3 - (b*c*(6*e^4*h + c^2*e^2*(e^2*f - 4*d*e*g - 5*d^2*h) + c^4*d^2*(2*e^2*f + d*e*g + 2*d^2*h))*Log[d + e*x])/((-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]) + (b*c*(6*e^4*h + c^2*e^2*(e^2*f - 4*d*e*g - 5*d^2*h) + c^4*d^2*(2*e^2*f + d*e*g + 2*d^2*h))*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/((-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]))/e^3","A",1
104,1,575,470,2.9213669,"\int \frac{\left(f+g x+h x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{(d+e x)^5} \, dx","Integrate[((f + g*x + h*x^2)*(a + b*ArcSin[c*x]))/(d + e*x)^5,x]","-\frac{\frac{6 a \left(d^2 h-d e g+e^2 f\right)}{(d+e x)^4}+\frac{8 a (e g-2 d h)}{(d+e x)^3}+\frac{12 a h}{(d+e x)^2}+\frac{b c e \sqrt{1-c^2 x^2} \left(c^4 d^2 \left(-2 d^4 h-d^3 e (2 g+5 h x)+d^2 e^2 (18 f+x (g-h x))+d e^3 x (27 f+g x)+11 e^4 f x^2\right)+c^2 e^2 \left(11 d^4 h+d^3 e (19 h x-15 g)+d^2 e^2 (x (4 h x-35 g)-5 f)+d e^3 x (3 f-16 g x)+4 e^4 f x^2\right)+2 e^4 \left(3 d^2 h+d e (g+8 h x)+e^2 (f+2 x (g+3 h x))\right)\right)}{\left(e^2-c^2 d^2\right)^3 (d+e x)^3}+\frac{b c^3 \log \left(\sqrt{1-c^2 x^2} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right) \left(2 c^4 d^3 \left(d^2 h+d e g+3 e^2 f\right)+c^2 d e^2 \left(-7 d^2 h-13 d e g+9 e^2 f\right)-4 e^4 (e g-5 d h)\right)}{(c d-e)^3 (c d+e)^3 \sqrt{e^2-c^2 d^2}}-\frac{b c^3 \log (d+e x) \left(2 c^4 d^3 \left(d^2 h+d e g+3 e^2 f\right)+c^2 d e^2 \left(-7 d^2 h-13 d e g+9 e^2 f\right)-4 e^4 (e g-5 d h)\right)}{(c d-e)^3 (c d+e)^3 \sqrt{e^2-c^2 d^2}}+\frac{2 b \sin ^{-1}(c x) \left(d^2 h+d e (g+4 h x)+e^2 \left(3 f+4 g x+6 h x^2\right)\right)}{(d+e x)^4}}{24 e^3}","-\frac{\left(a+b \sin ^{-1}(c x)\right) \left(d^2 h-d e g+e^2 f\right)}{4 e^3 (d+e x)^4}-\frac{(e g-2 d h) \left(a+b \sin ^{-1}(c x)\right)}{3 e^3 (d+e x)^3}-\frac{h \left(a+b \sin ^{-1}(c x)\right)}{2 e^3 (d+e x)^2}-\frac{b c \sqrt{1-c^2 x^2} \left(4 e^2 (e g-2 d h)-c^2 d \left(-3 d^2 h-d e g+5 e^2 f\right)\right)}{24 e^2 \left(c^2 d^2-e^2\right)^2 (d+e x)^2}+\frac{b c \sqrt{1-c^2 x^2} \left(d^2 h-d e g+e^2 f\right)}{12 e^2 \left(c^2 d^2-e^2\right) (d+e x)^3}+\frac{b c \sqrt{1-c^2 x^2} \left(c^4 d^2 \left(d^2 (-h)+d e g+11 e^2 f\right)+4 c^2 e^2 \left(d^2 h-4 d e g+e^2 f\right)+12 e^4 h\right)}{24 e^2 \left(c^2 d^2-e^2\right)^3 (d+e x)}-\frac{b c^3 \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right) \left(-2 c^4 d^3 \left(d^2 h+d e g+3 e^2 f\right)-c^2 d e^2 \left(-7 d^2 h-13 d e g+9 e^2 f\right)+4 e^4 (e g-5 d h)\right)}{24 e^3 \left(c^2 d^2-e^2\right)^{7/2}}",1,"-1/24*((6*a*(e^2*f - d*e*g + d^2*h))/(d + e*x)^4 + (8*a*(e*g - 2*d*h))/(d + e*x)^3 + (12*a*h)/(d + e*x)^2 + (b*c*e*Sqrt[1 - c^2*x^2]*(c^4*d^2*(-2*d^4*h + 11*e^4*f*x^2 + d*e^3*x*(27*f + g*x) - d^3*e*(2*g + 5*h*x) + d^2*e^2*(18*f + x*(g - h*x))) + 2*e^4*(3*d^2*h + d*e*(g + 8*h*x) + e^2*(f + 2*x*(g + 3*h*x))) + c^2*e^2*(11*d^4*h + 4*e^4*f*x^2 + d*e^3*x*(3*f - 16*g*x) + d^3*e*(-15*g + 19*h*x) + d^2*e^2*(-5*f + x*(-35*g + 4*h*x)))))/((-(c^2*d^2) + e^2)^3*(d + e*x)^3) + (2*b*(d^2*h + d*e*(g + 4*h*x) + e^2*(3*f + 4*g*x + 6*h*x^2))*ArcSin[c*x])/(d + e*x)^4 - (b*c^3*(-4*e^4*(e*g - 5*d*h) + c^2*d*e^2*(9*e^2*f - 13*d*e*g - 7*d^2*h) + 2*c^4*d^3*(3*e^2*f + d*e*g + d^2*h))*Log[d + e*x])/((c*d - e)^3*(c*d + e)^3*Sqrt[-(c^2*d^2) + e^2]) + (b*c^3*(-4*e^4*(e*g - 5*d*h) + c^2*d*e^2*(9*e^2*f - 13*d*e*g - 7*d^2*h) + 2*c^4*d^3*(3*e^2*f + d*e*g + d^2*h))*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/((c*d - e)^3*(c*d + e)^3*Sqrt[-(c^2*d^2) + e^2]))/e^3","A",1
105,1,682,593,2.5915212,"\int \frac{\left(f+g x+h x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{(d+e x)^6} \, dx","Integrate[((f + g*x + h*x^2)*(a + b*ArcSin[c*x]))/(d + e*x)^6,x]","-\frac{\frac{24 a \left(d^2 h-d e g+e^2 f\right)}{(d+e x)^5}+\frac{30 a (e g-2 d h)}{(d+e x)^4}+\frac{40 a h}{(d+e x)^3}-\frac{b c e \sqrt{1-c^2 x^2} \left(-2 \left(e^2-c^2 d^2\right)^2 (d+e x) \left(c^2 d \left(3 d^2 h+2 d e g-7 e^2 f\right)+5 e^2 (e g-2 d h)\right)+6 \left(c^2 d^2-e^2\right)^3 \left(d^2 h-d e g+e^2 f\right)-\left(e^2-c^2 d^2\right) (d+e x)^2 \left(c^4 \left(-d^2\right) \left(4 d^2 h+d e g-26 e^2 f\right)+c^2 e^2 \left(19 d^2 h-34 d e g+9 e^2 f\right)+20 e^4 h\right)+5 c^2 e (d+e x)^3 \left(c^4 d^3 (d g+10 e f)+c^2 d e \left(d^2 h-18 d e g+11 e^2 f\right)-4 e^3 (e g-5 d h)\right)\right)}{\left(e^2-c^2 d^2\right)^4 (d+e x)^4}+\frac{b c^3 \log \left(\sqrt{1-c^2 x^2} \sqrt{e^2-c^2 d^2}+c^2 d x+e\right) \left(2 c^6 d^4 \left(2 d^2 h+3 d e g+12 e^2 f\right)-3 c^4 d^2 e^2 \left(6 d^2 h+19 d e g-24 e^2 f\right)+9 c^2 e^4 \left(11 d^2 h-6 d e g+e^2 f\right)+20 e^6 h\right)}{(e-c d)^4 (c d+e)^4 \sqrt{e^2-c^2 d^2}}-\frac{b c^3 \log (d+e x) \left(2 c^6 d^4 \left(2 d^2 h+3 d e g+12 e^2 f\right)-3 c^4 d^2 e^2 \left(6 d^2 h+19 d e g-24 e^2 f\right)+9 c^2 e^4 \left(11 d^2 h-6 d e g+e^2 f\right)+20 e^6 h\right)}{(e-c d)^4 (c d+e)^4 \sqrt{e^2-c^2 d^2}}+\frac{2 b \sin ^{-1}(c x) \left(2 d^2 h+d e (3 g+10 h x)+e^2 (12 f+5 x (3 g+4 h x))\right)}{(d+e x)^5}}{120 e^3}","-\frac{\left(a+b \sin ^{-1}(c x)\right) \left(d^2 h-d e g+e^2 f\right)}{5 e^3 (d+e x)^5}-\frac{(e g-2 d h) \left(a+b \sin ^{-1}(c x)\right)}{4 e^3 (d+e x)^4}-\frac{h \left(a+b \sin ^{-1}(c x)\right)}{3 e^3 (d+e x)^3}-\frac{b c \sqrt{1-c^2 x^2} \left(5 e^2 (e g-2 d h)-c^2 d \left(-3 d^2 h-2 d e g+7 e^2 f\right)\right)}{60 e^2 \left(c^2 d^2-e^2\right)^2 (d+e x)^3}+\frac{b c \sqrt{1-c^2 x^2} \left(d^2 h-d e g+e^2 f\right)}{20 e^2 \left(c^2 d^2-e^2\right) (d+e x)^4}+\frac{b c \sqrt{1-c^2 x^2} \left(c^4 d^2 \left(-4 d^2 h-d e g+26 e^2 f\right)+c^2 e^2 \left(19 d^2 h-34 d e g+9 e^2 f\right)+20 e^4 h\right)}{120 e^2 \left(c^2 d^2-e^2\right)^3 (d+e x)^2}+\frac{b c^3 \sqrt{1-c^2 x^2} \left(c^4 d^3 (d g+10 e f)+c^2 d e \left(d^2 h-18 d e g+11 e^2 f\right)-4 e^3 (e g-5 d h)\right)}{24 e \left(c^2 d^2-e^2\right)^4 (d+e x)}+\frac{b c^3 \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right) \left(2 c^6 d^4 \left(2 d^2 h+3 d e g+12 e^2 f\right)+3 c^4 d^2 e^2 \left(-6 d^2 h-19 d e g+24 e^2 f\right)+9 c^2 e^4 \left(11 d^2 h-6 d e g+e^2 f\right)+20 e^6 h\right)}{120 e^3 \left(c^2 d^2-e^2\right)^{9/2}}",1,"-1/120*((24*a*(e^2*f - d*e*g + d^2*h))/(d + e*x)^5 + (30*a*(e*g - 2*d*h))/(d + e*x)^4 + (40*a*h)/(d + e*x)^3 - (b*c*e*Sqrt[1 - c^2*x^2]*(6*(c^2*d^2 - e^2)^3*(e^2*f - d*e*g + d^2*h) - 2*(-(c^2*d^2) + e^2)^2*(5*e^2*(e*g - 2*d*h) + c^2*d*(-7*e^2*f + 2*d*e*g + 3*d^2*h))*(d + e*x) - (-(c^2*d^2) + e^2)*(20*e^4*h - c^4*d^2*(-26*e^2*f + d*e*g + 4*d^2*h) + c^2*e^2*(9*e^2*f - 34*d*e*g + 19*d^2*h))*(d + e*x)^2 + 5*c^2*e*(c^4*d^3*(10*e*f + d*g) - 4*e^3*(e*g - 5*d*h) + c^2*d*e*(11*e^2*f - 18*d*e*g + d^2*h))*(d + e*x)^3))/((-(c^2*d^2) + e^2)^4*(d + e*x)^4) + (2*b*(2*d^2*h + d*e*(3*g + 10*h*x) + e^2*(12*f + 5*x*(3*g + 4*h*x)))*ArcSin[c*x])/(d + e*x)^5 - (b*c^3*(20*e^6*h + 2*c^6*d^4*(12*e^2*f + 3*d*e*g + 2*d^2*h) - 3*c^4*d^2*e^2*(-24*e^2*f + 19*d*e*g + 6*d^2*h) + 9*c^2*e^4*(e^2*f - 6*d*e*g + 11*d^2*h))*Log[d + e*x])/((-(c*d) + e)^4*(c*d + e)^4*Sqrt[-(c^2*d^2) + e^2]) + (b*c^3*(20*e^6*h + 2*c^6*d^4*(12*e^2*f + 3*d*e*g + 2*d^2*h) - 3*c^4*d^2*e^2*(-24*e^2*f + 19*d*e*g + 6*d^2*h) + 9*c^2*e^4*(e^2*f - 6*d*e*g + 11*d^2*h))*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/((-(c*d) + e)^4*(c*d + e)^4*Sqrt[-(c^2*d^2) + e^2]))/e^3","A",1
106,1,619,684,0.9490957,"\int (d+e x)^3 \left(f+g x+h x^2+i x^3\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^3*(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]),x]","a d^3 f x+\frac{1}{3} a d x^3 \left(d^2 h+3 d e g+3 e^2 f\right)+\frac{1}{5} a e x^5 \left(3 d^2 i+3 d e h+e^2 g\right)+\frac{1}{2} a d^2 x^2 (d g+3 e f)+\frac{1}{4} a x^4 \left(d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right)+\frac{1}{6} a e^2 x^6 (3 d i+e h)+\frac{1}{7} a e^3 i x^7-\frac{b \sin ^{-1}(c x) \left(24 c^4 d^2 (d g+3 e f)+9 c^2 \left(d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right)+5 e^2 (3 d i+e h)\right)}{96 c^6}+\frac{b \sqrt{1-c^2 x^2} \left(2 c^6 \left(1225 d^3 (144 f+x (36 g+x (16 h+9 i x)))+147 d^2 e x (900 f+x (400 g+9 x (25 h+16 i x)))+147 d e^2 x^2 (400 f+x (225 g+4 x (36 h+25 i x)))+e^3 x^3 (11025 f+4 x (1764 g+25 x (49 h+36 i x)))\right)+c^4 \left(1225 d^3 (64 h+27 i x)+147 d^2 e \left(1600 g+675 h x+384 i x^2\right)+147 d e^2 \left(1600 f+x \left(675 g+384 h x+250 i x^2\right)\right)+e^3 x \left(33075 f+2 x \left(9408 g+6125 h x+4320 i x^2\right)\right)\right)+3 c^2 e \left(37632 d^2 i+147 d e (256 h+125 i x)+e^2 (12544 g+5 x (1225 h+768 i x))\right)+23040 e^3 i\right)}{352800 c^7}+\frac{1}{420} b x \sin ^{-1}(c x) \left(35 d^3 (12 f+x (6 g+x (4 h+3 i x)))+21 d^2 e x (30 f+x (20 g+3 x (5 h+4 i x)))+21 d e^2 x^2 (20 f+x (15 g+2 x (6 h+5 i x)))+e^3 x^3 (105 f+2 x (42 g+5 x (7 h+6 i x)))\right)","d^3 f x \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} d x^3 \left(a+b \sin ^{-1}(c x)\right) \left(d^2 h+3 d e g+3 e^2 f\right)+\frac{1}{5} e x^5 \left(a+b \sin ^{-1}(c x)\right) \left(3 d^2 i+3 d e h+e^2 g\right)+\frac{1}{2} d^2 x^2 (d g+3 e f) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{4} x^4 \left(a+b \sin ^{-1}(c x)\right) \left(d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right)+\frac{1}{6} e^2 x^6 (3 d i+e h) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{7} e^3 i x^7 \left(a+b \sin ^{-1}(c x)\right)+\frac{b e^2 x^5 \sqrt{1-c^2 x^2} (3 d i+e h)}{36 c}+\frac{b e^3 i x^6 \sqrt{1-c^2 x^2}}{49 c}+\frac{b e x^4 \sqrt{1-c^2 x^2} \left(49 c^2 \left(3 d^2 i+3 d e h+e^2 g\right)+30 e^2 i\right)}{1225 c^3}+\frac{b x^3 \sqrt{1-c^2 x^2} \left(9 c^2 \left(d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right)+5 e^2 (3 d i+e h)\right)}{144 c^3}-\frac{b \sin ^{-1}(c x) \left(24 c^4 d^2 (d g+3 e f)+9 c^2 \left(d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right)+5 e^2 (3 d i+e h)\right)}{96 c^6}+\frac{b x^2 \sqrt{1-c^2 x^2} \left(1225 c^4 d \left(d^2 h+3 d e g+3 e^2 f\right)+588 c^2 e \left(3 d^2 i+3 d e h+e^2 g\right)+360 e^3 i\right)}{11025 c^5}+\frac{b \sqrt{1-c^2 x^2} \left(3675 c^2 x \left(24 c^4 d^2 (d g+3 e f)+9 c^2 \left(d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right)+5 e^2 (3 d i+e h)\right)+32 \left(11025 c^6 d^3 f+2450 c^4 d \left(d^2 h+3 d e g+3 e^2 f\right)+1176 c^2 e \left(3 d^2 i+3 d e h+e^2 g\right)+720 e^3 i\right)\right)}{352800 c^7}",1,"a*d^3*f*x + (a*d^2*(3*e*f + d*g)*x^2)/2 + (a*d*(3*e^2*f + 3*d*e*g + d^2*h)*x^3)/3 + (a*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i)*x^4)/4 + (a*e*(e^2*g + 3*d*e*h + 3*d^2*i)*x^5)/5 + (a*e^2*(e*h + 3*d*i)*x^6)/6 + (a*e^3*i*x^7)/7 + (b*Sqrt[1 - c^2*x^2]*(23040*e^3*i + 3*c^2*e*(37632*d^2*i + 147*d*e*(256*h + 125*i*x) + e^2*(12544*g + 5*x*(1225*h + 768*i*x))) + c^4*(1225*d^3*(64*h + 27*i*x) + 147*d^2*e*(1600*g + 675*h*x + 384*i*x^2) + 147*d*e^2*(1600*f + x*(675*g + 384*h*x + 250*i*x^2)) + e^3*x*(33075*f + 2*x*(9408*g + 6125*h*x + 4320*i*x^2))) + 2*c^6*(1225*d^3*(144*f + x*(36*g + x*(16*h + 9*i*x))) + 147*d^2*e*x*(900*f + x*(400*g + 9*x*(25*h + 16*i*x))) + 147*d*e^2*x^2*(400*f + x*(225*g + 4*x*(36*h + 25*i*x))) + e^3*x^3*(11025*f + 4*x*(1764*g + 25*x*(49*h + 36*i*x))))))/(352800*c^7) - (b*(24*c^4*d^2*(3*e*f + d*g) + 5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i))*ArcSin[c*x])/(96*c^6) + (b*x*(35*d^3*(12*f + x*(6*g + x*(4*h + 3*i*x))) + 21*d^2*e*x*(30*f + x*(20*g + 3*x*(5*h + 4*i*x))) + 21*d*e^2*x^2*(20*f + x*(15*g + 2*x*(6*h + 5*i*x))) + e^3*x^3*(105*f + 2*x*(42*g + 5*x*(7*h + 6*i*x))))*ArcSin[c*x])/420","A",1
107,1,380,484,0.9433074,"\int (d+e x)^2 \left(f+g x+h x^2+i x^3\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^2*(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]),x]","\frac{1}{3} x^3 \left(a+b \sin ^{-1}(c x)\right) \left(d^2 h+2 d e g+e^2 f\right)+\frac{1}{4} x^4 \left(a+b \sin ^{-1}(c x)\right) \left(d^2 i+2 d e h+e^2 g\right)+d^2 f x \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{2} d x^2 (d g+2 e f) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{5} e x^5 (2 d i+e h) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{6} e^2 i x^6 \left(a+b \sin ^{-1}(c x)\right)+\frac{b \left(c \sqrt{1-c^2 x^2} \left(2 c^4 \left(25 d^2 (144 f+x (36 g+x (16 h+9 i x)))+2 d e x (900 f+x (400 g+9 x (25 h+16 i x)))+e^2 x^2 (400 f+x (225 g+4 x (36 h+25 i x)))\right)+c^2 \left(25 d^2 (64 h+27 i x)+2 d e \left(1600 g+675 h x+384 i x^2\right)+e^2 \left(1600 f+x \left(675 g+384 h x+250 i x^2\right)\right)\right)+3 e (512 d i+256 e h+125 e i x)\right)-75 \sin ^{-1}(c x) \left(24 c^4 d (d g+2 e f)+9 c^2 \left(d^2 i+2 d e h+e^2 g\right)+5 e^2 i\right)\right)}{7200 c^6}","\frac{1}{3} x^3 \left(a+b \sin ^{-1}(c x)\right) \left(d^2 h+2 d e g+e^2 f\right)+\frac{1}{4} x^4 \left(a+b \sin ^{-1}(c x)\right) \left(d^2 i+2 d e h+e^2 g\right)+d^2 f x \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{2} d x^2 (d g+2 e f) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{5} e x^5 (2 d i+e h) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{6} e^2 i x^6 \left(a+b \sin ^{-1}(c x)\right)+\frac{b e x^4 \sqrt{1-c^2 x^2} (2 d i+e h)}{25 c}+\frac{b e^2 i x^5 \sqrt{1-c^2 x^2}}{36 c}+\frac{b x^2 \sqrt{1-c^2 x^2} \left(25 c^2 \left(d^2 h+2 d e g+e^2 f\right)+12 e (2 d i+e h)\right)}{225 c^3}+\frac{b x^3 \sqrt{1-c^2 x^2} \left(9 c^2 \left(d^2 i+2 d e h+e^2 g\right)+5 e^2 i\right)}{144 c^3}-\frac{b \sin ^{-1}(c x) \left(24 c^4 d (d g+2 e f)+9 c^2 \left(d^2 i+2 d e h+e^2 g\right)+5 e^2 i\right)}{96 c^6}+\frac{b \sqrt{1-c^2 x^2} \left(75 x \left(24 c^4 d (d g+2 e f)+9 c^2 \left(d^2 i+2 d e h+e^2 g\right)+5 e^2 i\right)+32 \left(225 c^4 d^2 f+50 c^2 \left(d^2 h+2 d e g+e^2 f\right)+24 e (2 d i+e h)\right)\right)}{7200 c^5}",1,"d^2*f*x*(a + b*ArcSin[c*x]) + (d*(2*e*f + d*g)*x^2*(a + b*ArcSin[c*x]))/2 + ((e^2*f + 2*d*e*g + d^2*h)*x^3*(a + b*ArcSin[c*x]))/3 + ((e^2*g + 2*d*e*h + d^2*i)*x^4*(a + b*ArcSin[c*x]))/4 + (e*(e*h + 2*d*i)*x^5*(a + b*ArcSin[c*x]))/5 + (e^2*i*x^6*(a + b*ArcSin[c*x]))/6 + (b*(c*Sqrt[1 - c^2*x^2]*(3*e*(256*e*h + 512*d*i + 125*e*i*x) + c^2*(25*d^2*(64*h + 27*i*x) + 2*d*e*(1600*g + 675*h*x + 384*i*x^2) + e^2*(1600*f + x*(675*g + 384*h*x + 250*i*x^2))) + 2*c^4*(25*d^2*(144*f + x*(36*g + x*(16*h + 9*i*x))) + 2*d*e*x*(900*f + x*(400*g + 9*x*(25*h + 16*i*x))) + e^2*x^2*(400*f + x*(225*g + 4*x*(36*h + 25*i*x))))) - 75*(24*c^4*d*(2*e*f + d*g) + 5*e^2*i + 9*c^2*(e^2*g + 2*d*e*h + d^2*i))*ArcSin[c*x]))/(7200*c^6)","A",1
108,1,253,308,0.4862713,"\int (d+e x) \left(f+g x+h x^2+i x^3\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)*(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]),x]","\frac{120 a c^5 x (5 d (12 f+x (6 g+x (4 h+3 i x)))+e x (30 f+x (20 g+3 x (5 h+4 i x))))+b \sqrt{1-c^2 x^2} \left(2 c^4 (25 d (144 f+x (36 g+x (16 h+9 i x)))+e x (900 f+x (400 g+9 x (25 h+16 i x))))+c^2 \left(25 d (64 h+27 i x)+e \left(1600 g+675 h x+384 i x^2\right)\right)+768 e i\right)+15 b c \sin ^{-1}(c x) \left(8 c^4 x (5 d (12 f+x (6 g+x (4 h+3 i x)))+e x (30 f+x (20 g+3 x (5 h+4 i x))))-120 c^2 (d g+e f)-45 (d i+e h)\right)}{7200 c^5}","\frac{1}{2} x^2 (d g+e f) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} x^3 (d h+e g) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{4} x^4 (d i+e h) \left(a+b \sin ^{-1}(c x)\right)+d f x \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{5} e i x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{b x^3 \sqrt{1-c^2 x^2} (d i+e h)}{16 c}+\frac{b e i x^4 \sqrt{1-c^2 x^2}}{25 c}-\frac{b \sin ^{-1}(c x) \left(8 c^2 (d g+e f)+3 (d i+e h)\right)}{32 c^4}+\frac{b x^2 \sqrt{1-c^2 x^2} \left(25 c^2 (d h+e g)+12 e i\right)}{225 c^3}+\frac{b \sqrt{1-c^2 x^2} \left(225 c^2 x \left(8 c^2 (d g+e f)+3 (d i+e h)\right)+32 \left(225 c^4 d f+50 c^2 (d h+e g)+24 e i\right)\right)}{7200 c^5}",1,"(120*a*c^5*x*(5*d*(12*f + x*(6*g + x*(4*h + 3*i*x))) + e*x*(30*f + x*(20*g + 3*x*(5*h + 4*i*x)))) + b*Sqrt[1 - c^2*x^2]*(768*e*i + c^2*(25*d*(64*h + 27*i*x) + e*(1600*g + 675*h*x + 384*i*x^2)) + 2*c^4*(25*d*(144*f + x*(36*g + x*(16*h + 9*i*x))) + e*x*(900*f + x*(400*g + 9*x*(25*h + 16*i*x))))) + 15*b*c*(-120*c^2*(e*f + d*g) - 45*(e*h + d*i) + 8*c^4*x*(5*d*(12*f + x*(6*g + x*(4*h + 3*i*x))) + e*x*(30*f + x*(20*g + 3*x*(5*h + 4*i*x)))))*ArcSin[c*x])/(7200*c^5)","A",1
109,1,498,623,1.0163713,"\int \frac{\left(f+g x+h x^2+i x^3\right) \left(a+b \sin ^{-1}(c x)\right)}{d+e x} \, dx","Integrate[((f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]))/(d + e*x),x]","\frac{6 e x \left(a+b \sin ^{-1}(c x)\right) \left(d^2 i-d e h+e^2 g\right)+6 \log (d+e x) \left(a+b \sin ^{-1}(c x)\right) \left(d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right)+3 e^2 x^2 (e h-d i) \left(a+b \sin ^{-1}(c x)\right)+2 e^3 i x^3 \left(a+b \sin ^{-1}(c x)\right)+\frac{b \left(-36 c^3 \sin ^{-1}(c x) \log (d+e x) \left(d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right)+36 c^2 e \sqrt{1-c^2 x^2} \left(d^2 i-d e h+e^2 g\right)+9 c^2 e^2 x \sqrt{1-c^2 x^2} (e h-d i)+4 e^3 i \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right)-18 i c^3 \left(d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right) \left(2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)+2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)+\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)\right)\right)\right)-9 c e^2 \sin ^{-1}(c x) (e h-d i)\right)}{6 c^3}}{6 e^4}","\frac{x \left(a+b \sin ^{-1}(c x)\right) \left(d^2 i-d e h+e^2 g\right)}{e^3}+\frac{\log (d+e x) \left(a+b \sin ^{-1}(c x)\right) \left(d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right)}{e^4}+\frac{x^2 (e h-d i) \left(a+b \sin ^{-1}(c x)\right)}{2 e^2}+\frac{i x^3 \left(a+b \sin ^{-1}(c x)\right)}{3 e}-\frac{i b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right) \left(d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right)}{e^4}-\frac{i b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right) \left(d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right)}{e^4}+\frac{b \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right) \left(d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right)}{e^4}+\frac{b \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right) \left(d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right)}{e^4}-\frac{b \sin ^{-1}(c x) (e h-d i)}{4 c^2 e^2}+\frac{b i x^2 \sqrt{1-c^2 x^2}}{9 c e}+\frac{b \sqrt{1-c^2 x^2} \left(4 \left(9 c^2 \left(d^2 i-d e h+e^2 g\right)+2 e^2 i\right)+9 c^2 e x (e h-d i)\right)}{36 c^3 e^3}-\frac{i b \sin ^{-1}(c x)^2 \left(d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right)}{2 e^4}-\frac{b \sin ^{-1}(c x) \log (d+e x) \left(d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right)}{e^4}",1,"(6*e*(e^2*g - d*e*h + d^2*i)*x*(a + b*ArcSin[c*x]) + 3*e^2*(e*h - d*i)*x^2*(a + b*ArcSin[c*x]) + 2*e^3*i*x^3*(a + b*ArcSin[c*x]) + 6*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x])*Log[d + e*x] + (b*(36*c^2*e*(e^2*g - d*e*h + d^2*i)*Sqrt[1 - c^2*x^2] + 9*c^2*e^2*(e*h - d*i)*x*Sqrt[1 - c^2*x^2] + 4*e^3*i*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) - 9*c*e^2*(e*h - d*i)*ArcSin[c*x] - 36*c^3*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcSin[c*x]*Log[d + e*x] - (18*I)*c^3*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])) + 2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + 2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])))/(6*c^3))/(6*e^4)","A",1
110,1,515,617,1.4828143,"\int \frac{\left(f+g x+h x^2+i x^3\right) \left(a+b \sin ^{-1}(c x)\right)}{(d+e x)^2} \, dx","Integrate[((f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]))/(d + e*x)^2,x]","\frac{2 \log (d+e x) \left(a+b \sin ^{-1}(c x)\right) \left(3 d^2 i-2 d e h+e^2 g\right)-\frac{2 \left(a+b \sin ^{-1}(c x)\right) \left(d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right)}{d+e x}+2 e x (e h-2 d i) \left(a+b \sin ^{-1}(c x)\right)+e^2 i x^2 \left(a+b \sin ^{-1}(c x)\right)-i b \left(3 d^2 i-2 d e h+e^2 g\right) \left(2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)+2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)+\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)\right)\right)\right)+\frac{2 b c \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right) \left(d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right)}{\sqrt{c^2 d^2-e^2}}+\frac{2 b e \sqrt{1-c^2 x^2} (e h-2 d i)}{c}+\frac{b e^2 i x \sqrt{1-c^2 x^2}}{2 c}-\frac{b e^2 i \sin ^{-1}(c x)}{2 c^2}-2 b \sin ^{-1}(c x) \log (d+e x) \left(3 d^2 i-2 d e h+e^2 g\right)}{2 e^4}","\frac{\log (d+e x) \left(a+b \sin ^{-1}(c x)\right) \left(3 d^2 i-2 d e h+e^2 g\right)}{e^4}-\frac{\left(a+b \sin ^{-1}(c x)\right) \left(d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right)}{e^4 (d+e x)}+\frac{x (e h-2 d i) \left(a+b \sin ^{-1}(c x)\right)}{e^3}+\frac{i x^2 \left(a+b \sin ^{-1}(c x)\right)}{2 e^2}-\frac{i b \left(3 d^2 i-2 d e h+e^2 g\right) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^4}-\frac{i b \left(3 d^2 i-2 d e h+e^2 g\right) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^4}+\frac{b \sin ^{-1}(c x) \left(3 d^2 i-2 d e h+e^2 g\right) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^4}+\frac{b \sin ^{-1}(c x) \left(3 d^2 i-2 d e h+e^2 g\right) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)}{e^4}+\frac{b c \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right) \left(d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right)}{e^4 \sqrt{c^2 d^2-e^2}}+\frac{b \sqrt{1-c^2 x^2} (e h-2 d i)}{c e^3}+\frac{b i x \sqrt{1-c^2 x^2}}{4 c e^2}-\frac{b i \sin ^{-1}(c x)}{4 c^2 e^2}-\frac{i b \sin ^{-1}(c x)^2 \left(3 d^2 i-2 d e h+e^2 g\right)}{2 e^4}-\frac{b \sin ^{-1}(c x) \log (d+e x) \left(3 d^2 i-2 d e h+e^2 g\right)}{e^4}",1,"((2*b*e*(e*h - 2*d*i)*Sqrt[1 - c^2*x^2])/c + (b*e^2*i*x*Sqrt[1 - c^2*x^2])/(2*c) - (b*e^2*i*ArcSin[c*x])/(2*c^2) + 2*e*(e*h - 2*d*i)*x*(a + b*ArcSin[c*x]) + e^2*i*x^2*(a + b*ArcSin[c*x]) - (2*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*(a + b*ArcSin[c*x]))/(d + e*x) + (2*b*c*(e^3*f - d*e^2*g + d^2*e*h - d^3*i)*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2] - 2*b*(e^2*g - 2*d*e*h + 3*d^2*i)*ArcSin[c*x]*Log[d + e*x] + 2*(e^2*g - 2*d*e*h + 3*d^2*i)*(a + b*ArcSin[c*x])*Log[d + e*x] - I*b*(e^2*g - 2*d*e*h + 3*d^2*i)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])) + 2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + 2*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/(2*e^4)","A",1
111,1,1556,1016,6.4712309,"\int \frac{\left(f+g x+h x^2+i x^3\right) \left(a+b \sin ^{-1}(c x)\right)}{(d+e x)^3} \, dx","Integrate[((f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]))/(d + e*x)^3,x]","\frac{-3 a i d^2+2 a e h d-a e^2 g}{e^4 (d+e x)}+\frac{a i x}{e^3}+b f \left(-\frac{c \sqrt{\frac{-d-\sqrt{\frac{1}{c^2}} e}{d+e x}+1} \sqrt{\frac{\sqrt{\frac{1}{c^2}} e-d}{d+e x}+1} F_1\left(2;\frac{1}{2},\frac{1}{2};3;-\frac{\sqrt{\frac{1}{c^2}} e-d}{d+e x},-\frac{-d-\sqrt{\frac{1}{c^2}} e}{d+e x}\right)}{4 e^2 (d+e x) \sqrt{1-c^2 x^2}}-\frac{\sin ^{-1}(c x)}{2 e (d+e x)^2}\right)+\frac{(a e h-3 a d i) \log (d+e x)}{e^4}+b g \left(\frac{\frac{c \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right)}{\sqrt{c^2 d^2-e^2}}-\frac{\sin ^{-1}(c x)}{d+e x}}{e^2}-\frac{d \left(-\frac{i d \left(\log \left(\frac{e^2 \sqrt{c^2 d^2-e^2} \left(i d x c^2+i e+\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}\right)}{c^3 d (d+e x)}\right)+\log (4)\right) c^3}{(c d-e) e (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{\sqrt{1-c^2 x^2} c}{\left(c^2 d^2-e^2\right) (d+e x)}-\frac{\sin ^{-1}(c x)}{e (d+e x)^2}\right)}{2 e}\right)+b i \left(-\frac{\left(-\frac{i d \left(\log \left(\frac{e^2 \sqrt{c^2 d^2-e^2} \left(i d x c^2+i e+\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}\right)}{c^3 d (d+e x)}\right)+\log (4)\right) c^3}{(c d-e) e (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{\sqrt{1-c^2 x^2} c}{\left(c^2 d^2-e^2\right) (d+e x)}-\frac{\sin ^{-1}(c x)}{e (d+e x)^2}\right) d^3}{2 e^3}+\frac{3 \left(\frac{c \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right)}{\sqrt{c^2 d^2-e^2}}-\frac{\sin ^{-1}(c x)}{d+e x}\right) d^2}{e^4}-\frac{3 \left(-\frac{i \sin ^{-1}(c x)^2}{2 e}+\frac{\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right) \sin ^{-1}(c x)}{e}+\frac{\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right) \sin ^{-1}(c x)}{e}-\frac{i \text{Li}_2\left(-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)}{e}-\frac{i \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e}\right) d}{e^3}+\frac{c x \sin ^{-1}(c x)+\sqrt{1-c^2 x^2}}{c e^3}\right)+b h \left(\frac{\left(-\frac{i d \left(\log \left(\frac{e^2 \sqrt{c^2 d^2-e^2} \left(i d x c^2+i e+\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}\right)}{c^3 d (d+e x)}\right)+\log (4)\right) c^3}{(c d-e) e (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{\sqrt{1-c^2 x^2} c}{\left(c^2 d^2-e^2\right) (d+e x)}-\frac{\sin ^{-1}(c x)}{e (d+e x)^2}\right) d^2}{2 e^2}-\frac{2 \left(\frac{c \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right)}{\sqrt{c^2 d^2-e^2}}-\frac{\sin ^{-1}(c x)}{d+e x}\right) d}{e^3}+\frac{-\frac{i \sin ^{-1}(c x)^2}{2 e}+\frac{\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right) \sin ^{-1}(c x)}{e}+\frac{\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right) \sin ^{-1}(c x)}{e}-\frac{i \text{Li}_2\left(-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)}{e}-\frac{i \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e}}{e^2}\right)+\frac{a i d^3-a e h d^2+a e^2 g d-a e^3 f}{2 e^4 (d+e x)^2}","\frac{5 b c^3 i \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right) d^4}{2 e^4 \left(c^2 d^2-e^2\right)^{3/2}}+\frac{5 b c i \sqrt{1-c^2 x^2} d^3}{2 e^3 \left(c^2 d^2-e^2\right) (d+e x)}-\frac{b c \left(3 d h c^2+4 e i\right) \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right) d^2}{2 e^3 \left(c^2 d^2-e^2\right)^{3/2}}-\frac{b c (3 e h+4 d i) \sqrt{1-c^2 x^2} d^2}{2 e^3 \left(c^2 d^2-e^2\right) (d+e x)}+\frac{b c \left(\left(4 i d^3+e^2 g d\right) c^2+4 e^2 (e h-2 d i)\right) \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right) d}{2 e^4 \left(c^2 d^2-e^2\right)^{3/2}}+\frac{b c \left(-4 i d^2+4 e h d+e^2 g\right) \sqrt{1-c^2 x^2} d}{2 e^3 \left(c^2 d^2-e^2\right) (d+e x)}-\frac{i b (e h-3 d i) \sin ^{-1}(c x)^2}{2 e^4}+\frac{i x \left(a+b \sin ^{-1}(c x)\right)}{e^3}-\frac{\left(3 i d^2-2 e h d+e^2 g\right) \left(a+b \sin ^{-1}(c x)\right)}{e^4 (d+e x)}-\frac{\left(-i d^3+e h d^2-e^2 g d+e^3 f\right) \left(a+b \sin ^{-1}(c x)\right)}{2 e^4 (d+e x)^2}-\frac{b c \left(2 g e^4-6 d^2 i e^2-c^2 \left(d e^3 f-4 d^4 i\right)\right) \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right)}{2 e^4 \left(c^2 d^2-e^2\right)^{3/2}}+\frac{b (e h-3 d i) \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^4}+\frac{b (e h-3 d i) \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^4}-\frac{b (e h-3 d i) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{(e h-3 d i) \left(a+b \sin ^{-1}(c x)\right) \log (d+e x)}{e^4}-\frac{i b (e h-3 d i) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^4}-\frac{i b (e h-3 d i) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^4}+\frac{b i \sqrt{1-c^2 x^2}}{c e^3}+\frac{b c \left(2 i d^3-2 e^2 g d+e^3 f\right) \sqrt{1-c^2 x^2}}{2 e^3 \left(c^2 d^2-e^2\right) (d+e x)}",1,"(a*i*x)/e^3 + (-(a*e^3*f) + a*d*e^2*g - a*d^2*e*h + a*d^3*i)/(2*e^4*(d + e*x)^2) + (-(a*e^2*g) + 2*a*d*e*h - 3*a*d^2*i)/(e^4*(d + e*x)) + b*f*(-1/4*(c*Sqrt[1 + (-d - Sqrt[c^(-2)]*e)/(d + e*x)]*Sqrt[1 + (-d + Sqrt[c^(-2)]*e)/(d + e*x)]*AppellF1[2, 1/2, 1/2, 3, -((-d + Sqrt[c^(-2)]*e)/(d + e*x)), -((-d - Sqrt[c^(-2)]*e)/(d + e*x))])/(e^2*(d + e*x)*Sqrt[1 - c^2*x^2]) - ArcSin[c*x]/(2*e*(d + e*x)^2)) + ((a*e*h - 3*a*d*i)*Log[d + e*x])/e^4 + b*g*((-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2])/e^2 - (d*((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2) - (I*c^3*d*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*(d + e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2])))/(2*e)) + b*i*((Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x])/(c*e^3) + (3*d^2*(-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2]))/e^4 - (d^3*((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2) - (I*c^3*d*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*(d + e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2])))/(2*e^3) - (3*d*(((-1/2*I)*ArcSin[c*x]^2)/e + (ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e + (ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e - (I*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])])/e - (I*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e))/e^3) + b*h*((-2*d*(-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2]))/e^3 + (d^2*((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2) - (I*c^3*d*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*(d + e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2])))/(2*e^2) + (((-1/2*I)*ArcSin[c*x]^2)/e + (ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e + (ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e - (I*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])])/e - (I*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e)/e^2)","C",0
112,1,1921,1278,6.9698432,"\int \frac{\left(f+g x+h x^2+i x^3\right) \left(a+b \sin ^{-1}(c x)\right)}{(d+e x)^4} \, dx","Integrate[((f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]))/(d + e*x)^4,x]","\frac{3 a d i-a e h}{e^4 (d+e x)}+b f \left(-\frac{c \sqrt{\frac{-d-\sqrt{\frac{1}{c^2}} e}{d+e x}+1} \sqrt{\frac{\sqrt{\frac{1}{c^2}} e-d}{d+e x}+1} F_1\left(3;\frac{1}{2},\frac{1}{2};4;-\frac{\sqrt{\frac{1}{c^2}} e-d}{d+e x},-\frac{-d-\sqrt{\frac{1}{c^2}} e}{d+e x}\right)}{9 e^2 (d+e x)^2 \sqrt{1-c^2 x^2}}-\frac{\sin ^{-1}(c x)}{3 e (d+e x)^3}\right)+\frac{a i \log (d+e x)}{e^4}+b h \left(\frac{\left(\frac{\left(2 c^2 d^2+e^2\right) \log (d+e x) c^3}{e (e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}-\frac{\left(2 c^2 d^2+e^2\right) \log \left(d x c^2+e+\sqrt{e^2-c^2 d^2} \sqrt{1-c^2 x^2}\right) c^3}{e (e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}+\frac{\sqrt{1-c^2 x^2} \left(c^3 d (4 d+3 e x)-c e^2\right)}{\left(e^2-c^2 d^2\right)^2 (d+e x)^2}-\frac{2 \sin ^{-1}(c x)}{e (d+e x)^3}\right) d^2}{6 e^2}-\frac{\left(-\frac{i d \left(\log \left(\frac{e^2 \sqrt{c^2 d^2-e^2} \left(i d x c^2+i e+\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}\right)}{c^3 d (d+e x)}\right)+\log (4)\right) c^3}{(c d-e) e (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{\sqrt{1-c^2 x^2} c}{\left(c^2 d^2-e^2\right) (d+e x)}-\frac{\sin ^{-1}(c x)}{e (d+e x)^2}\right) d}{e^2}+\frac{\frac{c \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right)}{\sqrt{c^2 d^2-e^2}}-\frac{\sin ^{-1}(c x)}{d+e x}}{e^3}\right)+b g \left(\frac{-\frac{i d \left(\log \left(\frac{e^2 \sqrt{c^2 d^2-e^2} \left(i d x c^2+i e+\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}\right)}{c^3 d (d+e x)}\right)+\log (4)\right) c^3}{(c d-e) e (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{\sqrt{1-c^2 x^2} c}{\left(c^2 d^2-e^2\right) (d+e x)}-\frac{\sin ^{-1}(c x)}{e (d+e x)^2}}{2 e}-\frac{d \left(\frac{\left(2 c^2 d^2+e^2\right) \log (d+e x) c^3}{e (e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}-\frac{\left(2 c^2 d^2+e^2\right) \log \left(d x c^2+e+\sqrt{e^2-c^2 d^2} \sqrt{1-c^2 x^2}\right) c^3}{e (e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}+\frac{\sqrt{1-c^2 x^2} \left(c^3 d (4 d+3 e x)-c e^2\right)}{\left(e^2-c^2 d^2\right)^2 (d+e x)^2}-\frac{2 \sin ^{-1}(c x)}{e (d+e x)^3}\right)}{6 e}\right)+b i \left(-\frac{\left(\frac{\left(2 c^2 d^2+e^2\right) \log (d+e x) c^3}{e (e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}-\frac{\left(2 c^2 d^2+e^2\right) \log \left(d x c^2+e+\sqrt{e^2-c^2 d^2} \sqrt{1-c^2 x^2}\right) c^3}{e (e-c d)^2 (c d+e)^2 \sqrt{e^2-c^2 d^2}}+\frac{\sqrt{1-c^2 x^2} \left(c^3 d (4 d+3 e x)-c e^2\right)}{\left(e^2-c^2 d^2\right)^2 (d+e x)^2}-\frac{2 \sin ^{-1}(c x)}{e (d+e x)^3}\right) d^3}{6 e^3}+\frac{3 \left(-\frac{i d \left(\log \left(\frac{e^2 \sqrt{c^2 d^2-e^2} \left(i d x c^2+i e+\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}\right)}{c^3 d (d+e x)}\right)+\log (4)\right) c^3}{(c d-e) e (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{\sqrt{1-c^2 x^2} c}{\left(c^2 d^2-e^2\right) (d+e x)}-\frac{\sin ^{-1}(c x)}{e (d+e x)^2}\right) d^2}{2 e^3}-\frac{3 \left(\frac{c \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right)}{\sqrt{c^2 d^2-e^2}}-\frac{\sin ^{-1}(c x)}{d+e x}\right) d}{e^4}+\frac{-\frac{i \sin ^{-1}(c x)^2}{2 e}+\frac{\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right) \sin ^{-1}(c x)}{e}+\frac{\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right) \sin ^{-1}(c x)}{e}-\frac{i \text{Li}_2\left(-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)}{e}-\frac{i \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e}}{e^3}\right)+\frac{-3 a i d^2+2 a e h d-a e^2 g}{2 e^4 (d+e x)^2}+\frac{a i d^3-a e h d^2+a e^2 g d-a e^3 f}{3 e^4 (d+e x)^3}","-\frac{11 b c^3 i \sqrt{1-c^2 x^2} d^4}{4 e^3 \left(c^2 d^2-e^2\right)^2 (d+e x)}-\frac{11 b c^3 \left(2 c^2 d^2+e^2\right) i \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right) d^3}{12 e^4 \left(c^2 d^2-e^2\right)^{5/2}}-\frac{11 b c i \sqrt{1-c^2 x^2} d^3}{12 e^3 \left(c^2 d^2-e^2\right) (d+e x)^2}+\frac{b c^3 \left(4 c^2 h d^2+e (2 e h+81 d i)\right) \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right) d^2}{12 e^3 \left(c^2 d^2-e^2\right)^{5/2}}+\frac{b c \left(d (2 e h+9 d i) c^2+18 e^2 i\right) \sqrt{1-c^2 x^2} d^2}{4 e^3 \left(c^2 d^2-e^2\right)^2 (d+e x)}+\frac{b c (2 e h+27 d i) \sqrt{1-c^2 x^2} d^2}{12 e^3 \left(c^2 d^2-e^2\right) (d+e x)^2}+\frac{b c \left(2 d^2 g c^4+\left(-18 i d^2-18 e h d+e^2 g\right) c^2-36 e^2 i\right) \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right) d}{12 e^2 \left(c^2 d^2-e^2\right)^{5/2}}-\frac{b c \left(4 e^2 (e h+6 d i)-c^2 d \left(6 i d^2-2 e h d+e^2 g\right)\right) \sqrt{1-c^2 x^2} d}{4 e^3 \left(c^2 d^2-e^2\right)^2 (d+e x)}+\frac{b c \left(-18 i d^2-6 e h d+e^2 g\right) \sqrt{1-c^2 x^2} d}{12 e^3 \left(c^2 d^2-e^2\right) (d+e x)^2}-\frac{i b i \sin ^{-1}(c x)^2}{2 e^4}-\frac{(e h-3 d i) \left(a+b \sin ^{-1}(c x)\right)}{e^4 (d+e x)}-\frac{\left(3 i d^2-2 e h d+e^2 g\right) \left(a+b \sin ^{-1}(c x)\right)}{2 e^4 (d+e x)^2}-\frac{\left(-i d^3+e h d^2-e^2 g d+e^3 f\right) \left(a+b \sin ^{-1}(c x)\right)}{3 e^4 (d+e x)^3}+\frac{b c \left(4 d^2 f c^4+\left(6 h d^2-9 e g d+2 e^2 f\right) c^2+12 e^2 h\right) \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right)}{12 e \left(c^2 d^2-e^2\right)^{5/2}}+\frac{b i \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^4}+\frac{b i \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^4}-\frac{b i \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac{i \left(a+b \sin ^{-1}(c x)\right) \log (d+e x)}{e^4}-\frac{i b i \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^4}-\frac{i b i \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^4}-\frac{b c \left(2 e^2 (e g-4 d h)-c^2 d \left(-2 h d^2-e g d+2 e^2 f\right)\right) \sqrt{1-c^2 x^2}}{4 e^2 \left(c^2 d^2-e^2\right)^2 (d+e x)}+\frac{b c \left(6 h d^2-3 e g d+2 e^2 f\right) \sqrt{1-c^2 x^2}}{12 e^2 \left(c^2 d^2-e^2\right) (d+e x)^2}",1,"(-(a*e^3*f) + a*d*e^2*g - a*d^2*e*h + a*d^3*i)/(3*e^4*(d + e*x)^3) + (-(a*e^2*g) + 2*a*d*e*h - 3*a*d^2*i)/(2*e^4*(d + e*x)^2) + (-(a*e*h) + 3*a*d*i)/(e^4*(d + e*x)) + b*f*(-1/9*(c*Sqrt[1 + (-d - Sqrt[c^(-2)]*e)/(d + e*x)]*Sqrt[1 + (-d + Sqrt[c^(-2)]*e)/(d + e*x)]*AppellF1[3, 1/2, 1/2, 4, -((-d + Sqrt[c^(-2)]*e)/(d + e*x)), -((-d - Sqrt[c^(-2)]*e)/(d + e*x))])/(e^2*(d + e*x)^2*Sqrt[1 - c^2*x^2]) - ArcSin[c*x]/(3*e*(d + e*x)^3)) + (a*i*Log[d + e*x])/e^4 + b*h*((-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2])/e^3 - (d*((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2) - (I*c^3*d*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*(d + e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2])))/e^2 + (d^2*((Sqrt[1 - c^2*x^2]*(-(c*e^2) + c^3*d*(4*d + 3*e*x)))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (2*ArcSin[c*x])/(e*(d + e*x)^3) + (c^3*(2*c^2*d^2 + e^2)*Log[d + e*x])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]) - (c^3*(2*c^2*d^2 + e^2)*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2])))/(6*e^2)) + b*g*(((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2) - (I*c^3*d*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*(d + e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2]))/(2*e) - (d*((Sqrt[1 - c^2*x^2]*(-(c*e^2) + c^3*d*(4*d + 3*e*x)))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (2*ArcSin[c*x])/(e*(d + e*x)^3) + (c^3*(2*c^2*d^2 + e^2)*Log[d + e*x])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]) - (c^3*(2*c^2*d^2 + e^2)*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2])))/(6*e)) + b*i*((-3*d*(-(ArcSin[c*x]/(d + e*x)) + (c*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2]))/e^4 + (3*d^2*((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) - ArcSin[c*x]/(e*(d + e*x)^2) - (I*c^3*d*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*(d + e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2])))/(2*e^3) - (d^3*((Sqrt[1 - c^2*x^2]*(-(c*e^2) + c^3*d*(4*d + 3*e*x)))/((-(c^2*d^2) + e^2)^2*(d + e*x)^2) - (2*ArcSin[c*x])/(e*(d + e*x)^3) + (c^3*(2*c^2*d^2 + e^2)*Log[d + e*x])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2]) - (c^3*(2*c^2*d^2 + e^2)*Log[e + c^2*d*x + Sqrt[-(c^2*d^2) + e^2]*Sqrt[1 - c^2*x^2]])/(e*(-(c*d) + e)^2*(c*d + e)^2*Sqrt[-(c^2*d^2) + e^2])))/(6*e^3) + (((-1/2*I)*ArcSin[c*x]^2)/e + (ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e + (ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e - (I*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])])/e - (I*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e)/e^3)","C",0
113,1,574,935,1.701642,"\int \frac{(f+g x) \left(a+b \sin ^{-1}(c x)\right)^2}{(d+e x)^3} \, dx","Integrate[((f + g*x)*(a + b*ArcSin[c*x])^2)/(d + e*x)^3,x]","\frac{\frac{2 b c (e f-d g) \left(-i c^2 d (d+e x) \left(\left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)-\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)\right)-i b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)+i b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)\right)+e \sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2} \left(a+b \sin ^{-1}(c x)\right)-b c \sqrt{c^2 d^2-e^2} (d+e x) \log (d+e x)\right)}{\left(c^2 d^2-e^2\right)^{3/2} (d+e x)}+\frac{4 b c g \left(-i \left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)-\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)\right)-b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)+b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)\right)}{\sqrt{c^2 d^2-e^2}}-\frac{(e f-d g) \left(a+b \sin ^{-1}(c x)\right)^2}{(d+e x)^2}-\frac{2 g \left(a+b \sin ^{-1}(c x)\right)^2}{d+e x}}{2 e^2}","-\frac{i b^2 d (e f-d g) \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right) c^3}{e^2 \left(c^2 d^2-e^2\right)^{3/2}}+\frac{i b^2 d (e f-d g) \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right) c^3}{e^2 \left(c^2 d^2-e^2\right)^{3/2}}-\frac{b^2 d (e f-d g) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right) c^3}{e^2 \left(c^2 d^2-e^2\right)^{3/2}}+\frac{b^2 d (e f-d g) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right) c^3}{e^2 \left(c^2 d^2-e^2\right)^{3/2}}-\frac{b^2 (e f-d g) \log (d+e x) c^2}{e^2 \left(c^2 d^2-e^2\right)}+\frac{b^2 (e f-d g) \sqrt{1-c^2 x^2} \sin ^{-1}(c x) c}{e \left(c^2 d^2-e^2\right) (d+e x)}-\frac{a b \left(2 e^2 g-c^2 d (e f+d g)\right) \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right) c}{e^2 \left(c^2 d^2-e^2\right)^{3/2}}-\frac{2 i b^2 g \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right) c}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 i b^2 g \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right) c}{e^2 \sqrt{c^2 d^2-e^2}}-\frac{2 b^2 g \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right) c}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 g \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right) c}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{a b (e f-d g) \sqrt{1-c^2 x^2} c}{e \left(c^2 d^2-e^2\right) (d+e x)}+\frac{b^2 g^2 \sin ^{-1}(c x)^2}{2 e^2 (e f-d g)}-\frac{(f+g x)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{2 (e f-d g) (d+e x)^2}+\frac{a b g^2 \sin ^{-1}(c x)}{e^2 (e f-d g)}",1,"(-(((e*f - d*g)*(a + b*ArcSin[c*x])^2)/(d + e*x)^2) - (2*g*(a + b*ArcSin[c*x])^2)/(d + e*x) + (4*b*c*g*((-I)*(a + b*ArcSin[c*x])*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) - b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/Sqrt[c^2*d^2 - e^2] + (2*b*c*(e*f - d*g)*(e*Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - b*c*Sqrt[c^2*d^2 - e^2]*(d + e*x)*Log[d + e*x] - I*c^2*d*(d + e*x)*((a + b*ArcSin[c*x])*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) - I*b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + I*b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])))/((c^2*d^2 - e^2)^(3/2)*(d + e*x)))/(2*e^2)","A",0
114,1,901,1678,4.9798262,"\int \frac{(f+g x)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{(d+e x)^3} \, dx","Integrate[((f + g*x)^2*(a + b*ArcSin[c*x])^2)/(d + e*x)^3,x]","\frac{-\frac{2 i g^2 \left(a+b \sin ^{-1}(c x)\right)^3}{b}+6 g^2 \log \left(\frac{i e^{i \sin ^{-1}(c x)} e}{\sqrt{c^2 d^2-e^2}-c d}+1\right) \left(a+b \sin ^{-1}(c x)\right)^2+6 g^2 \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{12 g (d g-e f) \left(a+b \sin ^{-1}(c x)\right)^2}{d+e x}-\frac{3 (e f-d g)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{(d+e x)^2}+\frac{24 b c g (d g-e f) \left(i \left(a+b \sin ^{-1}(c x)\right) \left(\log \left(\frac{i e^{i \sin ^{-1}(c x)} e}{\sqrt{c^2 d^2-e^2}-c d}+1\right)-\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)\right)+b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)-b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)\right)}{\sqrt{c^2 d^2-e^2}}+\frac{6 b c^2 (e f-d g)^2 \left(\frac{e \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c d+c e x}-b \log (d+e x)+\frac{c d \left(-i \left(a+b \sin ^{-1}(c x)\right) \left(\log \left(\frac{i e^{i \sin ^{-1}(c x)} e}{\sqrt{c^2 d^2-e^2}-c d}+1\right)-\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)\right)-b \text{Li}_2\left(-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)+b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)\right)}{\sqrt{c^2 d^2-e^2}}\right)}{c^2 d^2-e^2}-12 b g^2 \left(i \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)-b \text{Li}_3\left(-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)\right)-12 b g^2 \left(i \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)-b \text{Li}_3\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)\right)}{6 e^3}","-\frac{i b^2 d (e f-d g)^2 \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right) c^3}{e^3 \left(c^2 d^2-e^2\right)^{3/2}}+\frac{i b^2 d (e f-d g)^2 \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right) c^3}{e^3 \left(c^2 d^2-e^2\right)^{3/2}}-\frac{b^2 d (e f-d g)^2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right) c^3}{e^3 \left(c^2 d^2-e^2\right)^{3/2}}+\frac{b^2 d (e f-d g)^2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right) c^3}{e^3 \left(c^2 d^2-e^2\right)^{3/2}}-\frac{b^2 (e f-d g)^2 \log (d+e x) c^2}{e^3 \left(c^2 d^2-e^2\right)}+\frac{b^2 (e f-d g)^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) c}{e^2 \left(c^2 d^2-e^2\right) (d+e x)}-\frac{a b (e f-d g) \left(4 e^2 g-c^2 d (e f+3 d g)\right) \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right) c}{e^3 \left(c^2 d^2-e^2\right)^{3/2}}-\frac{4 i b^2 g (e f-d g) \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right) c}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{4 i b^2 g (e f-d g) \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right) c}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{4 b^2 g (e f-d g) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right) c}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{4 b^2 g (e f-d g) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right) c}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{a b (e f-d g)^2 \sqrt{1-c^2 x^2} c}{e^2 \left(c^2 d^2-e^2\right) (d+e x)}-\frac{i b^2 g^2 \sin ^{-1}(c x)^3}{3 e^3}-\frac{i a b g^2 \sin ^{-1}(c x)^2}{e^3}-\frac{2 b^2 g (e f-d g) \sin ^{-1}(c x)^2}{e^3 (d+e x)}-\frac{b^2 (e f-d g)^2 \sin ^{-1}(c x)^2}{2 e^3 (d+e x)^2}-\frac{4 a b g (e f-d g) \sin ^{-1}(c x)}{e^3 (d+e x)}-\frac{a b (e f-d g)^2 \sin ^{-1}(c x)}{e^3 (d+e x)^2}+\frac{b^2 g^2 \sin ^{-1}(c x)^2 \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^3}+\frac{2 a b g^2 \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^3}+\frac{b^2 g^2 \sin ^{-1}(c x)^2 \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^3}+\frac{2 a b g^2 \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^3}+\frac{a^2 g^2 \log (d+e x)}{e^3}-\frac{2 i a b g^2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^3}-\frac{2 i b^2 g^2 \sin ^{-1}(c x) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^3}-\frac{2 i a b g^2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^3}-\frac{2 i b^2 g^2 \sin ^{-1}(c x) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^3}+\frac{2 b^2 g^2 \text{Li}_3\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^3}+\frac{2 b^2 g^2 \text{Li}_3\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^3}-\frac{2 a^2 g (e f-d g)}{e^3 (d+e x)}-\frac{a^2 (e f-d g)^2}{2 e^3 (d+e x)^2}",1,"((-3*(e*f - d*g)^2*(a + b*ArcSin[c*x])^2)/(d + e*x)^2 + (12*g*(-(e*f) + d*g)*(a + b*ArcSin[c*x])^2)/(d + e*x) - ((2*I)*g^2*(a + b*ArcSin[c*x])^3)/b + 6*g^2*(a + b*ArcSin[c*x])^2*Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + 6*g^2*(a + b*ArcSin[c*x])^2*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] + (24*b*c*g*(-(e*f) + d*g)*(I*(a + b*ArcSin[c*x])*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) + b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] - b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/Sqrt[c^2*d^2 - e^2] + (6*b*c^2*(e*f - d*g)^2*((e*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(c*d + c*e*x) - b*Log[d + e*x] + (c*d*((-I)*(a + b*ArcSin[c*x])*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) - b*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/Sqrt[c^2*d^2 - e^2]))/(c^2*d^2 - e^2) - 12*b*g^2*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] - b*PolyLog[3, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])]) - 12*b*g^2*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] - b*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/(6*e^3)","A",0
115,1,734,1016,1.0894415,"\int (g+h x)^3 \left(d+e x+f x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(g + h*x)^3*(d + e*x + f*x^2)*(a + b*ArcSin[c*x])^2,x]","-\frac{f h^3 \left(45 a^2-6 a b c x \sqrt{1-c^2 x^2} \left(8 c^4 x^4+10 c^2 x^2+15\right)-6 b \sin ^{-1}(c x) \left(b c x \sqrt{1-c^2 x^2} \left(8 c^4 x^4+10 c^2 x^2+15\right)-15 a\right)+b^2 c^2 x^2 \left(8 c^4 x^4+15 c^2 x^2+45\right)+45 b^2 \sin ^{-1}(c x)^2\right)}{864 c^6}-\frac{1}{4} b g^2 (3 d h+e g) \left(-\frac{2 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b c^2}+b x^2\right)-2 b d g^3 \left(b x-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}\right)-\frac{2 b g \left(-3 a \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right)+b c x \left(c^2 x^2+6\right)-3 b \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right) \sin ^{-1}(c x)\right) \left(3 h (d h+e g)+f g^2\right)}{27 c^3}-\frac{2 b h^2 (e h+3 f g) \left(-15 a \sqrt{1-c^2 x^2} \left(3 c^4 x^4+4 c^2 x^2+8\right)+b c x \left(9 c^4 x^4+20 c^2 x^2+120\right)-15 b \sqrt{1-c^2 x^2} \left(3 c^4 x^4+4 c^2 x^2+8\right) \sin ^{-1}(c x)\right)}{1125 c^5}-\frac{1}{32} b h \left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)^2}{b c^4}-\frac{4 x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}-\frac{6 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c^3}+\frac{3 b x^2}{c^2}+b x^4\right) \left(h (d h+3 e g)+3 f g^2\right)+\frac{1}{4} h x^4 \left(a+b \sin ^{-1}(c x)\right)^2 \left(h (d h+3 e g)+3 f g^2\right)+\frac{1}{3} g x^3 \left(a+b \sin ^{-1}(c x)\right)^2 \left(3 h (d h+e g)+f g^2\right)+\frac{1}{2} g^2 x^2 (3 d h+e g) \left(a+b \sin ^{-1}(c x)\right)^2+d g^3 x \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{5} h^2 x^5 (e h+3 f g) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{6} f h^3 x^6 \left(a+b \sin ^{-1}(c x)\right)^2","-\frac{1}{108} b^2 f h^3 x^6+\frac{1}{6} f h^3 \left(a+b \sin ^{-1}(c x)\right)^2 x^6+\frac{1}{5} h^2 (3 f g+e h) \left(a+b \sin ^{-1}(c x)\right)^2 x^5-\frac{2}{125} b^2 h^2 (3 f g+e h) x^5+\frac{b f h^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) x^5}{18 c}-\frac{5 b^2 f h^3 x^4}{288 c^2}+\frac{1}{4} h \left(3 f g^2+h (3 e g+d h)\right) \left(a+b \sin ^{-1}(c x)\right)^2 x^4-\frac{1}{32} b^2 h \left(3 f g^2+h (3 e g+d h)\right) x^4+\frac{2 b h^2 (3 f g+e h) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) x^4}{25 c}+\frac{1}{3} g \left(f g^2+3 h (e g+d h)\right) \left(a+b \sin ^{-1}(c x)\right)^2 x^3-\frac{8 b^2 h^2 (3 f g+e h) x^3}{225 c^2}-\frac{2}{27} b^2 g \left(f g^2+3 h (e g+d h)\right) x^3+\frac{5 b f h^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) x^3}{72 c^3}+\frac{b h \left(3 f g^2+h (3 e g+d h)\right) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) x^3}{8 c}-\frac{5 b^2 f h^3 x^2}{96 c^4}+\frac{1}{2} g^2 (e g+3 d h) \left(a+b \sin ^{-1}(c x)\right)^2 x^2-\frac{1}{4} b^2 g^2 (e g+3 d h) x^2-\frac{3 b^2 h \left(3 f g^2+h (3 e g+d h)\right) x^2}{32 c^2}+\frac{8 b h^2 (3 f g+e h) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) x^2}{75 c^3}+\frac{2 b g \left(f g^2+3 h (e g+d h)\right) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) x^2}{9 c}-2 b^2 d g^3 x+d g^3 \left(a+b \sin ^{-1}(c x)\right)^2 x-\frac{16 b^2 h^2 (3 f g+e h) x}{75 c^4}-\frac{4 b^2 g \left(f g^2+3 h (e g+d h)\right) x}{9 c^2}+\frac{5 b f h^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) x}{48 c^5}+\frac{b g^2 (e g+3 d h) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) x}{2 c}+\frac{3 b h \left(3 f g^2+h (3 e g+d h)\right) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) x}{16 c^3}-\frac{5 f h^3 \left(a+b \sin ^{-1}(c x)\right)^2}{96 c^6}-\frac{g^2 (e g+3 d h) \left(a+b \sin ^{-1}(c x)\right)^2}{4 c^2}-\frac{3 h \left(3 f g^2+h (3 e g+d h)\right) \left(a+b \sin ^{-1}(c x)\right)^2}{32 c^4}+\frac{2 b d g^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{16 b h^2 (3 f g+e h) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{75 c^5}+\frac{4 b g \left(f g^2+3 h (e g+d h)\right) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 c^3}",1,"d*g^3*x*(a + b*ArcSin[c*x])^2 + (g^2*(e*g + 3*d*h)*x^2*(a + b*ArcSin[c*x])^2)/2 + (g*(f*g^2 + 3*h*(e*g + d*h))*x^3*(a + b*ArcSin[c*x])^2)/3 + (h*(3*f*g^2 + h*(3*e*g + d*h))*x^4*(a + b*ArcSin[c*x])^2)/4 + (h^2*(3*f*g + e*h)*x^5*(a + b*ArcSin[c*x])^2)/5 + (f*h^3*x^6*(a + b*ArcSin[c*x])^2)/6 - (2*b*g*(f*g^2 + 3*h*(e*g + d*h))*(-3*a*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + b*c*x*(6 + c^2*x^2) - 3*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2)*ArcSin[c*x]))/(27*c^3) - (2*b*h^2*(3*f*g + e*h)*(-15*a*Sqrt[1 - c^2*x^2]*(8 + 4*c^2*x^2 + 3*c^4*x^4) + b*c*x*(120 + 20*c^2*x^2 + 9*c^4*x^4) - 15*b*Sqrt[1 - c^2*x^2]*(8 + 4*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x]))/(1125*c^5) - (f*h^3*(45*a^2 - 6*a*b*c*x*Sqrt[1 - c^2*x^2]*(15 + 10*c^2*x^2 + 8*c^4*x^4) + b^2*c^2*x^2*(45 + 15*c^2*x^2 + 8*c^4*x^4) - 6*b*(-15*a + b*c*x*Sqrt[1 - c^2*x^2]*(15 + 10*c^2*x^2 + 8*c^4*x^4))*ArcSin[c*x] + 45*b^2*ArcSin[c*x]^2))/(864*c^6) - 2*b*d*g^3*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c) - (b*h*(3*f*g^2 + h*(3*e*g + d*h))*((3*b*x^2)/c^2 + b*x^4 - (6*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^3 - (4*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (3*(a + b*ArcSin[c*x])^2)/(b*c^4)))/32 - (b*g^2*(e*g + 3*d*h)*(b*x^2 - (2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (a + b*ArcSin[c*x])^2/(b*c^2)))/4","A",1
116,1,534,701,0.607155,"\int (g+h x)^2 \left(d+e x+f x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(g + h*x)^2*(d + e*x + f*x^2)*(a + b*ArcSin[c*x])^2,x]","-\frac{1}{4} b g (2 d h+e g) \left(-\frac{2 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b c^2}+b x^2\right)-2 b d g^2 \left(b x-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}\right)-\frac{2 b \left(-3 a \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right)+b c x \left(c^2 x^2+6\right)-3 b \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right) \sin ^{-1}(c x)\right) \left(h (d h+2 e g)+f g^2\right)}{27 c^3}-\frac{2 b f h^2 \left(-15 a \sqrt{1-c^2 x^2} \left(3 c^4 x^4+4 c^2 x^2+8\right)+b c x \left(9 c^4 x^4+20 c^2 x^2+120\right)-15 b \sqrt{1-c^2 x^2} \left(3 c^4 x^4+4 c^2 x^2+8\right) \sin ^{-1}(c x)\right)}{1125 c^5}-\frac{1}{32} b h (e h+2 f g) \left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)^2}{b c^4}-\frac{4 x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}-\frac{6 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c^3}+\frac{3 b x^2}{c^2}+b x^4\right)+\frac{1}{3} x^3 \left(a+b \sin ^{-1}(c x)\right)^2 \left(h (d h+2 e g)+f g^2\right)+\frac{1}{2} g x^2 (2 d h+e g) \left(a+b \sin ^{-1}(c x)\right)^2+d g^2 x \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{4} h x^4 (e h+2 f g) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{5} f h^2 x^5 \left(a+b \sin ^{-1}(c x)\right)^2","-\frac{3 h (e h+2 f g) \left(a+b \sin ^{-1}(c x)\right)^2}{32 c^4}+\frac{2 b x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \left(h (d h+2 e g)+f g^2\right)}{9 c}+\frac{b g x \sqrt{1-c^2 x^2} (2 d h+e g) \left(a+b \sin ^{-1}(c x)\right)}{2 c}-\frac{g (2 d h+e g) \left(a+b \sin ^{-1}(c x)\right)^2}{4 c^2}+\frac{2 b d g^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{b h x^3 \sqrt{1-c^2 x^2} (e h+2 f g) \left(a+b \sin ^{-1}(c x)\right)}{8 c}+\frac{2 b f h^2 x^4 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{25 c}+\frac{16 b f h^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{75 c^5}+\frac{4 b \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) \left(h (d h+2 e g)+f g^2\right)}{9 c^3}+\frac{3 b h x \sqrt{1-c^2 x^2} (e h+2 f g) \left(a+b \sin ^{-1}(c x)\right)}{16 c^3}+\frac{8 b f h^2 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{75 c^3}+\frac{1}{3} x^3 \left(a+b \sin ^{-1}(c x)\right)^2 \left(h (d h+2 e g)+f g^2\right)+\frac{1}{2} g x^2 (2 d h+e g) \left(a+b \sin ^{-1}(c x)\right)^2+d g^2 x \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{4} h x^4 (e h+2 f g) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{5} f h^2 x^5 \left(a+b \sin ^{-1}(c x)\right)^2-\frac{16 b^2 f h^2 x}{75 c^4}-\frac{4 b^2 x \left(h (d h+2 e g)+f g^2\right)}{9 c^2}-\frac{3 b^2 h x^2 (e h+2 f g)}{32 c^2}-\frac{8 b^2 f h^2 x^3}{225 c^2}-\frac{2}{27} b^2 x^3 \left(h (d h+2 e g)+f g^2\right)-\frac{1}{4} b^2 g x^2 (2 d h+e g)-2 b^2 d g^2 x-\frac{1}{32} b^2 h x^4 (e h+2 f g)-\frac{2}{125} b^2 f h^2 x^5",1,"d*g^2*x*(a + b*ArcSin[c*x])^2 + (g*(e*g + 2*d*h)*x^2*(a + b*ArcSin[c*x])^2)/2 + ((f*g^2 + h*(2*e*g + d*h))*x^3*(a + b*ArcSin[c*x])^2)/3 + (h*(2*f*g + e*h)*x^4*(a + b*ArcSin[c*x])^2)/4 + (f*h^2*x^5*(a + b*ArcSin[c*x])^2)/5 - (2*b*(f*g^2 + h*(2*e*g + d*h))*(-3*a*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + b*c*x*(6 + c^2*x^2) - 3*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2)*ArcSin[c*x]))/(27*c^3) - (2*b*f*h^2*(-15*a*Sqrt[1 - c^2*x^2]*(8 + 4*c^2*x^2 + 3*c^4*x^4) + b*c*x*(120 + 20*c^2*x^2 + 9*c^4*x^4) - 15*b*Sqrt[1 - c^2*x^2]*(8 + 4*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x]))/(1125*c^5) - 2*b*d*g^2*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c) - (b*h*(2*f*g + e*h)*((3*b*x^2)/c^2 + b*x^4 - (6*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^3 - (4*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (3*(a + b*ArcSin[c*x])^2)/(b*c^4)))/32 - (b*g*(e*g + 2*d*h)*(b*x^2 - (2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (a + b*ArcSin[c*x])^2/(b*c^2)))/4","A",1
