1,1,87,128,0.1470916,"\int x^4 \left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^4*(d - c^2*d*x^2)*(a + b*ArcSin[c*x]),x]","\frac{d \left(-105 a x^5 \left(5 c^2 x^2-7\right)-105 b x^5 \left(5 c^2 x^2-7\right) \sin ^{-1}(c x)+\frac{b \sqrt{1-c^2 x^2} \left(-75 c^6 x^6+57 c^4 x^4+76 c^2 x^2+152\right)}{c^5}\right)}{3675}","-\frac{1}{7} c^2 d x^7 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{5} d x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{b d \left(1-c^2 x^2\right)^{7/2}}{49 c^5}-\frac{8 b d \left(1-c^2 x^2\right)^{5/2}}{175 c^5}+\frac{b d \left(1-c^2 x^2\right)^{3/2}}{105 c^5}+\frac{2 b d \sqrt{1-c^2 x^2}}{35 c^5}",1,"(d*(-105*a*x^5*(-7 + 5*c^2*x^2) + (b*Sqrt[1 - c^2*x^2]*(152 + 76*c^2*x^2 + 57*c^4*x^4 - 75*c^6*x^6))/c^5 - 105*b*x^5*(-7 + 5*c^2*x^2)*ArcSin[c*x]))/3675","A",1
2,1,89,123,0.1160861,"\int x^3 \left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^3*(d - c^2*d*x^2)*(a + b*ArcSin[c*x]),x]","\frac{d \left(-6 a c^4 x^4 \left(2 c^2 x^2-3\right)-3 b \left(4 c^6 x^6-6 c^4 x^4+1\right) \sin ^{-1}(c x)+b c x \sqrt{1-c^2 x^2} \left(-2 c^4 x^4+2 c^2 x^2+3\right)\right)}{72 c^4}","-\frac{1}{6} c^2 d x^6 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{4} d x^4 \left(a+b \sin ^{-1}(c x)\right)-\frac{b d \sin ^{-1}(c x)}{24 c^4}-\frac{1}{36} b c d x^5 \sqrt{1-c^2 x^2}+\frac{b d x^3 \sqrt{1-c^2 x^2}}{36 c}+\frac{b d x \sqrt{1-c^2 x^2}}{24 c^3}",1,"(d*(-6*a*c^4*x^4*(-3 + 2*c^2*x^2) + b*c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2 - 2*c^4*x^4) - 3*b*(1 - 6*c^4*x^4 + 4*c^6*x^6)*ArcSin[c*x]))/(72*c^4)","A",1
3,1,85,105,0.1053855,"\int x^2 \left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^2*(d - c^2*d*x^2)*(a + b*ArcSin[c*x]),x]","\frac{d \left(a \left(75 c^3 x^3-45 c^5 x^5\right)+b \sqrt{1-c^2 x^2} \left(-9 c^4 x^4+13 c^2 x^2+26\right)+15 b c^3 x^3 \left(5-3 c^2 x^2\right) \sin ^{-1}(c x)\right)}{225 c^3}","-\frac{1}{5} c^2 d x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} d x^3 \left(a+b \sin ^{-1}(c x)\right)-\frac{b d \left(1-c^2 x^2\right)^{5/2}}{25 c^3}+\frac{b d \left(1-c^2 x^2\right)^{3/2}}{45 c^3}+\frac{2 b d \sqrt{1-c^2 x^2}}{15 c^3}",1,"(d*(b*Sqrt[1 - c^2*x^2]*(26 + 13*c^2*x^2 - 9*c^4*x^4) + a*(75*c^3*x^3 - 45*c^5*x^5) + 15*b*c^3*x^3*(5 - 3*c^2*x^2)*ArcSin[c*x]))/(225*c^3)","A",1
4,1,77,90,0.1129403,"\int x \left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x*(d - c^2*d*x^2)*(a + b*ArcSin[c*x]),x]","-\frac{d \left(c x \left(8 a c x \left(c^2 x^2-2\right)+b \sqrt{1-c^2 x^2} \left(2 c^2 x^2-5\right)\right)+b \left(8 c^4 x^4-16 c^2 x^2+5\right) \sin ^{-1}(c x)\right)}{32 c^2}","-\frac{d \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{4 c^2}+\frac{b d x \left(1-c^2 x^2\right)^{3/2}}{16 c}+\frac{3 b d x \sqrt{1-c^2 x^2}}{32 c}+\frac{3 b d \sin ^{-1}(c x)}{32 c^2}",1,"-1/32*(d*(c*x*(8*a*c*x*(-2 + c^2*x^2) + b*Sqrt[1 - c^2*x^2]*(-5 + 2*c^2*x^2)) + b*(5 - 16*c^2*x^2 + 8*c^4*x^4)*ArcSin[c*x]))/c^2","A",1
5,1,88,77,0.0977302,"\int \left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d - c^2*d*x^2)*(a + b*ArcSin[c*x]),x]","-\frac{1}{3} a c^2 d x^3+a d x-\frac{1}{3} b c^2 d x^3 \sin ^{-1}(c x)-\frac{1}{9} b c d x^2 \sqrt{1-c^2 x^2}+\frac{7 b d \sqrt{1-c^2 x^2}}{9 c}+b d x \sin ^{-1}(c x)","-\frac{1}{3} c^2 d x^3 \left(a+b \sin ^{-1}(c x)\right)+d x \left(a+b \sin ^{-1}(c x)\right)+\frac{b d \left(1-c^2 x^2\right)^{3/2}}{9 c}+\frac{2 b d \sqrt{1-c^2 x^2}}{3 c}",1,"a*d*x - (a*c^2*d*x^3)/3 + (7*b*d*Sqrt[1 - c^2*x^2])/(9*c) - (b*c*d*x^2*Sqrt[1 - c^2*x^2])/9 + b*d*x*ArcSin[c*x] - (b*c^2*d*x^3*ArcSin[c*x])/3","A",1
6,1,99,121,0.1331238,"\int \frac{\left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{x} \, dx","Integrate[((d - c^2*d*x^2)*(a + b*ArcSin[c*x]))/x,x]","-\frac{1}{4} d \left(2 a c^2 x^2-4 a \log (x)+b c x \sqrt{1-c^2 x^2}+b \sin ^{-1}(c x) \left(2 c^2 x^2-4 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-1\right)+2 i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+2 i b \sin ^{-1}(c x)^2\right)","\frac{1}{2} d \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{i d \left(a+b \sin ^{-1}(c x)\right)^2}{2 b}+d \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{1}{4} b c d x \sqrt{1-c^2 x^2}-\frac{1}{2} i b d \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-\frac{1}{4} b d \sin ^{-1}(c x)",1,"-1/4*(d*(2*a*c^2*x^2 + b*c*x*Sqrt[1 - c^2*x^2] + (2*I)*b*ArcSin[c*x]^2 + b*ArcSin[c*x]*(-1 + 2*c^2*x^2 - 4*Log[1 - E^((2*I)*ArcSin[c*x])]) - 4*a*Log[x] + (2*I)*b*PolyLog[2, E^((2*I)*ArcSin[c*x])]))","A",0
7,1,78,69,0.0504165,"\int \frac{\left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d - c^2*d*x^2)*(a + b*ArcSin[c*x]))/x^2,x]","-a c^2 d x-\frac{a d}{x}-b c d \sqrt{1-c^2 x^2}-b c d \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-b c^2 d x \sin ^{-1}(c x)-\frac{b d \sin ^{-1}(c x)}{x}","c^2 (-d) x \left(a+b \sin ^{-1}(c x)\right)-\frac{d \left(a+b \sin ^{-1}(c x)\right)}{x}-b c d \sqrt{1-c^2 x^2}-b c d \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)",1,"-((a*d)/x) - a*c^2*d*x - b*c*d*Sqrt[1 - c^2*x^2] - (b*d*ArcSin[c*x])/x - b*c^2*d*x*ArcSin[c*x] - b*c*d*ArcTanh[Sqrt[1 - c^2*x^2]]","A",1
8,1,110,139,0.121071,"\int \frac{\left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d - c^2*d*x^2)*(a + b*ArcSin[c*x]))/x^3,x]","-\frac{d \left(2 a c^2 x^2 \log (x)+a-i b c^2 x^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+b c x \sqrt{1-c^2 x^2}-i b c^2 x^2 \sin ^{-1}(c x)^2+b \sin ^{-1}(c x) \left(1+2 c^2 x^2 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)\right)\right)}{2 x^2}","-\frac{d \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{2 x^2}+\frac{i c^2 d \left(a+b \sin ^{-1}(c x)\right)^2}{2 b}-c^2 d \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{2} i b c^2 d \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-\frac{b c d \sqrt{1-c^2 x^2}}{2 x}-\frac{1}{2} b c^2 d \sin ^{-1}(c x)",1,"-1/2*(d*(a + b*c*x*Sqrt[1 - c^2*x^2] - I*b*c^2*x^2*ArcSin[c*x]^2 + b*ArcSin[c*x]*(1 + 2*c^2*x^2*Log[1 - E^((2*I)*ArcSin[c*x])]) + 2*a*c^2*x^2*Log[x] - I*b*c^2*x^2*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/x^2","A",0
9,1,93,81,0.0582541,"\int \frac{\left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d - c^2*d*x^2)*(a + b*ArcSin[c*x]))/x^4,x]","\frac{a c^2 d}{x}-\frac{a d}{3 x^3}-\frac{b c d \sqrt{1-c^2 x^2}}{6 x^2}+\frac{b c^2 d \sin ^{-1}(c x)}{x}+\frac{5}{6} b c^3 d \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-\frac{b d \sin ^{-1}(c x)}{3 x^3}","\frac{c^2 d \left(a+b \sin ^{-1}(c x)\right)}{x}-\frac{d \left(a+b \sin ^{-1}(c x)\right)}{3 x^3}-\frac{b c d \sqrt{1-c^2 x^2}}{6 x^2}+\frac{5}{6} b c^3 d \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)",1,"-1/3*(a*d)/x^3 + (a*c^2*d)/x - (b*c*d*Sqrt[1 - c^2*x^2])/(6*x^2) - (b*d*ArcSin[c*x])/(3*x^3) + (b*c^2*d*ArcSin[c*x])/x + (5*b*c^3*d*ArcTanh[Sqrt[1 - c^2*x^2]])/6","A",1
10,1,119,186,0.1268688,"\int x^4 \left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^4*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]),x]","\frac{d^2 \left(315 a c^5 x^5 \left(35 c^4 x^4-90 c^2 x^2+63\right)+315 b c^5 x^5 \left(35 c^4 x^4-90 c^2 x^2+63\right) \sin ^{-1}(c x)+b \sqrt{1-c^2 x^2} \left(1225 c^8 x^8-2650 c^6 x^6+789 c^4 x^4+1052 c^2 x^2+2104\right)\right)}{99225 c^5}","\frac{1}{9} c^4 d^2 x^9 \left(a+b \sin ^{-1}(c x)\right)-\frac{2}{7} c^2 d^2 x^7 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{5} d^2 x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{b d^2 \left(1-c^2 x^2\right)^{9/2}}{81 c^5}-\frac{10 b d^2 \left(1-c^2 x^2\right)^{7/2}}{441 c^5}+\frac{b d^2 \left(1-c^2 x^2\right)^{5/2}}{525 c^5}+\frac{4 b d^2 \left(1-c^2 x^2\right)^{3/2}}{945 c^5}+\frac{8 b d^2 \sqrt{1-c^2 x^2}}{315 c^5}",1,"(d^2*(315*a*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(2104 + 1052*c^2*x^2 + 789*c^4*x^4 - 2650*c^6*x^6 + 1225*c^8*x^8) + 315*b*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4)*ArcSin[c*x]))/(99225*c^5)","A",1
11,1,115,184,0.1017871,"\int x^3 \left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^3*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]),x]","\frac{d^2 \left(384 a c^4 x^4 \left(3 c^4 x^4-8 c^2 x^2+6\right)+3 b \left(384 c^8 x^8-1024 c^6 x^6+768 c^4 x^4-73\right) \sin ^{-1}(c x)+b c x \sqrt{1-c^2 x^2} \left(144 c^6 x^6-344 c^4 x^4+146 c^2 x^2+219\right)\right)}{9216 c^4}","\frac{1}{8} c^4 d^2 x^8 \left(a+b \sin ^{-1}(c x)\right)-\frac{1}{3} c^2 d^2 x^6 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{4} d^2 x^4 \left(a+b \sin ^{-1}(c x)\right)-\frac{73 b d^2 \sin ^{-1}(c x)}{3072 c^4}-\frac{43 b c d^2 x^5 \sqrt{1-c^2 x^2}}{1152}+\frac{73 b d^2 x^3 \sqrt{1-c^2 x^2}}{4608 c}+\frac{73 b d^2 x \sqrt{1-c^2 x^2}}{3072 c^3}+\frac{1}{64} b c^3 d^2 x^7 \sqrt{1-c^2 x^2}",1,"(d^2*(384*a*c^4*x^4*(6 - 8*c^2*x^2 + 3*c^4*x^4) + b*c*x*Sqrt[1 - c^2*x^2]*(219 + 146*c^2*x^2 - 344*c^4*x^4 + 144*c^6*x^6) + 3*b*(-73 + 768*c^4*x^4 - 1024*c^6*x^6 + 384*c^8*x^8)*ArcSin[c*x]))/(9216*c^4)","A",1
12,1,111,161,0.1191453,"\int x^2 \left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^2*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]),x]","\frac{d^2 \left(105 a c^3 x^3 \left(15 c^4 x^4-42 c^2 x^2+35\right)+b \sqrt{1-c^2 x^2} \left(225 c^6 x^6-612 c^4 x^4+409 c^2 x^2+818\right)+105 b c^3 x^3 \left(15 c^4 x^4-42 c^2 x^2+35\right) \sin ^{-1}(c x)\right)}{11025 c^3}","\frac{1}{7} c^4 d^2 x^7 \left(a+b \sin ^{-1}(c x)\right)-\frac{2}{5} c^2 d^2 x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} d^2 x^3 \left(a+b \sin ^{-1}(c x)\right)-\frac{b d^2 \left(1-c^2 x^2\right)^{7/2}}{49 c^3}+\frac{b d^2 \left(1-c^2 x^2\right)^{5/2}}{175 c^3}+\frac{4 b d^2 \left(1-c^2 x^2\right)^{3/2}}{315 c^3}+\frac{8 b d^2 \sqrt{1-c^2 x^2}}{105 c^3}",1,"(d^2*(105*a*c^3*x^3*(35 - 42*c^2*x^2 + 15*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(818 + 409*c^2*x^2 - 612*c^4*x^4 + 225*c^6*x^6) + 105*b*c^3*x^3*(35 - 42*c^2*x^2 + 15*c^4*x^4)*ArcSin[c*x]))/(11025*c^3)","A",1
13,1,94,124,0.0783222,"\int x \left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]),x]","\frac{d^2 \left(48 a \left(c^2 x^2-1\right)^3+b c x \sqrt{1-c^2 x^2} \left(8 c^4 x^4-26 c^2 x^2+33\right)+3 b \left(16 c^6 x^6-48 c^4 x^4+48 c^2 x^2-11\right) \sin ^{-1}(c x)\right)}{288 c^2}","-\frac{d^2 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)}{6 c^2}+\frac{b d^2 x \left(1-c^2 x^2\right)^{5/2}}{36 c}+\frac{5 b d^2 x \left(1-c^2 x^2\right)^{3/2}}{144 c}+\frac{5 b d^2 x \sqrt{1-c^2 x^2}}{96 c}+\frac{5 b d^2 \sin ^{-1}(c x)}{96 c^2}",1,"(d^2*(48*a*(-1 + c^2*x^2)^3 + b*c*x*Sqrt[1 - c^2*x^2]*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 3*b*(-11 + 48*c^2*x^2 - 48*c^4*x^4 + 16*c^6*x^6)*ArcSin[c*x]))/(288*c^2)","A",1
14,1,95,131,0.1123544,"\int \left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]),x]","\frac{d^2 \left(15 a c x \left(3 c^4 x^4-10 c^2 x^2+15\right)+b \sqrt{1-c^2 x^2} \left(9 c^4 x^4-38 c^2 x^2+149\right)+15 b c x \left(3 c^4 x^4-10 c^2 x^2+15\right) \sin ^{-1}(c x)\right)}{225 c}","\frac{1}{5} c^4 d^2 x^5 \left(a+b \sin ^{-1}(c x)\right)-\frac{2}{3} c^2 d^2 x^3 \left(a+b \sin ^{-1}(c x)\right)+d^2 x \left(a+b \sin ^{-1}(c x)\right)+\frac{b d^2 \left(1-c^2 x^2\right)^{5/2}}{25 c}+\frac{4 b d^2 \left(1-c^2 x^2\right)^{3/2}}{45 c}+\frac{8 b d^2 \sqrt{1-c^2 x^2}}{15 c}",1,"(d^2*(15*a*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(149 - 38*c^2*x^2 + 9*c^4*x^4) + 15*b*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x]))/(225*c)","A",1
15,1,142,184,0.1745736,"\int \frac{\left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{x} \, dx","Integrate[((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]))/x,x]","\frac{1}{32} d^2 \left(8 a c^4 x^4-32 a c^2 x^2+32 a \log (x)-13 b c x \sqrt{1-c^2 x^2}+b \sin ^{-1}(c x) \left(8 c^4 x^4-32 c^2 x^2+32 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+13\right)+2 b c^3 x^3 \sqrt{1-c^2 x^2}-16 i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-16 i b \sin ^{-1}(c x)^2\right)","\frac{1}{4} d^2 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{2} d^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{i d^2 \left(a+b \sin ^{-1}(c x)\right)^2}{2 b}+d^2 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{1}{16} b c d^2 x \left(1-c^2 x^2\right)^{3/2}-\frac{11}{32} b c d^2 x \sqrt{1-c^2 x^2}-\frac{1}{2} i b d^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-\frac{11}{32} b d^2 \sin ^{-1}(c x)",1,"(d^2*(-32*a*c^2*x^2 + 8*a*c^4*x^4 - 13*b*c*x*Sqrt[1 - c^2*x^2] + 2*b*c^3*x^3*Sqrt[1 - c^2*x^2] - (16*I)*b*ArcSin[c*x]^2 + b*ArcSin[c*x]*(13 - 32*c^2*x^2 + 8*c^4*x^4 + 32*Log[1 - E^((2*I)*ArcSin[c*x])]) + 32*a*Log[x] - (16*I)*b*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/32","A",0
16,1,126,123,0.1056929,"\int \frac{\left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]))/x^2,x]","\frac{d^2 \left(3 a c^4 x^4-18 a c^2 x^2-9 a-16 b c x \sqrt{1-c^2 x^2}-9 b c x \log \left(\sqrt{1-c^2 x^2}+1\right)+3 b \left(c^4 x^4-6 c^2 x^2-3\right) \sin ^{-1}(c x)+b c^3 x^3 \sqrt{1-c^2 x^2}+9 b c x \log (x)\right)}{9 x}","\frac{1}{3} c^4 d^2 x^3 \left(a+b \sin ^{-1}(c x)\right)-2 c^2 d^2 x \left(a+b \sin ^{-1}(c x)\right)-\frac{d^2 \left(a+b \sin ^{-1}(c x)\right)}{x}-\frac{1}{9} b c d^2 \left(1-c^2 x^2\right)^{3/2}-\frac{5}{3} b c d^2 \sqrt{1-c^2 x^2}-b c d^2 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)",1,"(d^2*(-9*a - 18*a*c^2*x^2 + 3*a*c^4*x^4 - 16*b*c*x*Sqrt[1 - c^2*x^2] + b*c^3*x^3*Sqrt[1 - c^2*x^2] + 3*b*(-3 - 6*c^2*x^2 + c^4*x^4)*ArcSin[c*x] + 9*b*c*x*Log[x] - 9*b*c*x*Log[1 + Sqrt[1 - c^2*x^2]]))/(9*x)","A",1
17,1,162,201,0.1787446,"\int \frac{\left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]))/x^3,x]","\frac{d^2 \left(2 a c^4 x^4-8 a c^2 x^2 \log (x)-2 a+4 i b c^2 x^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-2 b c x \sqrt{1-c^2 x^2}+4 i b c^2 x^2 \sin ^{-1}(c x)^2+b \sin ^{-1}(c x) \left(2 c^4 x^4-c^2 x^2-8 c^2 x^2 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-2\right)+b c^3 x^3 \sqrt{1-c^2 x^2}\right)}{4 x^2}","-c^2 d^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{d^2 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{2 x^2}+\frac{i c^2 d^2 \left(a+b \sin ^{-1}(c x)\right)^2}{b}-2 c^2 d^2 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+i b c^2 d^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-\frac{b c d^2 \left(1-c^2 x^2\right)^{3/2}}{2 x}-\frac{1}{4} b c^2 d^2 \sin ^{-1}(c x)-\frac{1}{4} b c^3 d^2 x \sqrt{1-c^2 x^2}",1,"(d^2*(-2*a + 2*a*c^4*x^4 - 2*b*c*x*Sqrt[1 - c^2*x^2] + b*c^3*x^3*Sqrt[1 - c^2*x^2] + (4*I)*b*c^2*x^2*ArcSin[c*x]^2 + b*ArcSin[c*x]*(-2 - c^2*x^2 + 2*c^4*x^4 - 8*c^2*x^2*Log[1 - E^((2*I)*ArcSin[c*x])]) - 8*a*c^2*x^2*Log[x] + (4*I)*b*c^2*x^2*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/(4*x^2)","A",0
18,1,136,128,0.1126863,"\int \frac{\left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]))/x^4,x]","\frac{d^2 \left(6 a c^4 x^4+12 a c^2 x^2-2 a-11 b c^3 x^3 \log (x)-b c x \sqrt{1-c^2 x^2}+2 b \left(3 c^4 x^4+6 c^2 x^2-1\right) \sin ^{-1}(c x)+6 b c^3 x^3 \sqrt{1-c^2 x^2}+11 b c^3 x^3 \log \left(\sqrt{1-c^2 x^2}+1\right)\right)}{6 x^3}","c^4 d^2 x \left(a+b \sin ^{-1}(c x)\right)+\frac{2 c^2 d^2 \left(a+b \sin ^{-1}(c x)\right)}{x}-\frac{d^2 \left(a+b \sin ^{-1}(c x)\right)}{3 x^3}-\frac{b c d^2 \sqrt{1-c^2 x^2}}{6 x^2}+b c^3 d^2 \sqrt{1-c^2 x^2}+\frac{11}{6} b c^3 d^2 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)",1,"(d^2*(-2*a + 12*a*c^2*x^2 + 6*a*c^4*x^4 - b*c*x*Sqrt[1 - c^2*x^2] + 6*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 2*b*(-1 + 6*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x] - 11*b*c^3*x^3*Log[x] + 11*b*c^3*x^3*Log[1 + Sqrt[1 - c^2*x^2]]))/(6*x^3)","A",1
19,1,143,232,0.2087819,"\int x^4 \left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^4*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]),x]","\frac{d^3 \left(-3465 a c^5 x^5 \left(105 c^6 x^6-385 c^4 x^4+495 c^2 x^2-231\right)-3465 b c^5 x^5 \left(105 c^6 x^6-385 c^4 x^4+495 c^2 x^2-231\right) \sin ^{-1}(c x)+b \sqrt{1-c^2 x^2} \left(-33075 c^{10} x^{10}+111475 c^8 x^8-117625 c^6 x^6+18933 c^4 x^4+25244 c^2 x^2+50488\right)\right)}{4002075 c^5}","-\frac{1}{11} c^6 d^3 x^{11} \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} c^4 d^3 x^9 \left(a+b \sin ^{-1}(c x)\right)-\frac{3}{7} c^2 d^3 x^7 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{5} d^3 x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{b d^3 \left(1-c^2 x^2\right)^{11/2}}{121 c^5}-\frac{4 b d^3 \left(1-c^2 x^2\right)^{9/2}}{297 c^5}+\frac{b d^3 \left(1-c^2 x^2\right)^{7/2}}{1617 c^5}+\frac{2 b d^3 \left(1-c^2 x^2\right)^{5/2}}{1925 c^5}+\frac{8 b d^3 \left(1-c^2 x^2\right)^{3/2}}{3465 c^5}+\frac{16 b d^3 \sqrt{1-c^2 x^2}}{1155 c^5}",1,"(d^3*(-3465*a*c^5*x^5*(-231 + 495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6) + b*Sqrt[1 - c^2*x^2]*(50488 + 25244*c^2*x^2 + 18933*c^4*x^4 - 117625*c^6*x^6 + 111475*c^8*x^8 - 33075*c^10*x^10) - 3465*b*c^5*x^5*(-231 + 495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6)*ArcSin[c*x]))/(4002075*c^5)","A",1
20,1,139,206,0.2099162,"\int x^3 \left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^3*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]),x]","\frac{d^3 \left(-1920 a c^4 x^4 \left(4 c^6 x^6-15 c^4 x^4+20 c^2 x^2-10\right)-15 b \left(512 c^{10} x^{10}-1920 c^8 x^8+2560 c^6 x^6-1280 c^4 x^4+79\right) \sin ^{-1}(c x)+b c x \sqrt{1-c^2 x^2} \left(-768 c^8 x^8+2736 c^6 x^6-3208 c^4 x^4+790 c^2 x^2+1185\right)\right)}{76800 c^4}","\frac{d^3 \left(1-c^2 x^2\right)^5 \left(a+b \sin ^{-1}(c x)\right)}{10 c^4}-\frac{d^3 \left(1-c^2 x^2\right)^4 \left(a+b \sin ^{-1}(c x)\right)}{8 c^4}+\frac{49 b d^3 \sin ^{-1}(c x)}{5120 c^4}-\frac{b d^3 x \left(1-c^2 x^2\right)^{9/2}}{100 c^3}+\frac{7 b d^3 x \left(1-c^2 x^2\right)^{7/2}}{1600 c^3}+\frac{49 b d^3 x \left(1-c^2 x^2\right)^{5/2}}{9600 c^3}+\frac{49 b d^3 x \left(1-c^2 x^2\right)^{3/2}}{7680 c^3}+\frac{49 b d^3 x \sqrt{1-c^2 x^2}}{5120 c^3}",1,"(d^3*(-1920*a*c^4*x^4*(-10 + 20*c^2*x^2 - 15*c^4*x^4 + 4*c^6*x^6) + b*c*x*Sqrt[1 - c^2*x^2]*(1185 + 790*c^2*x^2 - 3208*c^4*x^4 + 2736*c^6*x^6 - 768*c^8*x^8) - 15*b*(79 - 1280*c^4*x^4 + 2560*c^6*x^6 - 1920*c^8*x^8 + 512*c^10*x^10)*ArcSin[c*x]))/(76800*c^4)","A",1
21,1,135,207,0.1916072,"\int x^2 \left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^2*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]),x]","\frac{d^3 \left(-315 a c^3 x^3 \left(35 c^6 x^6-135 c^4 x^4+189 c^2 x^2-105\right)+b \sqrt{1-c^2 x^2} \left(-1225 c^8 x^8+4675 c^6 x^6-6297 c^4 x^4+2629 c^2 x^2+5258\right)-315 b c^3 x^3 \left(35 c^6 x^6-135 c^4 x^4+189 c^2 x^2-105\right) \sin ^{-1}(c x)\right)}{99225 c^3}","-\frac{1}{9} c^6 d^3 x^9 \left(a+b \sin ^{-1}(c x)\right)+\frac{3}{7} c^4 d^3 x^7 \left(a+b \sin ^{-1}(c x)\right)-\frac{3}{5} c^2 d^3 x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} d^3 x^3 \left(a+b \sin ^{-1}(c x)\right)-\frac{b d^3 \left(1-c^2 x^2\right)^{9/2}}{81 c^3}+\frac{b d^3 \left(1-c^2 x^2\right)^{7/2}}{441 c^3}+\frac{2 b d^3 \left(1-c^2 x^2\right)^{5/2}}{525 c^3}+\frac{8 b d^3 \left(1-c^2 x^2\right)^{3/2}}{945 c^3}+\frac{16 b d^3 \sqrt{1-c^2 x^2}}{315 c^3}",1,"(d^3*(-315*a*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6) + b*Sqrt[1 - c^2*x^2]*(5258 + 2629*c^2*x^2 - 6297*c^4*x^4 + 4675*c^6*x^6 - 1225*c^8*x^8) - 315*b*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6)*ArcSin[c*x]))/(99225*c^3)","A",1
22,1,110,150,0.0986203,"\int x \left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]),x]","-\frac{d^3 \left(384 a \left(c^2 x^2-1\right)^4+b c x \sqrt{1-c^2 x^2} \left(48 c^6 x^6-200 c^4 x^4+326 c^2 x^2-279\right)+3 b \left(128 c^8 x^8-512 c^6 x^6+768 c^4 x^4-512 c^2 x^2+93\right) \sin ^{-1}(c x)\right)}{3072 c^2}","-\frac{d^3 \left(1-c^2 x^2\right)^4 \left(a+b \sin ^{-1}(c x)\right)}{8 c^2}+\frac{b d^3 x \left(1-c^2 x^2\right)^{7/2}}{64 c}+\frac{7 b d^3 x \left(1-c^2 x^2\right)^{5/2}}{384 c}+\frac{35 b d^3 x \left(1-c^2 x^2\right)^{3/2}}{1536 c}+\frac{35 b d^3 x \sqrt{1-c^2 x^2}}{1024 c}+\frac{35 b d^3 \sin ^{-1}(c x)}{1024 c^2}",1,"-1/3072*(d^3*(384*a*(-1 + c^2*x^2)^4 + b*c*x*Sqrt[1 - c^2*x^2]*(-279 + 326*c^2*x^2 - 200*c^4*x^4 + 48*c^6*x^6) + 3*b*(93 - 512*c^2*x^2 + 768*c^4*x^4 - 512*c^6*x^6 + 128*c^8*x^8)*ArcSin[c*x]))/c^2","A",1
23,1,119,175,0.2662209,"\int \left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]),x]","-\frac{d^3 \left(105 a c x \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right)+b \sqrt{1-c^2 x^2} \left(75 c^6 x^6-351 c^4 x^4+757 c^2 x^2-2161\right)+105 b c x \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right) \sin ^{-1}(c x)\right)}{3675 c}","-\frac{1}{7} c^6 d^3 x^7 \left(a+b \sin ^{-1}(c x)\right)+\frac{3}{5} c^4 d^3 x^5 \left(a+b \sin ^{-1}(c x)\right)-c^2 d^3 x^3 \left(a+b \sin ^{-1}(c x)\right)+d^3 x \left(a+b \sin ^{-1}(c x)\right)+\frac{b d^3 \left(1-c^2 x^2\right)^{7/2}}{49 c}+\frac{6 b d^3 \left(1-c^2 x^2\right)^{5/2}}{175 c}+\frac{8 b d^3 \left(1-c^2 x^2\right)^{3/2}}{105 c}+\frac{16 b d^3 \sqrt{1-c^2 x^2}}{35 c}",1,"-1/3675*(d^3*(105*a*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + b*Sqrt[1 - c^2*x^2]*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6) + 105*b*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)*ArcSin[c*x]))/c","A",1
24,1,183,235,0.2315908,"\int \frac{\left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)}{x} \, dx","Integrate[((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]))/x,x]","-\frac{1}{144} d^3 \left(24 a c^6 x^6-108 a c^4 x^4+216 a c^2 x^2-144 a \log (x)+75 b c x \sqrt{1-c^2 x^2}+4 b c^5 x^5 \sqrt{1-c^2 x^2}-22 b c^3 x^3 \sqrt{1-c^2 x^2}+3 b \sin ^{-1}(c x) \left(8 c^6 x^6-36 c^4 x^4+72 c^2 x^2-48 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-25\right)+72 i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+72 i b \sin ^{-1}(c x)^2\right)","\frac{1}{6} d^3 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{4} d^3 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{2} d^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{i d^3 \left(a+b \sin ^{-1}(c x)\right)^2}{2 b}+d^3 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{1}{36} b c d^3 x \left(1-c^2 x^2\right)^{5/2}-\frac{7}{72} b c d^3 x \left(1-c^2 x^2\right)^{3/2}-\frac{19}{48} b c d^3 x \sqrt{1-c^2 x^2}-\frac{1}{2} i b d^3 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-\frac{19}{48} b d^3 \sin ^{-1}(c x)",1,"-1/144*(d^3*(216*a*c^2*x^2 - 108*a*c^4*x^4 + 24*a*c^6*x^6 + 75*b*c*x*Sqrt[1 - c^2*x^2] - 22*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 4*b*c^5*x^5*Sqrt[1 - c^2*x^2] + (72*I)*b*ArcSin[c*x]^2 + 3*b*ArcSin[c*x]*(-25 + 72*c^2*x^2 - 36*c^4*x^4 + 8*c^6*x^6 - 48*Log[1 - E^((2*I)*ArcSin[c*x])]) - 144*a*Log[x] + (72*I)*b*PolyLog[2, E^((2*I)*ArcSin[c*x])]))","A",0
25,1,166,164,0.1273318,"\int \frac{\left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]))/x^2,x]","-\frac{d^3 \left(5 a c^6 x^6-25 a c^4 x^4+75 a c^2 x^2+25 a+61 b c x \sqrt{1-c^2 x^2}+25 b c x \log \left(\sqrt{1-c^2 x^2}+1\right)+b c^5 x^5 \sqrt{1-c^2 x^2}-7 b c^3 x^3 \sqrt{1-c^2 x^2}+5 b \left(c^6 x^6-5 c^4 x^4+15 c^2 x^2+5\right) \sin ^{-1}(c x)-25 b c x \log (x)\right)}{25 x}","-\frac{1}{5} c^6 d^3 x^5 \left(a+b \sin ^{-1}(c x)\right)+c^4 d^3 x^3 \left(a+b \sin ^{-1}(c x)\right)-3 c^2 d^3 x \left(a+b \sin ^{-1}(c x)\right)-\frac{d^3 \left(a+b \sin ^{-1}(c x)\right)}{x}-\frac{1}{25} b c d^3 \left(1-c^2 x^2\right)^{5/2}-\frac{1}{5} b c d^3 \left(1-c^2 x^2\right)^{3/2}-\frac{11}{5} b c d^3 \sqrt{1-c^2 x^2}-b c d^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)",1,"-1/25*(d^3*(25*a + 75*a*c^2*x^2 - 25*a*c^4*x^4 + 5*a*c^6*x^6 + 61*b*c*x*Sqrt[1 - c^2*x^2] - 7*b*c^3*x^3*Sqrt[1 - c^2*x^2] + b*c^5*x^5*Sqrt[1 - c^2*x^2] + 5*b*(5 + 15*c^2*x^2 - 5*c^4*x^4 + c^6*x^6)*ArcSin[c*x] - 25*b*c*x*Log[x] + 25*b*c*x*Log[1 + Sqrt[1 - c^2*x^2]]))/x","A",1