117,1,364,425,0.3945567,"\int (g+h x) \left(d+e x+f x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(g + h*x)*(d + e*x + f*x^2)*(a + b*ArcSin[c*x])^2,x]","-\frac{1}{4} b (d h+e g) \left(-\frac{2 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b c^2}+b x^2\right)-2 b d g \left(b x-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}\right)-\frac{2 b (e h+f g) \left(-3 a \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right)+b c x \left(c^2 x^2+6\right)-3 b \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right) \sin ^{-1}(c x)\right)}{27 c^3}-\frac{1}{32} b f h \left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)^2}{b c^4}-\frac{4 x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}-\frac{6 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c^3}+\frac{3 b x^2}{c^2}+b x^4\right)+\frac{1}{2} x^2 (d h+e g) \left(a+b \sin ^{-1}(c x)\right)^2+d g x \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{3} x^3 (e h+f g) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{4} f h x^4 \left(a+b \sin ^{-1}(c x)\right)^2","-\frac{3 f h \left(a+b \sin ^{-1}(c x)\right)^2}{32 c^4}+\frac{b x \sqrt{1-c^2 x^2} (d h+e g) \left(a+b \sin ^{-1}(c x)\right)}{2 c}-\frac{(d h+e g) \left(a+b \sin ^{-1}(c x)\right)^2}{4 c^2}+\frac{2 b d g \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{2 b x^2 \sqrt{1-c^2 x^2} (e h+f g) \left(a+b \sin ^{-1}(c x)\right)}{9 c}+\frac{b f h x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 c}+\frac{4 b \sqrt{1-c^2 x^2} (e h+f g) \left(a+b \sin ^{-1}(c x)\right)}{9 c^3}+\frac{3 b f h x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{16 c^3}+\frac{1}{2} x^2 (d h+e g) \left(a+b \sin ^{-1}(c x)\right)^2+d g x \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{3} x^3 (e h+f g) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{4} f h x^4 \left(a+b \sin ^{-1}(c x)\right)^2-\frac{4 b^2 x (e h+f g)}{9 c^2}-\frac{3 b^2 f h x^2}{32 c^2}-\frac{1}{4} b^2 x^2 (d h+e g)-2 b^2 d g x-\frac{2}{27} b^2 x^3 (e h+f g)-\frac{1}{32} b^2 f h x^4",1,"d*g*x*(a + b*ArcSin[c*x])^2 + ((e*g + d*h)*x^2*(a + b*ArcSin[c*x])^2)/2 + ((f*g + e*h)*x^3*(a + b*ArcSin[c*x])^2)/3 + (f*h*x^4*(a + b*ArcSin[c*x])^2)/4 - (2*b*(f*g + e*h)*(-3*a*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + b*c*x*(6 + c^2*x^2) - 3*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2)*ArcSin[c*x]))/(27*c^3) - 2*b*d*g*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c) - (b*f*h*((3*b*x^2)/c^2 + b*x^4 - (6*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^3 - (4*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (3*(a + b*ArcSin[c*x])^2)/(b*c^4)))/32 - (b*(e*g + d*h)*(b*x^2 - (2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (a + b*ArcSin[c*x])^2/(b*c^2)))/4","A",1
118,1,556,1067,0.7468424,"\int \frac{\left(d+e x+f x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{g+h x} \, dx","Integrate[((d + e*x + f*x^2)*(a + b*ArcSin[c*x])^2)/(g + h*x),x]","\frac{-24 b \left(h (d h-e g)+f g^2\right) \left(i \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right)-b \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right)\right)-24 b \left(h (d h-e g)+f g^2\right) \left(i \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right)-b \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right)\right)+12 \left(a+b \sin ^{-1}(c x)\right)^2 \left(h (d h-e g)+f g^2\right) \log \left(1+\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}-c g}\right)+12 \left(a+b \sin ^{-1}(c x)\right)^2 \left(h (d h-e g)+f g^2\right) \log \left(1-\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}+c g}\right)+24 b h (f g-e h) \left(b x-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}\right)-3 b f h^2 \left(-\frac{2 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b c^2}+b x^2\right)-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)^3 \left(h (d h-e g)+f g^2\right)}{b}+12 h x (e h-f g) \left(a+b \sin ^{-1}(c x)\right)^2+6 f h^2 x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{12 h^3}","-\frac{i b^2 \left(f g^2-e h g+d h^2\right) \sin ^{-1}(c x)^3}{3 h^3}+\frac{b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac{i a b \left(f g^2-e h g+d h^2\right) \sin ^{-1}(c x)^2}{h^3}-\frac{b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac{b^2 \left(f g^2-e h g+d h^2\right) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right) \sin ^{-1}(c x)^2}{h^3}+\frac{b^2 \left(f g^2-e h g+d h^2\right) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right) \sin ^{-1}(c x)^2}{h^3}-\frac{b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}+\frac{a b f x^2 \sin ^{-1}(c x)}{h}-\frac{2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac{2 a b \left(f g^2-e h g+d h^2\right) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right) \sin ^{-1}(c x)}{h^3}+\frac{2 a b \left(f g^2-e h g+d h^2\right) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right) \sin ^{-1}(c x)}{h^3}-\frac{2 i b^2 \left(f g^2-e h g+d h^2\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right) \sin ^{-1}(c x)}{h^3}-\frac{2 i b^2 \left(f g^2-e h g+d h^2\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right) \sin ^{-1}(c x)}{h^3}-\frac{2 b^2 (f g-e h) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac{b^2 f x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac{a b f \sin ^{-1}(c x)}{2 c^2 h}+\frac{a^2 f x^2}{2 h}-\frac{b^2 f x^2}{4 h}-\frac{a^2 (f g-e h) x}{h^2}+\frac{2 b^2 (f g-e h) x}{h^2}+\frac{a^2 \left(f g^2-e h g+d h^2\right) \log (g+h x)}{h^3}-\frac{2 i a b \left(f g^2-e h g+d h^2\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right)}{h^3}-\frac{2 i a b \left(f g^2-e h g+d h^2\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right)}{h^3}+\frac{2 b^2 \left(f g^2-e h g+d h^2\right) \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right)}{h^3}+\frac{2 b^2 \left(f g^2-e h g+d h^2\right) \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right)}{h^3}-\frac{a b (4 (f g-e h)-f h x) \sqrt{1-c^2 x^2}}{2 c h^2}",1,"(12*h*(-(f*g) + e*h)*x*(a + b*ArcSin[c*x])^2 + 6*f*h^2*x^2*(a + b*ArcSin[c*x])^2 - ((4*I)*(f*g^2 + h*(-(e*g) + d*h))*(a + b*ArcSin[c*x])^3)/b + 24*b*h*(f*g - e*h)*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c) - 3*b*f*h^2*(b*x^2 - (2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (a + b*ArcSin[c*x])^2/(b*c^2)) + 12*(f*g^2 + h*(-(e*g) + d*h))*(a + b*ArcSin[c*x])^2*Log[1 + (I*E^(I*ArcSin[c*x])*h)/(-(c*g) + Sqrt[c^2*g^2 - h^2])] + 12*(f*g^2 + h*(-(e*g) + d*h))*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])] - 24*b*(f*g^2 + h*(-(e*g) + d*h))*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])] - b*PolyLog[3, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])]) - 24*b*(f*g^2 + h*(-(e*g) + d*h))*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])] - b*PolyLog[3, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])]))/(12*h^3)","A",0
119,1,688,1323,1.3917961,"\int \frac{\left(d+e x+f x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{(g+h x)^2} \, dx","Integrate[((d + e*x + f*x^2)*(a + b*ArcSin[c*x])^2)/(g + h*x)^2,x]","\frac{\frac{6 b c \left(h (d h-e g)+f g^2\right) \left(-i \left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}-c g}\right)-\log \left(1-\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}+c g}\right)\right)-b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right)+b \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right)\right)}{\sqrt{c^2 g^2-h^2}}+6 b (2 f g-e h) \left(i \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right)-b \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right)\right)+6 b (2 f g-e h) \left(i \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right)-b \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right)\right)-3 (2 f g-e h) \left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1+\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}-c g}\right)-3 (2 f g-e h) \left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}+c g}\right)-6 b f h \left(b x-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}\right)-\frac{3 \left(a+b \sin ^{-1}(c x)\right)^2 \left(h (d h-e g)+f g^2\right)}{g+h x}+\frac{i (2 f g-e h) \left(a+b \sin ^{-1}(c x)\right)^3}{b}+3 f h x \left(a+b \sin ^{-1}(c x)\right)^2}{3 h^3}","\frac{i b^2 (2 f g-e h) \sin ^{-1}(c x)^3}{3 h^3}+\frac{i a b (2 f g-e h) \sin ^{-1}(c x)^2}{h^3}+\frac{b^2 f x \sin ^{-1}(c x)^2}{h^2}-\frac{b^2 (2 f g-e h) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right) \sin ^{-1}(c x)^2}{h^3}-\frac{b^2 (2 f g-e h) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right) \sin ^{-1}(c x)^2}{h^3}-\frac{b^2 \left(f g^2-e h g+d h^2\right) \sin ^{-1}(c x)^2}{h^3 (g+h x)}+\frac{2 a b f x \sin ^{-1}(c x)}{h^2}-\frac{2 a b (2 f g-e h) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right) \sin ^{-1}(c x)}{h^3}-\frac{2 i b^2 c \left(f g^2-e h g+d h^2\right) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right) \sin ^{-1}(c x)}{h^3 \sqrt{c^2 g^2-h^2}}-\frac{2 a b (2 f g-e h) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right) \sin ^{-1}(c x)}{h^3}+\frac{2 i b^2 c \left(f g^2-e h g+d h^2\right) \log \left(1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right) \sin ^{-1}(c x)}{h^3 \sqrt{c^2 g^2-h^2}}+\frac{2 i b^2 (2 f g-e h) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right) \sin ^{-1}(c x)}{h^3}+\frac{2 i b^2 (2 f g-e h) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right) \sin ^{-1}(c x)}{h^3}+\frac{2 b^2 f \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}-\frac{2 a b \left(f g^2-e h g+d h^2\right) \sin ^{-1}(c x)}{h^3 (g+h x)}+\frac{a^2 f x}{h^2}-\frac{2 b^2 f x}{h^2}+\frac{2 a b c \left(f g^2-e h g+d h^2\right) \tan ^{-1}\left(\frac{g x c^2+h}{\sqrt{c^2 g^2-h^2} \sqrt{1-c^2 x^2}}\right)}{h^3 \sqrt{c^2 g^2-h^2}}-\frac{a^2 (2 f g-e h) \log (g+h x)}{h^3}+\frac{2 i a b (2 f g-e h) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right)}{h^3}-\frac{2 b^2 c \left(f g^2-e h g+d h^2\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right)}{h^3 \sqrt{c^2 g^2-h^2}}+\frac{2 i a b (2 f g-e h) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right)}{h^3}+\frac{2 b^2 c \left(f g^2-e h g+d h^2\right) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right)}{h^3 \sqrt{c^2 g^2-h^2}}-\frac{2 b^2 (2 f g-e h) \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right)}{h^3}-\frac{2 b^2 (2 f g-e h) \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right)}{h^3}+\frac{2 a b f \sqrt{1-c^2 x^2}}{c h^2}-\frac{a^2 \left(f g^2-e h g+d h^2\right)}{h^3 (g+h x)}",1,"(3*f*h*x*(a + b*ArcSin[c*x])^2 - (3*(f*g^2 + h*(-(e*g) + d*h))*(a + b*ArcSin[c*x])^2)/(g + h*x) + (I*(2*f*g - e*h)*(a + b*ArcSin[c*x])^3)/b - 6*b*f*h*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c) - 3*(2*f*g - e*h)*(a + b*ArcSin[c*x])^2*Log[1 + (I*E^(I*ArcSin[c*x])*h)/(-(c*g) + Sqrt[c^2*g^2 - h^2])] - 3*(2*f*g - e*h)*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])] + (6*b*c*(f*g^2 + h*(-(e*g) + d*h))*((-I)*(a + b*ArcSin[c*x])*(Log[1 + (I*E^(I*ArcSin[c*x])*h)/(-(c*g) + Sqrt[c^2*g^2 - h^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])]) - b*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])] + b*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])]))/Sqrt[c^2*g^2 - h^2] + 6*b*(2*f*g - e*h)*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])] - b*PolyLog[3, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])]) + 6*b*(2*f*g - e*h)*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])] - b*PolyLog[3, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])]))/(3*h^3)","A",0
120,1,307,520,0.4337266,"\int \frac{\left(e f+2 d h x+e h x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{(d+e x)^2} \, dx","Integrate[((e*f + 2*d*h*x + e*h*x^2)*(a + b*ArcSin[c*x])^2)/(d + e*x)^2,x]","\frac{2 b c \left(e^2 f-d^2 h\right) \left(-i \left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)-\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)\right)-b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)+b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)\right)}{e^2 \sqrt{c^2 d^2-e^2}}-\frac{2 b h \left(b x-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}\right)}{e}-\frac{\left(f-\frac{d^2 h}{e^2}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d+e x}+\frac{h x \left(a+b \sin ^{-1}(c x)\right)^2}{e}","\frac{2 a b c \left(e^2 f-d^2 h\right) \tan ^{-1}\left(\frac{c^2 d x+e}{\sqrt{1-c^2 x^2} \sqrt{c^2 d^2-e^2}}\right)}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 a b h \sqrt{1-c^2 x^2}}{c e}-\frac{\left(f-\frac{d^2 h}{e^2}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d+e x}+\frac{h x \left(a+b \sin ^{-1}(c x)\right)^2}{e}-\frac{2 b^2 c \left(e^2 f-d^2 h\right) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 c \left(e^2 f-d^2 h\right) \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^2 \sqrt{c^2 d^2-e^2}}-\frac{2 i b^2 c \sin ^{-1}(c x) \left(e^2 f-d^2 h\right) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 i b^2 c \sin ^{-1}(c x) \left(e^2 f-d^2 h\right) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)}{e^2 \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 h \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e}-\frac{2 b^2 h x}{e}",1,"(h*x*(a + b*ArcSin[c*x])^2)/e - ((f - (d^2*h)/e^2)*(a + b*ArcSin[c*x])^2)/(d + e*x) - (2*b*h*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c))/e + (2*b*c*(e^2*f - d^2*h)*((-I)*(a + b*ArcSin[c*x])*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) - b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/(e^2*Sqrt[c^2*d^2 - e^2])","A",0
121,1,526,920,0.8575373,"\int \frac{\left(e f+2 d h x+e h x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{(d+e x)^2} \, dx","Integrate[((e*f + 2*d*h*x + e*h*x^2)^2*(a + b*ArcSin[c*x])^2)/(d + e*x)^2,x]","-\frac{2 b h \left(2 e^2 f-d^2 h\right) \left(b x-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}\right)}{e^2}+\frac{2 b c \left(e^2 f-d^2 h\right)^2 \left(-i \left(a+b \sin ^{-1}(c x)\right) \left(\log \left(1+\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}-c d}\right)-\log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d^2-e^2}+c d}\right)\right)-b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)+b \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)\right)}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{b d h^2 \left(-\frac{2 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b c^2}+b x^2\right)}{2 e}-\frac{2 b h^2 \left(-3 a \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right)+b c x \left(c^2 x^2+6\right)-3 b \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right) \sin ^{-1}(c x)\right)}{27 c^3}+\frac{h x \left(2 e^2 f-d^2 h\right) \left(a+b \sin ^{-1}(c x)\right)^2}{e^2}-\frac{\left(e^2 f-d^2 h\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{e^3 (d+e x)}+\frac{d h^2 x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{e}+\frac{1}{3} h^2 x^3 \left(a+b \sin ^{-1}(c x)\right)^2","-\frac{b^2 h^2 \sin ^{-1}(c x)^2 d^3}{3 e^3}-\frac{b^2 h^2 x^2 d}{2 e}-\frac{b^2 h^2 \sin ^{-1}(c x)^2 d}{2 c^2 e}-\frac{a b \left(2 c^2 d^2+3 e^2\right) h^2 \sin ^{-1}(c x) d}{3 c^2 e^3}+\frac{b^2 h^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x) d}{c e}+\frac{5 a b h^2 (d+e x) \sqrt{1-c^2 x^2} d}{9 c e^2}-\frac{2}{27} b^2 h^2 x^3+\frac{h^2 (d+e x)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{3 e^3}+\frac{2 h \left(e^2 f-d^2 h\right) x \left(a+b \sin ^{-1}(c x)\right)^2}{e^2}-\frac{\left(e^2 f-d^2 h\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{e^3 (d+e x)}-\frac{4 b^2 h^2 x}{9 c^2}-\frac{2 b^2 h \left(2 e^2 f-d^2 h\right) x}{e^2}+\frac{4 b^2 h^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c^3}+\frac{2 b^2 h^2 x^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{9 c}+\frac{2 b^2 h \left(2 e^2 f-d^2 h\right) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c e^2}+\frac{2 a b c \left(e^2 f-d^2 h\right)^2 \tan ^{-1}\left(\frac{d x c^2+e}{\sqrt{c^2 d^2-e^2} \sqrt{1-c^2 x^2}}\right)}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{2 i b^2 c \left(e^2 f-d^2 h\right)^2 \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{2 i b^2 c \left(e^2 f-d^2 h\right)^2 \sin ^{-1}(c x) \log \left(1-\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^3 \sqrt{c^2 d^2-e^2}}-\frac{2 b^2 c \left(e^2 f-d^2 h\right)^2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt{c^2 d^2-e^2}}\right)}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{2 b^2 c \left(e^2 f-d^2 h\right)^2 \text{Li}_2\left(\frac{i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt{c^2 d^2-e^2}}\right)}{e^3 \sqrt{c^2 d^2-e^2}}+\frac{2 a b h^2 (d+e x)^2 \sqrt{1-c^2 x^2}}{9 c e^2}+\frac{a b h \left(\left(36 e^2 f-25 d^2 h\right) c^2+4 e^2 h\right) \sqrt{1-c^2 x^2}}{9 c^3 e^2}",1,"(h*(2*e^2*f - d^2*h)*x*(a + b*ArcSin[c*x])^2)/e^2 + (d*h^2*x^2*(a + b*ArcSin[c*x])^2)/e + (h^2*x^3*(a + b*ArcSin[c*x])^2)/3 - ((e^2*f - d^2*h)^2*(a + b*ArcSin[c*x])^2)/(e^3*(d + e*x)) - (2*b*h^2*(-3*a*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + b*c*x*(6 + c^2*x^2) - 3*b*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2)*ArcSin[c*x]))/(27*c^3) - (2*b*h*(2*e^2*f - d^2*h)*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c))/e^2 - (b*d*h^2*(b*x^2 - (2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (a + b*ArcSin[c*x])^2/(b*c^2)))/(2*e) + (2*b*c*(e^2*f - d^2*h)^2*((-I)*(a + b*ArcSin[c*x])*(Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) - b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])] + b*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/(e^3*Sqrt[c^2*d^2 - e^2])","A",0
122,1,99,137,0.1050543,"\int x^3 \sin ^{-1}(a+b x) \, dx","Integrate[x^3*ArcSin[a + b*x],x]","\frac{\sqrt{-a^2-2 a b x-b^2 x^2+1} \left(-50 a^3+26 a^2 b x-a \left(14 b^2 x^2+55\right)+6 b^3 x^3+9 b x\right)-3 \left(8 a^4+24 a^2-8 b^4 x^4+3\right) \sin ^{-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 \sin ^{-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]*(-50*a^3 + 9*b*x + 26*a^2*b*x + 6*b^3*x^3 - a*(55 + 14*b^2*x^2)) - 3*(3 + 24*a^2 + 8*a^4 - 8*b^4*x^4)*ArcSin[a + b*x])/(96*b^4)","A",1
123,1,77,94,0.0781674,"\int x^2 \sin ^{-1}(a+b x) \, dx","Integrate[x^2*ArcSin[a + b*x],x]","\frac{\left(6 a^3+9 a+6 b^3 x^3\right) \sin ^{-1}(a+b x)+\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)}{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 \sin ^{-1}(a+b x)+\frac{x^2 \sqrt{1-(a+b x)^2}}{9 b}",1,"(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) + (9*a + 6*a^3 + 6*b^3*x^3)*ArcSin[a + b*x])/(18*b^3)","A",1
124,1,62,80,0.0494668,"\int x \sin ^{-1}(a+b x) \, dx","Integrate[x*ArcSin[a + b*x],x]","\frac{\sqrt{-a^2-2 a b x-b^2 x^2+1} (b x-3 a)+\left(-2 a^2+2 b^2 x^2-1\right) \sin ^{-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 \sin ^{-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] + (-1 - 2*a^2 + 2*b^2*x^2)*ArcSin[a + b*x])/(4*b^2)","A",1
125,1,41,35,0.0408061,"\int \sin ^{-1}(a+b x) \, dx","Integrate[ArcSin[a + b*x],x]","\frac{\sqrt{-a^2-2 a b x-b^2 x^2+1}+(a+b x) \sin ^{-1}(a+b x)}{b}","\frac{\sqrt{1-(a+b x)^2}}{b}+\frac{(a+b x) \sin ^{-1}(a+b x)}{b}",1,"(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] + (a + b*x)*ArcSin[a + b*x])/b","A",1
126,1,197,181,0.01864,"\int \frac{\sin ^{-1}(a+b x)}{x} \, dx","Integrate[ArcSin[a + b*x]/x,x]","-i \text{Li}_2\left(-\frac{e^{i \sin ^{-1}(a+b x)}}{\sqrt{1-a^2}-i a}\right)-i \text{Li}_2\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt{1-a^2}}\right)+\sin ^{-1}(a+b x) \log \left(1+\frac{e^{i \sin ^{-1}(a+b x)}}{b \left(-\frac{\sqrt{1-a^2}}{b}-\frac{i a}{b}\right)}\right)+\sin ^{-1}(a+b x) \log \left(1+\frac{e^{i \sin ^{-1}(a+b x)}}{b \left(\frac{\sqrt{1-a^2}}{b}-\frac{i a}{b}\right)}\right)-\frac{1}{2} i \sin ^{-1}(a+b x)^2","-i \text{Li}_2\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt{1-a^2}}\right)-i \text{Li}_2\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt{1-a^2}}\right)+\sin ^{-1}(a+b x) \log \left(1-\frac{e^{i \sin ^{-1}(a+b x)}}{-\sqrt{1-a^2}+i a}\right)+\sin ^{-1}(a+b x) \log \left(1-\frac{e^{i \sin ^{-1}(a+b x)}}{\sqrt{1-a^2}+i a}\right)-\frac{1}{2} i \sin ^{-1}(a+b x)^2",1,"(-1/2*I)*ArcSin[a + b*x]^2 + ArcSin[a + b*x]*Log[1 + E^(I*ArcSin[a + b*x])/((((-I)*a)/b - Sqrt[1 - a^2]/b)*b)] + ArcSin[a + b*x]*Log[1 + E^(I*ArcSin[a + b*x])/((((-I)*a)/b + Sqrt[1 - a^2]/b)*b)] - I*PolyLog[2, -(E^(I*ArcSin[a + b*x])/((-I)*a + Sqrt[1 - a^2]))] - I*PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])]","A",1
127,1,66,64,0.057159,"\int \frac{\sin ^{-1}(a+b x)}{x^2} \, dx","Integrate[ArcSin[a + b*x]/x^2,x]","-\frac{b \tanh ^{-1}\left(\frac{-a^2-a b x+1}{\sqrt{1-a^2} \sqrt{1-(a+b x)^2}}\right)}{\sqrt{1-a^2}}-\frac{\sin ^{-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{\sin ^{-1}(a+b x)}{x}",1,"-(ArcSin[a + b*x]/x) - (b*ArcTanh[(1 - a^2 - a*b*x)/(Sqrt[1 - a^2]*Sqrt[1 - (a + b*x)^2])])/Sqrt[1 - a^2]","A",1
128,1,125,103,0.2165014,"\int \frac{\sin ^{-1}(a+b x)}{x^3} \, dx","Integrate[ArcSin[a + b*x]/x^3,x]","-\frac{\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}}+\sin ^{-1}(a+b x)}{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{\sin ^{-1}(a+b x)}{2 x^2}",1,"-1/2*(ArcSin[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
129,1,166,144,0.2371521,"\int \frac{\sin ^{-1}(a+b x)}{x^4} \, dx","Integrate[ArcSin[a + b*x]/x^4,x]","\frac{\left(2 a^2+1\right) b^3 x^3 \log (x)+\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} \sin ^{-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{\sin ^{-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)*ArcSin[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
130,1,194,186,0.2582446,"\int \frac{\sin ^{-1}(a+b x)}{x^5} \, dx","Integrate[ArcSin[a + b*x]/x^5,x]","\frac{1}{8} \left(\frac{a \left(2 a^2+3\right) b^4 \log (x)}{\left(1-a^2\right)^{7/2}}-\frac{a \left(2 a^2+3\right) b^4 \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)}{\left(1-a^2\right)^{7/2}}+\frac{b \sqrt{-a^2-2 a b x-b^2 x^2+1} \left(2 a^4-5 a^3 b x+a^2 \left(11 b^2 x^2-4\right)+5 a b x+4 b^2 x^2+2\right)}{3 \left(a^2-1\right)^3 x^3}-\frac{2 \sin ^{-1}(a+b x)}{x^4}\right)","-\frac{a \left(2 a^2+3\right) b^4 \tanh ^{-1}\left(\frac{1-a (a+b x)}{\sqrt{1-a^2} \sqrt{1-(a+b x)^2}}\right)}{8 \left(1-a^2\right)^{7/2}}-\frac{\left(11 a^2+4\right) b^3 \sqrt{1-(a+b x)^2}}{24 \left(1-a^2\right)^3 x}-\frac{5 a b^2 \sqrt{1-(a+b x)^2}}{24 \left(1-a^2\right)^2 x^2}-\frac{b \sqrt{1-(a+b x)^2}}{12 \left(1-a^2\right) x^3}-\frac{\sin ^{-1}(a+b x)}{4 x^4}",1,"((b*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(2 + 2*a^4 + 5*a*b*x - 5*a^3*b*x + 4*b^2*x^2 + a^2*(-4 + 11*b^2*x^2)))/(3*(-1 + a^2)^3*x^3) - (2*ArcSin[a + b*x])/x^4 + (a*(3 + 2*a^2)*b^4*Log[x])/(1 - a^2)^(7/2) - (a*(3 + 2*a^2)*b^4*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)^(7/2))/8","A",1
131,1,148,343,0.2318848,"\int x^3 \sin ^{-1}(a+b x)^2 \, dx","Integrate[x^3*ArcSin[a + b*x]^2,x]","\frac{-9 \left(8 a^4+24 a^2-8 b^4 x^4+3\right) \sin ^{-1}(a+b x)^2+b x \left(300 a^3-78 a^2 b x+a \left(28 b^2 x^2+330\right)-9 b x \left(b^2 x^2+3\right)\right)-6 \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) \sin ^{-1}(a+b x)}{288 b^4}","-\frac{a^4 \sin ^{-1}(a+b x)^2}{4 b^4}-\frac{2 a^3 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^4}+\frac{2 a^3 x}{b^3}-\frac{3 a^2 (a+b x)^2}{4 b^4}-\frac{3 a^2 \sin ^{-1}(a+b x)^2}{4 b^4}+\frac{3 a^2 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{2 b^4}+\frac{2 a (a+b x)^3}{9 b^4}-\frac{(a+b x)^4}{32 b^4}-\frac{3 (a+b x)^2}{32 b^4}-\frac{2 a (a+b x)^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{3 b^4}-\frac{4 a \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{3 b^4}-\frac{3 \sin ^{-1}(a+b x)^2}{32 b^4}+\frac{(a+b x)^3 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{8 b^4}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{16 b^4}+\frac{4 a x}{3 b^3}+\frac{1}{4} x^4 \sin ^{-1}(a+b x)^2",1,"(b*x*(300*a^3 - 78*a^2*b*x - 9*b*x*(3 + b^2*x^2) + a*(330 + 28*b^2*x^2)) - 6*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)*ArcSin[a + b*x] - 9*(3 + 24*a^2 + 8*a^4 - 8*b^4*x^4)*ArcSin[a + b*x]^2)/(288*b^4)","A",1
132,1,111,220,0.1828335,"\int x^2 \sin ^{-1}(a+b x)^2 \, dx","Integrate[x^2*ArcSin[a + b*x]^2,x]","\frac{9 \left(2 a^3+3 a+2 b^3 x^3\right) \sin ^{-1}(a+b x)^2-b x \left(66 a^2-15 a b x+4 b^2 x^2+24\right)+6 \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) \sin ^{-1}(a+b x)}{54 b^3}","\frac{a^3 \sin ^{-1}(a+b x)^2}{3 b^3}+\frac{2 a^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}-\frac{2 a^2 x}{b^2}+\frac{a (a+b x)^2}{2 b^3}-\frac{2 (a+b x)^3}{27 b^3}+\frac{a \sin ^{-1}(a+b x)^2}{2 b^3}-\frac{a (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^3}+\frac{2 (a+b x)^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac{4 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{9 b^3}+\frac{1}{3} x^3 \sin ^{-1}(a+b x)^2-\frac{4 x}{9 b^2}",1,"(-(b*x*(24 + 66*a^2 - 15*a*b*x + 4*b^2*x^2)) + 6*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)*ArcSin[a + b*x] + 9*(3*a + 2*a^3 + 2*b^3*x^3)*ArcSin[a + b*x]^2)/(54*b^3)","A",1
133,1,83,130,0.1028921,"\int x \sin ^{-1}(a+b x)^2 \, dx","Integrate[x*ArcSin[a + b*x]^2,x]","\frac{\left(-2 a^2+2 b^2 x^2-1\right) \sin ^{-1}(a+b x)^2-2 (3 a-b x) \sqrt{-a^2-2 a b x-b^2 x^2+1} \sin ^{-1}(a+b x)+b x (6 a-b x)}{4 b^2}","-\frac{a^2 \sin ^{-1}(a+b x)^2}{2 b^2}-\frac{(a+b x)^2}{4 b^2}+\frac{\sqrt{1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)}{2 b^2}-\frac{\sin ^{-1}(a+b x)^2}{4 b^2}-\frac{2 a \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^2}+\frac{1}{2} x^2 \sin ^{-1}(a+b x)^2+\frac{2 a x}{b}",1,"(b*x*(6*a - b*x) - 2*(3*a - b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x] + (-1 - 2*a^2 + 2*b^2*x^2)*ArcSin[a + b*x]^2)/(4*b^2)","A",1
134,1,49,47,0.0275795,"\int \sin ^{-1}(a+b x)^2 \, dx","Integrate[ArcSin[a + b*x]^2,x]","\frac{-2 (a+b x)+(a+b x) \sin ^{-1}(a+b x)^2+2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{b}","\frac{(a+b x) \sin ^{-1}(a+b x)^2}{b}+\frac{2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{b}-2 x",1,"(-2*(a + b*x) + 2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x] + (a + b*x)*ArcSin[a + b*x]^2)/b","A",1
135,1,309,271,0.0434324,"\int \frac{\sin ^{-1}(a+b x)^2}{x} \, dx","Integrate[ArcSin[a + b*x]^2/x,x]","-2 i \sin ^{-1}(a+b x) \text{Li}_2\left(-\frac{e^{i \sin ^{-1}(a+b x)}}{\left(-\frac{i a}{b}-\frac{\sqrt{1-a^2}}{b}\right) b}\right)-2 i \sin ^{-1}(a+b x) \text{Li}_2\left(-\frac{e^{i \sin ^{-1}(a+b x)}}{\left(\frac{\sqrt{1-a^2}}{b}-\frac{i a}{b}\right) b}\right)+2 \text{Li}_3\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt{1-a^2}}\right)+2 \text{Li}_3\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt{1-a^2}}\right)+\sin ^{-1}(a+b x)^2 \log \left(1+\frac{e^{i \sin ^{-1}(a+b x)}}{b \left(-\frac{\sqrt{1-a^2}}{b}-\frac{i a}{b}\right)}\right)+\sin ^{-1}(a+b x)^2 \log \left(1+\frac{e^{i \sin ^{-1}(a+b x)}}{b \left(\frac{\sqrt{1-a^2}}{b}-\frac{i a}{b}\right)}\right)-\frac{1}{3} i \sin ^{-1}(a+b x)^3","-2 i \sin ^{-1}(a+b x) \text{Li}_2\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt{1-a^2}}\right)-2 i \sin ^{-1}(a+b x) \text{Li}_2\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt{1-a^2}}\right)+2 \text{Li}_3\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt{1-a^2}}\right)+2 \text{Li}_3\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt{1-a^2}}\right)+\sin ^{-1}(a+b x)^2 \log \left(1-\frac{e^{i \sin ^{-1}(a+b x)}}{-\sqrt{1-a^2}+i a}\right)+\sin ^{-1}(a+b x)^2 \log \left(1-\frac{e^{i \sin ^{-1}(a+b x)}}{\sqrt{1-a^2}+i a}\right)-\frac{1}{3} i \sin ^{-1}(a+b x)^3",1,"(-1/3*I)*ArcSin[a + b*x]^3 + ArcSin[a + b*x]^2*Log[1 + E^(I*ArcSin[a + b*x])/((((-I)*a)/b - Sqrt[1 - a^2]/b)*b)] + ArcSin[a + b*x]^2*Log[1 + E^(I*ArcSin[a + b*x])/((((-I)*a)/b + Sqrt[1 - a^2]/b)*b)] - (2*I)*ArcSin[a + b*x]*PolyLog[2, -(E^(I*ArcSin[a + b*x])/((((-I)*a)/b - Sqrt[1 - a^2]/b)*b))] - (2*I)*ArcSin[a + b*x]*PolyLog[2, -(E^(I*ArcSin[a + b*x])/((((-I)*a)/b + Sqrt[1 - a^2]/b)*b))] + 2*PolyLog[3, E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])] + 2*PolyLog[3, E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])]","A",1
136,1,208,230,0.1510282,"\int \frac{\sin ^{-1}(a+b x)^2}{x^2} \, dx","Integrate[ArcSin[a + b*x]^2/x^2,x]","\frac{2 b x \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(a+b x)}}{\sqrt{a^2-1}-a}\right)-2 b x \text{Li}_2\left(-\frac{i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt{a^2-1}}\right)-\sqrt{a^2-1} \sin ^{-1}(a+b x)^2+2 i b x \sin ^{-1}(a+b x) \left(\log \left(\frac{-\sqrt{a^2-1}+i e^{i \sin ^{-1}(a+b x)}+a}{a-\sqrt{a^2-1}}\right)-\log \left(\frac{\sqrt{a^2-1}+i e^{i \sin ^{-1}(a+b x)}+a}{\sqrt{a^2-1}+a}\right)\right)}{\sqrt{a^2-1} x}","\frac{2 i b \text{Li}_2\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt{1-a^2}}\right)}{\sqrt{1-a^2}}-\frac{2 i b \text{Li}_2\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt{1-a^2}}\right)}{\sqrt{1-a^2}}-\frac{2 b \sin ^{-1}(a+b x) \log \left(1-\frac{e^{i \sin ^{-1}(a+b x)}}{-\sqrt{1-a^2}+i a}\right)}{\sqrt{1-a^2}}+\frac{2 b \sin ^{-1}(a+b x) \log \left(1-\frac{e^{i \sin ^{-1}(a+b x)}}{\sqrt{1-a^2}+i a}\right)}{\sqrt{1-a^2}}-\frac{\sin ^{-1}(a+b x)^2}{x}",1,"(-(Sqrt[-1 + a^2]*ArcSin[a + b*x]^2) + (2*I)*b*x*ArcSin[a + b*x]*(Log[(a - Sqrt[-1 + a^2] + I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])] - Log[(a + Sqrt[-1 + a^2] + I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])]) + 2*b*x*PolyLog[2, (I*E^(I*ArcSin[a + b*x]))/(-a + Sqrt[-1 + a^2])] - 2*b*x*PolyLog[2, ((-I)*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/(Sqrt[-1 + a^2]*x)","A",0
137,1,314,272,0.1425523,"\int \frac{\sin ^{-1}(a+b x)^2}{x^3} \, dx","Integrate[ArcSin[a + b*x]^2/x^3,x]","\frac{-2 a b^2 x^2 \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(a+b x)}}{\sqrt{a^2-1}-a}\right)+2 a b^2 x^2 \text{Li}_2\left(-\frac{i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt{a^2-1}}\right)-2 \sqrt{a^2-1} b^2 x^2 \log (x)-2 i a b^2 x^2 \sin ^{-1}(a+b x) \log \left(\frac{-\sqrt{a^2-1}+i e^{i \sin ^{-1}(a+b x)}+a}{a-\sqrt{a^2-1}}\right)+2 i a b^2 x^2 \sin ^{-1}(a+b x) \log \left(\frac{\sqrt{a^2-1}+i e^{i \sin ^{-1}(a+b x)}+a}{\sqrt{a^2-1}+a}\right)+2 \sqrt{a^2-1} b x \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)-a^2 \sqrt{a^2-1} \sin ^{-1}(a+b x)^2+\sqrt{a^2-1} \sin ^{-1}(a+b x)^2}{2 \left(a^2-1\right)^{3/2} x^2}","-\frac{a b^2 \text{Li}_2\left(-\frac{i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt{a^2-1}}\right)}{\left(a^2-1\right)^{3/2}}+\frac{a b^2 \text{Li}_2\left(-\frac{i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt{a^2-1}}\right)}{\left(a^2-1\right)^{3/2}}+\frac{b^2 \log (x)}{1-a^2}-\frac{i a b^2 \sin ^{-1}(a+b x) \log \left(1+\frac{i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt{a^2-1}}\right)}{\left(a^2-1\right)^{3/2}}+\frac{i a b^2 \sin ^{-1}(a+b x) \log \left(1+\frac{i e^{i \sin ^{-1}(a+b x)}}{\sqrt{a^2-1}+a}\right)}{\left(a^2-1\right)^{3/2}}-\frac{b \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{\left(1-a^2\right) x}-\frac{\sin ^{-1}(a+b x)^2}{2 x^2}",1,"(2*Sqrt[-1 + a^2]*b*x*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x] + Sqrt[-1 + a^2]*ArcSin[a + b*x]^2 - a^2*Sqrt[-1 + a^2]*ArcSin[a + b*x]^2 - (2*I)*a*b^2*x^2*ArcSin[a + b*x]*Log[(a - Sqrt[-1 + a^2] + I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])] + (2*I)*a*b^2*x^2*ArcSin[a + b*x]*Log[(a + Sqrt[-1 + a^2] + I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])] - 2*Sqrt[-1 + a^2]*b^2*x^2*Log[x] - 2*a*b^2*x^2*PolyLog[2, (I*E^(I*ArcSin[a + b*x]))/(-a + Sqrt[-1 + a^2])] + 2*a*b^2*x^2*PolyLog[2, ((-I)*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/(2*(-1 + a^2)^(3/2)*x^2)","A",1
138,1,181,371,0.2451599,"\int x^2 \sin ^{-1}(a+b x)^3 \, dx","Integrate[x^2*ArcSin[a + b*x]^3,x]","\frac{18 \left(2 a^3+3 a+2 b^3 x^3\right) \sin ^{-1}(a+b x)^3-\sqrt{-a^2-2 a b x-b^2 x^2+1} \left(575 a^2-65 a b x+8 b^2 x^2+160\right)+18 \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) \sin ^{-1}(a+b x)^2-3 \left(170 a^3+132 a^2 b x+a \left(75-30 b^2 x^2\right)+8 b x \left(b^2 x^2+6\right)\right) \sin ^{-1}(a+b x)}{108 b^3}","\frac{a^3 \sin ^{-1}(a+b x)^3}{3 b^3}-\frac{6 a^2 \sqrt{1-(a+b x)^2}}{b^3}-\frac{6 a^2 (a+b x) \sin ^{-1}(a+b x)}{b^3}+\frac{3 a^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^3}+\frac{3 a \sqrt{1-(a+b x)^2} (a+b x)}{4 b^3}+\frac{2 \left(1-(a+b x)^2\right)^{3/2}}{27 b^3}-\frac{14 \sqrt{1-(a+b x)^2}}{9 b^3}-\frac{2 (a+b x)^3 \sin ^{-1}(a+b x)}{9 b^3}+\frac{\sqrt{1-(a+b x)^2} (a+b x)^2 \sin ^{-1}(a+b x)^2}{3 b^3}+\frac{3 a (a+b x)^2 \sin ^{-1}(a+b x)}{2 b^3}-\frac{3 a \sqrt{1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)^2}{2 b^3}-\frac{4 (a+b x) \sin ^{-1}(a+b x)}{3 b^3}+\frac{a \sin ^{-1}(a+b x)^3}{2 b^3}+\frac{2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{3 b^3}-\frac{3 a \sin ^{-1}(a+b x)}{4 b^3}+\frac{1}{3} x^3 \sin ^{-1}(a+b x)^3",1,"(-(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(160 + 575*a^2 - 65*a*b*x + 8*b^2*x^2)) - 3*(170*a^3 + 132*a^2*b*x + a*(75 - 30*b^2*x^2) + 8*b*x*(6 + b^2*x^2))*ArcSin[a + b*x] + 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)*ArcSin[a + b*x]^2 + 18*(3*a + 2*a^3 + 2*b^3*x^3)*ArcSin[a + b*x]^3)/(108*b^3)","A",1
139,1,135,211,0.1663463,"\int x \sin ^{-1}(a+b x)^3 \, dx","Integrate[x*ArcSin[a + b*x]^3,x]","\frac{3 (15 a-b x) \sqrt{-a^2-2 a b x-b^2 x^2+1}+\left(-4 a^2+4 b^2 x^2-2\right) \sin ^{-1}(a+b x)^3-6 (3 a-b x) \sqrt{-a^2-2 a b x-b^2 x^2+1} \sin ^{-1}(a+b x)^2+\left(42 a^2+36 a b x-6 b^2 x^2+3\right) \sin ^{-1}(a+b x)}{8 b^2}","-\frac{a^2 \sin ^{-1}(a+b x)^3}{2 b^2}-\frac{3 (a+b x) \sqrt{1-(a+b x)^2}}{8 b^2}+\frac{6 a \sqrt{1-(a+b x)^2}}{b^2}-\frac{\sin ^{-1}(a+b x)^3}{4 b^2}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{4 b^2}-\frac{3 a \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b^2}-\frac{3 (a+b x)^2 \sin ^{-1}(a+b x)}{4 b^2}+\frac{6 a (a+b x) \sin ^{-1}(a+b x)}{b^2}+\frac{3 \sin ^{-1}(a+b x)}{8 b^2}+\frac{1}{2} x^2 \sin ^{-1}(a+b x)^3",1,"(3*(15*a - b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] + (3 + 42*a^2 + 36*a*b*x - 6*b^2*x^2)*ArcSin[a + b*x] - 6*(3*a - b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^2 + (-2 - 4*a^2 + 4*b^2*x^2)*ArcSin[a + b*x]^3)/(8*b^2)","A",1
140,1,74,82,0.0357595,"\int \sin ^{-1}(a+b x)^3 \, dx","Integrate[ArcSin[a + b*x]^3,x]","\frac{-6 \sqrt{1-(a+b x)^2}+(a+b x) \sin ^{-1}(a+b x)^3+3 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2-6 (a+b x) \sin ^{-1}(a+b x)}{b}","-\frac{6 \sqrt{1-(a+b x)^2}}{b}+\frac{(a+b x) \sin ^{-1}(a+b x)^3}{b}+\frac{3 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{b}-\frac{6 (a+b x) \sin ^{-1}(a+b x)}{b}",1,"(-6*Sqrt[1 - (a + b*x)^2] - 6*(a + b*x)*ArcSin[a + b*x] + 3*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2 + (a + b*x)*ArcSin[a + b*x]^3)/b","A",1
141,1,424,365,0.0591415,"\int \frac{\sin ^{-1}(a+b x)^3}{x} \, dx","Integrate[ArcSin[a + b*x]^3/x,x]","-3 i \sin ^{-1}(a+b x)^2 \text{Li}_2\left(-\frac{e^{i \sin ^{-1}(a+b x)}}{\left(-\frac{i a}{b}-\frac{\sqrt{1-a^2}}{b}\right) b}\right)-3 i \sin ^{-1}(a+b x)^2 \text{Li}_2\left(-\frac{e^{i \sin ^{-1}(a+b x)}}{\left(\frac{\sqrt{1-a^2}}{b}-\frac{i a}{b}\right) b}\right)+6 \sin ^{-1}(a+b x) \text{Li}_3\left(-\frac{e^{i \sin ^{-1}(a+b x)}}{\left(-\frac{i a}{b}-\frac{\sqrt{1-a^2}}{b}\right) b}\right)+6 \sin ^{-1}(a+b x) \text{Li}_3\left(-\frac{e^{i \sin ^{-1}(a+b x)}}{\left(\frac{\sqrt{1-a^2}}{b}-\frac{i a}{b}\right) b}\right)+6 i \text{Li}_4\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt{1-a^2}}\right)+6 i \text{Li}_4\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt{1-a^2}}\right)+\sin ^{-1}(a+b x)^3 \log \left(1+\frac{e^{i \sin ^{-1}(a+b x)}}{b \left(-\frac{\sqrt{1-a^2}}{b}-\frac{i a}{b}\right)}\right)+\sin ^{-1}(a+b x)^3 \log \left(1+\frac{e^{i \sin ^{-1}(a+b x)}}{b \left(\frac{\sqrt{1-a^2}}{b}-\frac{i a}{b}\right)}\right)-\frac{1}{4} i \sin ^{-1}(a+b x)^4","-3 i \sin ^{-1}(a+b x)^2 \text{Li}_2\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt{1-a^2}}\right)-3 i \sin ^{-1}(a+b x)^2 \text{Li}_2\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt{1-a^2}}\right)+6 \sin ^{-1}(a+b x) \text{Li}_3\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt{1-a^2}}\right)+6 \sin ^{-1}(a+b x) \text{Li}_3\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt{1-a^2}}\right)+6 i \text{Li}_4\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt{1-a^2}}\right)+6 i \text{Li}_4\left(\frac{e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt{1-a^2}}\right)+\sin ^{-1}(a+b x)^3 \log \left(1-\frac{e^{i \sin ^{-1}(a+b x)}}{-\sqrt{1-a^2}+i a}\right)+\sin ^{-1}(a+b x)^3 \log \left(1-\frac{e^{i \sin ^{-1}(a+b x)}}{\sqrt{1-a^2}+i a}\right)-\frac{1}{4} i \sin ^{-1}(a+b x)^4",1,"(-1/4*I)*ArcSin[a + b*x]^4 + ArcSin[a + b*x]^3*Log[1 + E^(I*ArcSin[a + b*x])/((((-I)*a)/b - Sqrt[1 - a^2]/b)*b)] + ArcSin[a + b*x]^3*Log[1 + E^(I*ArcSin[a + b*x])/((((-I)*a)/b + Sqrt[1 - a^2]/b)*b)] - (3*I)*ArcSin[a + b*x]^2*PolyLog[2, -(E^(I*ArcSin[a + b*x])/((((-I)*a)/b - Sqrt[1 - a^2]/b)*b))] - (3*I)*ArcSin[a + b*x]^2*PolyLog[2, -(E^(I*ArcSin[a + b*x])/((((-I)*a)/b + Sqrt[1 - a^2]/b)*b))] + 6*ArcSin[a + b*x]*PolyLog[3, -(E^(I*ArcSin[a + b*x])/((((-I)*a)/b - Sqrt[1 - a^2]/b)*b))] + 6*ArcSin[a + b*x]*PolyLog[3, -(E^(I*ArcSin[a + b*x])/((((-I)*a)/b + Sqrt[1 - a^2]/b)*b))] + (6*I)*PolyLog[4, E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])] + (6*I)*PolyLog[4, E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])]","A",1
142,1,309,316,0.14013,"\int \frac{\sin ^{-1}(a+b x)^3}{x^2} \, dx","Integrate[ArcSin[a + b*x]^3/x^2,x]","-\frac{-6 b x \sin ^{-1}(a+b x) \text{Li}_2\left(\frac{i e^{i \sin ^{-1}(a+b x)}}{\sqrt{a^2-1}-a}\right)+6 b x \sin ^{-1}(a+b x) \text{Li}_2\left(-\frac{i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt{a^2-1}}\right)-6 i b x \text{Li}_3\left(\frac{i e^{i \sin ^{-1}(a+b x)}}{\sqrt{a^2-1}-a}\right)+6 i b x \text{Li}_3\left(-\frac{i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt{a^2-1}}\right)+\sqrt{a^2-1} \sin ^{-1}(a+b x)^3-3 i b x \sin ^{-1}(a+b x)^2 \log \left(\frac{-\sqrt{a^2-1}+i e^{i \sin ^{-1}(a+b x)}+a}{a-\sqrt{a^2-1}}\right)+3 i b x \sin ^{-1}(a+b x)^2 \log \left(\frac{\sqrt{a^2-1}+i e^{i \sin ^{-1}(a+b x)}+a}{\sqrt{a^2-1}+a}\right)}{\sqrt{a^2-1} x}","\frac{6 b \sin ^{-1}(a+b x) \text{Li}_2\left(-\frac{i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt{a^2-1}}\right)}{\sqrt{a^2-1}}-\frac{6 b \sin ^{-1}(a+b x) \text{Li}_2\left(-\frac{i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt{a^2-1}}\right)}{\sqrt{a^2-1}}+\frac{6 i b \text{Li}_3\left(-\frac{i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt{a^2-1}}\right)}{\sqrt{a^2-1}}-\frac{6 i b \text{Li}_3\left(-\frac{i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt{a^2-1}}\right)}{\sqrt{a^2-1}}+\frac{3 i b \sin ^{-1}(a+b x)^2 \log \left(1+\frac{i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt{a^2-1}}\right)}{\sqrt{a^2-1}}-\frac{3 i b \sin ^{-1}(a+b x)^2 \log \left(1+\frac{i e^{i \sin ^{-1}(a+b x)}}{\sqrt{a^2-1}+a}\right)}{\sqrt{a^2-1}}-\frac{\sin ^{-1}(a+b x)^3}{x}",1,"-((Sqrt[-1 + a^2]*ArcSin[a + b*x]^3 - (3*I)*b*x*ArcSin[a + b*x]^2*Log[(a - Sqrt[-1 + a^2] + I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])] + (3*I)*b*x*ArcSin[a + b*x]^2*Log[(a + Sqrt[-1 + a^2] + I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])] - 6*b*x*ArcSin[a + b*x]*PolyLog[2, (I*E^(I*ArcSin[a + b*x]))/(-a + Sqrt[-1 + a^2])] + 6*b*x*ArcSin[a + b*x]*PolyLog[2, ((-I)*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])] - (6*I)*b*x*PolyLog[3, (I*E^(I*ArcSin[a + b*x]))/(-a + Sqrt[-1 + a^2])] + (6*I)*b*x*PolyLog[3, ((-I)*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/(Sqrt[-1 + a^2]*x))","A",1
143,1,45,60,0.1780929,"\int \frac{x^2}{\sin ^{-1}(a+b x)} \, dx","Integrate[x^2/ArcSin[a + b*x],x]","-\frac{-\left(\left(4 a^2+1\right) \text{Ci}\left(\sin ^{-1}(a+b x)\right)\right)+\text{Ci}\left(3 \sin ^{-1}(a+b x)\right)+4 a \text{Si}\left(2 \sin ^{-1}(a+b x)\right)}{4 b^3}","\frac{a^2 \text{Ci}\left(\sin ^{-1}(a+b x)\right)}{b^3}+\frac{\text{Ci}\left(\sin ^{-1}(a+b x)\right)}{4 b^3}-\frac{\text{Ci}\left(3 \sin ^{-1}(a+b x)\right)}{4 b^3}-\frac{a \text{Si}\left(2 \sin ^{-1}(a+b x)\right)}{b^3}",1,"-1/4*(-((1 + 4*a^2)*CosIntegral[ArcSin[a + b*x]]) + CosIntegral[3*ArcSin[a + b*x]] + 4*a*SinIntegral[2*ArcSin[a + b*x]])/b^3","A",1
144,1,30,30,0.0667449,"\int \frac{x}{\sin ^{-1}(a+b x)} \, dx","Integrate[x/ArcSin[a + b*x],x]","\frac{\text{Si}\left(2 \sin ^{-1}(a+b x)\right)}{2 b^2}-\frac{a \text{Ci}\left(\sin ^{-1}(a+b x)\right)}{b^2}","\frac{\text{Si}\left(2 \sin ^{-1}(a+b x)\right)}{2 b^2}-\frac{a \text{Ci}\left(\sin ^{-1}(a+b x)\right)}{b^2}",1,"-((a*CosIntegral[ArcSin[a + b*x]])/b^2) + SinIntegral[2*ArcSin[a + b*x]]/(2*b^2)","A",1
145,1,11,11,0.0178952,"\int \frac{1}{\sin ^{-1}(a+b x)} \, dx","Integrate[ArcSin[a + b*x]^(-1),x]","\frac{\text{Ci}\left(\sin ^{-1}(a+b x)\right)}{b}","\frac{\text{Ci}\left(\sin ^{-1}(a+b x)\right)}{b}",1,"CosIntegral[ArcSin[a + b*x]]/b","A",1
146,0,0,15,0.2419184,"\int \frac{1}{x \sin ^{-1}(a+b x)} \, dx","Integrate[1/(x*ArcSin[a + b*x]),x]","\int \frac{1}{x \sin ^{-1}(a+b x)} \, dx","\text{Int}\left(\frac{1}{x \sin ^{-1}(a+b x)},x\right)",0,"Integrate[1/(x*ArcSin[a + b*x]), x]","A",-1
147,1,86,84,0.6081059,"\int \frac{x^2}{\sin ^{-1}(a+b x)^2} \, dx","Integrate[x^2/ArcSin[a + b*x]^2,x]","-\frac{\frac{4 b^2 x^2 \sqrt{-a^2-2 a b x-b^2 x^2+1}}{\sin ^{-1}(a+b x)}+\left(4 a^2+1\right) \text{Si}\left(\sin ^{-1}(a+b x)\right)+8 a \text{Ci}\left(2 \sin ^{-1}(a+b x)\right)-3 \text{Si}\left(3 \sin ^{-1}(a+b x)\right)}{4 b^3}","-\frac{\left(4 a^2+1\right) \text{Si}\left(\sin ^{-1}(a+b x)\right)}{4 b^3}-\frac{2 a \text{Ci}\left(2 \sin ^{-1}(a+b x)\right)}{b^3}+\frac{3 \text{Si}\left(3 \sin ^{-1}(a+b x)\right)}{4 b^3}-\frac{x^2 \sqrt{1-(a+b x)^2}}{b \sin ^{-1}(a+b x)}",1,"-1/4*((4*b^2*x^2*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2])/ArcSin[a + b*x] + 8*a*CosIntegral[2*ArcSin[a + b*x]] + (1 + 4*a^2)*SinIntegral[ArcSin[a + b*x]] - 3*SinIntegral[3*ArcSin[a + b*x]])/b^3","A",1
148,1,63,55,0.1917327,"\int \frac{x}{\sin ^{-1}(a+b x)^2} \, dx","Integrate[x/ArcSin[a + b*x]^2,x]","\frac{\sin ^{-1}(a+b x) \text{Ci}\left(2 \sin ^{-1}(a+b x)\right)+a \sin ^{-1}(a+b x) \text{Si}\left(\sin ^{-1}(a+b x)\right)-b x \sqrt{1-(a+b x)^2}}{b^2 \sin ^{-1}(a+b x)}","\frac{\text{Ci}\left(2 \sin ^{-1}(a+b x)\right)}{b^2}+\frac{a \text{Si}\left(\sin ^{-1}(a+b x)\right)}{b^2}-\frac{x \sqrt{1-(a+b x)^2}}{b \sin ^{-1}(a+b x)}",1,"(-(b*x*Sqrt[1 - (a + b*x)^2]) + ArcSin[a + b*x]*CosIntegral[2*ArcSin[a + b*x]] + a*ArcSin[a + b*x]*SinIntegral[ArcSin[a + b*x]])/(b^2*ArcSin[a + b*x])","A",1
149,1,37,41,0.07641,"\int \frac{1}{\sin ^{-1}(a+b x)^2} \, dx","Integrate[ArcSin[a + b*x]^(-2),x]","-\frac{\text{Si}\left(\sin ^{-1}(a+b x)\right)+\frac{\sqrt{1-(a+b x)^2}}{\sin ^{-1}(a+b x)}}{b}","-\frac{\text{Si}\left(\sin ^{-1}(a+b x)\right)}{b}-\frac{\sqrt{1-(a+b x)^2}}{b \sin ^{-1}(a+b x)}",1,"-((Sqrt[1 - (a + b*x)^2]/ArcSin[a + b*x] + SinIntegral[ArcSin[a + b*x]])/b)","A",1
150,0,0,15,3.0904036,"\int \frac{1}{x \sin ^{-1}(a+b x)^2} \, dx","Integrate[1/(x*ArcSin[a + b*x]^2),x]","\int \frac{1}{x \sin ^{-1}(a+b x)^2} \, dx","\text{Int}\left(\frac{1}{x \sin ^{-1}(a+b x)^2},x\right)",0,"Integrate[1/(x*ArcSin[a + b*x]^2), x]","A",-1
151,1,115,176,0.5781362,"\int \frac{x^2}{\sin ^{-1}(a+b x)^3} \, dx","Integrate[x^2/ArcSin[a + b*x]^3,x]","\frac{\frac{4 b x \left(\left(2 a^2+5 a b x+3 b^2 x^2-2\right) \sin ^{-1}(a+b x)-b x \sqrt{-a^2-2 a b x-b^2 x^2+1}\right)}{\sin ^{-1}(a+b x)^2}-\left(4 a^2+1\right) \text{Ci}\left(\sin ^{-1}(a+b x)\right)+9 \text{Ci}\left(3 \sin ^{-1}(a+b x)\right)+16 a \text{Si}\left(2 \sin ^{-1}(a+b x)\right)}{8 b^3}","-\frac{\left(4 a^2+1\right) \text{Ci}\left(\sin ^{-1}(a+b x)\right)}{8 b^3}+\frac{a^2 (a+b x)}{2 b^3 \sin ^{-1}(a+b x)}+\frac{9 \text{Ci}\left(3 \sin ^{-1}(a+b x)\right)}{8 b^3}+\frac{2 a \text{Si}\left(2 \sin ^{-1}(a+b x)\right)}{b^3}-\frac{2 a (a+b x)^2}{b^3 \sin ^{-1}(a+b x)}-\frac{3 \sin \left(3 \sin ^{-1}(a+b x)\right)}{8 b^3 \sin ^{-1}(a+b x)}+\frac{9 a+b x}{8 b^3 \sin ^{-1}(a+b x)}-\frac{x^2 \sqrt{1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2}",1,"((4*b*x*(-(b*x*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]) + (-2 + 2*a^2 + 5*a*b*x + 3*b^2*x^2)*ArcSin[a + b*x]))/ArcSin[a + b*x]^2 - (1 + 4*a^2)*CosIntegral[ArcSin[a + b*x]] + 9*CosIntegral[3*ArcSin[a + b*x]] + 16*a*SinIntegral[2*ArcSin[a + b*x]])/(8*b^3)","A",1
152,1,121,108,0.117546,"\int \frac{x}{\sin ^{-1}(a+b x)^3} \, dx","Integrate[x/ArcSin[a + b*x]^3,x]","-\frac{x \sqrt{-a^2-2 a b x-b^2 x^2+1}}{2 b \sin ^{-1}(a+b x)^2}+\frac{a^2+3 a b x+2 b^2 x^2-1}{2 b^2 \sin ^{-1}(a+b x)}-2 \left(\frac{\text{Si}\left(2 \sin ^{-1}(a+b x)\right)}{2 b^2}-\frac{a \text{Ci}\left(\sin ^{-1}(a+b x)\right)}{b^2}\right)-\frac{3 a \text{Ci}\left(\sin ^{-1}(a+b x)\right)}{2 b^2}","\frac{a \text{Ci}\left(\sin ^{-1}(a+b x)\right)}{2 b^2}-\frac{\text{Si}\left(2 \sin ^{-1}(a+b x)\right)}{b^2}-\frac{a (a+b x)}{2 b^2 \sin ^{-1}(a+b x)}-\frac{1-2 (a+b x)^2}{2 b^2 \sin ^{-1}(a+b x)}-\frac{x \sqrt{1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2}",1,"-1/2*(x*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2])/(b*ArcSin[a + b*x]^2) + (-1 + a^2 + 3*a*b*x + 2*b^2*x^2)/(2*b^2*ArcSin[a + b*x]) - (3*a*CosIntegral[ArcSin[a + b*x]])/(2*b^2) - 2*(-((a*CosIntegral[ArcSin[a + b*x]])/b^2) + SinIntegral[2*ArcSin[a + b*x]]/(2*b^2))","A",1
153,1,65,65,0.0732468,"\int \frac{1}{\sin ^{-1}(a+b x)^3} \, dx","Integrate[ArcSin[a + b*x]^(-3),x]","-\frac{\text{Ci}\left(\sin ^{-1}(a+b x)\right)}{2 b}+\frac{a+b x}{2 b \sin ^{-1}(a+b x)}-\frac{\sqrt{1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2}","-\frac{\text{Ci}\left(\sin ^{-1}(a+b x)\right)}{2 b}+\frac{a+b x}{2 b \sin ^{-1}(a+b x)}-\frac{\sqrt{1-(a+b x)^2}}{2 b \sin ^{-1}(a+b x)^2}",1,"-1/2*Sqrt[1 - (a + b*x)^2]/(b*ArcSin[a + b*x]^2) + (a + b*x)/(2*b*ArcSin[a + b*x]) - CosIntegral[ArcSin[a + b*x]]/(2*b)","A",1
154,0,0,15,2.7130039,"\int \frac{1}{x \sin ^{-1}(a+b x)^3} \, dx","Integrate[1/(x*ArcSin[a + b*x]^3),x]","\int \frac{1}{x \sin ^{-1}(a+b x)^3} \, dx","\text{Int}\left(\frac{1}{x \sin ^{-1}(a+b x)^3},x\right)",0,"Integrate[1/(x*ArcSin[a + b*x]^3), x]","A",-1
155,1,473,535,1.9280018,"\int x^2 \sqrt{a+b \sin ^{-1}(c+d x)} \, dx","Integrate[x^2*Sqrt[a + b*ArcSin[c + d*x]],x]","\frac{36 \sqrt{2 \pi } c^2 \sin \left(\frac{a}{b}\right) C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)-9 \sqrt{2 \pi } \left(4 c^2+1\right) \cos \left(\frac{a}{b}\right) S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)+72 \sqrt{\frac{1}{b}} c^2 (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}+9 \sqrt{2 \pi } \sin \left(\frac{a}{b}\right) C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)-\sqrt{6 \pi } \sin \left(\frac{3 a}{b}\right) C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)-18 \sqrt{\pi } c \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)-18 \sqrt{\pi } c \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)+\sqrt{6 \pi } \cos \left(\frac{3 a}{b}\right) S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)-6 \sqrt{\frac{1}{b}} \sin \left(3 \sin ^{-1}(c+d x)\right) \sqrt{a+b \sin ^{-1}(c+d x)}+18 \sqrt{\frac{1}{b}} (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}+36 \sqrt{\frac{1}{b}} c \cos \left(2 \sin ^{-1}(c+d x)\right) \sqrt{a+b \sin ^{-1}(c+d x)}}{72 \sqrt{\frac{1}{b}} d^3}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} c^2 \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{d^3}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} c^2 \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{d^3}+\frac{c^2 (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{d^3}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d^3}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{12 d^3}-\frac{\sqrt{\pi } \sqrt{b} c \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{4 d^3}-\frac{\sqrt{\pi } \sqrt{b} c \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{4 d^3}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d^3}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{12 d^3}+\frac{(c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d^3}+\frac{c \cos \left(2 \sin ^{-1}(c+d x)\right) \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d^3}",1,"(18*Sqrt[b^(-1)]*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]] + 72*Sqrt[b^(-1)]*c^2*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]] + 36*Sqrt[b^(-1)]*c*Sqrt[a + b*ArcSin[c + d*x]]*Cos[2*ArcSin[c + d*x]] - 18*c*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]] - 9*(1 + 4*c^2)*Sqrt[2*Pi]*Cos[a/b]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]] + Sqrt[6*Pi]*Cos[(3*a)/b]*FresnelS[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]]] + 9*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*Sin[a/b] + 36*c^2*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*Sin[a/b] - 18*c*Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b] - Sqrt[6*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*Sin[(3*a)/b] - 6*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]]*Sin[3*ArcSin[c + d*x]])/(72*Sqrt[b^(-1)]*d^3)","A",1
156,1,256,269,3.2065203,"\int x \sqrt{a+b \sin ^{-1}(c+d x)} \, dx","Integrate[x*Sqrt[a + b*ArcSin[c + d*x]],x]","\frac{\frac{\sqrt{\pi } \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)}{\sqrt{\frac{1}{b}}}+\frac{\sqrt{\pi } \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)}{\sqrt{\frac{1}{b}}}+\frac{2 \left(-\left(\cos \left(2 \sin ^{-1}(c+d x)\right) \left(a+b \sin ^{-1}(c+d x)\right)\right)-2 b c e^{-\frac{i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-2 b c e^{\frac{i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{\sqrt{a+b \sin ^{-1}(c+d x)}}}{8 d^2}","-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} c \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{d^2}+\frac{\sqrt{\pi } \sqrt{b} \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{8 d^2}+\frac{\sqrt{\pi } \sqrt{b} \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{8 d^2}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} c \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{d^2}-\frac{c (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{d^2}-\frac{\cos \left(2 \sin ^{-1}(c+d x)\right) \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d^2}",1,"((Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]])/Sqrt[b^(-1)] + (2*(-((a + b*ArcSin[c + d*x])*Cos[2*ArcSin[c + d*x]]) - (2*b*c*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b])/E^((I*a)/b) - 2*b*c*E^((I*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/Sqrt[a + b*ArcSin[c + d*x]] + (Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b])/Sqrt[b^(-1)])/(8*d^2)","C",0
157,1,129,133,0.1104391,"\int \sqrt{a+b \sin ^{-1}(c+d x)} \, dx","Integrate[Sqrt[a + b*ArcSin[c + d*x]],x]","\frac{b e^{-\frac{i a}{b}} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{2 d \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{d}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{d}+\frac{(c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{d}",1,"(b*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(2*d*E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
158,1,635,343,7.6312023,"\int x \left(a+b \sin ^{-1}(c+d x)\right)^{3/2} \, dx","Integrate[x*(a + b*ArcSin[c + d*x])^(3/2),x]","-\frac{b c \left(-\sqrt{2 \pi } \sqrt{\frac{1}{b}} \left(2 a \sin \left(\frac{a}{b}\right)+3 b \cos \left(\frac{a}{b}\right)\right) C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)+\sqrt{2 \pi } \sqrt{\frac{1}{b}} \left(2 a \cos \left(\frac{a}{b}\right)-3 b \sin \left(\frac{a}{b}\right)\right) S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)+2 \left(3 \sqrt{1-(c+d x)^2}+2 (c+d x) \sin ^{-1}(c+d x)\right) \sqrt{a+b \sin ^{-1}(c+d x)}\right)}{4 d^2}+\frac{a \left(\sqrt{\pi } \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)-2 \sqrt{\frac{1}{b}} \cos \left(2 \sin ^{-1}(c+d x)\right) \sqrt{a+b \sin ^{-1}(c+d x)}\right)}{8 \sqrt{\frac{1}{b}} d^2}+\frac{b \left(\sqrt{\pi } \sqrt{\frac{1}{b}} \left(3 b \sin \left(\frac{2 a}{b}\right)-4 a \cos \left(\frac{2 a}{b}\right)\right) C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \left(-\sqrt{\frac{1}{b}}\right) \left(4 a \sin \left(\frac{2 a}{b}\right)+3 b \cos \left(\frac{2 a}{b}\right)\right) S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)+2 \left(3 \sin \left(2 \sin ^{-1}(c+d x)\right)-4 \sin ^{-1}(c+d x) \cos \left(2 \sin ^{-1}(c+d x)\right)\right) \sqrt{a+b \sin ^{-1}(c+d x)}\right)}{32 d^2}-\frac{a b c e^{-\frac{i a}{b}} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{2 d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{3 \sqrt{\pi } b^{3/2} \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{32 d^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} c \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 d^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} c \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 d^2}-\frac{3 \sqrt{\pi } b^{3/2} \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{32 d^2}+\frac{3 b \sin \left(2 \sin ^{-1}(c+d x)\right) \sqrt{a+b \sin ^{-1}(c+d x)}}{16 d^2}-\frac{3 b c \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d^2}-\frac{c (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{d^2}-\frac{\cos \left(2 \sin ^{-1}(c+d x)\right) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{4 d^2}",1,"-1/2*(a*b*c*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(d^2*E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]]) - (b*c*(2*Sqrt[a + b*ArcSin[c + d*x]]*(3*Sqrt[1 - (c + d*x)^2] + 2*(c + d*x)*ArcSin[c + d*x]) - Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) + Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*(2*a*Cos[a/b] - 3*b*Sin[a/b])))/(4*d^2) + (a*(-2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]]*Cos[2*ArcSin[c + d*x]] + Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]] + Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b]))/(8*Sqrt[b^(-1)]*d^2) + (b*(-(Sqrt[b^(-1)]*Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*(3*b*Cos[(2*a)/b] + 4*a*Sin[(2*a)/b])) + Sqrt[b^(-1)]*Sqrt[Pi]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*(-4*a*Cos[(2*a)/b] + 3*b*Sin[(2*a)/b]) + 2*Sqrt[a + b*ArcSin[c + d*x]]*(-4*ArcSin[c + d*x]*Cos[2*ArcSin[c + d*x]] + 3*Sin[2*ArcSin[c + d*x]])))/(32*d^2)","C",0