26,1,203,263,0.1895959,"\int \frac{\left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]))/x^3,x]","-\frac{d^3 \left(8 a c^6 x^6-48 a c^4 x^4+96 a c^2 x^2 \log (x)+16 a-48 i b c^2 x^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+16 b c x \sqrt{1-c^2 x^2}-48 i b c^2 x^2 \sin ^{-1}(c x)^2+2 b c^5 x^5 \sqrt{1-c^2 x^2}-21 b c^3 x^3 \sqrt{1-c^2 x^2}+b \sin ^{-1}(c x) \left(8 c^6 x^6-48 c^4 x^4+21 c^2 x^2+96 c^2 x^2 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+16\right)\right)}{32 x^2}","-\frac{d^3 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)}{2 x^2}-\frac{3}{4} c^2 d^3 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)-\frac{3}{2} c^2 d^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)+\frac{3 i c^2 d^3 \left(a+b \sin ^{-1}(c x)\right)^2}{2 b}-3 c^2 d^3 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+\frac{3}{2} i b c^2 d^3 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-\frac{b c d^3 \left(1-c^2 x^2\right)^{5/2}}{2 x}+\frac{3}{32} b c^2 d^3 \sin ^{-1}(c x)-\frac{7}{16} b c^3 d^3 x \left(1-c^2 x^2\right)^{3/2}+\frac{3}{32} b c^3 d^3 x \sqrt{1-c^2 x^2}",1,"-1/32*(d^3*(16*a - 48*a*c^4*x^4 + 8*a*c^6*x^6 + 16*b*c*x*Sqrt[1 - c^2*x^2] - 21*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 2*b*c^5*x^5*Sqrt[1 - c^2*x^2] - (48*I)*b*c^2*x^2*ArcSin[c*x]^2 + b*ArcSin[c*x]*(16 + 21*c^2*x^2 - 48*c^4*x^4 + 8*c^6*x^6 + 96*c^2*x^2*Log[1 - E^((2*I)*ArcSin[c*x])]) + 96*a*c^2*x^2*Log[x] - (48*I)*b*c^2*x^2*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/x^2","A",0
27,1,175,178,0.1583348,"\int \frac{\left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]))/x^4,x]","-\frac{d^3 \left(6 a c^6 x^6-54 a c^4 x^4-54 a c^2 x^2+6 a+51 b c^3 x^3 \log (x)+3 b c x \sqrt{1-c^2 x^2}+2 b c^5 x^5 \sqrt{1-c^2 x^2}-50 b c^3 x^3 \sqrt{1-c^2 x^2}-51 b c^3 x^3 \log \left(\sqrt{1-c^2 x^2}+1\right)+6 b \left(c^6 x^6-9 c^4 x^4-9 c^2 x^2+1\right) \sin ^{-1}(c x)\right)}{18 x^3}","-\frac{1}{3} c^6 d^3 x^3 \left(a+b \sin ^{-1}(c x)\right)+3 c^4 d^3 x \left(a+b \sin ^{-1}(c x)\right)+\frac{3 c^2 d^3 \left(a+b \sin ^{-1}(c x)\right)}{x}-\frac{d^3 \left(a+b \sin ^{-1}(c x)\right)}{3 x^3}-\frac{b c d^3 \sqrt{1-c^2 x^2}}{6 x^2}+\frac{1}{9} b c^3 d^3 \left(1-c^2 x^2\right)^{3/2}+\frac{8}{3} b c^3 d^3 \sqrt{1-c^2 x^2}+\frac{17}{6} b c^3 d^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)",1,"-1/18*(d^3*(6*a - 54*a*c^2*x^2 - 54*a*c^4*x^4 + 6*a*c^6*x^6 + 3*b*c*x*Sqrt[1 - c^2*x^2] - 50*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 2*b*c^5*x^5*Sqrt[1 - c^2*x^2] + 6*b*(1 - 9*c^2*x^2 - 9*c^4*x^4 + c^6*x^6)*ArcSin[c*x] + 51*b*c^3*x^3*Log[x] - 51*b*c^3*x^3*Log[1 + Sqrt[1 - c^2*x^2]]))/x^3","A",1
28,1,286,172,0.3601052,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2),x]","-\frac{6 a c^3 x^3+18 a c x+9 a \log (1-c x)-9 a \log (c x+1)+6 b c^3 x^3 \sin ^{-1}(c x)+2 b c^2 x^2 \sqrt{1-c^2 x^2}+22 b \sqrt{1-c^2 x^2}-18 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+18 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+18 b c x \sin ^{-1}(c x)+9 i \pi  b \sin ^{-1}(c x)-18 b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-9 \pi  b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+18 b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-9 \pi  b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+9 \pi  b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+9 \pi  b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{18 c^5 d}","-\frac{2 i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^5 d}-\frac{x \left(a+b \sin ^{-1}(c x)\right)}{c^4 d}-\frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{3 c^2 d}+\frac{i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c^5 d}-\frac{i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c^5 d}+\frac{b \left(1-c^2 x^2\right)^{3/2}}{9 c^5 d}-\frac{4 b \sqrt{1-c^2 x^2}}{3 c^5 d}",1,"-1/18*(18*a*c*x + 6*a*c^3*x^3 + 22*b*Sqrt[1 - c^2*x^2] + 2*b*c^2*x^2*Sqrt[1 - c^2*x^2] + (9*I)*b*Pi*ArcSin[c*x] + 18*b*c*x*ArcSin[c*x] + 6*b*c^3*x^3*ArcSin[c*x] - 9*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 18*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 9*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 18*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 9*a*Log[1 - c*x] - 9*a*Log[1 + c*x] + 9*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 9*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (18*I)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (18*I)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^5*d)","A",0
29,1,294,144,0.1435396,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Integrate[(x^3*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2),x]","-\frac{2 a c^2 x^2+2 a \log \left(1-c^2 x^2\right)+b c x \sqrt{1-c^2 x^2}+2 b c^2 x^2 \sin ^{-1}(c x)-4 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-4 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-2 i b \sin ^{-1}(c x)^2-b \sin ^{-1}(c x)+4 i \pi  b \sin ^{-1}(c x)+8 \pi  b \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+4 b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+2 \pi  b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+4 b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-2 \pi  b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-2 \pi  b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-8 \pi  b \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 \pi  b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{4 c^4 d}","\frac{i \left(a+b \sin ^{-1}(c x)\right)^2}{2 b c^4 d}-\frac{\log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^4 d}-\frac{x^2 \left(a+b \sin ^{-1}(c x)\right)}{2 c^2 d}+\frac{i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 c^4 d}+\frac{b \sin ^{-1}(c x)}{4 c^4 d}-\frac{b x \sqrt{1-c^2 x^2}}{4 c^3 d}",1,"-1/4*(2*a*c^2*x^2 + b*c*x*Sqrt[1 - c^2*x^2] - b*ArcSin[c*x] + (4*I)*b*Pi*ArcSin[c*x] + 2*b*c^2*x^2*ArcSin[c*x] - (2*I)*b*ArcSin[c*x]^2 + 8*b*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 2*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] + 4*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 2*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 4*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 2*a*Log[1 - c^2*x^2] - 8*b*Pi*Log[Cos[ArcSin[c*x]/2]] + 2*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 2*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (4*I)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (4*I)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^4*d)","B",0
30,1,238,124,0.1200372,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2),x]","-\frac{2 a c x+a \log (1-c x)-a \log (c x+1)+2 b \sqrt{1-c^2 x^2}-2 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+2 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+2 b c x \sin ^{-1}(c x)+i \pi  b \sin ^{-1}(c x)-2 b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-\pi  b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+2 b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-\pi  b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+\pi  b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+\pi  b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{2 c^3 d}","-\frac{2 i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^3 d}-\frac{x \left(a+b \sin ^{-1}(c x)\right)}{c^2 d}+\frac{i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c^3 d}-\frac{i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c^3 d}-\frac{b \sqrt{1-c^2 x^2}}{c^3 d}",1,"-1/2*(2*a*c*x + 2*b*Sqrt[1 - c^2*x^2] + I*b*Pi*ArcSin[c*x] + 2*b*c*x*ArcSin[c*x] - b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 2*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 2*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + a*Log[1 - c*x] - a*Log[1 + c*x] + b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (2*I)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^3*d)","A",0
31,1,244,82,0.0883503,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Integrate[(x*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2),x]","-\frac{a \log \left(1-c^2 x^2\right)-2 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-2 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-i b \sin ^{-1}(c x)^2+2 i \pi  b \sin ^{-1}(c x)+2 b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+2 b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+4 \pi  b \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+\pi  b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-\pi  b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-\pi  b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-4 \pi  b \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\pi  b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{2 c^2 d}","\frac{i \left(a+b \sin ^{-1}(c x)\right)^2}{2 b c^2 d}-\frac{\log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^2 d}+\frac{i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 c^2 d}",1,"-1/2*((2*I)*b*Pi*ArcSin[c*x] - I*b*ArcSin[c*x]^2 + 4*b*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] + 2*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 2*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + a*Log[1 - c^2*x^2] - 4*b*Pi*Log[Cos[ArcSin[c*x]/2]] + b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (2*I)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^2*d)","B",0
32,1,207,84,0.2435666,"\int \frac{a+b \sin ^{-1}(c x)}{d-c^2 d x^2} \, dx","Integrate[(a + b*ArcSin[c*x])/(d - c^2*d*x^2),x]","\frac{-a \log (1-c x)+a \log (c x+1)+2 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-2 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-i \pi  b \sin ^{-1}(c x)+2 b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-2 b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+\pi  b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+\pi  b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-\pi  b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-\pi  b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{2 c d}","-\frac{2 i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c d}+\frac{i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c d}-\frac{i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c d}",1,"((-I)*b*Pi*ArcSin[c*x] + b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] + 2*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] - 2*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - a*Log[1 - c*x] + a*Log[1 + c*x] - b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (2*I)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (2*I)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(2*c*d)","B",0
33,1,105,71,0.0981175,"\int \frac{a+b \sin ^{-1}(c x)}{x \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSin[c*x])/(x*(d - c^2*d*x^2)),x]","\frac{-a \log \left(1-c^2 x^2\right)+2 a \log (x)+i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)-i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+2 b \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-2 b \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)}{2 d}","-\frac{2 \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d}+\frac{i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 d}-\frac{i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d}",1,"(2*b*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - 2*b*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] + 2*a*Log[x] - a*Log[1 - c^2*x^2] + I*b*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - I*b*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(2*d)","A",0
34,1,259,116,0.3688734,"\int \frac{a+b \sin ^{-1}(c x)}{x^2 \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^2*(d - c^2*d*x^2)),x]","-\frac{a c x \log (1-c x)-a c x \log (c x+1)+2 a+2 b c x \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-2 i b c x \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+2 i b c x \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+2 b \sin ^{-1}(c x)+i \pi  b c x \sin ^{-1}(c x)-\pi  b c x \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-2 b c x \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-\pi  b c x \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+2 b c x \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+\pi  b c x \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+\pi  b c x \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{2 d x}","-\frac{a+b \sin ^{-1}(c x)}{d x}-\frac{2 i c \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d}-\frac{b c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{d}+\frac{i b c \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{d}-\frac{i b c \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{d}",1,"-1/2*(2*a + 2*b*ArcSin[c*x] + I*b*c*Pi*x*ArcSin[c*x] + 2*b*c*x*ArcTanh[Sqrt[1 - c^2*x^2]] - b*c*Pi*x*Log[1 - I*E^(I*ArcSin[c*x])] - 2*b*c*x*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - b*c*Pi*x*Log[1 + I*E^(I*ArcSin[c*x])] + 2*b*c*x*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + a*c*x*Log[1 - c*x] - a*c*x*Log[1 + c*x] + b*c*Pi*x*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + b*c*Pi*x*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*b*c*x*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (2*I)*b*c*x*PolyLog[2, I*E^(I*ArcSin[c*x])])/(d*x)","B",0
35,1,149,124,0.3618043,"\int \frac{a+b \sin ^{-1}(c x)}{x^3 \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)),x]","-\frac{a c^2 \log \left(1-c^2 x^2\right)-2 a c^2 \log (x)+\frac{a}{x^2}+b c^2 \left(\frac{\sqrt{1-c^2 x^2}}{c x}+\frac{\sin ^{-1}(c x)}{c^2 x^2}-i \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+i \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-2 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+2 \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)}{2 d}","-\frac{2 c^2 \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2}+\frac{i b c^2 \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 d}-\frac{i b c^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d}-\frac{b c \sqrt{1-c^2 x^2}}{2 d x}",1,"-1/2*(a/x^2 - 2*a*c^2*Log[x] + a*c^2*Log[1 - c^2*x^2] + b*c^2*(Sqrt[1 - c^2*x^2]/(c*x) + ArcSin[c*x]/(c^2*x^2) - 2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 2*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] - I*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + I*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/d","A",0
36,1,350,173,0.1524755,"\int \frac{a+b \sin ^{-1}(c x)}{x^4 \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)),x]","-\frac{3 a c^3 x^3 \log (1-c x)-3 a c^3 x^3 \log (c x+1)+6 a c^2 x^2+2 a-6 i b c^3 x^3 \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+6 i b c^3 x^3 \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+3 i \pi  b c^3 x^3 \sin ^{-1}(c x)-6 b c^3 x^3 \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-3 \pi  b c^3 x^3 \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+6 b c^3 x^3 \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-3 \pi  b c^3 x^3 \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+3 \pi  b c^3 x^3 \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+3 \pi  b c^3 x^3 \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+b c x \sqrt{1-c^2 x^2}+6 b c^2 x^2 \sin ^{-1}(c x)+7 b c^3 x^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+2 b \sin ^{-1}(c x)}{6 d x^3}","-\frac{2 i c^3 \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d}-\frac{c^2 \left(a+b \sin ^{-1}(c x)\right)}{d x}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}+\frac{i b c^3 \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{d}-\frac{i b c^3 \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{d}-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{7 b c^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{6 d}",1,"-1/6*(2*a + 6*a*c^2*x^2 + b*c*x*Sqrt[1 - c^2*x^2] + 2*b*ArcSin[c*x] + 6*b*c^2*x^2*ArcSin[c*x] + (3*I)*b*c^3*Pi*x^3*ArcSin[c*x] + 7*b*c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]] - 3*b*c^3*Pi*x^3*Log[1 - I*E^(I*ArcSin[c*x])] - 6*b*c^3*x^3*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 3*b*c^3*Pi*x^3*Log[1 + I*E^(I*ArcSin[c*x])] + 6*b*c^3*x^3*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 3*a*c^3*x^3*Log[1 - c*x] - 3*a*c^3*x^3*Log[1 + c*x] + 3*b*c^3*Pi*x^3*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 3*b*c^3*Pi*x^3*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (6*I)*b*c^3*x^3*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (6*I)*b*c^3*x^3*PolyLog[2, I*E^(I*ArcSin[c*x])])/(d*x^3)","B",0
37,1,332,187,0.4376047,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^2,x]","\frac{-\frac{2 a c x}{c^2 x^2-1}+4 a c x+3 a \log (1-c x)-3 a \log (c x+1)+\frac{b \sqrt{1-c^2 x^2}}{c x-1}-\frac{b \sqrt{1-c^2 x^2}}{c x+1}+4 b \sqrt{1-c^2 x^2}-6 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+6 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+4 b c x \sin ^{-1}(c x)+\frac{b \sin ^{-1}(c x)}{1-c x}-\frac{b \sin ^{-1}(c x)}{c x+1}+3 i \pi  b \sin ^{-1}(c x)-6 b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-3 \pi  b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+6 b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-3 \pi  b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+3 \pi  b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+3 \pi  b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{4 c^5 d^2}","\frac{3 i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^5 d^2}+\frac{3 x \left(a+b \sin ^{-1}(c x)\right)}{2 c^4 d^2}+\frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{3 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{2 c^5 d^2}+\frac{3 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{2 c^5 d^2}+\frac{b \sqrt{1-c^2 x^2}}{c^5 d^2}-\frac{b}{2 c^5 d^2 \sqrt{1-c^2 x^2}}",1,"(4*a*c*x + 4*b*Sqrt[1 - c^2*x^2] + (b*Sqrt[1 - c^2*x^2])/(-1 + c*x) - (b*Sqrt[1 - c^2*x^2])/(1 + c*x) - (2*a*c*x)/(-1 + c^2*x^2) + (3*I)*b*Pi*ArcSin[c*x] + 4*b*c*x*ArcSin[c*x] + (b*ArcSin[c*x])/(1 - c*x) - (b*ArcSin[c*x])/(1 + c*x) - 3*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 6*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 3*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 6*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 3*a*Log[1 - c*x] - 3*a*Log[1 + c*x] + 3*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 3*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (6*I)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (6*I)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(4*c^5*d^2)","A",0
38,1,334,155,0.5095754,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x^3*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^2,x]","\frac{-\frac{2 a}{c^2 x^2-1}+2 a \log \left(1-c^2 x^2\right)+\frac{b \sqrt{1-c^2 x^2}}{c x-1}+\frac{b \sqrt{1-c^2 x^2}}{c x+1}-4 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-4 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-2 i b \sin ^{-1}(c x)^2+\frac{b \sin ^{-1}(c x)}{1-c x}+\frac{b \sin ^{-1}(c x)}{c x+1}+4 i \pi  b \sin ^{-1}(c x)+4 b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+4 b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+8 \pi  b \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+2 \pi  b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-2 \pi  b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-2 \pi  b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-8 \pi  b \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 \pi  b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{4 c^4 d^2}","-\frac{i \left(a+b \sin ^{-1}(c x)\right)^2}{2 b c^4 d^2}+\frac{\log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^4 d^2}+\frac{x^2 \left(a+b \sin ^{-1}(c x)\right)}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 c^4 d^2}+\frac{b \sin ^{-1}(c x)}{2 c^4 d^2}-\frac{b x}{2 c^3 d^2 \sqrt{1-c^2 x^2}}",1,"((b*Sqrt[1 - c^2*x^2])/(-1 + c*x) + (b*Sqrt[1 - c^2*x^2])/(1 + c*x) - (2*a)/(-1 + c^2*x^2) + (4*I)*b*Pi*ArcSin[c*x] + (b*ArcSin[c*x])/(1 - c*x) + (b*ArcSin[c*x])/(1 + c*x) - (2*I)*b*ArcSin[c*x]^2 + 8*b*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 2*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] + 4*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 2*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 4*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 2*a*Log[1 - c^2*x^2] - 8*b*Pi*Log[Cos[ArcSin[c*x]/2]] + 2*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 2*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (4*I)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (4*I)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(4*c^4*d^2)","B",0
39,1,463,144,0.1927374,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^2,x]","\frac{a \log (1-c x)}{4 c^3 d^2}-\frac{a \log (c x+1)}{4 c^3 d^2}-\frac{a x}{2 c^2 d^2 \left(c^2 x^2-1\right)}+\frac{b \left(\frac{-\frac{2 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{i \sin ^{-1}(c x)^2}{2 c}+\frac{3 i \pi  \sin ^{-1}(c x)}{2 c}+\frac{2 \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)}{c}+\frac{2 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)}{c}-\frac{\pi  \log \left(1+i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{2 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{c}+\frac{\pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{c}}{4 c^2}-\frac{-\frac{2 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{i \sin ^{-1}(c x)^2}{2 c}+\frac{i \pi  \sin ^{-1}(c x)}{2 c}+\frac{2 \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)}{c}+\frac{2 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)}{c}+\frac{\pi  \log \left(1-i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{\pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{c}-\frac{2 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{c}}{4 c^2}-\frac{\sqrt{1-c^2 x^2}+\sin ^{-1}(c x)}{4 c^2 \left(c^2 x+c\right)}+\frac{\sqrt{1-c^2 x^2}-\sin ^{-1}(c x)}{4 c^3 (c x-1)}\right)}{d^2}","\frac{i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^3 d^2}+\frac{x \left(a+b \sin ^{-1}(c x)\right)}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{2 c^3 d^2}+\frac{i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{2 c^3 d^2}-\frac{b}{2 c^3 d^2 \sqrt{1-c^2 x^2}}",1,"-1/2*(a*x)/(c^2*d^2*(-1 + c^2*x^2)) + (a*Log[1 - c*x])/(4*c^3*d^2) - (a*Log[1 + c*x])/(4*c^3*d^2) + (b*((Sqrt[1 - c^2*x^2] - ArcSin[c*x])/(4*c^3*(-1 + c*x)) - (Sqrt[1 - c^2*x^2] + ArcSin[c*x])/(4*c^2*(c + c^2*x)) + ((((3*I)/2)*Pi*ArcSin[c*x])/c - ((I/2)*ArcSin[c*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])])/c - (Pi*Log[1 + I*E^(I*ArcSin[c*x])])/c + (2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c + (Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]])/c - ((2*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/c)/(4*c^2) - (((I/2)*Pi*ArcSin[c*x])/c - ((I/2)*ArcSin[c*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])])/c + (Pi*Log[1 - I*E^(I*ArcSin[c*x])])/c + (2*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c - (Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]])/c - ((2*I)*PolyLog[2, I*E^(I*ArcSin[c*x])])/c)/(4*c^2)))/d^2","B",0
40,1,50,57,0.0466801,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^2,x]","\frac{a-b c x \sqrt{1-c^2 x^2}+b \sin ^{-1}(c x)}{2 c^2 d^2-2 c^4 d^2 x^2}","\frac{a+b \sin ^{-1}(c x)}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{b x}{2 c d^2 \sqrt{1-c^2 x^2}}",1,"(a - b*c*x*Sqrt[1 - c^2*x^2] + b*ArcSin[c*x])/(2*c^2*d^2 - 2*c^4*d^2*x^2)","A",1
41,1,334,141,0.8401778,"\int \frac{a+b \sin ^{-1}(c x)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^2,x]","-\frac{\frac{2 a x}{c^2 x^2-1}+\frac{a \log (1-c x)}{c}-\frac{a \log (c x+1)}{c}+\frac{b \sqrt{1-c^2 x^2}}{c-c^2 x}+\frac{b \sqrt{1-c^2 x^2}}{c^2 x+c}+\frac{b \sin ^{-1}(c x)}{c^2 x+c}-\frac{2 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c}+\frac{2 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c}+\frac{b \sin ^{-1}(c x)}{c (c x-1)}+\frac{i \pi  b \sin ^{-1}(c x)}{c}-\frac{2 b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{\pi  b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)}{c}+\frac{2 b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{\pi  b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)}{c}+\frac{\pi  b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{c}+\frac{\pi  b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{c}}{4 d^2}","\frac{x \left(a+b \sin ^{-1}(c x)\right)}{2 d^2 \left(1-c^2 x^2\right)}-\frac{i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c d^2}-\frac{b}{2 c d^2 \sqrt{1-c^2 x^2}}+\frac{i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{2 c d^2}-\frac{i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{2 c d^2}",1,"-1/4*((b*Sqrt[1 - c^2*x^2])/(c - c^2*x) + (b*Sqrt[1 - c^2*x^2])/(c + c^2*x) + (2*a*x)/(-1 + c^2*x^2) + (I*b*Pi*ArcSin[c*x])/c + (b*ArcSin[c*x])/(c*(-1 + c*x)) + (b*ArcSin[c*x])/(c + c^2*x) - (b*Pi*Log[1 - I*E^(I*ArcSin[c*x])])/c - (2*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])])/c - (b*Pi*Log[1 + I*E^(I*ArcSin[c*x])])/c + (2*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])])/c + (a*Log[1 - c*x])/c - (a*Log[1 + c*x])/c + (b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]])/c + (b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]])/c - ((2*I)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/c + ((2*I)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/c)/d^2","B",0
42,1,153,122,0.4107032,"\int \frac{a+b \sin ^{-1}(c x)}{x \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])/(x*(d - c^2*d*x^2)^2),x]","\frac{\frac{a}{1-c^2 x^2}-a \log \left(1-c^2 x^2\right)+2 a \log (x)+b \left(-\frac{c x}{\sqrt{1-c^2 x^2}}+\frac{\sin ^{-1}(c x)}{1-c^2 x^2}+i \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)-i \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+2 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-2 \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)}{2 d^2}","\frac{a+b \sin ^{-1}(c x)}{2 d^2 \left(1-c^2 x^2\right)}-\frac{2 \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}-\frac{b c x}{2 d^2 \sqrt{1-c^2 x^2}}+\frac{i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 d^2}-\frac{i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d^2}",1,"(a/(1 - c^2*x^2) + 2*a*Log[x] - a*Log[1 - c^2*x^2] + b*(-((c*x)/Sqrt[1 - c^2*x^2]) + ArcSin[c*x]/(1 - c^2*x^2) + 2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - 2*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] + I*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - I*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/(2*d^2)","A",0
43,1,348,186,0.9101308,"\int \frac{a+b \sin ^{-1}(c x)}{x^2 \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^2*(d - c^2*d*x^2)^2),x]","-\frac{\frac{2 a c^2 x}{c^2 x^2-1}+3 a c \log (1-c x)-3 a c \log (c x+1)+\frac{4 a}{x}+\frac{b c \sqrt{1-c^2 x^2}}{1-c x}+\frac{b c \sqrt{1-c^2 x^2}}{c x+1}+4 b c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-6 i b c \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+6 i b c \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+\frac{b c \sin ^{-1}(c x)}{c x-1}+\frac{b c \sin ^{-1}(c x)}{c x+1}+3 i \pi  b c \sin ^{-1}(c x)+\frac{4 b \sin ^{-1}(c x)}{x}-6 b c \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-3 \pi  b c \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+6 b c \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-3 \pi  b c \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+3 \pi  b c \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+3 \pi  b c \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{4 d^2}","\frac{3 c^2 x \left(a+b \sin ^{-1}(c x)\right)}{2 d^2 \left(1-c^2 x^2\right)}-\frac{a+b \sin ^{-1}(c x)}{d^2 x \left(1-c^2 x^2\right)}-\frac{3 i c \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}-\frac{b c}{2 d^2 \sqrt{1-c^2 x^2}}-\frac{b c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{d^2}+\frac{3 i b c \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{2 d^2}-\frac{3 i b c \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{2 d^2}",1,"-1/4*((4*a)/x + (b*c*Sqrt[1 - c^2*x^2])/(1 - c*x) + (b*c*Sqrt[1 - c^2*x^2])/(1 + c*x) + (2*a*c^2*x)/(-1 + c^2*x^2) + (3*I)*b*c*Pi*ArcSin[c*x] + (4*b*ArcSin[c*x])/x + (b*c*ArcSin[c*x])/(-1 + c*x) + (b*c*ArcSin[c*x])/(1 + c*x) + 4*b*c*ArcTanh[Sqrt[1 - c^2*x^2]] - 3*b*c*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 6*b*c*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 3*b*c*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 6*b*c*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 3*a*c*Log[1 - c*x] - 3*a*c*Log[1 + c*x] + 3*b*c*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 3*b*c*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (6*I)*b*c*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (6*I)*b*c*PolyLog[2, I*E^(I*ArcSin[c*x])])/d^2","A",0
44,1,213,159,0.7990648,"\int \frac{a+b \sin ^{-1}(c x)}{x^3 \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)^2),x]","-\frac{\frac{a c^2}{c^2 x^2-1}+2 a c^2 \log \left(1-c^2 x^2\right)-4 a c^2 \log (x)+\frac{a}{x^2}-2 i b c^2 \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+2 i b c^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+\frac{b c \sqrt{1-c^2 x^2}}{x}+\frac{b c^2 \sin ^{-1}(c x)}{c^2 x^2-1}-4 b c^2 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+4 b c^2 \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)+\frac{b c^3 x}{\sqrt{1-c^2 x^2}}+\frac{b \sin ^{-1}(c x)}{x^2}}{2 d^2}","\frac{c^2 \left(a+b \sin ^{-1}(c x)\right)}{d^2 \left(1-c^2 x^2\right)}-\frac{a+b \sin ^{-1}(c x)}{2 d^2 x^2 \left(1-c^2 x^2\right)}-\frac{4 c^2 \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}+\frac{i b c^2 \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{d^2}-\frac{i b c^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{d^2}-\frac{b c}{2 d^2 x \sqrt{1-c^2 x^2}}",1,"-1/2*(a/x^2 + (b*c^3*x)/Sqrt[1 - c^2*x^2] + (b*c*Sqrt[1 - c^2*x^2])/x + (a*c^2)/(-1 + c^2*x^2) + (b*ArcSin[c*x])/x^2 + (b*c^2*ArcSin[c*x])/(-1 + c^2*x^2) - 4*b*c^2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 4*b*c^2*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] - 4*a*c^2*Log[x] + 2*a*c^2*Log[1 - c^2*x^2] - (2*I)*b*c^2*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + (2*I)*b*c^2*PolyLog[2, E^((2*I)*ArcSin[c*x])])/d^2","A",0
45,1,426,259,0.9042688,"\int \frac{a+b \sin ^{-1}(c x)}{x^4 \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^2),x]","-\frac{15 a c^3 \log (1-c x)-15 a c^3 \log (c x+1)+\frac{24 a c^2}{x}+\frac{6 a c^4 x}{c^2 x^2-1}+\frac{4 a}{x^3}-30 i b c^3 \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+30 i b c^3 \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+\frac{3 b c^3 \sin ^{-1}(c x)}{c x-1}+\frac{3 b c^3 \sin ^{-1}(c x)}{c x+1}+15 i \pi  b c^3 \sin ^{-1}(c x)-30 b c^3 \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-15 \pi  b c^3 \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+30 b c^3 \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-15 \pi  b c^3 \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+15 \pi  b c^3 \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+15 \pi  b c^3 \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+\frac{2 b c \sqrt{1-c^2 x^2}}{x^2}+\frac{24 b c^2 \sin ^{-1}(c x)}{x}-\frac{3 b c^3 \sqrt{1-c^2 x^2}}{c x-1}+\frac{3 b c^3 \sqrt{1-c^2 x^2}}{c x+1}+26 b c^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+\frac{4 b \sin ^{-1}(c x)}{x^3}}{12 d^2}","-\frac{5 i c^3 \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}-\frac{5 c^2 \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 x \left(1-c^2 x^2\right)}-\frac{a+b \sin ^{-1}(c x)}{3 d^2 x^3 \left(1-c^2 x^2\right)}+\frac{5 c^4 x \left(a+b \sin ^{-1}(c x)\right)}{2 d^2 \left(1-c^2 x^2\right)}+\frac{5 i b c^3 \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{2 d^2}-\frac{5 i b c^3 \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{2 d^2}-\frac{b c}{6 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{b c^3}{3 d^2 \sqrt{1-c^2 x^2}}-\frac{13 b c^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{6 d^2}",1,"-1/12*((4*a)/x^3 + (24*a*c^2)/x + (2*b*c*Sqrt[1 - c^2*x^2])/x^2 - (3*b*c^3*Sqrt[1 - c^2*x^2])/(-1 + c*x) + (3*b*c^3*Sqrt[1 - c^2*x^2])/(1 + c*x) + (6*a*c^4*x)/(-1 + c^2*x^2) + (15*I)*b*c^3*Pi*ArcSin[c*x] + (4*b*ArcSin[c*x])/x^3 + (24*b*c^2*ArcSin[c*x])/x + (3*b*c^3*ArcSin[c*x])/(-1 + c*x) + (3*b*c^3*ArcSin[c*x])/(1 + c*x) + 26*b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]] - 15*b*c^3*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 30*b*c^3*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 15*b*c^3*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 30*b*c^3*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 15*a*c^3*Log[1 - c*x] - 15*a*c^3*Log[1 + c*x] + 15*b*c^3*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 15*b*c^3*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (30*I)*b*c^3*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (30*I)*b*c^3*PolyLog[2, I*E^(I*ArcSin[c*x])])/d^2","A",0
46,1,445,204,0.936097,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^3,x]","\frac{\frac{30 a c x}{c^2 x^2-1}+\frac{12 a c x}{\left(c^2 x^2-1\right)^2}-9 a \log (1-c x)+9 a \log (c x+1)-\frac{15 b \sqrt{1-c^2 x^2}}{c x-1}+\frac{15 b \sqrt{1-c^2 x^2}}{c x+1}+\frac{b c x \sqrt{1-c^2 x^2}}{(c x-1)^2}-\frac{2 b \sqrt{1-c^2 x^2}}{(c x-1)^2}-\frac{b c x \sqrt{1-c^2 x^2}}{(c x+1)^2}-\frac{2 b \sqrt{1-c^2 x^2}}{(c x+1)^2}+18 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-18 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+\frac{15 b \sin ^{-1}(c x)}{c x-1}+\frac{15 b \sin ^{-1}(c x)}{c x+1}+\frac{3 b \sin ^{-1}(c x)}{(c x-1)^2}-\frac{3 b \sin ^{-1}(c x)}{(c x+1)^2}-9 i \pi  b \sin ^{-1}(c x)+18 b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+9 \pi  b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-18 b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+9 \pi  b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-9 \pi  b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-9 \pi  b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{48 c^5 d^3}","-\frac{3 i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 c^5 d^3}+\frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{4 c^2 d^3 \left(1-c^2 x^2\right)^2}-\frac{3 x \left(a+b \sin ^{-1}(c x)\right)}{8 c^4 d^3 \left(1-c^2 x^2\right)}+\frac{3 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{8 c^5 d^3}-\frac{3 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{8 c^5 d^3}+\frac{5 b}{8 c^5 d^3 \sqrt{1-c^2 x^2}}-\frac{b}{12 c^5 d^3 \left(1-c^2 x^2\right)^{3/2}}",1,"((-2*b*Sqrt[1 - c^2*x^2])/(-1 + c*x)^2 + (b*c*x*Sqrt[1 - c^2*x^2])/(-1 + c*x)^2 - (15*b*Sqrt[1 - c^2*x^2])/(-1 + c*x) - (2*b*Sqrt[1 - c^2*x^2])/(1 + c*x)^2 - (b*c*x*Sqrt[1 - c^2*x^2])/(1 + c*x)^2 + (15*b*Sqrt[1 - c^2*x^2])/(1 + c*x) + (12*a*c*x)/(-1 + c^2*x^2)^2 + (30*a*c*x)/(-1 + c^2*x^2) - (9*I)*b*Pi*ArcSin[c*x] + (3*b*ArcSin[c*x])/(-1 + c*x)^2 + (15*b*ArcSin[c*x])/(-1 + c*x) - (3*b*ArcSin[c*x])/(1 + c*x)^2 + (15*b*ArcSin[c*x])/(1 + c*x) + 9*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] + 18*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 9*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] - 18*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - 9*a*Log[1 - c*x] + 9*a*Log[1 + c*x] - 9*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 9*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (18*I)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (18*I)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(48*c^5*d^3)","B",0
47,1,79,100,0.0772408,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(x^3*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^3,x]","\frac{a \left(6 c^2 x^2-3\right)+b c x \sqrt{1-c^2 x^2} \left(3-4 c^2 x^2\right)+3 b \left(2 c^2 x^2-1\right) \sin ^{-1}(c x)}{12 c^4 d^3 \left(c^2 x^2-1\right)^2}","\frac{x^4 \left(a+b \sin ^{-1}(c x)\right)}{4 d^3 \left(1-c^2 x^2\right)^2}-\frac{b \sin ^{-1}(c x)}{4 c^4 d^3}-\frac{b x^3}{12 c d^3 \left(1-c^2 x^2\right)^{3/2}}+\frac{b x}{4 c^3 d^3 \sqrt{1-c^2 x^2}}",1,"(b*c*x*(3 - 4*c^2*x^2)*Sqrt[1 - c^2*x^2] + a*(-3 + 6*c^2*x^2) + 3*b*(-1 + 2*c^2*x^2)*ArcSin[c*x])/(12*c^4*d^3*(-1 + c^2*x^2)^2)","A",1
48,1,445,202,0.7734962,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^3,x]","\frac{\frac{6 a c x}{c^2 x^2-1}+\frac{12 a c x}{\left(c^2 x^2-1\right)^2}+3 a \log (1-c x)-3 a \log (c x+1)-\frac{3 b \sqrt{1-c^2 x^2}}{c x-1}+\frac{3 b \sqrt{1-c^2 x^2}}{c x+1}+\frac{b c x \sqrt{1-c^2 x^2}}{(c x-1)^2}-\frac{2 b \sqrt{1-c^2 x^2}}{(c x-1)^2}-\frac{b c x \sqrt{1-c^2 x^2}}{(c x+1)^2}-\frac{2 b \sqrt{1-c^2 x^2}}{(c x+1)^2}-6 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+6 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+\frac{3 b \sin ^{-1}(c x)}{c x-1}+\frac{3 b \sin ^{-1}(c x)}{c x+1}+\frac{3 b \sin ^{-1}(c x)}{(c x-1)^2}-\frac{3 b \sin ^{-1}(c x)}{(c x+1)^2}+3 i \pi  b \sin ^{-1}(c x)-6 b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-3 \pi  b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+6 b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-3 \pi  b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+3 \pi  b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+3 \pi  b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{48 c^3 d^3}","\frac{i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 c^3 d^3}-\frac{x \left(a+b \sin ^{-1}(c x)\right)}{8 c^2 d^3 \left(1-c^2 x^2\right)}+\frac{x \left(a+b \sin ^{-1}(c x)\right)}{4 c^2 d^3 \left(1-c^2 x^2\right)^2}-\frac{i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{8 c^3 d^3}+\frac{i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{8 c^3 d^3}+\frac{b}{8 c^3 d^3 \sqrt{1-c^2 x^2}}-\frac{b}{12 c^3 d^3 \left(1-c^2 x^2\right)^{3/2}}",1,"((-2*b*Sqrt[1 - c^2*x^2])/(-1 + c*x)^2 + (b*c*x*Sqrt[1 - c^2*x^2])/(-1 + c*x)^2 - (3*b*Sqrt[1 - c^2*x^2])/(-1 + c*x) - (2*b*Sqrt[1 - c^2*x^2])/(1 + c*x)^2 - (b*c*x*Sqrt[1 - c^2*x^2])/(1 + c*x)^2 + (3*b*Sqrt[1 - c^2*x^2])/(1 + c*x) + (12*a*c*x)/(-1 + c^2*x^2)^2 + (6*a*c*x)/(-1 + c^2*x^2) + (3*I)*b*Pi*ArcSin[c*x] + (3*b*ArcSin[c*x])/(-1 + c*x)^2 + (3*b*ArcSin[c*x])/(-1 + c*x) - (3*b*ArcSin[c*x])/(1 + c*x)^2 + (3*b*ArcSin[c*x])/(1 + c*x) - 3*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 6*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 3*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 6*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 3*a*Log[1 - c*x] - 3*a*Log[1 + c*x] + 3*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 3*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (6*I)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (6*I)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/(48*c^3*d^3)","B",0
49,1,62,83,0.1116885,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(x*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^3,x]","\frac{\frac{a+b \sin ^{-1}(c x)}{\left(c^2 x^2-1\right)^2}+\frac{b c x \left(2 c^2 x^2-3\right)}{3 \left(1-c^2 x^2\right)^{3/2}}}{4 c^2 d^3}","\frac{a+b \sin ^{-1}(c x)}{4 c^2 d^3 \left(1-c^2 x^2\right)^2}-\frac{b x}{6 c d^3 \sqrt{1-c^2 x^2}}-\frac{b x}{12 c d^3 \left(1-c^2 x^2\right)^{3/2}}",1,"((b*c*x*(-3 + 2*c^2*x^2))/(3*(1 - c^2*x^2)^(3/2)) + (a + b*ArcSin[c*x])/(-1 + c^2*x^2)^2)/(4*c^2*d^3)","A",1
50,1,501,196,1.6099218,"\int \frac{a+b \sin ^{-1}(c x)}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^3,x]","-\frac{\frac{6 a x}{c^2 x^2-1}-\frac{4 a x}{\left(c^2 x^2-1\right)^2}+\frac{3 a \log (1-c x)}{c}-\frac{3 a \log (c x+1)}{c}+\frac{3 b \sqrt{1-c^2 x^2}}{c-c^2 x}+\frac{3 b \sqrt{1-c^2 x^2}}{c^2 x+c}-\frac{b x \sqrt{1-c^2 x^2}}{3 (c x-1)^2}+\frac{2 b \sqrt{1-c^2 x^2}}{3 c (c x-1)^2}+\frac{b x \sqrt{1-c^2 x^2}}{3 (c x+1)^2}+\frac{2 b \sqrt{1-c^2 x^2}}{3 c (c x+1)^2}-\frac{3 b \sin ^{-1}(c x)}{c-c^2 x}+\frac{3 b \sin ^{-1}(c x)}{c^2 x+c}-\frac{6 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c}+\frac{6 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{b \sin ^{-1}(c x)}{c (c x-1)^2}+\frac{b \sin ^{-1}(c x)}{c (c x+1)^2}+\frac{3 i \pi  b \sin ^{-1}(c x)}{c}-\frac{6 b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{3 \pi  b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)}{c}+\frac{6 b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{3 \pi  b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)}{c}+\frac{3 \pi  b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{c}+\frac{3 \pi  b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{c}}{16 d^3}","\frac{3 x \left(a+b \sin ^{-1}(c x)\right)}{8 d^3 \left(1-c^2 x^2\right)}+\frac{x \left(a+b \sin ^{-1}(c x)\right)}{4 d^3 \left(1-c^2 x^2\right)^2}-\frac{3 i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 c d^3}-\frac{3 b}{8 c d^3 \sqrt{1-c^2 x^2}}-\frac{b}{12 c d^3 \left(1-c^2 x^2\right)^{3/2}}+\frac{3 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{8 c d^3}-\frac{3 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{8 c d^3}",1,"-1/16*((2*b*Sqrt[1 - c^2*x^2])/(3*c*(-1 + c*x)^2) - (b*x*Sqrt[1 - c^2*x^2])/(3*(-1 + c*x)^2) + (2*b*Sqrt[1 - c^2*x^2])/(3*c*(1 + c*x)^2) + (b*x*Sqrt[1 - c^2*x^2])/(3*(1 + c*x)^2) + (3*b*Sqrt[1 - c^2*x^2])/(c - c^2*x) + (3*b*Sqrt[1 - c^2*x^2])/(c + c^2*x) - (4*a*x)/(-1 + c^2*x^2)^2 + (6*a*x)/(-1 + c^2*x^2) + ((3*I)*b*Pi*ArcSin[c*x])/c - (b*ArcSin[c*x])/(c*(-1 + c*x)^2) + (b*ArcSin[c*x])/(c*(1 + c*x)^2) - (3*b*ArcSin[c*x])/(c - c^2*x) + (3*b*ArcSin[c*x])/(c + c^2*x) - (3*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])])/c - (6*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])])/c - (3*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])])/c + (6*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])])/c + (3*a*Log[1 - c*x])/c - (3*a*Log[1 + c*x])/c + (3*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]])/c + (3*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]])/c - ((6*I)*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/c + ((6*I)*b*PolyLog[2, I*E^(I*ArcSin[c*x])])/c)/d^3","B",0
51,1,201,173,1.0282476,"\int \frac{a+b \sin ^{-1}(c x)}{x \left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSin[c*x])/(x*(d - c^2*d*x^2)^3),x]","-\frac{\frac{6 a}{c^2 x^2-1}-\frac{3 a}{\left(c^2 x^2-1\right)^2}+6 a \log \left(1-c^2 x^2\right)-12 a \log (x)+b \left(\frac{8 c x}{\sqrt{1-c^2 x^2}}+\frac{c x}{\left(1-c^2 x^2\right)^{3/2}}+\frac{6 \sin ^{-1}(c x)}{c^2 x^2-1}-\frac{3 \sin ^{-1}(c x)}{\left(c^2 x^2-1\right)^2}-6 i \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+6 i \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-12 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+12 \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)}{12 d^3}","\frac{a+b \sin ^{-1}(c x)}{2 d^3 \left(1-c^2 x^2\right)}+\frac{a+b \sin ^{-1}(c x)}{4 d^3 \left(1-c^2 x^2\right)^2}-\frac{2 \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^3}-\frac{2 b c x}{3 d^3 \sqrt{1-c^2 x^2}}-\frac{b c x}{12 d^3 \left(1-c^2 x^2\right)^{3/2}}+\frac{i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 d^3}-\frac{i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d^3}",1,"-1/12*((-3*a)/(-1 + c^2*x^2)^2 + (6*a)/(-1 + c^2*x^2) - 12*a*Log[x] + 6*a*Log[1 - c^2*x^2] + b*((c*x)/(1 - c^2*x^2)^(3/2) + (8*c*x)/Sqrt[1 - c^2*x^2] - (3*ArcSin[c*x])/(-1 + c^2*x^2)^2 + (6*ArcSin[c*x])/(-1 + c^2*x^2) - 12*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 12*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] - (6*I)*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + (6*I)*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/d^3","A",0
52,1,512,242,1.550929,"\int \frac{a+b \sin ^{-1}(c x)}{x^2 \left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^2*(d - c^2*d*x^2)^3),x]","-\frac{\frac{14 a c^2 x}{c^2 x^2-1}-\frac{4 a c^2 x}{\left(c^2 x^2-1\right)^2}+15 a c \log (1-c x)-15 a c \log (c x+1)+\frac{16 a}{x}-\frac{b c^2 x \sqrt{1-c^2 x^2}}{3 (c x-1)^2}+\frac{b c^2 x \sqrt{1-c^2 x^2}}{3 (c x+1)^2}-\frac{7 b c \sqrt{1-c^2 x^2}}{c x-1}+\frac{7 b c \sqrt{1-c^2 x^2}}{c x+1}+\frac{2 b c \sqrt{1-c^2 x^2}}{3 (c x-1)^2}+\frac{2 b c \sqrt{1-c^2 x^2}}{3 (c x+1)^2}+16 b c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-30 i b c \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+30 i b c \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+\frac{7 b c \sin ^{-1}(c x)}{c x-1}+\frac{7 b c \sin ^{-1}(c x)}{c x+1}-\frac{b c \sin ^{-1}(c x)}{(c x-1)^2}+\frac{b c \sin ^{-1}(c x)}{(c x+1)^2}+15 i \pi  b c \sin ^{-1}(c x)+\frac{16 b \sin ^{-1}(c x)}{x}-30 b c \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-15 \pi  b c \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+30 b c \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-15 \pi  b c \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+15 \pi  b c \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+15 \pi  b c \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{16 d^3}","\frac{15 c^2 x \left(a+b \sin ^{-1}(c x)\right)}{8 d^3 \left(1-c^2 x^2\right)}+\frac{5 c^2 x \left(a+b \sin ^{-1}(c x)\right)}{4 d^3 \left(1-c^2 x^2\right)^2}-\frac{a+b \sin ^{-1}(c x)}{d^3 x \left(1-c^2 x^2\right)^2}-\frac{15 i c \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 d^3}-\frac{7 b c}{8 d^3 \sqrt{1-c^2 x^2}}-\frac{b c}{12 d^3 \left(1-c^2 x^2\right)^{3/2}}-\frac{b c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{d^3}+\frac{15 i b c \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{8 d^3}-\frac{15 i b c \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{8 d^3}",1,"-1/16*((16*a)/x + (2*b*c*Sqrt[1 - c^2*x^2])/(3*(-1 + c*x)^2) - (b*c^2*x*Sqrt[1 - c^2*x^2])/(3*(-1 + c*x)^2) - (7*b*c*Sqrt[1 - c^2*x^2])/(-1 + c*x) + (2*b*c*Sqrt[1 - c^2*x^2])/(3*(1 + c*x)^2) + (b*c^2*x*Sqrt[1 - c^2*x^2])/(3*(1 + c*x)^2) + (7*b*c*Sqrt[1 - c^2*x^2])/(1 + c*x) - (4*a*c^2*x)/(-1 + c^2*x^2)^2 + (14*a*c^2*x)/(-1 + c^2*x^2) + (15*I)*b*c*Pi*ArcSin[c*x] + (16*b*ArcSin[c*x])/x - (b*c*ArcSin[c*x])/(-1 + c*x)^2 + (7*b*c*ArcSin[c*x])/(-1 + c*x) + (b*c*ArcSin[c*x])/(1 + c*x)^2 + (7*b*c*ArcSin[c*x])/(1 + c*x) + 16*b*c*ArcTanh[Sqrt[1 - c^2*x^2]] - 15*b*c*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 30*b*c*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 15*b*c*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 30*b*c*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 15*a*c*Log[1 - c*x] - 15*a*c*Log[1 + c*x] + 15*b*c*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 15*b*c*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (30*I)*b*c*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (30*I)*b*c*PolyLog[2, I*E^(I*ArcSin[c*x])])/d^3","B",0