159,1,313,175,3.3413365,"\int \left(a+b \sin ^{-1}(c+d x)\right)^{3/2} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(3/2),x]","\frac{b \left(-\sqrt{2 \pi } \sqrt{\frac{1}{b}} \left(2 a \sin \left(\frac{a}{b}\right)+3 b \cos \left(\frac{a}{b}\right)\right) C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)+\sqrt{2 \pi } \sqrt{\frac{1}{b}} \left(2 a \cos \left(\frac{a}{b}\right)-3 b \sin \left(\frac{a}{b}\right)\right) S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)+2 \left(3 \sqrt{1-(c+d x)^2}+2 (c+d x) \sin ^{-1}(c+d x)\right) \sqrt{a+b \sin ^{-1}(c+d x)}+\frac{2 a e^{-\frac{i a}{b}} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{\sqrt{a+b \sin ^{-1}(c+d x)}}\right)}{4 d}","-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 d}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 d}+\frac{3 b \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{d}",1,"(b*(2*Sqrt[a + b*ArcSin[c + d*x]]*(3*Sqrt[1 - (c + d*x)^2] + 2*(c + d*x)*ArcSin[c + d*x]) + (2*a*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]]) - Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) + Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*(2*a*Cos[a/b] - 3*b*Sin[a/b])))/(4*d)","C",0
160,1,1083,406,9.7090764,"\int x \left(a+b \sin ^{-1}(c+d x)\right)^{5/2} \, dx","Integrate[x*(a + b*ArcSin[c + d*x])^(5/2),x]","-\frac{b c e^{-\frac{i a}{b}} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right) a^2}{2 d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{\left(-2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)} \cos \left(2 \sin ^{-1}(c+d x)\right)+\sqrt{\pi } \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)+\sqrt{\pi } S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right) \sin \left(\frac{2 a}{b}\right)\right) a^2}{8 \sqrt{\frac{1}{b}} d^2}-\frac{b c \left(2 \sqrt{a+b \sin ^{-1}(c+d x)} \left(2 (c+d x) \sin ^{-1}(c+d x)+3 \sqrt{1-(c+d x)^2}\right)-\sqrt{\frac{1}{b}} \sqrt{2 \pi } C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right) \left(3 b \cos \left(\frac{a}{b}\right)+2 a \sin \left(\frac{a}{b}\right)\right)+\sqrt{\frac{1}{b}} \sqrt{2 \pi } S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right) \left(2 a \cos \left(\frac{a}{b}\right)-3 b \sin \left(\frac{a}{b}\right)\right)\right) a}{2 d^2}+\frac{b \left(-\sqrt{\frac{1}{b}} \sqrt{\pi } S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right) \left(3 b \cos \left(\frac{2 a}{b}\right)+4 a \sin \left(\frac{2 a}{b}\right)\right)+\sqrt{\frac{1}{b}} \sqrt{\pi } C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right) \left(3 b \sin \left(\frac{2 a}{b}\right)-4 a \cos \left(\frac{2 a}{b}\right)\right)+2 \sqrt{a+b \sin ^{-1}(c+d x)} \left(3 \sin \left(2 \sin ^{-1}(c+d x)\right)-4 \sin ^{-1}(c+d x) \cos \left(2 \sin ^{-1}(c+d x)\right)\right)\right) a}{16 d^2}-\frac{c \left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)} \left(b (c+d x) \left(4 \sin ^{-1}(c+d x)^2-15\right)-2 \sqrt{1-(c+d x)^2} \left(a-5 b \sin ^{-1}(c+d x)\right)\right)}{\sqrt{\frac{1}{b}}}+\sqrt{2 \pi } S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right) \left(\left(15 b^2-4 a^2\right) \cos \left(\frac{a}{b}\right)+12 a b \sin \left(\frac{a}{b}\right)\right)+\sqrt{2 \pi } C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right) \left(12 a b \cos \left(\frac{a}{b}\right)+\left(4 a^2-15 b^2\right) \sin \left(\frac{a}{b}\right)\right)\right)}{8 \sqrt{\frac{1}{b}} d^2}+\frac{\sqrt{\pi } C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right) \left(\left(16 a^2-15 b^2\right) \cos \left(\frac{2 a}{b}\right)-24 a b \sin \left(\frac{2 a}{b}\right)\right)-\sqrt{\pi } S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right) \left(\left(15 b^2-16 a^2\right) \sin \left(\frac{2 a}{b}\right)-24 a b \cos \left(\frac{2 a}{b}\right)\right)-\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)} \left(b \left(16 \sin ^{-1}(c+d x)^2-15\right) \cos \left(2 \sin ^{-1}(c+d x)\right)+4 \left(a-5 b \sin ^{-1}(c+d x)\right) \sin \left(2 \sin ^{-1}(c+d x)\right)\right)}{\sqrt{\frac{1}{b}}}}{128 \sqrt{\frac{1}{b}} d^2}","\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} c \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d^2}-\frac{15 \sqrt{\pi } b^{5/2} \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{128 d^2}-\frac{15 \sqrt{\pi } b^{5/2} \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{128 d^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} c \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d^2}+\frac{15 b^2 c (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d^2}+\frac{15 b^2 \cos \left(2 \sin ^{-1}(c+d x)\right) \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d^2}-\frac{5 b c \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{2 d^2}+\frac{5 b \sin \left(2 \sin ^{-1}(c+d x)\right) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{16 d^2}-\frac{c (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{d^2}-\frac{\cos \left(2 \sin ^{-1}(c+d x)\right) \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{4 d^2}",1,"-1/2*(a^2*b*c*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(d^2*E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]]) - (a*b*c*(2*Sqrt[a + b*ArcSin[c + d*x]]*(3*Sqrt[1 - (c + d*x)^2] + 2*(c + d*x)*ArcSin[c + d*x]) - Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) + Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*(2*a*Cos[a/b] - 3*b*Sin[a/b])))/(2*d^2) - (c*((2*Sqrt[a + b*ArcSin[c + d*x]]*(-2*Sqrt[1 - (c + d*x)^2]*(a - 5*b*ArcSin[c + d*x]) + b*(c + d*x)*(-15 + 4*ArcSin[c + d*x]^2)))/Sqrt[b^(-1)] + Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*((-4*a^2 + 15*b^2)*Cos[a/b] + 12*a*b*Sin[a/b]) + Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*(12*a*b*Cos[a/b] + (4*a^2 - 15*b^2)*Sin[a/b])))/(8*Sqrt[b^(-1)]*d^2) + (a^2*(-2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]]*Cos[2*ArcSin[c + d*x]] + Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]] + Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b]))/(8*Sqrt[b^(-1)]*d^2) + (a*b*(-(Sqrt[b^(-1)]*Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*(3*b*Cos[(2*a)/b] + 4*a*Sin[(2*a)/b])) + Sqrt[b^(-1)]*Sqrt[Pi]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*(-4*a*Cos[(2*a)/b] + 3*b*Sin[(2*a)/b]) + 2*Sqrt[a + b*ArcSin[c + d*x]]*(-4*ArcSin[c + d*x]*Cos[2*ArcSin[c + d*x]] + 3*Sin[2*ArcSin[c + d*x]])))/(16*d^2) + (Sqrt[Pi]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*((16*a^2 - 15*b^2)*Cos[(2*a)/b] - 24*a*b*Sin[(2*a)/b]) - Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*(-24*a*b*Cos[(2*a)/b] + (-16*a^2 + 15*b^2)*Sin[(2*a)/b]) - (2*Sqrt[a + b*ArcSin[c + d*x]]*(b*(-15 + 16*ArcSin[c + d*x]^2)*Cos[2*ArcSin[c + d*x]] + 4*(a - 5*b*ArcSin[c + d*x])*Sin[2*ArcSin[c + d*x]]))/Sqrt[b^(-1)])/(128*Sqrt[b^(-1)]*d^2)","C",0
161,1,432,204,3.6233133,"\int \left(a+b \sin ^{-1}(c+d x)\right)^{5/2} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(5/2),x]","\frac{e^{-\frac{i a}{b}} \left(\frac{i \sqrt{\frac{\pi }{2}} \left(4 a^2+15 b^2\right) \left(-1+e^{\frac{2 i a}{b}}\right) \sqrt{a+b \sin ^{-1}(c+d x)} C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)}{\sqrt{\frac{1}{b}}}+\frac{\sqrt{\frac{\pi }{2}} \left(4 a^2+15 b^2\right) \left(1+e^{\frac{2 i a}{b}}\right) \sqrt{a+b \sin ^{-1}(c+d x)} S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)}{\sqrt{\frac{1}{b}}}+2 b \left(2 a^2 \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+2 a^2 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right) \left(2 \sin ^{-1}(c+d x) \left(4 a (c+d x)+5 b \sqrt{-c^2-2 c d x-d^2 x^2+1}\right)+10 a \sqrt{-c^2-2 c d x-d^2 x^2+1}-15 b (c+d x)+4 b (c+d x) \sin ^{-1}(c+d x)^2\right)\right)\right)}{8 d \sqrt{a+b \sin ^{-1}(c+d x)}}","-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d}+\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d}-\frac{15 b^2 (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}+\frac{5 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{2 d}+\frac{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{d}",1,"((I*(4*a^2 + 15*b^2)*(-1 + E^(((2*I)*a)/b))*Sqrt[Pi/2]*Sqrt[a + b*ArcSin[c + d*x]]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]])/Sqrt[b^(-1)] + ((4*a^2 + 15*b^2)*(1 + E^(((2*I)*a)/b))*Sqrt[Pi/2]*Sqrt[a + b*ArcSin[c + d*x]]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]])/Sqrt[b^(-1)] + 2*b*(E^((I*a)/b)*(a + b*ArcSin[c + d*x])*(-15*b*(c + d*x) + 10*a*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2] + 2*(4*a*(c + d*x) + 5*b*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])*ArcSin[c + d*x] + 4*b*(c + d*x)*ArcSin[c + d*x]^2) + 2*a^2*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + 2*a^2*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(8*d*E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
162,1,551,243,5.640857,"\int \left(a+b \sin ^{-1}(c+d x)\right)^{7/2} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(7/2),x]","\frac{e^{-\frac{i a}{b}} \left(\sqrt{2 \pi } \left(8 i a^3 \left(-1+e^{\frac{2 i a}{b}}\right)+105 b^3 \left(1+e^{\frac{2 i a}{b}}\right)\right) \sqrt{a+b \sin ^{-1}(c+d x)} C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)-i \sqrt{2 \pi } \left(8 i a^3 \left(1+e^{\frac{2 i a}{b}}\right)+105 b^3 \left(-1+e^{\frac{2 i a}{b}}\right)\right) \sqrt{a+b \sin ^{-1}(c+d x)} S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)+\frac{4 \left(4 a^3 \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+4 a^3 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right) \left(7 \left(4 a^2 \sqrt{-c^2-2 c d x-d^2 x^2+1}-10 a b (c+d x)-15 b^2 \sqrt{-c^2-2 c d x-d^2 x^2+1}\right)+\sin ^{-1}(c+d x) \left(24 a^2 (c+d x)+56 a b \sqrt{-c^2-2 c d x-d^2 x^2+1}-70 b^2 (c+d x)\right)+4 b \sin ^{-1}(c+d x)^2 \left(6 a (c+d x)+7 b \sqrt{-c^2-2 c d x-d^2 x^2+1}\right)+8 b^2 (c+d x) \sin ^{-1}(c+d x)^3\right)\right)}{\sqrt{\frac{1}{b}}}\right)}{32 \sqrt{\frac{1}{b}} d \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 d}+\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 d}-\frac{105 b^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{35 b^2 (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{4 d}+\frac{7 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{2 d}+\frac{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{7/2}}{d}",1,"(((8*I)*a^3*(-1 + E^(((2*I)*a)/b)) + 105*b^3*(1 + E^(((2*I)*a)/b)))*Sqrt[2*Pi]*Sqrt[a + b*ArcSin[c + d*x]]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]] - I*(105*b^3*(-1 + E^(((2*I)*a)/b)) + (8*I)*a^3*(1 + E^(((2*I)*a)/b)))*Sqrt[2*Pi]*Sqrt[a + b*ArcSin[c + d*x]]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]] + (4*(E^((I*a)/b)*(a + b*ArcSin[c + d*x])*(7*(-10*a*b*(c + d*x) + 4*a^2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2] - 15*b^2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]) + (24*a^2*(c + d*x) - 70*b^2*(c + d*x) + 56*a*b*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])*ArcSin[c + d*x] + 4*b*(6*a*(c + d*x) + 7*b*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])*ArcSin[c + d*x]^2 + 8*b^2*(c + d*x)*ArcSin[c + d*x]^3) + 4*a^3*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + 4*a^3*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/Sqrt[b^(-1)])/(32*Sqrt[b^(-1)]*d*E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
163,1,335,440,1.1746364,"\int \frac{x^2}{\sqrt{a+b \sin ^{-1}(c+d x)}} \, dx","Integrate[x^2/Sqrt[a + b*ArcSin[c + d*x]],x]","\frac{\sqrt{\pi } \sqrt{\frac{1}{b}} \left(3 \sqrt{2} \left(4 c^2+1\right) \cos \left(\frac{a}{b}\right) C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)+12 \sqrt{2} c^2 \sin \left(\frac{a}{b}\right) S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)+12 c \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)-\sqrt{6} \cos \left(\frac{3 a}{b}\right) C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)+3 \sqrt{2} \sin \left(\frac{a}{b}\right) S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)-\sqrt{6} \sin \left(\frac{3 a}{b}\right) S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)-12 c \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)\right)}{12 d^3}","\frac{\sqrt{2 \pi } c^2 \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d^3}+\frac{\sqrt{2 \pi } c^2 \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d^3}+\frac{\sqrt{\pi } c \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{\sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{2}} \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d^3}-\frac{\sqrt{\frac{\pi }{6}} \cos \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{2}} \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d^3}-\frac{\sqrt{\frac{\pi }{6}} \sin \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d^3}-\frac{\sqrt{\pi } c \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{\sqrt{b} d^3}",1,"(Sqrt[b^(-1)]*Sqrt[Pi]*(3*Sqrt[2]*(1 + 4*c^2)*Cos[a/b]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]] - Sqrt[6]*Cos[(3*a)/b]*FresnelC[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]]] - 12*c*Cos[(2*a)/b]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]] + 3*Sqrt[2]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*Sin[a/b] + 12*Sqrt[2]*c^2*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*Sin[a/b] + 12*c*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b] - Sqrt[6]*FresnelS[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*Sin[(3*a)/b]))/(12*d^3)","A",1
164,1,224,211,0.6803972,"\int \frac{x}{\sqrt{a+b \sin ^{-1}(c+d x)}} \, dx","Integrate[x/Sqrt[a + b*ArcSin[c + d*x]],x]","\frac{\sqrt{\pi } \sqrt{\frac{1}{b}} \left(\cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)-\sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)\right)+\frac{i c e^{-\frac{i a}{b}} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{\sqrt{a+b \sin ^{-1}(c+d x)}}}{2 d^2}","-\frac{\sqrt{\pi } \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{2 \sqrt{b} d^2}-\frac{\sqrt{2 \pi } c \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d^2}-\frac{\sqrt{2 \pi } c \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d^2}+\frac{\sqrt{\pi } \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{2 \sqrt{b} d^2}",1,"((I*c*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] - E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]]) + Sqrt[b^(-1)]*Sqrt[Pi]*(Cos[(2*a)/b]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]] - FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b]))/(2*d^2)","C",0
165,1,131,105,0.1040186,"\int \frac{1}{\sqrt{a+b \sin ^{-1}(c+d x)}} \, dx","Integrate[1/Sqrt[a + b*ArcSin[c + d*x]],x]","\frac{i e^{-\frac{i a}{b}} \left(e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{2 d \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{\sqrt{2 \pi } \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d}+\frac{\sqrt{2 \pi } \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d}",1,"((I/2)*(-(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b]) + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(d*E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
166,1,287,287,2.4860666,"\int \frac{x}{\left(a+b \sin ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[x/(a + b*ArcSin[c + d*x])^(3/2),x]","\frac{2 \sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)+\frac{2 \sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left(\frac{2 a}{b}\right) \sqrt{a+b \sin ^{-1}(c+d x)} S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)-c e^{-\frac{i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-c e^{\frac{i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+c e^{-i \sin ^{-1}(c+d x)}+c e^{i \sin ^{-1}(c+d x)}-\sin \left(2 \sin ^{-1}(c+d x)\right)}{b \sqrt{a+b \sin ^{-1}(c+d x)}}}{d^2}","-\frac{2 \sqrt{2 \pi } c \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d^2}+\frac{2 \sqrt{\pi } \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{b^{3/2} d^2}+\frac{2 \sqrt{\pi } \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{b^{3/2} d^2}+\frac{2 \sqrt{2 \pi } c \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d^2}+\frac{2 c \sqrt{1-(c+d x)^2}}{b d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{2 (c+d x) \sqrt{1-(c+d x)^2}}{b d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}",1,"(2*(b^(-1))^(3/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]] + (c/E^(I*ArcSin[c + d*x]) + c*E^(I*ArcSin[c + d*x]) - (c*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b])/E^((I*a)/b) - c*E^((I*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b] + 2*Sqrt[b^(-1)]*Sqrt[Pi]*Sqrt[a + b*ArcSin[c + d*x]]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b] - Sin[2*ArcSin[c + d*x]])/(b*Sqrt[a + b*ArcSin[c + d*x]]))/d^2","C",0
167,1,185,144,0.3184056,"\int \frac{1}{\left(a+b \sin ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(-3/2),x]","\frac{e^{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(e^{i \sin ^{-1}(c+d x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{i a}{b}} \left(e^{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-e^{2 i \sin ^{-1}(c+d x)}-1\right)\right)}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{2 \sqrt{2 \pi } \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}-\frac{2 \sqrt{2 \pi } \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}-\frac{2 \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}",1,"(E^(I*ArcSin[c + d*x])*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + E^((I*a)/b)*(-1 - E^((2*I)*ArcSin[c + d*x]) + E^((I*(a + b*ArcSin[c + d*x]))/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(b*d*E^((I*(a + b*ArcSin[c + d*x]))/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
168,1,392,384,3.1320779,"\int \frac{x}{\left(a+b \sin ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[x/(a + b*ArcSin[c + d*x])^(5/2),x]","\frac{8 \sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left(\frac{2 a}{b}\right) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2} C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)-8 \sqrt{\pi } \sqrt{\frac{1}{b}} \cos \left(\frac{2 a}{b}\right) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2} S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)+2 b c e^{-\frac{i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+c e^{-i \sin ^{-1}(c+d x)} \left(2 b e^{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+2 i a e^{2 i \sin ^{-1}(c+d x)}-2 i a+b e^{2 i \sin ^{-1}(c+d x)}+2 i b \left(-1+e^{2 i \sin ^{-1}(c+d x)}\right) \sin ^{-1}(c+d x)+b\right)-4 a \cos \left(2 \sin ^{-1}(c+d x)\right)-b \sin \left(2 \sin ^{-1}(c+d x)\right)-4 b \sin ^{-1}(c+d x) \cos \left(2 \sin ^{-1}(c+d x)\right)}{3 b^2 d^2 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}","\frac{8 \sqrt{\pi } \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{3 b^{5/2} d^2}+\frac{4 \sqrt{2 \pi } c \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d^2}+\frac{4 \sqrt{2 \pi } c \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d^2}-\frac{8 \sqrt{\pi } \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{3 b^{5/2} d^2}+\frac{8 (c+d x)^2}{3 b^2 d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{4 c (c+d x)}{3 b^2 d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{4}{3 b^2 d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{2 \sqrt{1-(c+d x)^2} (c+d x)}{3 b d^2 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}+\frac{2 c \sqrt{1-(c+d x)^2}}{3 b d^2 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}",1,"(-4*a*Cos[2*ArcSin[c + d*x]] - 4*b*ArcSin[c + d*x]*Cos[2*ArcSin[c + d*x]] - 8*Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcSin[c + d*x])^(3/2)*Cos[(2*a)/b]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]] + (2*b*c*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b])/E^((I*a)/b) + (c*((-2*I)*a + b + (2*I)*a*E^((2*I)*ArcSin[c + d*x]) + b*E^((2*I)*ArcSin[c + d*x]) + (2*I)*b*(-1 + E^((2*I)*ArcSin[c + d*x]))*ArcSin[c + d*x] + 2*b*E^((I*(a + b*ArcSin[c + d*x]))/b)*((I*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b]))/E^(I*ArcSin[c + d*x]) + 8*Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcSin[c + d*x])^(3/2)*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b] - b*Sin[2*ArcSin[c + d*x]])/(3*b^2*d^2*(a + b*ArcSin[c + d*x])^(3/2))","C",0
169,1,238,179,0.600171,"\int \frac{1}{\left(a+b \sin ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(-5/2),x]","\frac{e^{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(-2 b e^{i \sin ^{-1}(c+d x)} \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-i e^{\frac{i a}{b}} \left(-2 i b e^{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+2 a \left(-1+e^{2 i \sin ^{-1}(c+d x)}\right)+b \left(-2 \sin ^{-1}(c+d x)+e^{2 i \sin ^{-1}(c+d x)} \left(2 \sin ^{-1}(c+d x)-i\right)-i\right)\right)\right)}{3 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}","-\frac{4 \sqrt{2 \pi } \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}-\frac{4 \sqrt{2 \pi } \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}+\frac{4 (c+d x)}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{2 \sqrt{1-(c+d x)^2}}{3 b d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}",1,"(-2*b*E^(I*ArcSin[c + d*x])*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] - I*E^((I*a)/b)*(2*a*(-1 + E^((2*I)*ArcSin[c + d*x])) + b*(-I - 2*ArcSin[c + d*x] + E^((2*I)*ArcSin[c + d*x])*(-I + 2*ArcSin[c + d*x])) - (2*I)*b*E^((I*(a + b*ArcSin[c + d*x]))/b)*((I*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(3*b^2*d*E^((I*(a + b*ArcSin[c + d*x]))/b)*(a + b*ArcSin[c + d*x])^(3/2))","C",0
170,1,524,468,2.006838,"\int \frac{x}{\left(a+b \sin ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[x/(a + b*ArcSin[c + d*x])^(7/2),x]","\frac{-2 \left(-16 a^2 \sin \left(2 \sin ^{-1}(c+d x)\right)+32 \sqrt{\pi } \sqrt{\frac{1}{b}} \cos \left(\frac{2 a}{b}\right) \left(a+b \sin ^{-1}(c+d x)\right)^{5/2} C\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)+32 \sqrt{\pi } \sqrt{\frac{1}{b}} \sin \left(\frac{2 a}{b}\right) \left(a+b \sin ^{-1}(c+d x)\right)^{5/2} S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi }}\right)-32 a b \sin ^{-1}(c+d x) \sin \left(2 \sin ^{-1}(c+d x)\right)+4 a b \cos \left(2 \sin ^{-1}(c+d x)\right)+3 b^2 \sin \left(2 \sin ^{-1}(c+d x)\right)-16 b^2 \sin ^{-1}(c+d x)^2 \sin \left(2 \sin ^{-1}(c+d x)\right)+4 b^2 \sin ^{-1}(c+d x) \cos \left(2 \sin ^{-1}(c+d x)\right)\right)-c \left(e^{-i \sin ^{-1}(c+d x)} \left(8 a^2+4 a b \left(4 \sin ^{-1}(c+d x)+i\right)-8 e^{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^2 \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+2 b^2 \left(4 \sin ^{-1}(c+d x)^2+2 i \sin ^{-1}(c+d x)-3\right)\right)+4 e^{-\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right) \left(e^{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(2 a+2 b \sin ^{-1}(c+d x)-i b\right)-2 i b \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)-6 b^2 e^{i \sin ^{-1}(c+d x)}\right)}{30 b^3 d^2 \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}","\frac{8 \sqrt{2 \pi } c \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d^2}-\frac{32 \sqrt{\pi } \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{15 b^{7/2} d^2}-\frac{32 \sqrt{\pi } \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{15 b^{7/2} d^2}-\frac{8 \sqrt{2 \pi } c \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d^2}+\frac{32 \sqrt{1-(c+d x)^2} (c+d x)}{15 b^3 d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{8 c \sqrt{1-(c+d x)^2}}{15 b^3 d^2 \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{8 (c+d x)^2}{15 b^2 d^2 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}-\frac{4 c (c+d x)}{15 b^2 d^2 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}-\frac{4}{15 b^2 d^2 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}-\frac{2 \sqrt{1-(c+d x)^2} (c+d x)}{5 b d^2 \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}+\frac{2 c \sqrt{1-(c+d x)^2}}{5 b d^2 \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}",1,"(-(c*(-6*b^2*E^(I*ArcSin[c + d*x]) + (4*(a + b*ArcSin[c + d*x])*(E^((I*(a + b*ArcSin[c + d*x]))/b)*(2*a - I*b + 2*b*ArcSin[c + d*x]) - (2*I)*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b]))/E^((I*a)/b) + (8*a^2 + 4*a*b*(I + 4*ArcSin[c + d*x]) + 2*b^2*(-3 + (2*I)*ArcSin[c + d*x] + 4*ArcSin[c + d*x]^2) - 8*E^((I*(a + b*ArcSin[c + d*x]))/b)*(a + b*ArcSin[c + d*x])^2*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b])/E^(I*ArcSin[c + d*x]))) - 2*(4*a*b*Cos[2*ArcSin[c + d*x]] + 4*b^2*ArcSin[c + d*x]*Cos[2*ArcSin[c + d*x]] + 32*Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcSin[c + d*x])^(5/2)*Cos[(2*a)/b]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]] + 32*Sqrt[b^(-1)]*Sqrt[Pi]*(a + b*ArcSin[c + d*x])^(5/2)*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[Pi]]*Sin[(2*a)/b] - 16*a^2*Sin[2*ArcSin[c + d*x]] + 3*b^2*Sin[2*ArcSin[c + d*x]] - 32*a*b*ArcSin[c + d*x]*Sin[2*ArcSin[c + d*x]] - 16*b^2*ArcSin[c + d*x]^2*Sin[2*ArcSin[c + d*x]]))/(30*b^3*d^2*(a + b*ArcSin[c + d*x])^(5/2))","C",0
171,1,287,218,0.7386942,"\int \frac{1}{\left(a+b \sin ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(-7/2),x]","\frac{e^{-i \sin ^{-1}(c+d x)} \left(8 a^2+4 a b \left(4 \sin ^{-1}(c+d x)+i\right)-8 e^{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^2 \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+2 b^2 \left(4 \sin ^{-1}(c+d x)^2+2 i \sin ^{-1}(c+d x)-3\right)\right)+4 e^{-\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right) \left(e^{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(2 a+b \left(2 \sin ^{-1}(c+d x)-i\right)\right)-2 i b \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)-6 b^2 e^{i \sin ^{-1}(c+d x)}}{30 b^3 d \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}","-\frac{8 \sqrt{2 \pi } \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}+\frac{8 \sqrt{2 \pi } \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}+\frac{8 \sqrt{1-(c+d x)^2}}{15 b^3 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{4 (c+d x)}{15 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}-\frac{2 \sqrt{1-(c+d x)^2}}{5 b d \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}",1,"(-6*b^2*E^(I*ArcSin[c + d*x]) + (4*(a + b*ArcSin[c + d*x])*(E^((I*(a + b*ArcSin[c + d*x]))/b)*(2*a + b*(-I + 2*ArcSin[c + d*x])) - (2*I)*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b]))/E^((I*a)/b) + (8*a^2 + 4*a*b*(I + 4*ArcSin[c + d*x]) + 2*b^2*(-3 + (2*I)*ArcSin[c + d*x] + 4*ArcSin[c + d*x]^2) - 8*E^((I*(a + b*ArcSin[c + d*x]))/b)*(a + b*ArcSin[c + d*x])^2*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b])/E^(I*ArcSin[c + d*x]))/(30*b^3*d*(a + b*ArcSin[c + d*x])^(5/2))","C",0
172,0,0,19,0.5395896,"\int x^m \left(a+b \sin ^{-1}(c+d x)\right)^n \, dx","Integrate[x^m*(a + b*ArcSin[c + d*x])^n,x]","\int x^m \left(a+b \sin ^{-1}(c+d x)\right)^n \, dx","\text{Int}\left(x^m \left(a+b \sin ^{-1}(c+d x)\right)^n,x\right)",0,"Integrate[x^m*(a + b*ArcSin[c + d*x])^n, x]","A",-1
173,1,419,611,0.5717161,"\int x^2 \left(a+b \sin ^{-1}(c+d x)\right)^n \, dx","Integrate[x^2*(a + b*ArcSin[c + d*x])^n,x]","\frac{2^{-n-3} 3^{-n-1} e^{-\frac{3 i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{b^2}\right)^{-n} \left(i \left(4 c^2+1\right) 2^n 3^{n+1} e^{\frac{4 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^n \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-i \left(4 c^2+1\right) 2^n 3^{n+1} e^{\frac{2 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+2 c 3^{n+1} e^{\frac{5 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^n \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-i 2^n e^{\frac{6 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^n \Gamma \left(n+1,\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+2 c 3^{n+1} e^{\frac{i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+i 2^n \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{d^3}","-\frac{i c^2 e^{-\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{2 d^3}+\frac{i c^2 e^{\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{2 d^3}-\frac{i e^{-\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{8 d^3}+\frac{c 2^{-n-2} e^{-\frac{2 i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{d^3}+\frac{i 3^{-n-1} e^{-\frac{3 i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{8 d^3}+\frac{i e^{\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{8 d^3}+\frac{c 2^{-n-2} e^{\frac{2 i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{d^3}-\frac{i 3^{-n-1} e^{\frac{3 i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{8 d^3}",1,"(2^(-3 - n)*3^(-1 - n)*(a + b*ArcSin[c + d*x])^n*((-I)*2^n*3^(1 + n)*(1 + 4*c^2)*E^(((2*I)*a)/b)*((I*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c + d*x]))/b] + I*2^n*3^(1 + n)*(1 + 4*c^2)*E^(((4*I)*a)/b)*(((-I)*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, (I*(a + b*ArcSin[c + d*x]))/b] + 2*3^(1 + n)*c*E^((I*a)/b)*((I*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] + 2*3^(1 + n)*c*E^(((5*I)*a)/b)*(((-I)*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c + d*x]))/b] + I*2^n*((I*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((-3*I)*(a + b*ArcSin[c + d*x]))/b] - I*2^n*E^(((6*I)*a)/b)*(((-I)*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c + d*x]))/b]))/(d^3*E^(((3*I)*a)/b)*((a + b*ArcSin[c + d*x])^2/b^2)^n)","A",1
174,1,269,301,0.2652449,"\int x \left(a+b \sin ^{-1}(c+d x)\right)^n \, dx","Integrate[x*(a + b*ArcSin[c + d*x])^n,x]","-\frac{i 2^{-n-3} e^{-\frac{2 i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{b^2}\right)^{-n} \left(c 2^{n+2} e^{\frac{3 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^n \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-c 2^{n+2} e^{\frac{i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-i \left(e^{\frac{4 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^n \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+\left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)\right)}{d^2}","\frac{i c e^{-\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{2 d^2}-\frac{2^{-n-3} e^{-\frac{2 i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{d^2}-\frac{i c e^{\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{2 d^2}-\frac{2^{-n-3} e^{\frac{2 i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{d^2}",1,"((-I)*2^(-3 - n)*(a + b*ArcSin[c + d*x])^n*(-(2^(2 + n)*c*E^((I*a)/b)*((I*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((-I)*(a + b*ArcSin[c + d*x]))/b]) + 2^(2 + n)*c*E^(((3*I)*a)/b)*(((-I)*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, (I*(a + b*ArcSin[c + d*x]))/b] - I*(((I*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((4*I)*a)/b)*(((-I)*(a + b*ArcSin[c + d*x]))/b)^n*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c + d*x]))/b])))/(d^2*E^(((2*I)*a)/b)*((a + b*ArcSin[c + d*x])^2/b^2)^n)","A",1
175,1,129,147,0.1229272,"\int \left(a+b \sin ^{-1}(c+d x)\right)^n \, dx","Integrate[(a + b*ArcSin[c + d*x])^n,x]","-\frac{i e^{-\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(\left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-e^{\frac{2 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{2 d}","\frac{i e^{\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{2 d}-\frac{i e^{-\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{2 d}",1,"((-1/2*I)*(a + b*ArcSin[c + d*x])^n*(Gamma[1 + n, ((-I)*(a + b*ArcSin[c + d*x]))/b]/(((-I)*(a + b*ArcSin[c + d*x]))/b)^n - (E^(((2*I)*a)/b)*Gamma[1 + n, (I*(a + b*ArcSin[c + d*x]))/b])/((I*(a + b*ArcSin[c + d*x]))/b)^n))/(d*E^((I*a)/b))","A",1
176,0,0,19,0.2064673,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^n}{x} \, dx","Integrate[(a + b*ArcSin[c + d*x])^n/x,x]","\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^n}{x} \, dx","\text{Int}\left(\frac{\left(a+b \sin ^{-1}(c+d x)\right)^n}{x},x\right)",0,"Integrate[(a + b*ArcSin[c + d*x])^n/x, x]","A",-1
177,1,77,106,0.1277789,"\int (c e+d e x)^4 \left(a+b \sin ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^4*(a + b*ArcSin[c + d*x]),x]","\frac{e^4 \left(\frac{1}{5} (c+d x)^5 \left(a+b \sin ^{-1}(c+d x)\right)-\frac{1}{75} b \sqrt{1-(c+d x)^2} \left(-3 \left((c+d x)^2-1\right)^2+10 \left(1-(c+d x)^2\right)-15\right)\right)}{d}","\frac{e^4 (c+d x)^5 \left(a+b \sin ^{-1}(c+d x)\right)}{5 d}+\frac{b e^4 \left(1-(c+d x)^2\right)^{5/2}}{25 d}-\frac{2 b e^4 \left(1-(c+d x)^2\right)^{3/2}}{15 d}+\frac{b e^4 \sqrt{1-(c+d x)^2}}{5 d}",1,"(e^4*(-1/75*(b*Sqrt[1 - (c + d*x)^2]*(-15 + 10*(1 - (c + d*x)^2) - 3*(-1 + (c + d*x)^2)^2)) + ((c + d*x)^5*(a + b*ArcSin[c + d*x]))/5))/d","A",1
178,1,87,109,0.1002528,"\int (c e+d e x)^3 \left(a+b \sin ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x]),x]","\frac{e^3 \left(8 (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)+2 b \sqrt{1-(c+d x)^2} (c+d x)^3+3 b \sqrt{1-(c+d x)^2} (c+d x)-3 b \sin ^{-1}(c+d x)\right)}{32 d}","\frac{e^3 (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)}{4 d}+\frac{b e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{16 d}+\frac{3 b e^3 \sqrt{1-(c+d x)^2} (c+d x)}{32 d}-\frac{3 b e^3 \sin ^{-1}(c+d x)}{32 d}",1,"(e^3*(3*b*(c + d*x)*Sqrt[1 - (c + d*x)^2] + 2*b*(c + d*x)^3*Sqrt[1 - (c + d*x)^2] - 3*b*ArcSin[c + d*x] + 8*(c + d*x)^4*(a + b*ArcSin[c + d*x])))/(32*d)","A",1
179,1,64,80,0.068704,"\int (c e+d e x)^2 \left(a+b \sin ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcSin[c + d*x]),x]","\frac{e^2 \left(3 (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)+b \left(c^2+2 c d x+d^2 x^2+2\right) \sqrt{1-(c+d x)^2}\right)}{9 d}","\frac{e^2 (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)}{3 d}-\frac{b e^2 \left(1-(c+d x)^2\right)^{3/2}}{9 d}+\frac{b e^2 \sqrt{1-(c+d x)^2}}{3 d}",1,"(e^2*(b*(2 + c^2 + 2*c*d*x + d^2*x^2)*Sqrt[1 - (c + d*x)^2] + 3*(c + d*x)^3*(a + b*ArcSin[c + d*x])))/(9*d)","A",1
180,1,59,70,0.0895491,"\int (c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcSin[c + d*x]),x]","\frac{e \left(2 (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)+b \sqrt{1-(c+d x)^2} (c+d x)-b \sin ^{-1}(c+d x)\right)}{4 d}","\frac{e (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)}{2 d}+\frac{b e \sqrt{1-(c+d x)^2} (c+d x)}{4 d}-\frac{b e \sin ^{-1}(c+d x)}{4 d}",1,"(e*(b*(c + d*x)*Sqrt[1 - (c + d*x)^2] - b*ArcSin[c + d*x] + 2*(c + d*x)^2*(a + b*ArcSin[c + d*x])))/(4*d)","A",1
181,1,51,40,0.0507683,"\int \left(a+b \sin ^{-1}(c+d x)\right) \, dx","Integrate[a + b*ArcSin[c + d*x],x]","a x+\frac{b \left(\sqrt{-c^2-2 c d x-d^2 x^2+1}+c \sin ^{-1}(c+d x)\right)}{d}+b x \sin ^{-1}(c+d x)","a x+\frac{b \sqrt{1-(c+d x)^2}}{d}+\frac{b (c+d x) \sin ^{-1}(c+d x)}{d}",1,"a*x + b*x*ArcSin[c + d*x] + (b*(Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2] + c*ArcSin[c + d*x]))/d","A",1
182,1,71,89,0.0714079,"\int \frac{a+b \sin ^{-1}(c+d x)}{c e+d e x} \, dx","Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x),x]","\frac{a \log (c+d x)-\frac{1}{2} i b \left(\sin ^{-1}(c+d x)^2+\text{Li}_2\left(e^{2 i \sin ^{-1}(c+d x)}\right)\right)+b \sin ^{-1}(c+d x) \log \left(1-e^{2 i \sin ^{-1}(c+d x)}\right)}{d e}","-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)^2}{2 b d e}+\frac{\log \left(1-e^{2 i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e}-\frac{i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c+d x)}\right)}{2 d e}",1,"(b*ArcSin[c + d*x]*Log[1 - E^((2*I)*ArcSin[c + d*x])] + a*Log[c + d*x] - (I/2)*b*(ArcSin[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c + d*x])]))/(d*e)","A",0
183,1,45,51,0.0404149,"\int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^2} \, dx","Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^2,x]","-\frac{\frac{a+b \sin ^{-1}(c+d x)}{c+d x}+b \tanh ^{-1}\left(\sqrt{1-(c+d x)^2}\right)}{d e^2}","-\frac{a+b \sin ^{-1}(c+d x)}{d e^2 (c+d x)}-\frac{b \tanh ^{-1}\left(\sqrt{1-(c+d x)^2}\right)}{d e^2}",1,"-(((a + b*ArcSin[c + d*x])/(c + d*x) + b*ArcTanh[Sqrt[1 - (c + d*x)^2]])/(d*e^2))","A",1
184,1,49,61,0.0986545,"\int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^3} \, dx","Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^3,x]","-\frac{a+b (c+d x) \sqrt{1-(c+d x)^2}+b \sin ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}","-\frac{a+b \sin ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac{b \sqrt{1-(c+d x)^2}}{2 d e^3 (c+d x)}",1,"-1/2*(a + b*(c + d*x)*Sqrt[1 - (c + d*x)^2] + b*ArcSin[c + d*x])/(d*e^3*(c + d*x)^2)","A",1
185,1,86,88,0.1010636,"\int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^4} \, dx","Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^4,x]","-\frac{2 a+b \sqrt{-c^2-2 c d x-d^2 x^2+1} (c+d x)+2 b \sin ^{-1}(c+d x)+b (c+d x)^3 \tanh ^{-1}\left(\sqrt{1-(c+d x)^2}\right)}{6 d e^4 (c+d x)^3}","-\frac{a+b \sin ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}-\frac{b \sqrt{1-(c+d x)^2}}{6 d e^4 (c+d x)^2}-\frac{b \tanh ^{-1}\left(\sqrt{1-(c+d x)^2}\right)}{6 d e^4}",1,"-1/6*(2*a + b*(c + d*x)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2] + 2*b*ArcSin[c + d*x] + b*(c + d*x)^3*ArcTanh[Sqrt[1 - (c + d*x)^2]])/(d*e^4*(c + d*x)^3)","A",1
186,1,63,94,0.0881508,"\int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^5} \, dx","Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^5,x]","-\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)+b (c+d x) \sqrt{1-(c+d x)^2} \left(2 (c+d x)^2+1\right)}{12 d e^5 (c+d x)^4}","-\frac{a+b \sin ^{-1}(c+d x)}{4 d e^5 (c+d x)^4}-\frac{b \sqrt{1-(c+d x)^2}}{6 d e^5 (c+d x)}-\frac{b \sqrt{1-(c+d x)^2}}{12 d e^5 (c+d x)^3}",1,"-1/12*(b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(1 + 2*(c + d*x)^2) + 3*(a + b*ArcSin[c + d*x]))/(d*e^5*(c + d*x)^4)","A",1
187,1,65,121,0.1001105,"\int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^6} \, dx","Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^6,x]","-\frac{\frac{a+b \sin ^{-1}(c+d x)}{(c+d x)^5}+b \sqrt{1-(c+d x)^2} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-(c+d x)^2\right)}{5 d e^6}","-\frac{a+b \sin ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}-\frac{3 b \sqrt{1-(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac{b \sqrt{1-(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac{3 b \tanh ^{-1}\left(\sqrt{1-(c+d x)^2}\right)}{40 d e^6}",1,"-1/5*((a + b*ArcSin[c + d*x])/(c + d*x)^5 + b*Sqrt[1 - (c + d*x)^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 - (c + d*x)^2])/(d*e^6)","C",1
188,1,164,203,0.5019657,"\int (c e+d e x)^4 \left(a+b \sin ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^4*(a + b*ArcSin[c + d*x])^2,x]","\frac{e^4 \left((c+d x)^5 \left(a+b \sin ^{-1}(c+d x)\right)^2-\frac{2}{25} b \left(-5 \sqrt{1-(c+d x)^2} (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)-\frac{20}{3} \sqrt{1-(c+d x)^2} (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)+\frac{40}{3} \left(b d x-\sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)\right)+b (c+d x)^5+\frac{20}{9} b (c+d x)^3\right)\right)}{5 d}","\frac{e^4 (c+d x)^5 \left(a+b \sin ^{-1}(c+d x)\right)^2}{5 d}+\frac{2 b e^4 \sqrt{1-(c+d x)^2} (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)}{25 d}+\frac{8 b e^4 \sqrt{1-(c+d x)^2} (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)}{75 d}+\frac{16 b e^4 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)}{75 d}-\frac{2 b^2 e^4 (c+d x)^5}{125 d}-\frac{8 b^2 e^4 (c+d x)^3}{225 d}-\frac{16}{75} b^2 e^4 x",1,"(e^4*((c + d*x)^5*(a + b*ArcSin[c + d*x])^2 - (2*b*((20*b*(c + d*x)^3)/9 + b*(c + d*x)^5 - (20*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/3 - 5*(c + d*x)^4*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) + (40*(b*d*x - Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])))/3))/25))/(5*d)","A",1
189,1,142,176,0.2477117,"\int (c e+d e x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^2,x]","\frac{e^3 \left(\frac{1}{8} \left(-3 \left(-2 b \sqrt{1-(c+d x)^2} (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)+\left(a+b \sin ^{-1}(c+d x)\right)^2+b^2 (c+d x)^2\right)+4 b \sqrt{1-(c+d x)^2} (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)-b^2 (c+d x)^4\right)+(c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)^2\right)}{4 d}","\frac{e^3 (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)^2}{4 d}+\frac{b e^3 \sqrt{1-(c+d x)^2} (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)}{8 d}+\frac{3 b e^3 \sqrt{1-(c+d x)^2} (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)}{16 d}-\frac{3 e^3 \left(a+b \sin ^{-1}(c+d x)\right)^2}{32 d}-\frac{b^2 e^3 (c+d x)^4}{32 d}-\frac{3 b^2 e^3 (c+d x)^2}{32 d}",1,"(e^3*((c + d*x)^4*(a + b*ArcSin[c + d*x])^2 + (-(b^2*(c + d*x)^4) + 4*b*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) - 3*(b^2*(c + d*x)^2 - 2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) + (a + b*ArcSin[c + d*x])^2))/8))/(4*d)","A",1
190,1,112,140,0.2499637,"\int (c e+d e x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcSin[c + d*x])^2,x]","\frac{e^2 \left((c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^2-\frac{2}{9} b \left(-3 \sqrt{1-(c+d x)^2} (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)-6 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)+b (c+d x)^3+6 b d x\right)\right)}{3 d}","\frac{e^2 (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^2}{3 d}+\frac{2 b e^2 \sqrt{1-(c+d x)^2} (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)}{9 d}+\frac{4 b e^2 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)}{9 d}-\frac{2 b^2 e^2 (c+d x)^3}{27 d}-\frac{4}{9} b^2 e^2 x",1,"(e^2*((c + d*x)^3*(a + b*ArcSin[c + d*x])^2 - (2*b*(6*b*d*x + b*(c + d*x)^3 - 6*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) - 3*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])))/9))/(3*d)","A",1
191,1,86,105,0.0801109,"\int (c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcSin[c + d*x])^2,x]","-\frac{e \left(-2 (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^2-2 b \sqrt{1-(c+d x)^2} (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)+\left(a+b \sin ^{-1}(c+d x)\right)^2+b^2 (c+d x)^2\right)}{4 d}","\frac{e (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^2}{2 d}+\frac{b e \sqrt{1-(c+d x)^2} (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)}{2 d}-\frac{e \left(a+b \sin ^{-1}(c+d x)\right)^2}{4 d}-\frac{b^2 e (c+d x)^2}{4 d}",1,"-1/4*(e*(b^2*(c + d*x)^2 - 2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) + (a + b*ArcSin[c + d*x])^2 - 2*(c + d*x)^2*(a + b*ArcSin[c + d*x])^2))/d","A",1
192,1,61,59,0.1240324,"\int \left(a+b \sin ^{-1}(c+d x)\right)^2 \, dx","Integrate[(a + b*ArcSin[c + d*x])^2,x]","\frac{2 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)+(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^2-2 b^2 (c+d x)}{d}","\frac{2 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)}{d}+\frac{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^2}{d}-2 b^2 x",1,"(-2*b^2*(c + d*x) + 2*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) + (c + d*x)*(a + b*ArcSin[c + d*x])^2)/d","A",1
193,1,170,126,0.1898243,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{c e+d e x} \, dx","Integrate[(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x),x]","\frac{a^2 \log (c+d x)-i a b \left(\sin ^{-1}(c+d x)^2+\text{Li}_2\left(e^{2 i \sin ^{-1}(c+d x)}\right)\right)+2 a b \sin ^{-1}(c+d x) \log \left(1-e^{2 i \sin ^{-1}(c+d x)}\right)+b^2 \left(i \sin ^{-1}(c+d x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c+d x)}\right)+\frac{1}{2} \text{Li}_3\left(e^{-2 i \sin ^{-1}(c+d x)}\right)+\frac{1}{3} i \sin ^{-1}(c+d x)^3+\sin ^{-1}(c+d x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c+d x)}\right)-\frac{i \pi ^3}{24}\right)}{d e}","-\frac{i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e}-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)^3}{3 b d e}+\frac{\log \left(1-e^{2 i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^2}{d e}+\frac{b^2 \text{Li}_3\left(e^{2 i \sin ^{-1}(c+d x)}\right)}{2 d e}",1,"(2*a*b*ArcSin[c + d*x]*Log[1 - E^((2*I)*ArcSin[c + d*x])] + a^2*Log[c + d*x] - I*a*b*(ArcSin[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c + d*x])]) + b^2*((-1/24*I)*Pi^3 + (I/3)*ArcSin[c + d*x]^3 + ArcSin[c + d*x]^2*Log[1 - E^((-2*I)*ArcSin[c + d*x])] + I*ArcSin[c + d*x]*PolyLog[2, E^((-2*I)*ArcSin[c + d*x])] + PolyLog[3, E^((-2*I)*ArcSin[c + d*x])]/2))/(d*e)","A",0
194,1,176,116,0.6354047,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{(c e+d e x)^2} \, dx","Integrate[(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^2,x]","\frac{-\frac{a^2}{c+d x}-2 a b \left(\frac{\sin ^{-1}(c+d x)}{c+d x}-\log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)\right)+\log \left(\frac{1}{2} (c+d x) \csc \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)\right)\right)+b^2 \left(2 i \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right)-2 i \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right)+\sin ^{-1}(c+d x) \left(-\frac{\sin ^{-1}(c+d x)}{c+d x}+2 \log \left(1-e^{i \sin ^{-1}(c+d x)}\right)-2 \log \left(1+e^{i \sin ^{-1}(c+d x)}\right)\right)\right)}{d e^2}","-\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{d e^2 (c+d x)}-\frac{4 b \tanh ^{-1}\left(e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e^2}+\frac{2 i b^2 \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right)}{d e^2}-\frac{2 i b^2 \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right)}{d e^2}",1,"(-(a^2/(c + d*x)) - 2*a*b*(ArcSin[c + d*x]/(c + d*x) + Log[((c + d*x)*Csc[ArcSin[c + d*x]/2])/2] - Log[Sin[ArcSin[c + d*x]/2]]) + b^2*(ArcSin[c + d*x]*(-(ArcSin[c + d*x]/(c + d*x)) + 2*Log[1 - E^(I*ArcSin[c + d*x])] - 2*Log[1 + E^(I*ArcSin[c + d*x])]) + (2*I)*PolyLog[2, -E^(I*ArcSin[c + d*x])] - (2*I)*PolyLog[2, E^(I*ArcSin[c + d*x])]))/(d*e^2)","A",0
195,1,126,87,0.2559276,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{(c e+d e x)^3} \, dx","Integrate[(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^3,x]","-\frac{a \left(a+2 b (c+d x) \sqrt{-c^2-2 c d x-d^2 x^2+1}\right)+2 b \sin ^{-1}(c+d x) \left(a+b (c+d x) \sqrt{-c^2-2 c d x-d^2 x^2+1}\right)-2 b^2 (c+d x)^2 \log (c+d x)+b^2 \sin ^{-1}(c+d x)^2}{2 d e^3 (c+d x)^2}","-\frac{b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)}{d e^3 (c+d x)}-\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{2 d e^3 (c+d x)^2}+\frac{b^2 \log (c+d x)}{d e^3}",1,"-1/2*(a*(a + 2*b*(c + d*x)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]) + 2*b*(a + b*(c + d*x)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])*ArcSin[c + d*x] + b^2*ArcSin[c + d*x]^2 - 2*b^2*(c + d*x)^2*Log[c + d*x])/(d*e^3*(c + d*x)^2)","A",1
196,1,246,187,2.2980663,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{(c e+d e x)^4} \, dx","Integrate[(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^4,x]","-\frac{4 a^2+8 a b \sin ^{-1}(c+d x)+2 a b \sin \left(2 \sin ^{-1}(c+d x)\right)+a b \left(3 (c+d x)-\sin \left(3 \sin ^{-1}(c+d x)\right)\right) \left(\log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)\right)\right)-4 i b^2 (c+d x)^3 \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right)+b^2 \left(4 i (c+d x)^3 \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right)+4 (c+d x)^2+4 \sin ^{-1}(c+d x)^2+\sin ^{-1}(c+d x) \left(2 \sin \left(2 \sin ^{-1}(c+d x)\right)+\left(\sin \left(3 \sin ^{-1}(c+d x)\right)-3 (c+d x)\right) \left(\log \left(1-e^{i \sin ^{-1}(c+d x)}\right)-\log \left(1+e^{i \sin ^{-1}(c+d x)}\right)\right)\right)\right)}{12 d e^4 (c+d x)^3}","-\frac{b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)}{3 d e^4 (c+d x)^2}-\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{3 d e^4 (c+d x)^3}-\frac{2 b \tanh ^{-1}\left(e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{3 d e^4}+\frac{i b^2 \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right)}{3 d e^4}-\frac{i b^2 \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right)}{3 d e^4}-\frac{b^2}{3 d e^4 (c+d x)}",1,"-1/12*(4*a^2 + 8*a*b*ArcSin[c + d*x] - (4*I)*b^2*(c + d*x)^3*PolyLog[2, -E^(I*ArcSin[c + d*x])] + 2*a*b*Sin[2*ArcSin[c + d*x]] + a*b*(Log[Cos[ArcSin[c + d*x]/2]] - Log[Sin[ArcSin[c + d*x]/2]])*(3*(c + d*x) - Sin[3*ArcSin[c + d*x]]) + b^2*(4*(c + d*x)^2 + 4*ArcSin[c + d*x]^2 + (4*I)*(c + d*x)^3*PolyLog[2, E^(I*ArcSin[c + d*x])] + ArcSin[c + d*x]*(2*Sin[2*ArcSin[c + d*x]] + (Log[1 - E^(I*ArcSin[c + d*x])] - Log[1 + E^(I*ArcSin[c + d*x])])*(-3*(c + d*x) + Sin[3*ArcSin[c + d*x]]))))/(d*e^4*(c + d*x)^3)","A",0
197,1,307,338,1.1039259,"\int (c e+d e x)^4 \left(a+b \sin ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)^4*(a + b*ArcSin[c + d*x])^3,x]","\frac{e^4 \left((c+d x)^5 \left(a+b \sin ^{-1}(c+d x)\right)^3-\frac{1}{25} b \left(6 b (c+d x)^5 \left(a+b \sin ^{-1}(c+d x)\right)-15 \sqrt{1-(c+d x)^2} (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)^2+\frac{40}{3} b (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)-20 \sqrt{1-(c+d x)^2} (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^2-40 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^2+80 b \left(a d x+b \sqrt{1-(c+d x)^2}+b (c+d x) \sin ^{-1}(c+d x)\right)+\frac{40}{9} b^2 \left(c^2+2 c d x+d^2 x^2+2\right) \sqrt{1-(c+d x)^2}-\frac{2}{5} b^2 \sqrt{1-(c+d x)^2} \left(-3 \left((c+d x)^2-1\right)^2+10 \left(1-(c+d x)^2\right)-15\right)\right)\right)}{5 d}","-\frac{6 b^2 e^4 (c+d x)^5 \left(a+b \sin ^{-1}(c+d x)\right)}{125 d}-\frac{8 b^2 e^4 (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)}{75 d}-\frac{16}{25} a b^2 e^4 x+\frac{e^4 (c+d x)^5 \left(a+b \sin ^{-1}(c+d x)\right)^3}{5 d}+\frac{3 b e^4 \sqrt{1-(c+d x)^2} (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)^2}{25 d}+\frac{4 b e^4 \sqrt{1-(c+d x)^2} (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^2}{25 d}+\frac{8 b e^4 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^2}{25 d}-\frac{6 b^3 e^4 \left(1-(c+d x)^2\right)^{5/2}}{625 d}+\frac{76 b^3 e^4 \left(1-(c+d x)^2\right)^{3/2}}{1125 d}-\frac{298 b^3 e^4 \sqrt{1-(c+d x)^2}}{375 d}-\frac{16 b^3 e^4 (c+d x) \sin ^{-1}(c+d x)}{25 d}",1,"(e^4*((c + d*x)^5*(a + b*ArcSin[c + d*x])^3 - (b*((40*b^2*(2 + c^2 + 2*c*d*x + d^2*x^2)*Sqrt[1 - (c + d*x)^2])/9 - (2*b^2*Sqrt[1 - (c + d*x)^2]*(-15 + 10*(1 - (c + d*x)^2) - 3*(-1 + (c + d*x)^2)^2))/5 + (40*b*(c + d*x)^3*(a + b*ArcSin[c + d*x]))/3 + 6*b*(c + d*x)^5*(a + b*ArcSin[c + d*x]) - 40*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 - 20*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 - 15*(c + d*x)^4*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 + 80*b*(a*d*x + b*Sqrt[1 - (c + d*x)^2] + b*(c + d*x)*ArcSin[c + d*x])))/25))/(5*d)","A",1
198,1,232,287,0.597141,"\int (c e+d e x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^3,x]","\frac{e^3 \left((c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)^3-\frac{3}{8} \left(b^2 (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)+3 b^2 (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)-2 b \sqrt{1-(c+d x)^2} (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^2-3 b \sqrt{1-(c+d x)^2} (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^2+\left(a+b \sin ^{-1}(c+d x)\right)^3+\frac{1}{4} b^3 \sqrt{1-(c+d x)^2} (c+d x)^3+\frac{15}{8} b^3 \sqrt{1-(c+d x)^2} (c+d x)-\frac{15}{8} b^3 \sin ^{-1}(c+d x)\right)\right)}{4 d}","-\frac{3 b^2 e^3 (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)}{32 d}-\frac{9 b^2 e^3 (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)}{32 d}+\frac{e^3 (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)^3}{4 d}+\frac{3 b e^3 \sqrt{1-(c+d x)^2} (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^2}{16 d}+\frac{9 b e^3 \sqrt{1-(c+d x)^2} (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^2}{32 d}-\frac{3 e^3 \left(a+b \sin ^{-1}(c+d x)\right)^3}{32 d}-\frac{3 b^3 e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{128 d}-\frac{45 b^3 e^3 \sqrt{1-(c+d x)^2} (c+d x)}{256 d}+\frac{45 b^3 e^3 \sin ^{-1}(c+d x)}{256 d}",1,"(e^3*((c + d*x)^4*(a + b*ArcSin[c + d*x])^3 - (3*((15*b^3*(c + d*x)*Sqrt[1 - (c + d*x)^2])/8 + (b^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/4 - (15*b^3*ArcSin[c + d*x])/8 + 3*b^2*(c + d*x)^2*(a + b*ArcSin[c + d*x]) + b^2*(c + d*x)^4*(a + b*ArcSin[c + d*x]) - 3*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 - 2*b*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 + (a + b*ArcSin[c + d*x])^3))/8))/(4*d)","A",1
199,1,199,235,0.3882123,"\int (c e+d e x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcSin[c + d*x])^3,x]","\frac{e^2 \left((c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^3-b \left(\frac{2}{3} b (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)-\sqrt{1-(c+d x)^2} (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^2-2 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^2+4 b \left(a d x+b \sqrt{1-(c+d x)^2}+b (c+d x) \sin ^{-1}(c+d x)\right)+\frac{2}{9} b^2 \left(c^2+2 c d x+d^2 x^2+2\right) \sqrt{1-(c+d x)^2}\right)\right)}{3 d}","-\frac{2 b^2 e^2 (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)}{9 d}-\frac{4}{3} a b^2 e^2 x+\frac{2 b e^2 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^2}{3 d}+\frac{b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^2}{3 d}+\frac{e^2 (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^3}{3 d}+\frac{2 b^3 e^2 \left(1-(c+d x)^2\right)^{3/2}}{27 d}-\frac{14 b^3 e^2 \sqrt{1-(c+d x)^2}}{9 d}-\frac{4 b^3 e^2 (c+d x) \sin ^{-1}(c+d x)}{3 d}",1,"(e^2*((c + d*x)^3*(a + b*ArcSin[c + d*x])^3 - b*((2*b^2*(2 + c^2 + 2*c*d*x + d^2*x^2)*Sqrt[1 - (c + d*x)^2])/9 + (2*b*(c + d*x)^3*(a + b*ArcSin[c + d*x]))/3 - 2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 - (c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 + 4*b*(a*d*x + b*Sqrt[1 - (c + d*x)^2] + b*(c + d*x)*ArcSin[c + d*x]))))/(3*d)","A",1