53,1,256,248,1.6858361,"\int \frac{a+b \sin ^{-1}(c x)}{x^3 \left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)^3),x]","-\frac{\frac{12 a c^2}{c^2 x^2-1}-\frac{3 a c^2}{\left(c^2 x^2-1\right)^2}+18 a c^2 \log \left(1-c^2 x^2\right)-36 a c^2 \log (x)+\frac{6 a}{x^2}+b c^2 \left(\frac{14 c x}{\sqrt{1-c^2 x^2}}+\frac{c x}{\left(1-c^2 x^2\right)^{3/2}}+\frac{6 \sqrt{1-c^2 x^2}}{c x}+\frac{12 \sin ^{-1}(c x)}{c^2 x^2-1}-\frac{3 \sin ^{-1}(c x)}{\left(c^2 x^2-1\right)^2}+\frac{6 \sin ^{-1}(c x)}{c^2 x^2}-18 i \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+18 i \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-36 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+36 \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)}{12 d^3}","\frac{3 c^2 \left(a+b \sin ^{-1}(c x)\right)}{2 d^3 \left(1-c^2 x^2\right)}+\frac{3 c^2 \left(a+b \sin ^{-1}(c x)\right)}{4 d^3 \left(1-c^2 x^2\right)^2}-\frac{a+b \sin ^{-1}(c x)}{2 d^3 x^2 \left(1-c^2 x^2\right)^2}-\frac{6 c^2 \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^3}+\frac{3 i b c^2 \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 d^3}-\frac{3 i b c^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d^3}-\frac{b c}{2 d^3 x \left(1-c^2 x^2\right)^{3/2}}-\frac{2 b c^3 x}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 x}{12 d^3 \left(1-c^2 x^2\right)^{3/2}}",1,"-1/12*((6*a)/x^2 - (3*a*c^2)/(-1 + c^2*x^2)^2 + (12*a*c^2)/(-1 + c^2*x^2) - 36*a*c^2*Log[x] + 18*a*c^2*Log[1 - c^2*x^2] + b*c^2*((c*x)/(1 - c^2*x^2)^(3/2) + (14*c*x)/Sqrt[1 - c^2*x^2] + (6*Sqrt[1 - c^2*x^2])/(c*x) + (6*ArcSin[c*x])/(c^2*x^2) - (3*ArcSin[c*x])/(-1 + c^2*x^2)^2 + (12*ArcSin[c*x])/(-1 + c^2*x^2) - 36*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 36*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] - (18*I)*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + (18*I)*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/d^3","A",0
54,1,587,317,1.6567115,"\int \frac{a+b \sin ^{-1}(c x)}{x^4 \left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^3),x]","-\frac{105 a c^3 \log (1-c x)-105 a c^3 \log (c x+1)+\frac{144 a c^2}{x}+\frac{66 a c^4 x}{c^2 x^2-1}-\frac{12 a c^4 x}{\left(c^2 x^2-1\right)^2}+\frac{16 a}{x^3}-210 i b c^3 \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+210 i b c^3 \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+\frac{33 b c^3 \sin ^{-1}(c x)}{c x-1}+\frac{33 b c^3 \sin ^{-1}(c x)}{c x+1}-\frac{3 b c^3 \sin ^{-1}(c x)}{(c x-1)^2}+\frac{3 b c^3 \sin ^{-1}(c x)}{(c x+1)^2}+105 i \pi  b c^3 \sin ^{-1}(c x)-210 b c^3 \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-105 \pi  b c^3 \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+210 b c^3 \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-105 \pi  b c^3 \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+105 \pi  b c^3 \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+105 \pi  b c^3 \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+\frac{8 b c \sqrt{1-c^2 x^2}}{x^2}+\frac{144 b c^2 \sin ^{-1}(c x)}{x}-\frac{b c^4 x \sqrt{1-c^2 x^2}}{(c x-1)^2}+\frac{b c^4 x \sqrt{1-c^2 x^2}}{(c x+1)^2}-\frac{33 b c^3 \sqrt{1-c^2 x^2}}{c x-1}+\frac{33 b c^3 \sqrt{1-c^2 x^2}}{c x+1}+\frac{2 b c^3 \sqrt{1-c^2 x^2}}{(c x-1)^2}+\frac{2 b c^3 \sqrt{1-c^2 x^2}}{(c x+1)^2}+152 b c^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+\frac{16 b \sin ^{-1}(c x)}{x^3}}{48 d^3}","-\frac{35 i c^3 \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 d^3}-\frac{7 c^2 \left(a+b \sin ^{-1}(c x)\right)}{3 d^3 x \left(1-c^2 x^2\right)^2}-\frac{a+b \sin ^{-1}(c x)}{3 d^3 x^3 \left(1-c^2 x^2\right)^2}+\frac{35 c^4 x \left(a+b \sin ^{-1}(c x)\right)}{8 d^3 \left(1-c^2 x^2\right)}+\frac{35 c^4 x \left(a+b \sin ^{-1}(c x)\right)}{12 d^3 \left(1-c^2 x^2\right)^2}+\frac{35 i b c^3 \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{8 d^3}-\frac{35 i b c^3 \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{8 d^3}-\frac{b c}{6 d^3 x^2 \left(1-c^2 x^2\right)^{3/2}}-\frac{29 b c^3}{24 d^3 \sqrt{1-c^2 x^2}}+\frac{b c^3}{12 d^3 \left(1-c^2 x^2\right)^{3/2}}-\frac{19 b c^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{6 d^3}",1,"-1/48*((16*a)/x^3 + (144*a*c^2)/x + (8*b*c*Sqrt[1 - c^2*x^2])/x^2 + (2*b*c^3*Sqrt[1 - c^2*x^2])/(-1 + c*x)^2 - (b*c^4*x*Sqrt[1 - c^2*x^2])/(-1 + c*x)^2 - (33*b*c^3*Sqrt[1 - c^2*x^2])/(-1 + c*x) + (2*b*c^3*Sqrt[1 - c^2*x^2])/(1 + c*x)^2 + (b*c^4*x*Sqrt[1 - c^2*x^2])/(1 + c*x)^2 + (33*b*c^3*Sqrt[1 - c^2*x^2])/(1 + c*x) - (12*a*c^4*x)/(-1 + c^2*x^2)^2 + (66*a*c^4*x)/(-1 + c^2*x^2) + (105*I)*b*c^3*Pi*ArcSin[c*x] + (16*b*ArcSin[c*x])/x^3 + (144*b*c^2*ArcSin[c*x])/x - (3*b*c^3*ArcSin[c*x])/(-1 + c*x)^2 + (33*b*c^3*ArcSin[c*x])/(-1 + c*x) + (3*b*c^3*ArcSin[c*x])/(1 + c*x)^2 + (33*b*c^3*ArcSin[c*x])/(1 + c*x) + 152*b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]] - 105*b*c^3*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 210*b*c^3*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 105*b*c^3*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 210*b*c^3*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 105*a*c^3*Log[1 - c*x] - 105*a*c^3*Log[1 + c*x] + 105*b*c^3*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 105*b*c^3*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (210*I)*b*c^3*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (210*I)*b*c^3*PolyLog[2, I*E^(I*ArcSin[c*x])])/d^3","A",0
55,1,169,262,0.1523791,"\int x^4 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]),x]","\frac{\sqrt{d-c^2 d x^2} \left(9 a^2+6 a b c x \sqrt{1-c^2 x^2} \left(8 c^4 x^4-2 c^2 x^2-3\right)+6 b \sin ^{-1}(c x) \left(3 a+b c x \sqrt{1-c^2 x^2} \left(8 c^4 x^4-2 c^2 x^2-3\right)\right)+b^2 c^2 x^2 \left(-8 c^4 x^4+3 c^2 x^2+9\right)+9 b^2 \sin ^{-1}(c x)^2\right)}{288 b c^5 \sqrt{1-c^2 x^2}}","\frac{1}{6} x^5 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{24 c^2}+\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{32 b c^5 \sqrt{1-c^2 x^2}}-\frac{x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{16 c^4}-\frac{b c x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{1-c^2 x^2}}+\frac{b x^4 \sqrt{d-c^2 d x^2}}{96 c \sqrt{1-c^2 x^2}}+\frac{b x^2 \sqrt{d-c^2 d x^2}}{32 c^3 \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(9*a^2 + b^2*c^2*x^2*(9 + 3*c^2*x^2 - 8*c^4*x^4) + 6*a*b*c*x*Sqrt[1 - c^2*x^2]*(-3 - 2*c^2*x^2 + 8*c^4*x^4) + 6*b*(3*a + b*c*x*Sqrt[1 - c^2*x^2]*(-3 - 2*c^2*x^2 + 8*c^4*x^4))*ArcSin[c*x] + 9*b^2*ArcSin[c*x]^2))/(288*b*c^5*Sqrt[1 - c^2*x^2])","A",1
56,1,140,189,0.1132303,"\int x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]),x]","\frac{\sqrt{d-c^2 d x^2} \left(a^2+2 a b c x \sqrt{1-c^2 x^2} \left(2 c^2 x^2-1\right)+2 b \sin ^{-1}(c x) \left(a+b c x \sqrt{1-c^2 x^2} \left(2 c^2 x^2-1\right)\right)+b^2 c^2 x^2 \left(1-c^2 x^2\right)+b^2 \sin ^{-1}(c x)^2\right)}{16 b c^3 \sqrt{1-c^2 x^2}}","-\frac{x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 c^2}+\frac{1}{4} x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{\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 x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{1-c^2 x^2}}-\frac{b c x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(a^2 + b^2*c^2*x^2*(1 - c^2*x^2) + 2*a*b*c*x*Sqrt[1 - c^2*x^2]*(-1 + 2*c^2*x^2) + 2*b*(a + b*c*x*Sqrt[1 - c^2*x^2]*(-1 + 2*c^2*x^2))*ArcSin[c*x] + b^2*ArcSin[c*x]^2))/(16*b*c^3*Sqrt[1 - c^2*x^2])","A",1
57,1,111,116,0.0589114,"\int \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]),x]","\frac{\sqrt{d-c^2 d x^2} \left(a^2+2 a b c x \sqrt{1-c^2 x^2}+2 b \sin ^{-1}(c x) \left(a+b c x \sqrt{1-c^2 x^2}\right)-b^2 c^2 x^2+b^2 \sin ^{-1}(c x)^2\right)}{4 b c \sqrt{1-c^2 x^2}}","\frac{1}{2} x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{\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{b c x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(a^2 - b^2*c^2*x^2 + 2*a*b*c*x*Sqrt[1 - c^2*x^2] + 2*b*(a + b*c*x*Sqrt[1 - c^2*x^2])*ArcSin[c*x] + b^2*ArcSin[c*x]^2))/(4*b*c*Sqrt[1 - c^2*x^2])","A",1
58,1,142,110,0.4666085,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{x^2} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x^2,x]","-\frac{a \sqrt{-d \left(c^2 x^2-1\right)}}{x}+a c \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{-d \left(c^2 x^2-1\right)}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-\frac{b c \sqrt{d \left(1-c^2 x^2\right)} \left(\frac{2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c x}-2 \log (c x)+\sin ^{-1}(c x)^2\right)}{2 \sqrt{1-c^2 x^2}}","-\frac{c \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 b \sqrt{1-c^2 x^2}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{x}+\frac{b c \log (x) \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}",1,"-((a*Sqrt[-(d*(-1 + c^2*x^2))])/x) + a*c*Sqrt[d]*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))] - (b*c*Sqrt[d*(1 - c^2*x^2)]*((2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*x) + ArcSin[c*x]^2 - 2*Log[c*x]))/(2*Sqrt[1 - c^2*x^2])","A",1
59,1,134,111,0.2346875,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{x^4} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x^4,x]","\frac{\sqrt{d-c^2 d x^2} \left(2 a \left(c^2 x^2-1\right)^2+b c x \left(1-3 c^2 x^2\right) \sqrt{1-c^2 x^2}+2 b \left(c^2 x^2-1\right)^2 \sin ^{-1}(c x)\right)}{6 x^3 \left(c^2 x^2-1\right)}-\frac{b c^3 \log (x) \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}","-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 d x^3}-\frac{b c \sqrt{d-c^2 d x^2}}{6 x^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 \log (x) \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(b*c*x*(1 - 3*c^2*x^2)*Sqrt[1 - c^2*x^2] + 2*a*(-1 + c^2*x^2)^2 + 2*b*(-1 + c^2*x^2)^2*ArcSin[c*x]))/(6*x^3*(-1 + c^2*x^2)) - (b*c^3*Sqrt[d - c^2*d*x^2]*Log[x])/(3*Sqrt[1 - c^2*x^2])","A",1
60,1,162,187,0.1577273,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{x^6} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x^6,x]","\frac{\sqrt{d-c^2 d x^2} \left(12 a \left(2 c^2 x^2+3\right) \left(c^2 x^2-1\right)^2+12 b \left(2 c^2 x^2+3\right) \left(c^2 x^2-1\right)^2 \sin ^{-1}(c x)+b c x \sqrt{1-c^2 x^2} \left(-50 c^4 x^4-6 c^2 x^2+9\right)\right)}{180 x^5 \left(c^2 x^2-1\right)}-\frac{2 b c^5 \log (x) \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}}","-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{5 d x^5}-\frac{2 c^2 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{15 d x^3}-\frac{b c \sqrt{d-c^2 d x^2}}{20 x^4 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 \log (x) \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{30 x^2 \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(12*a*(-1 + c^2*x^2)^2*(3 + 2*c^2*x^2) + b*c*x*Sqrt[1 - c^2*x^2]*(9 - 6*c^2*x^2 - 50*c^4*x^4) + 12*b*(-1 + c^2*x^2)^2*(3 + 2*c^2*x^2)*ArcSin[c*x]))/(180*x^5*(-1 + c^2*x^2)) - (2*b*c^5*Sqrt[d - c^2*d*x^2]*Log[x])/(15*Sqrt[1 - c^2*x^2])","A",1
61,1,187,263,0.1866004,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{x^8} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x^8,x]","\frac{\sqrt{d-c^2 d x^2} \left(20 a \left(8 c^4 x^4+12 c^2 x^2+15\right) \left(c^2 x^2-1\right)^2+20 b \left(8 c^4 x^4+12 c^2 x^2+15\right) \left(c^2 x^2-1\right)^2 \sin ^{-1}(c x)-b c x \sqrt{1-c^2 x^2} \left(392 c^6 x^6+40 c^4 x^4+15 c^2 x^2-50\right)\right)}{2100 x^7 \left(c^2 x^2-1\right)}-\frac{8 b c^7 \log (x) \sqrt{d-c^2 d x^2}}{105 \sqrt{1-c^2 x^2}}","-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{7 d x^7}-\frac{4 c^2 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{35 d x^5}-\frac{8 c^4 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{105 d x^3}-\frac{b c \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{1-c^2 x^2}}-\frac{8 b c^7 \log (x) \sqrt{d-c^2 d x^2}}{105 \sqrt{1-c^2 x^2}}+\frac{2 b c^5 \sqrt{d-c^2 d x^2}}{105 x^2 \sqrt{1-c^2 x^2}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{140 x^4 \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(20*a*(-1 + c^2*x^2)^2*(15 + 12*c^2*x^2 + 8*c^4*x^4) - b*c*x*Sqrt[1 - c^2*x^2]*(-50 + 15*c^2*x^2 + 40*c^4*x^4 + 392*c^6*x^6) + 20*b*(-1 + c^2*x^2)^2*(15 + 12*c^2*x^2 + 8*c^4*x^4)*ArcSin[c*x]))/(2100*x^7*(-1 + c^2*x^2)) - (8*b*c^7*Sqrt[d - c^2*d*x^2]*Log[x])/(105*Sqrt[1 - c^2*x^2])","A",1
62,1,157,256,0.1911255,"\int x^5 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]),x]","\frac{\sqrt{d-c^2 d x^2} \left(105 a \sqrt{1-c^2 x^2} \left(15 c^6 x^6-3 c^4 x^4-4 c^2 x^2-8\right)+b c x \left(-225 c^6 x^6+63 c^4 x^4+140 c^2 x^2+840\right)+105 b \sqrt{1-c^2 x^2} \left(15 c^6 x^6-3 c^4 x^4-4 c^2 x^2-8\right) \sin ^{-1}(c x)\right)}{11025 c^6 \sqrt{1-c^2 x^2}}","-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{7 c^6 d^3}+\frac{2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{5 c^6 d^2}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 c^6 d}-\frac{b c x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}+\frac{b x^5 \sqrt{d-c^2 d x^2}}{175 c \sqrt{1-c^2 x^2}}+\frac{8 b x \sqrt{d-c^2 d x^2}}{105 c^5 \sqrt{1-c^2 x^2}}+\frac{4 b x^3 \sqrt{d-c^2 d x^2}}{315 c^3 \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(b*c*x*(840 + 140*c^2*x^2 + 63*c^4*x^4 - 225*c^6*x^6) + 105*a*Sqrt[1 - c^2*x^2]*(-8 - 4*c^2*x^2 - 3*c^4*x^4 + 15*c^6*x^6) + 105*b*Sqrt[1 - c^2*x^2]*(-8 - 4*c^2*x^2 - 3*c^4*x^4 + 15*c^6*x^6)*ArcSin[c*x]))/(11025*c^6*Sqrt[1 - c^2*x^2])","A",1
63,1,134,183,0.1269828,"\int x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]),x]","\frac{\sqrt{d-c^2 d x^2} \left(15 a \sqrt{1-c^2 x^2} \left(3 c^4 x^4-c^2 x^2-2\right)+b \left(-9 c^5 x^5+5 c^3 x^3+30 c x\right)+15 b \sqrt{1-c^2 x^2} \left(3 c^4 x^4-c^2 x^2-2\right) \sin ^{-1}(c x)\right)}{225 c^4 \sqrt{1-c^2 x^2}}","\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{5 c^4 d^2}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 c^4 d}-\frac{b c x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}+\frac{b x^3 \sqrt{d-c^2 d x^2}}{45 c \sqrt{1-c^2 x^2}}+\frac{2 b 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]*(15*a*Sqrt[1 - c^2*x^2]*(-2 - c^2*x^2 + 3*c^4*x^4) + b*(30*c*x + 5*c^3*x^3 - 9*c^5*x^5) + 15*b*Sqrt[1 - c^2*x^2]*(-2 - c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x]))/(225*c^4*Sqrt[1 - c^2*x^2])","A",1
64,1,70,110,0.0930068,"\int x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]),x]","\frac{\sqrt{d-c^2 d x^2} \left(\left(c^2 x^2-1\right) \left(a+b \sin ^{-1}(c x)\right)+\frac{b c \left(x-\frac{c^2 x^3}{3}\right)}{\sqrt{1-c^2 x^2}}\right)}{3 c^2}","-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 c^2 d}+\frac{b x \sqrt{d-c^2 d x^2}}{3 c \sqrt{1-c^2 x^2}}-\frac{b c x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*((b*c*(x - (c^2*x^3)/3))/Sqrt[1 - c^2*x^2] + (-1 + c^2*x^2)*(a + b*ArcSin[c*x])))/(3*c^2)","A",1
65,1,187,203,0.6183923,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{x} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x,x]","a \sqrt{d-c^2 d x^2}-a \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+a \sqrt{d} \log (x)+\frac{b \sqrt{d-c^2 d x^2} \left(\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-c x+\sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-\sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}","\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}+\frac{i b \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{i b \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{b c x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}",1,"a*Sqrt[d - c^2*d*x^2] + a*Sqrt[d]*Log[x] - a*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + I*PolyLog[2, -E^(I*ArcSin[c*x])] - I*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2]","A",0
66,1,239,225,2.3550825,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{x^3} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x^3,x]","\frac{1}{8} \left(-\frac{4 a \sqrt{d-c^2 d x^2}}{x^2}+4 a c^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-4 a c^2 \sqrt{d} \log (x)+\frac{b c^2 d \sqrt{1-c^2 x^2} \left(-4 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+4 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-4 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+4 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)-2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)-2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{\sqrt{d-c^2 d x^2}}\right)","-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 x^2}+\frac{c^2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{i b c^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{2 \sqrt{1-c^2 x^2}}+\frac{i b c^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{2 \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{d-c^2 d x^2}}{2 x \sqrt{1-c^2 x^2}}",1,"((-4*a*Sqrt[d - c^2*d*x^2])/x^2 - 4*a*c^2*Sqrt[d]*Log[x] + 4*a*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*c^2*d*Sqrt[1 - c^2*x^2]*(-2*Cot[ArcSin[c*x]/2] - ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (4*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 - 2*Tan[ArcSin[c*x]/2]))/Sqrt[d - c^2*d*x^2])/8","A",1
67,1,321,301,4.5636696,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{x^5} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x^5,x]","-\frac{1}{8} a c^4 \sqrt{d} \log (x)+\frac{a \left(c^2 x^2-2\right) \sqrt{d-c^2 d x^2}}{8 x^4}+\frac{1}{8} a c^4 \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+\frac{b c^4 \sqrt{d-c^2 d x^2} \left(-\frac{16 \sin ^4\left(\frac{1}{2} \sin ^{-1}(c x)\right)}{c^3 x^3}-24 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+24 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-24 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+24 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)+8 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)+8 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)-c x \csc ^4\left(\frac{1}{2} \sin ^{-1}(c x)\right)-3 \sin ^{-1}(c x) \csc ^4\left(\frac{1}{2} \sin ^{-1}(c x)\right)+6 \sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+3 \sin ^{-1}(c x) \sec ^4\left(\frac{1}{2} \sin ^{-1}(c x)\right)-6 \sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{192 \sqrt{1-c^2 x^2}}","\frac{c^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 x^2}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{4 x^4}+\frac{c^4 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{1-c^2 x^2}}-\frac{i b c^4 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{8 \sqrt{1-c^2 x^2}}+\frac{i b c^4 \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{8 \sqrt{1-c^2 x^2}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{8 x \sqrt{1-c^2 x^2}}",1,"(a*(-2 + c^2*x^2)*Sqrt[d - c^2*d*x^2])/(8*x^4) - (a*c^4*Sqrt[d]*Log[x])/8 + (a*c^4*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/8 + (b*c^4*Sqrt[d - c^2*d*x^2]*(8*Cot[ArcSin[c*x]/2] + 6*ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 - c*x*Csc[ArcSin[c*x]/2]^4 - 3*ArcSin[c*x]*Csc[ArcSin[c*x]/2]^4 - 24*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 24*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (24*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (24*I)*PolyLog[2, E^(I*ArcSin[c*x])] - 6*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 + 3*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^4 - (16*Sin[ArcSin[c*x]/2]^4)/(c^3*x^3) + 8*Tan[ArcSin[c*x]/2]))/(192*Sqrt[1 - c^2*x^2])","A",0
68,1,193,340,0.2075063,"\int x^4 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]","\frac{d \sqrt{d-c^2 d x^2} \left(3 a^2-2 a b c x \sqrt{1-c^2 x^2} \left(16 c^6 x^6-24 c^4 x^4+2 c^2 x^2+3\right)-2 b \sin ^{-1}(c x) \left(b c x \sqrt{1-c^2 x^2} \left(16 c^6 x^6-24 c^4 x^4+2 c^2 x^2+3\right)-3 a\right)+b^2 c^2 x^2 \left(4 c^6 x^6-8 c^4 x^4+c^2 x^2+3\right)+3 b^2 \sin ^{-1}(c x)^2\right)}{256 b c^5 \sqrt{1-c^2 x^2}}","\frac{1}{8} x^5 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{16} d x^5 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{d x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{64 c^2}+\frac{3 d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{256 b c^5 \sqrt{1-c^2 x^2}}-\frac{3 d x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{128 c^4}-\frac{b c d x^6 \sqrt{d-c^2 d x^2}}{32 \sqrt{1-c^2 x^2}}+\frac{b d x^4 \sqrt{d-c^2 d x^2}}{256 c \sqrt{1-c^2 x^2}}+\frac{3 b d x^2 \sqrt{d-c^2 d x^2}}{256 c^3 \sqrt{1-c^2 x^2}}+\frac{b c^3 d x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}",1,"(d*Sqrt[d - c^2*d*x^2]*(3*a^2 + b^2*c^2*x^2*(3 + c^2*x^2 - 8*c^4*x^4 + 4*c^6*x^6) - 2*a*b*c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6) - 2*b*(-3*a + b*c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6))*ArcSin[c*x] + 3*b^2*ArcSin[c*x]^2))/(256*b*c^5*Sqrt[1 - c^2*x^2])","A",1
69,1,170,265,0.1783716,"\int x^2 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[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(9 a^2-6 a b c x \sqrt{1-c^2 x^2} \left(8 c^4 x^4-14 c^2 x^2+3\right)+6 b \sin ^{-1}(c x) \left(3 a+b c x \sqrt{1-c^2 x^2} \left(-8 c^4 x^4+14 c^2 x^2-3\right)\right)+b^2 c^2 x^2 \left(8 c^4 x^4-21 c^2 x^2+9\right)+9 b^2 \sin ^{-1}(c x)^2\right)}{288 b c^3 \sqrt{1-c^2 x^2}}","-\frac{d x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{16 c^2}+\frac{1}{6} x^3 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{d \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{b d x^2 \sqrt{d-c^2 d x^2}}{32 c \sqrt{1-c^2 x^2}}-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}+\frac{b c^3 d 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]*(9*a^2 + b^2*c^2*x^2*(9 - 21*c^2*x^2 + 8*c^4*x^4) - 6*a*b*c*x*Sqrt[1 - c^2*x^2]*(3 - 14*c^2*x^2 + 8*c^4*x^4) + 6*b*(3*a + b*c*x*Sqrt[1 - c^2*x^2]*(-3 + 14*c^2*x^2 - 8*c^4*x^4))*ArcSin[c*x] + 9*b^2*ArcSin[c*x]^2))/(288*b*c^3*Sqrt[1 - c^2*x^2])","A",1
70,1,210,188,0.6153578,"\int \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]","\frac{d \sqrt{d-c^2 d x^2} \left(16 a c x \sqrt{1-c^2 x^2} \left(5-2 c^2 x^2\right)+16 b \cos \left(2 \sin ^{-1}(c x)\right)+b \cos \left(4 \sin ^{-1}(c x)\right)\right)-48 a d^{3/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+24 b d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2+4 b d \sqrt{d-c^2 d x^2} \left(8 \sin \left(2 \sin ^{-1}(c x)\right)+\sin \left(4 \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)}{128 c \sqrt{1-c^2 x^2}}","\frac{1}{4} x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)+\frac{3}{8} d x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{3 d \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{5 b c d x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{b c^3 d x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}",1,"(24*b*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2 - 48*a*d^(3/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + d*Sqrt[d - c^2*d*x^2]*(16*a*c*x*(5 - 2*c^2*x^2)*Sqrt[1 - c^2*x^2] + 16*b*Cos[2*ArcSin[c*x]] + b*Cos[4*ArcSin[c*x]]) + 4*b*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*(8*Sin[2*ArcSin[c*x]] + Sin[4*ArcSin[c*x]]))/(128*c*Sqrt[1 - c^2*x^2])","A",1
71,1,222,185,0.6188832,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^2,x]","\frac{3}{2} a c d^{3/2} \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)+\sqrt{-d \left(c^2 x^2-1\right)} \left(-\frac{1}{2} a c^2 d x-\frac{a d}{x}\right)-\frac{b c d \sqrt{d \left(1-c^2 x^2\right)} \left(\frac{2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c x}-2 \log (c x)+\sin ^{-1}(c x)^2\right)}{2 \sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d \left(1-c^2 x^2\right)} \left(2 \sin ^{-1}(c x) \left(\sin ^{-1}(c x)+\sin \left(2 \sin ^{-1}(c x)\right)\right)+\cos \left(2 \sin ^{-1}(c x)\right)\right)}{8 \sqrt{1-c^2 x^2}}","-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{3 c d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{4 b \sqrt{1-c^2 x^2}}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{x}+\frac{b c d \log (x) \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}+\frac{b c^3 d x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}",1,"(-((a*d)/x) - (a*c^2*d*x)/2)*Sqrt[-(d*(-1 + c^2*x^2))] + (3*a*c*d^(3/2)*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/2 - (b*c*d*Sqrt[d*(1 - c^2*x^2)]*((2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*x) + ArcSin[c*x]^2 - 2*Log[c*x]))/(2*Sqrt[1 - c^2*x^2]) - (b*c*d*Sqrt[d*(1 - c^2*x^2)]*(Cos[2*ArcSin[c*x]] + 2*ArcSin[c*x]*(ArcSin[c*x] + Sin[2*ArcSin[c*x]])))/(8*Sqrt[1 - c^2*x^2])","A",1
72,1,211,191,0.8464598,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^4,x]","-\frac{d \sqrt{d-c^2 d x^2} \left(2 a \left(1-4 c^2 x^2\right) \sqrt{1-c^2 x^2}+8 b c^3 x^3 \log (c x)+b c x\right)}{6 x^3 \sqrt{1-c^2 x^2}}-a c^3 d^{3/2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+\frac{b d \left(4 c^2 x^2-1\right) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{3 x^3}+\frac{b c^3 d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2}{2 \sqrt{1-c^2 x^2}}","\frac{c^2 d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{x}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 x^3}+\frac{c^3 d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 b \sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2}}{6 x^2 \sqrt{1-c^2 x^2}}-\frac{4 b c^3 d \log (x) \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}",1,"(b*d*(-1 + 4*c^2*x^2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(3*x^3) + (b*c^3*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/(2*Sqrt[1 - c^2*x^2]) - a*c^3*d^(3/2)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - (d*Sqrt[d - c^2*d*x^2]*(b*c*x + 2*a*(1 - 4*c^2*x^2)*Sqrt[1 - c^2*x^2] + 8*b*c^3*x^3*Log[c*x]))/(6*x^3*Sqrt[1 - c^2*x^2])","A",1
73,1,144,154,0.2576884,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^6,x]","\frac{b c^5 d \log (x) \sqrt{d-c^2 d x^2}}{5 \sqrt{1-c^2 x^2}}-\frac{d \sqrt{d-c^2 d x^2} \left(12 a \left(c^2 x^2-1\right)^3+12 b \left(c^2 x^2-1\right)^3 \sin ^{-1}(c x)+b c x \sqrt{1-c^2 x^2} \left(-25 c^4 x^4+12 c^2 x^2-3\right)\right)}{60 x^5 \left(c^2 x^2-1\right)}","-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{5 d x^5}-\frac{b c d \sqrt{d-c^2 d x^2}}{20 x^4 \sqrt{1-c^2 x^2}}+\frac{b c^5 d \log (x) \sqrt{d-c^2 d x^2}}{5 \sqrt{1-c^2 x^2}}+\frac{b c^3 d \sqrt{d-c^2 d x^2}}{5 x^2 \sqrt{1-c^2 x^2}}",1,"-1/60*(d*Sqrt[d - c^2*d*x^2]*(12*a*(-1 + c^2*x^2)^3 + b*c*x*Sqrt[1 - c^2*x^2]*(-3 + 12*c^2*x^2 - 25*c^4*x^4) + 12*b*(-1 + c^2*x^2)^3*ArcSin[c*x]))/(x^5*(-1 + c^2*x^2)) + (b*c^5*d*Sqrt[d - c^2*d*x^2]*Log[x])/(5*Sqrt[1 - c^2*x^2])","A",1
74,1,173,231,0.2163795,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^8,x]","\frac{2 b c^7 d \log (x) \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}-\frac{d \sqrt{d-c^2 d x^2} \left(30 a \left(2 c^2 x^2+5\right) \left(c^2 x^2-1\right)^3+30 b \left(2 c^2 x^2+5\right) \left(c^2 x^2-1\right)^3 \sin ^{-1}(c x)-b c x \sqrt{1-c^2 x^2} \left(147 c^6 x^6+15 c^4 x^4-60 c^2 x^2+25\right)\right)}{1050 x^7 \left(c^2 x^2-1\right)}","-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{7 d x^7}-\frac{2 c^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{35 d x^5}-\frac{b c d \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{1-c^2 x^2}}+\frac{2 b c^7 d \log (x) \sqrt{d-c^2 d x^2}}{35 \sqrt{1-c^2 x^2}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{70 x^2 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d \sqrt{d-c^2 d x^2}}{35 x^4 \sqrt{1-c^2 x^2}}",1,"-1/1050*(d*Sqrt[d - c^2*d*x^2]*(30*a*(-1 + c^2*x^2)^3*(5 + 2*c^2*x^2) - b*c*x*Sqrt[1 - c^2*x^2]*(25 - 60*c^2*x^2 + 15*c^4*x^4 + 147*c^6*x^6) + 30*b*(-1 + c^2*x^2)^3*(5 + 2*c^2*x^2)*ArcSin[c*x]))/(x^7*(-1 + c^2*x^2)) + (2*b*c^7*d*Sqrt[d - c^2*d*x^2]*Log[x])/(35*Sqrt[1 - c^2*x^2])","A",1
75,1,197,308,0.2547233,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{x^{10}} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^10,x]","\frac{8 b c^9 d \log (x) \sqrt{d-c^2 d x^2}}{315 \sqrt{1-c^2 x^2}}-\frac{d \sqrt{d-c^2 d x^2} \left(840 a \left(8 c^4 x^4+20 c^2 x^2+35\right) \left(c^2 x^2-1\right)^3+840 b \left(8 c^4 x^4+20 c^2 x^2+35\right) \left(c^2 x^2-1\right)^3 \sin ^{-1}(c x)-b c x \sqrt{1-c^2 x^2} \left(18264 c^8 x^8+1680 c^6 x^6+630 c^4 x^4-7000 c^2 x^2+3675\right)\right)}{264600 x^9 \left(c^2 x^2-1\right)}","-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{9 d x^9}-\frac{4 c^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{63 d x^7}-\frac{8 c^4 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{315 d x^5}-\frac{b c d \sqrt{d-c^2 d x^2}}{72 x^8 \sqrt{1-c^2 x^2}}+\frac{8 b c^9 d \log (x) \sqrt{d-c^2 d x^2}}{315 \sqrt{1-c^2 x^2}}-\frac{2 b c^7 d \sqrt{d-c^2 d x^2}}{315 x^2 \sqrt{1-c^2 x^2}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{420 x^4 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{1-c^2 x^2}}",1,"-1/264600*(d*Sqrt[d - c^2*d*x^2]*(840*a*(-1 + c^2*x^2)^3*(35 + 20*c^2*x^2 + 8*c^4*x^4) - b*c*x*Sqrt[1 - c^2*x^2]*(3675 - 7000*c^2*x^2 + 630*c^4*x^4 + 1680*c^6*x^6 + 18264*c^8*x^8) + 840*b*(-1 + c^2*x^2)^3*(35 + 20*c^2*x^2 + 8*c^4*x^4)*ArcSin[c*x]))/(x^9*(-1 + c^2*x^2)) + (8*b*c^9*d*Sqrt[d - c^2*d*x^2]*Log[x])/(315*Sqrt[1 - c^2*x^2])","A",1
76,1,221,385,0.270688,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{x^{12}} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^12,x]","\frac{16 b c^{11} d \log (x) \sqrt{d-c^2 d x^2}}{1155 \sqrt{1-c^2 x^2}}-\frac{d \sqrt{d-c^2 d x^2} \left(630 a \left(16 c^6 x^6+40 c^4 x^4+70 c^2 x^2+105\right) \left(c^2 x^2-1\right)^3+630 b \left(16 c^6 x^6+40 c^4 x^4+70 c^2 x^2+105\right) \left(c^2 x^2-1\right)^3 \sin ^{-1}(c x)-b c x \sqrt{1-c^2 x^2} \left(29524 c^{10} x^{10}+2520 c^8 x^8+945 c^6 x^6+525 c^4 x^4-11025 c^2 x^2+6615\right)\right)}{727650 x^{11} \left(c^2 x^2-1\right)}","-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{11 d x^{11}}-\frac{2 c^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{33 d x^9}-\frac{16 c^6 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{1155 d x^5}-\frac{8 c^4 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{231 d x^7}-\frac{b c d \sqrt{d-c^2 d x^2}}{110 x^{10} \sqrt{1-c^2 x^2}}+\frac{16 b c^{11} d \log (x) \sqrt{d-c^2 d x^2}}{1155 \sqrt{1-c^2 x^2}}-\frac{4 b c^9 d \sqrt{d-c^2 d x^2}}{1155 x^2 \sqrt{1-c^2 x^2}}-\frac{b c^7 d \sqrt{d-c^2 d x^2}}{770 x^4 \sqrt{1-c^2 x^2}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{1386 x^6 \sqrt{1-c^2 x^2}}+\frac{b c^3 d \sqrt{d-c^2 d x^2}}{66 x^8 \sqrt{1-c^2 x^2}}",1,"-1/727650*(d*Sqrt[d - c^2*d*x^2]*(630*a*(-1 + c^2*x^2)^3*(105 + 70*c^2*x^2 + 40*c^4*x^4 + 16*c^6*x^6) - b*c*x*Sqrt[1 - c^2*x^2]*(6615 - 11025*c^2*x^2 + 525*c^4*x^4 + 945*c^6*x^6 + 2520*c^8*x^8 + 29524*c^10*x^10) + 630*b*(-1 + c^2*x^2)^3*(105 + 70*c^2*x^2 + 40*c^4*x^4 + 16*c^6*x^6)*ArcSin[c*x]))/(x^11*(-1 + c^2*x^2)) + (16*b*c^11*d*Sqrt[d - c^2*d*x^2]*Log[x])/(1155*Sqrt[1 - c^2*x^2])","A",1
77,1,174,375,0.2195822,"\int x^7 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^7*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]","\frac{d \sqrt{d-c^2 d x^2} \left(-3465 a \left(105 c^6 x^6+70 c^4 x^4+40 c^2 x^2+16\right) \left(1-c^2 x^2\right)^{5/2}-3465 b \left(105 c^6 x^6+70 c^4 x^4+40 c^2 x^2+16\right) \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x)+b c x \left(33075 c^{10} x^{10}-53900 c^8 x^8+2475 c^6 x^6+4158 c^4 x^4+9240 c^2 x^2+55440\right)\right)}{4002075 c^8 \sqrt{1-c^2 x^2}}","\frac{\left(d-c^2 d x^2\right)^{11/2} \left(a+b \sin ^{-1}(c x)\right)}{11 c^8 d^4}-\frac{\left(d-c^2 d x^2\right)^{9/2} \left(a+b \sin ^{-1}(c x)\right)}{3 c^8 d^3}+\frac{3 \left(d-c^2 d x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{7 c^8 d^2}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{5 c^8 d}-\frac{4 b c d x^9 \sqrt{d-c^2 d x^2}}{297 \sqrt{1-c^2 x^2}}+\frac{b d x^7 \sqrt{d-c^2 d x^2}}{1617 c \sqrt{1-c^2 x^2}}+\frac{16 b d x \sqrt{d-c^2 d x^2}}{1155 c^7 \sqrt{1-c^2 x^2}}+\frac{8 b d x^3 \sqrt{d-c^2 d x^2}}{3465 c^5 \sqrt{1-c^2 x^2}}+\frac{b c^3 d x^{11} \sqrt{d-c^2 d x^2}}{121 \sqrt{1-c^2 x^2}}+\frac{2 b d x^5 \sqrt{d-c^2 d x^2}}{1925 c^3 \sqrt{1-c^2 x^2}}",1,"(d*Sqrt[d - c^2*d*x^2]*(-3465*a*(1 - c^2*x^2)^(5/2)*(16 + 40*c^2*x^2 + 70*c^4*x^4 + 105*c^6*x^6) + b*c*x*(55440 + 9240*c^2*x^2 + 4158*c^4*x^4 + 2475*c^6*x^6 - 53900*c^8*x^8 + 33075*c^10*x^10) - 3465*b*(1 - c^2*x^2)^(5/2)*(16 + 40*c^2*x^2 + 70*c^4*x^4 + 105*c^6*x^6)*ArcSin[c*x]))/(4002075*c^8*Sqrt[1 - c^2*x^2])","A",1
78,1,150,301,0.1998695,"\int x^5 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]","\frac{d \sqrt{d-c^2 d x^2} \left(-315 a \left(35 c^4 x^4+20 c^2 x^2+8\right) \left(1-c^2 x^2\right)^{5/2}-315 b \left(35 c^4 x^4+20 c^2 x^2+8\right) \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x)+b c x \left(1225 c^8 x^8-2250 c^6 x^6+189 c^4 x^4+420 c^2 x^2+2520\right)\right)}{99225 c^6 \sqrt{1-c^2 x^2}}","-\frac{\left(d-c^2 d x^2\right)^{9/2} \left(a+b \sin ^{-1}(c x)\right)}{9 c^6 d^3}+\frac{2 \left(d-c^2 d x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{7 c^6 d^2}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{5 c^6 d}-\frac{10 b c d x^7 \sqrt{d-c^2 d x^2}}{441 \sqrt{1-c^2 x^2}}+\frac{b d x^5 \sqrt{d-c^2 d x^2}}{525 c \sqrt{1-c^2 x^2}}+\frac{8 b d x \sqrt{d-c^2 d x^2}}{315 c^5 \sqrt{1-c^2 x^2}}+\frac{b c^3 d x^9 \sqrt{d-c^2 d x^2}}{81 \sqrt{1-c^2 x^2}}+\frac{4 b d x^3 \sqrt{d-c^2 d x^2}}{945 c^3 \sqrt{1-c^2 x^2}}",1,"(d*Sqrt[d - c^2*d*x^2]*(-315*a*(1 - c^2*x^2)^(5/2)*(8 + 20*c^2*x^2 + 35*c^4*x^4) + b*c*x*(2520 + 420*c^2*x^2 + 189*c^4*x^4 - 2250*c^6*x^6 + 1225*c^8*x^8) - 315*b*(1 - c^2*x^2)^(5/2)*(8 + 20*c^2*x^2 + 35*c^4*x^4)*ArcSin[c*x]))/(99225*c^6*Sqrt[1 - c^2*x^2])","A",1
79,1,126,227,0.1656372,"\int x^3 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[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(-105 a \left(5 c^2 x^2+2\right) \left(1-c^2 x^2\right)^{5/2}-105 b \left(5 c^2 x^2+2\right) \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x)+b c x \left(75 c^6 x^6-168 c^4 x^4+35 c^2 x^2+210\right)\right)}{3675 c^4 \sqrt{1-c^2 x^2}}","\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{7 c^4 d^2}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{5 c^4 d}-\frac{8 b c d x^5 \sqrt{d-c^2 d x^2}}{175 \sqrt{1-c^2 x^2}}+\frac{b d x^3 \sqrt{d-c^2 d x^2}}{105 c \sqrt{1-c^2 x^2}}+\frac{2 b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{b c^3 d x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}",1,"(d*Sqrt[d - c^2*d*x^2]*(-105*a*(1 - c^2*x^2)^(5/2)*(2 + 5*c^2*x^2) + b*c*x*(210 + 35*c^2*x^2 - 168*c^4*x^4 + 75*c^6*x^6) - 105*b*(1 - c^2*x^2)^(5/2)*(2 + 5*c^2*x^2)*ArcSin[c*x]))/(3675*c^4*Sqrt[1 - c^2*x^2])","A",1
80,1,84,153,0.0664413,"\int x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[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(\frac{b c \left(\frac{c^4 x^5}{5}-\frac{2 c^2 x^3}{3}+x\right)}{\sqrt{1-c^2 x^2}}-\left(c^2 x^2-1\right)^2 \left(a+b \sin ^{-1}(c x)\right)\right)}{5 c^2}","-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{5 c^2 d}+\frac{b d x \sqrt{d-c^2 d x^2}}{5 c \sqrt{1-c^2 x^2}}-\frac{2 b c d x^3 \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}}+\frac{b c^3 d 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]*((b*c*(x - (2*c^2*x^3)/3 + (c^4*x^5)/5))/Sqrt[1 - c^2*x^2] - (-1 + c^2*x^2)^2*(a + b*ArcSin[c*x])))/(5*c^2)","A",1
81,1,278,278,1.2072946,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{x} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x,x]","-a d^{3/2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-\frac{1}{3} a d \left(c^2 x^2-4\right) \sqrt{d-c^2 d x^2}+a d^{3/2} \log (x)+\frac{b d \sqrt{d-c^2 d x^2} \left(\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-c x+\sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-\sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}-\frac{b d \sqrt{d-c^2 d x^2} \left(-3 \sin ^{-1}(c x) \left(3 \sqrt{1-c^2 x^2}+\cos \left(3 \sin ^{-1}(c x)\right)\right)+9 c x+\sin \left(3 \sin ^{-1}(c x)\right)\right)}{36 \sqrt{1-c^2 x^2}}","\frac{1}{3} \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)+d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{2 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}+\frac{i b d \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{i b d \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{4 b c d x \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}+\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}",1,"-1/3*(a*d*(-4 + c^2*x^2)*Sqrt[d - c^2*d*x^2]) + a*d^(3/2)*Log[x] - a*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*d*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + I*PolyLog[2, -E^(I*ArcSin[c*x])] - I*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] - (b*d*Sqrt[d - c^2*d*x^2]*(9*c*x - 3*ArcSin[c*x]*(3*Sqrt[1 - c^2*x^2] + Cos[3*ArcSin[c*x]]) + Sin[3*ArcSin[c*x]]))/(36*Sqrt[1 - c^2*x^2])","A",0
82,1,389,297,2.0413264,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^3,x]","\frac{3}{2} a c^2 d^{3/2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-\frac{3}{2} a c^2 d^{3/2} \log (x)-\frac{a d \left(2 c^2 x^2+1\right) \sqrt{d-c^2 d x^2}}{2 x^2}+\frac{b c^2 d^2 \sqrt{1-c^2 x^2} \left(-4 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+4 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-4 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+4 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)-2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)-2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{8 \sqrt{d-c^2 d x^2}}+\frac{b c^2 d \sqrt{d-c^2 d x^2} \left(-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)-i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+c x-\sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+\sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}","-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{2 x^2}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{2 \sqrt{1-c^2 x^2}}+\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{2 \sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2}}{2 x \sqrt{1-c^2 x^2}}+\frac{b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}",1,"-1/2*(a*d*(1 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2])/x^2 - (3*a*c^2*d^(3/2)*Log[x])/2 + (3*a*c^2*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/2 + (b*c^2*d*Sqrt[d - c^2*d*x^2]*(c*x - Sqrt[1 - c^2*x^2]*ArcSin[c*x] - ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - I*PolyLog[2, -E^(I*ArcSin[c*x])] + I*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (b*c^2*d^2*Sqrt[1 - c^2*x^2]*(-2*Cot[ArcSin[c*x]/2] - ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (4*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 - 2*Tan[ArcSin[c*x]/2]))/(8*Sqrt[d - c^2*d*x^2])","A",0
83,1,494,307,5.9686038,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/x^5,x]","\frac{3}{8} a c^4 d^{3/2} \log (x)+\frac{a d \left(5 c^2 x^2-2\right) \sqrt{d-c^2 d x^2}}{8 x^4}-\frac{3}{8} a c^4 d^{3/2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-\frac{b c^4 d^2 \sqrt{1-c^2 x^2} \left(-4 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+4 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-4 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+4 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)-2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)-2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{8 \sqrt{d-c^2 d x^2}}+\frac{b c^4 d \sqrt{d-c^2 d x^2} \left(-\frac{16 \sin ^4\left(\frac{1}{2} \sin ^{-1}(c x)\right)}{c^3 x^3}-24 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+24 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-24 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+24 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)+8 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)+8 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)-c x \csc ^4\left(\frac{1}{2} \sin ^{-1}(c x)\right)-3 \sin ^{-1}(c x) \csc ^4\left(\frac{1}{2} \sin ^{-1}(c x)\right)+6 \sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+3 \sin ^{-1}(c x) \sec ^4\left(\frac{1}{2} \sin ^{-1}(c x)\right)-6 \sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{192 \sqrt{1-c^2 x^2}}","\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 x^2}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{4 x^4}-\frac{3 c^4 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 \sqrt{1-c^2 x^2}}-\frac{b c d \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{1-c^2 x^2}}+\frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{8 \sqrt{1-c^2 x^2}}-\frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{8 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{8 x \sqrt{1-c^2 x^2}}",1,"(a*d*(-2 + 5*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(8*x^4) + (3*a*c^4*d^(3/2)*Log[x])/8 - (3*a*c^4*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/8 - (b*c^4*d^2*Sqrt[1 - c^2*x^2]*(-2*Cot[ArcSin[c*x]/2] - ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (4*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 - 2*Tan[ArcSin[c*x]/2]))/(8*Sqrt[d - c^2*d*x^2]) + (b*c^4*d*Sqrt[d - c^2*d*x^2]*(8*Cot[ArcSin[c*x]/2] + 6*ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 - c*x*Csc[ArcSin[c*x]/2]^4 - 3*ArcSin[c*x]*Csc[ArcSin[c*x]/2]^4 - 24*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 24*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (24*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (24*I)*PolyLog[2, E^(I*ArcSin[c*x])] - 6*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 + 3*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^4 - (16*Sin[ArcSin[c*x]/2]^4)/(c^3*x^3) + 8*Tan[ArcSin[c*x]/2]))/(192*Sqrt[1 - c^2*x^2])","A",0
84,1,220,430,0.2790562,"\int x^4 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^4*(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(225 a^2+30 a b c x \sqrt{1-c^2 x^2} \left(128 c^8 x^8-336 c^6 x^6+248 c^4 x^4-10 c^2 x^2-15\right)+30 b \sin ^{-1}(c x) \left(15 a+b c x \sqrt{1-c^2 x^2} \left(128 c^8 x^8-336 c^6 x^6+248 c^4 x^4-10 c^2 x^2-15\right)\right)+b^2 c^2 x^2 \left(-384 c^8 x^8+1260 c^6 x^6-1240 c^4 x^4+75 c^2 x^2+225\right)+225 b^2 \sin ^{-1}(c x)^2\right)}{38400 b c^5 \sqrt{1-c^2 x^2}}","\frac{1}{32} d^2 x^5 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{d^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{128 c^2}+\frac{1}{10} x^5 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{16} d x^5 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)+\frac{3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{512 b c^5 \sqrt{1-c^2 x^2}}-\frac{3 d^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{256 c^4}-\frac{31 b c d^2 x^6 \sqrt{d-c^2 d x^2}}{960 \sqrt{1-c^2 x^2}}+\frac{b d^2 x^4 \sqrt{d-c^2 d x^2}}{512 c \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^{10} \sqrt{d-c^2 d x^2}}{100 \sqrt{1-c^2 x^2}}+\frac{3 b d^2 x^2 \sqrt{d-c^2 d x^2}}{512 c^3 \sqrt{1-c^2 x^2}}+\frac{21 b c^3 d^2 x^8 \sqrt{d-c^2 d x^2}}{640 \sqrt{1-c^2 x^2}}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(225*a^2 + b^2*c^2*x^2*(225 + 75*c^2*x^2 - 1240*c^4*x^4 + 1260*c^6*x^6 - 384*c^8*x^8) + 30*a*b*c*x*Sqrt[1 - c^2*x^2]*(-15 - 10*c^2*x^2 + 248*c^4*x^4 - 336*c^6*x^6 + 128*c^8*x^8) + 30*b*(15*a + b*c*x*Sqrt[1 - c^2*x^2]*(-15 - 10*c^2*x^2 + 248*c^4*x^4 - 336*c^6*x^6 + 128*c^8*x^8))*ArcSin[c*x] + 225*b^2*ArcSin[c*x]^2))/(38400*b*c^5*Sqrt[1 - c^2*x^2])","A",1
85,1,196,351,0.2289308,"\int x^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[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(45 a^2+6 a b c x \sqrt{1-c^2 x^2} \left(48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right)+6 b \sin ^{-1}(c x) \left(15 a+b c x \sqrt{1-c^2 x^2} \left(48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right)\right)+b^2 c^2 x^2 \left(-36 c^6 x^6+136 c^4 x^4-177 c^2 x^2+45\right)+45 b^2 \sin ^{-1}(c x)^2\right)}{2304 b c^3 \sqrt{1-c^2 x^2}}","-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{8} x^3 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)+\frac{5}{48} d x^3 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)+\frac{5 d^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{5 b d^2 x^2 \sqrt{d-c^2 d x^2}}{256 c \sqrt{1-c^2 x^2}}-\frac{59 b c d^2 x^4 \sqrt{d-c^2 d x^2}}{768 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{1-c^2 x^2}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2}}{288 \sqrt{1-c^2 x^2}}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(45*a^2 + b^2*c^2*x^2*(45 - 177*c^2*x^2 + 136*c^4*x^4 - 36*c^6*x^6) + 6*a*b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6) + 6*b*(15*a + b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6))*ArcSin[c*x] + 45*b^2*ArcSin[c*x]^2))/(2304*b*c^3*Sqrt[1 - c^2*x^2])","A",1
86,1,266,265,1.0658527,"\int \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]","\frac{d^2 \left(\sqrt{d-c^2 d x^2} \left(1584 a c x \sqrt{1-c^2 x^2}+384 a c^5 x^5 \sqrt{1-c^2 x^2}-1248 a c^3 x^3 \sqrt{1-c^2 x^2}+270 b \cos \left(2 \sin ^{-1}(c x)\right)+27 b \cos \left(4 \sin ^{-1}(c x)\right)+2 b \cos \left(6 \sin ^{-1}(c x)\right)\right)-720 a \sqrt{d} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+360 b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2+12 b \sqrt{d-c^2 d x^2} \left(45 \sin \left(2 \sin ^{-1}(c x)\right)+9 \sin \left(4 \sin ^{-1}(c x)\right)+\sin \left(6 \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)\right)}{2304 c \sqrt{1-c^2 x^2}}","\frac{5}{16} d^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{5 d^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{1}{6} x \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)+\frac{5}{24} d x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{25 b c d^2 x^2 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}+\frac{b d^2 \left(1-c^2 x^2\right)^{5/2} \sqrt{d-c^2 d x^2}}{36 c}+\frac{5 b c^3 d^2 x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{1-c^2 x^2}}",1,"(d^2*(360*b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2 - 720*a*Sqrt[d]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d - c^2*d*x^2]*(1584*a*c*x*Sqrt[1 - c^2*x^2] - 1248*a*c^3*x^3*Sqrt[1 - c^2*x^2] + 384*a*c^5*x^5*Sqrt[1 - c^2*x^2] + 270*b*Cos[2*ArcSin[c*x]] + 27*b*Cos[4*ArcSin[c*x]] + 2*b*Cos[6*ArcSin[c*x]]) + 12*b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*(45*Sin[2*ArcSin[c*x]] + 9*Sin[4*ArcSin[c*x]] + Sin[6*ArcSin[c*x]])))/(2304*c*Sqrt[1 - c^2*x^2])","A",1