200,1,137,165,0.2427464,"\int (c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcSin[c + d*x])^3,x]","\frac{e \left(-3 b^2 (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)+3 b (c+d x) \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^2+2 (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^3-\left(a+b \sin ^{-1}(c+d x)\right)^3+\frac{3}{2} b^3 \left(\sin ^{-1}(c+d x)-(c+d x) \sqrt{1-(c+d x)^2}\right)\right)}{4 d}","-\frac{3 b^2 e (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)}{4 d}+\frac{3 b e (c+d x) \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^2}{4 d}+\frac{e (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^3}{2 d}-\frac{e \left(a+b \sin ^{-1}(c+d x)\right)^3}{4 d}-\frac{3 b^3 e (c+d x) \sqrt{1-(c+d x)^2}}{8 d}+\frac{3 b^3 e \sin ^{-1}(c+d x)}{8 d}",1,"(e*((3*b^3*(-((c + d*x)*Sqrt[1 - (c + d*x)^2]) + ArcSin[c + d*x]))/2 - 3*b^2*(c + d*x)^2*(a + b*ArcSin[c + d*x]) + 3*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 - (a + b*ArcSin[c + d*x])^3 + 2*(c + d*x)^2*(a + b*ArcSin[c + d*x])^3))/(4*d)","A",1
201,1,96,104,0.0861637,"\int \left(a+b \sin ^{-1}(c+d x)\right)^3 \, dx","Integrate[(a + b*ArcSin[c + d*x])^3,x]","\frac{-6 b^2 \left(a (c+d x)+b \sqrt{1-(c+d x)^2}+b (c+d x) \sin ^{-1}(c+d x)\right)+(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^3+3 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^2}{d}","-6 a b^2 x+\frac{3 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^2}{d}+\frac{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^3}{d}-\frac{6 b^3 \sqrt{1-(c+d x)^2}}{d}-\frac{6 b^3 (c+d x) \sin ^{-1}(c+d x)}{d}",1,"(3*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 + (c + d*x)*(a + b*ArcSin[c + d*x])^3 - 6*b^2*(a*(c + d*x) + b*Sqrt[1 - (c + d*x)^2] + b*(c + d*x)*ArcSin[c + d*x]))/d","A",1
202,1,304,169,0.2428039,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{c e+d e x} \, dx","Integrate[(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x),x]","-\frac{i \left(64 i a^3 \log (c+d x)+96 a^2 b \text{Li}_2\left(e^{2 i \sin ^{-1}(c+d x)}\right)+96 a^2 b \sin ^{-1}(c+d x)^2+192 i a^2 b \sin ^{-1}(c+d x) \log \left(1-e^{2 i \sin ^{-1}(c+d x)}\right)-96 b^2 \sin ^{-1}(c+d x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c+d x)}\right) \left(2 a+b \sin ^{-1}(c+d x)\right)+96 i a b^2 \text{Li}_3\left(e^{-2 i \sin ^{-1}(c+d x)}\right)-64 a b^2 \sin ^{-1}(c+d x)^3+192 i a b^2 \sin ^{-1}(c+d x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c+d x)}\right)+8 \pi ^3 a b^2+96 i b^3 \sin ^{-1}(c+d x) \text{Li}_3\left(e^{-2 i \sin ^{-1}(c+d x)}\right)+48 b^3 \text{Li}_4\left(e^{-2 i \sin ^{-1}(c+d x)}\right)-16 b^3 \sin ^{-1}(c+d x)^4+64 i b^3 \sin ^{-1}(c+d x)^3 \log \left(1-e^{-2 i \sin ^{-1}(c+d x)}\right)+\pi ^4 b^3\right)}{64 d e}","\frac{3 b^2 \text{Li}_3\left(e^{2 i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{2 d e}-\frac{3 i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^2}{2 d e}-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)^4}{4 b d e}+\frac{\log \left(1-e^{2 i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^3}{d e}+\frac{3 i b^3 \text{Li}_4\left(e^{2 i \sin ^{-1}(c+d x)}\right)}{4 d e}",1,"((-1/64*I)*(8*a*b^2*Pi^3 + b^3*Pi^4 + 96*a^2*b*ArcSin[c + d*x]^2 - 64*a*b^2*ArcSin[c + d*x]^3 - 16*b^3*ArcSin[c + d*x]^4 + (192*I)*a*b^2*ArcSin[c + d*x]^2*Log[1 - E^((-2*I)*ArcSin[c + d*x])] + (64*I)*b^3*ArcSin[c + d*x]^3*Log[1 - E^((-2*I)*ArcSin[c + d*x])] + (192*I)*a^2*b*ArcSin[c + d*x]*Log[1 - E^((2*I)*ArcSin[c + d*x])] + (64*I)*a^3*Log[c + d*x] - 96*b^2*ArcSin[c + d*x]*(2*a + b*ArcSin[c + d*x])*PolyLog[2, E^((-2*I)*ArcSin[c + d*x])] + 96*a^2*b*PolyLog[2, E^((2*I)*ArcSin[c + d*x])] + (96*I)*a*b^2*PolyLog[3, E^((-2*I)*ArcSin[c + d*x])] + (96*I)*b^3*ArcSin[c + d*x]*PolyLog[3, E^((-2*I)*ArcSin[c + d*x])] + 48*b^3*PolyLog[4, E^((-2*I)*ArcSin[c + d*x])]))/(d*e)","A",0
203,1,342,190,1.0338985,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{(c e+d e x)^2} \, dx","Integrate[(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^2,x]","-\frac{\frac{a^3}{c+d x}+3 a^2 b \log \left(\sqrt{-c^2-2 c d x-d^2 x^2+1}+1\right)-3 a^2 b \log (c+d x)+\frac{3 a^2 b \sin ^{-1}(c+d x)}{c+d x}-6 i b^2 \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)+6 i b^2 \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)+\frac{3 a b^2 \sin ^{-1}(c+d x)^2}{c+d x}-6 a b^2 \sin ^{-1}(c+d x) \log \left(1-e^{i \sin ^{-1}(c+d x)}\right)+6 a b^2 \sin ^{-1}(c+d x) \log \left(1+e^{i \sin ^{-1}(c+d x)}\right)+6 b^3 \text{Li}_3\left(-e^{i \sin ^{-1}(c+d x)}\right)-6 b^3 \text{Li}_3\left(e^{i \sin ^{-1}(c+d x)}\right)+\frac{b^3 \sin ^{-1}(c+d x)^3}{c+d x}-3 b^3 \sin ^{-1}(c+d x)^2 \log \left(1-e^{i \sin ^{-1}(c+d x)}\right)+3 b^3 \sin ^{-1}(c+d x)^2 \log \left(1+e^{i \sin ^{-1}(c+d x)}\right)}{d e^2}","\frac{6 i b^2 \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e^2}-\frac{6 i b^2 \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e^2}-\frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{d e^2 (c+d x)}-\frac{6 b \tanh ^{-1}\left(e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^2}{d e^2}-\frac{6 b^3 \text{Li}_3\left(-e^{i \sin ^{-1}(c+d x)}\right)}{d e^2}+\frac{6 b^3 \text{Li}_3\left(e^{i \sin ^{-1}(c+d x)}\right)}{d e^2}",1,"-((a^3/(c + d*x) + (3*a^2*b*ArcSin[c + d*x])/(c + d*x) + (3*a*b^2*ArcSin[c + d*x]^2)/(c + d*x) + (b^3*ArcSin[c + d*x]^3)/(c + d*x) - 6*a*b^2*ArcSin[c + d*x]*Log[1 - E^(I*ArcSin[c + d*x])] - 3*b^3*ArcSin[c + d*x]^2*Log[1 - E^(I*ArcSin[c + d*x])] + 6*a*b^2*ArcSin[c + d*x]*Log[1 + E^(I*ArcSin[c + d*x])] + 3*b^3*ArcSin[c + d*x]^2*Log[1 + E^(I*ArcSin[c + d*x])] - 3*a^2*b*Log[c + d*x] + 3*a^2*b*Log[1 + Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]] - (6*I)*b^2*(a + b*ArcSin[c + d*x])*PolyLog[2, -E^(I*ArcSin[c + d*x])] + (6*I)*b^2*(a + b*ArcSin[c + d*x])*PolyLog[2, E^(I*ArcSin[c + d*x])] + 6*b^3*PolyLog[3, -E^(I*ArcSin[c + d*x])] - 6*b^3*PolyLog[3, E^(I*ArcSin[c + d*x])])/(d*e^2))","A",0
204,1,248,167,0.8854521,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{(c e+d e x)^3} \, dx","Integrate[(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^3,x]","-\frac{a \left(a \left(a+3 b (c+d x) \sqrt{-c^2-2 c d x-d^2 x^2+1}\right)-6 b^2 (c+d x)^2 \log (c+d x)\right)+3 b^2 \sin ^{-1}(c+d x)^2 \left(a+b (c+d x) \left(\sqrt{-c^2-2 c d x-d^2 x^2+1}+i c+i d x\right)\right)+3 b \sin ^{-1}(c+d x) \left(a \left(a+2 b (c+d x) \sqrt{-c^2-2 c d x-d^2 x^2+1}\right)-2 b^2 (c+d x)^2 \log \left(1-e^{2 i \sin ^{-1}(c+d x)}\right)\right)+3 i b^3 (c+d x)^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c+d x)}\right)+b^3 \sin ^{-1}(c+d x)^3}{2 d e^3 (c+d x)^2}","\frac{3 b^2 \log \left(1-e^{2 i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e^3}-\frac{3 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^2}{2 d e^3 (c+d x)}-\frac{3 i b \left(a+b \sin ^{-1}(c+d x)\right)^2}{2 d e^3}-\frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{2 d e^3 (c+d x)^2}-\frac{3 i b^3 \text{Li}_2\left(e^{2 i \sin ^{-1}(c+d x)}\right)}{2 d e^3}",1,"-1/2*(3*b^2*(a + b*(c + d*x)*(I*c + I*d*x + Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]))*ArcSin[c + d*x]^2 + b^3*ArcSin[c + d*x]^3 + 3*b*ArcSin[c + d*x]*(a*(a + 2*b*(c + d*x)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]) - 2*b^2*(c + d*x)^2*Log[1 - E^((2*I)*ArcSin[c + d*x])]) + a*(a*(a + 3*b*(c + d*x)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]) - 6*b^2*(c + d*x)^2*Log[c + d*x]) + (3*I)*b^3*(c + d*x)^2*PolyLog[2, E^((2*I)*ArcSin[c + d*x])])/(d*e^3*(c + d*x)^2)","A",0
205,1,732,291,8.0170027,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{(c e+d e x)^4} \, dx","Integrate[(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^4,x]","-\frac{a^3}{3 d e^4 (c+d x)^3}-\frac{a^2 b \sqrt{-c^2-2 c d x-d^2 x^2+1}}{2 d e^4 (c+d x)^2}-\frac{a^2 b \log \left(\sqrt{-c^2-2 c d x-d^2 x^2+1}+1\right)}{2 d e^4}+\frac{a^2 b \log (c+d x)}{2 d e^4}-\frac{a^2 b \sin ^{-1}(c+d x)}{d e^4 (c+d x)^3}+\frac{a b^2 \left(8 i \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right)-\frac{2 \left(4 i (c+d x)^3 \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right)+4 \sin ^{-1}(c+d x)^2+2 \sin ^{-1}(c+d x) \sin \left(2 \sin ^{-1}(c+d x)\right)-3 (c+d x) \sin ^{-1}(c+d x) \log \left(1-e^{i \sin ^{-1}(c+d x)}\right)+3 (c+d x) \sin ^{-1}(c+d x) \log \left(1+e^{i \sin ^{-1}(c+d x)}\right)+\sin ^{-1}(c+d x) \sin \left(3 \sin ^{-1}(c+d x)\right) \log \left(1-e^{i \sin ^{-1}(c+d x)}\right)-\sin ^{-1}(c+d x) \sin \left(3 \sin ^{-1}(c+d x)\right) \log \left(1+e^{i \sin ^{-1}(c+d x)}\right)-2 \cos \left(2 \sin ^{-1}(c+d x)\right)+2\right)}{(c+d x)^3}\right)}{8 d e^4}+\frac{b^3 \left(48 i \sin ^{-1}(c+d x) \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right)-48 i \sin ^{-1}(c+d x) \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right)-48 \text{Li}_3\left(-e^{i \sin ^{-1}(c+d x)}\right)+48 \text{Li}_3\left(e^{i \sin ^{-1}(c+d x)}\right)-\frac{16 \sin ^{-1}(c+d x)^3 \sin ^4\left(\frac{1}{2} \sin ^{-1}(c+d x)\right)}{(c+d x)^3}+24 \sin ^{-1}(c+d x)^2 \log \left(1-e^{i \sin ^{-1}(c+d x)}\right)-24 \sin ^{-1}(c+d x)^2 \log \left(1+e^{i \sin ^{-1}(c+d x)}\right)-4 \sin ^{-1}(c+d x)^3 \tan \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)-24 \sin ^{-1}(c+d x) \tan \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)-4 \sin ^{-1}(c+d x)^3 \cot \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)-24 \sin ^{-1}(c+d x) \cot \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)-\left((c+d x) \sin ^{-1}(c+d x)^3 \csc ^4\left(\frac{1}{2} \sin ^{-1}(c+d x)\right)\right)-6 \sin ^{-1}(c+d x)^2 \csc ^2\left(\frac{1}{2} \sin ^{-1}(c+d x)\right)+6 \sin ^{-1}(c+d x)^2 \sec ^2\left(\frac{1}{2} \sin ^{-1}(c+d x)\right)+48 \log \left(\tan \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)\right)\right)}{48 d e^4}","\frac{i b^2 \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e^4}-\frac{i b^2 \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e^4}-\frac{b^2 \left(a+b \sin ^{-1}(c+d x)\right)}{d e^4 (c+d x)}-\frac{b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^2}{2 d e^4 (c+d x)^2}-\frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{3 d e^4 (c+d x)^3}-\frac{b \tanh ^{-1}\left(e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^2}{d e^4}-\frac{b^3 \text{Li}_3\left(-e^{i \sin ^{-1}(c+d x)}\right)}{d e^4}+\frac{b^3 \text{Li}_3\left(e^{i \sin ^{-1}(c+d x)}\right)}{d e^4}-\frac{b^3 \tanh ^{-1}\left(\sqrt{1-(c+d x)^2}\right)}{d e^4}",1,"-1/3*a^3/(d*e^4*(c + d*x)^3) - (a^2*b*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(2*d*e^4*(c + d*x)^2) - (a^2*b*ArcSin[c + d*x])/(d*e^4*(c + d*x)^3) + (a^2*b*Log[c + d*x])/(2*d*e^4) - (a^2*b*Log[1 + Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]])/(2*d*e^4) + (a*b^2*((8*I)*PolyLog[2, -E^(I*ArcSin[c + d*x])] - (2*(2 + 4*ArcSin[c + d*x]^2 - 2*Cos[2*ArcSin[c + d*x]] - 3*(c + d*x)*ArcSin[c + d*x]*Log[1 - E^(I*ArcSin[c + d*x])] + 3*(c + d*x)*ArcSin[c + d*x]*Log[1 + E^(I*ArcSin[c + d*x])] + (4*I)*(c + d*x)^3*PolyLog[2, E^(I*ArcSin[c + d*x])] + 2*ArcSin[c + d*x]*Sin[2*ArcSin[c + d*x]] + ArcSin[c + d*x]*Log[1 - E^(I*ArcSin[c + d*x])]*Sin[3*ArcSin[c + d*x]] - ArcSin[c + d*x]*Log[1 + E^(I*ArcSin[c + d*x])]*Sin[3*ArcSin[c + d*x]]))/(c + d*x)^3))/(8*d*e^4) + (b^3*(-24*ArcSin[c + d*x]*Cot[ArcSin[c + d*x]/2] - 4*ArcSin[c + d*x]^3*Cot[ArcSin[c + d*x]/2] - 6*ArcSin[c + d*x]^2*Csc[ArcSin[c + d*x]/2]^2 - (c + d*x)*ArcSin[c + d*x]^3*Csc[ArcSin[c + d*x]/2]^4 + 24*ArcSin[c + d*x]^2*Log[1 - E^(I*ArcSin[c + d*x])] - 24*ArcSin[c + d*x]^2*Log[1 + E^(I*ArcSin[c + d*x])] + 48*Log[Tan[ArcSin[c + d*x]/2]] + (48*I)*ArcSin[c + d*x]*PolyLog[2, -E^(I*ArcSin[c + d*x])] - (48*I)*ArcSin[c + d*x]*PolyLog[2, E^(I*ArcSin[c + d*x])] - 48*PolyLog[3, -E^(I*ArcSin[c + d*x])] + 48*PolyLog[3, E^(I*ArcSin[c + d*x])] + 6*ArcSin[c + d*x]^2*Sec[ArcSin[c + d*x]/2]^2 - (16*ArcSin[c + d*x]^3*Sin[ArcSin[c + d*x]/2]^4)/(c + d*x)^3 - 24*ArcSin[c + d*x]*Tan[ArcSin[c + d*x]/2] - 4*ArcSin[c + d*x]^3*Tan[ArcSin[c + d*x]/2]))/(48*d*e^4)","B",0
206,1,287,357,0.5953902,"\int (c e+d e x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^4 \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^4,x]","\frac{e^3 \left(-3 b^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)-\frac{45}{2} b^3 (c+d x) \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)-6 b^2 (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)^2-18 b^2 (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^2+\frac{45}{4} b^2 \left(a+b \sin ^{-1}(c+d x)\right)^2+8 b (c+d x)^3 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^3+12 b (c+d x) \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^3+8 (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)^4-3 \left(a+b \sin ^{-1}(c+d x)\right)^4+\frac{3}{4} b^4 (c+d x)^4+\frac{45}{4} b^4 (c+d x)^2\right)}{32 d}","-\frac{3 b^3 e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)}{32 d}-\frac{45 b^3 e^3 (c+d x) \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)}{64 d}-\frac{3 b^2 e^3 (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)^2}{16 d}+\frac{45 b^2 e^3 \left(a+b \sin ^{-1}(c+d x)\right)^2}{128 d}-\frac{9 b^2 e^3 (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^2}{16 d}+\frac{b e^3 (c+d x)^3 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^3}{4 d}+\frac{3 b e^3 (c+d x) \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^3}{8 d}+\frac{e^3 (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)^4}{4 d}-\frac{3 e^3 \left(a+b \sin ^{-1}(c+d x)\right)^4}{32 d}+\frac{3 b^4 e^3 (c+d x)^4}{128 d}+\frac{45 b^4 e^3 (c+d x)^2}{128 d}",1,"(e^3*((45*b^4*(c + d*x)^2)/4 + (3*b^4*(c + d*x)^4)/4 - (45*b^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/2 - 3*b^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) + (45*b^2*(a + b*ArcSin[c + d*x])^2)/4 - 18*b^2*(c + d*x)^2*(a + b*ArcSin[c + d*x])^2 - 6*b^2*(c + d*x)^4*(a + b*ArcSin[c + d*x])^2 + 12*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3 + 8*b*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3 - 3*(a + b*ArcSin[c + d*x])^4 + 8*(c + d*x)^4*(a + b*ArcSin[c + d*x])^4))/(32*d)","A",1
207,1,235,289,0.777972,"\int (c e+d e x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^4 \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcSin[c + d*x])^4,x]","\frac{e^2 \left(\frac{1}{3} (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^4-\frac{4}{9} b \left(\frac{2}{3} b^2 \sqrt{1-(c+d x)^2} (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)-\frac{40}{3} b^2 \left(b d x-\sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)\right)+b (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^2-\sqrt{1-(c+d x)^2} (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^3+6 b (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^2-2 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^3-\frac{2}{9} b^3 (c+d x)^3\right)\right)}{d}","-\frac{160 b^3 e^2 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)}{27 d}-\frac{8 b^3 e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)}{27 d}-\frac{4 b^2 e^2 (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^2}{9 d}-\frac{8 b^2 e^2 (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^2}{3 d}+\frac{8 b e^2 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^3}{9 d}+\frac{4 b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^3}{9 d}+\frac{e^2 (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^4}{3 d}+\frac{8 b^4 e^2 (c+d x)^3}{81 d}+\frac{160}{27} b^4 e^2 x",1,"(e^2*(((c + d*x)^3*(a + b*ArcSin[c + d*x])^4)/3 - (4*b*((-2*b^3*(c + d*x)^3)/9 + (2*b^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]))/3 + 6*b*(c + d*x)*(a + b*ArcSin[c + d*x])^2 + b*(c + d*x)^3*(a + b*ArcSin[c + d*x])^2 - 2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3 - (c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3 - (40*b^2*(b*d*x - Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])))/3))/9))/d","A",1
208,1,163,198,0.3151543,"\int (c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^4 \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcSin[c + d*x])^4,x]","-\frac{e \left(3 b^2 \left(2 (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^2+2 b \sqrt{1-(c+d x)^2} (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)-\left(a+b \sin ^{-1}(c+d x)\right)^2-b^2 (c+d x)^2\right)-2 (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^4+\left(a+b \sin ^{-1}(c+d x)\right)^4-4 b (c+d x) \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^3\right)}{4 d}","-\frac{3 b^3 e (c+d x) \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)}{2 d}-\frac{3 b^2 e (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^2}{2 d}+\frac{3 b^2 e \left(a+b \sin ^{-1}(c+d x)\right)^2}{4 d}+\frac{b e (c+d x) \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^3}{d}+\frac{e (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^4}{2 d}-\frac{e \left(a+b \sin ^{-1}(c+d x)\right)^4}{4 d}+\frac{3 b^4 e (c+d x)^2}{4 d}",1,"-1/4*(e*(-4*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3 + (a + b*ArcSin[c + d*x])^4 - 2*(c + d*x)^2*(a + b*ArcSin[c + d*x])^4 + 3*b^2*(-(b^2*(c + d*x)^2) + 2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) - (a + b*ArcSin[c + d*x])^2 + 2*(c + d*x)^2*(a + b*ArcSin[c + d*x])^2)))/d","A",1
209,1,115,119,0.1784852,"\int \left(a+b \sin ^{-1}(c+d x)\right)^4 \, dx","Integrate[(a + b*ArcSin[c + d*x])^4,x]","\frac{-12 b^2 \left(2 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)+(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^2-2 b^2 (c+d x)\right)+(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^4+4 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^3}{d}","-\frac{24 b^3 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)}{d}-\frac{12 b^2 (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^2}{d}+\frac{4 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^3}{d}+\frac{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^4}{d}+24 b^4 x",1,"(4*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^3 + (c + d*x)*(a + b*ArcSin[c + d*x])^4 - 12*b^2*(-2*b^2*(c + d*x) + 2*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x]) + (c + d*x)*(a + b*ArcSin[c + d*x])^2))/d","A",1
210,1,439,202,0.470621,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^4}{c e+d e x} \, dx","Integrate[(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x),x]","\frac{16 a^4 \log (c+d x)+64 a^3 b \left(\sin ^{-1}(c+d x) \log \left(1-e^{2 i \sin ^{-1}(c+d x)}\right)-\frac{1}{2} i \left(\sin ^{-1}(c+d x)^2+\text{Li}_2\left(e^{2 i \sin ^{-1}(c+d x)}\right)\right)\right)+4 a^2 b^2 \left(24 i \sin ^{-1}(c+d x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c+d x)}\right)+12 \text{Li}_3\left(e^{-2 i \sin ^{-1}(c+d x)}\right)+8 i \sin ^{-1}(c+d x)^3+24 \sin ^{-1}(c+d x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c+d x)}\right)-i \pi ^3\right)-i a b^3 \left(-96 \sin ^{-1}(c+d x)^2 \text{Li}_2\left(e^{-2 i \sin ^{-1}(c+d x)}\right)+96 i \sin ^{-1}(c+d x) \text{Li}_3\left(e^{-2 i \sin ^{-1}(c+d x)}\right)+48 \text{Li}_4\left(e^{-2 i \sin ^{-1}(c+d x)}\right)-16 \sin ^{-1}(c+d x)^4+64 i \sin ^{-1}(c+d x)^3 \log \left(1-e^{-2 i \sin ^{-1}(c+d x)}\right)+\pi ^4\right)+16 b^4 \left(2 i \sin ^{-1}(c+d x)^3 \text{Li}_2\left(e^{-2 i \sin ^{-1}(c+d x)}\right)+3 \sin ^{-1}(c+d x)^2 \text{Li}_3\left(e^{-2 i \sin ^{-1}(c+d x)}\right)-3 i \sin ^{-1}(c+d x) \text{Li}_4\left(e^{-2 i \sin ^{-1}(c+d x)}\right)-\frac{3}{2} \text{Li}_5\left(e^{-2 i \sin ^{-1}(c+d x)}\right)+\frac{1}{5} i \sin ^{-1}(c+d x)^5+\sin ^{-1}(c+d x)^4 \log \left(1-e^{-2 i \sin ^{-1}(c+d x)}\right)-\frac{i \pi ^5}{160}\right)}{16 d e}","\frac{3 i b^3 \text{Li}_4\left(e^{2 i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e}+\frac{3 b^2 \text{Li}_3\left(e^{2 i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^2}{d e}-\frac{2 i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^3}{d e}-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)^5}{5 b d e}+\frac{\log \left(1-e^{2 i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^4}{d e}-\frac{3 b^4 \text{Li}_5\left(e^{2 i \sin ^{-1}(c+d x)}\right)}{2 d e}",1,"(16*a^4*Log[c + d*x] + 64*a^3*b*(ArcSin[c + d*x]*Log[1 - E^((2*I)*ArcSin[c + d*x])] - (I/2)*(ArcSin[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c + d*x])])) + 4*a^2*b^2*((-I)*Pi^3 + (8*I)*ArcSin[c + d*x]^3 + 24*ArcSin[c + d*x]^2*Log[1 - E^((-2*I)*ArcSin[c + d*x])] + (24*I)*ArcSin[c + d*x]*PolyLog[2, E^((-2*I)*ArcSin[c + d*x])] + 12*PolyLog[3, E^((-2*I)*ArcSin[c + d*x])]) - I*a*b^3*(Pi^4 - 16*ArcSin[c + d*x]^4 + (64*I)*ArcSin[c + d*x]^3*Log[1 - E^((-2*I)*ArcSin[c + d*x])] - 96*ArcSin[c + d*x]^2*PolyLog[2, E^((-2*I)*ArcSin[c + d*x])] + (96*I)*ArcSin[c + d*x]*PolyLog[3, E^((-2*I)*ArcSin[c + d*x])] + 48*PolyLog[4, E^((-2*I)*ArcSin[c + d*x])]) + 16*b^4*((-1/160*I)*Pi^5 + (I/5)*ArcSin[c + d*x]^5 + ArcSin[c + d*x]^4*Log[1 - E^((-2*I)*ArcSin[c + d*x])] + (2*I)*ArcSin[c + d*x]^3*PolyLog[2, E^((-2*I)*ArcSin[c + d*x])] + 3*ArcSin[c + d*x]^2*PolyLog[3, E^((-2*I)*ArcSin[c + d*x])] - (3*I)*ArcSin[c + d*x]*PolyLog[4, E^((-2*I)*ArcSin[c + d*x])] - (3*PolyLog[5, E^((-2*I)*ArcSin[c + d*x])])/2))/(16*d*e)","B",0
211,1,575,270,2.0890961,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^4}{(c e+d e x)^2} \, dx","Integrate[(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^2,x]","\frac{-\frac{a^4}{c+d x}-4 a^3 b \left(\frac{\sin ^{-1}(c+d x)}{c+d x}-\log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)\right)+\log \left(\frac{1}{2} (c+d x) \csc \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)\right)\right)+6 a^2 b^2 \left(2 i \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right)-2 i \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right)+\sin ^{-1}(c+d x) \left(-\frac{\sin ^{-1}(c+d x)}{c+d x}+2 \log \left(1-e^{i \sin ^{-1}(c+d x)}\right)-2 \log \left(1+e^{i \sin ^{-1}(c+d x)}\right)\right)\right)+4 a b^3 \left(6 i \sin ^{-1}(c+d x) \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right)-6 i \sin ^{-1}(c+d x) \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right)-6 \text{Li}_3\left(-e^{i \sin ^{-1}(c+d x)}\right)+6 \text{Li}_3\left(e^{i \sin ^{-1}(c+d x)}\right)-\frac{\sin ^{-1}(c+d x)^3}{c+d x}+3 \sin ^{-1}(c+d x)^2 \log \left(1-e^{i \sin ^{-1}(c+d x)}\right)-3 \sin ^{-1}(c+d x)^2 \log \left(1+e^{i \sin ^{-1}(c+d x)}\right)\right)+b^4 \left(12 i \sin ^{-1}(c+d x)^2 \text{Li}_2\left(e^{-i \sin ^{-1}(c+d x)}\right)+12 i \sin ^{-1}(c+d x)^2 \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right)+24 \sin ^{-1}(c+d x) \text{Li}_3\left(e^{-i \sin ^{-1}(c+d x)}\right)-24 \sin ^{-1}(c+d x) \text{Li}_3\left(-e^{i \sin ^{-1}(c+d x)}\right)-24 i \text{Li}_4\left(e^{-i \sin ^{-1}(c+d x)}\right)-24 i \text{Li}_4\left(-e^{i \sin ^{-1}(c+d x)}\right)-\frac{\sin ^{-1}(c+d x)^4}{c+d x}+i \sin ^{-1}(c+d x)^4+4 \sin ^{-1}(c+d x)^3 \log \left(1-e^{-i \sin ^{-1}(c+d x)}\right)-4 \sin ^{-1}(c+d x)^3 \log \left(1+e^{i \sin ^{-1}(c+d x)}\right)-\frac{i \pi ^4}{2}\right)}{d e^2}","-\frac{24 b^3 \text{Li}_3\left(-e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e^2}+\frac{24 b^3 \text{Li}_3\left(e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e^2}+\frac{12 i b^2 \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^2}{d e^2}-\frac{12 i b^2 \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^2}{d e^2}-\frac{\left(a+b \sin ^{-1}(c+d x)\right)^4}{d e^2 (c+d x)}-\frac{8 b \tanh ^{-1}\left(e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^3}{d e^2}-\frac{24 i b^4 \text{Li}_4\left(-e^{i \sin ^{-1}(c+d x)}\right)}{d e^2}+\frac{24 i b^4 \text{Li}_4\left(e^{i \sin ^{-1}(c+d x)}\right)}{d e^2}",1,"(-(a^4/(c + d*x)) - 4*a^3*b*(ArcSin[c + d*x]/(c + d*x) + Log[((c + d*x)*Csc[ArcSin[c + d*x]/2])/2] - Log[Sin[ArcSin[c + d*x]/2]]) + 6*a^2*b^2*(ArcSin[c + d*x]*(-(ArcSin[c + d*x]/(c + d*x)) + 2*Log[1 - E^(I*ArcSin[c + d*x])] - 2*Log[1 + E^(I*ArcSin[c + d*x])]) + (2*I)*PolyLog[2, -E^(I*ArcSin[c + d*x])] - (2*I)*PolyLog[2, E^(I*ArcSin[c + d*x])]) + 4*a*b^3*(-(ArcSin[c + d*x]^3/(c + d*x)) + 3*ArcSin[c + d*x]^2*Log[1 - E^(I*ArcSin[c + d*x])] - 3*ArcSin[c + d*x]^2*Log[1 + E^(I*ArcSin[c + d*x])] + (6*I)*ArcSin[c + d*x]*PolyLog[2, -E^(I*ArcSin[c + d*x])] - (6*I)*ArcSin[c + d*x]*PolyLog[2, E^(I*ArcSin[c + d*x])] - 6*PolyLog[3, -E^(I*ArcSin[c + d*x])] + 6*PolyLog[3, E^(I*ArcSin[c + d*x])]) + b^4*((-1/2*I)*Pi^4 + I*ArcSin[c + d*x]^4 - ArcSin[c + d*x]^4/(c + d*x) + 4*ArcSin[c + d*x]^3*Log[1 - E^((-I)*ArcSin[c + d*x])] - 4*ArcSin[c + d*x]^3*Log[1 + E^(I*ArcSin[c + d*x])] + (12*I)*ArcSin[c + d*x]^2*PolyLog[2, E^((-I)*ArcSin[c + d*x])] + (12*I)*ArcSin[c + d*x]^2*PolyLog[2, -E^(I*ArcSin[c + d*x])] + 24*ArcSin[c + d*x]*PolyLog[3, E^((-I)*ArcSin[c + d*x])] - 24*ArcSin[c + d*x]*PolyLog[3, -E^(I*ArcSin[c + d*x])] - (24*I)*PolyLog[4, E^((-I)*ArcSin[c + d*x])] - (24*I)*PolyLog[4, -E^(I*ArcSin[c + d*x])]))/(d*e^2)","B",0
212,1,385,198,1.4543545,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^4}{(c e+d e x)^3} \, dx","Integrate[(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^3,x]","\frac{-\frac{2 a^4}{(c+d x)^2}-\frac{8 a^3 b \sqrt{1-(c+d x)^2}}{c+d x}-\frac{8 a^3 b \sin ^{-1}(c+d x)}{(c+d x)^2}+24 a^2 b^2 \left(\log (c+d x)-\frac{\sin ^{-1}(c+d x)^2}{2 (c+d x)^2}-\frac{\sqrt{1-(c+d x)^2} \sin ^{-1}(c+d x)}{c+d x}\right)+8 a b^3 \left(-3 i \left(\sin ^{-1}(c+d x)^2+\text{Li}_2\left(e^{2 i \sin ^{-1}(c+d x)}\right)\right)-\frac{\sin ^{-1}(c+d x)^3}{(c+d x)^2}-\frac{3 \sqrt{1-(c+d x)^2} \sin ^{-1}(c+d x)^2}{c+d x}+6 \sin ^{-1}(c+d x) \log \left(1-e^{2 i \sin ^{-1}(c+d x)}\right)\right)+b^4 \left(24 i \sin ^{-1}(c+d x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c+d x)}\right)+12 \text{Li}_3\left(e^{-2 i \sin ^{-1}(c+d x)}\right)-\frac{8 \sqrt{1-(c+d x)^2} \sin ^{-1}(c+d x)^3}{c+d x}+8 i \sin ^{-1}(c+d x)^3+24 \sin ^{-1}(c+d x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c+d x)}\right)-i \pi ^3\right)-\frac{2 b^4 \sin ^{-1}(c+d x)^4}{(c+d x)^2}}{4 d e^3}","-\frac{6 i b^3 \text{Li}_2\left(e^{2 i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e^3}+\frac{6 b^2 \log \left(1-e^{2 i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^2}{d e^3}-\frac{2 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^3}{d e^3 (c+d x)}-\frac{2 i b \left(a+b \sin ^{-1}(c+d x)\right)^3}{d e^3}-\frac{\left(a+b \sin ^{-1}(c+d x)\right)^4}{2 d e^3 (c+d x)^2}+\frac{3 b^4 \text{Li}_3\left(e^{2 i \sin ^{-1}(c+d x)}\right)}{d e^3}",1,"((-2*a^4)/(c + d*x)^2 - (8*a^3*b*Sqrt[1 - (c + d*x)^2])/(c + d*x) - (8*a^3*b*ArcSin[c + d*x])/(c + d*x)^2 - (2*b^4*ArcSin[c + d*x]^4)/(c + d*x)^2 + 24*a^2*b^2*(-((Sqrt[1 - (c + d*x)^2]*ArcSin[c + d*x])/(c + d*x)) - ArcSin[c + d*x]^2/(2*(c + d*x)^2) + Log[c + d*x]) + 8*a*b^3*((-3*Sqrt[1 - (c + d*x)^2]*ArcSin[c + d*x]^2)/(c + d*x) - ArcSin[c + d*x]^3/(c + d*x)^2 + 6*ArcSin[c + d*x]*Log[1 - E^((2*I)*ArcSin[c + d*x])] - (3*I)*(ArcSin[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c + d*x])])) + b^4*((-I)*Pi^3 + (8*I)*ArcSin[c + d*x]^3 - (8*Sqrt[1 - (c + d*x)^2]*ArcSin[c + d*x]^3)/(c + d*x) + 24*ArcSin[c + d*x]^2*Log[1 - E^((-2*I)*ArcSin[c + d*x])] + (24*I)*ArcSin[c + d*x]*PolyLog[2, E^((-2*I)*ArcSin[c + d*x])] + 12*PolyLog[3, E^((-2*I)*ArcSin[c + d*x])]))/(4*d*e^3)","A",0
213,1,1274,439,11.3893061,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^4}{(c e+d e x)^4} \, dx","Integrate[(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^4,x]","-\frac{a^4}{3 d e^4 (c+d x)^3}+\frac{4 b \left(-\frac{1}{24} \sin ^{-1}(c+d x) \cot \left(\frac{1}{2} \sin ^{-1}(c+d x)\right) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c+d x)\right)-\frac{1}{24} \csc ^2\left(\frac{1}{2} \sin ^{-1}(c+d x)\right)+\frac{1}{24} \sec ^2\left(\frac{1}{2} \sin ^{-1}(c+d x)\right)-\frac{1}{12} \sin ^{-1}(c+d x) \cot \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)-\frac{1}{6} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)\right)+\frac{1}{6} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)\right)-\frac{1}{24} \sin ^{-1}(c+d x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c+d x)\right) \tan \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)-\frac{1}{12} \sin ^{-1}(c+d x) \tan \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)\right) a^3}{d e^4}+\frac{b^2 \left(8 i \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right)-\frac{2 \left(4 i \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right) (c+d x)^3-3 \sin ^{-1}(c+d x) \log \left(1-e^{i \sin ^{-1}(c+d x)}\right) (c+d x)+3 \sin ^{-1}(c+d x) \log \left(1+e^{i \sin ^{-1}(c+d x)}\right) (c+d x)+4 \sin ^{-1}(c+d x)^2-2 \cos \left(2 \sin ^{-1}(c+d x)\right)+2 \sin ^{-1}(c+d x) \sin \left(2 \sin ^{-1}(c+d x)\right)+\sin ^{-1}(c+d x) \log \left(1-e^{i \sin ^{-1}(c+d x)}\right) \sin \left(3 \sin ^{-1}(c+d x)\right)-\sin ^{-1}(c+d x) \log \left(1+e^{i \sin ^{-1}(c+d x)}\right) \sin \left(3 \sin ^{-1}(c+d x)\right)+2\right)}{(c+d x)^3}\right) a^2}{4 d e^4}+\frac{b^3 \left(-\left((c+d x) \sin ^{-1}(c+d x)^3 \csc ^4\left(\frac{1}{2} \sin ^{-1}(c+d x)\right)\right)-6 \sin ^{-1}(c+d x)^2 \csc ^2\left(\frac{1}{2} \sin ^{-1}(c+d x)\right)-\frac{16 \sin ^{-1}(c+d x)^3 \sin ^4\left(\frac{1}{2} \sin ^{-1}(c+d x)\right)}{(c+d x)^3}+6 \sin ^{-1}(c+d x)^2 \sec ^2\left(\frac{1}{2} \sin ^{-1}(c+d x)\right)-4 \sin ^{-1}(c+d x)^3 \cot \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)-24 \sin ^{-1}(c+d x) \cot \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)+24 \sin ^{-1}(c+d x)^2 \log \left(1-e^{i \sin ^{-1}(c+d x)}\right)-24 \sin ^{-1}(c+d x)^2 \log \left(1+e^{i \sin ^{-1}(c+d x)}\right)+48 \log \left(\tan \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)\right)+48 i \sin ^{-1}(c+d x) \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right)-48 i \sin ^{-1}(c+d x) \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right)-48 \text{Li}_3\left(-e^{i \sin ^{-1}(c+d x)}\right)+48 \text{Li}_3\left(e^{i \sin ^{-1}(c+d x)}\right)-4 \sin ^{-1}(c+d x)^3 \tan \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)-24 \sin ^{-1}(c+d x) \tan \left(\frac{1}{2} \sin ^{-1}(c+d x)\right)\right) a}{12 d e^4}+\frac{b^4 \left(-\frac{1}{2} (c+d x) \csc ^4\left(\frac{1}{2} \sin ^{-1}(c+d x)\right) \sin ^{-1}(c+d x)^4-\frac{8 \sin ^4\left(\frac{1}{2} \sin ^{-1}(c+d x)\right) \sin ^{-1}(c+d x)^4}{(c+d x)^3}-2 \cot \left(\frac{1}{2} \sin ^{-1}(c+d x)\right) \sin ^{-1}(c+d x)^4-2 \tan \left(\frac{1}{2} \sin ^{-1}(c+d x)\right) \sin ^{-1}(c+d x)^4+4 i \sin ^{-1}(c+d x)^4-4 \csc ^2\left(\frac{1}{2} \sin ^{-1}(c+d x)\right) \sin ^{-1}(c+d x)^3+4 \sec ^2\left(\frac{1}{2} \sin ^{-1}(c+d x)\right) \sin ^{-1}(c+d x)^3+16 \log \left(1-e^{-i \sin ^{-1}(c+d x)}\right) \sin ^{-1}(c+d x)^3-16 \log \left(1+e^{i \sin ^{-1}(c+d x)}\right) \sin ^{-1}(c+d x)^3-24 \cot \left(\frac{1}{2} \sin ^{-1}(c+d x)\right) \sin ^{-1}(c+d x)^2+48 i \text{Li}_2\left(e^{-i \sin ^{-1}(c+d x)}\right) \sin ^{-1}(c+d x)^2-24 \tan \left(\frac{1}{2} \sin ^{-1}(c+d x)\right) \sin ^{-1}(c+d x)^2+96 \log \left(1-e^{i \sin ^{-1}(c+d x)}\right) \sin ^{-1}(c+d x)-96 \log \left(1+e^{i \sin ^{-1}(c+d x)}\right) \sin ^{-1}(c+d x)+96 \text{Li}_3\left(e^{-i \sin ^{-1}(c+d x)}\right) \sin ^{-1}(c+d x)-96 \text{Li}_3\left(-e^{i \sin ^{-1}(c+d x)}\right) \sin ^{-1}(c+d x)+48 i \left(\sin ^{-1}(c+d x)^2+2\right) \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right)-96 i \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right)-96 i \text{Li}_4\left(e^{-i \sin ^{-1}(c+d x)}\right)-96 i \text{Li}_4\left(-e^{i \sin ^{-1}(c+d x)}\right)-2 i \pi ^4\right)}{24 d e^4}","-\frac{4 b^3 \text{Li}_3\left(-e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e^4}+\frac{4 b^3 \text{Li}_3\left(e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e^4}-\frac{8 b^3 \tanh ^{-1}\left(e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e^4}+\frac{2 i b^2 \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^2}{d e^4}-\frac{2 i b^2 \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^2}{d e^4}-\frac{2 b^2 \left(a+b \sin ^{-1}(c+d x)\right)^2}{d e^4 (c+d x)}-\frac{2 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^3}{3 d e^4 (c+d x)^2}-\frac{\left(a+b \sin ^{-1}(c+d x)\right)^4}{3 d e^4 (c+d x)^3}-\frac{4 b \tanh ^{-1}\left(e^{i \sin ^{-1}(c+d x)}\right) \left(a+b \sin ^{-1}(c+d x)\right)^3}{3 d e^4}+\frac{4 i b^4 \text{Li}_2\left(-e^{i \sin ^{-1}(c+d x)}\right)}{d e^4}-\frac{4 i b^4 \text{Li}_2\left(e^{i \sin ^{-1}(c+d x)}\right)}{d e^4}-\frac{4 i b^4 \text{Li}_4\left(-e^{i \sin ^{-1}(c+d x)}\right)}{d e^4}+\frac{4 i b^4 \text{Li}_4\left(e^{i \sin ^{-1}(c+d x)}\right)}{d e^4}",1,"-1/3*a^4/(d*e^4*(c + d*x)^3) + (a^2*b^2*((8*I)*PolyLog[2, -E^(I*ArcSin[c + d*x])] - (2*(2 + 4*ArcSin[c + d*x]^2 - 2*Cos[2*ArcSin[c + d*x]] - 3*(c + d*x)*ArcSin[c + d*x]*Log[1 - E^(I*ArcSin[c + d*x])] + 3*(c + d*x)*ArcSin[c + d*x]*Log[1 + E^(I*ArcSin[c + d*x])] + (4*I)*(c + d*x)^3*PolyLog[2, E^(I*ArcSin[c + d*x])] + 2*ArcSin[c + d*x]*Sin[2*ArcSin[c + d*x]] + ArcSin[c + d*x]*Log[1 - E^(I*ArcSin[c + d*x])]*Sin[3*ArcSin[c + d*x]] - ArcSin[c + d*x]*Log[1 + E^(I*ArcSin[c + d*x])]*Sin[3*ArcSin[c + d*x]]))/(c + d*x)^3))/(4*d*e^4) + (a*b^3*(-24*ArcSin[c + d*x]*Cot[ArcSin[c + d*x]/2] - 4*ArcSin[c + d*x]^3*Cot[ArcSin[c + d*x]/2] - 6*ArcSin[c + d*x]^2*Csc[ArcSin[c + d*x]/2]^2 - (c + d*x)*ArcSin[c + d*x]^3*Csc[ArcSin[c + d*x]/2]^4 + 24*ArcSin[c + d*x]^2*Log[1 - E^(I*ArcSin[c + d*x])] - 24*ArcSin[c + d*x]^2*Log[1 + E^(I*ArcSin[c + d*x])] + 48*Log[Tan[ArcSin[c + d*x]/2]] + (48*I)*ArcSin[c + d*x]*PolyLog[2, -E^(I*ArcSin[c + d*x])] - (48*I)*ArcSin[c + d*x]*PolyLog[2, E^(I*ArcSin[c + d*x])] - 48*PolyLog[3, -E^(I*ArcSin[c + d*x])] + 48*PolyLog[3, E^(I*ArcSin[c + d*x])] + 6*ArcSin[c + d*x]^2*Sec[ArcSin[c + d*x]/2]^2 - (16*ArcSin[c + d*x]^3*Sin[ArcSin[c + d*x]/2]^4)/(c + d*x)^3 - 24*ArcSin[c + d*x]*Tan[ArcSin[c + d*x]/2] - 4*ArcSin[c + d*x]^3*Tan[ArcSin[c + d*x]/2]))/(12*d*e^4) + (b^4*((-2*I)*Pi^4 + (4*I)*ArcSin[c + d*x]^4 - 24*ArcSin[c + d*x]^2*Cot[ArcSin[c + d*x]/2] - 2*ArcSin[c + d*x]^4*Cot[ArcSin[c + d*x]/2] - 4*ArcSin[c + d*x]^3*Csc[ArcSin[c + d*x]/2]^2 - ((c + d*x)*ArcSin[c + d*x]^4*Csc[ArcSin[c + d*x]/2]^4)/2 + 16*ArcSin[c + d*x]^3*Log[1 - E^((-I)*ArcSin[c + d*x])] + 96*ArcSin[c + d*x]*Log[1 - E^(I*ArcSin[c + d*x])] - 96*ArcSin[c + d*x]*Log[1 + E^(I*ArcSin[c + d*x])] - 16*ArcSin[c + d*x]^3*Log[1 + E^(I*ArcSin[c + d*x])] + (48*I)*ArcSin[c + d*x]^2*PolyLog[2, E^((-I)*ArcSin[c + d*x])] + (48*I)*(2 + ArcSin[c + d*x]^2)*PolyLog[2, -E^(I*ArcSin[c + d*x])] - (96*I)*PolyLog[2, E^(I*ArcSin[c + d*x])] + 96*ArcSin[c + d*x]*PolyLog[3, E^((-I)*ArcSin[c + d*x])] - 96*ArcSin[c + d*x]*PolyLog[3, -E^(I*ArcSin[c + d*x])] - (96*I)*PolyLog[4, E^((-I)*ArcSin[c + d*x])] - (96*I)*PolyLog[4, -E^(I*ArcSin[c + d*x])] + 4*ArcSin[c + d*x]^3*Sec[ArcSin[c + d*x]/2]^2 - (8*ArcSin[c + d*x]^4*Sin[ArcSin[c + d*x]/2]^4)/(c + d*x)^3 - 24*ArcSin[c + d*x]^2*Tan[ArcSin[c + d*x]/2] - 2*ArcSin[c + d*x]^4*Tan[ArcSin[c + d*x]/2]))/(24*d*e^4) + (4*a^3*b*(-1/12*(ArcSin[c + d*x]*Cot[ArcSin[c + d*x]/2]) - Csc[ArcSin[c + d*x]/2]^2/24 - (ArcSin[c + d*x]*Cot[ArcSin[c + d*x]/2]*Csc[ArcSin[c + d*x]/2]^2)/24 - Log[Cos[ArcSin[c + d*x]/2]]/6 + Log[Sin[ArcSin[c + d*x]/2]]/6 + Sec[ArcSin[c + d*x]/2]^2/24 - (ArcSin[c + d*x]*Tan[ArcSin[c + d*x]/2])/12 - (ArcSin[c + d*x]*Sec[ArcSin[c + d*x]/2]^2*Tan[ArcSin[c + d*x]/2])/24))/(d*e^4)","B",0
214,1,150,164,0.2327524,"\int \left(a+b \sin ^{-1}(c+d x)\right)^5 \, dx","Integrate[(a + b*ArcSin[c + d*x])^5,x]","\frac{-20 b^2 \left(-6 b^2 \left(a (c+d x)+b \sqrt{1-(c+d x)^2}+b (c+d x) \sin ^{-1}(c+d x)\right)+(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^3+3 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^2\right)+(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^5+5 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^4}{d}","120 a b^4 x-\frac{60 b^3 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^2}{d}-\frac{20 b^2 (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^3}{d}+\frac{5 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^4}{d}+\frac{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^5}{d}+\frac{120 b^5 \sqrt{1-(c+d x)^2}}{d}+\frac{120 b^5 (c+d x) \sin ^{-1}(c+d x)}{d}",1,"(5*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^4 + (c + d*x)*(a + b*ArcSin[c + d*x])^5 - 20*b^2*(3*b*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^2 + (c + d*x)*(a + b*ArcSin[c + d*x])^3 - 6*b^2*(a*(c + d*x) + b*Sqrt[1 - (c + d*x)^2] + b*(c + d*x)*ArcSin[c + d*x])))/d","A",1
215,1,150,213,0.366369,"\int \frac{(c e+d e x)^4}{a+b \sin ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)^4/(a + b*ArcSin[c + d*x]),x]","\frac{e^4 \left(2 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-3 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\cos \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-3 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\sin \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)\right)}{16 b d}","\frac{e^4 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{8 b d}-\frac{3 e^4 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{16 b d}+\frac{e^4 \cos \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{16 b d}+\frac{e^4 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{8 b d}-\frac{3 e^4 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{16 b d}+\frac{e^4 \sin \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{16 b d}",1,"(e^4*(2*Cos[a/b]*CosIntegral[a/b + ArcSin[c + d*x]] - 3*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c + d*x])] + Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c + d*x])] + 2*Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] - 3*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])] + Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c + d*x])]))/(16*b*d)","A",1
216,1,109,145,0.2608643,"\int \frac{(c e+d e x)^3}{a+b \sin ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x]),x]","\frac{e^3 \left(-2 \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\sin \left(\frac{4 a}{b}\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+2 \cos \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-\cos \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)\right)}{8 b d}","-\frac{e^3 \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{4 b d}+\frac{e^3 \sin \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{8 b d}+\frac{e^3 \cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{4 b d}-\frac{e^3 \cos \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{8 b d}",1,"(e^3*(-2*CosIntegral[2*(a/b + ArcSin[c + d*x])]*Sin[(2*a)/b] + CosIntegral[4*(a/b + ArcSin[c + d*x])]*Sin[(4*a)/b] + 2*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c + d*x])] - Cos[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c + d*x])]))/(8*b*d)","A",1
217,1,102,141,0.2117101,"\int \frac{(c e+d e x)^2}{a+b \sin ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)^2/(a + b*ArcSin[c + d*x]),x]","\frac{e^2 \left(\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-\cos \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-\sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)\right)}{4 b d}","\frac{e^2 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{4 b d}-\frac{e^2 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{4 b d}+\frac{e^2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{4 b d}-\frac{e^2 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{4 b d}",1,"(e^2*(Cos[a/b]*CosIntegral[a/b + ArcSin[c + d*x]] - Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c + d*x])] + Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] - Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])]))/(4*b*d)","A",1
218,1,61,69,0.0993802,"\int \frac{c e+d e x}{a+b \sin ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)/(a + b*ArcSin[c + d*x]),x]","\frac{e \left(\cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 a}{b}+2 \sin ^{-1}(c+d x)\right)-\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 a}{b}+2 \sin ^{-1}(c+d x)\right)\right)}{2 b d}","\frac{e \cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{2 b d}-\frac{e \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{2 b d}",1,"(e*(-(CosIntegral[(2*a)/b + 2*ArcSin[c + d*x]]*Sin[(2*a)/b]) + Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSin[c + d*x]]))/(2*b*d)","A",1
219,1,48,57,0.0887124,"\int \frac{1}{a+b \sin ^{-1}(c+d x)} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(-1),x]","\frac{\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)}{b d}","\frac{\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{b d}+\frac{\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{b d}",1,"(Cos[a/b]*CosIntegral[a/b + ArcSin[c + d*x]] + Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]])/(b*d)","A",1
220,0,0,27,0.8483497,"\int \frac{1}{(c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)} \, dx","Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])),x]","\int \frac{1}{(c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])), x]","A",-1
221,1,283,258,1.2853919,"\int \frac{(c e+d e x)^4}{\left(a+b \sin ^{-1}(c+d x)\right)^2} \, dx","Integrate[(c*e + d*e*x)^4/(a + b*ArcSin[c + d*x])^2,x]","\frac{e^4 \left(16 \left(-3 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+\sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+3 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-\cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)\right)+5 \left(10 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-5 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\sin \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-10 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+5 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-\cos \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)\right)-\frac{16 b \sqrt{1-(c+d x)^2} (c+d x)^4}{a+b \sin ^{-1}(c+d x)}\right)}{16 b^2 d}","\frac{e^4 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{8 b^2 d}-\frac{9 e^4 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{16 b^2 d}+\frac{5 e^4 \sin \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{16 b^2 d}-\frac{e^4 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{8 b^2 d}+\frac{9 e^4 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{16 b^2 d}-\frac{5 e^4 \cos \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{16 b^2 d}-\frac{e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{b d \left(a+b \sin ^{-1}(c+d x)\right)}",1,"(e^4*((-16*b*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x]) + 16*(-3*CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b] + CosIntegral[3*(a/b + ArcSin[c + d*x])]*Sin[(3*a)/b] + 3*Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] - Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])]) + 5*(10*CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b] - 5*CosIntegral[3*(a/b + ArcSin[c + d*x])]*Sin[(3*a)/b] + CosIntegral[5*(a/b + ArcSin[c + d*x])]*Sin[(5*a)/b] - 10*Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] + 5*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])] - Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c + d*x])])))/(16*b^2*d)","A",1
222,1,220,190,0.9453608,"\int \frac{(c e+d e x)^3}{\left(a+b \sin ^{-1}(c+d x)\right)^2} \, dx","Integrate[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^2,x]","-\frac{e^3 \left(3 \left(\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\sin \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-\log \left(a+b \sin ^{-1}(c+d x)\right)\right)-4 \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\cos \left(\frac{4 a}{b}\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-4 \sin \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\sin \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\frac{2 b \sqrt{1-(c+d x)^2} (c+d x)^3}{a+b \sin ^{-1}(c+d x)}+3 \log \left(a+b \sin ^{-1}(c+d x)\right)\right)}{2 b^2 d}","\frac{e^3 \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{2 b^2 d}-\frac{e^3 \cos \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{2 b^2 d}+\frac{e^3 \sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{2 b^2 d}-\frac{e^3 \sin \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{2 b^2 d}-\frac{e^3 (c+d x)^3 \sqrt{1-(c+d x)^2}}{b d \left(a+b \sin ^{-1}(c+d x)\right)}",1,"-1/2*(e^3*((2*b*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x]) - 4*Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c + d*x])] + Cos[(4*a)/b]*CosIntegral[4*(a/b + ArcSin[c + d*x])] + 3*Log[a + b*ArcSin[c + d*x]] - 4*Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c + d*x])] + 3*(Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c + d*x])] - Log[a + b*ArcSin[c + d*x]] + Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c + d*x])]) + Sin[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c + d*x])]))/(b^2*d)","A",1
223,1,140,186,0.8509504,"\int \frac{(c e+d e x)^2}{\left(a+b \sin ^{-1}(c+d x)\right)^2} \, dx","Integrate[(c*e + d*e*x)^2/(a + b*ArcSin[c + d*x])^2,x]","\frac{e^2 \left(\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-3 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+3 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-\frac{4 b \sqrt{1-(c+d x)^2} (c+d x)^2}{a+b \sin ^{-1}(c+d x)}\right)}{4 b^2 d}","\frac{e^2 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{4 b^2 d}-\frac{3 e^2 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{4 b^2 d}-\frac{e^2 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{4 b^2 d}+\frac{3 e^2 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{4 b^2 d}-\frac{e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{b d \left(a+b \sin ^{-1}(c+d x)\right)}",1,"(e^2*((-4*b*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x]) + CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b] - 3*CosIntegral[3*(a/b + ArcSin[c + d*x])]*Sin[(3*a)/b] - Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] + 3*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])]))/(4*b^2*d)","A",1
224,1,99,104,0.345041,"\int \frac{c e+d e x}{\left(a+b \sin ^{-1}(c+d x)\right)^2} \, dx","Integrate[(c*e + d*e*x)/(a + b*ArcSin[c + d*x])^2,x]","\frac{e \left(-\frac{b \sqrt{-c^2-2 c d x-d^2 x^2+1} (c+d x)}{a+b \sin ^{-1}(c+d x)}+\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\sin \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)\right)}{b^2 d}","\frac{e \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{b^2 d}+\frac{e \sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{b^2 d}-\frac{e \sqrt{1-(c+d x)^2} (c+d x)}{b d \left(a+b \sin ^{-1}(c+d x)\right)}",1,"(e*(-((b*(c + d*x)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])/(a + b*ArcSin[c + d*x])) + Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c + d*x])] + Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c + d*x])]))/(b^2*d)","A",1
225,1,79,93,0.0997961,"\int \frac{1}{\left(a+b \sin ^{-1}(c+d x)\right)^2} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(-2),x]","\frac{\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-\frac{b \sqrt{1-(c+d x)^2}}{a+b \sin ^{-1}(c+d x)}}{b^2 d}","\frac{\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{b^2 d}-\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{b^2 d}-\frac{\sqrt{1-(c+d x)^2}}{b d \left(a+b \sin ^{-1}(c+d x)\right)}",1,"(-((b*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])) + CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b] - Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]])/(b^2*d)","A",1
226,0,0,27,2.7995039,"\int \frac{1}{(c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^2} \, dx","Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^2),x]","\int \frac{1}{(c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^2} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^2},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^2), x]","A",-1
227,1,317,322,1.6008238,"\int \frac{(c e+d e x)^4}{\left(a+b \sin ^{-1}(c+d x)\right)^3} \, dx","Integrate[(c*e + d*e*x)^4/(a + b*ArcSin[c + d*x])^3,x]","\frac{e^4 \left(-\frac{16 b^2 \sqrt{1-(c+d x)^2} (c+d x)^4}{\left(a+b \sin ^{-1}(c+d x)\right)^2}+48 \left(\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-\cos \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-\sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)\right)-25 \left(2 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-3 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\cos \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-3 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\sin \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)\right)+\frac{16 b \left(5 (c+d x)^5-4 (c+d x)^3\right)}{a+b \sin ^{-1}(c+d x)}\right)}{32 b^3 d}","-\frac{e^4 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{16 b^3 d}+\frac{27 e^4 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{32 b^3 d}-\frac{25 e^4 \cos \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{32 b^3 d}-\frac{e^4 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{16 b^3 d}+\frac{27 e^4 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{32 b^3 d}-\frac{25 e^4 \sin \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{32 b^3 d}+\frac{5 e^4 (c+d x)^5}{2 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)}-\frac{2 e^4 (c+d x)^3}{b^2 d \left(a+b \sin ^{-1}(c+d x)\right)}-\frac{e^4 \sqrt{1-(c+d x)^2} (c+d x)^4}{2 b d \left(a+b \sin ^{-1}(c+d x)\right)^2}",1,"(e^4*((-16*b^2*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^2 + (16*b*(-4*(c + d*x)^3 + 5*(c + d*x)^5))/(a + b*ArcSin[c + d*x]) + 48*(Cos[a/b]*CosIntegral[a/b + ArcSin[c + d*x]] - Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c + d*x])] + Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] - Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])]) - 25*(2*Cos[a/b]*CosIntegral[a/b + ArcSin[c + d*x]] - 3*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c + d*x])] + Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c + d*x])] + 2*Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] - 3*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])] + Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c + d*x])])))/(32*b^3*d)","A",1
228,1,181,249,0.7467717,"\int \frac{(c e+d e x)^3}{\left(a+b \sin ^{-1}(c+d x)\right)^3} \, dx","Integrate[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^3,x]","\frac{e^3 \left(-\frac{b^2 \sqrt{1-(c+d x)^2} (c+d x)^3}{\left(a+b \sin ^{-1}(c+d x)\right)^2}+\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-2 \sin \left(\frac{4 a}{b}\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-\cos \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+2 \cos \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\frac{b \left(4 (c+d x)^4-3 (c+d x)^2\right)}{a+b \sin ^{-1}(c+d x)}\right)}{2 b^3 d}","\frac{e^3 \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{2 b^3 d}-\frac{e^3 \sin \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{b^3 d}-\frac{e^3 \cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{2 b^3 d}+\frac{e^3 \cos \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{b^3 d}+\frac{2 e^3 (c+d x)^4}{b^2 d \left(a+b \sin ^{-1}(c+d x)\right)}-\frac{3 e^3 (c+d x)^2}{2 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)}-\frac{e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{2 b d \left(a+b \sin ^{-1}(c+d x)\right)^2}",1,"(e^3*(-((b^2*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^2) + (b*(-3*(c + d*x)^2 + 4*(c + d*x)^4))/(a + b*ArcSin[c + d*x]) + CosIntegral[2*(a/b + ArcSin[c + d*x])]*Sin[(2*a)/b] - 2*CosIntegral[4*(a/b + ArcSin[c + d*x])]*Sin[(4*a)/b] - Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c + d*x])] + 2*Cos[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c + d*x])]))/(2*b^3*d)","A",1
229,1,219,248,0.8266771,"\int \frac{(c e+d e x)^2}{\left(a+b \sin ^{-1}(c+d x)\right)^3} \, dx","Integrate[(c*e + d*e*x)^2/(a + b*ArcSin[c + d*x])^3,x]","\frac{e^2 \left(-\frac{4 b^2 \sqrt{1-(c+d x)^2} (c+d x)^2}{\left(a+b \sin ^{-1}(c+d x)\right)^2}+8 \left(\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+9 \left(-\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+\cos \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+\sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)\right)+\frac{4 b \left(3 (c+d x)^3-2 (c+d x)\right)}{a+b \sin ^{-1}(c+d x)}\right)}{8 b^3 d}","-\frac{e^2 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{8 b^3 d}+\frac{9 e^2 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{8 b^3 d}-\frac{e^2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{8 b^3 d}+\frac{9 e^2 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{8 b^3 d}+\frac{3 e^2 (c+d x)^3}{2 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)}-\frac{e^2 (c+d x)}{b^2 d \left(a+b \sin ^{-1}(c+d x)\right)}-\frac{e^2 \sqrt{1-(c+d x)^2} (c+d x)^2}{2 b d \left(a+b \sin ^{-1}(c+d x)\right)^2}",1,"(e^2*((-4*b^2*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^2 + (4*b*(-2*(c + d*x) + 3*(c + d*x)^3))/(a + b*ArcSin[c + d*x]) + 8*(Cos[a/b]*CosIntegral[a/b + ArcSin[c + d*x]] + Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]]) + 9*(-(Cos[a/b]*CosIntegral[a/b + ArcSin[c + d*x]]) + Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c + d*x])] - Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] + Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])])))/(8*b^3*d)","A",1
230,1,107,157,0.6211032,"\int \frac{c e+d e x}{\left(a+b \sin ^{-1}(c+d x)\right)^3} \, dx","Integrate[(c*e + d*e*x)/(a + b*ArcSin[c + d*x])^3,x]","-\frac{e \left(-4 \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+4 \cos \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\frac{b \left(2 \cos \left(2 \sin ^{-1}(c+d x)\right) \left(a+b \sin ^{-1}(c+d x)\right)+b \sin \left(2 \sin ^{-1}(c+d x)\right)\right)}{\left(a+b \sin ^{-1}(c+d x)\right)^2}\right)}{4 b^3 d}","\frac{e \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{b^3 d}-\frac{e \cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{b^3 d}+\frac{e (c+d x)^2}{b^2 d \left(a+b \sin ^{-1}(c+d x)\right)}-\frac{e}{2 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)}-\frac{e \sqrt{1-(c+d x)^2} (c+d x)}{2 b d \left(a+b \sin ^{-1}(c+d x)\right)^2}",1,"-1/4*(e*(-4*CosIntegral[2*(a/b + ArcSin[c + d*x])]*Sin[(2*a)/b] + (b*(2*(a + b*ArcSin[c + d*x])*Cos[2*ArcSin[c + d*x]] + b*Sin[2*ArcSin[c + d*x]]))/(a + b*ArcSin[c + d*x])^2 + 4*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c + d*x])]))/(b^3*d)","A",1
231,1,100,127,0.6810674,"\int \frac{1}{\left(a+b \sin ^{-1}(c+d x)\right)^3} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(-3),x]","-\frac{\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+\frac{b \left(b \sqrt{1-(c+d x)^2}-(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)\right)}{\left(a+b \sin ^{-1}(c+d x)\right)^2}}{2 b^3 d}","-\frac{\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{2 b^3 d}-\frac{\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{2 b^3 d}+\frac{c+d x}{2 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)}-\frac{\sqrt{1-(c+d x)^2}}{2 b d \left(a+b \sin ^{-1}(c+d x)\right)^2}",1,"-1/2*((b*(b*Sqrt[1 - (c + d*x)^2] - (c + d*x)*(a + b*ArcSin[c + d*x])))/(a + b*ArcSin[c + d*x])^2 + Cos[a/b]*CosIntegral[a/b + ArcSin[c + d*x]] + Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]])/(b^3*d)","A",1
232,0,0,27,1.1712103,"\int \frac{1}{(c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^3} \, dx","Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^3),x]","\int \frac{1}{(c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^3} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^3},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^3), x]","A",-1
233,1,414,416,1.7159318,"\int \frac{(c e+d e x)^4}{\left(a+b \sin ^{-1}(c+d x)\right)^4} \, dx","Integrate[(c*e + d*e*x)^4/(a + b*ArcSin[c + d*x])^4,x]","\frac{e^4 \left(-\frac{32 b^3 \sqrt{1-(c+d x)^2} (c+d x)^4}{\left(a+b \sin ^{-1}(c+d x)\right)^3}+\frac{16 b^2 \left(5 (c+d x)^5-4 (c+d x)^3\right)}{\left(a+b \sin ^{-1}(c+d x)\right)^2}+384 \left(\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+544 \left(3 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-\sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-3 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+\cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)\right)-125 \left(10 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-5 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\sin \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-10 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+5 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-\cos \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)\right)+\frac{16 b \sqrt{1-(c+d x)^2} \left(25 (c+d x)^4-12 (c+d x)^2\right)}{a+b \sin ^{-1}(c+d x)}\right)}{96 b^4 d}","-\frac{e^4 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{48 b^4 d}+\frac{27 e^4 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{32 b^4 d}-\frac{125 e^4 \sin \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{96 b^4 d}+\frac{e^4 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{48 b^4 d}-\frac{27 e^4 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{32 b^4 d}+\frac{125 e^4 \cos \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{96 b^4 d}+\frac{25 e^4 \sqrt{1-(c+d x)^2} (c+d x)^4}{6 b^3 d \left(a+b \sin ^{-1}(c+d x)\right)}-\frac{2 e^4 \sqrt{1-(c+d x)^2} (c+d x)^2}{b^3 d \left(a+b \sin ^{-1}(c+d x)\right)}+\frac{5 e^4 (c+d x)^5}{6 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^2}-\frac{2 e^4 (c+d x)^3}{3 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^2}-\frac{e^4 \sqrt{1-(c+d x)^2} (c+d x)^4}{3 b d \left(a+b \sin ^{-1}(c+d x)\right)^3}",1,"(e^4*((-32*b^3*(c + d*x)^4*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^3 + (16*b^2*(-4*(c + d*x)^3 + 5*(c + d*x)^5))/(a + b*ArcSin[c + d*x])^2 + (16*b*Sqrt[1 - (c + d*x)^2]*(-12*(c + d*x)^2 + 25*(c + d*x)^4))/(a + b*ArcSin[c + d*x]) + 384*(-(CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b]) + Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]]) + 544*(3*CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b] - CosIntegral[3*(a/b + ArcSin[c + d*x])]*Sin[(3*a)/b] - 3*Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] + Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])]) - 125*(10*CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b] - 5*CosIntegral[3*(a/b + ArcSin[c + d*x])]*Sin[(3*a)/b] + CosIntegral[5*(a/b + ArcSin[c + d*x])]*Sin[(5*a)/b] - 10*Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] + 5*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])] - Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c + d*x])])))/(96*b^4*d)","A",1
234,1,320,346,1.0684831,"\int \frac{(c e+d e x)^3}{\left(a+b \sin ^{-1}(c+d x)\right)^4} \, dx","Integrate[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^4,x]","\frac{e^3 \left(-\frac{2 b^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{\left(a+b \sin ^{-1}(c+d x)\right)^3}+\frac{b^2 \left(4 (c+d x)^4-3 (c+d x)^2\right)}{\left(a+b \sin ^{-1}(c+d x)\right)^2}+30 \left(\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\sin \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-\log \left(a+b \sin ^{-1}(c+d x)\right)\right)+8 \left(-4 \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\cos \left(\frac{4 a}{b}\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-4 \sin \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\sin \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+3 \log \left(a+b \sin ^{-1}(c+d x)\right)\right)+\frac{2 b \sqrt{1-(c+d x)^2} \left(8 (c+d x)^3-3 (c+d x)\right)}{a+b \sin ^{-1}(c+d x)}+6 \log \left(a+b \sin ^{-1}(c+d x)\right)\right)}{6 b^4 d}","-\frac{e^3 \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{3 b^4 d}+\frac{4 e^3 \cos \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{3 b^4 d}-\frac{e^3 \sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{3 b^4 d}+\frac{4 e^3 \sin \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{3 b^4 d}+\frac{8 e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{3 b^3 d \left(a+b \sin ^{-1}(c+d x)\right)}-\frac{e^3 \sqrt{1-(c+d x)^2} (c+d x)}{b^3 d \left(a+b \sin ^{-1}(c+d x)\right)}+\frac{2 e^3 (c+d x)^4}{3 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^2}-\frac{e^3 (c+d x)^2}{2 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^2}-\frac{e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{3 b d \left(a+b \sin ^{-1}(c+d x)\right)^3}",1,"(e^3*((-2*b^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^3 + (b^2*(-3*(c + d*x)^2 + 4*(c + d*x)^4))/(a + b*ArcSin[c + d*x])^2 + (2*b*Sqrt[1 - (c + d*x)^2]*(-3*(c + d*x) + 8*(c + d*x)^3))/(a + b*ArcSin[c + d*x]) + 6*Log[a + b*ArcSin[c + d*x]] + 30*(Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c + d*x])] - Log[a + b*ArcSin[c + d*x]] + Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c + d*x])]) + 8*(-4*Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c + d*x])] + Cos[(4*a)/b]*CosIntegral[4*(a/b + ArcSin[c + d*x])] + 3*Log[a + b*ArcSin[c + d*x]] - 4*Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c + d*x])] + Sin[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c + d*x])])))/(6*b^4*d)","A",1
235,1,264,337,1.1481734,"\int \frac{(c e+d e x)^2}{\left(a+b \sin ^{-1}(c+d x)\right)^4} \, dx","Integrate[(c*e + d*e*x)^2/(a + b*ArcSin[c + d*x])^4,x]","\frac{e^2 \left(-\frac{8 b^3 (c+d x)^2 \sqrt{1-(c+d x)^2}}{\left(a+b \sin ^{-1}(c+d x)\right)^3}+\frac{4 b^2 \left(3 (c+d x)^3-2 (c+d x)\right)}{\left(a+b \sin ^{-1}(c+d x)\right)^2}+80 \left(\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+27 \left(-3 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+\sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+3 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-\cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)\right)+\frac{4 b \sqrt{1-(c+d x)^2} \left(9 (c+d x)^2-2\right)}{a+b \sin ^{-1}(c+d x)}\right)}{24 b^4 d}","-\frac{e^2 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{24 b^4 d}+\frac{9 e^2 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{8 b^4 d}+\frac{e^2 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{24 b^4 d}-\frac{9 e^2 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{8 b^4 d}+\frac{3 e^2 \sqrt{1-(c+d x)^2} (c+d x)^2}{2 b^3 d \left(a+b \sin ^{-1}(c+d x)\right)}-\frac{e^2 \sqrt{1-(c+d x)^2}}{3 b^3 d \left(a+b \sin ^{-1}(c+d x)\right)}+\frac{e^2 (c+d x)^3}{2 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^2}-\frac{e^2 (c+d x)}{3 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^2}-\frac{e^2 \sqrt{1-(c+d x)^2} (c+d x)^2}{3 b d \left(a+b \sin ^{-1}(c+d x)\right)^3}",1,"(e^2*((-8*b^3*(c + d*x)^2*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^3 + (4*b^2*(-2*(c + d*x) + 3*(c + d*x)^3))/(a + b*ArcSin[c + d*x])^2 + (4*b*Sqrt[1 - (c + d*x)^2]*(-2 + 9*(c + d*x)^2))/(a + b*ArcSin[c + d*x]) + 80*(CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b] - Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]]) + 27*(-3*CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b] + CosIntegral[3*(a/b + ArcSin[c + d*x])]*Sin[(3*a)/b] + 3*Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]] - Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c + d*x])])))/(24*b^4*d)","A",1
236,1,186,208,0.7890753,"\int \frac{c e+d e x}{\left(a+b \sin ^{-1}(c+d x)\right)^4} \, dx","Integrate[(c*e + d*e*x)/(a + b*ArcSin[c + d*x])^4,x]","\frac{e \left(-\frac{2 b^3 (c+d x) \sqrt{1-(c+d x)^2}}{\left(a+b \sin ^{-1}(c+d x)\right)^3}+\frac{b^2 \left(2 (c+d x)^2-1\right)}{\left(a+b \sin ^{-1}(c+d x)\right)^2}-4 \left(\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)+\sin \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)\right)-\log \left(a+b \sin ^{-1}(c+d x)\right)\right)+\frac{4 b (c+d x) \sqrt{1-(c+d x)^2}}{a+b \sin ^{-1}(c+d x)}-4 \log \left(a+b \sin ^{-1}(c+d x)\right)\right)}{6 b^4 d}","-\frac{2 e \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{3 b^4 d}-\frac{2 e \sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)}{3 b^4 d}+\frac{2 e \sqrt{1-(c+d x)^2} (c+d x)}{3 b^3 d \left(a+b \sin ^{-1}(c+d x)\right)}+\frac{e (c+d x)^2}{3 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^2}-\frac{e}{6 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^2}-\frac{e \sqrt{1-(c+d x)^2} (c+d x)}{3 b d \left(a+b \sin ^{-1}(c+d x)\right)^3}",1,"(e*((-2*b^3*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^3 + (b^2*(-1 + 2*(c + d*x)^2))/(a + b*ArcSin[c + d*x])^2 + (4*b*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x]) - 4*Log[a + b*ArcSin[c + d*x]] - 4*(Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c + d*x])] - Log[a + b*ArcSin[c + d*x]] + Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c + d*x])])))/(6*b^4*d)","A",1
237,1,134,164,0.3421196,"\int \frac{1}{\left(a+b \sin ^{-1}(c+d x)\right)^4} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(-4),x]","\frac{-\frac{2 b^3 \sqrt{1-(c+d x)^2}}{\left(a+b \sin ^{-1}(c+d x)\right)^3}+\frac{b^2 (c+d x)}{\left(a+b \sin ^{-1}(c+d x)\right)^2}-\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+\frac{b \sqrt{1-(c+d x)^2}}{a+b \sin ^{-1}(c+d x)}}{6 b^4 d}","-\frac{\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{6 b^4 d}+\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{6 b^4 d}+\frac{\sqrt{1-(c+d x)^2}}{6 b^3 d \left(a+b \sin ^{-1}(c+d x)\right)}+\frac{c+d x}{6 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^2}-\frac{\sqrt{1-(c+d x)^2}}{3 b d \left(a+b \sin ^{-1}(c+d x)\right)^3}",1,"((-2*b^3*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^3 + (b^2*(c + d*x))/(a + b*ArcSin[c + d*x])^2 + (b*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x]) - CosIntegral[a/b + ArcSin[c + d*x]]*Sin[a/b] + Cos[a/b]*SinIntegral[a/b + ArcSin[c + d*x]])/(6*b^4*d)","A",1
238,0,0,27,6.4940248,"\int \frac{1}{(c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^4} \, dx","Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^4),x]","\int \frac{1}{(c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^4} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^4},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^4), x]","A",-1
239,1,156,191,0.4491618,"\int \frac{1}{\left(a+b \sin ^{-1}(c+d x)\right)^5} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(-5),x]","\frac{-\frac{6 b^4 \sqrt{1-(c+d x)^2}}{\left(a+b \sin ^{-1}(c+d x)\right)^4}+\frac{2 b^3 (c+d x)}{\left(a+b \sin ^{-1}(c+d x)\right)^3}+\frac{b^2 \sqrt{1-(c+d x)^2}}{\left(a+b \sin ^{-1}(c+d x)\right)^2}+\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)+\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c+d x)\right)-\frac{b (c+d x)}{a+b \sin ^{-1}(c+d x)}}{24 b^5 d}","\frac{\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{24 b^5 d}+\frac{\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c+d x)}{b}\right)}{24 b^5 d}-\frac{c+d x}{24 b^4 d \left(a+b \sin ^{-1}(c+d x)\right)}+\frac{\sqrt{1-(c+d x)^2}}{24 b^3 d \left(a+b \sin ^{-1}(c+d x)\right)^2}+\frac{c+d x}{12 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^3}-\frac{\sqrt{1-(c+d x)^2}}{4 b d \left(a+b \sin ^{-1}(c+d x)\right)^4}",1,"((-6*b^4*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^4 + (2*b^3*(c + d*x))/(a + b*ArcSin[c + d*x])^3 + (b^2*Sqrt[1 - (c + d*x)^2])/(a + b*ArcSin[c + d*x])^2 - (b*(c + d*x))/(a + b*ArcSin[c + d*x]) + Cos[a/b]*CosIntegral[a/b + ArcSin[c + d*x]] + Sin[a/b]*SinIntegral[a/b + ArcSin[c + d*x]])/(24*b^5*d)","A",1
240,1,269,288,0.1980572,"\int (c e+d e x)^3 \sqrt{a+b \sin ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)^3*Sqrt[a + b*ArcSin[c + d*x]],x]","\frac{e^3 e^{-\frac{4 i a}{b}} \sqrt{a+b \sin ^{-1}(c+d x)} \left(-4 \sqrt{2} e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-4 \sqrt{2} e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+\sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{4 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{8 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{4 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{128 d \sqrt{\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{b^2}}}","-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^3 \cos \left(\frac{4 a}{b}\right) C\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64 d}+\frac{\sqrt{\pi } \sqrt{b} e^3 \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{16 d}+\frac{\sqrt{\pi } \sqrt{b} e^3 \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{16 d}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^3 \sin \left(\frac{4 a}{b}\right) S\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64 d}+\frac{e^3 (c+d x)^4 \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}-\frac{3 e^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}",1,"(e^3*Sqrt[a + b*ArcSin[c + d*x]]*(-4*Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] - 4*Sqrt[2]*E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b] + Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-4*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((8*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((4*I)*(a + b*ArcSin[c + d*x]))/b]))/(128*d*E^(((4*I)*a)/b)*Sqrt[(a + b*ArcSin[c + d*x])^2/b^2])","C",0
241,1,269,274,0.3360681,"\int (c e+d e x)^2 \sqrt{a+b \sin ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)^2*Sqrt[a + b*ArcSin[c + d*x]],x]","-\frac{i e^2 e^{-\frac{3 i a}{b}} \sqrt{a+b \sin ^{-1}(c+d x)} \left(9 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-9 e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+\sqrt{3} \left(e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-\sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)\right)}{72 d \sqrt{\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{b^2}}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^2 \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} e^2 \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{12 d}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^2 \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} e^2 \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{12 d}+\frac{e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}",1,"((-1/72*I)*e^2*Sqrt[a + b*ArcSin[c + d*x]]*(9*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] - 9*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b] + Sqrt[3]*(-(Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b]) + E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b])))/(d*E^(((3*I)*a)/b)*Sqrt[(a + b*ArcSin[c + d*x])^2/b^2])","C",0
242,1,154,156,0.0795759,"\int (c e+d e x) \sqrt{a+b \sin ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)*Sqrt[a + b*ArcSin[c + d*x]],x]","-\frac{e e^{-\frac{2 i a}{b}} \sqrt{a+b \sin ^{-1}(c+d x)} \left(\sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{8 \sqrt{2} d \sqrt{\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{b^2}}}","\frac{\sqrt{\pi } \sqrt{b} e \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{8 d}+\frac{\sqrt{\pi } \sqrt{b} e \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{8 d}+\frac{e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}-\frac{e \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}",1,"-1/8*(e*Sqrt[a + b*ArcSin[c + d*x]]*(Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]))/(Sqrt[2]*d*E^(((2*I)*a)/b)*Sqrt[(a + b*ArcSin[c + d*x])^2/b^2])","C",0
243,1,129,133,0.0647898,"\int \sqrt{a+b \sin ^{-1}(c+d x)} \, dx","Integrate[Sqrt[a + b*ArcSin[c + d*x]],x]","\frac{b e^{-\frac{i a}{b}} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{2 d \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{d}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{d}+\frac{(c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{d}",1,"(b*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(2*d*E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
244,0,0,29,2.6544937,"\int \frac{\sqrt{a+b \sin ^{-1}(c+d x)}}{c e+d e x} \, dx","Integrate[Sqrt[a + b*ArcSin[c + d*x]]/(c*e + d*e*x),x]","\int \frac{\sqrt{a+b \sin ^{-1}(c+d x)}}{c e+d e x} \, dx","\frac{\text{Int}\left(\frac{\sqrt{a+b \sin ^{-1}(c+d x)}}{c+d x},x\right)}{e}",0,"Integrate[Sqrt[a + b*ArcSin[c + d*x]]/(c*e + d*e*x), x]","A",-1
245,1,273,380,0.1975197,"\int (c e+d e x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2} \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^(3/2),x]","-\frac{i b e^3 e^{-\frac{4 i a}{b}} \sqrt{a+b \sin ^{-1}(c+d x)} \left(8 \sqrt{2} e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{5}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-8 \sqrt{2} e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{5}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-\sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{5}{2},-\frac{4 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{8 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{5}{2},\frac{4 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{512 d \sqrt{\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{b^2}}}","\frac{3 \sqrt{\pi } b^{3/2} e^3 \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{64 d}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^3 \sin \left(\frac{4 a}{b}\right) C\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{512 d}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^3 \cos \left(\frac{4 a}{b}\right) S\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{512 d}-\frac{3 \sqrt{\pi } b^{3/2} e^3 \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{64 d}+\frac{e^3 (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{4 d}+\frac{3 b e^3 \sqrt{1-(c+d x)^2} (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{9 b e^3 \sqrt{1-(c+d x)^2} (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}-\frac{3 e^3 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{32 d}",1,"((-1/512*I)*b*e^3*Sqrt[a + b*ArcSin[c + d*x]]*(8*Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] - 8*Sqrt[2]*E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b] - Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((-4*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((8*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((4*I)*(a + b*ArcSin[c + d*x]))/b]))/(d*E^(((4*I)*a)/b)*Sqrt[(a + b*ArcSin[c + d*x])^2/b^2])","C",0
246,1,268,361,0.2916636,"\int (c e+d e x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2} \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcSin[c + d*x])^(3/2),x]","\frac{b e^2 e^{-\frac{3 i a}{b}} \sqrt{a+b \sin ^{-1}(c+d x)} \left(27 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{5}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+27 e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{5}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-\sqrt{3} \left(\sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{5}{2},-\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{5}{2},\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)\right)}{216 d \sqrt{\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{b^2}}}","-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^2 \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 d}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} e^2 \cos \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{24 d}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^2 \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 d}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} e^2 \sin \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{24 d}+\frac{e^2 (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{3 d}+\frac{b e^2 \sqrt{1-(c+d x)^2} (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{6 d}+\frac{b e^2 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{3 d}",1,"(b*e^2*Sqrt[a + b*ArcSin[c + d*x]]*(27*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + 27*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, (I*(a + b*ArcSin[c + d*x]))/b] - Sqrt[3]*(Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b])))/(216*d*E^(((3*I)*a)/b)*Sqrt[(a + b*ArcSin[c + d*x])^2/b^2])","C",0
247,1,137,199,0.0727319,"\int (c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2} \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcSin[c + d*x])^(3/2),x]","\frac{b^2 e e^{-\frac{2 i a}{b}} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{5}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{4 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{5}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{16 \sqrt{2} d \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{3 \sqrt{\pi } b^{3/2} e \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{32 d}-\frac{3 \sqrt{\pi } b^{3/2} e \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{32 d}+\frac{e (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{2 d}+\frac{3 b e \sqrt{1-(c+d x)^2} (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{e \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{4 d}",1,"(b^2*e*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]))/(16*Sqrt[2]*d*E^(((2*I)*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
248,1,313,175,3.256429,"\int \left(a+b \sin ^{-1}(c+d x)\right)^{3/2} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(3/2),x]","\frac{b \left(-\sqrt{2 \pi } \sqrt{\frac{1}{b}} \left(2 a \sin \left(\frac{a}{b}\right)+3 b \cos \left(\frac{a}{b}\right)\right) C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)+\sqrt{2 \pi } \sqrt{\frac{1}{b}} \left(2 a \cos \left(\frac{a}{b}\right)-3 b \sin \left(\frac{a}{b}\right)\right) S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)+2 \left(3 \sqrt{1-(c+d x)^2}+2 (c+d x) \sin ^{-1}(c+d x)\right) \sqrt{a+b \sin ^{-1}(c+d x)}+\frac{2 a e^{-\frac{i a}{b}} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{\sqrt{a+b \sin ^{-1}(c+d x)}}\right)}{4 d}","-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 d}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 d}+\frac{3 b \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{d}",1,"(b*(2*Sqrt[a + b*ArcSin[c + d*x]]*(3*Sqrt[1 - (c + d*x)^2] + 2*(c + d*x)*ArcSin[c + d*x]) + (2*a*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]]) - Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) + Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]]*(2*a*Cos[a/b] - 3*b*Sin[a/b])))/(4*d)","C",0
249,0,0,29,2.2432142,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{c e+d e x} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(3/2)/(c*e + d*e*x),x]","\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{c e+d e x} \, dx","\frac{\text{Int}\left(\frac{\left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{c+d x},x\right)}{e}",0,"Integrate[(a + b*ArcSin[c + d*x])^(3/2)/(c*e + d*e*x), x]","A",-1
250,1,269,475,0.3209057,"\int (c e+d e x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^{5/2} \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^(5/2),x]","-\frac{e^3 e^{-\frac{4 i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^{5/2} \left(-16 \sqrt{2} e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{7}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-16 \sqrt{2} e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{7}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+\sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{7}{2},-\frac{4 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{8 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{7}{2},\frac{4 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{2048 d \left(\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{b^2}\right)^{3/2}}","\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^3 \cos \left(\frac{4 a}{b}\right) C\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4096 d}-\frac{15 \sqrt{\pi } b^{5/2} e^3 \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{256 d}-\frac{15 \sqrt{\pi } b^{5/2} e^3 \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{256 d}+\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^3 \sin \left(\frac{4 a}{b}\right) S\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4096 d}-\frac{15 b^2 e^3 (c+d x)^4 \sqrt{a+b \sin ^{-1}(c+d x)}}{256 d}-\frac{45 b^2 e^3 (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{256 d}+\frac{225 b^2 e^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{2048 d}+\frac{e^3 (c+d x)^4 \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{4 d}+\frac{5 b e^3 \sqrt{1-(c+d x)^2} (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{32 d}+\frac{15 b e^3 \sqrt{1-(c+d x)^2} (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{64 d}-\frac{3 e^3 \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{32 d}",1,"-1/2048*(e^3*(a + b*ArcSin[c + d*x])^(5/2)*(-16*Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] - 16*Sqrt[2]*E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b] + Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((-4*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((8*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((4*I)*(a + b*ArcSin[c + d*x]))/b]))/(d*E^(((4*I)*a)/b)*((a + b*ArcSin[c + d*x])^2/b^2)^(3/2))","C",0
251,1,249,427,0.2957304,"\int (c e+d e x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^{5/2} \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcSin[c + d*x])^(5/2),x]","\frac{b^3 e^2 e^{-\frac{3 i a}{b}} \left(-81 e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{7}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-81 e^{\frac{4 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{7}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+\sqrt{3} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{7}{2},-\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{6 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{7}{2},\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)\right)}{648 d \sqrt{a+b \sin ^{-1}(c+d x)}}","-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^2 \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d}+\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} e^2 \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{144 d}+\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^2 \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d}-\frac{5 \sqrt{\frac{\pi }{6}} b^{5/2} e^2 \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{144 d}-\frac{5 b^2 e^2 (c+d x)^3 \sqrt{a+b \sin ^{-1}(c+d x)}}{36 d}-\frac{5 b^2 e^2 (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{6 d}+\frac{e^2 (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{3 d}+\frac{5 b e^2 \sqrt{1-(c+d x)^2} (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{18 d}+\frac{5 b e^2 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{9 d}",1,"(b^3*e^2*(-81*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] - 81*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, (I*(a + b*ArcSin[c + d*x]))/b] + Sqrt[3]*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b])))/(648*d*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
252,1,154,256,0.1012927,"\int (c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^{5/2} \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcSin[c + d*x])^(5/2),x]","\frac{e e^{-\frac{2 i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^{5/2} \left(\sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{7}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{7}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{32 \sqrt{2} d \left(\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{b^2}\right)^{3/2}}","-\frac{15 \sqrt{\pi } b^{5/2} e \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{128 d}-\frac{15 \sqrt{\pi } b^{5/2} e \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{128 d}-\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sin ^{-1}(c+d x)}}{32 d}+\frac{15 b^2 e \sqrt{a+b \sin ^{-1}(c+d x)}}{64 d}+\frac{5 b e (c+d x) \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{8 d}+\frac{e (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{2 d}-\frac{e \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{4 d}",1,"(e*(a + b*ArcSin[c + d*x])^(5/2)*(Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]))/(32*Sqrt[2]*d*E^(((2*I)*a)/b)*((a + b*ArcSin[c + d*x])^2/b^2)^(3/2))","C",0
253,1,432,204,1.807467,"\int \left(a+b \sin ^{-1}(c+d x)\right)^{5/2} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(5/2),x]","\frac{e^{-\frac{i a}{b}} \left(\frac{i \sqrt{\frac{\pi }{2}} \left(4 a^2+15 b^2\right) \left(-1+e^{\frac{2 i a}{b}}\right) \sqrt{a+b \sin ^{-1}(c+d x)} C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)}{\sqrt{\frac{1}{b}}}+\frac{\sqrt{\frac{\pi }{2}} \left(4 a^2+15 b^2\right) \left(1+e^{\frac{2 i a}{b}}\right) \sqrt{a+b \sin ^{-1}(c+d x)} S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)}{\sqrt{\frac{1}{b}}}+2 b \left(2 a^2 \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+2 a^2 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right) \left(2 \sin ^{-1}(c+d x) \left(4 a (c+d x)+5 b \sqrt{-c^2-2 c d x-d^2 x^2+1}\right)+10 a \sqrt{-c^2-2 c d x-d^2 x^2+1}-15 b (c+d x)+4 b (c+d x) \sin ^{-1}(c+d x)^2\right)\right)\right)}{8 d \sqrt{a+b \sin ^{-1}(c+d x)}}","-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d}+\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d}-\frac{15 b^2 (c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}}{4 d}+\frac{5 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{2 d}+\frac{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{d}",1,"((I*(4*a^2 + 15*b^2)*(-1 + E^(((2*I)*a)/b))*Sqrt[Pi/2]*Sqrt[a + b*ArcSin[c + d*x]]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]])/Sqrt[b^(-1)] + ((4*a^2 + 15*b^2)*(1 + E^(((2*I)*a)/b))*Sqrt[Pi/2]*Sqrt[a + b*ArcSin[c + d*x]]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]])/Sqrt[b^(-1)] + 2*b*(E^((I*a)/b)*(a + b*ArcSin[c + d*x])*(-15*b*(c + d*x) + 10*a*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2] + 2*(4*a*(c + d*x) + 5*b*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])*ArcSin[c + d*x] + 4*b*(c + d*x)*ArcSin[c + d*x]^2) + 2*a^2*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + 2*a^2*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(8*d*E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
254,0,0,29,1.563758,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{c e+d e x} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(5/2)/(c*e + d*e*x),x]","\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{c e+d e x} \, dx","\frac{\text{Int}\left(\frac{\left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{c+d x},x\right)}{e}",0,"Integrate[(a + b*ArcSin[c + d*x])^(5/2)/(c*e + d*e*x), x]","A",-1
255,1,267,518,0.3439026,"\int (c e+d e x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^{7/2} \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcSin[c + d*x])^(7/2),x]","\frac{b e^2 e^{-\frac{3 i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^{5/2} \left(-243 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{9}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-243 e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{9}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+\sqrt{3} \left(\sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{9}{2},-\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{9}{2},\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)\right)}{1944 d \left(\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{b^2}\right)^{3/2}}","\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} e^2 \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 d}-\frac{35 \sqrt{\frac{\pi }{6}} b^{7/2} e^2 \cos \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{864 d}+\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} e^2 \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 d}-\frac{35 \sqrt{\frac{\pi }{6}} b^{7/2} e^2 \sin \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{864 d}-\frac{175 b^3 e^2 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{216 d}-\frac{35 b^2 e^2 (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{108 d}-\frac{35 b^2 e^2 (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{18 d}+\frac{7 b e^2 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{9 d}+\frac{7 b e^2 (c+d x)^2 \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left(a+b \sin ^{-1}(c+d x)\right)^{7/2}}{3 d}",1,"(b*e^2*(a + b*ArcSin[c + d*x])^(5/2)*(-243*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[9/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] - 243*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[9/2, (I*(a + b*ArcSin[c + d*x]))/b] + Sqrt[3]*(Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[9/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[9/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b])))/(1944*d*E^(((3*I)*a)/b)*((a + b*ArcSin[c + d*x])^2/b^2)^(3/2))","C",0
256,1,137,301,0.0700148,"\int (c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^{7/2} \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcSin[c + d*x])^(7/2),x]","-\frac{b^4 e e^{-\frac{2 i a}{b}} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{9}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{4 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{9}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{64 \sqrt{2} d \sqrt{a+b \sin ^{-1}(c+d x)}}","-\frac{105 \sqrt{\pi } b^{7/2} e \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{512 d}+\frac{105 \sqrt{\pi } b^{7/2} e \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{512 d}-\frac{105 b^3 e (c+d x) \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{128 d}-\frac{35 b^2 e (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{32 d}+\frac{35 b^2 e \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{64 d}+\frac{7 b e (c+d x) \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{8 d}+\frac{e (c+d x)^2 \left(a+b \sin ^{-1}(c+d x)\right)^{7/2}}{2 d}-\frac{e \left(a+b \sin ^{-1}(c+d x)\right)^{7/2}}{4 d}",1,"-1/64*(b^4*e*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[9/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[9/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]))/(Sqrt[2]*d*E^(((2*I)*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
257,1,551,243,2.2914415,"\int \left(a+b \sin ^{-1}(c+d x)\right)^{7/2} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(7/2),x]","\frac{e^{-\frac{i a}{b}} \left(\sqrt{2 \pi } \left(8 i a^3 \left(-1+e^{\frac{2 i a}{b}}\right)+105 b^3 \left(1+e^{\frac{2 i a}{b}}\right)\right) \sqrt{a+b \sin ^{-1}(c+d x)} C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)-i \sqrt{2 \pi } \left(8 i a^3 \left(1+e^{\frac{2 i a}{b}}\right)+105 b^3 \left(-1+e^{\frac{2 i a}{b}}\right)\right) \sqrt{a+b \sin ^{-1}(c+d x)} S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}\right)+\frac{4 \left(4 a^3 \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+4 a^3 e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right) \left(7 \left(4 a^2 \sqrt{-c^2-2 c d x-d^2 x^2+1}-10 a b (c+d x)-15 b^2 \sqrt{-c^2-2 c d x-d^2 x^2+1}\right)+\sin ^{-1}(c+d x) \left(24 a^2 (c+d x)+56 a b \sqrt{-c^2-2 c d x-d^2 x^2+1}-70 b^2 (c+d x)\right)+4 b \sin ^{-1}(c+d x)^2 \left(6 a (c+d x)+7 b \sqrt{-c^2-2 c d x-d^2 x^2+1}\right)+8 b^2 (c+d x) \sin ^{-1}(c+d x)^3\right)\right)}{\sqrt{\frac{1}{b}}}\right)}{32 \sqrt{\frac{1}{b}} d \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 d}+\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 d}-\frac{105 b^3 \sqrt{1-(c+d x)^2} \sqrt{a+b \sin ^{-1}(c+d x)}}{8 d}-\frac{35 b^2 (c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}{4 d}+\frac{7 b \sqrt{1-(c+d x)^2} \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}{2 d}+\frac{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{7/2}}{d}",1,"(((8*I)*a^3*(-1 + E^(((2*I)*a)/b)) + 105*b^3*(1 + E^(((2*I)*a)/b)))*Sqrt[2*Pi]*Sqrt[a + b*ArcSin[c + d*x]]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]] - I*(105*b^3*(-1 + E^(((2*I)*a)/b)) + (8*I)*a^3*(1 + E^(((2*I)*a)/b)))*Sqrt[2*Pi]*Sqrt[a + b*ArcSin[c + d*x]]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]]] + (4*(E^((I*a)/b)*(a + b*ArcSin[c + d*x])*(7*(-10*a*b*(c + d*x) + 4*a^2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2] - 15*b^2*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]) + (24*a^2*(c + d*x) - 70*b^2*(c + d*x) + 56*a*b*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])*ArcSin[c + d*x] + 4*b*(6*a*(c + d*x) + 7*b*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])*ArcSin[c + d*x]^2 + 8*b^2*(c + d*x)*ArcSin[c + d*x]^3) + 4*a^3*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + 4*a^3*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/Sqrt[b^(-1)])/(32*Sqrt[b^(-1)]*d*E^((I*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
258,0,0,29,1.5554646,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^{7/2}}{c e+d e x} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(7/2)/(c*e + d*e*x),x]","\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^{7/2}}{c e+d e x} \, dx","\frac{\text{Int}\left(\frac{\left(a+b \sin ^{-1}(c+d x)\right)^{7/2}}{c+d x},x\right)}{e}",0,"Integrate[(a + b*ArcSin[c + d*x])^(7/2)/(c*e + d*e*x), x]","A",-1
259,1,370,365,0.2711053,"\int \frac{(c e+d e x)^4}{\sqrt{a+b \sin ^{-1}(c+d x)}} \, dx","Integrate[(c*e + d*e*x)^4/Sqrt[a + b*ArcSin[c + d*x]],x]","\frac{i e^4 e^{-\frac{5 i a}{b}} \left(-10 e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+10 e^{\frac{6 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+5 \sqrt{3} e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-5 \sqrt{3} e^{\frac{8 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-\sqrt{5} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+\sqrt{5} e^{\frac{10 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{5 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{160 d \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{\sqrt{\frac{\pi }{2}} e^4 \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 \sqrt{b} d}-\frac{\sqrt{\frac{3 \pi }{2}} e^4 \cos \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{10}} e^4 \cos \left(\frac{5 a}{b}\right) C\left(\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{2}} e^4 \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 \sqrt{b} d}-\frac{\sqrt{\frac{3 \pi }{2}} e^4 \sin \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{10}} e^4 \sin \left(\frac{5 a}{b}\right) S\left(\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d}",1,"((I/160)*e^4*(-10*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + 10*E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b] + 5*Sqrt[3]*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b] - 5*Sqrt[3]*E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b] - Sqrt[5]*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-5*I)*(a + b*ArcSin[c + d*x]))/b] + Sqrt[5]*E^(((10*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((5*I)*(a + b*ArcSin[c + d*x]))/b]))/(d*E^(((5*I)*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
260,1,249,233,0.1459863,"\int \frac{(c e+d e x)^3}{\sqrt{a+b \sin ^{-1}(c+d x)}} \, dx","Integrate[(c*e + d*e*x)^3/Sqrt[a + b*ArcSin[c + d*x]],x]","\frac{e^3 e^{-\frac{4 i a}{b}} \left(-2 \sqrt{2} e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-2 \sqrt{2} e^{\frac{6 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{4 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{8 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{4 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{32 d \sqrt{a+b \sin ^{-1}(c+d x)}}","-\frac{\sqrt{\pi } e^3 \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{4 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{2}} e^3 \sin \left(\frac{4 a}{b}\right) C\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d}-\frac{\sqrt{\frac{\pi }{2}} e^3 \cos \left(\frac{4 a}{b}\right) S\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d}+\frac{\sqrt{\pi } e^3 \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{4 \sqrt{b} d}",1,"(e^3*(-2*Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] - 2*Sqrt[2]*E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b] + Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-4*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((4*I)*(a + b*ArcSin[c + d*x]))/b]))/(32*d*E^(((4*I)*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
261,1,249,243,0.2795193,"\int \frac{(c e+d e x)^2}{\sqrt{a+b \sin ^{-1}(c+d x)}} \, dx","Integrate[(c*e + d*e*x)^2/Sqrt[a + b*ArcSin[c + d*x]],x]","-\frac{i e^2 e^{-\frac{3 i a}{b}} \left(3 e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-3 e^{\frac{4 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+\sqrt{3} \left(e^{\frac{6 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)\right)}{24 d \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{\sqrt{\frac{\pi }{2}} e^2 \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d}-\frac{\sqrt{\frac{\pi }{6}} e^2 \cos \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{2}} e^2 \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d}-\frac{\sqrt{\frac{\pi }{6}} e^2 \sin \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d}",1,"((-1/24*I)*e^2*(3*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] - 3*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b] + Sqrt[3]*(-(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b]) + E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b])))/(d*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
262,1,134,105,0.0755671,"\int \frac{c e+d e x}{\sqrt{a+b \sin ^{-1}(c+d x)}} \, dx","Integrate[(c*e + d*e*x)/Sqrt[a + b*ArcSin[c + d*x]],x]","-\frac{e e^{-\frac{2 i a}{b}} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{4 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{4 \sqrt{2} d \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{\sqrt{\pi } e \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{2 \sqrt{b} d}-\frac{\sqrt{\pi } e \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{2 \sqrt{b} d}",1,"-1/4*(e*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]))/(Sqrt[2]*d*E^(((2*I)*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
263,0,0,105,0.0417049,"\int \frac{1}{\sqrt{a+b \sin ^{-1}(c+d x)}} \, dx","Integrate[1/Sqrt[a + b*ArcSin[c + d*x]],x]","\int \frac{1}{\sqrt{a+b \sin ^{-1}(c+d x)}} \, dx","\frac{\sqrt{2 \pi } \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d}+\frac{\sqrt{2 \pi } \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{\sqrt{b} d}",1,"Integrate[1/Sqrt[a + b*ArcSin[c + d*x]], x]","F",-1
264,0,0,29,0.0841916,"\int \frac{1}{(c e+d e x) \sqrt{a+b \sin ^{-1}(c+d x)}} \, dx","Integrate[1/((c*e + d*e*x)*Sqrt[a + b*ArcSin[c + d*x]]),x]","\int \frac{1}{(c e+d e x) \sqrt{a+b \sin ^{-1}(c+d x)}} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \sqrt{a+b \sin ^{-1}(c+d x)}},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*Sqrt[a + b*ArcSin[c + d*x]]), x]","A",-1
265,1,572,412,0.7437118,"\int \frac{(c e+d e x)^4}{\left(a+b \sin ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[(c*e + d*e*x)^4/(a + b*ArcSin[c + d*x])^(3/2),x]","\frac{e^4 e^{-\frac{5 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(3 e^{\frac{5 i a}{b}+2 i \sin ^{-1}(c+d x)}-2 e^{\frac{5 i a}{b}+4 i \sin ^{-1}(c+d x)}-2 e^{\frac{5 i a}{b}+6 i \sin ^{-1}(c+d x)}+3 e^{\frac{5 i a}{b}+8 i \sin ^{-1}(c+d x)}-e^{\frac{5 i \left(a+2 b \sin ^{-1}(c+d x)\right)}{b}}+2 e^{\frac{4 i a}{b}+5 i \sin ^{-1}(c+d x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+2 e^{\frac{6 i a}{b}+5 i \sin ^{-1}(c+d x)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-3 \sqrt{3} e^{\frac{2 i a}{b}+5 i \sin ^{-1}(c+d x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-3 \sqrt{3} e^{\frac{8 i a}{b}+5 i \sin ^{-1}(c+d x)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+\sqrt{5} e^{5 i \sin ^{-1}(c+d x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+\sqrt{5} e^{\frac{5 i \left(2 a+b \sin ^{-1}(c+d x)\right)}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{5 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-e^{\frac{5 i a}{b}}\right)}{16 b d \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{\sqrt{\frac{\pi }{2}} e^4 \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 b^{3/2} d}-\frac{3 \sqrt{\frac{3 \pi }{2}} e^4 \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 b^{3/2} d}+\frac{\sqrt{\frac{5 \pi }{2}} e^4 \sin \left(\frac{5 a}{b}\right) C\left(\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 b^{3/2} d}-\frac{\sqrt{\frac{\pi }{2}} e^4 \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 b^{3/2} d}+\frac{3 \sqrt{\frac{3 \pi }{2}} e^4 \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 b^{3/2} d}-\frac{\sqrt{\frac{5 \pi }{2}} e^4 \cos \left(\frac{5 a}{b}\right) S\left(\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 b^{3/2} d}-\frac{2 e^4 (c+d x)^4 \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}",1,"(e^4*(-E^(((5*I)*a)/b) + 3*E^(((5*I)*a)/b + (2*I)*ArcSin[c + d*x]) - 2*E^(((5*I)*a)/b + (4*I)*ArcSin[c + d*x]) - 2*E^(((5*I)*a)/b + (6*I)*ArcSin[c + d*x]) + 3*E^(((5*I)*a)/b + (8*I)*ArcSin[c + d*x]) - E^(((5*I)*(a + 2*b*ArcSin[c + d*x]))/b) + 2*E^(((4*I)*a)/b + (5*I)*ArcSin[c + d*x])*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + 2*E^(((6*I)*a)/b + (5*I)*ArcSin[c + d*x])*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b] - 3*Sqrt[3]*E^(((2*I)*a)/b + (5*I)*ArcSin[c + d*x])*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b] - 3*Sqrt[3]*E^(((8*I)*a)/b + (5*I)*ArcSin[c + d*x])*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b] + Sqrt[5]*E^((5*I)*ArcSin[c + d*x])*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-5*I)*(a + b*ArcSin[c + d*x]))/b] + Sqrt[5]*E^(((5*I)*(2*a + b*ArcSin[c + d*x]))/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((5*I)*(a + b*ArcSin[c + d*x]))/b]))/(16*b*d*E^(((5*I)*(a + b*ArcSin[c + d*x]))/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
266,1,300,270,0.3528301,"\int \frac{(c e+d e x)^3}{\left(a+b \sin ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^(3/2),x]","-\frac{i e^3 e^{-\frac{4 i a}{b}} \left(-2 i e^{\frac{4 i a}{b}} \sin \left(2 \sin ^{-1}(c+d x)\right)+i e^{\frac{4 i a}{b}} \sin \left(4 \sin ^{-1}(c+d x)\right)+\sqrt{2} e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-\sqrt{2} e^{\frac{6 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-\sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{4 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{8 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{4 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{4 b d \sqrt{a+b \sin ^{-1}(c+d x)}}","-\frac{\sqrt{\frac{\pi }{2}} e^3 \cos \left(\frac{4 a}{b}\right) C\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}+\frac{\sqrt{\pi } e^3 \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{b^{3/2} d}+\frac{\sqrt{\pi } e^3 \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{b^{3/2} d}-\frac{\sqrt{\frac{\pi }{2}} e^3 \sin \left(\frac{4 a}{b}\right) S\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}-\frac{2 e^3 (c+d x)^3 \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}",1,"((-1/4*I)*e^3*(Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] - Sqrt[2]*E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b] - Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-4*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((4*I)*(a + b*ArcSin[c + d*x]))/b] - (2*I)*E^(((4*I)*a)/b)*Sin[2*ArcSin[c + d*x]] + I*E^(((4*I)*a)/b)*Sin[4*ArcSin[c + d*x]]))/(b*d*E^(((4*I)*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
267,1,380,280,0.4392714,"\int \frac{(c e+d e x)^2}{\left(a+b \sin ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[(c*e + d*e*x)^2/(a + b*ArcSin[c + d*x])^(3/2),x]","\frac{e^2 e^{-\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(-e^{\frac{3 i a}{b}+2 i \sin ^{-1}(c+d x)}-e^{\frac{3 i a}{b}+4 i \sin ^{-1}(c+d x)}+e^{\frac{3 i \left(a+2 b \sin ^{-1}(c+d x)\right)}{b}}+e^{\frac{2 i a}{b}+3 i \sin ^{-1}(c+d x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{4 i a}{b}+3 i \sin ^{-1}(c+d x)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-\sqrt{3} e^{3 i \sin ^{-1}(c+d x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-\sqrt{3} e^{3 i \left(\frac{2 a}{b}+\sin ^{-1}(c+d x)\right)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{3 i a}{b}}\right)}{4 b d \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{\sqrt{\frac{\pi }{2}} e^2 \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}-\frac{\sqrt{\frac{3 \pi }{2}} e^2 \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}-\frac{\sqrt{\frac{\pi }{2}} e^2 \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}+\frac{\sqrt{\frac{3 \pi }{2}} e^2 \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}-\frac{2 e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}",1,"(e^2*(E^(((3*I)*a)/b) - E^(((3*I)*a)/b + (2*I)*ArcSin[c + d*x]) - E^(((3*I)*a)/b + (4*I)*ArcSin[c + d*x]) + E^(((3*I)*(a + 2*b*ArcSin[c + d*x]))/b) + E^(((2*I)*a)/b + (3*I)*ArcSin[c + d*x])*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + E^(((4*I)*a)/b + (3*I)*ArcSin[c + d*x])*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b] - Sqrt[3]*E^((3*I)*ArcSin[c + d*x])*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b] - Sqrt[3]*E^((3*I)*((2*a)/b + ArcSin[c + d*x]))*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b]))/(4*b*d*E^(((3*I)*(a + b*ArcSin[c + d*x]))/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
268,1,168,144,0.1623722,"\int \frac{c e+d e x}{\left(a+b \sin ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[(c*e + d*e*x)/(a + b*ArcSin[c + d*x])^(3/2),x]","\frac{i e e^{-\frac{2 i a}{b}} \left(2 i e^{\frac{2 i a}{b}} \sin \left(2 \sin ^{-1}(c+d x)\right)-\sqrt{2} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+\sqrt{2} e^{\frac{4 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)}{2 b d \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{2 \sqrt{\pi } e \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{b^{3/2} d}+\frac{2 \sqrt{\pi } e \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{b^{3/2} d}-\frac{2 e \sqrt{1-(c+d x)^2} (c+d x)}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}",1,"((I/2)*e*(-(Sqrt[2]*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b]) + Sqrt[2]*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b] + (2*I)*E^(((2*I)*a)/b)*Sin[2*ArcSin[c + d*x]]))/(b*d*E^(((2*I)*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
269,1,185,144,0.1855301,"\int \frac{1}{\left(a+b \sin ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(-3/2),x]","\frac{e^{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(e^{i \sin ^{-1}(c+d x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{i a}{b}} \left(e^{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-e^{2 i \sin ^{-1}(c+d x)}-1\right)\right)}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}","\frac{2 \sqrt{2 \pi } \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}-\frac{2 \sqrt{2 \pi } \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}-\frac{2 \sqrt{1-(c+d x)^2}}{b d \sqrt{a+b \sin ^{-1}(c+d x)}}",1,"(E^(I*ArcSin[c + d*x])*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] + E^((I*a)/b)*(-1 - E^((2*I)*ArcSin[c + d*x]) + E^((I*(a + b*ArcSin[c + d*x]))/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(b*d*E^((I*(a + b*ArcSin[c + d*x]))/b)*Sqrt[a + b*ArcSin[c + d*x]])","C",0
270,0,0,29,0.0907752,"\int \frac{1}{(c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^(3/2)),x]","\int \frac{1}{(c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^(3/2)), x]","A",-1
271,1,351,344,2.3396977,"\int \frac{(c e+d e x)^3}{\left(a+b \sin ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^(5/2),x]","\frac{e^3 \left(-4 \left(a+b \sin ^{-1}(c+d x)\right) \left(-\sqrt{2} e^{-\frac{2 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-\sqrt{2} e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{-2 i \sin ^{-1}(c+d x)}+e^{2 i \sin ^{-1}(c+d x)}\right)+4 \left(a+b \sin ^{-1}(c+d x)\right) \left(-2 e^{-\frac{4 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{4 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-2 e^{\frac{4 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{4 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{-4 i \sin ^{-1}(c+d x)}+e^{4 i \sin ^{-1}(c+d x)}\right)-2 b \sin \left(2 \sin ^{-1}(c+d x)\right)+b \sin \left(4 \sin ^{-1}(c+d x)\right)\right)}{12 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}","\frac{4 \sqrt{\pi } e^3 \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{3 b^{5/2} d}-\frac{4 \sqrt{2 \pi } e^3 \sin \left(\frac{4 a}{b}\right) C\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}+\frac{4 \sqrt{2 \pi } e^3 \cos \left(\frac{4 a}{b}\right) S\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}-\frac{4 \sqrt{\pi } e^3 \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{3 b^{5/2} d}+\frac{16 e^3 (c+d x)^4}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{4 e^3 (c+d x)^2}{b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{2 e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{3 b d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}",1,"(e^3*(-4*(a + b*ArcSin[c + d*x])*(E^((-2*I)*ArcSin[c + d*x]) + E^((2*I)*ArcSin[c + d*x]) - (Sqrt[2]*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((2*I)*a)/b) - Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]) + 4*(a + b*ArcSin[c + d*x])*(E^((-4*I)*ArcSin[c + d*x]) + E^((4*I)*ArcSin[c + d*x]) - (2*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-4*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((4*I)*a)/b) - 2*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((4*I)*(a + b*ArcSin[c + d*x]))/b]) - 2*b*Sin[2*ArcSin[c + d*x]] + b*Sin[4*ArcSin[c + d*x]]))/(12*b^2*d*(a + b*ArcSin[c + d*x])^(3/2))","C",0