87,1,257,268,1.3647419,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^2,x]","\frac{d^2 \left(\sqrt{d-c^2 d x^2} \left(16 \left(a \sqrt{1-c^2 x^2} \left(2 c^4 x^4-9 c^2 x^2-8\right)+8 b c x \log (c x)\right)-32 b c x \cos \left(2 \sin ^{-1}(c x)\right)-b c x \cos \left(4 \sin ^{-1}(c x)\right)\right)+240 a c \sqrt{d} x \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-120 b c x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2-4 b \sqrt{d-c^2 d x^2} \left(32 \sqrt{1-c^2 x^2}+16 c x \sin \left(2 \sin ^{-1}(c x)\right)+c x \sin \left(4 \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)\right)}{128 x \sqrt{1-c^2 x^2}}","-\frac{15}{8} c^2 d^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{15 c d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{16 b \sqrt{1-c^2 x^2}}-\frac{5}{4} c^2 d x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{x}+\frac{b c d^2 \log (x) \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}+\frac{9 b c^3 d^2 x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{1-c^2 x^2}}",1,"(d^2*(-120*b*c*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2 + 240*a*c*Sqrt[d]*x*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + Sqrt[d - c^2*d*x^2]*(-32*b*c*x*Cos[2*ArcSin[c*x]] - b*c*x*Cos[4*ArcSin[c*x]] + 16*(a*Sqrt[1 - c^2*x^2]*(-8 - 9*c^2*x^2 + 2*c^4*x^4) + 8*b*c*x*Log[c*x])) - 4*b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*(32*Sqrt[1 - c^2*x^2] + 16*c*x*Sin[2*ArcSin[c*x]] + c*x*Sin[4*ArcSin[c*x]])))/(128*x*Sqrt[1 - c^2*x^2])","A",1
88,1,243,277,1.649871,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^4,x]","\frac{1}{24} d^2 \left(\frac{\sqrt{d-c^2 d x^2} \left(4 a \sqrt{1-c^2 x^2} \left(3 c^4 x^4+14 c^2 x^2-2\right)-56 b c^3 x^3 \log (c x)+b \left(-6 c^5 x^5+3 c^3 x^3-4 c x\right)\right)}{x^3 \sqrt{1-c^2 x^2}}-60 a c^3 \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+\frac{4 b \left(3 c^4 x^4+14 c^2 x^2-2\right) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{x^3}+\frac{30 b c^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}\right)","\frac{5 c^2 d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 x}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{3 x^3}+\frac{5}{2} c^4 d^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{5 c^3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{4 b \sqrt{1-c^2 x^2}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{6 x^2 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{1-c^2 x^2}}-\frac{7 b c^3 d^2 \log (x) \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}",1,"(d^2*((4*b*Sqrt[d - c^2*d*x^2]*(-2 + 14*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x])/x^3 + (30*b*c^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] - 60*a*c^3*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (Sqrt[d - c^2*d*x^2]*(4*a*Sqrt[1 - c^2*x^2]*(-2 + 14*c^2*x^2 + 3*c^4*x^4) + b*(-4*c*x + 3*c^3*x^3 - 6*c^5*x^5) - 56*b*c^3*x^3*Log[c*x]))/(x^3*Sqrt[1 - c^2*x^2])))/24","A",1
89,1,234,277,1.5979942,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^6,x]","\frac{1}{60} d^2 \left(\frac{\sqrt{d-c^2 d x^2} \left(-4 a \sqrt{1-c^2 x^2} \left(23 c^4 x^4-11 c^2 x^2+3\right)+92 b c^5 x^5 \log (c x)+b c x \left(22 c^2 x^2-3\right)\right)}{x^5 \sqrt{1-c^2 x^2}}+60 a c^5 \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-\frac{30 b c^5 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}-\frac{4 b \left(23 c^4 x^4-11 c^2 x^2+3\right) \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{x^5}\right)","-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{5 x^5}+\frac{c^2 d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 x^3}-\frac{c^5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 b \sqrt{1-c^2 x^2}}-\frac{c^4 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{x}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{20 x^4 \sqrt{1-c^2 x^2}}+\frac{23 b c^5 d^2 \log (x) \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}}+\frac{11 b c^3 d^2 \sqrt{d-c^2 d x^2}}{30 x^2 \sqrt{1-c^2 x^2}}",1,"(d^2*((-4*b*Sqrt[d - c^2*d*x^2]*(3 - 11*c^2*x^2 + 23*c^4*x^4)*ArcSin[c*x])/x^5 - (30*b*c^5*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] + 60*a*c^5*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (Sqrt[d - c^2*d*x^2]*(b*c*x*(-3 + 22*c^2*x^2) - 4*a*Sqrt[1 - c^2*x^2]*(3 - 11*c^2*x^2 + 23*c^4*x^4) + 92*b*c^5*x^5*Log[c*x]))/(x^5*Sqrt[1 - c^2*x^2])))/60","A",1
90,1,156,203,0.3113925,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^8,x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(60 a \left(c^2 x^2-1\right)^4+60 b \left(c^2 x^2-1\right)^4 \sin ^{-1}(c x)+b c x \sqrt{1-c^2 x^2} \left(-147 c^6 x^6+90 c^4 x^4-45 c^2 x^2+10\right)\right)}{420 x^7 \left(c^2 x^2-1\right)}-\frac{b c^7 d^2 \log (x) \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}","-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{7 d x^7}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{1-c^2 x^2}}-\frac{b c^7 d^2 \log (x) \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}-\frac{3 b c^5 d^2 \sqrt{d-c^2 d x^2}}{14 x^2 \sqrt{1-c^2 x^2}}+\frac{3 b c^3 d^2 \sqrt{d-c^2 d x^2}}{28 x^4 \sqrt{1-c^2 x^2}}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(60*a*(-1 + c^2*x^2)^4 + b*c*x*Sqrt[1 - c^2*x^2]*(10 - 45*c^2*x^2 + 90*c^4*x^4 - 147*c^6*x^6) + 60*b*(-1 + c^2*x^2)^4*ArcSin[c*x]))/(420*x^7*(-1 + c^2*x^2)) - (b*c^7*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(7*Sqrt[1 - c^2*x^2])","A",1
91,1,184,282,0.2633829,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{x^{10}} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^10,x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(840 a \left(2 c^2 x^2+7\right) \left(c^2 x^2-1\right)^4+840 b \left(2 c^2 x^2+7\right) \left(c^2 x^2-1\right)^4 \sin ^{-1}(c x)+b c x \sqrt{1-c^2 x^2} \left(-4566 c^8 x^8-420 c^6 x^6+3150 c^4 x^4-2660 c^2 x^2+735\right)\right)}{52920 x^9 \left(c^2 x^2-1\right)}-\frac{2 b c^9 d^2 \log (x) \sqrt{d-c^2 d x^2}}{63 \sqrt{1-c^2 x^2}}","-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{9 d x^9}-\frac{2 c^2 \left(d-c^2 d x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{63 d x^7}-\frac{b c d^2 \left(1-c^2 x^2\right)^{7/2} \sqrt{d-c^2 d x^2}}{72 x^8}-\frac{2 b c^9 d^2 \log (x) \sqrt{d-c^2 d x^2}}{63 \sqrt{1-c^2 x^2}}-\frac{b c^7 d^2 \sqrt{d-c^2 d x^2}}{21 x^2 \sqrt{1-c^2 x^2}}+\frac{b c^5 d^2 \sqrt{d-c^2 d x^2}}{42 x^4 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{1-c^2 x^2}}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(840*a*(-1 + c^2*x^2)^4*(7 + 2*c^2*x^2) + b*c*x*Sqrt[1 - c^2*x^2]*(735 - 2660*c^2*x^2 + 3150*c^4*x^4 - 420*c^6*x^6 - 4566*c^8*x^8) + 840*b*(-1 + c^2*x^2)^4*(7 + 2*c^2*x^2)*ArcSin[c*x]))/(52920*x^9*(-1 + c^2*x^2)) - (2*b*c^9*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(63*Sqrt[1 - c^2*x^2])","A",1
92,1,209,361,0.3001363,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{x^{12}} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^12,x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(2520 a \left(8 c^4 x^4+28 c^2 x^2+63\right) \left(c^2 x^2-1\right)^4+2520 b \left(8 c^4 x^4+28 c^2 x^2+63\right) \left(c^2 x^2-1\right)^4 \sin ^{-1}(c x)-b c x \sqrt{1-c^2 x^2} \left(59048 c^{10} x^{10}+5040 c^8 x^8+1890 c^6 x^6-47460 c^4 x^4+50715 c^2 x^2-15876\right)\right)}{1746360 x^{11} \left(c^2 x^2-1\right)}-\frac{8 b c^{11} d^2 \log (x) \sqrt{d-c^2 d x^2}}{693 \sqrt{1-c^2 x^2}}","-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{11 d x^{11}}-\frac{4 c^2 \left(d-c^2 d x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{99 d x^9}-\frac{8 c^4 \left(d-c^2 d x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{693 d x^7}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{110 x^{10} \sqrt{1-c^2 x^2}}-\frac{8 b c^{11} d^2 \log (x) \sqrt{d-c^2 d x^2}}{693 \sqrt{1-c^2 x^2}}+\frac{2 b c^9 d^2 \sqrt{d-c^2 d x^2}}{693 x^2 \sqrt{1-c^2 x^2}}+\frac{b c^7 d^2 \sqrt{d-c^2 d x^2}}{924 x^4 \sqrt{1-c^2 x^2}}-\frac{113 b c^5 d^2 \sqrt{d-c^2 d x^2}}{4158 x^6 \sqrt{1-c^2 x^2}}+\frac{23 b c^3 d^2 \sqrt{d-c^2 d x^2}}{792 x^8 \sqrt{1-c^2 x^2}}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(2520*a*(-1 + c^2*x^2)^4*(63 + 28*c^2*x^2 + 8*c^4*x^4) - b*c*x*Sqrt[1 - c^2*x^2]*(-15876 + 50715*c^2*x^2 - 47460*c^4*x^4 + 1890*c^6*x^6 + 5040*c^8*x^8 + 59048*c^10*x^10) + 2520*b*(-1 + c^2*x^2)^4*(63 + 28*c^2*x^2 + 8*c^4*x^4)*ArcSin[c*x]))/(1746360*x^11*(-1 + c^2*x^2)) - (8*b*c^11*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(693*Sqrt[1 - c^2*x^2])","A",1
93,1,160,354,0.2455628,"\int x^5 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^5*(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(3465 a \left(63 c^4 x^4+28 c^2 x^2+8\right) \left(1-c^2 x^2\right)^{7/2}+3465 b \left(63 c^4 x^4+28 c^2 x^2+8\right) \left(1-c^2 x^2\right)^{7/2} \sin ^{-1}(c x)+b c x \left(19845 c^{10} x^{10}-61985 c^8 x^8+55935 c^6 x^6-2079 c^4 x^4-4620 c^2 x^2-27720\right)\right)}{2401245 c^6 \sqrt{1-c^2 x^2}}","-\frac{\left(d-c^2 d x^2\right)^{11/2} \left(a+b \sin ^{-1}(c x)\right)}{11 c^6 d^3}+\frac{2 \left(d-c^2 d x^2\right)^{9/2} \left(a+b \sin ^{-1}(c x)\right)}{9 c^6 d^2}-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{7 c^6 d}-\frac{113 b c d^2 x^7 \sqrt{d-c^2 d x^2}}{4851 \sqrt{1-c^2 x^2}}+\frac{b d^2 x^5 \sqrt{d-c^2 d x^2}}{1155 c \sqrt{1-c^2 x^2}}+\frac{8 b d^2 x \sqrt{d-c^2 d x^2}}{693 c^5 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^{11} \sqrt{d-c^2 d x^2}}{121 \sqrt{1-c^2 x^2}}+\frac{23 b c^3 d^2 x^9 \sqrt{d-c^2 d x^2}}{891 \sqrt{1-c^2 x^2}}+\frac{4 b d^2 x^3 \sqrt{d-c^2 d x^2}}{2079 c^3 \sqrt{1-c^2 x^2}}",1,"-1/2401245*(d^2*Sqrt[d - c^2*d*x^2]*(3465*a*(1 - c^2*x^2)^(7/2)*(8 + 28*c^2*x^2 + 63*c^4*x^4) + b*c*x*(-27720 - 4620*c^2*x^2 - 2079*c^4*x^4 + 55935*c^6*x^6 - 61985*c^8*x^8 + 19845*c^10*x^10) + 3465*b*(1 - c^2*x^2)^(7/2)*(8 + 28*c^2*x^2 + 63*c^4*x^4)*ArcSin[c*x]))/(c^6*Sqrt[1 - c^2*x^2])","A",1
94,1,137,278,0.2322004,"\int x^3 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[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(-63 a \left(7 c^2 x^2+2\right) \left(1-c^2 x^2\right)^{7/2}-63 b \left(7 c^2 x^2+2\right) \left(1-c^2 x^2\right)^{7/2} \sin ^{-1}(c x)+b \left(-49 c^9 x^9+171 c^7 x^7-189 c^5 x^5+21 c^3 x^3+126 c x\right)\right)}{3969 c^4 \sqrt{1-c^2 x^2}}","\frac{\left(d-c^2 d x^2\right)^{9/2} \left(a+b \sin ^{-1}(c x)\right)}{9 c^4 d^2}-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{7 c^4 d}-\frac{b c d^2 x^5 \sqrt{d-c^2 d x^2}}{21 \sqrt{1-c^2 x^2}}+\frac{b d^2 x^3 \sqrt{d-c^2 d x^2}}{189 c \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^9 \sqrt{d-c^2 d x^2}}{81 \sqrt{1-c^2 x^2}}+\frac{2 b d^2 x \sqrt{d-c^2 d x^2}}{63 c^3 \sqrt{1-c^2 x^2}}+\frac{19 b c^3 d^2 x^7 \sqrt{d-c^2 d x^2}}{441 \sqrt{1-c^2 x^2}}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(-63*a*(1 - c^2*x^2)^(7/2)*(2 + 7*c^2*x^2) + b*(126*c*x + 21*c^3*x^3 - 189*c^5*x^5 + 171*c^7*x^7 - 49*c^9*x^9) - 63*b*(1 - c^2*x^2)^(7/2)*(2 + 7*c^2*x^2)*ArcSin[c*x]))/(3969*c^4*Sqrt[1 - c^2*x^2])","A",1
95,1,93,202,0.0870543,"\int x \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[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(\left(c^2 x^2-1\right)^3 \left(a+b \sin ^{-1}(c x)\right)+\frac{b c \left(-\frac{1}{7} c^6 x^7+\frac{3 c^4 x^5}{5}-c^2 x^3+x\right)}{\sqrt{1-c^2 x^2}}\right)}{7 c^2}","-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{7 c^2 d}+\frac{b d^2 x \sqrt{d-c^2 d x^2}}{7 c \sqrt{1-c^2 x^2}}-\frac{b c d^2 x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{1-c^2 x^2}}+\frac{3 b c^3 d^2 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]*((b*c*(x - c^2*x^3 + (3*c^4*x^5)/5 - (c^6*x^7)/7))/Sqrt[1 - c^2*x^2] + (-1 + c^2*x^2)^3*(a + b*ArcSin[c*x])))/(7*c^2)","A",1
96,1,394,361,2.0324925,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{x} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x,x]","-a d^{5/2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+\frac{1}{15} a d^2 \left(3 c^4 x^4-11 c^2 x^2+23\right) \sqrt{d-c^2 d x^2}+a d^{5/2} \log (x)+\frac{b d^2 \sqrt{d-c^2 d x^2} \left(\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-c x+\sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-\sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}-\frac{b d^2 \sqrt{d-c^2 d x^2} \left(-3 \sin ^{-1}(c x) \left(3 \sqrt{1-c^2 x^2}+\cos \left(3 \sin ^{-1}(c x)\right)\right)+9 c x+\sin \left(3 \sin ^{-1}(c x)\right)\right)}{18 \sqrt{1-c^2 x^2}}+\frac{b d^2 \sqrt{d-c^2 d x^2} \left(-15 \sin ^{-1}(c x) \left(30 \sqrt{1-c^2 x^2}+5 \cos \left(3 \sin ^{-1}(c x)\right)-3 \cos \left(5 \sin ^{-1}(c x)\right)\right)+450 c x+25 \sin \left(3 \sin ^{-1}(c x)\right)-9 \sin \left(5 \sin ^{-1}(c x)\right)\right)}{3600 \sqrt{1-c^2 x^2}}","d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{2 d^2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}+\frac{1}{5} \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)+\frac{i b d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{i b d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{23 b c d^2 x \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{1-c^2 x^2}}+\frac{11 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 \sqrt{1-c^2 x^2}}",1,"(a*d^2*Sqrt[d - c^2*d*x^2]*(23 - 11*c^2*x^2 + 3*c^4*x^4))/15 + a*d^(5/2)*Log[x] - a*d^(5/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*d^2*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + I*PolyLog[2, -E^(I*ArcSin[c*x])] - I*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] - (b*d^2*Sqrt[d - c^2*d*x^2]*(9*c*x - 3*ArcSin[c*x]*(3*Sqrt[1 - c^2*x^2] + Cos[3*ArcSin[c*x]]) + Sin[3*ArcSin[c*x]]))/(18*Sqrt[1 - c^2*x^2]) + (b*d^2*Sqrt[d - c^2*d*x^2]*(450*c*x - 15*ArcSin[c*x]*(30*Sqrt[1 - c^2*x^2] + 5*Cos[3*ArcSin[c*x]] - 3*Cos[5*ArcSin[c*x]]) + 25*Sin[3*ArcSin[c*x]] - 9*Sin[5*ArcSin[c*x]]))/(3600*Sqrt[1 - c^2*x^2])","A",0
97,1,484,386,5.6987522,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^3,x]","\frac{-180 a c^2 d^{5/2} x^2 \log (x) \sqrt{d-c^2 d x^2}+180 a c^2 d^{5/2} x^2 \sqrt{d-c^2 d x^2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-12 a d^3 \left(c^2 x^2-1\right) \left(2 c^4 x^4-14 c^2 x^2-3\right)+144 b c^2 d^3 x^2 \sqrt{1-c^2 x^2} \left(-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)-i \left(\text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right)+c x-\sin ^{-1}(c x) \left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)\right)-9 b c^2 d^3 x^2 \sqrt{1-c^2 x^2} \left(4 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-4 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+4 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-4 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)+2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)+2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 b c^2 d^3 x^2 \sqrt{1-c^2 x^2} \left(-3 \sin ^{-1}(c x) \left(3 \sqrt{1-c^2 x^2}+\cos \left(3 \sin ^{-1}(c x)\right)\right)+9 c x+\sin \left(3 \sin ^{-1}(c x)\right)\right)}{72 x^2 \sqrt{d-c^2 d x^2}}","-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{5 c^2 d^2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{5}{6} c^2 d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{2 x^2}-\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{2 \sqrt{1-c^2 x^2}}+\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{2 \sqrt{1-c^2 x^2}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{2 x \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}+\frac{7 b c^3 d^2 x \sqrt{d-c^2 d x^2}}{3 \sqrt{1-c^2 x^2}}",1,"(-12*a*d^3*(-1 + c^2*x^2)*(-3 - 14*c^2*x^2 + 2*c^4*x^4) - 180*a*c^2*d^(5/2)*x^2*Sqrt[d - c^2*d*x^2]*Log[x] + 180*a*c^2*d^(5/2)*x^2*Sqrt[d - c^2*d*x^2]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + 144*b*c^2*d^3*x^2*Sqrt[1 - c^2*x^2]*(c*x - Sqrt[1 - c^2*x^2]*ArcSin[c*x] - ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) - I*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])])) + 2*b*c^2*d^3*x^2*Sqrt[1 - c^2*x^2]*(9*c*x - 3*ArcSin[c*x]*(3*Sqrt[1 - c^2*x^2] + Cos[3*ArcSin[c*x]]) + Sin[3*ArcSin[c*x]]) - 9*b*c^2*d^3*x^2*Sqrt[1 - c^2*x^2]*(2*Cot[ArcSin[c*x]/2] + ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 + 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, -E^(I*ArcSin[c*x])] - (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] - ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 + 2*Tan[ArcSin[c*x]/2]))/(72*x^2*Sqrt[d - c^2*d*x^2])","A",0
98,1,640,389,6.4578484,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/x^5,x]","\frac{15}{8} a c^4 d^{5/2} \log (x)-\frac{15}{8} a c^4 d^{5/2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+\frac{a d^2 \left(8 c^4 x^4+9 c^2 x^2-2\right) \sqrt{d-c^2 d x^2}}{8 x^4}-\frac{b c^4 d^3 \sqrt{1-c^2 x^2} \left(-4 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+4 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-4 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+4 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)-2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)-2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{4 \sqrt{d-c^2 d x^2}}+\frac{b c^4 d^2 \sqrt{d-c^2 d x^2} \left(\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-c x+\sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-\sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}+\frac{b c^4 d^2 \sqrt{d-c^2 d x^2} \left(-\frac{16 \sin ^4\left(\frac{1}{2} \sin ^{-1}(c x)\right)}{c^3 x^3}-24 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+24 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-24 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+24 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)+8 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)+8 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)-c x \csc ^4\left(\frac{1}{2} \sin ^{-1}(c x)\right)-3 \sin ^{-1}(c x) \csc ^4\left(\frac{1}{2} \sin ^{-1}(c x)\right)+6 \sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+3 \sin ^{-1}(c x) \sec ^4\left(\frac{1}{2} \sin ^{-1}(c x)\right)-6 \sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{192 \sqrt{1-c^2 x^2}}","\frac{5 c^2 d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{8 x^2}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{4 x^4}+\frac{15}{8} c^4 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{15 c^4 d^2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 \sqrt{1-c^2 x^2}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}+\frac{15 i b c^4 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{8 \sqrt{1-c^2 x^2}}-\frac{15 i b c^4 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{8 \sqrt{1-c^2 x^2}}+\frac{9 b c^3 d^2 \sqrt{d-c^2 d x^2}}{8 x \sqrt{1-c^2 x^2}}",1,"(a*d^2*Sqrt[d - c^2*d*x^2]*(-2 + 9*c^2*x^2 + 8*c^4*x^4))/(8*x^4) + (15*a*c^4*d^(5/2)*Log[x])/8 - (15*a*c^4*d^(5/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/8 + (b*c^4*d^2*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + I*PolyLog[2, -E^(I*ArcSin[c*x])] - I*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] - (b*c^4*d^3*Sqrt[1 - c^2*x^2]*(-2*Cot[ArcSin[c*x]/2] - ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (4*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 - 2*Tan[ArcSin[c*x]/2]))/(4*Sqrt[d - c^2*d*x^2]) + (b*c^4*d^2*Sqrt[d - c^2*d*x^2]*(8*Cot[ArcSin[c*x]/2] + 6*ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 - c*x*Csc[ArcSin[c*x]/2]^4 - 3*ArcSin[c*x]*Csc[ArcSin[c*x]/2]^4 - 24*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 24*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (24*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (24*I)*PolyLog[2, E^(I*ArcSin[c*x])] - 6*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 + 3*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^4 - (16*Sin[ArcSin[c*x]/2]^4)/(c^3*x^3) + 8*Tan[ArcSin[c*x]/2]))/(192*Sqrt[1 - c^2*x^2])","A",0
99,1,30,34,0.0117581,"\int \sqrt{1-x^2} \sin ^{-1}(x) \, dx","Integrate[Sqrt[1 - x^2]*ArcSin[x],x]","\frac{1}{4} \left(-x^2+2 \sqrt{1-x^2} x \sin ^{-1}(x)+\sin ^{-1}(x)^2\right)","-\frac{x^2}{4}+\frac{1}{2} \sqrt{1-x^2} x \sin ^{-1}(x)+\frac{1}{4} \sin ^{-1}(x)^2",1,"(-x^2 + 2*x*Sqrt[1 - x^2]*ArcSin[x] + ArcSin[x]^2)/4","A",1
100,1,87,68,0.0491559,"\int \sqrt{\pi -c^2 \pi  x^2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[Sqrt[Pi - c^2*Pi*x^2]*(a + b*ArcSin[c*x]),x]","\frac{\sqrt{\pi } \left(a^2+2 a b c x \sqrt{1-c^2 x^2}+2 b \sin ^{-1}(c x) \left(a+b c x \sqrt{1-c^2 x^2}\right)-b^2 c^2 x^2+b^2 \sin ^{-1}(c x)^2\right)}{4 b c}","\frac{1}{2} x \sqrt{\pi -\pi  c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{\sqrt{\pi } \left(a+b \sin ^{-1}(c x)\right)^2}{4 b c}-\frac{1}{4} \sqrt{\pi } b c x^2",1,"(Sqrt[Pi]*(a^2 - b^2*c^2*x^2 + 2*a*b*c*x*Sqrt[1 - c^2*x^2] + 2*b*(a + b*c*x*Sqrt[1 - c^2*x^2])*ArcSin[c*x] + b^2*ArcSin[c*x]^2))/(4*b*c)","A",1
101,1,64,88,0.0334488,"\int \frac{x^4 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^4*ArcSin[a*x])/Sqrt[1 - a^2*x^2],x]","\frac{a^2 x^2 \left(a^2 x^2+3\right)-2 a x \sqrt{1-a^2 x^2} \left(2 a^2 x^2+3\right) \sin ^{-1}(a x)+3 \sin ^{-1}(a x)^2}{16 a^5}","\frac{3 \sin ^{-1}(a x)^2}{16 a^5}+\frac{3 x^2}{16 a^3}-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{8 a^4}+\frac{x^4}{16 a}",1,"(a^2*x^2*(3 + a^2*x^2) - 2*a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)*ArcSin[a*x] + 3*ArcSin[a*x]^2)/(16*a^5)","A",1
102,1,49,72,0.03965,"\int \frac{x^3 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^3*ArcSin[a*x])/Sqrt[1 - a^2*x^2],x]","\frac{a x \left(a^2 x^2+6\right)-3 \sqrt{1-a^2 x^2} \left(a^2 x^2+2\right) \sin ^{-1}(a x)}{9 a^4}","\frac{2 x}{3 a^3}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{3 a^4}+\frac{x^3}{9 a}",1,"(a*x*(6 + a^2*x^2) - 3*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcSin[a*x])/(9*a^4)","A",1
103,1,43,50,0.0132217,"\int \frac{x^2 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^2*ArcSin[a*x])/Sqrt[1 - a^2*x^2],x]","\frac{a^2 x^2-2 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+\sin ^{-1}(a x)^2}{4 a^3}","\frac{\sin ^{-1}(a x)^2}{4 a^3}-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{2 a^2}+\frac{x^2}{4 a}",1,"(a^2*x^2 - 2*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + ArcSin[a*x]^2)/(4*a^3)","A",1
104,1,29,29,0.0089728,"\int \frac{x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x*ArcSin[a*x])/Sqrt[1 - a^2*x^2],x]","\frac{x}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2}","\frac{x}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2}",1,"x/a - (Sqrt[1 - a^2*x^2]*ArcSin[a*x])/a^2","A",1
105,1,13,13,0.0035216,"\int \frac{\sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]/Sqrt[1 - a^2*x^2],x]","\frac{\sin ^{-1}(a x)^2}{2 a}","\frac{\sin ^{-1}(a x)^2}{2 a}",1,"ArcSin[a*x]^2/(2*a)","A",1
106,1,71,52,0.127999,"\int \frac{\sin ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]/(x*Sqrt[1 - a^2*x^2]),x]","i \text{Li}_2\left(-e^{i \sin ^{-1}(a x)}\right)-i \text{Li}_2\left(e^{i \sin ^{-1}(a x)}\right)+\sin ^{-1}(a x) \left(\log \left(1-e^{i \sin ^{-1}(a x)}\right)-\log \left(1+e^{i \sin ^{-1}(a x)}\right)\right)","i \text{Li}_2\left(-e^{i \sin ^{-1}(a x)}\right)-i \text{Li}_2\left(e^{i \sin ^{-1}(a x)}\right)-2 \sin ^{-1}(a x) \tanh ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)",1,"ArcSin[a*x]*(Log[1 - E^(I*ArcSin[a*x])] - Log[1 + E^(I*ArcSin[a*x])]) + I*PolyLog[2, -E^(I*ArcSin[a*x])] - I*PolyLog[2, E^(I*ArcSin[a*x])]","A",0