272,1,411,342,1.969204,"\int \frac{(c e+d e x)^2}{\left(a+b \sin ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[(c*e + d*e*x)^2/(a + b*ArcSin[c + d*x])^(5/2),x]","\frac{e^2 \left(e^{3 i \sin ^{-1}(c+d x)} \left(6 i a+6 i b \sin ^{-1}(c+d x)+b\right)-i e^{i \sin ^{-1}(c+d x)} \left(2 a+2 b \sin ^{-1}(c+d x)-i b\right)-2 b e^{-\frac{i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+6 \sqrt{3} b e^{-\frac{3 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+i e^{-i \sin ^{-1}(c+d x)} \left(2 i b e^{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+2 a+2 b \sin ^{-1}(c+d x)+i b\right)+6 \sqrt{3} b e^{\frac{3 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-6 i a e^{-3 i \sin ^{-1}(c+d x)}+b e^{-3 i \sin ^{-1}(c+d x)} \left(1-6 i \sin ^{-1}(c+d x)\right)\right)}{12 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}","-\frac{\sqrt{2 \pi } e^2 \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}+\frac{\sqrt{6 \pi } e^2 \cos \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{5/2} d}-\frac{\sqrt{2 \pi } e^2 \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}+\frac{\sqrt{6 \pi } e^2 \sin \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{5/2} d}+\frac{4 e^2 (c+d x)^3}{b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{8 e^2 (c+d x)}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{2 e^2 \sqrt{1-(c+d x)^2} (c+d x)^2}{3 b d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}",1,"(e^2*(((-6*I)*a)/E^((3*I)*ArcSin[c + d*x]) + (b*(1 - (6*I)*ArcSin[c + d*x]))/E^((3*I)*ArcSin[c + d*x]) + E^((3*I)*ArcSin[c + d*x])*((6*I)*a + b + (6*I)*b*ArcSin[c + d*x]) - I*E^(I*ArcSin[c + d*x])*(2*a - I*b + 2*b*ArcSin[c + d*x]) - (2*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b])/E^((I*a)/b) + (I*(2*a + I*b + 2*b*ArcSin[c + d*x] + (2*I)*b*E^((I*(a + b*ArcSin[c + d*x]))/b)*((I*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b]))/E^(I*ArcSin[c + d*x]) + (6*Sqrt[3]*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((3*I)*a)/b) + 6*Sqrt[3]*b*E^(((3*I)*a)/b)*((I*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b]))/(12*b^2*d*(a + b*ArcSin[c + d*x])^(3/2))","C",0
273,1,192,207,1.2901816,"\int \frac{c e+d e x}{\left(a+b \sin ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[(c*e + d*e*x)/(a + b*ArcSin[c + d*x])^(5/2),x]","-\frac{e \left(b \sin \left(2 \sin ^{-1}(c+d x)\right)+2 \left(a+b \sin ^{-1}(c+d x)\right) \left(-\sqrt{2} e^{-\frac{2 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-\sqrt{2} e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{-2 i \sin ^{-1}(c+d x)}+e^{2 i \sin ^{-1}(c+d x)}\right)\right)}{3 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}","\frac{8 \sqrt{\pi } e \sin \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{3 b^{5/2} d}-\frac{8 \sqrt{\pi } e \cos \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{3 b^{5/2} d}+\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{2 e \sqrt{1-(c+d x)^2} (c+d x)}{3 b d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}",1,"-1/3*(e*(2*(a + b*ArcSin[c + d*x])*(E^((-2*I)*ArcSin[c + d*x]) + E^((2*I)*ArcSin[c + d*x]) - (Sqrt[2]*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((2*I)*a)/b) - Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]) + b*Sin[2*ArcSin[c + d*x]]))/(b^2*d*(a + b*ArcSin[c + d*x])^(3/2))","C",0
274,1,238,179,0.6008975,"\int \frac{1}{\left(a+b \sin ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(-5/2),x]","\frac{e^{-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(-2 b e^{i \sin ^{-1}(c+d x)} \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-i e^{\frac{i a}{b}} \left(-2 i b e^{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+2 a \left(-1+e^{2 i \sin ^{-1}(c+d x)}\right)+b \left(-2 \sin ^{-1}(c+d x)+e^{2 i \sin ^{-1}(c+d x)} \left(2 \sin ^{-1}(c+d x)-i\right)-i\right)\right)\right)}{3 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}","-\frac{4 \sqrt{2 \pi } \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}-\frac{4 \sqrt{2 \pi } \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}+\frac{4 (c+d x)}{3 b^2 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{2 \sqrt{1-(c+d x)^2}}{3 b d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}",1,"(-2*b*E^(I*ArcSin[c + d*x])*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b] - I*E^((I*a)/b)*(2*a*(-1 + E^((2*I)*ArcSin[c + d*x])) + b*(-I - 2*ArcSin[c + d*x] + E^((2*I)*ArcSin[c + d*x])*(-I + 2*ArcSin[c + d*x])) - (2*I)*b*E^((I*(a + b*ArcSin[c + d*x]))/b)*((I*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(3*b^2*d*E^((I*(a + b*ArcSin[c + d*x]))/b)*(a + b*ArcSin[c + d*x])^(3/2))","C",0
275,0,0,29,0.0919209,"\int \frac{1}{(c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^(5/2)),x]","\int \frac{1}{(c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^(5/2)), x]","A",-1
276,1,445,442,2.6903464,"\int \frac{(c e+d e x)^3}{\left(a+b \sin ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[(c*e + d*e*x)^3/(a + b*ArcSin[c + d*x])^(7/2),x]","\frac{e^3 \left(-4 \left(a+b \sin ^{-1}(c+d x)\right) \left(e^{2 i \sin ^{-1}(c+d x)} \left(4 i a+4 i b \sin ^{-1}(c+d x)+b\right)+4 \sqrt{2} b e^{-\frac{2 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+e^{-2 i \sin ^{-1}(c+d x)} \left(4 \sqrt{2} b e^{\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-4 i a-4 i b \sin ^{-1}(c+d x)+b\right)\right)+4 \left(a+b \sin ^{-1}(c+d x)\right) \left(e^{-4 i \sin ^{-1}(c+d x)} \left(-8 i a-8 i b \sin ^{-1}(c+d x)+b\right)+e^{4 i \sin ^{-1}(c+d x)} \left(8 i a+8 i b \sin ^{-1}(c+d x)+b\right)+16 b e^{-\frac{4 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{4 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+16 b e^{\frac{4 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},\frac{4 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)-6 b^2 \sin \left(2 \sin ^{-1}(c+d x)\right)+3 b^2 \sin \left(4 \sin ^{-1}(c+d x)\right)\right)}{60 b^3 d \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}","\frac{32 \sqrt{2 \pi } e^3 \cos \left(\frac{4 a}{b}\right) C\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}-\frac{16 \sqrt{\pi } e^3 \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{15 b^{7/2} d}-\frac{16 \sqrt{\pi } e^3 \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{15 b^{7/2} d}+\frac{32 \sqrt{2 \pi } e^3 \sin \left(\frac{4 a}{b}\right) S\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}+\frac{128 e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{15 b^3 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{16 e^3 \sqrt{1-(c+d x)^2} (c+d x)}{5 b^3 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{16 e^3 (c+d x)^4}{15 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}-\frac{4 e^3 (c+d x)^2}{5 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}-\frac{2 e^3 \sqrt{1-(c+d x)^2} (c+d x)^3}{5 b d \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}",1,"(e^3*(-4*(a + b*ArcSin[c + d*x])*(E^((2*I)*ArcSin[c + d*x])*((4*I)*a + b + (4*I)*b*ArcSin[c + d*x]) + (4*Sqrt[2]*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((2*I)*a)/b) + ((-4*I)*a + b - (4*I)*b*ArcSin[c + d*x] + 4*Sqrt[2]*b*E^(((2*I)*(a + b*ArcSin[c + d*x]))/b)*((I*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b])/E^((2*I)*ArcSin[c + d*x])) + 4*(a + b*ArcSin[c + d*x])*(((-8*I)*a + b - (8*I)*b*ArcSin[c + d*x])/E^((4*I)*ArcSin[c + d*x]) + E^((4*I)*ArcSin[c + d*x])*((8*I)*a + b + (8*I)*b*ArcSin[c + d*x]) + (16*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-4*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((4*I)*a)/b) + 16*b*E^(((4*I)*a)/b)*((I*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((4*I)*(a + b*ArcSin[c + d*x]))/b]) - 6*b^2*Sin[2*ArcSin[c + d*x]] + 3*b^2*Sin[4*ArcSin[c + d*x]]))/(60*b^3*d*(a + b*ArcSin[c + d*x])^(5/2))","C",0
277,1,538,441,1.991064,"\int \frac{(c e+d e x)^2}{\left(a+b \sin ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[(c*e + d*e*x)^2/(a + b*ArcSin[c + d*x])^(7/2),x]","\frac{e^2 \left(e^{-i \sin ^{-1}(c+d x)} \left(4 a^2+2 a b \left(4 \sin ^{-1}(c+d x)+i\right)-4 e^{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^2 \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+b^2 \left(4 \sin ^{-1}(c+d x)^2+2 i \sin ^{-1}(c+d x)-3\right)\right)+3 e^{-3 i \sin ^{-1}(c+d x)} \left(b^2-2 \left(a+b \sin ^{-1}(c+d x)\right) \left(6 i \sqrt{3} b e^{\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+6 a+6 b \sin ^{-1}(c+d x)+i b\right)\right)+2 e^{-\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right) \left(e^{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(2 a+2 b \sin ^{-1}(c+d x)-i b\right)-2 i b \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)-6 e^{-\frac{3 i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right) \left(e^{\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(6 a+6 b \sin ^{-1}(c+d x)-i b\right)-6 i \sqrt{3} b \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)-3 b^2 e^{i \sin ^{-1}(c+d x)}+3 b^2 e^{3 i \sin ^{-1}(c+d x)}\right)}{60 b^3 d \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}","-\frac{2 \sqrt{2 \pi } e^2 \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}+\frac{6 \sqrt{6 \pi } e^2 \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{5 b^{7/2} d}+\frac{2 \sqrt{2 \pi } e^2 \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}-\frac{6 \sqrt{6 \pi } e^2 \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{5 b^{7/2} d}+\frac{24 e^2 \sqrt{1-(c+d x)^2} (c+d x)^2}{5 b^3 d \sqrt{a+b \sin ^{-1}(c+d x)}}-\frac{16 e^2 \sqrt{1-(c+d x)^2}}{15 b^3 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{4 e^2 (c+d x)^3}{5 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}-\frac{8 e^2 (c+d x)}{15 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}-\frac{2 e^2 \sqrt{1-(c+d x)^2} (c+d x)^2}{5 b d \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}",1,"(e^2*(-3*b^2*E^(I*ArcSin[c + d*x]) + 3*b^2*E^((3*I)*ArcSin[c + d*x]) + (2*(a + b*ArcSin[c + d*x])*(E^((I*(a + b*ArcSin[c + d*x]))/b)*(2*a - I*b + 2*b*ArcSin[c + d*x]) - (2*I)*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b]))/E^((I*a)/b) + (4*a^2 + 2*a*b*(I + 4*ArcSin[c + d*x]) + b^2*(-3 + (2*I)*ArcSin[c + d*x] + 4*ArcSin[c + d*x]^2) - 4*E^((I*(a + b*ArcSin[c + d*x]))/b)*(a + b*ArcSin[c + d*x])^2*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b])/E^(I*ArcSin[c + d*x]) - (6*(a + b*ArcSin[c + d*x])*(E^(((3*I)*(a + b*ArcSin[c + d*x]))/b)*(6*a - I*b + 6*b*ArcSin[c + d*x]) - (6*I)*Sqrt[3]*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b]))/E^(((3*I)*a)/b) + (3*(b^2 - 2*(a + b*ArcSin[c + d*x])*(6*a + I*b + 6*b*ArcSin[c + d*x] + (6*I)*Sqrt[3]*b*E^(((3*I)*(a + b*ArcSin[c + d*x]))/b)*((I*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b])))/E^((3*I)*ArcSin[c + d*x])))/(60*b^3*d*(a + b*ArcSin[c + d*x])^(5/2))","C",0
278,1,254,252,1.0500894,"\int \frac{c e+d e x}{\left(a+b \sin ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[(c*e + d*e*x)/(a + b*ArcSin[c + d*x])^(7/2),x]","-\frac{e \left(3 b^2 \sin \left(2 \sin ^{-1}(c+d x)\right)+\left(a+b \sin ^{-1}(c+d x)\right) \left(e^{-\frac{2 i a}{b}} \left(2 e^{\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(4 i a+4 i b \sin ^{-1}(c+d x)+b\right)+8 \sqrt{2} b \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)+2 e^{-2 i \sin ^{-1}(c+d x)} \left(4 \sqrt{2} b e^{\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},\frac{2 i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)-4 i a-4 i b \sin ^{-1}(c+d x)+b\right)\right)\right)}{15 b^3 d \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}","-\frac{32 \sqrt{\pi } e \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{15 b^{7/2} d}-\frac{32 \sqrt{\pi } e \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right)}{15 b^{7/2} d}+\frac{32 e \sqrt{1-(c+d x)^2} (c+d x)}{15 b^3 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{8 e (c+d x)^2}{15 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}-\frac{4 e}{15 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}-\frac{2 e \sqrt{1-(c+d x)^2} (c+d x)}{5 b d \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}",1,"-1/15*(e*((a + b*ArcSin[c + d*x])*((2*E^(((2*I)*(a + b*ArcSin[c + d*x]))/b)*((4*I)*a + b + (4*I)*b*ArcSin[c + d*x]) + 8*Sqrt[2]*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((2*I)*a)/b) + (2*((-4*I)*a + b - (4*I)*b*ArcSin[c + d*x] + 4*Sqrt[2]*b*E^(((2*I)*(a + b*ArcSin[c + d*x]))/b)*((I*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]))/E^((2*I)*ArcSin[c + d*x])) + 3*b^2*Sin[2*ArcSin[c + d*x]]))/(b^3*d*(a + b*ArcSin[c + d*x])^(5/2))","C",0
279,1,287,218,0.2442135,"\int \frac{1}{\left(a+b \sin ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^(-7/2),x]","\frac{e^{-i \sin ^{-1}(c+d x)} \left(8 a^2+4 a b \left(4 \sin ^{-1}(c+d x)+i\right)-8 e^{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(a+b \sin ^{-1}(c+d x)\right)^2 \sqrt{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)+2 b^2 \left(4 \sin ^{-1}(c+d x)^2+2 i \sin ^{-1}(c+d x)-3\right)\right)+4 e^{-\frac{i a}{b}} \left(a+b \sin ^{-1}(c+d x)\right) \left(e^{\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}} \left(2 a+b \left(2 \sin ^{-1}(c+d x)-i\right)\right)-2 i b \left(-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c+d x)\right)}{b}\right)\right)-6 b^2 e^{i \sin ^{-1}(c+d x)}}{30 b^3 d \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}","-\frac{8 \sqrt{2 \pi } \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}+\frac{8 \sqrt{2 \pi } \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}+\frac{8 \sqrt{1-(c+d x)^2}}{15 b^3 d \sqrt{a+b \sin ^{-1}(c+d x)}}+\frac{4 (c+d x)}{15 b^2 d \left(a+b \sin ^{-1}(c+d x)\right)^{3/2}}-\frac{2 \sqrt{1-(c+d x)^2}}{5 b d \left(a+b \sin ^{-1}(c+d x)\right)^{5/2}}",1,"(-6*b^2*E^(I*ArcSin[c + d*x]) + (4*(a + b*ArcSin[c + d*x])*(E^((I*(a + b*ArcSin[c + d*x]))/b)*(2*a + b*(-I + 2*ArcSin[c + d*x])) - (2*I)*b*(((-I)*(a + b*ArcSin[c + d*x]))/b)^(3/2)*Gamma[1/2, ((-I)*(a + b*ArcSin[c + d*x]))/b]))/E^((I*a)/b) + (8*a^2 + 4*a*b*(I + 4*ArcSin[c + d*x]) + 2*b^2*(-3 + (2*I)*ArcSin[c + d*x] + 4*ArcSin[c + d*x]^2) - 8*E^((I*(a + b*ArcSin[c + d*x]))/b)*(a + b*ArcSin[c + d*x])^2*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c + d*x]))/b])/E^(I*ArcSin[c + d*x]))/(30*b^3*d*(a + b*ArcSin[c + d*x])^(5/2))","C",0
280,0,0,29,0.09853,"\int \frac{1}{(c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^(7/2)),x]","\int \frac{1}{(c e+d e x) \left(a+b \sin ^{-1}(c+d x)\right)^{7/2}} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \left(a+b \sin ^{-1}(c+d x)\right)^{7/2}},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*(a + b*ArcSin[c + d*x])^(7/2)), x]","A",-1
281,1,115,156,0.2612573,"\int (c e+d e x)^{7/2} \left(a+b \sin ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSin[c + d*x]),x]","\frac{2 (e (c+d x))^{7/2} \left(45 a (c+d x)^3-14 b \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};(c+d x)^2\right)+10 b \sqrt{1-(c+d x)^2} (c+d x)^2+14 b \sqrt{1-(c+d x)^2}+45 b (c+d x)^3 \sin ^{-1}(c+d x)\right)}{405 d (c+d x)^2}","\frac{2 (e (c+d x))^{9/2} \left(a+b \sin ^{-1}(c+d x)\right)}{9 d e}+\frac{28 b e^3 \sqrt{e (c+d x)} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{-c-d x+1}}{\sqrt{2}}\right)\right|2\right)}{135 d \sqrt{c+d x}}+\frac{28 b e^2 \sqrt{1-(c+d x)^2} (e (c+d x))^{3/2}}{405 d}+\frac{4 b \sqrt{1-(c+d x)^2} (e (c+d x))^{7/2}}{81 d}",1,"(2*(e*(c + d*x))^(7/2)*(45*a*(c + d*x)^3 + 14*b*Sqrt[1 - (c + d*x)^2] + 10*b*(c + d*x)^2*Sqrt[1 - (c + d*x)^2] + 45*b*(c + d*x)^3*ArcSin[c + d*x] - 14*b*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2]))/(405*d*(c + d*x)^2)","C",1
282,1,115,136,0.200452,"\int (c e+d e x)^{5/2} \left(a+b \sin ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcSin[c + d*x]),x]","\frac{2 (e (c+d x))^{5/2} \left(21 a (c+d x)^3-10 b \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};(c+d x)^2\right)+6 b \sqrt{1-(c+d x)^2} (c+d x)^2+10 b \sqrt{1-(c+d x)^2}+21 b (c+d x)^3 \sin ^{-1}(c+d x)\right)}{147 d (c+d x)^2}","\frac{2 (e (c+d x))^{7/2} \left(a+b \sin ^{-1}(c+d x)\right)}{7 d e}-\frac{20 b e^{5/2} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)\right|-1\right)}{147 d}+\frac{20 b e^2 \sqrt{1-(c+d x)^2} \sqrt{e (c+d x)}}{147 d}+\frac{4 b \sqrt{1-(c+d x)^2} (e (c+d x))^{5/2}}{49 d}",1,"(2*(e*(c + d*x))^(5/2)*(21*a*(c + d*x)^3 + 10*b*Sqrt[1 - (c + d*x)^2] + 6*b*(c + d*x)^2*Sqrt[1 - (c + d*x)^2] + 21*b*(c + d*x)^3*ArcSin[c + d*x] - 10*b*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]))/(147*d*(c + d*x)^2)","C",1
283,1,87,117,0.0557542,"\int (c e+d e x)^{3/2} \left(a+b \sin ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^(3/2)*(a + b*ArcSin[c + d*x]),x]","\frac{2 (e (c+d x))^{3/2} \left(5 a c+5 a d x-2 b \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};(c+d x)^2\right)+2 b \sqrt{1-(c+d x)^2}+5 b c \sin ^{-1}(c+d x)+5 b d x \sin ^{-1}(c+d x)\right)}{25 d}","\frac{2 (e (c+d x))^{5/2} \left(a+b \sin ^{-1}(c+d x)\right)}{5 d e}+\frac{4 b \sqrt{1-(c+d x)^2} (e (c+d x))^{3/2}}{25 d}+\frac{12 b e \sqrt{e (c+d x)} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{-c-d x+1}}{\sqrt{2}}\right)\right|2\right)}{25 d \sqrt{c+d x}}",1,"(2*(e*(c + d*x))^(3/2)*(5*a*c + 5*a*d*x + 2*b*Sqrt[1 - (c + d*x)^2] + 5*b*c*ArcSin[c + d*x] + 5*b*d*x*ArcSin[c + d*x] - 2*b*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2]))/(25*d)","C",1
284,1,87,99,0.0379153,"\int \sqrt{c e+d e x} \left(a+b \sin ^{-1}(c+d x)\right) \, dx","Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcSin[c + d*x]),x]","\frac{2 \sqrt{e (c+d x)} \left(3 a c+3 a d x-2 b \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};(c+d x)^2\right)+2 b \sqrt{1-(c+d x)^2}+3 b c \sin ^{-1}(c+d x)+3 b d x \sin ^{-1}(c+d x)\right)}{9 d}","\frac{2 (e (c+d x))^{3/2} \left(a+b \sin ^{-1}(c+d x)\right)}{3 d e}+\frac{4 b \sqrt{1-(c+d x)^2} \sqrt{e (c+d x)}}{9 d}-\frac{4 b \sqrt{e} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)\right|-1\right)}{9 d}",1,"(2*Sqrt[e*(c + d*x)]*(3*a*c + 3*a*d*x + 2*b*Sqrt[1 - (c + d*x)^2] + 3*b*c*ArcSin[c + d*x] + 3*b*d*x*ArcSin[c + d*x] - 2*b*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]))/(9*d)","C",1
285,1,59,81,0.034715,"\int \frac{a+b \sin ^{-1}(c+d x)}{\sqrt{c e+d e x}} \, dx","Integrate[(a + b*ArcSin[c + d*x])/Sqrt[c*e + d*e*x],x]","-\frac{2 \sqrt{e (c+d x)} \left(2 b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};(c+d x)^2\right)-3 \left(a+b \sin ^{-1}(c+d x)\right)\right)}{3 d e}","\frac{2 \sqrt{e (c+d x)} \left(a+b \sin ^{-1}(c+d x)\right)}{d e}+\frac{4 b \sqrt{e (c+d x)} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{-c-d x+1}}{\sqrt{2}}\right)\right|2\right)}{d e \sqrt{c+d x}}",1,"(-2*Sqrt[e*(c + d*x)]*(-3*(a + b*ArcSin[c + d*x]) + 2*b*(c + d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2]))/(3*d*e)","C",1
286,1,54,61,0.0268308,"\int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(3/2),x]","-\frac{2 \left(a-2 b (c+d x) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};(c+d x)^2\right)+b \sin ^{-1}(c+d x)\right)}{d e \sqrt{e (c+d x)}}","\frac{4 b F\left(\left.\sin ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)\right|-1\right)}{d e^{3/2}}-\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{d e \sqrt{e (c+d x)}}",1,"(-2*(a + b*ArcSin[c + d*x] - 2*b*(c + d*x)*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]))/(d*e*Sqrt[e*(c + d*x)])","C",1
287,1,56,122,0.0381528,"\int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(5/2),x]","-\frac{2 \left(a+2 b (c+d x) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};(c+d x)^2\right)+b \sin ^{-1}(c+d x)\right)}{3 d e (e (c+d x))^{3/2}}","-\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{3 d e (e (c+d x))^{3/2}}+\frac{4 b \sqrt{e (c+d x)} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{-c-d x+1}}{\sqrt{2}}\right)\right|2\right)}{3 d e^3 \sqrt{c+d x}}-\frac{4 b \sqrt{1-(c+d x)^2}}{3 d e^2 \sqrt{e (c+d x)}}",1,"(-2*(a + b*ArcSin[c + d*x] + 2*b*(c + d*x)*Hypergeometric2F1[-1/4, 1/2, 3/4, (c + d*x)^2]))/(3*d*e*(e*(c + d*x))^(3/2))","C",1
288,1,59,102,0.0440564,"\int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^{7/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(7/2),x]","\frac{-6 \left(a+b \sin ^{-1}(c+d x)\right)-4 b (c+d x) \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};(c+d x)^2\right)}{15 d e (e (c+d x))^{5/2}}","-\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{5 d e (e (c+d x))^{5/2}}+\frac{4 b F\left(\left.\sin ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)\right|-1\right)}{15 d e^{7/2}}-\frac{4 b \sqrt{1-(c+d x)^2}}{15 d e^2 (e (c+d x))^{3/2}}",1,"(-6*(a + b*ArcSin[c + d*x]) - 4*b*(c + d*x)*Hypergeometric2F1[-3/4, 1/2, 1/4, (c + d*x)^2])/(15*d*e*(e*(c + d*x))^(5/2))","C",1
289,1,66,159,0.0534882,"\int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^{9/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(9/2),x]","-\frac{2 \sqrt{e (c+d x)} \left(5 \left(a+b \sin ^{-1}(c+d x)\right)+2 b (c+d x) \, _2F_1\left(-\frac{5}{4},\frac{1}{2};-\frac{1}{4};(c+d x)^2\right)\right)}{35 d e^5 (c+d x)^4}","-\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{7 d e (e (c+d x))^{7/2}}+\frac{12 b \sqrt{e (c+d x)} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{-c-d x+1}}{\sqrt{2}}\right)\right|2\right)}{35 d e^5 \sqrt{c+d x}}-\frac{12 b \sqrt{1-(c+d x)^2}}{35 d e^4 \sqrt{e (c+d x)}}-\frac{4 b \sqrt{1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}",1,"(-2*Sqrt[e*(c + d*x)]*(5*(a + b*ArcSin[c + d*x]) + 2*b*(c + d*x)*Hypergeometric2F1[-5/4, 1/2, -1/4, (c + d*x)^2]))/(35*d*e^5*(c + d*x)^4)","C",1
290,1,66,139,0.0592183,"\int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^{11/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])/(c*e + d*e*x)^(11/2),x]","-\frac{2 \sqrt{e (c+d x)} \left(7 \left(a+b \sin ^{-1}(c+d x)\right)+2 b (c+d x) \, _2F_1\left(-\frac{7}{4},\frac{1}{2};-\frac{3}{4};(c+d x)^2\right)\right)}{63 d e^6 (c+d x)^5}","-\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)}{9 d e (e (c+d x))^{9/2}}+\frac{20 b F\left(\left.\sin ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)\right|-1\right)}{189 d e^{11/2}}-\frac{20 b \sqrt{1-(c+d x)^2}}{189 d e^4 (e (c+d x))^{3/2}}-\frac{4 b \sqrt{1-(c+d x)^2}}{63 d e^2 (e (c+d x))^{7/2}}",1,"(-2*Sqrt[e*(c + d*x)]*(7*(a + b*ArcSin[c + d*x]) + 2*b*(c + d*x)*Hypergeometric2F1[-7/4, 1/2, -3/4, (c + d*x)^2]))/(63*d*e^6*(c + d*x)^5)","C",1
291,1,114,130,0.128735,"\int (c e+d e x)^{7/2} \left(a+b \sin ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSin[c + d*x])^2,x]","\frac{2 e^3 (c+d x)^4 \sqrt{e (c+d x)} \left(8 b^2 (c+d x)^2 \, _3F_2\left(1,\frac{13}{4},\frac{13}{4};\frac{15}{4},\frac{17}{4};(c+d x)^2\right)+13 \left(a+b \sin ^{-1}(c+d x)\right) \left(11 \left(a+b \sin ^{-1}(c+d x)\right)-4 b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{11}{4};\frac{15}{4};(c+d x)^2\right)\right)\right)}{1287 d}","\frac{16 b^2 (e (c+d x))^{13/2} \, _3F_2\left(1,\frac{13}{4},\frac{13}{4};\frac{15}{4},\frac{17}{4};(c+d x)^2\right)}{1287 d e^3}-\frac{8 b (e (c+d x))^{11/2} \, _2F_1\left(\frac{1}{2},\frac{11}{4};\frac{15}{4};(c+d x)^2\right) \left(a+b \sin ^{-1}(c+d x)\right)}{99 d e^2}+\frac{2 (e (c+d x))^{9/2} \left(a+b \sin ^{-1}(c+d x)\right)^2}{9 d e}",1,"(2*e^3*(c + d*x)^4*Sqrt[e*(c + d*x)]*(13*(a + b*ArcSin[c + d*x])*(11*(a + b*ArcSin[c + d*x]) - 4*b*(c + d*x)*Hypergeometric2F1[1/2, 11/4, 15/4, (c + d*x)^2]) + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 13/4, 13/4}, {15/4, 17/4}, (c + d*x)^2]))/(1287*d)","A",1
292,1,106,130,0.1452293,"\int (c e+d e x)^{5/2} \left(a+b \sin ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcSin[c + d*x])^2,x]","\frac{2 (e (c+d x))^{7/2} \left(8 b^2 (c+d x)^2 \, _3F_2\left(1,\frac{11}{4},\frac{11}{4};\frac{13}{4},\frac{15}{4};(c+d x)^2\right)-44 b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{9}{4};\frac{13}{4};(c+d x)^2\right) \left(a+b \sin ^{-1}(c+d x)\right)+99 \left(a+b \sin ^{-1}(c+d x)\right)^2\right)}{693 d e}","\frac{16 b^2 (e (c+d x))^{11/2} \, _3F_2\left(1,\frac{11}{4},\frac{11}{4};\frac{13}{4},\frac{15}{4};(c+d x)^2\right)}{693 d e^3}-\frac{8 b (e (c+d x))^{9/2} \, _2F_1\left(\frac{1}{2},\frac{9}{4};\frac{13}{4};(c+d x)^2\right) \left(a+b \sin ^{-1}(c+d x)\right)}{63 d e^2}+\frac{2 (e (c+d x))^{7/2} \left(a+b \sin ^{-1}(c+d x)\right)^2}{7 d e}",1,"(2*(e*(c + d*x))^(7/2)*(99*(a + b*ArcSin[c + d*x])^2 - 44*b*(c + d*x)*(a + b*ArcSin[c + d*x])*Hypergeometric2F1[1/2, 9/4, 13/4, (c + d*x)^2] + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 11/4, 11/4}, {13/4, 15/4}, (c + d*x)^2]))/(693*d*e)","A",1
293,1,107,130,0.1232454,"\int (c e+d e x)^{3/2} \left(a+b \sin ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^(3/2)*(a + b*ArcSin[c + d*x])^2,x]","\frac{2 (e (c+d x))^{5/2} \left(8 b^2 (c+d x)^2 \, _3F_2\left(1,\frac{9}{4},\frac{9}{4};\frac{11}{4},\frac{13}{4};(c+d x)^2\right)+9 \left(a+b \sin ^{-1}(c+d x)\right) \left(7 \left(a+b \sin ^{-1}(c+d x)\right)-4 b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{4};\frac{11}{4};(c+d x)^2\right)\right)\right)}{315 d e}","\frac{16 b^2 (e (c+d x))^{9/2} \, _3F_2\left(1,\frac{9}{4},\frac{9}{4};\frac{11}{4},\frac{13}{4};(c+d x)^2\right)}{315 d e^3}-\frac{8 b (e (c+d x))^{7/2} \, _2F_1\left(\frac{1}{2},\frac{7}{4};\frac{11}{4};(c+d x)^2\right) \left(a+b \sin ^{-1}(c+d x)\right)}{35 d e^2}+\frac{2 (e (c+d x))^{5/2} \left(a+b \sin ^{-1}(c+d x)\right)^2}{5 d e}",1,"(2*(e*(c + d*x))^(5/2)*(9*(a + b*ArcSin[c + d*x])*(7*(a + b*ArcSin[c + d*x]) - 4*b*(c + d*x)*Hypergeometric2F1[1/2, 7/4, 11/4, (c + d*x)^2]) + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 9/4, 9/4}, {11/4, 13/4}, (c + d*x)^2]))/(315*d*e)","A",1
294,1,107,130,0.1000221,"\int \sqrt{c e+d e x} \left(a+b \sin ^{-1}(c+d x)\right)^2 \, dx","Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcSin[c + d*x])^2,x]","\frac{2 (e (c+d x))^{3/2} \left(8 b^2 (c+d x)^2 \, _3F_2\left(1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};(c+d x)^2\right)+7 \left(a+b \sin ^{-1}(c+d x)\right) \left(5 \left(a+b \sin ^{-1}(c+d x)\right)-4 b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{4};\frac{9}{4};(c+d x)^2\right)\right)\right)}{105 d e}","\frac{16 b^2 (e (c+d x))^{7/2} \, _3F_2\left(1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};(c+d x)^2\right)}{105 d e^3}-\frac{8 b (e (c+d x))^{5/2} \, _2F_1\left(\frac{1}{2},\frac{5}{4};\frac{9}{4};(c+d x)^2\right) \left(a+b \sin ^{-1}(c+d x)\right)}{15 d e^2}+\frac{2 (e (c+d x))^{3/2} \left(a+b \sin ^{-1}(c+d x)\right)^2}{3 d e}",1,"(2*(e*(c + d*x))^(3/2)*(7*(a + b*ArcSin[c + d*x])*(5*(a + b*ArcSin[c + d*x]) - 4*b*(c + d*x)*Hypergeometric2F1[1/2, 5/4, 9/4, (c + d*x)^2]) + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, (c + d*x)^2]))/(105*d*e)","A",1
295,1,107,128,0.094705,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{\sqrt{c e+d e x}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^2/Sqrt[c*e + d*e*x],x]","\frac{2 \sqrt{e (c+d x)} \left(8 b^2 (c+d x)^2 \, _3F_2\left(1,\frac{5}{4},\frac{5}{4};\frac{7}{4},\frac{9}{4};(c+d x)^2\right)+5 \left(a+b \sin ^{-1}(c+d x)\right) \left(3 \left(a+b \sin ^{-1}(c+d x)\right)-4 b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};(c+d x)^2\right)\right)\right)}{15 d e}","\frac{16 b^2 (e (c+d x))^{5/2} \, _3F_2\left(1,\frac{5}{4},\frac{5}{4};\frac{7}{4},\frac{9}{4};(c+d x)^2\right)}{15 d e^3}-\frac{8 b (e (c+d x))^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};(c+d x)^2\right) \left(a+b \sin ^{-1}(c+d x)\right)}{3 d e^2}+\frac{2 \sqrt{e (c+d x)} \left(a+b \sin ^{-1}(c+d x)\right)^2}{d e}",1,"(2*Sqrt[e*(c + d*x)]*(5*(a + b*ArcSin[c + d*x])*(3*(a + b*ArcSin[c + d*x]) - 4*b*(c + d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2]) + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 5/4, 5/4}, {7/4, 9/4}, (c + d*x)^2]))/(15*d*e)","A",1
296,1,104,126,0.0882146,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{(c e+d e x)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^(3/2),x]","-\frac{2 \left(8 b^2 (c+d x)^2 \, _3F_2\left(\frac{3}{4},\frac{3}{4},1;\frac{5}{4},\frac{7}{4};(c+d x)^2\right)+3 \left(a+b \sin ^{-1}(c+d x)\right) \left(a-4 b (c+d x) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};(c+d x)^2\right)+b \sin ^{-1}(c+d x)\right)\right)}{3 d e \sqrt{e (c+d x)}}","-\frac{16 b^2 (e (c+d x))^{3/2} \, _3F_2\left(\frac{3}{4},\frac{3}{4},1;\frac{5}{4},\frac{7}{4};(c+d x)^2\right)}{3 d e^3}+\frac{8 b \sqrt{e (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};(c+d x)^2\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e^2}-\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)^2}{d e \sqrt{e (c+d x)}}",1,"(-2*(3*(a + b*ArcSin[c + d*x])*(a + b*ArcSin[c + d*x] - 4*b*(c + d*x)*Hypergeometric2F1[1/4, 1/2, 5/4, (c + d*x)^2]) + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, (c + d*x)^2]))/(3*d*e*Sqrt[e*(c + d*x)])","A",1
297,1,102,130,0.0914556,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{(c e+d e x)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^(5/2),x]","-\frac{2 \left(4 b (c+d x) \left(\, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};(c+d x)^2\right) \left(a+b \sin ^{-1}(c+d x)\right)-2 b (c+d x) \, _3F_2\left(\frac{1}{4},\frac{1}{4},1;\frac{3}{4},\frac{5}{4};(c+d x)^2\right)\right)+\left(a+b \sin ^{-1}(c+d x)\right)^2\right)}{3 d e (e (c+d x))^{3/2}}","\frac{16 b^2 \sqrt{e (c+d x)} \, _3F_2\left(\frac{1}{4},\frac{1}{4},1;\frac{3}{4},\frac{5}{4};(c+d x)^2\right)}{3 d e^3}-\frac{8 b \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};(c+d x)^2\right) \left(a+b \sin ^{-1}(c+d x)\right)}{3 d e^2 \sqrt{e (c+d x)}}-\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)^2}{3 d e (e (c+d x))^{3/2}}",1,"(-2*((a + b*ArcSin[c + d*x])^2 + 4*b*(c + d*x)*((a + b*ArcSin[c + d*x])*Hypergeometric2F1[-1/4, 1/2, 3/4, (c + d*x)^2] - 2*b*(c + d*x)*HypergeometricPFQ[{1/4, 1/4, 1}, {3/4, 5/4}, (c + d*x)^2])))/(3*d*e*(e*(c + d*x))^(3/2))","A",1
298,1,106,130,0.0959327,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{(c e+d e x)^{7/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^(7/2),x]","-\frac{2 \left(8 b^2 (c+d x)^2 \, _3F_2\left(-\frac{1}{4},-\frac{1}{4},1;\frac{1}{4},\frac{3}{4};(c+d x)^2\right)+\left(a+b \sin ^{-1}(c+d x)\right) \left(3 \left(a+b \sin ^{-1}(c+d x)\right)+4 b (c+d x) \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};(c+d x)^2\right)\right)\right)}{15 d e (e (c+d x))^{5/2}}","-\frac{16 b^2 \, _3F_2\left(-\frac{1}{4},-\frac{1}{4},1;\frac{1}{4},\frac{3}{4};(c+d x)^2\right)}{15 d e^3 \sqrt{e (c+d x)}}-\frac{8 b \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};(c+d x)^2\right) \left(a+b \sin ^{-1}(c+d x)\right)}{15 d e^2 (e (c+d x))^{3/2}}-\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)^2}{5 d e (e (c+d x))^{5/2}}",1,"(-2*((a + b*ArcSin[c + d*x])*(3*(a + b*ArcSin[c + d*x]) + 4*b*(c + d*x)*Hypergeometric2F1[-3/4, 1/2, 1/4, (c + d*x)^2]) + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{-1/4, -1/4, 1}, {1/4, 3/4}, (c + d*x)^2]))/(15*d*e*(e*(c + d*x))^(5/2))","A",1
299,1,114,130,0.1014565,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{(c e+d e x)^{9/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^2/(c*e + d*e*x)^(9/2),x]","-\frac{2 \sqrt{e (c+d x)} \left(8 b^2 (c+d x)^2 \, _3F_2\left(-\frac{3}{4},-\frac{3}{4},1;-\frac{1}{4},\frac{1}{4};(c+d x)^2\right)+3 \left(a+b \sin ^{-1}(c+d x)\right) \left(5 \left(a+b \sin ^{-1}(c+d x)\right)+4 b (c+d x) \, _2F_1\left(-\frac{5}{4},\frac{1}{2};-\frac{1}{4};(c+d x)^2\right)\right)\right)}{105 d e^5 (c+d x)^4}","-\frac{16 b^2 \, _3F_2\left(-\frac{3}{4},-\frac{3}{4},1;-\frac{1}{4},\frac{1}{4};(c+d x)^2\right)}{105 d e^3 (e (c+d x))^{3/2}}-\frac{8 b \, _2F_1\left(-\frac{5}{4},\frac{1}{2};-\frac{1}{4};(c+d x)^2\right) \left(a+b \sin ^{-1}(c+d x)\right)}{35 d e^2 (e (c+d x))^{5/2}}-\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)^2}{7 d e (e (c+d x))^{7/2}}",1,"(-2*Sqrt[e*(c + d*x)]*(3*(a + b*ArcSin[c + d*x])*(5*(a + b*ArcSin[c + d*x]) + 4*b*(c + d*x)*Hypergeometric2F1[-5/4, 1/2, -1/4, (c + d*x)^2]) + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{-3/4, -3/4, 1}, {-1/4, 1/4}, (c + d*x)^2]))/(105*d*e^5*(c + d*x)^4)","A",1
300,-1,0,82,180.0006936,"\int \sqrt{c e+d e x} \left(a+b \sin ^{-1}(c+d x)\right)^3 \, dx","Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcSin[c + d*x])^3,x]","\text{\$Aborted}","\frac{2 (e (c+d x))^{3/2} \left(a+b \sin ^{-1}(c+d x)\right)^3}{3 d e}-\frac{2 b \text{Int}\left(\frac{(e (c+d x))^{3/2} \left(a+b \sin ^{-1}(c+d x)\right)^2}{\sqrt{1-(c+d x)^2}},x\right)}{e}",0,"$Aborted","F",-1
301,0,0,80,10.9153197,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{\sqrt{c e+d e x}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^3/Sqrt[c*e + d*e*x],x]","\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{\sqrt{c e+d e x}} \, dx","\frac{2 \sqrt{e (c+d x)} \left(a+b \sin ^{-1}(c+d x)\right)^3}{d e}-\frac{6 b \text{Int}\left(\frac{\sqrt{e (c+d x)} \left(a+b \sin ^{-1}(c+d x)\right)^2}{\sqrt{1-(c+d x)^2}},x\right)}{e}",0,"Integrate[(a + b*ArcSin[c + d*x])^3/Sqrt[c*e + d*e*x], x]","A",-1
302,0,0,80,15.1914717,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{(c e+d e x)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^(3/2),x]","\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{(c e+d e x)^{3/2}} \, dx","\frac{6 b \text{Int}\left(\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{\sqrt{1-(c+d x)^2} \sqrt{e (c+d x)}},x\right)}{e}-\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)^3}{d e \sqrt{e (c+d x)}}",0,"Integrate[(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^(3/2), x]","A",-1
303,0,0,82,28.7966551,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{(c e+d e x)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^(5/2),x]","\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{(c e+d e x)^{5/2}} \, dx","\frac{2 b \text{Int}\left(\frac{\left(a+b \sin ^{-1}(c+d x)\right)^2}{\sqrt{1-(c+d x)^2} (e (c+d x))^{3/2}},x\right)}{e}-\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)^3}{3 d e (e (c+d x))^{3/2}}",0,"Integrate[(a + b*ArcSin[c + d*x])^3/(c*e + d*e*x)^(5/2), x]","A",-1
304,-1,0,84,180.0013688,"\int \sqrt{c e+d e x} \left(a+b \sin ^{-1}(c+d x)\right)^4 \, dx","Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcSin[c + d*x])^4,x]","\text{\$Aborted}","\frac{2 (e (c+d x))^{3/2} \left(a+b \sin ^{-1}(c+d x)\right)^4}{3 d e}-\frac{8 b \text{Int}\left(\frac{(e (c+d x))^{3/2} \left(a+b \sin ^{-1}(c+d x)\right)^3}{\sqrt{1-(c+d x)^2}},x\right)}{3 e}",0,"$Aborted","F",-1
305,0,0,80,10.5306312,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^4}{\sqrt{c e+d e x}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^4/Sqrt[c*e + d*e*x],x]","\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^4}{\sqrt{c e+d e x}} \, dx","\frac{2 \sqrt{e (c+d x)} \left(a+b \sin ^{-1}(c+d x)\right)^4}{d e}-\frac{8 b \text{Int}\left(\frac{\sqrt{e (c+d x)} \left(a+b \sin ^{-1}(c+d x)\right)^3}{\sqrt{1-(c+d x)^2}},x\right)}{e}",0,"Integrate[(a + b*ArcSin[c + d*x])^4/Sqrt[c*e + d*e*x], x]","A",-1
306,0,0,80,29.4966008,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^4}{(c e+d e x)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^(3/2),x]","\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^4}{(c e+d e x)^{3/2}} \, dx","\frac{8 b \text{Int}\left(\frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{\sqrt{1-(c+d x)^2} \sqrt{e (c+d x)}},x\right)}{e}-\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)^4}{d e \sqrt{e (c+d x)}}",0,"Integrate[(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^(3/2), x]","A",-1
307,0,0,84,46.9122164,"\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^4}{(c e+d e x)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^(5/2),x]","\int \frac{\left(a+b \sin ^{-1}(c+d x)\right)^4}{(c e+d e x)^{5/2}} \, dx","\frac{8 b \text{Int}\left(\frac{\left(a+b \sin ^{-1}(c+d x)\right)^3}{\sqrt{1-(c+d x)^2} (e (c+d x))^{3/2}},x\right)}{3 e}-\frac{2 \left(a+b \sin ^{-1}(c+d x)\right)^4}{3 d e (e (c+d x))^{3/2}}",0,"Integrate[(a + b*ArcSin[c + d*x])^4/(c*e + d*e*x)^(5/2), x]","A",-1
308,0,0,89,3.2211543,"\int (c e+d e x)^m \left(a+b \sin ^{-1}(c+d x)\right)^4 \, dx","Integrate[(c*e + d*e*x)^m*(a + b*ArcSin[c + d*x])^4,x]","\int (c e+d e x)^m \left(a+b \sin ^{-1}(c+d x)\right)^4 \, dx","\frac{(e (c+d x))^{m+1} \left(a+b \sin ^{-1}(c+d x)\right)^4}{d e (m+1)}-\frac{4 b \text{Int}\left(\frac{(e (c+d x))^{m+1} \left(a+b \sin ^{-1}(c+d x)\right)^3}{\sqrt{1-(c+d x)^2}},x\right)}{e (m+1)}",0,"Integrate[(c*e + d*e*x)^m*(a + b*ArcSin[c + d*x])^4, x]","A",-1
309,0,0,89,2.0802257,"\int (c e+d e x)^m \left(a+b \sin ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)^m*(a + b*ArcSin[c + d*x])^3,x]","\int (c e+d e x)^m \left(a+b \sin ^{-1}(c+d x)\right)^3 \, dx","\frac{(e (c+d x))^{m+1} \left(a+b \sin ^{-1}(c+d x)\right)^3}{d e (m+1)}-\frac{3 b \text{Int}\left(\frac{(e (c+d x))^{m+1} \left(a+b \sin ^{-1}(c+d x)\right)^2}{\sqrt{1-(c+d x)^2}},x\right)}{e (m+1)}",0,"Integrate[(c*e + d*e*x)^m*(a + b*ArcSin[c + d*x])^3, x]","A",-1
310,1,151,183,0.1463274,"\int (c e+d e x)^m \left(a+b \sin ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^m*(a + b*ArcSin[c + d*x])^2,x]","\frac{(c+d x) (e (c+d x))^m \left(\frac{2 b^2 (c+d x)^2 \, _3F_2\left(1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2};\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2};(c+d x)^2\right)}{(m+2) (m+3)}-\frac{2 b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};(c+d x)^2\right) \left(a+b \sin ^{-1}(c+d x)\right)}{m+2}+\left(a+b \sin ^{-1}(c+d x)\right)^2\right)}{d (m+1)}","\frac{2 b^2 (e (c+d x))^{m+3} \, _3F_2\left(1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2};\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2};(c+d x)^2\right)}{d e^3 (m+1) (m+2) (m+3)}-\frac{2 b (e (c+d x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};(c+d x)^2\right) \left(a+b \sin ^{-1}(c+d x)\right)}{d e^2 (m+1) (m+2)}+\frac{(e (c+d x))^{m+1} \left(a+b \sin ^{-1}(c+d x)\right)^2}{d e (m+1)}",1,"((c + d*x)*(e*(c + d*x))^m*((a + b*ArcSin[c + d*x])^2 - (2*b*(c + d*x)*(a + b*ArcSin[c + d*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2])/(2 + m) + (2*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, (c + d*x)^2])/((2 + m)*(3 + m))))/(d*(1 + m))","A",1
311,1,77,89,0.0491403,"\int (c e+d e x)^m \left(a+b \sin ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^m*(a + b*ArcSin[c + d*x]),x]","-\frac{(c+d x) (e (c+d x))^m \left(b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};(c+d x)^2\right)-(m+2) \left(a+b \sin ^{-1}(c+d x)\right)\right)}{d (m+1) (m+2)}","\frac{(e (c+d x))^{m+1} \left(a+b \sin ^{-1}(c+d x)\right)}{d e (m+1)}-\frac{b (e (c+d x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};(c+d x)^2\right)}{d e^2 (m+1) (m+2)}",1,"-(((c + d*x)*(e*(c + d*x))^m*(-((2 + m)*(a + b*ArcSin[c + d*x])) + b*(c + d*x)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, (c + d*x)^2]))/(d*(1 + m)*(2 + m)))","A",1
312,0,0,25,1.599211,"\int \frac{(c e+d e x)^m}{a+b \sin ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)^m/(a + b*ArcSin[c + d*x]),x]","\int \frac{(c e+d e x)^m}{a+b \sin ^{-1}(c+d x)} \, dx","\text{Int}\left(\frac{(e (c+d x))^m}{a+b \sin ^{-1}(c+d x)},x\right)",0,"Integrate[(c*e + d*e*x)^m/(a + b*ArcSin[c + d*x]), x]","A",-1
313,1,133,135,0.1350877,"\int \sqrt{1-a^2-2 a b x-b^2 x^2} \sin ^{-1}(a+b x)^3 \, dx","Integrate[Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^3,x]","\frac{4 (a+b x) \sqrt{-a^2-2 a b x-b^2 x^2+1} \sin ^{-1}(a+b x)^3-3 \left(2 a^2+4 a b x+2 b^2 x^2-1\right) \sin ^{-1}(a+b x)^2-6 (a+b x) \sqrt{-a^2-2 a b x-b^2 x^2+1} \sin ^{-1}(a+b x)+3 b x (2 a+b x)+\sin ^{-1}(a+b x)^4}{8 b}","\frac{3 (a+b x)^2}{8 b}+\frac{\sin ^{-1}(a+b x)^4}{8 b}+\frac{(a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^3}{2 b}-\frac{3 (a+b x)^2 \sin ^{-1}(a+b x)^2}{4 b}+\frac{3 \sin ^{-1}(a+b x)^2}{8 b}-\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{4 b}",1,"(3*b*x*(2*a + b*x) - 6*(a + b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x] - 3*(-1 + 2*a^2 + 4*a*b*x + 2*b^2*x^2)*ArcSin[a + b*x]^2 + 4*(a + b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^3 + ArcSin[a + b*x]^4)/(8*b)","A",1
314,1,116,111,0.1097552,"\int \sqrt{1-a^2-2 a b x-b^2 x^2} \sin ^{-1}(a+b x)^2 \, dx","Integrate[Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^2,x]","\frac{-3 (a+b x) \sqrt{-a^2-2 a b x-b^2 x^2+1}+6 (a+b x) \sqrt{-a^2-2 a b x-b^2 x^2+1} \sin ^{-1}(a+b x)^2-3 \left(2 a^2+4 a b x+2 b^2 x^2-1\right) \sin ^{-1}(a+b x)+2 \sin ^{-1}(a+b x)^3}{12 b}","-\frac{(a+b x) \sqrt{1-(a+b x)^2}}{4 b}+\frac{\sin ^{-1}(a+b x)^3}{6 b}+\frac{(a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b}-\frac{(a+b x)^2 \sin ^{-1}(a+b x)}{2 b}+\frac{\sin ^{-1}(a+b x)}{4 b}",1,"(-3*(a + b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] - 3*(-1 + 2*a^2 + 4*a*b*x + 2*b^2*x^2)*ArcSin[a + b*x] + 6*(a + b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^2 + 2*ArcSin[a + b*x]^3)/(12*b)","A",1
315,1,64,63,0.0639389,"\int \sqrt{1-a^2-2 a b x-b^2 x^2} \sin ^{-1}(a+b x) \, dx","Integrate[Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x],x]","\frac{2 (a+b x) \sqrt{-a^2-2 a b x-b^2 x^2+1} \sin ^{-1}(a+b x)-b x (2 a+b x)+\sin ^{-1}(a+b x)^2}{4 b}","-\frac{(a+b x)^2}{4 b}+\frac{\sqrt{1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)}{2 b}+\frac{\sin ^{-1}(a+b x)^2}{4 b}",1,"(-(b*x*(2*a + b*x)) + 2*(a + b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x] + ArcSin[a + b*x]^2)/(4*b)","A",1
316,1,24,31,0.075031,"\int \frac{\sqrt{1-a^2-2 a b x-b^2 x^2}}{\sin ^{-1}(a+b x)} \, dx","Integrate[Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]/ArcSin[a + b*x],x]","\frac{\text{Ci}\left(2 \sin ^{-1}(a+b x)\right)+\log \left(\sin ^{-1}(a+b x)\right)}{2 b}","\frac{\text{Ci}\left(2 \sin ^{-1}(a+b x)\right)}{2 b}+\frac{\log \left(\sin ^{-1}(a+b x)\right)}{2 b}",1,"(CosIntegral[2*ArcSin[a + b*x]] + Log[ArcSin[a + b*x]])/(2*b)","A",1
317,1,46,39,0.0712155,"\int \frac{\sqrt{1-a^2-2 a b x-b^2 x^2}}{\sin ^{-1}(a+b x)^2} \, dx","Integrate[Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]/ArcSin[a + b*x]^2,x]","\frac{a^2-\sin ^{-1}(a+b x) \text{Si}\left(2 \sin ^{-1}(a+b x)\right)+2 a b x+b^2 x^2-1}{b \sin ^{-1}(a+b x)}","-\frac{\text{Si}\left(2 \sin ^{-1}(a+b x)\right)}{b}-\frac{1-(a+b x)^2}{b \sin ^{-1}(a+b x)}",1,"(-1 + a^2 + 2*a*b*x + b^2*x^2 - ArcSin[a + b*x]*SinIntegral[2*ArcSin[a + b*x]])/(b*ArcSin[a + b*x])","A",1
318,1,88,71,0.328753,"\int \frac{\sqrt{1-a^2-2 a b x-b^2 x^2}}{\sin ^{-1}(a+b x)^3} \, dx","Integrate[Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]/ArcSin[a + b*x]^3,x]","\frac{2 (a+b x) \sqrt{-a^2-2 a b x-b^2 x^2+1} \sin ^{-1}(a+b x)+a^2-2 \sin ^{-1}(a+b x)^2 \text{Ci}\left(2 \sin ^{-1}(a+b x)\right)+2 a b x+b^2 x^2-1}{2 b \sin ^{-1}(a+b x)^2}","-\frac{\text{Ci}\left(2 \sin ^{-1}(a+b x)\right)}{b}+\frac{\sqrt{1-(a+b x)^2} (a+b x)}{b \sin ^{-1}(a+b x)}+\frac{(a+b x)^2-1}{2 b \sin ^{-1}(a+b x)^2}",1,"(-1 + a^2 + 2*a*b*x + b^2*x^2 + 2*(a + b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x] - 2*ArcSin[a + b*x]^2*CosIntegral[2*ArcSin[a + b*x]])/(2*b*ArcSin[a + b*x]^2)","A",1
319,1,117,115,0.1228041,"\int \frac{\sqrt{1-a^2-2 a b x-b^2 x^2}}{\sin ^{-1}(a+b x)^4} \, dx","Integrate[Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]/ArcSin[a + b*x]^4,x]","\frac{-\left(2 a^2+4 a b x+2 b^2 x^2-1\right) \sin ^{-1}(a+b x)^2+(a+b x) \sqrt{-a^2-2 a b x-b^2 x^2+1} \sin ^{-1}(a+b x)+a^2+2 \sin ^{-1}(a+b x)^3 \text{Si}\left(2 \sin ^{-1}(a+b x)\right)+2 a b x+b^2 x^2-1}{3 b \sin ^{-1}(a+b x)^3}","\frac{2 \text{Si}\left(2 \sin ^{-1}(a+b x)\right)}{3 b}-\frac{2 (a+b x)^2}{3 b \sin ^{-1}(a+b x)}+\frac{\sqrt{1-(a+b x)^2} (a+b x)}{3 b \sin ^{-1}(a+b x)^2}+\frac{1}{3 b \sin ^{-1}(a+b x)}-\frac{1-(a+b x)^2}{3 b \sin ^{-1}(a+b x)^3}",1,"(-1 + a^2 + 2*a*b*x + b^2*x^2 + (a + b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x] - (-1 + 2*a^2 + 4*a*b*x + 2*b^2*x^2)*ArcSin[a + b*x]^2 + 2*ArcSin[a + b*x]^3*SinIntegral[2*ArcSin[a + b*x]])/(3*b*ArcSin[a + b*x]^3)","A",1
320,1,272,245,0.2702492,"\int \left(1-a^2-2 a b x-b^2 x^2\right)^{3/2} \sin ^{-1}(a+b x)^3 \, dx","Integrate[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^3,x]","\frac{3 \left(17-6 a^2\right) b^2 x^2+6 a \left(17-2 a^2\right) b x-16 \sqrt{-a^2-2 a b x-b^2 x^2+1} \left(2 a^3+6 a^2 b x+6 a b^2 x^2-5 a+2 b^3 x^3-5 b x\right) \sin ^{-1}(a+b x)^3+6 \sqrt{-a^2-2 a b x-b^2 x^2+1} \left(2 a^3+6 a^2 b x+6 a b^2 x^2-17 a+2 b^3 x^3-17 b x\right) \sin ^{-1}(a+b x)+3 \left(8 a^4+32 a^3 b x+8 a^2 \left(6 b^2 x^2-5\right)+16 a b x \left(2 b^2 x^2-5\right)+8 b^4 x^4-40 b^2 x^2+17\right) \sin ^{-1}(a+b x)^2-12 a b^3 x^3+12 \sin ^{-1}(a+b x)^4-3 b^4 x^4}{128 b}","-\frac{3 (a+b x)^4}{128 b}+\frac{51 (a+b x)^2}{128 b}-\frac{9 (a+b x)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac{\left(1-(a+b x)^2\right)^{3/2} (a+b x) \sin ^{-1}(a+b x)^3}{4 b}+\frac{3 \sqrt{1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)^3}{8 b}-\frac{3 \left(1-(a+b x)^2\right)^{3/2} (a+b x) \sin ^{-1}(a+b x)}{32 b}-\frac{45 \sqrt{1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)}{64 b}+\frac{3 \sin ^{-1}(a+b x)^4}{32 b}+\frac{3 \left(1-(a+b x)^2\right)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac{27 \sin ^{-1}(a+b x)^2}{128 b}",1,"(6*a*(17 - 2*a^2)*b*x + 3*(17 - 6*a^2)*b^2*x^2 - 12*a*b^3*x^3 - 3*b^4*x^4 + 6*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(-17*a + 2*a^3 - 17*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSin[a + b*x] + 3*(17 + 8*a^4 + 32*a^3*b*x - 40*b^2*x^2 + 8*b^4*x^4 + 16*a*b*x*(-5 + 2*b^2*x^2) + 8*a^2*(-5 + 6*b^2*x^2))*ArcSin[a + b*x]^2 - 16*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(-5*a + 2*a^3 - 5*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSin[a + b*x]^3 + 12*ArcSin[a + b*x]^4)/(128*b)","A",1
321,1,216,199,0.1974489,"\int \left(1-a^2-2 a b x-b^2 x^2\right)^{3/2} \sin ^{-1}(a+b x)^2 \, dx","Integrate[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^2,x]","\frac{\left(8 a^4-40 a^2+17\right) \sin ^{-1}(a+b x)+\sqrt{-a^2-2 a b x-b^2 x^2+1} \left(2 a^3+6 a^2 b x+6 a b^2 x^2-17 a+2 b^3 x^3-17 b x\right)-8 \sqrt{-a^2-2 a b x-b^2 x^2+1} \left(2 a^3+6 a^2 b x+6 a b^2 x^2-5 a+2 b^3 x^3-5 b x\right) \sin ^{-1}(a+b x)^2+8 b x \left(4 a^3+6 a^2 b x+4 a b^2 x^2-10 a+b^3 x^3-5 b x\right) \sin ^{-1}(a+b x)+8 \sin ^{-1}(a+b x)^3}{64 b}","-\frac{(a+b x) \left(1-(a+b x)^2\right)^{3/2}}{32 b}-\frac{15 (a+b x) \sqrt{1-(a+b x)^2}}{64 b}+\frac{\sin ^{-1}(a+b x)^3}{8 b}+\frac{(a+b x) \left(1-(a+b x)^2\right)^{3/2} \sin ^{-1}(a+b x)^2}{4 b}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{8 b}-\frac{3 (a+b x)^2 \sin ^{-1}(a+b x)}{8 b}+\frac{\left(1-(a+b x)^2\right)^2 \sin ^{-1}(a+b x)}{8 b}+\frac{9 \sin ^{-1}(a+b x)}{64 b}",1,"(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(-17*a + 2*a^3 - 17*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3) + (17 - 40*a^2 + 8*a^4)*ArcSin[a + b*x] + 8*b*x*(-10*a + 4*a^3 - 5*b*x + 6*a^2*b*x + 4*a*b^2*x^2 + b^3*x^3)*ArcSin[a + b*x] - 8*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(-5*a + 2*a^3 - 5*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSin[a + b*x]^2 + 8*ArcSin[a + b*x]^3)/(64*b)","A",1
322,1,129,110,0.077835,"\int \left(1-a^2-2 a b x-b^2 x^2\right)^{3/2} \sin ^{-1}(a+b x) \, dx","Integrate[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x],x]","\frac{1}{16} \left(\left(6 a^2-5\right) b x^2+2 a \left(2 a^2-5\right) x-\frac{2 \sqrt{-a^2-2 a b x-b^2 x^2+1} \left(2 a^3+6 a^2 b x+6 a b^2 x^2-5 a+2 b^3 x^3-5 b x\right) \sin ^{-1}(a+b x)}{b}+4 a b^2 x^3+\frac{3 \sin ^{-1}(a+b x)^2}{b}+b^3 x^4\right)","\frac{(a+b x)^4}{16 b}-\frac{5 (a+b x)^2}{16 b}+\frac{\left(1-(a+b x)^2\right)^{3/2} (a+b x) \sin ^{-1}(a+b x)}{4 b}+\frac{3 \sqrt{1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)}{8 b}+\frac{3 \sin ^{-1}(a+b x)^2}{16 b}",1,"(2*a*(-5 + 2*a^2)*x + (-5 + 6*a^2)*b*x^2 + 4*a*b^2*x^3 + b^3*x^4 - (2*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*(-5*a + 2*a^3 - 5*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSin[a + b*x])/b + (3*ArcSin[a + b*x]^2)/b)/16","A",1
323,1,37,47,0.3402421,"\int \frac{\left(1-a^2-2 a b x-b^2 x^2\right)^{3/2}}{\sin ^{-1}(a+b x)} \, dx","Integrate[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)/ArcSin[a + b*x],x]","\frac{4 \text{Ci}\left(2 \sin ^{-1}(a+b x)\right)+\text{Ci}\left(4 \sin ^{-1}(a+b x)\right)+3 \log \left(\sin ^{-1}(a+b x)\right)}{8 b}","\frac{\text{Ci}\left(2 \sin ^{-1}(a+b x)\right)}{2 b}+\frac{\text{Ci}\left(4 \sin ^{-1}(a+b x)\right)}{8 b}+\frac{3 \log \left(\sin ^{-1}(a+b x)\right)}{8 b}",1,"(4*CosIntegral[2*ArcSin[a + b*x]] + CosIntegral[4*ArcSin[a + b*x]] + 3*Log[ArcSin[a + b*x]])/(8*b)","A",1
324,1,70,57,0.334112,"\int \frac{\left(1-a^2-2 a b x-b^2 x^2\right)^{3/2}}{\sin ^{-1}(a+b x)^2} \, dx","Integrate[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)/ArcSin[a + b*x]^2,x]","-\frac{2 \left(a^2+2 a b x+b^2 x^2-1\right)^2+2 \sin ^{-1}(a+b x) \text{Si}\left(2 \sin ^{-1}(a+b x)\right)+\sin ^{-1}(a+b x) \text{Si}\left(4 \sin ^{-1}(a+b x)\right)}{2 b \sin ^{-1}(a+b x)}","-\frac{\text{Si}\left(2 \sin ^{-1}(a+b x)\right)}{b}-\frac{\text{Si}\left(4 \sin ^{-1}(a+b x)\right)}{2 b}-\frac{\left(1-(a+b x)^2\right)^2}{b \sin ^{-1}(a+b x)}",1,"-1/2*(2*(-1 + a^2 + 2*a*b*x + b^2*x^2)^2 + 2*ArcSin[a + b*x]*SinIntegral[2*ArcSin[a + b*x]] + ArcSin[a + b*x]*SinIntegral[4*ArcSin[a + b*x]])/(b*ArcSin[a + b*x])","A",1
325,1,110,90,0.4727996,"\int \frac{\left(1-a^2-2 a b x-b^2 x^2\right)^{3/2}}{\sin ^{-1}(a+b x)^3} \, dx","Integrate[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)/ArcSin[a + b*x]^3,x]","-\frac{\frac{\left(a^2+2 a b x+b^2 x^2-1\right) \left(4 (a+b x) \sqrt{-a^2-2 a b x-b^2 x^2+1} \sin ^{-1}(a+b x)+a^2+2 a b x+b^2 x^2-1\right)}{\sin ^{-1}(a+b x)^2}+2 \text{Ci}\left(2 \sin ^{-1}(a+b x)\right)+2 \text{Ci}\left(4 \sin ^{-1}(a+b x)\right)}{2 b}","-\frac{\text{Ci}\left(2 \sin ^{-1}(a+b x)\right)}{b}-\frac{\text{Ci}\left(4 \sin ^{-1}(a+b x)\right)}{b}-\frac{\left(1-(a+b x)^2\right)^2}{2 b \sin ^{-1}(a+b x)^2}+\frac{2 (a+b x) \left(1-(a+b x)^2\right)^{3/2}}{b \sin ^{-1}(a+b x)}",1,"-1/2*(((-1 + a^2 + 2*a*b*x + b^2*x^2)*(-1 + a^2 + 2*a*b*x + b^2*x^2 + 4*(a + b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]))/ArcSin[a + b*x]^2 + 2*CosIntegral[2*ArcSin[a + b*x]] + 2*CosIntegral[4*ArcSin[a + b*x]])/b","A",1
326,1,143,155,0.504907,"\int \frac{\left(1-a^2-2 a b x-b^2 x^2\right)^{3/2}}{\sin ^{-1}(a+b x)^4} \, dx","Integrate[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)/ArcSin[a + b*x]^4,x]","\frac{\frac{\left(a^2+2 a b x+b^2 x^2-1\right) \left(2 \left(4 a^2+8 a b x+4 b^2 x^2-1\right) \sin ^{-1}(a+b x)^2-2 (a+b x) \sqrt{-a^2-2 a b x-b^2 x^2+1} \sin ^{-1}(a+b x)-a^2-2 a b x-b^2 x^2+1\right)}{\sin ^{-1}(a+b x)^3}+2 \text{Si}\left(2 \sin ^{-1}(a+b x)\right)+4 \text{Si}\left(4 \sin ^{-1}(a+b x)\right)}{3 b}","\frac{2 \text{Si}\left(2 \sin ^{-1}(a+b x)\right)}{3 b}+\frac{4 \text{Si}\left(4 \sin ^{-1}(a+b x)\right)}{3 b}-\frac{8 \left(1-(a+b x)^2\right) (a+b x)^2}{3 b \sin ^{-1}(a+b x)}+\frac{2 \left(1-(a+b x)^2\right)^{3/2} (a+b x)}{3 b \sin ^{-1}(a+b x)^2}+\frac{2 \left(1-(a+b x)^2\right)}{3 b \sin ^{-1}(a+b x)}-\frac{\left(1-(a+b x)^2\right)^2}{3 b \sin ^{-1}(a+b x)^3}",1,"(((-1 + a^2 + 2*a*b*x + b^2*x^2)*(1 - a^2 - 2*a*b*x - b^2*x^2 - 2*(a + b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x] + 2*(-1 + 4*a^2 + 8*a*b*x + 4*b^2*x^2)*ArcSin[a + b*x]^2))/ArcSin[a + b*x]^3 + 2*SinIntegral[2*ArcSin[a + b*x]] + 4*SinIntegral[4*ArcSin[a + b*x]])/(3*b)","A",1
327,1,19,19,0.0323795,"\int \frac{\sin ^{-1}(a+b x)^n}{\sqrt{1-a^2-2 a b x-b^2 x^2}} \, dx","Integrate[ArcSin[a + b*x]^n/Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2],x]","\frac{\sin ^{-1}(a+b x)^{n+1}}{b (n+1)}","\frac{\sin ^{-1}(a+b x)^{n+1}}{b (n+1)}",1,"ArcSin[a + b*x]^(1 + n)/(b*(1 + n))","A",1
328,1,15,15,0.0225907,"\int \frac{\sin ^{-1}(a+b x)^2}{\sqrt{1-a^2-2 a b x-b^2 x^2}} \, dx","Integrate[ArcSin[a + b*x]^2/Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2],x]","\frac{\sin ^{-1}(a+b x)^3}{3 b}","\frac{\sin ^{-1}(a+b x)^3}{3 b}",1,"ArcSin[a + b*x]^3/(3*b)","A",1
329,1,15,15,0.0216858,"\int \frac{\sin ^{-1}(a+b x)}{\sqrt{1-a^2-2 a b x-b^2 x^2}} \, dx","Integrate[ArcSin[a + b*x]/Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2],x]","\frac{\sin ^{-1}(a+b x)^2}{2 b}","\frac{\sin ^{-1}(a+b x)^2}{2 b}",1,"ArcSin[a + b*x]^2/(2*b)","A",1
330,1,11,11,0.0381864,"\int \frac{1}{\sqrt{1-a^2-2 a b x-b^2 x^2} \sin ^{-1}(a+b x)} \, dx","Integrate[1/(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]),x]","\frac{\log \left(\sin ^{-1}(a+b x)\right)}{b}","\frac{\log \left(\sin ^{-1}(a+b x)\right)}{b}",1,"Log[ArcSin[a + b*x]]/b","A",1
331,1,13,13,0.0159564,"\int \frac{1}{\sqrt{1-a^2-2 a b x-b^2 x^2} \sin ^{-1}(a+b x)^2} \, dx","Integrate[1/(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^2),x]","-\frac{1}{b \sin ^{-1}(a+b x)}","-\frac{1}{b \sin ^{-1}(a+b x)}",1,"-(1/(b*ArcSin[a + b*x]))","A",1
332,1,15,15,0.0158913,"\int \frac{1}{\sqrt{1-a^2-2 a b x-b^2 x^2} \sin ^{-1}(a+b x)^3} \, dx","Integrate[1/(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^3),x]","-\frac{1}{2 b \sin ^{-1}(a+b x)^2}","-\frac{1}{2 b \sin ^{-1}(a+b x)^2}",1,"-1/2*1/(b*ArcSin[a + b*x]^2)","A",1
333,1,144,128,0.6280869,"\int \frac{\sin ^{-1}(a+b x)^3}{\left(1-a^2-2 a b x-b^2 x^2\right)^{3/2}} \, dx","Integrate[ArcSin[a + b*x]^3/(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2),x]","\frac{2 \sin ^{-1}(a+b x)^2 \left(\frac{\left(-i \sqrt{-a^2-2 a b x-b^2 x^2+1}+a+b x\right) \sin ^{-1}(a+b x)}{\sqrt{-a^2-2 a b x-b^2 x^2+1}}+3 \log \left(1+e^{2 i \sin ^{-1}(a+b x)}\right)\right)-6 i \sin ^{-1}(a+b x) \text{Li}_2\left(-e^{2 i \sin ^{-1}(a+b x)}\right)+3 \text{Li}_3\left(-e^{2 i \sin ^{-1}(a+b x)}\right)}{2 b}","-\frac{3 i \sin ^{-1}(a+b x) \text{Li}_2\left(-e^{2 i \sin ^{-1}(a+b x)}\right)}{b}+\frac{3 \text{Li}_3\left(-e^{2 i \sin ^{-1}(a+b x)}\right)}{2 b}+\frac{(a+b x) \sin ^{-1}(a+b x)^3}{b \sqrt{1-(a+b x)^2}}-\frac{i \sin ^{-1}(a+b x)^3}{b}+\frac{3 \sin ^{-1}(a+b x)^2 \log \left(1+e^{2 i \sin ^{-1}(a+b x)}\right)}{b}",1,"(2*ArcSin[a + b*x]^2*(((a + b*x - I*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2])*ArcSin[a + b*x])/Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] + 3*Log[1 + E^((2*I)*ArcSin[a + b*x])]) - (6*I)*ArcSin[a + b*x]*PolyLog[2, -E^((2*I)*ArcSin[a + b*x])] + 3*PolyLog[3, -E^((2*I)*ArcSin[a + b*x])])/(2*b)","A",1