107,1,28,28,0.0298726,"\int \frac{\sin ^{-1}(a x)}{x^2 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]/(x^2*Sqrt[1 - a^2*x^2]),x]","a \log (x)-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{x}","a \log (x)-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{x}",1,"-((Sqrt[1 - a^2*x^2]*ArcSin[a*x])/x) + a*Log[x]","A",1
108,1,137,98,0.9171302,"\int \frac{\sin ^{-1}(a x)}{x^3 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]/(x^3*Sqrt[1 - a^2*x^2]),x]","\frac{1}{8} a^2 \left(4 i \text{Li}_2\left(-e^{i \sin ^{-1}(a x)}\right)-4 i \text{Li}_2\left(e^{i \sin ^{-1}(a x)}\right)+4 \sin ^{-1}(a x) \log \left(1-e^{i \sin ^{-1}(a x)}\right)-4 \sin ^{-1}(a x) \log \left(1+e^{i \sin ^{-1}(a x)}\right)-2 \tan \left(\frac{1}{2} \sin ^{-1}(a x)\right)-2 \cot \left(\frac{1}{2} \sin ^{-1}(a x)\right)-\sin ^{-1}(a x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(a x)\right)+\sin ^{-1}(a x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(a x)\right)\right)","\frac{1}{2} i a^2 \text{Li}_2\left(-e^{i \sin ^{-1}(a x)}\right)-\frac{1}{2} i a^2 \text{Li}_2\left(e^{i \sin ^{-1}(a x)}\right)-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{2 x^2}+a^2 \left(-\sin ^{-1}(a x)\right) \tanh ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)-\frac{a}{2 x}",1,"(a^2*(-2*Cot[ArcSin[a*x]/2] - ArcSin[a*x]*Csc[ArcSin[a*x]/2]^2 + 4*ArcSin[a*x]*Log[1 - E^(I*ArcSin[a*x])] - 4*ArcSin[a*x]*Log[1 + E^(I*ArcSin[a*x])] + (4*I)*PolyLog[2, -E^(I*ArcSin[a*x])] - (4*I)*PolyLog[2, E^(I*ArcSin[a*x])] + ArcSin[a*x]*Sec[ArcSin[a*x]/2]^2 - 2*Tan[ArcSin[a*x]/2]))/8","A",0
109,1,119,224,0.1052112,"\int \frac{x^5 \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^5*(a + b*ArcSin[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{15 a \left(3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right)+b c x \sqrt{1-c^2 x^2} \left(9 c^4 x^4+20 c^2 x^2+120\right)+15 b \left(3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right) \sin ^{-1}(c x)}{225 c^6 \sqrt{d-c^2 d x^2}}","-\frac{x^4 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{5 c^2 d}-\frac{8 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{15 c^6 d}-\frac{4 x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{15 c^4 d}+\frac{b x^5 \sqrt{1-c^2 x^2}}{25 c \sqrt{d-c^2 d x^2}}+\frac{8 b x \sqrt{1-c^2 x^2}}{15 c^5 \sqrt{d-c^2 d x^2}}+\frac{4 b x^3 \sqrt{1-c^2 x^2}}{45 c^3 \sqrt{d-c^2 d x^2}}",1,"(b*c*x*Sqrt[1 - c^2*x^2]*(120 + 20*c^2*x^2 + 9*c^4*x^4) + 15*a*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6) + 15*b*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6)*ArcSin[c*x])/(225*c^6*Sqrt[d - c^2*d*x^2])","A",1
110,1,161,200,0.9747894,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{-\frac{16 a c x \left(2 c^2 x^2+3\right) \sqrt{d-c^2 d x^2}}{d}-\frac{48 a \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)}{\sqrt{d}}+\frac{b \sqrt{1-c^2 x^2} \left(4 \sin ^{-1}(c x) \left(6 \sin ^{-1}(c x)-8 \sin \left(2 \sin ^{-1}(c x)\right)+\sin \left(4 \sin ^{-1}(c x)\right)\right)-16 \cos \left(2 \sin ^{-1}(c x)\right)+\cos \left(4 \sin ^{-1}(c x)\right)\right)}{\sqrt{d-c^2 d x^2}}}{128 c^5}","-\frac{x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{4 c^2 d}+\frac{3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{16 b c^5 \sqrt{d-c^2 d x^2}}-\frac{3 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 c^4 d}+\frac{b x^4 \sqrt{1-c^2 x^2}}{16 c \sqrt{d-c^2 d x^2}}+\frac{3 b x^2 \sqrt{1-c^2 x^2}}{16 c^3 \sqrt{d-c^2 d x^2}}",1,"((-16*a*c*x*(3 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2])/d - (48*a*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/Sqrt[d] + (b*Sqrt[1 - c^2*x^2]*(-16*Cos[2*ArcSin[c*x]] + Cos[4*ArcSin[c*x]] + 4*ArcSin[c*x]*(6*ArcSin[c*x] - 8*Sin[2*ArcSin[c*x]] + Sin[4*ArcSin[c*x]])))/Sqrt[d - c^2*d*x^2])/(128*c^5)","A",1
111,1,92,148,0.0582643,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^3*(a + b*ArcSin[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{3 a \left(c^4 x^4+c^2 x^2-2\right)+b c x \sqrt{1-c^2 x^2} \left(c^2 x^2+6\right)+3 b \left(c^4 x^4+c^2 x^2-2\right) \sin ^{-1}(c x)}{9 c^4 \sqrt{d-c^2 d x^2}}","-\frac{x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 c^2 d}-\frac{2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 c^4 d}+\frac{b x^3 \sqrt{1-c^2 x^2}}{9 c \sqrt{d-c^2 d x^2}}+\frac{2 b x \sqrt{1-c^2 x^2}}{3 c^3 \sqrt{d-c^2 d x^2}}",1,"(b*c*x*Sqrt[1 - c^2*x^2]*(6 + c^2*x^2) + 3*a*(-2 + c^2*x^2 + c^4*x^4) + 3*b*(-2 + c^2*x^2 + c^4*x^4)*ArcSin[c*x])/(9*c^4*Sqrt[d - c^2*d*x^2])","A",1
112,1,134,124,1.4067823,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x]))/Sqrt[d - c^2*d*x^2],x]","-\frac{\frac{4 a c x \sqrt{d-c^2 d x^2}}{d}+\frac{4 a \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)}{\sqrt{d}}+\frac{b \sqrt{1-c^2 x^2} \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)}{\sqrt{d-c^2 d x^2}}}{8 c^3}","-\frac{x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 c^2 d}+\frac{\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{b x^2 \sqrt{1-c^2 x^2}}{4 c \sqrt{d-c^2 d x^2}}",1,"-1/8*((4*a*c*x*Sqrt[d - c^2*d*x^2])/d + (4*a*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/Sqrt[d] + (b*Sqrt[1 - c^2*x^2]*(-2*ArcSin[c*x]^2 + Cos[2*ArcSin[c*x]] + 2*ArcSin[c*x]*Sin[2*ArcSin[c*x]]))/Sqrt[d - c^2*d*x^2])/c^3","A",1
113,1,64,67,0.0328899,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x*(a + b*ArcSin[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{a \left(c^2 x^2-1\right)+b c x \sqrt{1-c^2 x^2}+b \left(c^2 x^2-1\right) \sin ^{-1}(c x)}{c^2 \sqrt{d-c^2 d x^2}}","\frac{b x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{c^2 d}",1,"(b*c*x*Sqrt[1 - c^2*x^2] + a*(-1 + c^2*x^2) + b*(-1 + c^2*x^2)*ArcSin[c*x])/(c^2*Sqrt[d - c^2*d*x^2])","A",1
114,1,50,49,0.07392,"\int \frac{a+b \sin ^{-1}(c x)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])/Sqrt[d - c^2*d*x^2],x]","\frac{\sqrt{1-c^2 x^2} \sin ^{-1}(c x) \left(2 a+b \sin ^{-1}(c x)\right)}{2 c \sqrt{d-c^2 d x^2}}","\frac{\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}}",1,"(Sqrt[1 - c^2*x^2]*ArcSin[c*x]*(2*a + b*ArcSin[c*x]))/(2*c*Sqrt[d - c^2*d*x^2])","A",1
115,1,146,145,0.3436771,"\int \frac{a+b \sin ^{-1}(c x)}{x \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(x*Sqrt[d - c^2*d*x^2]),x]","-\frac{a \log \left(\sqrt{d} \sqrt{-d \left(c^2 x^2-1\right)}+d\right)}{\sqrt{d}}+\frac{a \log (x)}{\sqrt{d}}+\frac{b \sqrt{1-c^2 x^2} \left(i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+\sin ^{-1}(c x) \left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)\right)}{\sqrt{d \left(1-c^2 x^2\right)}}","-\frac{2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}}+\frac{i b \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{d-c^2 d x^2}}-\frac{i b \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{d-c^2 d x^2}}",1,"(a*Log[x])/Sqrt[d] - (a*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/Sqrt[d] + (b*Sqrt[1 - c^2*x^2]*(ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + I*PolyLog[2, -E^(I*ArcSin[c*x])] - I*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[d*(1 - c^2*x^2)]","A",0
116,1,69,66,0.1658521,"\int \frac{a+b \sin ^{-1}(c x)}{x^2 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^2*Sqrt[d - c^2*d*x^2]),x]","\frac{b c \log (x) \sqrt{d-c^2 d x^2}}{d \sqrt{1-c^2 x^2}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{d x}","\frac{b c \sqrt{1-c^2 x^2} \log (x)}{\sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{d x}",1,"-((Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(d*x)) + (b*c*Sqrt[d - c^2*d*x^2]*Log[x])/(d*Sqrt[1 - c^2*x^2])","A",1
117,1,244,229,2.7801492,"\int \frac{a+b \sin ^{-1}(c x)}{x^3 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^3*Sqrt[d - c^2*d*x^2]),x]","\frac{-\frac{4 a \sqrt{d-c^2 d x^2}}{x^2}-4 a c^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+4 a c^2 \sqrt{d} \log (x)+\frac{b c^2 d^2 \left(1-c^2 x^2\right)^{3/2} \left(4 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-4 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+4 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-4 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)-2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)-2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{\left(d-c^2 d x^2\right)^{3/2}}}{8 d}","-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 d x^2}-\frac{c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}}+\frac{i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{2 \sqrt{d-c^2 d x^2}}-\frac{i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{2 \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2}}{2 x \sqrt{d-c^2 d x^2}}",1,"((-4*a*Sqrt[d - c^2*d*x^2])/x^2 + 4*a*c^2*Sqrt[d]*Log[x] - 4*a*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*c^2*d^2*(1 - c^2*x^2)^(3/2)*(-2*Cot[ArcSin[c*x]/2] - ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 + 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, -E^(I*ArcSin[c*x])] - (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 - 2*Tan[ArcSin[c*x]/2]))/(d - c^2*d*x^2)^(3/2))/(8*d)","A",0
118,1,152,147,0.2584565,"\int \frac{a+b \sin ^{-1}(c x)}{x^4 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^4*Sqrt[d - c^2*d*x^2]),x]","\frac{\sqrt{d-c^2 d x^2} \left(a \left(-4 c^4 x^4+2 c^2 x^2+2\right)+b c x \sqrt{1-c^2 x^2} \left(6 c^2 x^2+1\right)+2 b \left(-2 c^4 x^4+c^2 x^2+1\right) \sin ^{-1}(c x)\right)}{6 d x^3 \left(c^2 x^2-1\right)}+\frac{2 b c^3 \log (x) \sqrt{d-c^2 d x^2}}{3 d \sqrt{1-c^2 x^2}}","-\frac{2 c^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 d x}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 d x^3}-\frac{b c \sqrt{1-c^2 x^2}}{6 x^2 \sqrt{d-c^2 d x^2}}+\frac{2 b c^3 \sqrt{1-c^2 x^2} \log (x)}{3 \sqrt{d-c^2 d x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(b*c*x*Sqrt[1 - c^2*x^2]*(1 + 6*c^2*x^2) + a*(2 + 2*c^2*x^2 - 4*c^4*x^4) + 2*b*(1 + c^2*x^2 - 2*c^4*x^4)*ArcSin[c*x]))/(6*d*x^3*(-1 + c^2*x^2)) + (2*b*c^3*Sqrt[d - c^2*d*x^2]*Log[x])/(3*d*Sqrt[1 - c^2*x^2])","A",1
119,1,166,221,0.3056102,"\int \frac{x^5 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^5*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(3/2),x]","\frac{\sqrt{d-c^2 d x^2} \left(\sqrt{-c^2} \left(3 a \left(c^4 x^4+4 c^2 x^2-8\right)+b c x \sqrt{1-c^2 x^2} \left(c^2 x^2+15\right)+3 b \left(c^4 x^4+4 c^2 x^2-8\right) \sin ^{-1}(c x)\right)-9 i b c \sqrt{1-c^2 x^2} F\left(\left.i \sinh ^{-1}\left(\sqrt{-c^2} x\right)\right|1\right)\right)}{9 c^6 \sqrt{-c^2} d^2 \left(c^2 x^2-1\right)}","-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 c^6 d^3}+\frac{2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{c^6 d^2}+\frac{a+b \sin ^{-1}(c x)}{c^6 d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{d-c^2 d x^2} \tanh ^{-1}(c x)}{c^6 d^2 \sqrt{1-c^2 x^2}}-\frac{5 b x \sqrt{d-c^2 d x^2}}{3 c^5 d^2 \sqrt{1-c^2 x^2}}-\frac{b x^3 \sqrt{d-c^2 d x^2}}{9 c^3 d^2 \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(Sqrt[-c^2]*(b*c*x*Sqrt[1 - c^2*x^2]*(15 + c^2*x^2) + 3*a*(-8 + 4*c^2*x^2 + c^4*x^4) + 3*b*(-8 + 4*c^2*x^2 + c^4*x^4)*ArcSin[c*x]) - (9*I)*b*c*Sqrt[1 - c^2*x^2]*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], 1]))/(9*c^6*Sqrt[-c^2]*d^2*(-1 + c^2*x^2))","C",1
120,1,173,214,0.5223737,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(3/2),x]","\frac{-4 a c \sqrt{d} x \left(c^2 x^2-3\right)+12 a \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 \sqrt{d} \left(\sqrt{1-c^2 x^2} \left(4 \log \left(1-c^2 x^2\right)-6 \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)+8 c x \sin ^{-1}(c x)\right)}{8 c^5 d^{3/2} \sqrt{d-c^2 d x^2}}","\frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{4 b c^5 d \sqrt{d-c^2 d x^2}}+\frac{3 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 c^4 d^2}+\frac{b \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)}{2 c^5 d \sqrt{d-c^2 d x^2}}-\frac{b x^2 \sqrt{1-c^2 x^2}}{4 c^3 d \sqrt{d-c^2 d x^2}}",1,"(-4*a*c*Sqrt[d]*x*(-3 + c^2*x^2) + 12*a*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + b*Sqrt[d]*(8*c*x*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*(-6*ArcSin[c*x]^2 + Cos[2*ArcSin[c*x]] + 4*Log[1 - c^2*x^2] + 2*ArcSin[c*x]*Sin[2*ArcSin[c*x]])))/(8*c^5*d^(3/2)*Sqrt[d - c^2*d*x^2])","A",1
121,1,136,142,0.2416013,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^3*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(3/2),x]","\frac{\sqrt{d-c^2 d x^2} \left(\sqrt{-c^2} \left(a c^2 x^2-2 a+b c x \sqrt{1-c^2 x^2}+b \left(c^2 x^2-2\right) \sin ^{-1}(c x)\right)-i b c \sqrt{1-c^2 x^2} F\left(\left.i \sinh ^{-1}\left(\sqrt{-c^2} x\right)\right|1\right)\right)}{c^4 \sqrt{-c^2} d^2 \left(c^2 x^2-1\right)}","\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{c^4 d^2}+\frac{a+b \sin ^{-1}(c x)}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{d-c^2 d x^2} \tanh ^{-1}(c x)}{c^4 d^2 \sqrt{1-c^2 x^2}}-\frac{b x \sqrt{d-c^2 d x^2}}{c^3 d^2 \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(Sqrt[-c^2]*(-2*a + a*c^2*x^2 + b*c*x*Sqrt[1 - c^2*x^2] + b*(-2 + c^2*x^2)*ArcSin[c*x]) - I*b*c*Sqrt[1 - c^2*x^2]*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], 1]))/(c^4*Sqrt[-c^2]*d^2*(-1 + c^2*x^2))","C",1
122,1,160,135,0.2208811,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(3/2),x]","-\frac{a x \sqrt{-d \left(c^2 x^2-1\right)}}{c^2 d^2 \left(c^2 x^2-1\right)}+\frac{a \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^3 d^{3/2}}+\frac{b \left(2 c x \sin ^{-1}(c x)-\sqrt{1-c^2 x^2} \left(\sin ^{-1}(c x)^2-2 \log \left(\sqrt{1-c^2 x^2}\right)\right)\right)}{2 c^3 d \sqrt{d \left(1-c^2 x^2\right)}}","\frac{x \left(a+b \sin ^{-1}(c x)\right)}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\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} \log \left(1-c^2 x^2\right)}{2 c^3 d \sqrt{d-c^2 d x^2}}",1,"-((a*x*Sqrt[-(d*(-1 + c^2*x^2))])/(c^2*d^2*(-1 + c^2*x^2))) + (a*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(c^3*d^(3/2)) + (b*(2*c*x*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*(ArcSin[c*x]^2 - 2*Log[Sqrt[1 - c^2*x^2]])))/(2*c^3*d*Sqrt[d*(1 - c^2*x^2)])","A",1
123,1,51,73,0.0218129,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(3/2),x]","\frac{a-b \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)+b \sin ^{-1}(c x)}{c^2 d \sqrt{d-c^2 d x^2}}","\frac{a+b \sin ^{-1}(c x)}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{c^2 d \sqrt{d-c^2 d x^2}}",1,"(a + b*ArcSin[c*x] - b*Sqrt[1 - c^2*x^2]*ArcTanh[c*x])/(c^2*d*Sqrt[d - c^2*d*x^2])","A",1
124,1,77,80,0.2221483,"\int \frac{a+b \sin ^{-1}(c x)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(3/2),x]","-\frac{\sqrt{d-c^2 d x^2} \left(2 a c x+b \sqrt{1-c^2 x^2} \log \left(c^2 x^2-1\right)+2 b c x \sin ^{-1}(c x)\right)}{2 c d^2 \left(c^2 x^2-1\right)}","\frac{x \left(a+b \sin ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)}{2 c d \sqrt{d-c^2 d x^2}}",1,"-1/2*(Sqrt[d - c^2*d*x^2]*(2*a*c*x + 2*b*c*x*ArcSin[c*x] + b*Sqrt[1 - c^2*x^2]*Log[-1 + c^2*x^2]))/(c*d^2*(-1 + c^2*x^2))","A",1
125,1,300,220,1.1609967,"\int \frac{a+b \sin ^{-1}(c x)}{x \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(x*(d - c^2*d*x^2)^(3/2)),x]","\frac{-\frac{a \sqrt{d-c^2 d x^2}}{c^2 x^2-1}-a \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+a \sqrt{d} \log (x)+\frac{b d \left(i \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-i \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+\sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-\sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)+\sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-\sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}}}{d^2}","\frac{a+b \sin ^{-1}(c x)}{d \sqrt{d-c^2 d x^2}}-\frac{2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}+\frac{i b \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}-\frac{i b \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{d \sqrt{d-c^2 d x^2}}",1,"(-((a*Sqrt[d - c^2*d*x^2])/(-1 + c^2*x^2)) + a*Sqrt[d]*Log[x] - a*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*d*(ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + I*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])] - I*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[d - c^2*d*x^2])/d^2","A",1
126,1,117,150,0.2694924,"\int \frac{a+b \sin ^{-1}(c x)}{x^2 \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^2*(d - c^2*d*x^2)^(3/2)),x]","-\frac{\sqrt{d-c^2 d x^2} \left(4 a c^2 x^2-2 a+b c x \sqrt{1-c^2 x^2} \log \left(x^2\right)+b c x \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)+2 b \left(2 c^2 x^2-1\right) \sin ^{-1}(c x)\right)}{2 d^2 x \left(c^2 x^2-1\right)}","\frac{2 c^2 x \left(a+b \sin ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}-\frac{a+b \sin ^{-1}(c x)}{d x \sqrt{d-c^2 d x^2}}+\frac{b c \log (x) \sqrt{d-c^2 d x^2}}{d^2 \sqrt{1-c^2 x^2}}+\frac{b c \sqrt{d-c^2 d x^2} \log \left(1-c^2 x^2\right)}{2 d^2 \sqrt{1-c^2 x^2}}",1,"-1/2*(Sqrt[d - c^2*d*x^2]*(-2*a + 4*a*c^2*x^2 + 2*b*(-1 + 2*c^2*x^2)*ArcSin[c*x] + b*c*x*Sqrt[1 - c^2*x^2]*Log[x^2] + b*c*x*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2]))/(d^2*x*(-1 + c^2*x^2))","A",1
127,1,404,316,2.4722819,"\int \frac{a+b \sin ^{-1}(c x)}{x^3 \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)^(3/2)),x]","\frac{\frac{4 a \sqrt{d} \left(3 c^2 x^2-1\right)}{x^2 \sqrt{d-c^2 d x^2}}-12 a c^2 \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+12 a c^2 \log (x)+\frac{b \sqrt{d} \left(\sqrt{1-c^2 x^2} \left(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(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)+3 \sin ^{-1}(c x) \left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)\right)+6 i c x \sin \left(2 \sin ^{-1}(c x)\right) \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-6 i c x \sin \left(2 \sin ^{-1}(c x)\right) \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+2 \sin ^{-1}(c x)-2 \sin \left(2 \sin ^{-1}(c x)\right)-6 \sin ^{-1}(c x) \cos \left(2 \sin ^{-1}(c x)\right)-3 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)+3 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)-2 \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 \cos \left(3 \sin ^{-1}(c x)\right) \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)}{x^2 \sqrt{d-c^2 d x^2}}}{8 d^{3/2}}","\frac{3 c^2 \left(a+b \sin ^{-1}(c x)\right)}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \sqrt{d-c^2 d x^2}}-\frac{3 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}+\frac{3 i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{2 d \sqrt{d-c^2 d x^2}}-\frac{3 i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{2 d \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2}}{2 d x \sqrt{d-c^2 d x^2}}-\frac{b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{d \sqrt{d-c^2 d x^2}}",1,"((4*a*Sqrt[d]*(-1 + 3*c^2*x^2))/(x^2*Sqrt[d - c^2*d*x^2]) + 12*a*c^2*Log[x] - 12*a*c^2*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*Sqrt[d]*(2*ArcSin[c*x] - 6*ArcSin[c*x]*Cos[2*ArcSin[c*x]] - 3*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 - E^(I*ArcSin[c*x])] + 3*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + E^(I*ArcSin[c*x])] - 2*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + Sqrt[1 - c^2*x^2]*(3*ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + 2*(Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])) + 2*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 2*Sin[2*ArcSin[c*x]] + (6*I)*c*x*PolyLog[2, -E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] - (6*I)*c*x*PolyLog[2, E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]]))/(x^2*Sqrt[d - c^2*d*x^2]))/(8*d^(3/2))","A",1
128,1,162,238,0.3492566,"\int \frac{a+b \sin ^{-1}(c x)}{x^4 \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^(3/2)),x]","\frac{\sqrt{d-c^2 d x^2} \left(-16 a c^4 x^4+8 a c^2 x^2+2 a+b c x \sqrt{1-c^2 x^2}+2 b \left(-8 c^4 x^4+4 c^2 x^2+1\right) \sin ^{-1}(c x)-5 b c^3 x^3 \sqrt{1-c^2 x^2} \log \left(x^2\right)-3 b c^3 x^3 \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)\right)}{6 d^2 x^3 \left(c^2 x^2-1\right)}","-\frac{4 c^2 \left(a+b \sin ^{-1}(c x)\right)}{3 d x \sqrt{d-c^2 d x^2}}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3 \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left(a+b \sin ^{-1}(c x)\right)}{3 d \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{d-c^2 d x^2}}{6 d^2 x^2 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 \log (x) \sqrt{d-c^2 d x^2}}{3 d^2 \sqrt{1-c^2 x^2}}+\frac{b c^3 \sqrt{d-c^2 d x^2} \log \left(1-c^2 x^2\right)}{2 d^2 \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(2*a + 8*a*c^2*x^2 - 16*a*c^4*x^4 + b*c*x*Sqrt[1 - c^2*x^2] + 2*b*(1 + 4*c^2*x^2 - 8*c^4*x^4)*ArcSin[c*x] - 5*b*c^3*x^3*Sqrt[1 - c^2*x^2]*Log[x^2] - 3*b*c^3*x^3*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2]))/(6*d^2*x^3*(-1 + c^2*x^2))","A",1
129,1,253,293,0.6888131,"\int \frac{x^6 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^6*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]","\frac{\sqrt{d} \left(4 a c x \left(3 c^4 x^4-20 c^2 x^2+15\right)+28 b \left(1-c^2 x^2\right)^{3/2} \log \left(1-c^2 x^2\right)+b \left(6 c^4 x^4-9 c^2 x^2+7\right) \sqrt{1-c^2 x^2}\right)-60 a \left(c^2 x^2-1\right) \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)-30 b \sqrt{d} \left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x)^2+4 b c \sqrt{d} x \left(3 c^4 x^4-20 c^2 x^2+15\right) \sin ^{-1}(c x)}{24 c^7 d^{5/2} \left(c^2 x^2-1\right) \sqrt{d-c^2 d x^2}}","\frac{x^5 \left(a+b \sin ^{-1}(c x)\right)}{3 c^2 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{5 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{4 b c^7 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 c^6 d^3}-\frac{5 x^3 \left(a+b \sin ^{-1}(c x)\right)}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b}{6 c^7 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{7 b \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)}{6 c^7 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x^2 \sqrt{1-c^2 x^2}}{4 c^5 d^2 \sqrt{d-c^2 d x^2}}",1,"(4*b*c*Sqrt[d]*x*(15 - 20*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x] - 30*b*Sqrt[d]*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]^2 - 60*a*(-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]*(4*a*c*x*(15 - 20*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(7 - 9*c^2*x^2 + 6*c^4*x^4) + 28*b*(1 - c^2*x^2)^(3/2)*Log[1 - c^2*x^2]))/(24*c^7*d^(5/2)*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])","A",1
130,1,169,219,0.3364099,"\int \frac{x^5 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^5*(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(2 a \left(3 c^4 x^4-12 c^2 x^2+8\right)+b c x \sqrt{1-c^2 x^2} \left(6 c^2 x^2-5\right)+2 b \left(3 c^4 x^4-12 c^2 x^2+8\right) \sin ^{-1}(c x)\right)+11 i b c \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 c^4 \left(-c^2\right)^{3/2} d^3 \left(c^2 x^2-1\right)^2}","-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{c^6 d^3}-\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \sin ^{-1}(c x)}{3 c^6 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{11 b \sqrt{d-c^2 d x^2} \tanh ^{-1}(c x)}{6 c^6 d^3 \sqrt{1-c^2 x^2}}+\frac{b x \sqrt{d-c^2 d x^2}}{c^5 d^3 \sqrt{1-c^2 x^2}}-\frac{b x \sqrt{d-c^2 d x^2}}{6 c^5 d^3 \left(1-c^2 x^2\right)^{3/2}}",1,"(Sqrt[d - c^2*d*x^2]*(Sqrt[-c^2]*(b*c*x*Sqrt[1 - c^2*x^2]*(-5 + 6*c^2*x^2) + 2*a*(8 - 12*c^2*x^2 + 3*c^4*x^4) + 2*b*(8 - 12*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x]) + (11*I)*b*c*(1 - c^2*x^2)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], 1]))/(6*c^4*(-c^2)^(3/2)*d^3*(-1 + c^2*x^2)^2)","C",1