334,1,114,97,0.3897824,"\int \frac{\sin ^{-1}(a+b x)^2}{\left(1-a^2-2 a b x-b^2 x^2\right)^{3/2}} \, dx","Integrate[ArcSin[a + b*x]^2/(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2),x]","\frac{\sin ^{-1}(a+b x) \left(\frac{\left(-i \sqrt{-a^2-2 a b x-b^2 x^2+1}+a+b x\right) \sin ^{-1}(a+b x)}{\sqrt{-a^2-2 a b x-b^2 x^2+1}}+2 \log \left(1+e^{2 i \sin ^{-1}(a+b x)}\right)\right)-i \text{Li}_2\left(-e^{2 i \sin ^{-1}(a+b x)}\right)}{b}","-\frac{i \text{Li}_2\left(-e^{2 i \sin ^{-1}(a+b x)}\right)}{b}+\frac{(a+b x) \sin ^{-1}(a+b x)^2}{b \sqrt{1-(a+b x)^2}}-\frac{i \sin ^{-1}(a+b x)^2}{b}+\frac{2 \sin ^{-1}(a+b x) \log \left(1+e^{2 i \sin ^{-1}(a+b x)}\right)}{b}",1,"(ArcSin[a + b*x]*(((a + b*x - I*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2])*ArcSin[a + b*x])/Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] + 2*Log[1 + E^((2*I)*ArcSin[a + b*x])]) - I*PolyLog[2, -E^((2*I)*ArcSin[a + b*x])])/b","A",1
335,1,66,50,0.1095288,"\int \frac{\sin ^{-1}(a+b x)}{\left(1-a^2-2 a b x-b^2 x^2\right)^{3/2}} \, dx","Integrate[ArcSin[a + b*x]/(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2),x]","\frac{\log \left(-a^2-2 a b x-b^2 x^2+1\right)+\frac{2 (a+b x) \sin ^{-1}(a+b x)}{\sqrt{-a^2-2 a b x-b^2 x^2+1}}}{2 b}","\frac{\log \left(1-(a+b x)^2\right)}{2 b}+\frac{(a+b x) \sin ^{-1}(a+b x)}{b \sqrt{1-(a+b x)^2}}",1,"((2*(a + b*x)*ArcSin[a + b*x])/Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] + Log[1 - a^2 - 2*a*b*x - b^2*x^2])/(2*b)","A",1
336,0,0,27,0.8083598,"\int \frac{1}{\left(1-a^2-2 a b x-b^2 x^2\right)^{3/2} \sin ^{-1}(a+b x)} \, dx","Integrate[1/((1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]),x]","\int \frac{1}{\left(1-a^2-2 a b x-b^2 x^2\right)^{3/2} \sin ^{-1}(a+b x)} \, dx","\text{Int}\left(\frac{1}{\left(1-(a+b x)^2\right)^{3/2} \sin ^{-1}(a+b x)},x\right)",0,"Integrate[1/((1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]), x]","A",-1
337,0,0,59,11.9014221,"\int \frac{1}{\left(1-a^2-2 a b x-b^2 x^2\right)^{3/2} \sin ^{-1}(a+b x)^2} \, dx","Integrate[1/((1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^2),x]","\int \frac{1}{\left(1-a^2-2 a b x-b^2 x^2\right)^{3/2} \sin ^{-1}(a+b x)^2} \, dx","2 \text{Int}\left(\frac{a+b x}{\left(1-(a+b x)^2\right)^2 \sin ^{-1}(a+b x)},x\right)-\frac{1}{b \left(1-(a+b x)^2\right) \sin ^{-1}(a+b x)}",0,"Integrate[1/((1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^2), x]","A",-1
338,1,46,46,0.1255419,"\int \frac{\sin ^{-1}(a+b x)}{\sqrt{c-c (a+b x)^2}} \, dx","Integrate[ArcSin[a + b*x]/Sqrt[c - c*(a + b*x)^2],x]","\frac{\sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt{-c \left((a+b x)^2-1\right)}}","\frac{\sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt{c-c (a+b x)^2}}",1,"(Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(2*b*Sqrt[-(c*(-1 + (a + b*x)^2))])","A",1
339,1,54,46,0.0497361,"\int \frac{\sin ^{-1}(a+b x)}{\sqrt{\left(1-a^2\right) c-2 a b c x-b^2 c x^2}} \, dx","Integrate[ArcSin[a + b*x]/Sqrt[(1 - a^2)*c - 2*a*b*c*x - b^2*c*x^2],x]","\frac{\sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt{-c \left(a^2+2 a b x+b^2 x^2-1\right)}}","\frac{\sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt{c-c (a+b x)^2}}",1,"(Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(2*b*Sqrt[-(c*(-1 + a^2 + 2*a*b*x + b^2*x^2))])","A",1
340,1,60,84,0.0765633,"\int x^9 \left(a+b \sin ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^9*(a + b*ArcSin[c*x^2]),x]","\frac{1}{150} \left(15 a x^{10}+\frac{b \sqrt{1-c^2 x^4} \left(3 c^4 x^8+4 c^2 x^4+8\right)}{c^5}+15 b x^{10} \sin ^{-1}\left(c x^2\right)\right)","\frac{1}{10} x^{10} \left(a+b \sin ^{-1}\left(c x^2\right)\right)+\frac{b \left(1-c^2 x^4\right)^{5/2}}{50 c^5}-\frac{b \left(1-c^2 x^4\right)^{3/2}}{15 c^5}+\frac{b \sqrt{1-c^2 x^4}}{10 c^5}",1,"(15*a*x^10 + (b*Sqrt[1 - c^2*x^4]*(8 + 4*c^2*x^4 + 3*c^4*x^8))/c^5 + 15*b*x^10*ArcSin[c*x^2])/150","A",1
341,1,87,82,0.0393756,"\int x^7 \left(a+b \sin ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^7*(a + b*ArcSin[c*x^2]),x]","\frac{a x^8}{8}-\frac{3 b \sin ^{-1}\left(c x^2\right)}{64 c^4}+\frac{b x^6 \sqrt{1-c^2 x^4}}{32 c}+\frac{3 b x^2 \sqrt{1-c^2 x^4}}{64 c^3}+\frac{1}{8} b x^8 \sin ^{-1}\left(c x^2\right)","\frac{1}{8} x^8 \left(a+b \sin ^{-1}\left(c x^2\right)\right)-\frac{3 b \sin ^{-1}\left(c x^2\right)}{64 c^4}+\frac{b x^6 \sqrt{1-c^2 x^4}}{32 c}+\frac{3 b x^2 \sqrt{1-c^2 x^4}}{64 c^3}",1,"(a*x^8)/8 + (3*b*x^2*Sqrt[1 - c^2*x^4])/(64*c^3) + (b*x^6*Sqrt[1 - c^2*x^4])/(32*c) - (3*b*ArcSin[c*x^2])/(64*c^4) + (b*x^8*ArcSin[c*x^2])/8","A",1
342,1,70,62,0.0455679,"\int x^5 \left(a+b \sin ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^5*(a + b*ArcSin[c*x^2]),x]","\frac{a x^6}{6}+\frac{b x^4 \sqrt{1-c^2 x^4}}{18 c}+\frac{b \sqrt{1-c^2 x^4}}{9 c^3}+\frac{1}{6} b x^6 \sin ^{-1}\left(c x^2\right)","\frac{1}{6} x^6 \left(a+b \sin ^{-1}\left(c x^2\right)\right)-\frac{b \left(1-c^2 x^4\right)^{3/2}}{18 c^3}+\frac{b \sqrt{1-c^2 x^4}}{6 c^3}",1,"(a*x^6)/6 + (b*Sqrt[1 - c^2*x^4])/(9*c^3) + (b*x^4*Sqrt[1 - c^2*x^4])/(18*c) + (b*x^6*ArcSin[c*x^2])/6","A",1
343,1,62,57,0.0272367,"\int x^3 \left(a+b \sin ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^3*(a + b*ArcSin[c*x^2]),x]","\frac{a x^4}{4}-\frac{b \sin ^{-1}\left(c x^2\right)}{8 c^2}+\frac{b x^2 \sqrt{1-c^2 x^4}}{8 c}+\frac{1}{4} b x^4 \sin ^{-1}\left(c x^2\right)","\frac{1}{4} x^4 \left(a+b \sin ^{-1}\left(c x^2\right)\right)-\frac{b \sin ^{-1}\left(c x^2\right)}{8 c^2}+\frac{b x^2 \sqrt{1-c^2 x^4}}{8 c}",1,"(a*x^4)/4 + (b*x^2*Sqrt[1 - c^2*x^4])/(8*c) - (b*ArcSin[c*x^2])/(8*c^2) + (b*x^4*ArcSin[c*x^2])/4","A",1
344,1,43,45,0.008908,"\int x \left(a+b \sin ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x*(a + b*ArcSin[c*x^2]),x]","\frac{a x^2}{2}+\frac{1}{2} b \left(\frac{\sqrt{1-c^2 x^4}}{c}+x^2 \sin ^{-1}\left(c x^2\right)\right)","\frac{a x^2}{2}+\frac{b \sqrt{1-c^2 x^4}}{2 c}+\frac{1}{2} b x^2 \sin ^{-1}\left(c x^2\right)",1,"(a*x^2)/2 + (b*(Sqrt[1 - c^2*x^4]/c + x^2*ArcSin[c*x^2]))/2","A",1
345,1,64,69,0.0386096,"\int \frac{a+b \sin ^{-1}\left(c x^2\right)}{x} \, dx","Integrate[(a + b*ArcSin[c*x^2])/x,x]","a \log (x)+\frac{1}{2} b \left(\sin ^{-1}\left(c x^2\right) \log \left(1-e^{2 i \sin ^{-1}\left(c x^2\right)}\right)-\frac{1}{2} i \left(\sin ^{-1}\left(c x^2\right)^2+\text{Li}_2\left(e^{2 i \sin ^{-1}\left(c x^2\right)}\right)\right)\right)","a \log (x)-\frac{1}{4} i b \text{Li}_2\left(e^{2 i \sin ^{-1}\left(c x^2\right)}\right)-\frac{1}{4} i b \sin ^{-1}\left(c x^2\right)^2+\frac{1}{2} b \sin ^{-1}\left(c x^2\right) \log \left(1-e^{2 i \sin ^{-1}\left(c x^2\right)}\right)",1,"a*Log[x] + (b*(ArcSin[c*x^2]*Log[1 - E^((2*I)*ArcSin[c*x^2])] - (I/2)*(ArcSin[c*x^2]^2 + PolyLog[2, E^((2*I)*ArcSin[c*x^2])])))/2","A",1
346,1,44,39,0.0076864,"\int \frac{a+b \sin ^{-1}\left(c x^2\right)}{x^3} \, dx","Integrate[(a + b*ArcSin[c*x^2])/x^3,x]","-\frac{a}{2 x^2}-\frac{1}{2} b c \tanh ^{-1}\left(\sqrt{1-c^2 x^4}\right)-\frac{b \sin ^{-1}\left(c x^2\right)}{2 x^2}","-\frac{a+b \sin ^{-1}\left(c x^2\right)}{2 x^2}-\frac{1}{2} b c \tanh ^{-1}\left(\sqrt{1-c^2 x^4}\right)",1,"-1/2*a/x^2 - (b*ArcSin[c*x^2])/(2*x^2) - (b*c*ArcTanh[Sqrt[1 - c^2*x^4]])/2","A",1
347,1,46,41,0.0187616,"\int \frac{a+b \sin ^{-1}\left(c x^2\right)}{x^5} \, dx","Integrate[(a + b*ArcSin[c*x^2])/x^5,x]","-\frac{a}{4 x^4}-\frac{b c \sqrt{1-c^2 x^4}}{4 x^2}-\frac{b \sin ^{-1}\left(c x^2\right)}{4 x^4}","-\frac{a+b \sin ^{-1}\left(c x^2\right)}{4 x^4}-\frac{b c \sqrt{1-c^2 x^4}}{4 x^2}",1,"-1/4*a/x^4 - (b*c*Sqrt[1 - c^2*x^4])/(4*x^2) - (b*ArcSin[c*x^2])/(4*x^4)","A",1
348,1,69,64,0.0238977,"\int \frac{a+b \sin ^{-1}\left(c x^2\right)}{x^7} \, dx","Integrate[(a + b*ArcSin[c*x^2])/x^7,x]","-\frac{a}{6 x^6}-\frac{b c \sqrt{1-c^2 x^4}}{12 x^4}-\frac{1}{12} b c^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^4}\right)-\frac{b \sin ^{-1}\left(c x^2\right)}{6 x^6}","-\frac{a+b \sin ^{-1}\left(c x^2\right)}{6 x^6}-\frac{b c \sqrt{1-c^2 x^4}}{12 x^4}-\frac{1}{12} b c^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^4}\right)",1,"-1/6*a/x^6 - (b*c*Sqrt[1 - c^2*x^4])/(12*x^4) - (b*ArcSin[c*x^2])/(6*x^6) - (b*c^3*ArcTanh[Sqrt[1 - c^2*x^4]])/12","A",1
349,1,60,66,0.0418592,"\int \frac{a+b \sin ^{-1}\left(c x^2\right)}{x^9} \, dx","Integrate[(a + b*ArcSin[c*x^2])/x^9,x]","\frac{1}{2} b \left(-\frac{c \sqrt{1-c^2 x^4} \left(2 c^2 x^4+1\right)}{12 x^6}-\frac{\sin ^{-1}\left(c x^2\right)}{4 x^8}\right)-\frac{a}{8 x^8}","-\frac{a+b \sin ^{-1}\left(c x^2\right)}{8 x^8}-\frac{b c \sqrt{1-c^2 x^4}}{24 x^6}-\frac{b c^3 \sqrt{1-c^2 x^4}}{12 x^2}",1,"-1/8*a/x^8 + (b*(-1/12*(c*Sqrt[1 - c^2*x^4]*(1 + 2*c^2*x^4))/x^6 - ArcSin[c*x^2]/(4*x^8)))/2","A",1
350,1,63,89,0.0217341,"\int \frac{a+b \sin ^{-1}\left(c x^2\right)}{x^{11}} \, dx","Integrate[(a + b*ArcSin[c*x^2])/x^11,x]","-\frac{a}{10 x^{10}}-\frac{1}{10} b c^5 \sqrt{1-c^2 x^4} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-c^2 x^4\right)-\frac{b \sin ^{-1}\left(c x^2\right)}{10 x^{10}}","-\frac{a+b \sin ^{-1}\left(c x^2\right)}{10 x^{10}}-\frac{b c \sqrt{1-c^2 x^4}}{40 x^8}-\frac{3}{80} b c^5 \tanh ^{-1}\left(\sqrt{1-c^2 x^4}\right)-\frac{3 b c^3 \sqrt{1-c^2 x^4}}{80 x^4}",1,"-1/10*a/x^10 - (b*ArcSin[c*x^2])/(10*x^10) - (b*c^5*Sqrt[1 - c^2*x^4]*Hypergeometric2F1[1/2, 3, 3/2, 1 - c^2*x^4])/10","C",1
351,1,68,91,0.0527453,"\int \frac{a+b \sin ^{-1}\left(c x^2\right)}{x^{13}} \, dx","Integrate[(a + b*ArcSin[c*x^2])/x^13,x]","\frac{1}{2} b \left(-\frac{c \sqrt{1-c^2 x^4} \left(8 c^4 x^8+4 c^2 x^4+3\right)}{90 x^{10}}-\frac{\sin ^{-1}\left(c x^2\right)}{6 x^{12}}\right)-\frac{a}{12 x^{12}}","-\frac{a+b \sin ^{-1}\left(c x^2\right)}{12 x^{12}}-\frac{b c \sqrt{1-c^2 x^4}}{60 x^{10}}-\frac{2 b c^5 \sqrt{1-c^2 x^4}}{45 x^2}-\frac{b c^3 \sqrt{1-c^2 x^4}}{45 x^6}",1,"-1/12*a/x^12 + (b*(-1/90*(c*Sqrt[1 - c^2*x^4]*(3 + 4*c^2*x^4 + 8*c^4*x^8))/x^10 - ArcSin[c*x^2]/(6*x^12)))/2","A",1
352,1,82,86,0.2266738,"\int x^6 \left(a+b \sin ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^6*(a + b*ArcSin[c*x^2]),x]","\frac{1}{147} \left(21 a x^7+\frac{2 b x \sqrt{1-c^2 x^4} \left(3 c^2 x^4+5\right)}{c^3}+21 b x^7 \sin ^{-1}\left(c x^2\right)-\frac{10 i b F\left(\left.i \sinh ^{-1}\left(\sqrt{-c} x\right)\right|-1\right)}{(-c)^{7/2}}\right)","\frac{1}{7} x^7 \left(a+b \sin ^{-1}\left(c x^2\right)\right)-\frac{10 b F\left(\left.\sin ^{-1}\left(\sqrt{c} x\right)\right|-1\right)}{147 c^{7/2}}+\frac{2 b x^5 \sqrt{1-c^2 x^4}}{49 c}+\frac{10 b x \sqrt{1-c^2 x^4}}{147 c^3}",1,"(21*a*x^7 + (2*b*x*Sqrt[1 - c^2*x^4]*(5 + 3*c^2*x^4))/c^3 + 21*b*x^7*ArcSin[c*x^2] - ((10*I)*b*EllipticF[I*ArcSinh[Sqrt[-c]*x], -1])/(-c)^(7/2))/147","C",1
353,1,93,83,0.2851578,"\int x^4 \left(a+b \sin ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^4*(a + b*ArcSin[c*x^2]),x]","\frac{1}{25} \left(5 a x^5+\frac{2 b x^3 \sqrt{1-c^2 x^4}}{c}+5 b x^5 \sin ^{-1}\left(c x^2\right)+\frac{6 i b \left(E\left(\left.i \sinh ^{-1}\left(\sqrt{-c} x\right)\right|-1\right)-F\left(\left.i \sinh ^{-1}\left(\sqrt{-c} x\right)\right|-1\right)\right)}{(-c)^{5/2}}\right)","\frac{1}{5} x^5 \left(a+b \sin ^{-1}\left(c x^2\right)\right)+\frac{6 b F\left(\left.\sin ^{-1}\left(\sqrt{c} x\right)\right|-1\right)}{25 c^{5/2}}-\frac{6 b E\left(\left.\sin ^{-1}\left(\sqrt{c} x\right)\right|-1\right)}{25 c^{5/2}}+\frac{2 b x^3 \sqrt{1-c^2 x^4}}{25 c}",1,"(5*a*x^5 + (2*b*x^3*Sqrt[1 - c^2*x^4])/c + 5*b*x^5*ArcSin[c*x^2] + ((6*I)*b*(EllipticE[I*ArcSinh[Sqrt[-c]*x], -1] - EllipticF[I*ArcSinh[Sqrt[-c]*x], -1]))/(-c)^(5/2))/25","C",1
354,1,72,61,0.1657176,"\int x^2 \left(a+b \sin ^{-1}\left(c x^2\right)\right) \, dx","Integrate[x^2*(a + b*ArcSin[c*x^2]),x]","\frac{1}{9} \left(3 a x^3+\frac{2 b x \sqrt{1-c^2 x^4}}{c}+3 b x^3 \sin ^{-1}\left(c x^2\right)-\frac{2 i b F\left(\left.i \sinh ^{-1}\left(\sqrt{-c} x\right)\right|-1\right)}{(-c)^{3/2}}\right)","\frac{1}{3} x^3 \left(a+b \sin ^{-1}\left(c x^2\right)\right)-\frac{2 b F\left(\left.\sin ^{-1}\left(\sqrt{c} x\right)\right|-1\right)}{9 c^{3/2}}+\frac{2 b x \sqrt{1-c^2 x^4}}{9 c}",1,"(3*a*x^3 + (2*b*x*Sqrt[1 - c^2*x^4])/c + 3*b*x^3*ArcSin[c*x^2] - ((2*I)*b*EllipticF[I*ArcSinh[Sqrt[-c]*x], -1])/(-c)^(3/2))/9","C",1
355,1,39,49,0.0055715,"\int \left(a+b \sin ^{-1}\left(c x^2\right)\right) \, dx","Integrate[a + b*ArcSin[c*x^2],x]","a x-\frac{2}{3} b c x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};c^2 x^4\right)+b x \sin ^{-1}\left(c x^2\right)","a x+b x \sin ^{-1}\left(c x^2\right)+\frac{2 b F\left(\left.\sin ^{-1}\left(\sqrt{c} x\right)\right|-1\right)}{\sqrt{c}}-\frac{2 b E\left(\left.\sin ^{-1}\left(\sqrt{c} x\right)\right|-1\right)}{\sqrt{c}}",1,"a*x + b*x*ArcSin[c*x^2] - (2*b*c*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, c^2*x^4])/3","C",1
356,1,44,34,0.0588756,"\int \frac{a+b \sin ^{-1}\left(c x^2\right)}{x^2} \, dx","Integrate[(a + b*ArcSin[c*x^2])/x^2,x]","-\frac{a+b \sin ^{-1}\left(c x^2\right)-2 i b \sqrt{-c} x F\left(\left.i \sinh ^{-1}\left(\sqrt{-c} x\right)\right|-1\right)}{x}","2 b \sqrt{c} F\left(\left.\sin ^{-1}\left(\sqrt{c} x\right)\right|-1\right)-\frac{a+b \sin ^{-1}\left(c x^2\right)}{x}",1,"-((a + b*ArcSin[c*x^2] - (2*I)*b*Sqrt[-c]*x*EllipticF[I*ArcSinh[Sqrt[-c]*x], -1])/x)","C",1
357,1,89,81,0.2063889,"\int \frac{a+b \sin ^{-1}\left(c x^2\right)}{x^4} \, dx","Integrate[(a + b*ArcSin[c*x^2])/x^4,x]","-\frac{a+2 b c x^2 \sqrt{1-c^2 x^4}+2 i b \sqrt{-c} c x^3 \left(E\left(\left.i \sinh ^{-1}\left(\sqrt{-c} x\right)\right|-1\right)-F\left(\left.i \sinh ^{-1}\left(\sqrt{-c} x\right)\right|-1\right)\right)+b \sin ^{-1}\left(c x^2\right)}{3 x^3}","-\frac{a+b \sin ^{-1}\left(c x^2\right)}{3 x^3}+\frac{2}{3} b c^{3/2} F\left(\left.\sin ^{-1}\left(\sqrt{c} x\right)\right|-1\right)-\frac{2}{3} b c^{3/2} E\left(\left.\sin ^{-1}\left(\sqrt{c} x\right)\right|-1\right)-\frac{2 b c \sqrt{1-c^2 x^4}}{3 x}",1,"-1/3*(a + 2*b*c*x^2*Sqrt[1 - c^2*x^4] + b*ArcSin[c*x^2] + (2*I)*b*Sqrt[-c]*c*x^3*(EllipticE[I*ArcSinh[Sqrt[-c]*x], -1] - EllipticF[I*ArcSinh[Sqrt[-c]*x], -1]))/x^3","C",1
358,1,72,61,0.1481315,"\int \frac{a+b \sin ^{-1}\left(c x^2\right)}{x^6} \, dx","Integrate[(a + b*ArcSin[c*x^2])/x^6,x]","-\frac{3 a+2 b c x^2 \sqrt{1-c^2 x^4}-2 i b (-c)^{5/2} x^5 F\left(\left.i \sinh ^{-1}\left(\sqrt{-c} x\right)\right|-1\right)+3 b \sin ^{-1}\left(c x^2\right)}{15 x^5}","-\frac{a+b \sin ^{-1}\left(c x^2\right)}{5 x^5}+\frac{2}{15} b c^{5/2} F\left(\left.\sin ^{-1}\left(\sqrt{c} x\right)\right|-1\right)-\frac{2 b c \sqrt{1-c^2 x^4}}{15 x^3}",1,"-1/15*(3*a + 2*b*c*x^2*Sqrt[1 - c^2*x^4] + 3*b*ArcSin[c*x^2] - (2*I)*b*(-c)^(5/2)*x^5*EllipticF[I*ArcSinh[Sqrt[-c]*x], -1])/x^5","C",1
359,1,100,106,0.2503085,"\int \frac{a+b \sin ^{-1}\left(c x^2\right)}{x^8} \, dx","Integrate[(a + b*ArcSin[c*x^2])/x^8,x]","-\frac{5 a+2 b x^2 \sqrt{1-c^2 x^4} \left(3 c^3 x^4+c\right)-6 i b (-c)^{7/2} x^7 \left(E\left(\left.i \sinh ^{-1}\left(\sqrt{-c} x\right)\right|-1\right)-F\left(\left.i \sinh ^{-1}\left(\sqrt{-c} x\right)\right|-1\right)\right)+5 b \sin ^{-1}\left(c x^2\right)}{35 x^7}","-\frac{a+b \sin ^{-1}\left(c x^2\right)}{7 x^7}+\frac{6}{35} b c^{7/2} F\left(\left.\sin ^{-1}\left(\sqrt{c} x\right)\right|-1\right)-\frac{6}{35} b c^{7/2} E\left(\left.\sin ^{-1}\left(\sqrt{c} x\right)\right|-1\right)-\frac{2 b c \sqrt{1-c^2 x^4}}{35 x^5}-\frac{6 b c^3 \sqrt{1-c^2 x^4}}{35 x}",1,"-1/35*(5*a + 2*b*x^2*Sqrt[1 - c^2*x^4]*(c + 3*c^3*x^4) + 5*b*ArcSin[c*x^2] - (6*I)*b*(-c)^(7/2)*x^7*(EllipticE[I*ArcSinh[Sqrt[-c]*x], -1] - EllipticF[I*ArcSinh[Sqrt[-c]*x], -1]))/x^7","C",1
360,1,58,62,0.0405207,"\int \frac{\sin ^{-1}\left(a x^5\right)}{x} \, dx","Integrate[ArcSin[a*x^5]/x,x]","\frac{1}{5} \left(\sin ^{-1}\left(a x^5\right) \log \left(1-e^{2 i \sin ^{-1}\left(a x^5\right)}\right)-\frac{1}{2} i \left(\sin ^{-1}\left(a x^5\right)^2+\text{Li}_2\left(e^{2 i \sin ^{-1}\left(a x^5\right)}\right)\right)\right)","-\frac{1}{10} i \text{Li}_2\left(e^{2 i \sin ^{-1}\left(a x^5\right)}\right)-\frac{1}{10} i \sin ^{-1}\left(a x^5\right)^2+\frac{1}{5} \sin ^{-1}\left(a x^5\right) \log \left(1-e^{2 i \sin ^{-1}\left(a x^5\right)}\right)",1,"(ArcSin[a*x^5]*Log[1 - E^((2*I)*ArcSin[a*x^5])] - (I/2)*(ArcSin[a*x^5]^2 + PolyLog[2, E^((2*I)*ArcSin[a*x^5])]))/5","A",1
361,1,64,78,0.0400668,"\int x^2 \sin ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x^2*ArcSin[Sqrt[x]],x]","\frac{1}{144} \left(8 \sqrt{1-x} x^{5/2}+10 \sqrt{1-x} x^{3/2}+3 \left(16 x^3-5\right) \sin ^{-1}\left(\sqrt{x}\right)+15 \sqrt{-((x-1) x)}\right)","\frac{1}{18} \sqrt{1-x} x^{5/2}+\frac{5}{72} \sqrt{1-x} x^{3/2}+\frac{1}{3} x^3 \sin ^{-1}\left(\sqrt{x}\right)+\frac{5}{48} \sqrt{1-x} \sqrt{x}+\frac{5}{96} \sin ^{-1}(1-2 x)",1,"(10*Sqrt[1 - x]*x^(3/2) + 8*Sqrt[1 - x]*x^(5/2) + 15*Sqrt[-((-1 + x)*x)] + 3*(-5 + 16*x^3)*ArcSin[Sqrt[x]])/144","A",1
362,1,47,60,0.0311817,"\int x \sin ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x*ArcSin[Sqrt[x]],x]","\frac{1}{16} \left(2 \sqrt{1-x} x^{3/2}+\left(8 x^2-3\right) \sin ^{-1}\left(\sqrt{x}\right)+3 \sqrt{-((x-1) x)}\right)","\frac{1}{8} \sqrt{1-x} x^{3/2}+\frac{1}{2} x^2 \sin ^{-1}\left(\sqrt{x}\right)+\frac{3}{16} \sqrt{1-x} \sqrt{x}+\frac{3}{32} \sin ^{-1}(1-2 x)",1,"(2*Sqrt[1 - x]*x^(3/2) + 3*Sqrt[-((-1 + x)*x)] + (-3 + 8*x^2)*ArcSin[Sqrt[x]])/16","A",1
363,1,34,37,0.0159705,"\int \sin ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[ArcSin[Sqrt[x]],x]","\frac{1}{2} \left(\sqrt{-((x-1) x)}+\sin ^{-1}\left(\sqrt{1-x}\right)\right)+x \sin ^{-1}\left(\sqrt{x}\right)","\frac{1}{2} \sqrt{1-x} \sqrt{x}+\frac{1}{4} \sin ^{-1}(1-2 x)+x \sin ^{-1}\left(\sqrt{x}\right)",1,"(Sqrt[-((-1 + x)*x)] + ArcSin[Sqrt[1 - x]])/2 + x*ArcSin[Sqrt[x]]","A",1
364,1,53,56,0.0373411,"\int \frac{\sin ^{-1}\left(\sqrt{x}\right)}{x} \, dx","Integrate[ArcSin[Sqrt[x]]/x,x]","2 \sin ^{-1}\left(\sqrt{x}\right) \log \left(1-e^{2 i \sin ^{-1}\left(\sqrt{x}\right)}\right)-i \left(\sin ^{-1}\left(\sqrt{x}\right)^2+\text{Li}_2\left(e^{2 i \sin ^{-1}\left(\sqrt{x}\right)}\right)\right)","-i \text{Li}_2\left(e^{2 i \sin ^{-1}\left(\sqrt{x}\right)}\right)-i \sin ^{-1}\left(\sqrt{x}\right)^2+2 \sin ^{-1}\left(\sqrt{x}\right) \log \left(1-e^{2 i \sin ^{-1}\left(\sqrt{x}\right)}\right)",1,"2*ArcSin[Sqrt[x]]*Log[1 - E^((2*I)*ArcSin[Sqrt[x]])] - I*(ArcSin[Sqrt[x]]^2 + PolyLog[2, E^((2*I)*ArcSin[Sqrt[x]])])","A",1
365,1,23,28,0.0179015,"\int \frac{\sin ^{-1}\left(\sqrt{x}\right)}{x^2} \, dx","Integrate[ArcSin[Sqrt[x]]/x^2,x]","-\frac{\sqrt{x-x^2}+\sin ^{-1}\left(\sqrt{x}\right)}{x}","-\frac{\sqrt{1-x}}{\sqrt{x}}-\frac{\sin ^{-1}\left(\sqrt{x}\right)}{x}",1,"-((Sqrt[x - x^2] + ArcSin[Sqrt[x]])/x)","A",1
366,1,32,50,0.0400008,"\int \frac{\sin ^{-1}\left(\sqrt{x}\right)}{x^3} \, dx","Integrate[ArcSin[Sqrt[x]]/x^3,x]","-\frac{\sqrt{-((x-1) x)} (2 x+1)+3 \sin ^{-1}\left(\sqrt{x}\right)}{6 x^2}","-\frac{\sqrt{1-x}}{6 x^{3/2}}-\frac{\sin ^{-1}\left(\sqrt{x}\right)}{2 x^2}-\frac{\sqrt{1-x}}{3 \sqrt{x}}",1,"-1/6*(Sqrt[-((-1 + x)*x)]*(1 + 2*x) + 3*ArcSin[Sqrt[x]])/x^2","A",1
367,1,44,68,0.0318516,"\int \frac{\sin ^{-1}\left(\sqrt{x}\right)}{x^4} \, dx","Integrate[ArcSin[Sqrt[x]]/x^4,x]","2 \left(-\frac{\sin ^{-1}\left(\sqrt{x}\right)}{6 x^3}-\frac{\sqrt{1-x} \left(8 x^2+4 x+3\right)}{90 x^{5/2}}\right)","-\frac{4 \sqrt{1-x}}{45 x^{3/2}}-\frac{\sqrt{1-x}}{15 x^{5/2}}-\frac{\sin ^{-1}\left(\sqrt{x}\right)}{3 x^3}-\frac{8 \sqrt{1-x}}{45 \sqrt{x}}",1,"2*(-1/90*(Sqrt[1 - x]*(3 + 4*x + 8*x^2))/x^(5/2) - ArcSin[Sqrt[x]]/(6*x^3))","A",1
368,1,49,86,0.0377929,"\int \frac{\sin ^{-1}\left(\sqrt{x}\right)}{x^5} \, dx","Integrate[ArcSin[Sqrt[x]]/x^5,x]","2 \left(-\frac{\sin ^{-1}\left(\sqrt{x}\right)}{8 x^4}-\frac{\sqrt{1-x} \left(16 x^3+8 x^2+6 x+5\right)}{280 x^{7/2}}\right)","-\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{\sin ^{-1}\left(\sqrt{x}\right)}{4 x^4}-\frac{4 \sqrt{1-x}}{35 \sqrt{x}}",1,"2*(-1/280*(Sqrt[1 - x]*(5 + 6*x + 8*x^2 + 16*x^3))/x^(7/2) - ArcSin[Sqrt[x]]/(8*x^4))","A",1
369,1,91,89,0.0850878,"\int x^4 \left(a+b \sin ^{-1}\left(\frac{c}{x}\right)\right) \, dx","Integrate[x^4*(a + b*ArcSin[c/x]),x]","\frac{a x^5}{5}+\frac{3}{40} b c^5 \log \left(x \left(\sqrt{\frac{x^2-c^2}{x^2}}+1\right)\right)+b \sqrt{\frac{x^2-c^2}{x^2}} \left(\frac{3 c^3 x^2}{40}+\frac{c x^4}{20}\right)+\frac{1}{5} b x^5 \sin ^{-1}\left(\frac{c}{x}\right)","\frac{1}{5} x^5 \left(a+b \sin ^{-1}\left(\frac{c}{x}\right)\right)+\frac{1}{20} b c x^4 \sqrt{1-\frac{c^2}{x^2}}+\frac{3}{40} b c^5 \tanh ^{-1}\left(\sqrt{1-\frac{c^2}{x^2}}\right)+\frac{3}{40} b c^3 x^2 \sqrt{1-\frac{c^2}{x^2}}",1,"(a*x^5)/5 + b*Sqrt[(-c^2 + x^2)/x^2]*((3*c^3*x^2)/40 + (c*x^4)/20) + (b*x^5*ArcSin[c/x])/5 + (3*b*c^5*Log[x*(1 + Sqrt[(-c^2 + x^2)/x^2])])/40","A",1
370,1,59,64,0.0541957,"\int x^3 \left(a+b \sin ^{-1}\left(\frac{c}{x}\right)\right) \, dx","Integrate[x^3*(a + b*ArcSin[c/x]),x]","\frac{a x^4}{4}+b \sqrt{\frac{x^2-c^2}{x^2}} \left(\frac{c^3 x}{6}+\frac{c x^3}{12}\right)+\frac{1}{4} b x^4 \sin ^{-1}\left(\frac{c}{x}\right)","\frac{1}{4} x^4 \left(a+b \sin ^{-1}\left(\frac{c}{x}\right)\right)+\frac{1}{12} b c x^3 \sqrt{1-\frac{c^2}{x^2}}+\frac{1}{6} b c^3 x \sqrt{1-\frac{c^2}{x^2}}",1,"(a*x^4)/4 + b*Sqrt[(-c^2 + x^2)/x^2]*((c^3*x)/6 + (c*x^3)/12) + (b*x^4*ArcSin[c/x])/4","A",1
371,1,79,64,0.0376402,"\int x^2 \left(a+b \sin ^{-1}\left(\frac{c}{x}\right)\right) \, dx","Integrate[x^2*(a + b*ArcSin[c/x]),x]","\frac{a x^3}{3}+\frac{1}{6} b c x^2 \sqrt{\frac{x^2-c^2}{x^2}}+\frac{1}{6} b c^3 \log \left(x \left(\sqrt{\frac{x^2-c^2}{x^2}}+1\right)\right)+\frac{1}{3} b x^3 \sin ^{-1}\left(\frac{c}{x}\right)","\frac{1}{3} x^3 \left(a+b \sin ^{-1}\left(\frac{c}{x}\right)\right)+\frac{1}{6} b c x^2 \sqrt{1-\frac{c^2}{x^2}}+\frac{1}{6} b c^3 \tanh ^{-1}\left(\sqrt{1-\frac{c^2}{x^2}}\right)",1,"(a*x^3)/3 + (b*c*x^2*Sqrt[(-c^2 + x^2)/x^2])/6 + (b*x^3*ArcSin[c/x])/3 + (b*c^3*Log[x*(1 + Sqrt[(-c^2 + x^2)/x^2])])/6","A",1
372,1,47,39,0.0322125,"\int x \left(a+b \sin ^{-1}\left(\frac{c}{x}\right)\right) \, dx","Integrate[x*(a + b*ArcSin[c/x]),x]","\frac{a x^2}{2}+\frac{1}{2} b c x \sqrt{\frac{x^2-c^2}{x^2}}+\frac{1}{2} b x^2 \sin ^{-1}\left(\frac{c}{x}\right)","\frac{1}{2} x^2 \left(a+b \sin ^{-1}\left(\frac{c}{x}\right)\right)+\frac{1}{2} b c x \sqrt{1-\frac{c^2}{x^2}}",1,"(a*x^2)/2 + (b*c*x*Sqrt[(-c^2 + x^2)/x^2])/2 + (b*x^2*ArcSin[c/x])/2","A",1
373,1,89,31,0.1005332,"\int \left(a+b \sin ^{-1}\left(\frac{c}{x}\right)\right) \, dx","Integrate[a + b*ArcSin[c/x],x]","a x+\frac{b c \sqrt{x^2-c^2} \left(\log \left(\frac{x}{\sqrt{x^2-c^2}}+1\right)-\log \left(1-\frac{x}{\sqrt{x^2-c^2}}\right)\right)}{2 x \sqrt{1-\frac{c^2}{x^2}}}+b x \sin ^{-1}\left(\frac{c}{x}\right)","a x+b c \tanh ^{-1}\left(\sqrt{1-\frac{c^2}{x^2}}\right)+b x \csc ^{-1}\left(\frac{x}{c}\right)",1,"a*x + b*x*ArcSin[c/x] + (b*c*Sqrt[-c^2 + x^2]*(-Log[1 - x/Sqrt[-c^2 + x^2]] + Log[1 + x/Sqrt[-c^2 + x^2]]))/(2*Sqrt[1 - c^2/x^2]*x)","B",1
374,1,61,67,0.0389157,"\int \frac{a+b \sin ^{-1}\left(\frac{c}{x}\right)}{x} \, dx","Integrate[(a + b*ArcSin[c/x])/x,x]","a \log (x)+\frac{1}{2} i b \left(\sin ^{-1}\left(\frac{c}{x}\right)^2+\text{Li}_2\left(e^{2 i \sin ^{-1}\left(\frac{c}{x}\right)}\right)\right)-b \sin ^{-1}\left(\frac{c}{x}\right) \log \left(1-e^{2 i \sin ^{-1}\left(\frac{c}{x}\right)}\right)","a \log (x)+\frac{1}{2} i b \text{Li}_2\left(e^{2 i \sin ^{-1}\left(\frac{c}{x}\right)}\right)+\frac{1}{2} i b \sin ^{-1}\left(\frac{c}{x}\right)^2-b \sin ^{-1}\left(\frac{c}{x}\right) \log \left(1-e^{2 i \sin ^{-1}\left(\frac{c}{x}\right)}\right)",1,"-(b*ArcSin[c/x]*Log[1 - E^((2*I)*ArcSin[c/x])]) + a*Log[x] + (I/2)*b*(ArcSin[c/x]^2 + PolyLog[2, E^((2*I)*ArcSin[c/x])])","A",1
375,1,39,39,0.0278883,"\int \frac{a+b \sin ^{-1}\left(\frac{c}{x}\right)}{x^2} \, dx","Integrate[(a + b*ArcSin[c/x])/x^2,x]","-\frac{a}{x}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{c}-\frac{b \sin ^{-1}\left(\frac{c}{x}\right)}{x}","-\frac{a}{x}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{c}-\frac{b \csc ^{-1}\left(\frac{x}{c}\right)}{x}",1,"-((b*Sqrt[1 - c^2/x^2])/c) - a/x - (b*ArcSin[c/x])/x","A",1
376,1,65,57,0.0387829,"\int \frac{a+b \sin ^{-1}\left(\frac{c}{x}\right)}{x^3} \, dx","Integrate[(a + b*ArcSin[c/x])/x^3,x]","-\frac{a}{2 x^2}-\frac{b \sqrt{\frac{x^2-c^2}{x^2}}}{4 c x}+\frac{b \sin ^{-1}\left(\frac{c}{x}\right)}{4 c^2}-\frac{b \sin ^{-1}\left(\frac{c}{x}\right)}{2 x^2}","-\frac{a+b \sin ^{-1}\left(\frac{c}{x}\right)}{2 x^2}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{4 c x}+\frac{b \csc ^{-1}\left(\frac{x}{c}\right)}{4 c^2}",1,"-1/2*a/x^2 - (b*Sqrt[(-c^2 + x^2)/x^2])/(4*c*x) + (b*ArcSin[c/x])/(4*c^2) - (b*ArcSin[c/x])/(2*x^2)","A",1
377,1,60,62,0.0674615,"\int \frac{a+b \sin ^{-1}\left(\frac{c}{x}\right)}{x^4} \, dx","Integrate[(a + b*ArcSin[c/x])/x^4,x]","-\frac{a}{3 x^3}+b \left(-\frac{2}{9 c^3}-\frac{1}{9 c x^2}\right) \sqrt{\frac{x^2-c^2}{x^2}}-\frac{b \sin ^{-1}\left(\frac{c}{x}\right)}{3 x^3}","-\frac{a+b \sin ^{-1}\left(\frac{c}{x}\right)}{3 x^3}+\frac{b \left(1-\frac{c^2}{x^2}\right)^{3/2}}{9 c^3}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{3 c^3}",1,"-1/3*a/x^3 + b*(-2/(9*c^3) - 1/(9*c*x^2))*Sqrt[(-c^2 + x^2)/x^2] - (b*ArcSin[c/x])/(3*x^3)","A",1
378,1,77,82,0.0531903,"\int \frac{a+b \sin ^{-1}\left(\frac{c}{x}\right)}{x^5} \, dx","Integrate[(a + b*ArcSin[c/x])/x^5,x]","-\frac{a}{4 x^4}+\frac{3 b \sin ^{-1}\left(\frac{c}{x}\right)}{32 c^4}+b \left(-\frac{3}{32 c^3 x}-\frac{1}{16 c x^3}\right) \sqrt{\frac{x^2-c^2}{x^2}}-\frac{b \sin ^{-1}\left(\frac{c}{x}\right)}{4 x^4}","-\frac{a+b \sin ^{-1}\left(\frac{c}{x}\right)}{4 x^4}+\frac{3 b \csc ^{-1}\left(\frac{x}{c}\right)}{32 c^4}-\frac{b \sqrt{1-\frac{c^2}{x^2}}}{16 c x^3}-\frac{3 b \sqrt{1-\frac{c^2}{x^2}}}{32 c^3 x}",1,"-1/4*a/x^4 + b*(-1/16*1/(c*x^3) - 3/(32*c^3*x))*Sqrt[(-c^2 + x^2)/x^2] + (3*b*ArcSin[c/x])/(32*c^4) - (b*ArcSin[c/x])/(4*x^4)","A",1
379,1,78,81,0.134555,"\int x^m \left(a+b \sin ^{-1}\left(c x^n\right)\right) \, dx","Integrate[x^m*(a + b*ArcSin[c*x^n]),x]","\frac{x^{m+1} \left((m+n+1) \left(a+b \sin ^{-1}\left(c x^n\right)\right)-b c n x^n \, _2F_1\left(\frac{1}{2},\frac{m+n+1}{2 n};\frac{m+3 n+1}{2 n};c^2 x^{2 n}\right)\right)}{(m+1) (m+n+1)}","\frac{x^{m+1} \left(a+b \sin ^{-1}\left(c x^n\right)\right)}{m+1}-\frac{b c n x^{m+n+1} \, _2F_1\left(\frac{1}{2},\frac{m+n+1}{2 n};\frac{m+3 n+1}{2 n};c^2 x^{2 n}\right)}{(m+1) (m+n+1)}",1,"(x^(1 + m)*((1 + m + n)*(a + b*ArcSin[c*x^n]) - b*c*n*x^n*Hypergeometric2F1[1/2, (1 + m + n)/(2*n), (1 + m + 3*n)/(2*n), c^2*x^(2*n)]))/((1 + m)*(1 + m + n))","A",1
380,1,75,68,0.0836875,"\int x^2 \left(a+b \sin ^{-1}\left(c x^n\right)\right) \, dx","Integrate[x^2*(a + b*ArcSin[c*x^n]),x]","\frac{a x^3}{3}-\frac{b c n x^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2 n};\frac{n+3}{2 n}+1;c^2 x^{2 n}\right)}{3 (n+3)}+\frac{1}{3} b x^3 \sin ^{-1}\left(c x^n\right)","\frac{1}{3} x^3 \left(a+b \sin ^{-1}\left(c x^n\right)\right)-\frac{b c n x^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2 n};\frac{3 (n+1)}{2 n};c^2 x^{2 n}\right)}{3 (n+3)}",1,"(a*x^3)/3 + (b*x^3*ArcSin[c*x^n])/3 - (b*c*n*x^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/(2*n), 1 + (3 + n)/(2*n), c^2*x^(2*n)])/(3*(3 + n))","A",1
381,1,75,69,0.0759503,"\int x \left(a+b \sin ^{-1}\left(c x^n\right)\right) \, dx","Integrate[x*(a + b*ArcSin[c*x^n]),x]","\frac{a x^2}{2}-\frac{b c n x^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2 n};\frac{n+2}{2 n}+1;c^2 x^{2 n}\right)}{2 (n+2)}+\frac{1}{2} b x^2 \sin ^{-1}\left(c x^n\right)","\frac{1}{2} x^2 \left(a+b \sin ^{-1}\left(c x^n\right)\right)-\frac{b c n x^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2 n};\frac{1}{2} \left(3+\frac{2}{n}\right);c^2 x^{2 n}\right)}{2 (n+2)}",1,"(a*x^2)/2 + (b*x^2*ArcSin[c*x^n])/2 - (b*c*n*x^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/(2*n), 1 + (2 + n)/(2*n), c^2*x^(2*n)])/(2*(2 + n))","A",1
382,1,60,60,0.0417861,"\int \left(a+b \sin ^{-1}\left(c x^n\right)\right) \, dx","Integrate[a + b*ArcSin[c*x^n],x]","a x-\frac{b c n x^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);c^2 x^{2 n}\right)}{n+1}+b x \sin ^{-1}\left(c x^n\right)","a x-\frac{b c n x^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);c^2 x^{2 n}\right)}{n+1}+b x \sin ^{-1}\left(c x^n\right)",1,"a*x + b*x*ArcSin[c*x^n] - (b*c*n*x^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/(2*n), (3 + n^(-1))/2, c^2*x^(2*n)])/(1 + n)","A",1
383,1,157,75,0.2113358,"\int \frac{a+b \sin ^{-1}\left(c x^n\right)}{x} \, dx","Integrate[(a + b*ArcSin[c*x^n])/x,x]","a \log (x)-\frac{b c \left(\log (x) \log \left(\sqrt{1-c^2 x^{2 n}}+\sqrt{-c^2} x^n\right)+\frac{i \left(i \sinh ^{-1}\left(\sqrt{-c^2} x^n\right) \log \left(1-e^{-2 \sinh ^{-1}\left(\sqrt{-c^2} x^n\right)}\right)-\frac{1}{2} i \left(\text{Li}_2\left(e^{-2 \sinh ^{-1}\left(\sqrt{-c^2} x^n\right)}\right)-\sinh ^{-1}\left(\sqrt{-c^2} x^n\right)^2\right)\right)}{n}\right)}{\sqrt{-c^2}}+b \log (x) \sin ^{-1}\left(c x^n\right)","a \log (x)-\frac{i b \text{Li}_2\left(e^{2 i \sin ^{-1}\left(c x^n\right)}\right)}{2 n}-\frac{i b \sin ^{-1}\left(c x^n\right)^2}{2 n}+\frac{b \sin ^{-1}\left(c x^n\right) \log \left(1-e^{2 i \sin ^{-1}\left(c x^n\right)}\right)}{n}",1,"a*Log[x] + b*ArcSin[c*x^n]*Log[x] - (b*c*(Log[x]*Log[Sqrt[-c^2]*x^n + Sqrt[1 - c^2*x^(2*n)]] + (I*(I*ArcSinh[Sqrt[-c^2]*x^n]*Log[1 - E^(-2*ArcSinh[Sqrt[-c^2]*x^n])] - (I/2)*(-ArcSinh[Sqrt[-c^2]*x^n]^2 + PolyLog[2, E^(-2*ArcSinh[Sqrt[-c^2]*x^n])])))/n))/Sqrt[-c^2]","B",0
384,1,68,69,0.0844673,"\int \frac{a+b \sin ^{-1}\left(c x^n\right)}{x^2} \, dx","Integrate[(a + b*ArcSin[c*x^n])/x^2,x]","-\frac{a}{x}+\frac{b c n x^{n-1} \, _2F_1\left(\frac{1}{2},\frac{n-1}{2 n};\frac{n-1}{2 n}+1;c^2 x^{2 n}\right)}{n-1}-\frac{b \sin ^{-1}\left(c x^n\right)}{x}","-\frac{a+b \sin ^{-1}\left(c x^n\right)}{x}-\frac{b c n x^{n-1} \, _2F_1\left(\frac{1}{2},-\frac{1-n}{2 n};\frac{1}{2} \left(3-\frac{1}{n}\right);c^2 x^{2 n}\right)}{1-n}",1,"-(a/x) - (b*ArcSin[c*x^n])/x + (b*c*n*x^(-1 + n)*Hypergeometric2F1[1/2, (-1 + n)/(2*n), 1 + (-1 + n)/(2*n), c^2*x^(2*n)])/(-1 + n)","A",1
385,1,75,72,0.060018,"\int \frac{a+b \sin ^{-1}\left(c x^n\right)}{x^3} \, dx","Integrate[(a + b*ArcSin[c*x^n])/x^3,x]","-\frac{a}{2 x^2}+\frac{b c n x^{n-2} \, _2F_1\left(\frac{1}{2},\frac{n-2}{2 n};\frac{n-2}{2 n}+1;c^2 x^{2 n}\right)}{2 (n-2)}-\frac{b \sin ^{-1}\left(c x^n\right)}{2 x^2}","-\frac{a+b \sin ^{-1}\left(c x^n\right)}{2 x^2}-\frac{b c n x^{n-2} \, _2F_1\left(\frac{1}{2},\frac{1}{2} \left(1-\frac{2}{n}\right);\frac{1}{2} \left(3-\frac{2}{n}\right);c^2 x^{2 n}\right)}{2 (2-n)}",1,"-1/2*a/x^2 - (b*ArcSin[c*x^n])/(2*x^2) + (b*c*n*x^(-2 + n)*Hypergeometric2F1[1/2, (-2 + n)/(2*n), 1 + (-2 + n)/(2*n), c^2*x^(2*n)])/(2*(-2 + n))","A",1
386,1,116,129,0.1390743,"\int x^5 \left(a+b \sin ^{-1}\left(c+d x^2\right)\right) \, dx","Integrate[x^5*(a + b*ArcSin[c + d*x^2]),x]","\frac{a x^6}{6}+\frac{b c \left(2 c^2+3\right) \sin ^{-1}\left(c+d x^2\right)}{12 d^3}+\frac{1}{2} b \left(\frac{11 c^2+4}{18 d^3}-\frac{5 c x^2}{18 d^2}+\frac{x^4}{9 d}\right) \sqrt{-c^2-2 c d x^2-d^2 x^4+1}+\frac{1}{6} b x^6 \sin ^{-1}\left(c+d x^2\right)","\frac{1}{6} x^6 \left(a+b \sin ^{-1}\left(c+d x^2\right)\right)+\frac{b c \left(2 c^2+3\right) \sin ^{-1}\left(c+d x^2\right)}{12 d^3}+\frac{b x^4 \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{18 d}+\frac{b \left(11 c^2-5 c d x^2+4\right) \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{36 d^3}",1,"(a*x^6)/6 + (b*((4 + 11*c^2)/(18*d^3) - (5*c*x^2)/(18*d^2) + x^4/(9*d))*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/2 + (b*c*(3 + 2*c^2)*ArcSin[c + d*x^2])/(12*d^3) + (b*x^6*ArcSin[c + d*x^2])/6","A",1
387,1,98,115,0.0982184,"\int x^3 \left(a+b \sin ^{-1}\left(c+d x^2\right)\right) \, dx","Integrate[x^3*(a + b*ArcSin[c + d*x^2]),x]","\frac{a x^4}{4}-\frac{b \left(2 c^2+1\right) \sin ^{-1}\left(c+d x^2\right)}{8 d^2}+\frac{1}{2} b \left(\frac{x^2}{4 d}-\frac{3 c}{4 d^2}\right) \sqrt{-c^2-2 c d x^2-d^2 x^4+1}+\frac{1}{4} b x^4 \sin ^{-1}\left(c+d x^2\right)","\frac{1}{4} x^4 \left(a+b \sin ^{-1}\left(c+d x^2\right)\right)-\frac{b \left(2 c^2+1\right) \sin ^{-1}\left(c+d x^2\right)}{8 d^2}+\frac{b x^2 \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{8 d}-\frac{3 b c \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{8 d^2}",1,"(a*x^4)/4 + (b*((-3*c)/(4*d^2) + x^2/(4*d))*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/2 - (b*(1 + 2*c^2)*ArcSin[c + d*x^2])/(8*d^2) + (b*x^4*ArcSin[c + d*x^2])/4","A",1
388,1,70,57,0.0636637,"\int x \left(a+b \sin ^{-1}\left(c+d x^2\right)\right) \, dx","Integrate[x*(a + b*ArcSin[c + d*x^2]),x]","\frac{a x^2}{2}+\frac{b \left(\sqrt{-c^2-2 c d x^2-d^2 x^4+1}+c \sin ^{-1}\left(c+d x^2\right)\right)}{2 d}+\frac{1}{2} b x^2 \sin ^{-1}\left(c+d x^2\right)","\frac{a x^2}{2}+\frac{b \sqrt{1-\left(c+d x^2\right)^2}}{2 d}+\frac{b \left(c+d x^2\right) \sin ^{-1}\left(c+d x^2\right)}{2 d}",1,"(a*x^2)/2 + (b*x^2*ArcSin[c + d*x^2])/2 + (b*(Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4] + c*ArcSin[c + d*x^2]))/(2*d)","A",1
389,1,230,214,0.0267669,"\int \frac{a+b \sin ^{-1}\left(c+d x^2\right)}{x} \, dx","Integrate[(a + b*ArcSin[c + d*x^2])/x,x]","a \log (x)-\frac{1}{2} i b \text{Li}_2\left(-\frac{e^{i \sin ^{-1}\left(d x^2+c\right)}}{\sqrt{1-c^2}-i c}\right)-\frac{1}{2} i b \text{Li}_2\left(\frac{e^{i \sin ^{-1}\left(d x^2+c\right)}}{i c+\sqrt{1-c^2}}\right)+\frac{1}{2} b \sin ^{-1}\left(c+d x^2\right) \log \left(1+\frac{e^{i \sin ^{-1}\left(c+d x^2\right)}}{d \left(-\frac{\sqrt{1-c^2}}{d}-\frac{i c}{d}\right)}\right)+\frac{1}{2} b \sin ^{-1}\left(c+d x^2\right) \log \left(1+\frac{e^{i \sin ^{-1}\left(c+d x^2\right)}}{d \left(\frac{\sqrt{1-c^2}}{d}-\frac{i c}{d}\right)}\right)-\frac{1}{4} i b \sin ^{-1}\left(c+d x^2\right)^2","a \log (x)-\frac{1}{2} i b \text{Li}_2\left(\frac{e^{i \sin ^{-1}\left(d x^2+c\right)}}{i c-\sqrt{1-c^2}}\right)-\frac{1}{2} i b \text{Li}_2\left(\frac{e^{i \sin ^{-1}\left(d x^2+c\right)}}{i c+\sqrt{1-c^2}}\right)+\frac{1}{2} b \sin ^{-1}\left(c+d x^2\right) \log \left(1-\frac{e^{i \sin ^{-1}\left(c+d x^2\right)}}{-\sqrt{1-c^2}+i c}\right)+\frac{1}{2} b \sin ^{-1}\left(c+d x^2\right) \log \left(1-\frac{e^{i \sin ^{-1}\left(c+d x^2\right)}}{\sqrt{1-c^2}+i c}\right)-\frac{1}{4} i b \sin ^{-1}\left(c+d x^2\right)^2",1,"(-1/4*I)*b*ArcSin[c + d*x^2]^2 + (b*ArcSin[c + d*x^2]*Log[1 + E^(I*ArcSin[c + d*x^2])/((((-I)*c)/d - Sqrt[1 - c^2]/d)*d)])/2 + (b*ArcSin[c + d*x^2]*Log[1 + E^(I*ArcSin[c + d*x^2])/((((-I)*c)/d + Sqrt[1 - c^2]/d)*d)])/2 + a*Log[x] - (I/2)*b*PolyLog[2, -(E^(I*ArcSin[c + d*x^2])/((-I)*c + Sqrt[1 - c^2]))] - (I/2)*b*PolyLog[2, E^(I*ArcSin[c + d*x^2])/(I*c + Sqrt[1 - c^2])]","A",1
390,1,81,90,0.0941642,"\int \frac{a+b \sin ^{-1}\left(c+d x^2\right)}{x^3} \, dx","Integrate[(a + b*ArcSin[c + d*x^2])/x^3,x]","\frac{1}{2} \left(-\frac{a+b \sin ^{-1}\left(c+d x^2\right)}{x^2}-\frac{b d \tanh ^{-1}\left(\frac{-c^2-c d x^2+1}{\sqrt{1-c^2} \sqrt{1-\left(c+d x^2\right)^2}}\right)}{\sqrt{1-c^2}}\right)","-\frac{a+b \sin ^{-1}\left(c+d x^2\right)}{2 x^2}-\frac{b d \tanh ^{-1}\left(\frac{-c^2-c d x^2+1}{\sqrt{1-c^2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}\right)}{2 \sqrt{1-c^2}}",1,"(-((a + b*ArcSin[c + d*x^2])/x^2) - (b*d*ArcTanh[(1 - c^2 - c*d*x^2)/(Sqrt[1 - c^2]*Sqrt[1 - (c + d*x^2)^2])])/Sqrt[1 - c^2])/2","A",1
391,1,150,137,0.3190327,"\int \frac{a+b \sin ^{-1}\left(c+d x^2\right)}{x^5} \, dx","Integrate[(a + b*ArcSin[c + d*x^2])/x^5,x]","-\frac{a}{4 x^4}+\frac{b d \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{4 \left(c^2-1\right) x^2}+\frac{b c d^2 \tanh ^{-1}\left(\frac{-c^2-c d x^2+1}{\sqrt{1-c^2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}\right)}{4 (c-1) (c+1) \sqrt{1-c^2}}-\frac{b \sin ^{-1}\left(c+d x^2\right)}{4 x^4}","-\frac{a+b \sin ^{-1}\left(c+d x^2\right)}{4 x^4}-\frac{b d \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{4 \left(1-c^2\right) x^2}-\frac{b c d^2 \tanh ^{-1}\left(\frac{-c^2-c d x^2+1}{\sqrt{1-c^2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}\right)}{4 \left(1-c^2\right)^{3/2}}",1,"-1/4*a/x^4 + (b*d*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(4*(-1 + c^2)*x^2) - (b*ArcSin[c + d*x^2])/(4*x^4) + (b*c*d^2*ArcTanh[(1 - c^2 - c*d*x^2)/(Sqrt[1 - c^2]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])])/(4*(-1 + c)*(1 + c)*Sqrt[1 - c^2])","A",1
392,1,176,190,0.2780579,"\int \frac{a+b \sin ^{-1}\left(c+d x^2\right)}{x^7} \, dx","Integrate[(a + b*ArcSin[c + d*x^2])/x^7,x]","-\frac{a}{6 x^6}+b \left(\frac{d}{12 \left(c^2-1\right) x^4}-\frac{c d^2}{4 \left(c^2-1\right)^2 x^2}\right) \sqrt{-c^2-2 c d x^2-d^2 x^4+1}-\frac{b \left(2 c^2+1\right) d^3 \tanh ^{-1}\left(\frac{-c^2-c d x^2+1}{\sqrt{1-c^2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}\right)}{12 (c-1)^2 (c+1)^2 \sqrt{1-c^2}}-\frac{b \sin ^{-1}\left(c+d x^2\right)}{6 x^6}","-\frac{a+b \sin ^{-1}\left(c+d x^2\right)}{6 x^6}-\frac{b c d^2 \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{4 \left(1-c^2\right)^2 x^2}-\frac{b d \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{12 \left(1-c^2\right) x^4}-\frac{b \left(2 c^2+1\right) d^3 \tanh ^{-1}\left(\frac{-c^2-c d x^2+1}{\sqrt{1-c^2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}\right)}{12 \left(1-c^2\right)^{5/2}}",1,"-1/6*a/x^6 + b*(d/(12*(-1 + c^2)*x^4) - (c*d^2)/(4*(-1 + c^2)^2*x^2))*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4] - (b*ArcSin[c + d*x^2])/(6*x^6) - (b*(1 + 2*c^2)*d^3*ArcTanh[(1 - c^2 - c*d*x^2)/(Sqrt[1 - c^2]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])])/(12*(-1 + c)^2*(1 + c)^2*Sqrt[1 - c^2])","A",1
393,1,349,336,0.6588471,"\int x^4 \left(a+b \sin ^{-1}\left(c+d x^2\right)\right) \, dx","Integrate[x^4*(a + b*ArcSin[c + d*x^2]),x]","\frac{x \sqrt{\frac{d}{c+1}} \left(15 a d^2 x^4 \sqrt{-c^2-2 c d x^2-d^2 x^4+1}+15 b d^2 x^4 \sqrt{-c^2-2 c d x^2-d^2 x^4+1} \sin ^{-1}\left(c+d x^2\right)+2 b \left(8 c^3+13 c^2 d x^2+2 c d^2 x^4-8 c-3 d^3 x^6+3 d x^2\right)\right)-2 i b \left(15 c^3-23 c^2+17 c-9\right) \sqrt{\frac{c+d x^2-1}{c-1}} \sqrt{\frac{c+d x^2+1}{c+1}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c+1}} x\right)|\frac{c+1}{c-1}\right)+2 i b \left(23 c^3-23 c^2+9 c-9\right) \sqrt{\frac{c+d x^2-1}{c-1}} \sqrt{\frac{c+d x^2+1}{c+1}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c+1}} x\right)|\frac{c+1}{c-1}\right)}{75 d^2 \sqrt{\frac{d}{c+1}} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}","\frac{1}{5} x^5 \left(a+b \sin ^{-1}\left(c+d x^2\right)\right)-\frac{16 b c x \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{75 d^2}+\frac{2 b x^3 \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{25 d}+\frac{2 b \sqrt{1-c} (c+1) \left(15 c^2+8 c+9\right) \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{1-c}}\right)|-\frac{1-c}{c+1}\right)}{75 d^{5/2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}-\frac{2 b \sqrt{1-c} (c+1) \left(23 c^2+9\right) \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{1-c}}\right)|-\frac{1-c}{c+1}\right)}{75 d^{5/2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}",1,"(Sqrt[d/(1 + c)]*x*(15*a*d^2*x^4*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4] + 2*b*(-8*c + 8*c^3 + 3*d*x^2 + 13*c^2*d*x^2 + 2*c*d^2*x^4 - 3*d^3*x^6) + 15*b*d^2*x^4*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4]*ArcSin[c + d*x^2]) + (2*I)*b*(-9 + 9*c - 23*c^2 + 23*c^3)*Sqrt[(-1 + c + d*x^2)/(-1 + c)]*Sqrt[(1 + c + d*x^2)/(1 + c)]*EllipticE[I*ArcSinh[Sqrt[d/(1 + c)]*x], (1 + c)/(-1 + c)] - (2*I)*b*(-9 + 17*c - 23*c^2 + 15*c^3)*Sqrt[(-1 + c + d*x^2)/(-1 + c)]*Sqrt[(1 + c + d*x^2)/(1 + c)]*EllipticF[I*ArcSinh[Sqrt[d/(1 + c)]*x], (1 + c)/(-1 + c)])/(75*d^2*Sqrt[d/(1 + c)]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])","C",1
394,1,307,287,0.5897715,"\int x^2 \left(a+b \sin ^{-1}\left(c+d x^2\right)\right) \, dx","Integrate[x^2*(a + b*ArcSin[c + d*x^2]),x]","\frac{x \sqrt{\frac{d}{c+1}} \left(3 a d x^2 \sqrt{-c^2-2 c d x^2-d^2 x^4+1}-2 b \left(c^2+2 c d x^2+d^2 x^4-1\right)+3 b d x^2 \sqrt{-c^2-2 c d x^2-d^2 x^4+1} \sin ^{-1}\left(c+d x^2\right)\right)+2 i b \left(3 c^2-4 c+1\right) \sqrt{\frac{c+d x^2-1}{c-1}} \sqrt{\frac{c+d x^2+1}{c+1}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c+1}} x\right)|\frac{c+1}{c-1}\right)-8 i b (c-1) c \sqrt{\frac{c+d x^2-1}{c-1}} \sqrt{\frac{c+d x^2+1}{c+1}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c+1}} x\right)|\frac{c+1}{c-1}\right)}{9 d \sqrt{\frac{d}{c+1}} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}","\frac{1}{3} x^3 \left(a+b \sin ^{-1}\left(c+d x^2\right)\right)+\frac{2 b x \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{9 d}-\frac{2 b \sqrt{1-c} (c+1) (3 c+1) \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{1-c}}\right)|-\frac{1-c}{c+1}\right)}{9 d^{3/2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}+\frac{8 b \sqrt{1-c} c (c+1) \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{1-c}}\right)|-\frac{1-c}{c+1}\right)}{9 d^{3/2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}",1,"(Sqrt[d/(1 + c)]*x*(3*a*d*x^2*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4] - 2*b*(-1 + c^2 + 2*c*d*x^2 + d^2*x^4) + 3*b*d*x^2*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4]*ArcSin[c + d*x^2]) - (8*I)*b*(-1 + c)*c*Sqrt[(-1 + c + d*x^2)/(-1 + c)]*Sqrt[(1 + c + d*x^2)/(1 + c)]*EllipticE[I*ArcSinh[Sqrt[d/(1 + c)]*x], (1 + c)/(-1 + c)] + (2*I)*b*(1 - 4*c + 3*c^2)*Sqrt[(-1 + c + d*x^2)/(-1 + c)]*Sqrt[(1 + c + d*x^2)/(1 + c)]*EllipticF[I*ArcSinh[Sqrt[d/(1 + c)]*x], (1 + c)/(-1 + c)])/(9*d*Sqrt[d/(1 + c)]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])","C",1
395,1,155,237,0.1530176,"\int \left(a+b \sin ^{-1}\left(c+d x^2\right)\right) \, dx","Integrate[a + b*ArcSin[c + d*x^2],x]","a x+\frac{2 i b (c-1) \sqrt{\frac{c+d x^2-1}{c-1}} \sqrt{\frac{c+d x^2+1}{c+1}} \left(E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c+1}} x\right)|\frac{c+1}{c-1}\right)-F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c+1}} x\right)|\frac{c+1}{c-1}\right)\right)}{\sqrt{\frac{d}{c+1}} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}+b x \sin ^{-1}\left(c+d x^2\right)","a x+\frac{2 b \sqrt{1-c} (c+1) \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{1-c}}\right)|-\frac{1-c}{c+1}\right)}{\sqrt{d} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}-\frac{2 b \sqrt{1-c} (c+1) \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{1-c}}\right)|-\frac{1-c}{c+1}\right)}{\sqrt{d} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}+b x \sin ^{-1}\left(c+d x^2\right)",1,"a*x + b*x*ArcSin[c + d*x^2] + ((2*I)*b*(-1 + c)*Sqrt[(-1 + c + d*x^2)/(-1 + c)]*Sqrt[(1 + c + d*x^2)/(1 + c)]*(EllipticE[I*ArcSinh[Sqrt[d/(1 + c)]*x], (1 + c)/(-1 + c)] - EllipticF[I*ArcSinh[Sqrt[d/(1 + c)]*x], (1 + c)/(-1 + c)]))/(Sqrt[d/(1 + c)]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])","C",1
396,1,140,126,0.2436103,"\int \frac{a+b \sin ^{-1}\left(c+d x^2\right)}{x^2} \, dx","Integrate[(a + b*ArcSin[c + d*x^2])/x^2,x]","-\frac{a}{x}-\frac{2 i b d \sqrt{1-\frac{d x^2}{-c-1}} \sqrt{1-\frac{d x^2}{1-c}} F\left(i \sinh ^{-1}\left(\sqrt{-\frac{d}{-c-1}} x\right)|\frac{-c-1}{1-c}\right)}{\sqrt{-\frac{d}{-c-1}} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}-\frac{b \sin ^{-1}\left(c+d x^2\right)}{x}","\frac{2 b \sqrt{1-c} \sqrt{d} \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{1-c}}\right)|-\frac{1-c}{c+1}\right)}{\sqrt{-c^2-2 c d x^2-d^2 x^4+1}}-\frac{a+b \sin ^{-1}\left(c+d x^2\right)}{x}",1,"-(a/x) - (b*ArcSin[c + d*x^2])/x - ((2*I)*b*d*Sqrt[1 - (d*x^2)/(-1 - c)]*Sqrt[1 - (d*x^2)/(1 - c)]*EllipticF[I*ArcSinh[Sqrt[-(d/(-1 - c))]*x], (-1 - c)/(1 - c)])/(Sqrt[-(d/(-1 - c))]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])","C",1
397,1,243,284,0.420306,"\int \frac{a+b \sin ^{-1}\left(c+d x^2\right)}{x^4} \, dx","Integrate[(a + b*ArcSin[c + d*x^2])/x^4,x]","-\frac{a}{3 x^3}+\frac{2 b d \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{3 \left(c^2-1\right) x}+\frac{2 i b (1-c) d^2 \sqrt{1-\frac{d x^2}{-c-1}} \sqrt{1-\frac{d x^2}{1-c}} \left(E\left(i \sinh ^{-1}\left(\sqrt{-\frac{d}{-c-1}} x\right)|\frac{-c-1}{1-c}\right)-F\left(i \sinh ^{-1}\left(\sqrt{-\frac{d}{-c-1}} x\right)|\frac{-c-1}{1-c}\right)\right)}{3 (c-1) (c+1) \sqrt{-\frac{d}{-c-1}} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}-\frac{b \sin ^{-1}\left(c+d x^2\right)}{3 x^3}","-\frac{a+b \sin ^{-1}\left(c+d x^2\right)}{3 x^3}-\frac{2 b d \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{3 \left(1-c^2\right) x}+\frac{2 b d^{3/2} \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{1-c}}\right)|-\frac{1-c}{c+1}\right)}{3 \sqrt{1-c} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}-\frac{2 b d^{3/2} \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{1-c}}\right)|-\frac{1-c}{c+1}\right)}{3 \sqrt{1-c} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}",1,"-1/3*a/x^3 + (2*b*d*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(3*(-1 + c^2)*x) - (b*ArcSin[c + d*x^2])/(3*x^3) + (((2*I)/3)*b*(1 - c)*d^2*Sqrt[1 - (d*x^2)/(-1 - c)]*Sqrt[1 - (d*x^2)/(1 - c)]*(EllipticE[I*ArcSinh[Sqrt[-(d/(-1 - c))]*x], (-1 - c)/(1 - c)] - EllipticF[I*ArcSinh[Sqrt[-(d/(-1 - c))]*x], (-1 - c)/(1 - c)]))/((-1 + c)*(1 + c)*Sqrt[-(d/(-1 - c))]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])","C",1
398,1,370,355,0.8511093,"\int \frac{a+b \sin ^{-1}\left(c+d x^2\right)}{x^6} \, dx","Integrate[(a + b*ArcSin[c + d*x^2])/x^6,x]","\frac{\sqrt{\frac{d}{c+1}} \left(-3 a \left(c^2-1\right)^2 \sqrt{-c^2-2 c d x^2-d^2 x^4+1}-3 b \left(c^2-1\right)^2 \sqrt{-c^2-2 c d x^2-d^2 x^4+1} \sin ^{-1}\left(c+d x^2\right)+2 b d x^2 \left(-c^4+2 c^3 d x^2+c^2 \left(7 d^2 x^4+2\right)+c \left(4 d^3 x^6-2 d x^2\right)+d^2 x^4-1\right)\right)-2 i b \left(3 c^2-4 c+1\right) d^3 x^5 \sqrt{\frac{c+d x^2-1}{c-1}} \sqrt{\frac{c+d x^2+1}{c+1}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c+1}} x\right)|\frac{c+1}{c-1}\right)+8 i b (c-1) c d^3 x^5 \sqrt{\frac{c+d x^2-1}{c-1}} \sqrt{\frac{c+d x^2+1}{c+1}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{d}{c+1}} x\right)|\frac{c+1}{c-1}\right)}{15 \left(c^2-1\right)^2 x^5 \sqrt{\frac{d}{c+1}} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}","-\frac{a+b \sin ^{-1}\left(c+d x^2\right)}{5 x^5}-\frac{8 b c d^2 \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{15 \left(1-c^2\right)^2 x}-\frac{2 b d \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{15 \left(1-c^2\right) x^3}+\frac{2 b (3 c+1) d^{5/2} \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} F\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{1-c}}\right)|-\frac{1-c}{c+1}\right)}{15 \sqrt{1-c} \left(1-c^2\right) \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}-\frac{8 b c d^{5/2} \sqrt{1-\frac{d x^2}{1-c}} \sqrt{\frac{d x^2}{c+1}+1} E\left(\sin ^{-1}\left(\frac{\sqrt{d} x}{\sqrt{1-c}}\right)|-\frac{1-c}{c+1}\right)}{15 \sqrt{1-c} \left(1-c^2\right) \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}",1,"(Sqrt[d/(1 + c)]*(-3*a*(-1 + c^2)^2*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4] + 2*b*d*x^2*(-1 - c^4 + 2*c^3*d*x^2 + d^2*x^4 + c^2*(2 + 7*d^2*x^4) + c*(-2*d*x^2 + 4*d^3*x^6)) - 3*b*(-1 + c^2)^2*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4]*ArcSin[c + d*x^2]) + (8*I)*b*(-1 + c)*c*d^3*x^5*Sqrt[(-1 + c + d*x^2)/(-1 + c)]*Sqrt[(1 + c + d*x^2)/(1 + c)]*EllipticE[I*ArcSinh[Sqrt[d/(1 + c)]*x], (1 + c)/(-1 + c)] - (2*I)*b*(1 - 4*c + 3*c^2)*d^3*x^5*Sqrt[(-1 + c + d*x^2)/(-1 + c)]*Sqrt[(1 + c + d*x^2)/(1 + c)]*EllipticF[I*ArcSinh[Sqrt[d/(1 + c)]*x], (1 + c)/(-1 + c)])/(15*(-1 + c^2)^2*Sqrt[d/(1 + c)]*x^5*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])","C",1
399,1,41,47,0.0292592,"\int x^3 \sin ^{-1}\left(a+b x^4\right) \, dx","Integrate[x^3*ArcSin[a + b*x^4],x]","\frac{\sqrt{1-\left(a+b x^4\right)^2}+\left(a+b x^4\right) \sin ^{-1}\left(a+b x^4\right)}{4 b}","\frac{\sqrt{1-\left(a+b x^4\right)^2}}{4 b}+\frac{\left(a+b x^4\right) \sin ^{-1}\left(a+b x^4\right)}{4 b}",1,"(Sqrt[1 - (a + b*x^4)^2] + (a + b*x^4)*ArcSin[a + b*x^4])/(4*b)","A",1
400,1,47,47,0.0399299,"\int x^{-1+n} \sin ^{-1}\left(a+b x^n\right) \, dx","Integrate[x^(-1 + n)*ArcSin[a + b*x^n],x]","\frac{\sqrt{1-\left(a+b x^n\right)^2}}{b n}+\frac{\left(a+b x^n\right) \sin ^{-1}\left(a+b x^n\right)}{b n}","\frac{\sqrt{1-\left(a+b x^n\right)^2}}{b n}+\frac{\left(a+b x^n\right) \sin ^{-1}\left(a+b x^n\right)}{b n}",1,"Sqrt[1 - (a + b*x^n)^2]/(b*n) + ((a + b*x^n)*ArcSin[a + b*x^n])/(b*n)","A",1
401,1,123,127,0.1177232,"\int \left(a+b \sin ^{-1}\left(1+d x^2\right)\right)^4 \, dx","Integrate[(a + b*ArcSin[1 + d*x^2])^4,x]","-48 b^2 \left(\frac{4 b \sqrt{-d x^2 \left(d x^2+2\right)} \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)}{d x}+x \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^2-8 b^2 x\right)+x \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^4+\frac{8 b \sqrt{-d x^2 \left(d x^2+2\right)} \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^3}{d x}","-\frac{192 b^3 \sqrt{-d^2 x^4-2 d x^2} \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)}{d x}-48 b^2 x \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^2+\frac{8 b \sqrt{-d^2 x^4-2 d x^2} \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^3}{d x}+x \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^4+384 b^4 x",1,"(8*b*Sqrt[-(d*x^2*(2 + d*x^2))]*(a + b*ArcSin[1 + d*x^2])^3)/(d*x) + x*(a + b*ArcSin[1 + d*x^2])^4 - 48*b^2*(-8*b^2*x + (4*b*Sqrt[-(d*x^2*(2 + d*x^2))]*(a + b*ArcSin[1 + d*x^2]))/(d*x) + x*(a + b*ArcSin[1 + d*x^2])^2)","A",1
402,1,162,110,0.1253798,"\int \left(a+b \sin ^{-1}\left(1+d x^2\right)\right)^3 \, dx","Integrate[(a + b*ArcSin[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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-1}\left(d x^2+1\right)\right)^2}{d x}+x \left(a+b \sin ^{-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 \sin ^{-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))])*ArcSin[1 + d*x^2] + 3*b^2*(a*d*x^2 + 2*b*Sqrt[-(d*x^2*(2 + d*x^2))])*ArcSin[1 + d*x^2]^2 + b^3*d*x^2*ArcSin[1 + d*x^2]^3)/(d*x)","A",1
403,1,63,63,0.0169587,"\int \left(a+b \sin ^{-1}\left(1+d x^2\right)\right)^2 \, dx","Integrate[(a + b*ArcSin[1 + d*x^2])^2,x]","\frac{4 b \sqrt{-d^2 x^4-2 d x^2} \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)}{d x}+x \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^2-8 b^2 x","\frac{4 b \sqrt{-d^2 x^4-2 d x^2} \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)}{d x}+x \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^2-8 b^2 x",1,"-8*b^2*x + (4*b*Sqrt[-2*d*x^2 - d^2*x^4]*(a + b*ArcSin[1 + d*x^2]))/(d*x) + x*(a + b*ArcSin[1 + d*x^2])^2","A",1
404,1,41,43,0.0311568,"\int \left(a+b \sin ^{-1}\left(1+d x^2\right)\right) \, dx","Integrate[a + b*ArcSin[1 + d*x^2],x]","a x+\frac{2 b \sqrt{-d x^2 \left(d x^2+2\right)}}{d x}+b x \sin ^{-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 \sin ^{-1}\left(d x^2+1\right)",1,"a*x + (2*b*Sqrt[-(d*x^2*(2 + d*x^2))])/(d*x) + b*x*ArcSin[1 + d*x^2]","A",1
405,1,120,159,0.7903979,"\int \frac{1}{a+b \sin ^{-1}\left(1+d x^2\right)} \, dx","Integrate[(a + b*ArcSin[1 + d*x^2])^(-1),x]","-\frac{x \left(\left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{1}{2} \left(\frac{a}{b}+\sin ^{-1}\left(d x^2+1\right)\right)\right)+\left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{1}{2} \left(\frac{a}{b}+\sin ^{-1}\left(d x^2+1\right)\right)\right)\right)}{2 b \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}","-\frac{x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{a+b \sin ^{-1}\left(d x^2+1\right)}{2 b}\right)}{2 b \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}-\frac{x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{a+b \sin ^{-1}\left(d x^2+1\right)}{2 b}\right)}{2 b \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}",1,"-1/2*(x*(CosIntegral[(a/b + ArcSin[1 + d*x^2])/2]*(Cos[a/(2*b)] - Sin[a/(2*b)]) + (Cos[a/(2*b)] + Sin[a/(2*b)])*SinIntegral[(a/b + ArcSin[1 + d*x^2])/2]))/(b*(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2]))","A",1
406,1,164,205,1.4660438,"\int \frac{1}{\left(a+b \sin ^{-1}\left(1+d x^2\right)\right)^2} \, dx","Integrate[(a + b*ArcSin[1 + d*x^2])^(-2),x]","-\frac{\frac{x^2 \left(\left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{1}{2} \left(\frac{a}{b}+\sin ^{-1}\left(d x^2+1\right)\right)\right)+\left(\sin \left(\frac{a}{2 b}\right)-\cos \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{1}{2} \left(\frac{a}{b}+\sin ^{-1}\left(d x^2+1\right)\right)\right)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)}+\frac{2 b \sqrt{-d x^2 \left(d x^2+2\right)}}{d \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)}}{4 b^2 x}","-\frac{x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{a+b \sin ^{-1}\left(d x^2+1\right)}{2 b}\right)}{4 b^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}+\frac{x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{a+b \sin ^{-1}\left(d x^2+1\right)}{2 b}\right)}{4 b^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}-\frac{\sqrt{-d^2 x^4-2 d x^2}}{2 b d x \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)}",1,"-1/4*((2*b*Sqrt[-(d*x^2*(2 + d*x^2))])/(d*(a + b*ArcSin[1 + d*x^2])) + (x^2*(CosIntegral[(a/b + ArcSin[1 + d*x^2])/2]*(Cos[a/(2*b)] + Sin[a/(2*b)]) + (-Cos[a/(2*b)] + Sin[a/(2*b)])*SinIntegral[(a/b + ArcSin[1 + d*x^2])/2]))/(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2]))/(b^2*x)","A",1
407,1,187,227,0.5586747,"\int \frac{1}{\left(a+b \sin ^{-1}\left(1+d x^2\right)\right)^3} \, dx","Integrate[(a + b*ArcSin[1 + d*x^2])^(-3),x]","\frac{x \left(\left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{1}{2} \left(\frac{a}{b}+\sin ^{-1}\left(d x^2+1\right)\right)\right)+\left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{1}{2} \left(\frac{a}{b}+\sin ^{-1}\left(d x^2+1\right)\right)\right)\right)}{16 b^3 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}+\frac{x}{8 b^2 \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)}-\frac{\sqrt{-d x^2 \left(d x^2+2\right)}}{4 b d x \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^2}","\frac{x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{a+b \sin ^{-1}\left(d x^2+1\right)}{2 b}\right)}{16 b^3 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}+\frac{x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{a+b \sin ^{-1}\left(d x^2+1\right)}{2 b}\right)}{16 b^3 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}+\frac{x}{8 b^2 \left(a+b \sin ^{-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 \sin ^{-1}\left(d x^2+1\right)\right)^2}",1,"-1/4*Sqrt[-(d*x^2*(2 + d*x^2))]/(b*d*x*(a + b*ArcSin[1 + d*x^2])^2) + x/(8*b^2*(a + b*ArcSin[1 + d*x^2])) + (x*(CosIntegral[(a/b + ArcSin[1 + d*x^2])/2]*(Cos[a/(2*b)] - Sin[a/(2*b)]) + (Cos[a/(2*b)] + Sin[a/(2*b)])*SinIntegral[(a/b + ArcSin[1 + d*x^2])/2]))/(16*b^3*(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2]))","A",1
408,1,131,135,0.1286462,"\int \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^4 \, dx","Integrate[(a - b*ArcSin[1 - d*x^2])^4,x]","-48 b^2 \left(\frac{4 b \sqrt{-d x^2 \left(d x^2-2\right)} \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)}{d x}+x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^2-8 b^2 x\right)+x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^4+\frac{8 b \sqrt{-d x^2 \left(d x^2-2\right)} \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^3}{d x}","-\frac{192 b^3 \sqrt{2 d x^2-d^2 x^4} \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)}{d x}-48 b^2 x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^2+\frac{8 b \sqrt{2 d x^2-d^2 x^4} \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^3}{d x}+x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^4+384 b^4 x",1,"(8*b*Sqrt[-(d*x^2*(-2 + d*x^2))]*(a - b*ArcSin[1 - d*x^2])^3)/(d*x) + x*(a - b*ArcSin[1 - d*x^2])^4 - 48*b^2*(-8*b^2*x + (4*b*Sqrt[-(d*x^2*(-2 + d*x^2))]*(a - b*ArcSin[1 - d*x^2]))/(d*x) + x*(a - b*ArcSin[1 - d*x^2])^2)","A",1
409,1,166,115,0.1308106,"\int \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^3 \, dx","Integrate[(a - b*ArcSin[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 \sin ^{-1}\left(1-d x^2\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 \sin ^{-1}\left(1-d x^2\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 \sin ^{-1}\left(1-d x^2\right)^3}{d x}","-24 a b^2 x+\frac{6 b \sqrt{2 d x^2-d^2 x^4} \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^2}{d x}+x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^3-\frac{48 b^3 \sqrt{2 d x^2-d^2 x^4}}{d x}+24 b^3 x \sin ^{-1}\left(1-d x^2\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))])*ArcSin[1 - d*x^2] + 3*b^2*(a*d*x^2 + 2*b*Sqrt[-(d*x^2*(-2 + d*x^2))])*ArcSin[1 - d*x^2]^2 - b^3*d*x^2*ArcSin[1 - d*x^2]^3)/(d*x)","A",1
410,1,67,67,0.0247733,"\int \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^2 \, dx","Integrate[(a - b*ArcSin[1 - d*x^2])^2,x]","\frac{4 b \sqrt{2 d x^2-d^2 x^4} \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)}{d x}+x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^2-8 b^2 x","\frac{4 b \sqrt{2 d x^2-d^2 x^4} \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)}{d x}+x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^2-8 b^2 x",1,"-8*b^2*x + (4*b*Sqrt[2*d*x^2 - d^2*x^4]*(a - b*ArcSin[1 - d*x^2]))/(d*x) + x*(a - b*ArcSin[1 - d*x^2])^2","A",1
411,1,43,45,0.0299852,"\int \left(a-b \sin ^{-1}\left(1-d x^2\right)\right) \, dx","Integrate[a - b*ArcSin[1 - d*x^2],x]","a x+\frac{2 b \sqrt{-d x^2 \left(d x^2-2\right)}}{d x}+b (-x) \sin ^{-1}\left(1-d x^2\right)","a x+\frac{2 b \sqrt{2 d x^2-d^2 x^4}}{d x}+b (-x) \sin ^{-1}\left(1-d x^2\right)",1,"a*x + (2*b*Sqrt[-(d*x^2*(-2 + d*x^2))])/(d*x) - b*x*ArcSin[1 - d*x^2]","A",1
412,1,130,168,0.2195906,"\int \frac{1}{a-b \sin ^{-1}\left(1-d x^2\right)} \, dx","Integrate[(a - b*ArcSin[1 - d*x^2])^(-1),x]","\frac{\left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right) \left(\left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{1}{2} \left(\sin ^{-1}\left(1-d x^2\right)-\frac{a}{b}\right)\right)+\left(\sin \left(\frac{a}{2 b}\right)-\cos \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{a-b \sin ^{-1}\left(1-d x^2\right)}{2 b}\right)\right)}{2 b d x}","\frac{x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(-\frac{a-b \sin ^{-1}\left(1-d x^2\right)}{2 b}\right)}{2 b \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}-\frac{x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{a}{2 b}-\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)}{2 b \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}",1,"((Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2])*(CosIntegral[(-(a/b) + ArcSin[1 - d*x^2])/2]*(Cos[a/(2*b)] + Sin[a/(2*b)]) + (-Cos[a/(2*b)] + Sin[a/(2*b)])*SinIntegral[(a - b*ArcSin[1 - d*x^2])/(2*b)]))/(2*b*d*x)","A",1
413,1,183,216,0.4087555,"\int \frac{1}{\left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^2} \, dx","Integrate[(a - b*ArcSin[1 - d*x^2])^(-2),x]","\frac{\left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right) \left(a-b \sin ^{-1}\left(1-d x^2\right)\right) \left(\left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{1}{2} \left(\sin ^{-1}\left(1-d x^2\right)-\frac{a}{b}\right)\right)+\left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{a-b \sin ^{-1}\left(1-d x^2\right)}{2 b}\right)\right)+2 b \sqrt{d x^2 \left(2-d x^2\right)}}{4 b^2 d x \left(b \sin ^{-1}\left(1-d x^2\right)-a\right)}","-\frac{x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(-\frac{a-b \sin ^{-1}\left(1-d x^2\right)}{2 b}\right)}{4 b^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}-\frac{x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{a}{2 b}-\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)}{4 b^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}-\frac{\sqrt{2 d x^2-d^2 x^4}}{2 b d x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)}",1,"(2*b*Sqrt[d*x^2*(2 - d*x^2)] + (a - b*ArcSin[1 - d*x^2])*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2])*(CosIntegral[(-(a/b) + ArcSin[1 - d*x^2])/2]*(Cos[a/(2*b)] - Sin[a/(2*b)]) + (Cos[a/(2*b)] + Sin[a/(2*b)])*SinIntegral[(a - b*ArcSin[1 - d*x^2])/(2*b)]))/(4*b^2*d*x*(-a + b*ArcSin[1 - d*x^2]))","A",0
414,1,195,240,0.5673862,"\int \frac{1}{\left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^3} \, dx","Integrate[(a - b*ArcSin[1 - d*x^2])^(-3),x]","-\frac{\frac{4 b^2 \sqrt{-d x^2 \left(d x^2-2\right)}}{d \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^2}+\frac{\left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right) \left(\left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{1}{2} \left(\sin ^{-1}\left(1-d x^2\right)-\frac{a}{b}\right)\right)+\left(\sin \left(\frac{a}{2 b}\right)-\cos \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{a-b \sin ^{-1}\left(1-d x^2\right)}{2 b}\right)\right)}{d}-\frac{2 b x^2}{a-b \sin ^{-1}\left(1-d x^2\right)}}{16 b^3 x}","-\frac{x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(-\frac{a-b \sin ^{-1}\left(1-d x^2\right)}{2 b}\right)}{16 b^3 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}+\frac{x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{a}{2 b}-\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)}{16 b^3 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}+\frac{x}{8 b^2 \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)}-\frac{\sqrt{2 d x^2-d^2 x^4}}{4 b d x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^2}",1,"-1/16*((4*b^2*Sqrt[-(d*x^2*(-2 + d*x^2))])/(d*(a - b*ArcSin[1 - d*x^2])^2) - (2*b*x^2)/(a - b*ArcSin[1 - d*x^2]) + ((Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2])*(CosIntegral[(-(a/b) + ArcSin[1 - d*x^2])/2]*(Cos[a/(2*b)] + Sin[a/(2*b)]) + (-Cos[a/(2*b)] + Sin[a/(2*b)])*SinIntegral[(a - b*ArcSin[1 - d*x^2])/(2*b)]))/d)/(b^3*x)","A",1
415,1,40,40,0.0160189,"\int \sin ^{-1}\left(1+x^2\right)^2 \, dx","Integrate[ArcSin[1 + x^2]^2,x]","x \sin ^{-1}\left(x^2+1\right)^2+\frac{4 \sqrt{-x^4-2 x^2} \sin ^{-1}\left(x^2+1\right)}{x}-8 x","x \sin ^{-1}\left(x^2+1\right)^2+\frac{4 \sqrt{-x^4-2 x^2} \sin ^{-1}\left(x^2+1\right)}{x}-8 x",1,"-8*x + (4*Sqrt[-2*x^2 - x^4]*ArcSin[1 + x^2])/x + x*ArcSin[1 + x^2]^2","A",1
416,1,44,44,0.0167713,"\int \sin ^{-1}\left(1-x^2\right)^2 \, dx","Integrate[ArcSin[1 - x^2]^2,x]","x \sin ^{-1}\left(1-x^2\right)^2-\frac{4 \sqrt{2 x^2-x^4} \sin ^{-1}\left(1-x^2\right)}{x}-8 x","x \sin ^{-1}\left(1-x^2\right)^2-\frac{4 \sqrt{2 x^2-x^4} \sin ^{-1}\left(1-x^2\right)}{x}-8 x",1,"-8*x - (4*Sqrt[2*x^2 - x^4]*ArcSin[1 - x^2])/x + x*ArcSin[1 - x^2]^2","A",1
417,1,269,277,0.3105561,"\int \left(a+b \sin ^{-1}\left(1+d x^2\right)\right)^{5/2} \, dx","Integrate[(a + b*ArcSin[1 + d*x^2])^(5/2),x]","-\frac{15 x \left(-\sqrt{\pi } \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)+\sqrt{\frac{1}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right) \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}\right)}{\left(\frac{1}{b}\right)^{5/2} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}+x \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^{5/2}+\frac{5 b \sqrt{-d x^2 \left(d x^2+2\right)} \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^{3/2}}{d x}","-15 b^2 x \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}+\frac{5 b \sqrt{-d^2 x^4-2 d x^2} \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^{3/2}}{d x}+\frac{15 \sqrt{\pi } x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\left(\frac{1}{b}\right)^{5/2} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}-\frac{15 \sqrt{\pi } x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\left(\frac{1}{b}\right)^{5/2} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}+x \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^{5/2}",1,"(5*b*Sqrt[-(d*x^2*(2 + d*x^2))]*(a + b*ArcSin[1 + d*x^2])^(3/2))/(d*x) + x*(a + b*ArcSin[1 + d*x^2])^(5/2) - (15*x*(Sqrt[Pi]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]) - Sqrt[Pi]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]) + Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[1 + d*x^2]]*(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2])))/((b^(-1))^(5/2)*(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2]))","A",1
418,1,249,247,0.4303785,"\int \left(a+b \sin ^{-1}\left(1+d x^2\right)\right)^{3/2} \, dx","Integrate[(a + b*ArcSin[1 + d*x^2])^(3/2),x]","\frac{3 \sqrt{\pi } b^{3/2} x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{b} \sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)}+\frac{3 \sqrt{\pi } b^{3/2} x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{b} \sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)}+\frac{\sqrt{a+b \sin ^{-1}\left(d x^2+1\right)} \left(a d x^2+3 b \sqrt{-d x^2 \left(d x^2+2\right)}+b d x^2 \sin ^{-1}\left(d x^2+1\right)\right)}{d x}","\frac{3 \sqrt{\pi } b^{3/2} x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{b} \sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)}+\frac{3 \sqrt{\pi } b^{3/2} x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{b} \sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)}+\frac{3 b \sqrt{-d^2 x^4-2 d x^2} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{d x}+x \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^{3/2}",1,"(Sqrt[a + b*ArcSin[1 + d*x^2]]*(a*d*x^2 + 3*b*Sqrt[-(d*x^2*(2 + d*x^2))] + b*d*x^2*ArcSin[1 + d*x^2]))/(d*x) + (3*b^(3/2)*Sqrt[Pi]*x*FresnelC[Sqrt[a + b*ArcSin[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2]) + (3*b^(3/2)*Sqrt[Pi]*x*FresnelS[Sqrt[a + b*ArcSin[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2])","A",1
419,1,207,210,0.0645542,"\int \sqrt{a+b \sin ^{-1}\left(1+d x^2\right)} \, dx","Integrate[Sqrt[a + b*ArcSin[1 + d*x^2]],x]","\frac{x \left(-\sqrt{\pi } \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)+\sqrt{\frac{1}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right) \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}\right)}{\sqrt{\frac{1}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}","-\frac{\sqrt{\pi } x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\sqrt{\frac{1}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}+\frac{\sqrt{\pi } x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\sqrt{\frac{1}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}+x \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}",1,"(x*(Sqrt[Pi]*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]) - Sqrt[Pi]*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]) + Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[1 + d*x^2]]*(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2])))/(Sqrt[b^(-1)]*(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2]))","A",1
420,1,143,185,0.0465577,"\int \frac{1}{\sqrt{a+b \sin ^{-1}\left(1+d x^2\right)}} \, dx","Integrate[1/Sqrt[a + b*ArcSin[1 + d*x^2]],x]","-\frac{\sqrt{\pi } x \left(\left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{b} \sqrt{\pi }}\right)+\left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{b} \sqrt{\pi }}\right)\right)}{\sqrt{b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}","-\frac{\sqrt{\pi } x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{b} \sqrt{\pi }}\right)}{\sqrt{b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}-\frac{\sqrt{\pi } x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{b} \sqrt{\pi }}\right)}{\sqrt{b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}",1,"-((Sqrt[Pi]*x*(FresnelC[Sqrt[a + b*ArcSin[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]) + FresnelS[Sqrt[a + b*ArcSin[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)])))/(Sqrt[b]*(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2])))","A",1
421,1,238,238,0.6929153,"\int \frac{1}{\left(a+b \sin ^{-1}\left(1+d x^2\right)\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[1 + d*x^2])^(-3/2),x]","-\frac{\sqrt{-d^2 x^4-2 d x^2}}{b d x \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}-\frac{\sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)}+\frac{\sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)}","-\frac{\sqrt{-d^2 x^4-2 d x^2}}{b d x \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}-\frac{\sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)}+\frac{\sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)}",1,"-(Sqrt[-2*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a + b*ArcSin[1 + d*x^2]])) + ((b^(-1))^(3/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2]) - ((b^(-1))^(3/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2])","A",1
422,1,247,261,0.5220699,"\int \frac{1}{\left(a+b \sin ^{-1}\left(1+d x^2\right)\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[1 + d*x^2])^(-5/2),x]","\frac{x \left(\frac{\sqrt{\pi } \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{b} \sqrt{\pi }}\right)}{\sqrt{b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}+\frac{\sqrt{\pi } \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{b} \sqrt{\pi }}\right)}{\sqrt{b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}+\frac{b \left(d x^2+2\right)}{\sqrt{-d x^2 \left(d x^2+2\right)} \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^{3/2}}+\frac{1}{\sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}\right)}{3 b^2}","\frac{\sqrt{\pi } x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{b} \sqrt{\pi }}\right)}{3 b^{5/2} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}+\frac{\sqrt{\pi } x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{b} \sqrt{\pi }}\right)}{3 b^{5/2} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}+\frac{x}{3 b^2 \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}-\frac{\sqrt{-d^2 x^4-2 d x^2}}{3 b d x \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^{3/2}}",1,"(x*((b*(2 + d*x^2))/(Sqrt[-(d*x^2*(2 + d*x^2))]*(a + b*ArcSin[1 + d*x^2])^(3/2)) + 1/Sqrt[a + b*ArcSin[1 + d*x^2]] + (Sqrt[Pi]*FresnelC[Sqrt[a + b*ArcSin[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Sqrt[b]*(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2])) + (Sqrt[Pi]*FresnelS[Sqrt[a + b*ArcSin[1 + d*x^2]]/(Sqrt[b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Sqrt[b]*(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2]))))/(3*b^2)","A",1
423,1,297,317,0.9296808,"\int \frac{1}{\left(a+b \sin ^{-1}\left(1+d x^2\right)\right)^{7/2}} \, dx","Integrate[(a + b*ArcSin[1 + d*x^2])^(-7/2),x]","\frac{\frac{\sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)}-\frac{\sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)}+\frac{x^2 \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)+\frac{\sqrt{-d x^2 \left(d x^2+2\right)} \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^2}{b d}-\frac{3 b \sqrt{-d x^2 \left(d x^2+2\right)}}{d}}{x \left(a+b \sin ^{-1}\left(d x^2+1\right)\right)^{5/2}}}{15 b^2}","\frac{\sqrt{-d^2 x^4-2 d x^2}}{15 b^3 d x \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}+\frac{x}{15 b^2 \left(a+b \sin ^{-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 \sin ^{-1}\left(d x^2+1\right)\right)^{5/2}}+\frac{\sqrt{\pi } \left(\frac{1}{b}\right)^{7/2} x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{15 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}-\frac{\sqrt{\pi } \left(\frac{1}{b}\right)^{7/2} x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}\left(d x^2+1\right)}}{\sqrt{\pi }}\right)}{15 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(d x^2+1\right)\right)\right)}",1,"(((-3*b*Sqrt[-(d*x^2*(2 + d*x^2))])/d + x^2*(a + b*ArcSin[1 + d*x^2]) + (Sqrt[-(d*x^2*(2 + d*x^2))]*(a + b*ArcSin[1 + d*x^2])^2)/(b*d))/(x*(a + b*ArcSin[1 + d*x^2])^(5/2)) - ((b^(-1))^(3/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2]) + ((b^(-1))^(3/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[1 + d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Cos[ArcSin[1 + d*x^2]/2] - Sin[ArcSin[1 + d*x^2]/2]))/(15*b^2)","A",1
424,1,292,299,0.3443075,"\int \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{5/2} \, dx","Integrate[(a - b*ArcSin[1 - d*x^2])^(5/2),x]","\frac{15 b x \left(-\sqrt{\pi } \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)+\sqrt{-\frac{1}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right) \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}\right)}{\left(-\frac{1}{b}\right)^{3/2} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}+x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{5/2}+\frac{5 b \sqrt{-d x^2 \left(d x^2-2\right)} \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{3/2}}{d x}","-15 b^2 x \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}+\frac{5 b \sqrt{2 d x^2-d^2 x^4} \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{3/2}}{d x}+\frac{15 \sqrt{\pi } x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)}{\left(-\frac{1}{b}\right)^{5/2} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}-\frac{15 \sqrt{\pi } x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)}{\left(-\frac{1}{b}\right)^{5/2} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}+x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{5/2}",1,"(5*b*Sqrt[-(d*x^2*(-2 + d*x^2))]*(a - b*ArcSin[1 - d*x^2])^(3/2))/(d*x) + x*(a - b*ArcSin[1 - d*x^2])^(5/2) + (15*b*x*(-(Sqrt[Pi]*FresnelC[(Sqrt[-b^(-1)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)])) + Sqrt[Pi]*FresnelS[(Sqrt[-b^(-1)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]) + Sqrt[-b^(-1)]*Sqrt[a - b*ArcSin[1 - d*x^2]]*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2])))/((-b^(-1))^(3/2)*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]))","A",1
425,1,265,267,0.5484526,"\int \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{3/2} \, dx","Integrate[(a - b*ArcSin[1 - d*x^2])^(3/2),x]","\frac{3 \sqrt{\pi } (-b)^{3/2} x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{-b} \sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)}+\frac{3 \sqrt{\pi } (-b)^{3/2} x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{-b} \sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)}+x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{3/2}+\frac{3 b \sqrt{-d x^2 \left(d x^2-2\right)} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{d x}","\frac{3 b \sqrt{2 d x^2-d^2 x^4} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{d x}+\frac{3 \sqrt{\pi } (-b)^{3/2} x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{-b} \sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)}+\frac{3 \sqrt{\pi } (-b)^{3/2} x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{-b} \sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)}+x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{3/2}",1,"(3*b*Sqrt[-(d*x^2*(-2 + d*x^2))]*Sqrt[a - b*ArcSin[1 - d*x^2]])/(d*x) + x*(a - b*ArcSin[1 - d*x^2])^(3/2) + (3*(-b)^(3/2)*Sqrt[Pi]*x*FresnelS[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]) + (3*(-b)^(3/2)*Sqrt[Pi]*x*FresnelC[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2])","A",1
426,1,225,228,0.0737305,"\int \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)} \, dx","Integrate[Sqrt[a - b*ArcSin[1 - d*x^2]],x]","\frac{x \left(-\sqrt{\pi } \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)+\sqrt{-\frac{1}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right) \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}\right)}{\sqrt{-\frac{1}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}","-\frac{\sqrt{\pi } x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)}{\sqrt{-\frac{1}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}+\frac{\sqrt{\pi } x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)}{\sqrt{-\frac{1}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}+x \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}",1,"(x*(-(Sqrt[Pi]*FresnelC[(Sqrt[-b^(-1)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)])) + Sqrt[Pi]*FresnelS[(Sqrt[-b^(-1)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]) + Sqrt[-b^(-1)]*Sqrt[a - b*ArcSin[1 - d*x^2]]*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2])))/(Sqrt[-b^(-1)]*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]))","A",1
427,1,155,201,0.0669031,"\int \frac{1}{\sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}} \, dx","Integrate[1/Sqrt[a - b*ArcSin[1 - d*x^2]],x]","\frac{\sqrt{\pi } b x \left(\left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{-b} \sqrt{\pi }}\right)+\left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{-b} \sqrt{\pi }}\right)\right)}{(-b)^{3/2} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}","-\frac{\sqrt{\pi } x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{-b} \sqrt{\pi }}\right)}{\sqrt{-b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}-\frac{\sqrt{\pi } x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{-b} \sqrt{\pi }}\right)}{\sqrt{-b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}",1,"(b*Sqrt[Pi]*x*(FresnelS[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]) + FresnelC[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)])))/((-b)^(3/2)*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]))","A",1
428,1,256,256,0.5418662,"\int \frac{1}{\left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{3/2}} \, dx","Integrate[(a - b*ArcSin[1 - d*x^2])^(-3/2),x]","-\frac{\sqrt{2 d x^2-d^2 x^4}}{b d x \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}-\frac{\sqrt{\pi } \left(-\frac{1}{b}\right)^{3/2} x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)}+\frac{\sqrt{\pi } \left(-\frac{1}{b}\right)^{3/2} x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)}","-\frac{\sqrt{2 d x^2-d^2 x^4}}{b d x \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}-\frac{\sqrt{\pi } \left(-\frac{1}{b}\right)^{3/2} x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)}+\frac{\sqrt{\pi } \left(-\frac{1}{b}\right)^{3/2} x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)}",1,"-(Sqrt[2*d*x^2 - d^2*x^4]/(b*d*x*Sqrt[a - b*ArcSin[1 - d*x^2]])) - ((-b^(-1))^(3/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[-b^(-1)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]) + ((-b^(-1))^(3/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[-b^(-1)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2])","A",1
429,1,270,281,0.9063995,"\int \frac{1}{\left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{5/2}} \, dx","Integrate[(a - b*ArcSin[1 - d*x^2])^(-5/2),x]","\frac{\frac{\sqrt{\pi } x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{-b} \sqrt{\pi }}\right)}{\sqrt{-b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}+\frac{\sqrt{\pi } x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{-b} \sqrt{\pi }}\right)}{\sqrt{-b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}+\frac{x^2 \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)-\frac{b \sqrt{-d x^2 \left(d x^2-2\right)}}{d}}{x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{3/2}}}{3 b^2}","\frac{x}{3 b^2 \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}-\frac{\sqrt{2 d x^2-d^2 x^4}}{3 b d x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{3/2}}+\frac{\sqrt{\pi } x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{-b} \sqrt{\pi }}\right)}{3 (-b)^{5/2} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}+\frac{\sqrt{\pi } x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{-b} \sqrt{\pi }}\right)}{3 (-b)^{5/2} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}",1,"((-((b*Sqrt[-(d*x^2*(-2 + d*x^2))])/d) + x^2*(a - b*ArcSin[1 - d*x^2]))/(x*(a - b*ArcSin[1 - d*x^2])^(3/2)) + (Sqrt[Pi]*x*FresnelS[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Sqrt[-b]*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2])) + (Sqrt[Pi]*x*FresnelC[Sqrt[a - b*ArcSin[1 - d*x^2]]/(Sqrt[-b]*Sqrt[Pi])]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Sqrt[-b]*(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2])))/(3*b^2)","A",1
430,1,319,339,1.0742743,"\int \frac{1}{\left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{7/2}} \, dx","Integrate[(a - b*ArcSin[1 - d*x^2])^(-7/2),x]","\frac{\frac{\sqrt{\pi } \left(-\frac{1}{b}\right)^{3/2} x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)}+\frac{\sqrt{\pi } b \left(-\frac{1}{b}\right)^{5/2} x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)}+\frac{x^2 \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)+\frac{\sqrt{d x^2 \left(2-d x^2\right)} \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^2}{b d}-\frac{3 b \sqrt{d x^2 \left(2-d x^2\right)}}{d}}{x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{5/2}}}{15 b^2}","\frac{\sqrt{2 d x^2-d^2 x^4}}{15 b^3 d x \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}+\frac{x}{15 b^2 \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{3/2}}-\frac{\sqrt{2 d x^2-d^2 x^4}}{5 b d x \left(a-b \sin ^{-1}\left(1-d x^2\right)\right)^{5/2}}+\frac{\sqrt{\pi } \left(-\frac{1}{b}\right)^{7/2} x \left(\cos \left(\frac{a}{2 b}\right)-\sin \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)}{15 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}-\frac{\sqrt{\pi } \left(-\frac{1}{b}\right)^{7/2} x \left(\sin \left(\frac{a}{2 b}\right)+\cos \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{1}{b}} \sqrt{a-b \sin ^{-1}\left(1-d x^2\right)}}{\sqrt{\pi }}\right)}{15 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-d x^2\right)\right)\right)}",1,"(((-3*b*Sqrt[d*x^2*(2 - d*x^2)])/d + x^2*(a - b*ArcSin[1 - d*x^2]) + (Sqrt[d*x^2*(2 - d*x^2)]*(a - b*ArcSin[1 - d*x^2])^2)/(b*d))/(x*(a - b*ArcSin[1 - d*x^2])^(5/2)) + ((-b^(-1))^(3/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[-b^(-1)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] - Sin[a/(2*b)]))/(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]) + ((-b^(-1))^(5/2)*b*Sqrt[Pi]*x*FresnelS[(Sqrt[-b^(-1)]*Sqrt[a - b*ArcSin[1 - d*x^2]])/Sqrt[Pi]]*(Cos[a/(2*b)] + Sin[a/(2*b)]))/(Cos[ArcSin[1 - d*x^2]/2] - Sin[ArcSin[1 - d*x^2]/2]))/(15*b^2)","A",1
431,0,0,43,0.1150433,"\int \frac{\left(a+b \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^n}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2),x]","\int \frac{\left(a+b \sin ^{-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 \sin ^{-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*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]","A",-1
432,0,0,275,0.4230343,"\int \frac{\left(a+b \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^3}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2),x]","\int \frac{\left(a+b \sin ^{-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 \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sin ^{-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 \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 c}+\frac{i \left(a+b \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^4}{4 b c}-\frac{\log \left(1-e^{2 i \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sin ^{-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 \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right)}{4 c}",1,"Integrate[(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2), x]","F",-1
433,0,0,205,0.8565473,"\int \frac{\left(a+b \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2),x]","\int \frac{\left(a+b \sin ^{-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 \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}+\frac{i \left(a+b \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^3}{3 b c}-\frac{\log \left(1-e^{2 i \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sin ^{-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 \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right)}{2 c}",1,"Integrate[(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2), x]","F",-1
434,0,0,141,1.266746,"\int \frac{a+b \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(1 - c^2*x^2),x]","\int \frac{a+b \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)}{1-c^2 x^2} \, dx","\frac{i \left(a+b \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 b c}-\frac{\log \left(1-e^{2 i \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}+\frac{i b \text{Li}_2\left(e^{2 i \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right)}{2 c}",1,"Integrate[(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(1 - c^2*x^2), x]","F",-1
435,0,0,43,0.1102506,"\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)} \, dx","Integrate[1/((1 - c^2*x^2)*(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])),x]","\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \sin ^{-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 \sin ^{-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*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]","A",-1
436,0,0,43,1.9105535,"\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2} \, dx","Integrate[1/((1 - c^2*x^2)*(a + b*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2),x]","\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \sin ^{-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 \sin ^{-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*ArcSin[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]","A",-1
437,1,22,22,0.0097514,"\int e^x \sin ^{-1}\left(e^x\right) \, dx","Integrate[E^x*ArcSin[E^x],x]","\sqrt{1-e^{2 x}}+e^x \sin ^{-1}\left(e^x\right)","\sqrt{1-e^{2 x}}+e^x \sin ^{-1}\left(e^x\right)",1,"Sqrt[1 - E^(2*x)] + E^x*ArcSin[E^x]","A",1
438,0,0,84,1.1355591,"\int \sin ^{-1}\left(c e^{a+b x}\right) \, dx","Integrate[ArcSin[c*E^(a + b*x)],x]","\int \sin ^{-1}\left(c e^{a+b x}\right) \, dx","-\frac{i \text{Li}_2\left(e^{2 i \sin ^{-1}\left(c e^{a+b x}\right)}\right)}{2 b}-\frac{i \sin ^{-1}\left(c e^{a+b x}\right)^2}{2 b}+\frac{\sin ^{-1}\left(c e^{a+b x}\right) \log \left(1-e^{2 i \sin ^{-1}\left(c e^{a+b x}\right)}\right)}{b}",1,"Integrate[ArcSin[c*E^(a + b*x)], x]","F",-1
439,1,50,81,0.1449706,"\int e^{\sin ^{-1}(a x)} x^3 \, dx","Integrate[E^ArcSin[a*x]*x^3,x]","\frac{e^{\sin ^{-1}(a x)} \left(34 \sin \left(2 \sin ^{-1}(a x)\right)-5 \sin \left(4 \sin ^{-1}(a x)\right)-68 \cos \left(2 \sin ^{-1}(a x)\right)+20 \cos \left(4 \sin ^{-1}(a x)\right)\right)}{680 a^4}","\frac{e^{\sin ^{-1}(a x)} \sin \left(2 \sin ^{-1}(a x)\right)}{20 a^4}-\frac{e^{\sin ^{-1}(a x)} \sin \left(4 \sin ^{-1}(a x)\right)}{136 a^4}-\frac{e^{\sin ^{-1}(a x)} \cos \left(2 \sin ^{-1}(a x)\right)}{10 a^4}+\frac{e^{\sin ^{-1}(a x)} \cos \left(4 \sin ^{-1}(a x)\right)}{34 a^4}",1,"(E^ArcSin[a*x]*(-68*Cos[2*ArcSin[a*x]] + 20*Cos[4*ArcSin[a*x]] + 34*Sin[2*ArcSin[a*x]] - 5*Sin[4*ArcSin[a*x]]))/(680*a^4)","A",1
440,1,50,82,0.1346406,"\int e^{\sin ^{-1}(a x)} x^2 \, dx","Integrate[E^ArcSin[a*x]*x^2,x]","-\frac{e^{\sin ^{-1}(a x)} \left(-5 \sqrt{1-a^2 x^2}-5 a x+3 \sin \left(3 \sin ^{-1}(a x)\right)+\cos \left(3 \sin ^{-1}(a x)\right)\right)}{40 a^3}","-\frac{3 e^{\sin ^{-1}(a x)} \sin \left(3 \sin ^{-1}(a x)\right)}{40 a^3}-\frac{e^{\sin ^{-1}(a x)} \cos \left(3 \sin ^{-1}(a x)\right)}{40 a^3}+\frac{x e^{\sin ^{-1}(a x)}}{8 a^2}+\frac{\sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}}{8 a^3}",1,"-1/40*(E^ArcSin[a*x]*(-5*a*x - 5*Sqrt[1 - a^2*x^2] + Cos[3*ArcSin[a*x]] + 3*Sin[3*ArcSin[a*x]]))/a^3","A",1
441,1,30,41,0.048689,"\int e^{\sin ^{-1}(a x)} x \, dx","Integrate[E^ArcSin[a*x]*x,x]","\frac{e^{\sin ^{-1}(a x)} \left(\sin \left(2 \sin ^{-1}(a x)\right)-2 \cos \left(2 \sin ^{-1}(a x)\right)\right)}{10 a^2}","\frac{e^{\sin ^{-1}(a x)} \sin \left(2 \sin ^{-1}(a x)\right)}{10 a^2}-\frac{e^{\sin ^{-1}(a x)} \cos \left(2 \sin ^{-1}(a x)\right)}{5 a^2}",1,"(E^ArcSin[a*x]*(-2*Cos[2*ArcSin[a*x]] + Sin[2*ArcSin[a*x]]))/(10*a^2)","A",1
442,1,31,39,0.028808,"\int e^{\sin ^{-1}(a x)} \, dx","Integrate[E^ArcSin[a*x],x]","\frac{\left(\sqrt{1-a^2 x^2}+a x\right) e^{\sin ^{-1}(a x)}}{2 a}","\frac{\sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}}{2 a}+\frac{1}{2} x e^{\sin ^{-1}(a x)}",1,"(E^ArcSin[a*x]*(a*x + Sqrt[1 - a^2*x^2]))/(2*a)","A",1
443,1,75,43,0.060402,"\int \frac{e^{\sin ^{-1}(a x)}}{x} \, dx","Integrate[E^ArcSin[a*x]/x,x]","i \left(-e^{\sin ^{-1}(a x)} \, _2F_1\left(-\frac{i}{2},1;1-\frac{i}{2};e^{2 i \sin ^{-1}(a x)}\right)-\left(\frac{1}{5}-\frac{2 i}{5}\right) e^{(1+2 i) \sin ^{-1}(a x)} \, _2F_1\left(1,1-\frac{i}{2};2-\frac{i}{2};e^{2 i \sin ^{-1}(a x)}\right)\right)","i e^{\sin ^{-1}(a x)}-2 i e^{\sin ^{-1}(a x)} \, _2F_1\left(-\frac{i}{2},1;1-\frac{i}{2};e^{2 i \sin ^{-1}(a x)}\right)",1,"I*(-(E^ArcSin[a*x]*Hypergeometric2F1[-1/2*I, 1, 1 - I/2, E^((2*I)*ArcSin[a*x])]) - (1/5 - (2*I)/5)*E^((1 + 2*I)*ArcSin[a*x])*Hypergeometric2F1[1, 1 - I/2, 2 - I/2, E^((2*I)*ArcSin[a*x])])","A",0
444,1,54,83,0.1097123,"\int \frac{e^{\sin ^{-1}(a x)}}{x^2} \, dx","Integrate[E^ArcSin[a*x]/x^2,x]","-\frac{e^{\sin ^{-1}(a x)}+(1+i) a x e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};e^{2 i \sin ^{-1}(a x)}\right)}{x}","(1-i) a e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};e^{2 i \sin ^{-1}(a x)}\right)-(2-2 i) a e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left(\frac{1}{2}-\frac{i}{2},2;\frac{3}{2}-\frac{i}{2};e^{2 i \sin ^{-1}(a x)}\right)",1,"-((E^ArcSin[a*x] + (1 + I)*a*E^((1 + I)*ArcSin[a*x])*x*Hypergeometric2F1[1/2 - I/2, 1, 3/2 - I/2, E^((2*I)*ArcSin[a*x])])/x)","A",0
445,1,67,101,0.1509772,"\int e^{\sin ^{-1}(a x)^2} x^3 \, dx","Integrate[E^ArcSin[a*x]^2*x^3,x]","\frac{e \sqrt{\pi } \left(2 \left(\text{erf}\left(1-i \sin ^{-1}(a x)\right)+\text{erf}\left(1+i \sin ^{-1}(a x)\right)\right)-e^3 \left(\text{erf}\left(2-i \sin ^{-1}(a x)\right)+\text{erf}\left(2+i \sin ^{-1}(a x)\right)\right)\right)}{32 a^4}","\frac{e \sqrt{\pi } \text{erf}\left(1-i \sin ^{-1}(a x)\right)}{16 a^4}-\frac{e^4 \sqrt{\pi } \text{erf}\left(2-i \sin ^{-1}(a x)\right)}{32 a^4}+\frac{e \sqrt{\pi } \text{erf}\left(1+i \sin ^{-1}(a x)\right)}{16 a^4}-\frac{e^4 \sqrt{\pi } \text{erf}\left(2+i \sin ^{-1}(a x)\right)}{32 a^4}",1,"(E*Sqrt[Pi]*(2*(Erf[1 - I*ArcSin[a*x]] + Erf[1 + I*ArcSin[a*x]]) - E^3*(Erf[2 - I*ArcSin[a*x]] + Erf[2 + I*ArcSin[a*x]])))/(32*a^4)","A",1
446,1,84,129,0.1108506,"\int e^{\sin ^{-1}(a x)^2} x^2 \, dx","Integrate[E^ArcSin[a*x]^2*x^2,x]","\frac{\sqrt[4]{e} \sqrt{\pi } \left(\text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a x)-i\right)\right)+\text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a x)+i\right)\right)-e^2 \left(\text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a x)-3 i\right)\right)+\text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a x)+3 i\right)\right)\right)\right)}{16 a^3}","\frac{\sqrt[4]{e} \sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a x)-i\right)\right)}{16 a^3}+\frac{\sqrt[4]{e} \sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a x)+i\right)\right)}{16 a^3}-\frac{e^{9/4} \sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a x)-3 i\right)\right)}{16 a^3}-\frac{e^{9/4} \sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a x)+3 i\right)\right)}{16 a^3}",1,"(E^(1/4)*Sqrt[Pi]*(Erfi[(-I + 2*ArcSin[a*x])/2] + Erfi[(I + 2*ArcSin[a*x])/2] - E^2*(Erfi[(-3*I + 2*ArcSin[a*x])/2] + Erfi[(3*I + 2*ArcSin[a*x])/2])))/(16*a^3)","A",1
447,1,36,49,0.0390974,"\int e^{\sin ^{-1}(a x)^2} x \, dx","Integrate[E^ArcSin[a*x]^2*x,x]","\frac{e \sqrt{\pi } \left(\text{erf}\left(1-i \sin ^{-1}(a x)\right)+\text{erf}\left(1+i \sin ^{-1}(a x)\right)\right)}{8 a^2}","\frac{e \sqrt{\pi } \text{erf}\left(1-i \sin ^{-1}(a x)\right)}{8 a^2}+\frac{e \sqrt{\pi } \text{erf}\left(1+i \sin ^{-1}(a x)\right)}{8 a^2}",1,"(E*Sqrt[Pi]*(Erf[1 - I*ArcSin[a*x]] + Erf[1 + I*ArcSin[a*x]]))/(8*a^2)","A",1
448,1,48,65,0.036805,"\int e^{\sin ^{-1}(a x)^2} \, dx","Integrate[E^ArcSin[a*x]^2,x]","\frac{\sqrt[4]{e} \sqrt{\pi } \left(\text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a x)-i\right)\right)+\text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a x)+i\right)\right)\right)}{4 a}","\frac{\sqrt[4]{e} \sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a x)-i\right)\right)}{4 a}+\frac{\sqrt[4]{e} \sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a x)+i\right)\right)}{4 a}",1,"(E^(1/4)*Sqrt[Pi]*(Erfi[(-I + 2*ArcSin[a*x])/2] + Erfi[(I + 2*ArcSin[a*x])/2]))/(4*a)","A",1
449,0,0,20,0.2179182,"\int \frac{e^{\sin ^{-1}(a x)^2}}{x} \, dx","Integrate[E^ArcSin[a*x]^2/x,x]","\int \frac{e^{\sin ^{-1}(a x)^2}}{x} \, dx","a \text{Int}\left(\frac{e^{\sin ^{-1}(a x)^2}}{a x},x\right)",0,"Integrate[E^ArcSin[a*x]^2/x, x]","A",-1
450,0,0,22,1.8305286,"\int \frac{e^{\sin ^{-1}(a x)^2}}{x^2} \, dx","Integrate[E^ArcSin[a*x]^2/x^2,x]","\int \frac{e^{\sin ^{-1}(a x)^2}}{x^2} \, dx","a^2 \text{Int}\left(\frac{e^{\sin ^{-1}(a x)^2}}{a^2 x^2},x\right)",0,"Integrate[E^ArcSin[a*x]^2/x^2, x]","A",-1
451,1,148,309,0.4658117,"\int e^{\sin ^{-1}(a+b x)} x^3 \, dx","Integrate[E^ArcSin[a + b*x]*x^3,x]","\frac{e^{\sin ^{-1}(a+b x)} \left(-340 a^3 (a+b x)-85 \left(4 a^2+3\right) a \sqrt{1-(a+b x)^2}+204 a^2 \sin \left(2 \sin ^{-1}(a+b x)\right)-68 \left(6 a^2+1\right) \cos \left(2 \sin ^{-1}(a+b x)\right)-255 a (a+b x)+153 a \sin \left(3 \sin ^{-1}(a+b x)\right)+34 \sin \left(2 \sin ^{-1}(a+b x)\right)-5 \sin \left(4 \sin ^{-1}(a+b x)\right)+51 a \cos \left(3 \sin ^{-1}(a+b x)\right)+20 \cos \left(4 \sin ^{-1}(a+b x)\right)\right)}{680 b^4}","-\frac{a^3 (a+b x) e^{\sin ^{-1}(a+b x)}}{2 b^4}-\frac{a^3 \sqrt{1-(a+b x)^2} e^{\sin ^{-1}(a+b x)}}{2 b^4}+\frac{3 a^2 e^{\sin ^{-1}(a+b x)} \sin \left(2 \sin ^{-1}(a+b x)\right)}{10 b^4}-\frac{3 a^2 e^{\sin ^{-1}(a+b x)} \cos \left(2 \sin ^{-1}(a+b x)\right)}{5 b^4}-\frac{3 a (a+b x) e^{\sin ^{-1}(a+b x)}}{8 b^4}+\frac{9 a e^{\sin ^{-1}(a+b x)} \sin \left(3 \sin ^{-1}(a+b x)\right)}{40 b^4}-\frac{3 a \sqrt{1-(a+b x)^2} e^{\sin ^{-1}(a+b x)}}{8 b^4}+\frac{e^{\sin ^{-1}(a+b x)} \sin \left(2 \sin ^{-1}(a+b x)\right)}{20 b^4}-\frac{e^{\sin ^{-1}(a+b x)} \sin \left(4 \sin ^{-1}(a+b x)\right)}{136 b^4}+\frac{3 a e^{\sin ^{-1}(a+b x)} \cos \left(3 \sin ^{-1}(a+b x)\right)}{40 b^4}-\frac{e^{\sin ^{-1}(a+b x)} \cos \left(2 \sin ^{-1}(a+b x)\right)}{10 b^4}+\frac{e^{\sin ^{-1}(a+b x)} \cos \left(4 \sin ^{-1}(a+b x)\right)}{34 b^4}",1,"(E^ArcSin[a + b*x]*(-255*a*(a + b*x) - 340*a^3*(a + b*x) - 85*a*(3 + 4*a^2)*Sqrt[1 - (a + b*x)^2] - 68*(1 + 6*a^2)*Cos[2*ArcSin[a + b*x]] + 51*a*Cos[3*ArcSin[a + b*x]] + 20*Cos[4*ArcSin[a + b*x]] + 34*Sin[2*ArcSin[a + b*x]] + 204*a^2*Sin[2*ArcSin[a + b*x]] + 153*a*Sin[3*ArcSin[a + b*x]] - 5*Sin[4*ArcSin[a + b*x]]))/(680*b^4)","A",1
452,1,103,205,0.2558329,"\int e^{\sin ^{-1}(a+b x)} x^2 \, dx","Integrate[E^ArcSin[a + b*x]*x^2,x]","\frac{e^{\sin ^{-1}(a+b x)} \left(20 a^2 (a+b x)+5 \left(4 a^2+1\right) \sqrt{1-(a+b x)^2}+5 (a+b x)-8 a \sin \left(2 \sin ^{-1}(a+b x)\right)-3 \sin \left(3 \sin ^{-1}(a+b x)\right)+16 a \cos \left(2 \sin ^{-1}(a+b x)\right)-\cos \left(3 \sin ^{-1}(a+b x)\right)\right)}{40 b^3}","\frac{a^2 (a+b x) e^{\sin ^{-1}(a+b x)}}{2 b^3}+\frac{a^2 \sqrt{1-(a+b x)^2} e^{\sin ^{-1}(a+b x)}}{2 b^3}-\frac{a e^{\sin ^{-1}(a+b x)} \sin \left(2 \sin ^{-1}(a+b x)\right)}{5 b^3}+\frac{(a+b x) e^{\sin ^{-1}(a+b x)}}{8 b^3}-\frac{3 e^{\sin ^{-1}(a+b x)} \sin \left(3 \sin ^{-1}(a+b x)\right)}{40 b^3}+\frac{\sqrt{1-(a+b x)^2} e^{\sin ^{-1}(a+b x)}}{8 b^3}+\frac{2 a e^{\sin ^{-1}(a+b x)} \cos \left(2 \sin ^{-1}(a+b x)\right)}{5 b^3}-\frac{e^{\sin ^{-1}(a+b x)} \cos \left(3 \sin ^{-1}(a+b x)\right)}{40 b^3}",1,"(E^ArcSin[a + b*x]*(5*(a + b*x) + 20*a^2*(a + b*x) + 5*(1 + 4*a^2)*Sqrt[1 - (a + b*x)^2] + 16*a*Cos[2*ArcSin[a + b*x]] - Cos[3*ArcSin[a + b*x]] - 8*a*Sin[2*ArcSin[a + b*x]] - 3*Sin[3*ArcSin[a + b*x]]))/(40*b^3)","A",1
453,1,59,101,0.1692726,"\int e^{\sin ^{-1}(a+b x)} x \, dx","Integrate[E^ArcSin[a + b*x]*x,x]","-\frac{e^{\sin ^{-1}(a+b x)} \left(\sqrt{1-(a+b x)^2} (3 a-2 b x)+5 a (a+b x)+2 \cos \left(2 \sin ^{-1}(a+b x)\right)\right)}{10 b^2}","-\frac{a (a+b x) e^{\sin ^{-1}(a+b x)}}{2 b^2}-\frac{a \sqrt{1-(a+b x)^2} e^{\sin ^{-1}(a+b x)}}{2 b^2}+\frac{e^{\sin ^{-1}(a+b x)} \sin \left(2 \sin ^{-1}(a+b x)\right)}{10 b^2}-\frac{e^{\sin ^{-1}(a+b x)} \cos \left(2 \sin ^{-1}(a+b x)\right)}{5 b^2}",1,"-1/10*(E^ArcSin[a + b*x]*(5*a*(a + b*x) + (3*a - 2*b*x)*Sqrt[1 - (a + b*x)^2] + 2*Cos[2*ArcSin[a + b*x]]))/b^2","A",0
454,1,35,51,0.0184813,"\int e^{\sin ^{-1}(a+b x)} \, dx","Integrate[E^ArcSin[a + b*x],x]","\frac{\left(\sqrt{1-(a+b x)^2}+a+b x\right) e^{\sin ^{-1}(a+b x)}}{2 b}","\frac{(a+b x) e^{\sin ^{-1}(a+b x)}}{2 b}+\frac{\sqrt{1-(a+b x)^2} e^{\sin ^{-1}(a+b x)}}{2 b}",1,"(E^ArcSin[a + b*x]*(a + b*x + Sqrt[1 - (a + b*x)^2]))/(2*b)","A",1
455,0,0,20,0.1335685,"\int \frac{e^{\sin ^{-1}(a+b x)}}{x} \, dx","Integrate[E^ArcSin[a + b*x]/x,x]","\int \frac{e^{\sin ^{-1}(a+b x)}}{x} \, dx","b \text{Int}\left(\frac{e^{\sin ^{-1}(a+b x)}}{b x},x\right)",0,"Integrate[E^ArcSin[a + b*x]/x, x]","A",-1
456,0,0,22,0.269199,"\int \frac{e^{\sin ^{-1}(a+b x)}}{x^2} \, dx","Integrate[E^ArcSin[a + b*x]/x^2,x]","\int \frac{e^{\sin ^{-1}(a+b x)}}{x^2} \, dx","b^2 \text{Int}\left(\frac{e^{\sin ^{-1}(a+b x)}}{b^2 x^2},x\right)",0,"Integrate[E^ArcSin[a + b*x]/x^2, x]","A",-1
457,1,221,381,0.4102354,"\int e^{\sin ^{-1}(a+b x)^2} x^3 \, dx","Integrate[E^ArcSin[a + b*x]^2*x^3,x]","-\frac{\sqrt{\pi } \left(-2 \left(6 e a^2+e\right) \text{erf}\left(1-i \sin ^{-1}(a+b x)\right)+\sqrt[4]{e} \left(8 a^3 \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)+i\right)\right)-2 i \left(4 a^2+3\right) a \text{erf}\left(\frac{1}{2}+i \sin ^{-1}(a+b x)\right)-2 e^{3/4} \left(6 a^2+1\right) \text{erf}\left(1+i \sin ^{-1}(a+b x)\right)+6 i e^2 a \text{erf}\left(\frac{3}{2}+i \sin ^{-1}(a+b x)\right)+e^{15/4} \text{erf}\left(2+i \sin ^{-1}(a+b x)\right)+6 a \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)+i\right)\right)-6 e^2 a \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)+3 i\right)\right)\right)+e^4 \text{erf}\left(2-i \sin ^{-1}(a+b x)\right)\right)}{32 b^4}","-\frac{\sqrt[4]{e} \sqrt{\pi } a^3 \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)-i\right)\right)}{4 b^4}-\frac{\sqrt[4]{e} \sqrt{\pi } a^3 \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)+i\right)\right)}{4 b^4}+\frac{3 e \sqrt{\pi } a^2 \text{erf}\left(1-i \sin ^{-1}(a+b x)\right)}{8 b^4}+\frac{3 e \sqrt{\pi } a^2 \text{erf}\left(1+i \sin ^{-1}(a+b x)\right)}{8 b^4}+\frac{e \sqrt{\pi } \text{erf}\left(1-i \sin ^{-1}(a+b x)\right)}{16 b^4}-\frac{e^4 \sqrt{\pi } \text{erf}\left(2-i \sin ^{-1}(a+b x)\right)}{32 b^4}+\frac{e \sqrt{\pi } \text{erf}\left(1+i \sin ^{-1}(a+b x)\right)}{16 b^4}-\frac{e^4 \sqrt{\pi } \text{erf}\left(2+i \sin ^{-1}(a+b x)\right)}{32 b^4}-\frac{3 \sqrt[4]{e} \sqrt{\pi } a \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)-i\right)\right)}{16 b^4}-\frac{3 \sqrt[4]{e} \sqrt{\pi } a \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)+i\right)\right)}{16 b^4}+\frac{3 e^{9/4} \sqrt{\pi } a \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)-3 i\right)\right)}{16 b^4}+\frac{3 e^{9/4} \sqrt{\pi } a \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)+3 i\right)\right)}{16 b^4}",1,"-1/32*(Sqrt[Pi]*(-2*(E + 6*a^2*E)*Erf[1 - I*ArcSin[a + b*x]] + E^4*Erf[2 - I*ArcSin[a + b*x]] + E^(1/4)*((-2*I)*a*(3 + 4*a^2)*Erf[1/2 + I*ArcSin[a + b*x]] - 2*(1 + 6*a^2)*E^(3/4)*Erf[1 + I*ArcSin[a + b*x]] + (6*I)*a*E^2*Erf[3/2 + I*ArcSin[a + b*x]] + E^(15/4)*Erf[2 + I*ArcSin[a + b*x]] + 6*a*Erfi[(I + 2*ArcSin[a + b*x])/2] + 8*a^3*Erfi[(I + 2*ArcSin[a + b*x])/2] - 6*a*E^2*Erfi[(3*I + 2*ArcSin[a + b*x])/2])))/b^4","A",1
458,1,161,265,0.2457573,"\int e^{\sin ^{-1}(a+b x)^2} x^2 \, dx","Integrate[E^ArcSin[a + b*x]^2*x^2,x]","-\frac{\sqrt{\pi } \left(i \sqrt[4]{e} \left(4 a^2 \text{erf}\left(\frac{1}{2}+i \sin ^{-1}(a+b x)\right)-\left(4 a^2+1\right) \text{erf}\left(\frac{1}{2}-i \sin ^{-1}(a+b x)\right)-4 i e^{3/4} a \text{erf}\left(1+i \sin ^{-1}(a+b x)\right)+e^2 \text{erf}\left(\frac{3}{2}-i \sin ^{-1}(a+b x)\right)+\text{erf}\left(\frac{1}{2}+i \sin ^{-1}(a+b x)\right)-e^2 \text{erf}\left(\frac{3}{2}+i \sin ^{-1}(a+b x)\right)\right)+4 e a \text{erf}\left(1-i \sin ^{-1}(a+b x)\right)\right)}{16 b^3}","\frac{\sqrt[4]{e} \sqrt{\pi } a^2 \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)-i\right)\right)}{4 b^3}+\frac{\sqrt[4]{e} \sqrt{\pi } a^2 \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)+i\right)\right)}{4 b^3}-\frac{e \sqrt{\pi } a \text{erf}\left(1-i \sin ^{-1}(a+b x)\right)}{4 b^3}-\frac{e \sqrt{\pi } a \text{erf}\left(1+i \sin ^{-1}(a+b x)\right)}{4 b^3}+\frac{\sqrt[4]{e} \sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)-i\right)\right)}{16 b^3}+\frac{\sqrt[4]{e} \sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)+i\right)\right)}{16 b^3}-\frac{e^{9/4} \sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)-3 i\right)\right)}{16 b^3}-\frac{e^{9/4} \sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)+3 i\right)\right)}{16 b^3}",1,"-1/16*(Sqrt[Pi]*(4*a*E*Erf[1 - I*ArcSin[a + b*x]] + I*E^(1/4)*(-((1 + 4*a^2)*Erf[1/2 - I*ArcSin[a + b*x]]) + E^2*Erf[3/2 - I*ArcSin[a + b*x]] + Erf[1/2 + I*ArcSin[a + b*x]] + 4*a^2*Erf[1/2 + I*ArcSin[a + b*x]] - (4*I)*a*E^(3/4)*Erf[1 + I*ArcSin[a + b*x]] - E^2*Erf[3/2 + I*ArcSin[a + b*x]])))/b^3","A",1
459,1,93,123,0.1329979,"\int e^{\sin ^{-1}(a+b x)^2} x \, dx","Integrate[E^ArcSin[a + b*x]^2*x,x]","\frac{\sqrt{\pi } \left(e \text{erf}\left(1-i \sin ^{-1}(a+b x)\right)+e \text{erf}\left(1+i \sin ^{-1}(a+b x)\right)-2 \sqrt[4]{e} a \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)-i\right)\right)-2 \sqrt[4]{e} a \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)+i\right)\right)\right)}{8 b^2}","\frac{e \sqrt{\pi } \text{erf}\left(1-i \sin ^{-1}(a+b x)\right)}{8 b^2}+\frac{e \sqrt{\pi } \text{erf}\left(1+i \sin ^{-1}(a+b x)\right)}{8 b^2}-\frac{\sqrt[4]{e} \sqrt{\pi } a \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)-i\right)\right)}{4 b^2}-\frac{\sqrt[4]{e} \sqrt{\pi } a \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)+i\right)\right)}{4 b^2}",1,"(Sqrt[Pi]*(E*Erf[1 - I*ArcSin[a + b*x]] + E*Erf[1 + I*ArcSin[a + b*x]] - 2*a*E^(1/4)*Erfi[(-I + 2*ArcSin[a + b*x])/2] - 2*a*E^(1/4)*Erfi[(I + 2*ArcSin[a + b*x])/2]))/(8*b^2)","A",1
460,1,52,69,0.0380975,"\int e^{\sin ^{-1}(a+b x)^2} \, dx","Integrate[E^ArcSin[a + b*x]^2,x]","\frac{\sqrt[4]{e} \sqrt{\pi } \left(\text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)-i\right)\right)+\text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)+i\right)\right)\right)}{4 b}","\frac{\sqrt[4]{e} \sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)-i\right)\right)}{4 b}+\frac{\sqrt[4]{e} \sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sin ^{-1}(a+b x)+i\right)\right)}{4 b}",1,"(E^(1/4)*Sqrt[Pi]*(Erfi[(-I + 2*ArcSin[a + b*x])/2] + Erfi[(I + 2*ArcSin[a + b*x])/2]))/(4*b)","A",1
461,0,0,22,0.1528144,"\int \frac{e^{\sin ^{-1}(a+b x)^2}}{x} \, dx","Integrate[E^ArcSin[a + b*x]^2/x,x]","\int \frac{e^{\sin ^{-1}(a+b x)^2}}{x} \, dx","b \text{Int}\left(\frac{e^{\sin ^{-1}(a+b x)^2}}{b x},x\right)",0,"Integrate[E^ArcSin[a + b*x]^2/x, x]","A",-1
462,0,0,24,0.5075564,"\int \frac{e^{\sin ^{-1}(a+b x)^2}}{x^2} \, dx","Integrate[E^ArcSin[a + b*x]^2/x^2,x]","\int \frac{e^{\sin ^{-1}(a+b x)^2}}{x^2} \, dx","b^2 \text{Int}\left(\frac{e^{\sin ^{-1}(a+b x)^2}}{b^2 x^2},x\right)",0,"Integrate[E^ArcSin[a + b*x]^2/x^2, x]","A",-1
463,1,69,162,0.3998573,"\int e^{\sin ^{-1}(a x)} \left(1-a^2 x^2\right)^{5/2} \, dx","Integrate[E^ArcSin[a*x]*(1 - a^2*x^2)^(5/2),x]","\frac{e^{\sin ^{-1}(a x)} \left(3774 \sin \left(2 \sin ^{-1}(a x)\right)+888 \sin \left(4 \sin ^{-1}(a x)\right)+102 \sin \left(6 \sin ^{-1}(a x)\right)+1887 \cos \left(2 \sin ^{-1}(a x)\right)+222 \cos \left(4 \sin ^{-1}(a x)\right)+17 \cos \left(6 \sin ^{-1}(a x)\right)+6290\right)}{20128 a}","\frac{\left(1-a^2 x^2\right)^3 e^{\sin ^{-1}(a x)}}{37 a}+\frac{6}{37} x \left(1-a^2 x^2\right)^{5/2} e^{\sin ^{-1}(a x)}+\frac{30 \left(1-a^2 x^2\right)^2 e^{\sin ^{-1}(a x)}}{629 a}+\frac{120}{629} x \left(1-a^2 x^2\right)^{3/2} e^{\sin ^{-1}(a x)}+\frac{72 \left(1-a^2 x^2\right) e^{\sin ^{-1}(a x)}}{629 a}+\frac{144}{629} x \sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac{144 e^{\sin ^{-1}(a x)}}{629 a}",1,"(E^ArcSin[a*x]*(6290 + 1887*Cos[2*ArcSin[a*x]] + 222*Cos[4*ArcSin[a*x]] + 17*Cos[6*ArcSin[a*x]] + 3774*Sin[2*ArcSin[a*x]] + 888*Sin[4*ArcSin[a*x]] + 102*Sin[6*ArcSin[a*x]]))/(20128*a)","A",1
464,1,51,112,0.1676639,"\int e^{\sin ^{-1}(a x)} \left(1-a^2 x^2\right)^{3/2} \, dx","Integrate[E^ArcSin[a*x]*(1 - a^2*x^2)^(3/2),x]","\frac{e^{\sin ^{-1}(a x)} \left(136 \sin \left(2 \sin ^{-1}(a x)\right)+20 \sin \left(4 \sin ^{-1}(a x)\right)+68 \cos \left(2 \sin ^{-1}(a x)\right)+5 \cos \left(4 \sin ^{-1}(a x)\right)+255\right)}{680 a}","\frac{\left(1-a^2 x^2\right)^2 e^{\sin ^{-1}(a x)}}{17 a}+\frac{4}{17} x \left(1-a^2 x^2\right)^{3/2} e^{\sin ^{-1}(a x)}+\frac{12 \left(1-a^2 x^2\right) e^{\sin ^{-1}(a x)}}{85 a}+\frac{24}{85} x \sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac{24 e^{\sin ^{-1}(a x)}}{85 a}",1,"(E^ArcSin[a*x]*(255 + 68*Cos[2*ArcSin[a*x]] + 5*Cos[4*ArcSin[a*x]] + 136*Sin[2*ArcSin[a*x]] + 20*Sin[4*ArcSin[a*x]]))/(680*a)","A",1
465,1,31,62,0.0870817,"\int e^{\sin ^{-1}(a x)} \sqrt{1-a^2 x^2} \, dx","Integrate[E^ArcSin[a*x]*Sqrt[1 - a^2*x^2],x]","\frac{e^{\sin ^{-1}(a x)} \left(2 \sin \left(2 \sin ^{-1}(a x)\right)+\cos \left(2 \sin ^{-1}(a x)\right)+5\right)}{10 a}","\frac{2}{5} x \sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac{\left(1-a^2 x^2\right) e^{\sin ^{-1}(a x)}}{5 a}+\frac{2 e^{\sin ^{-1}(a x)}}{5 a}",1,"(E^ArcSin[a*x]*(5 + Cos[2*ArcSin[a*x]] + 2*Sin[2*ArcSin[a*x]]))/(10*a)","A",1
466,1,10,10,0.0201001,"\int \frac{e^{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx","Integrate[E^ArcSin[a*x]/Sqrt[1 - a^2*x^2],x]","\frac{e^{\sin ^{-1}(a x)}}{a}","\frac{e^{\sin ^{-1}(a x)}}{a}",1,"E^ArcSin[a*x]/a","A",1
467,1,45,45,0.0818312,"\int \frac{e^{\sin ^{-1}(a x)}}{\left(1-a^2 x^2\right)^{3/2}} \, dx","Integrate[E^ArcSin[a*x]/(1 - a^2*x^2)^(3/2),x]","\frac{\left(\frac{4}{5}-\frac{8 i}{5}\right) e^{(1+2 i) \sin ^{-1}(a x)} \, _2F_1\left(1-\frac{i}{2},2;2-\frac{i}{2};-e^{2 i \sin ^{-1}(a x)}\right)}{a}","\frac{\left(\frac{4}{5}-\frac{8 i}{5}\right) e^{(1+2 i) \sin ^{-1}(a x)} \, _2F_1\left(1-\frac{i}{2},2;2-\frac{i}{2};-e^{2 i \sin ^{-1}(a x)}\right)}{a}",1,"((4/5 - (8*I)/5)*E^((1 + 2*I)*ArcSin[a*x])*Hypergeometric2F1[1 - I/2, 2, 2 - I/2, -E^((2*I)*ArcSin[a*x])])/a","A",1
468,1,84,96,0.1989749,"\int \frac{e^{\sin ^{-1}(a x)}}{\left(1-a^2 x^2\right)^{5/2}} \, dx","Integrate[E^ArcSin[a*x]/(1 - a^2*x^2)^(5/2),x]","\frac{e^{\sin ^{-1}(a x)} \left(\frac{2 a x}{\sqrt{1-a^2 x^2}}+(1-2 i) \left(1+e^{2 i \sin ^{-1}(a x)}\right)^2 \, _2F_1\left(1-\frac{i}{2},2;2-\frac{i}{2};-e^{2 i \sin ^{-1}(a x)}\right)-1\right)}{6 \left(a-a^3 x^2\right)}","\frac{x e^{\sin ^{-1}(a x)}}{3 \left(1-a^2 x^2\right)^{3/2}}-\frac{e^{\sin ^{-1}(a x)}}{6 a \left(1-a^2 x^2\right)}+\frac{\left(\frac{2}{3}-\frac{4 i}{3}\right) e^{(1+2 i) \sin ^{-1}(a x)} \, _2F_1\left(1-\frac{i}{2},2;2-\frac{i}{2};-e^{2 i \sin ^{-1}(a x)}\right)}{a}",1,"(E^ArcSin[a*x]*(-1 + (2*a*x)/Sqrt[1 - a^2*x^2] + (1 - 2*I)*(1 + E^((2*I)*ArcSin[a*x]))^2*Hypergeometric2F1[1 - I/2, 2, 2 - I/2, -E^((2*I)*ArcSin[a*x])]))/(6*(a - a^3*x^2))","A",0
469,1,140,47,0.160249,"\int \sin ^{-1}\left(\frac{c}{a+b x}\right) \, dx","Integrate[ArcSin[c/(a + b*x)],x]","\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}}+x \sin ^{-1}\left(\frac{c}{a+b x}\right)","\frac{c \tanh ^{-1}\left(\sqrt{1-\frac{c^2}{(a+b x)^2}}\right)}{b}+\frac{(a+b x) \csc ^{-1}\left(\frac{a}{c}+\frac{b x}{c}\right)}{b}",1,"x*ArcSin[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
470,1,28,27,0.6526641,"\int \frac{x}{\sin ^{-1}(\sin (x))} \, dx","Integrate[x/ArcSin[Sin[x]],x]","x \sqrt{\cos ^2(x)} \sec (x) \log \left(\sin ^{-1}(\sin (x))\right)-\sin ^{-1}(\sin (x)) \left(\log \left(\sin ^{-1}(\sin (x))\right)-1\right)","\sin ^{-1}(\sin (x))+\log \left(\sin ^{-1}(\sin (x))\right) \left(x \sqrt{\cos ^2(x)} \sec (x)-\sin ^{-1}(\sin (x))\right)",1,"-(ArcSin[Sin[x]]*(-1 + Log[ArcSin[Sin[x]]])) + x*Sqrt[Cos[x]^2]*Log[ArcSin[Sin[x]]]*Sec[x]","A",1
471,1,38,38,0.0707139,"\int \frac{\sin ^{-1}\left(\sqrt{1+b x^2}\right)^n}{\sqrt{1+b x^2}} \, dx","Integrate[ArcSin[Sqrt[1 + b*x^2]]^n/Sqrt[1 + b*x^2],x]","\frac{\sqrt{-b x^2} \sin ^{-1}\left(\sqrt{b x^2+1}\right)^{n+1}}{b (n+1) x}","\frac{\sqrt{-b x^2} \sin ^{-1}\left(\sqrt{b x^2+1}\right)^{n+1}}{b (n+1) x}",1,"(Sqrt[-(b*x^2)]*ArcSin[Sqrt[1 + b*x^2]]^(1 + n))/(b*(1 + n)*x)","A",1
472,1,26,30,0.0309946,"\int \frac{1}{\sqrt{1+b x^2} \sin ^{-1}\left(\sqrt{1+b x^2}\right)} \, dx","Integrate[1/(Sqrt[1 + b*x^2]*ArcSin[Sqrt[1 + b*x^2]]),x]","-\frac{x \log \left(\sin ^{-1}\left(\sqrt{b x^2+1}\right)\right)}{\sqrt{-b x^2}}","\frac{\sqrt{-b x^2} \log \left(\sin ^{-1}\left(\sqrt{b x^2+1}\right)\right)}{b x}",1,"-((x*Log[ArcSin[Sqrt[1 + b*x^2]]])/Sqrt[-(b*x^2)])","A",1
473,1,16,16,0.0295017,"\int \left(\frac{x}{1-x^2}+\frac{1}{\sqrt{1-x^2} \sin ^{-1}(x)}\right) \, dx","Integrate[x/(1 - x^2) + 1/(Sqrt[1 - x^2]*ArcSin[x]),x]","\log \left(\sin ^{-1}(x)\right)-\frac{1}{2} \log \left(1-x^2\right)","\log \left(\sin ^{-1}(x)\right)-\frac{1}{2} \log \left(1-x^2\right)",1,"-1/2*Log[1 - x^2] + Log[ArcSin[x]]","A",1
474,1,16,16,0.1227607,"\int \frac{\sqrt{1-x^2}+x \sin ^{-1}(x)}{\sin ^{-1}(x)-x^2 \sin ^{-1}(x)} \, dx","Integrate[(Sqrt[1 - x^2] + x*ArcSin[x])/(ArcSin[x] - x^2*ArcSin[x]),x]","\log \left(\sin ^{-1}(x)\right)-\frac{1}{2} \log \left(1-x^2\right)","\log \left(\sin ^{-1}(x)\right)-\frac{1}{2} \log \left(1-x^2\right)",1,"-1/2*Log[1 - x^2] + Log[ArcSin[x]]","A",1