131,1,213,212,0.4622122,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]","\frac{\sqrt{d} \left(-8 a c^3 x^3+6 a c x+b \sqrt{1-c^2 x^2}+4 b \left(1-c^2 x^2\right)^{3/2} \log \left(1-c^2 x^2\right)\right)-6 a \left(c^2 x^2-1\right) \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(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x)^2+2 b \sqrt{d} \sin \left(3 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)}{6 c^5 d^{5/2} \left(c^2 x^2-1\right) \sqrt{d-c^2 d x^2}}","\frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{3 c^2 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 b c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{x \left(a+b \sin ^{-1}(c x)\right)}{c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b}{6 c^5 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{2 b \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}",1,"(-3*b*Sqrt[d]*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]^2 - 6*a*(-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]*(6*a*c*x - 8*a*c^3*x^3 + b*Sqrt[1 - c^2*x^2] + 4*b*(1 - c^2*x^2)^(3/2)*Log[1 - c^2*x^2]) + 2*b*Sqrt[d]*ArcSin[c*x]*Sin[3*ArcSin[c*x]])/(6*c^5*d^(5/2)*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])","A",1
132,1,143,150,0.249912,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(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(6 a c^2 x^2-4 a-b c x \sqrt{1-c^2 x^2}+2 b \left(3 c^2 x^2-2\right) \sin ^{-1}(c x)\right)-5 i b c \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 c^4 \sqrt{-c^2} d^3 \left(c^2 x^2-1\right)^2}","-\frac{a+b \sin ^{-1}(c x)}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \sin ^{-1}(c x)}{3 c^4 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{5 b \sqrt{d-c^2 d x^2} \tanh ^{-1}(c x)}{6 c^4 d^3 \sqrt{1-c^2 x^2}}-\frac{b x \sqrt{d-c^2 d x^2}}{6 c^3 d^3 \left(1-c^2 x^2\right)^{3/2}}",1,"(Sqrt[d - c^2*d*x^2]*(Sqrt[-c^2]*(-4*a + 6*a*c^2*x^2 - b*c*x*Sqrt[1 - c^2*x^2] + 2*b*(-2 + 3*c^2*x^2)*ArcSin[c*x]) - (5*I)*b*c*(1 - c^2*x^2)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], 1]))/(6*c^4*Sqrt[-c^2]*d^3*(-1 + c^2*x^2)^2)","C",1
133,1,103,125,0.2255528,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]","\frac{\sqrt{d-c^2 d x^2} \left(2 a c^3 x^3+2 b c^3 x^3 \sin ^{-1}(c x)-b \sqrt{1-c^2 x^2}-b \left(1-c^2 x^2\right)^{3/2} \log \left(c^2 x^2-1\right)\right)}{6 c^3 d^3 \left(c^2 x^2-1\right)^2}","\frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{b}{6 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(2*a*c^3*x^3 - b*Sqrt[1 - c^2*x^2] + 2*b*c^3*x^3*ArcSin[c*x] - b*(1 - c^2*x^2)^(3/2)*Log[-1 + c^2*x^2]))/(6*c^3*d^3*(-1 + c^2*x^2)^2)","A",1
134,1,85,119,0.0538934,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]","\frac{-2 a+b c x \sqrt{1-c^2 x^2}+b \left(1-c^2 x^2\right)^{3/2} \tanh ^{-1}(c x)-2 b \sin ^{-1}(c x)}{6 c^2 d^2 \left(c^2 x^2-1\right) \sqrt{d-c^2 d x^2}}","\frac{a+b \sin ^{-1}(c x)}{3 c^2 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{b x}{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} \tanh ^{-1}(c x)}{6 c^2 d^2 \sqrt{d-c^2 d x^2}}",1,"(-2*a + b*c*x*Sqrt[1 - c^2*x^2] - 2*b*ArcSin[c*x] + b*(1 - c^2*x^2)^(3/2)*ArcTanh[c*x])/(6*c^2*d^2*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])","A",1
135,1,113,154,0.2520253,"\int \frac{a+b \sin ^{-1}(c x)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(d - c^2*d*x^2)^(5/2),x]","-\frac{\sqrt{d-c^2 d x^2} \left(4 a c^3 x^3-6 a c x+b \sqrt{1-c^2 x^2}-2 b \left(1-c^2 x^2\right)^{3/2} \log \left(c^2 x^2-1\right)+2 b c x \left(2 c^2 x^2-3\right) \sin ^{-1}(c x)\right)}{6 c d^3 \left(c^2 x^2-1\right)^2}","\frac{2 x \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{x \left(a+b \sin ^{-1}(c x)\right)}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{b}{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} \log \left(1-c^2 x^2\right)}{3 c d^2 \sqrt{d-c^2 d x^2}}",1,"-1/6*(Sqrt[d - c^2*d*x^2]*(-6*a*c*x + 4*a*c^3*x^3 + b*Sqrt[1 - c^2*x^2] + 2*b*c*x*(-3 + 2*c^2*x^2)*ArcSin[c*x] - 2*b*(1 - c^2*x^2)^(3/2)*Log[-1 + c^2*x^2]))/(c*d^3*(-1 + c^2*x^2)^2)","A",1
136,1,456,291,2.2409673,"\int \frac{a+b \sin ^{-1}(c x)}{x \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(x*(d - c^2*d*x^2)^(5/2)),x]","-\frac{a \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)}{d^{5/2}}-\frac{a \left(3 c^2 x^2-4\right) \sqrt{d-c^2 d x^2}}{3 d^3 \left(c^2 x^2-1\right)^2}+\frac{a \log (x)}{d^{5/2}}+\frac{b \left(24 i \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-24 i \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+18 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-18 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)+21 \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-21 \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+20 \sin ^{-1}(c x)-2 \sin \left(2 \sin ^{-1}(c x)\right)+12 \sin ^{-1}(c x) \cos \left(2 \sin ^{-1}(c x)\right)+6 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)-6 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)+7 \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)-7 \cos \left(3 \sin ^{-1}(c x)\right) \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)}{24 d \left(d-c^2 d x^2\right)^{3/2}}","\frac{a+b \sin ^{-1}(c x)}{d^2 \sqrt{d-c^2 d x^2}}-\frac{2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \sin ^{-1}(c x)}{3 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{i b \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{d^2 \sqrt{d-c^2 d x^2}}-\frac{i b \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b c x}{6 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{7 b \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}",1,"-1/3*(a*(-4 + 3*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(d^3*(-1 + c^2*x^2)^2) + (a*Log[x])/d^(5/2) - (a*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/d^(5/2) + (b*(20*ArcSin[c*x] + 12*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + 18*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 6*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 - E^(I*ArcSin[c*x])] - 18*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - 6*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + E^(I*ArcSin[c*x])] + 21*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 7*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 21*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 7*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + (24*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^(I*ArcSin[c*x])] - (24*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, E^(I*ArcSin[c*x])] - 2*Sin[2*ArcSin[c*x]]))/(24*d*(d - c^2*d*x^2)^(3/2))","A",0
137,1,188,224,0.3413137,"\int \frac{a+b \sin ^{-1}(c x)}{x^2 \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^2*(d - c^2*d*x^2)^(5/2)),x]","-\frac{\sqrt{d-c^2 d x^2} \left(16 a c^4 x^4-24 a c^2 x^2+6 a+b c x \sqrt{1-c^2 x^2}-3 b c x \left(1-c^2 x^2\right)^{3/2} \log \left(x^2\right)-5 b c x \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)+2 b \left(8 c^4 x^4-12 c^2 x^2+3\right) \sin ^{-1}(c x)+5 b c^3 x^3 \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)\right)}{6 d^3 x \left(c^2 x^2-1\right)^2}","\frac{8 c^2 x \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 c^2 x \left(a+b \sin ^{-1}(c x)\right)}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{d x \left(d-c^2 d x^2\right)^{3/2}}-\frac{b c \sqrt{d-c^2 d x^2}}{6 d^3 \left(1-c^2 x^2\right)^{3/2}}+\frac{b c \log (x) \sqrt{d-c^2 d x^2}}{d^3 \sqrt{1-c^2 x^2}}+\frac{5 b c \sqrt{d-c^2 d x^2} \log \left(1-c^2 x^2\right)}{6 d^3 \sqrt{1-c^2 x^2}}",1,"-1/6*(Sqrt[d - c^2*d*x^2]*(6*a - 24*a*c^2*x^2 + 16*a*c^4*x^4 + b*c*x*Sqrt[1 - c^2*x^2] + 2*b*(3 - 12*c^2*x^2 + 8*c^4*x^4)*ArcSin[c*x] - 3*b*c*x*(1 - c^2*x^2)^(3/2)*Log[x^2] - 5*b*c*x*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2] + 5*b*c^3*x^3*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2]))/(d^3*x*(-1 + c^2*x^2)^2)","A",1
138,1,537,433,8.2419485,"\int \frac{a+b \sin ^{-1}(c x)}{x^3 \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^3*(d - c^2*d*x^2)^(5/2)),x]","-\frac{5 a c^2 \log \left(\sqrt{d} \sqrt{-d \left(c^2 x^2-1\right)}+d\right)}{2 d^{5/2}}+\frac{5 a c^2 \log (x)}{2 d^{5/2}}+\sqrt{-d \left(c^2 x^2-1\right)} \left(-\frac{2 a c^2}{d^3 \left(c^2 x^2-1\right)}+\frac{a c^2}{3 d^3 \left(c^2 x^2-1\right)^2}-\frac{a}{2 d^3 x^2}\right)+\frac{b c^2 \sqrt{1-c^2 x^2} \left(60 i \left(\text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right)-\frac{2 \left(\sin ^{-1}(c x)-1\right)}{c x-1}+52 \sin ^{-1}(c x)+60 \sin ^{-1}(c x) \left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)-6 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\frac{52 \sin ^{-1}(c x) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+\frac{4 \sin ^{-1}(c x) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}-\frac{52 \sin ^{-1}(c x) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+\frac{2 \left(\sin ^{-1}(c x)+1\right)}{\left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-\frac{4 \sin ^{-1}(c x) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}-6 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)-3 \sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+3 \sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+52 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-52 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)}{24 d^2 \sqrt{d \left(1-c^2 x^2\right)}}","\frac{5 c^2 \left(a+b \sin ^{-1}(c x)\right)}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left(a+b \sin ^{-1}(c x)\right)}{6 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2 \left(d-c^2 d x^2\right)^{3/2}}+\frac{5 i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{3 b c \sqrt{1-c^2 x^2}}{4 d^2 x \sqrt{d-c^2 d x^2}}+\frac{b c}{4 d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{13 b c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 b c^3 x}{12 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*(-1/2*a/(d^3*x^2) + (a*c^2)/(3*d^3*(-1 + c^2*x^2)^2) - (2*a*c^2)/(d^3*(-1 + c^2*x^2))) + (5*a*c^2*Log[x])/(2*d^(5/2)) - (5*a*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(5/2)) + (b*c^2*Sqrt[1 - c^2*x^2]*((-2*(-1 + ArcSin[c*x]))/(-1 + c*x) + 52*ArcSin[c*x] - 6*Cot[ArcSin[c*x]/2] - 3*ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 + 60*ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + 52*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 52*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + (60*I)*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]) + 3*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 + (4*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (52*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - (4*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 + (2*(1 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (52*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 6*Tan[ArcSin[c*x]/2]))/(24*d^2*Sqrt[d*(1 - c^2*x^2)])","A",0
139,1,213,310,0.3370209,"\int \frac{a+b \sin ^{-1}(c x)}{x^4 \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^4*(d - c^2*d*x^2)^(5/2)),x]","-\frac{\sqrt{d-c^2 d x^2} \left(32 a c^6 x^6-48 a c^4 x^4+12 a c^2 x^2+2 a+b c x \sqrt{1-c^2 x^2}+8 b c^5 x^5 \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)-8 b c^3 x^3 \left(1-c^2 x^2\right)^{3/2} \log \left(x^2\right)-8 b c^3 x^3 \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)+2 b \left(16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1\right) \sin ^{-1}(c x)\right)}{6 d^3 x^3 \left(c^2 x^2-1\right)^2}","-\frac{2 c^2 \left(a+b \sin ^{-1}(c x)\right)}{d x \left(d-c^2 d x^2\right)^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3 \left(d-c^2 d x^2\right)^{3/2}}+\frac{16 c^4 x \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left(a+b \sin ^{-1}(c x)\right)}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{b c \sqrt{d-c^2 d x^2}}{6 d^3 x^2 \sqrt{1-c^2 x^2}}-\frac{b c^3 \sqrt{d-c^2 d x^2}}{6 d^3 \left(1-c^2 x^2\right)^{3/2}}+\frac{8 b c^3 \log (x) \sqrt{d-c^2 d x^2}}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{4 b c^3 \sqrt{d-c^2 d x^2} \log \left(1-c^2 x^2\right)}{3 d^3 \sqrt{1-c^2 x^2}}",1,"-1/6*(Sqrt[d - c^2*d*x^2]*(2*a + 12*a*c^2*x^2 - 48*a*c^4*x^4 + 32*a*c^6*x^6 + b*c*x*Sqrt[1 - c^2*x^2] + 2*b*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6)*ArcSin[c*x] - 8*b*c^3*x^3*(1 - c^2*x^2)^(3/2)*Log[x^2] - 8*b*c^3*x^3*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2] + 8*b*c^5*x^5*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2]))/(d^3*x^3*(-1 + c^2*x^2)^2)","A",1
140,1,111,210,0.2170906,"\int \frac{\sin ^{-1}(a x)}{\left(c-a^2 c x^2\right)^{7/2}} \, dx","Integrate[ArcSin[a*x]/(c - a^2*c*x^2)^(7/2),x]","-\frac{\sqrt{c-a^2 c x^2} \left(\sqrt{1-a^2 x^2} \left(8 a^2 x^2+16 \left(a^2 x^2-1\right)^2 \log \left(a^2 x^2-1\right)-11\right)+4 a x \left(8 a^4 x^4-20 a^2 x^2+15\right) \sin ^{-1}(a x)\right)}{60 a c^4 \left(a^2 x^2-1\right)^3}","-\frac{2}{15 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}-\frac{1}{20 a c^3 \left(1-a^2 x^2\right)^{3/2} \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{1-a^2 x^2} \log \left(1-a^2 x^2\right)}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 x \sin ^{-1}(a x)}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \sin ^{-1}(a x)}{15 c^2 \left(c-a^2 c x^2\right)^{3/2}}+\frac{x \sin ^{-1}(a x)}{5 c \left(c-a^2 c x^2\right)^{5/2}}",1,"-1/60*(Sqrt[c - a^2*c*x^2]*(4*a*x*(15 - 20*a^2*x^2 + 8*a^4*x^4)*ArcSin[a*x] + Sqrt[1 - a^2*x^2]*(-11 + 8*a^2*x^2 + 16*(-1 + a^2*x^2)^2*Log[-1 + a^2*x^2])))/(a*c^4*(-1 + a^2*x^2)^3)","A",1
141,1,68,79,0.0453816,"\int \frac{(f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}} \, dx","Integrate[((f*x)^(3/2)*(a + b*ArcSin[c*x]))/Sqrt[1 - c^2*x^2],x]","\frac{2}{35} x (f x)^{3/2} \left(7 \, _2F_1\left(\frac{1}{2},\frac{5}{4};\frac{9}{4};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)-2 b c x \, _3F_2\left(1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};c^2 x^2\right)\right)","\frac{2 (f x)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{5}{4};\frac{9}{4};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{5 f}-\frac{4 b c (f x)^{7/2} \, _3F_2\left(1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};c^2 x^2\right)}{35 f^2}",1,"(2*x*(f*x)^(3/2)*(7*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, 5/4, 9/4, c^2*x^2] - 2*b*c*x*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, c^2*x^2]))/35","A",1
142,1,97,137,0.0405681,"\int \frac{(f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[((f*x)^(3/2)*(a + b*ArcSin[c*x]))/Sqrt[d - c^2*d*x^2],x]","-\frac{2 x \sqrt{1-c^2 x^2} (f x)^{3/2} \left(2 b c x \, _3F_2\left(1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};c^2 x^2\right)-7 \, _2F_1\left(\frac{1}{2},\frac{5}{4};\frac{9}{4};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)\right)}{35 \sqrt{d-c^2 d x^2}}","\frac{2 \sqrt{1-c^2 x^2} (f x)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{5}{4};\frac{9}{4};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{5 f \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{1-c^2 x^2} (f x)^{7/2} \, _3F_2\left(1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};c^2 x^2\right)}{35 f^2 \sqrt{d-c^2 d x^2}}",1,"(-2*x*(f*x)^(3/2)*Sqrt[1 - c^2*x^2]*(-7*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, 5/4, 9/4, c^2*x^2] + 2*b*c*x*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, c^2*x^2]))/(35*Sqrt[d - c^2*d*x^2])","A",1
143,1,256,315,0.4726103,"\int x^m \left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^m*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]),x]","\frac{x^{m+1} \left(\frac{6 d \left(-\frac{4 d^2 \left((m+2) \left(m \left(c^2 x^2-1\right)+c^2 x^2-3\right) \left(a+b \sin ^{-1}(c x)\right)+b c (m+1) x \, _2F_1\left(-\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)+2 b c x \, _2F_1\left(\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)\right)}{(m+1) (m+2) (m+3)}+\left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)-\frac{b c d^2 x \, _2F_1\left(-\frac{3}{2},\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)}{m+2}\right)}{m+5}+\left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)-\frac{b c d^3 x \, _2F_1\left(-\frac{5}{2},\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)}{m+2}\right)}{m+7}","-\frac{c^6 d^3 x^{m+7} \left(a+b \sin ^{-1}(c x)\right)}{m+7}+\frac{3 c^4 d^3 x^{m+5} \left(a+b \sin ^{-1}(c x)\right)}{m+5}-\frac{3 c^2 d^3 x^{m+3} \left(a+b \sin ^{-1}(c x)\right)}{m+3}+\frac{d^3 x^{m+1} \left(a+b \sin ^{-1}(c x)\right)}{m+1}-\frac{3 b c d^3 \left(35 m^3+455 m^2+1813 m+2161\right) x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{(m+1) (m+2) (m+3)^2 (m+5)^2 (m+7)^2}-\frac{b c d^3 \left(m^4+27 m^3+284 m^2+1329 m+2271\right) \sqrt{1-c^2 x^2} x^{m+2}}{(m+3)^2 (m+5)^2 (m+7)^2}-\frac{b c^5 d^3 \sqrt{1-c^2 x^2} x^{m+6}}{(m+7)^2}+\frac{b c^3 d^3 (m+9) (2 m+13) \sqrt{1-c^2 x^2} x^{m+4}}{(m+5)^2 (m+7)^2}",1,"(x^(1 + m)*((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]) - (b*c*d^3*x*Hypergeometric2F1[-5/2, 1 + m/2, 2 + m/2, c^2*x^2])/(2 + m) + (6*d*((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]) - (b*c*d^2*x*Hypergeometric2F1[-3/2, 1 + m/2, 2 + m/2, c^2*x^2])/(2 + m) - (4*d^2*((2 + m)*(-3 + c^2*x^2 + m*(-1 + c^2*x^2))*(a + b*ArcSin[c*x]) + b*c*(1 + m)*x*Hypergeometric2F1[-1/2, 1 + m/2, 2 + m/2, c^2*x^2] + 2*b*c*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, c^2*x^2]))/((1 + m)*(2 + m)*(3 + m))))/(5 + m)))/(7 + m)","A",1
144,1,187,217,0.015916,"\int x^m \left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^m*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]),x]","\frac{x^{m+1} \left(-\frac{4 d^2 \left((m+2) \left(m \left(c^2 x^2-1\right)+c^2 x^2-3\right) \left(a+b \sin ^{-1}(c x)\right)+b c (m+1) x \, _2F_1\left(-\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)+2 b c x \, _2F_1\left(\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)\right)}{(m+1) (m+2) (m+3)}+\left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)-\frac{b c d^2 x \, _2F_1\left(-\frac{3}{2},\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)}{m+2}\right)}{m+5}","\frac{c^4 d^2 x^{m+5} \left(a+b \sin ^{-1}(c x)\right)}{m+5}-\frac{2 c^2 d^2 x^{m+3} \left(a+b \sin ^{-1}(c x)\right)}{m+3}+\frac{d^2 x^{m+1} \left(a+b \sin ^{-1}(c x)\right)}{m+1}-\frac{b c d^2 \left(15 m^2+100 m+149\right) x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{(m+1) (m+2) (m+3)^2 (m+5)^2}-\frac{b c d^2 \left(m^2+13 m+38\right) \sqrt{1-c^2 x^2} x^{m+2}}{(m+3)^2 (m+5)^2}+\frac{b c^3 d^2 \sqrt{1-c^2 x^2} x^{m+4}}{(m+5)^2}",1,"(x^(1 + m)*((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]) - (b*c*d^2*x*Hypergeometric2F1[-3/2, 1 + m/2, 2 + m/2, c^2*x^2])/(2 + m) - (4*d^2*((2 + m)*(-3 + c^2*x^2 + m*(-1 + c^2*x^2))*(a + b*ArcSin[c*x]) + b*c*(1 + m)*x*Hypergeometric2F1[-1/2, 1 + m/2, 2 + m/2, c^2*x^2] + 2*b*c*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, c^2*x^2]))/((1 + m)*(2 + m)*(3 + m))))/(5 + m)","A",1
145,1,118,129,0.0565762,"\int x^m \left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^m*(d - c^2*d*x^2)*(a + b*ArcSin[c*x]),x]","-\frac{d x^{m+1} \left((m+2) \left(m \left(c^2 x^2-1\right)+c^2 x^2-3\right) \left(a+b \sin ^{-1}(c x)\right)+b c (m+1) x \, _2F_1\left(-\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)+2 b c x \, _2F_1\left(\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)\right)}{(m+1) (m+2) (m+3)}","-\frac{c^2 d x^{m+3} \left(a+b \sin ^{-1}(c x)\right)}{m+3}+\frac{d x^{m+1} \left(a+b \sin ^{-1}(c x)\right)}{m+1}-\frac{b c d (3 m+7) x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{(m+1) (m+2) (m+3)^2}-\frac{b c d \sqrt{1-c^2 x^2} x^{m+2}}{(m+3)^2}",1,"-((d*x^(1 + m)*((2 + m)*(-3 + c^2*x^2 + m*(-1 + c^2*x^2))*(a + b*ArcSin[c*x]) + b*c*(1 + m)*x*Hypergeometric2F1[-1/2, 1 + m/2, 2 + m/2, c^2*x^2] + 2*b*c*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, c^2*x^2]))/((1 + m)*(2 + m)*(3 + m)))","A",1
146,0,0,28,4.0102919,"\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Integrate[(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2),x]","\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","\text{Int}\left(\frac{x^m \left(a+b \sin ^{-1}(c x)\right)}{d-c^2 d x^2},x\right)",0,"Integrate[(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2), x]","A",-1
147,0,0,117,5.957175,"\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^2,x]","\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","\frac{(1-m) \text{Int}\left(\frac{x^m \left(a+b \sin ^{-1}(c x)\right)}{d-c^2 d x^2},x\right)}{2 d}+\frac{x^{m+1} \left(a+b \sin ^{-1}(c x)\right)}{2 d^2 \left(1-c^2 x^2\right)}-\frac{b c x^{m+2} \, _2F_1\left(\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{2 d^2 (m+2)}",0,"Integrate[(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^2, x]","A",-1
148,0,0,208,6.4096991,"\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^3,x]","\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^3} \, dx","\frac{(1-m) (3-m) \text{Int}\left(\frac{x^m \left(a+b \sin ^{-1}(c x)\right)}{d-c^2 d x^2},x\right)}{8 d^2}+\frac{(3-m) x^{m+1} \left(a+b \sin ^{-1}(c x)\right)}{8 d^3 \left(1-c^2 x^2\right)}+\frac{x^{m+1} \left(a+b \sin ^{-1}(c x)\right)}{4 d^3 \left(1-c^2 x^2\right)^2}-\frac{b c (3-m) x^{m+2} \, _2F_1\left(\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{8 d^3 (m+2)}-\frac{b c x^{m+2} \, _2F_1\left(\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{4 d^3 (m+2)}",0,"Integrate[(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^3, x]","A",-1
149,1,338,635,1.2676648,"\int x^m \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]","\frac{d^2 x^{m+1} \sqrt{d-c^2 d x^2} \left(-5 (m+6) \left(3 (m+4) \left(b c x \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)-(m+2) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)-(m+1) (m+2) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+b c (m+1) x\right)-\left((m+1) (m+4) (m+2)^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)\right)+b c (m+1) (m+2) x \left(-c^2 (m+2) x^2+m+4\right)\right)+(m+1) (m+2)^2 (m+4)^2 (m+6) \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)-b c (m+1) (m+2) (m+4) x \left(c^4 (m+2) (m+4) x^4-2 c^2 (m+2) (m+6) x^2+(m+4) (m+6)\right)\right)}{(m+1) (m+2)^2 (m+4)^2 (m+6)^2 \sqrt{1-c^2 x^2}}","-\frac{15 b c d^2 x^{m+2} \sqrt{d-c^2 d x^2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{(m+1) (m+2)^2 (m+4) (m+6) \sqrt{1-c^2 x^2}}+\frac{15 d^2 x^{m+1} \sqrt{d-c^2 d x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{(m+6) \left(m^3+7 m^2+14 m+8\right) \sqrt{1-c^2 x^2}}+\frac{15 d^2 x^{m+1} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{(m+6) \left(m^2+6 m+8\right)}+\frac{x^{m+1} \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{m+6}+\frac{5 d x^{m+1} \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{(m+4) (m+6)}-\frac{5 b c d^2 x^{m+2} \sqrt{d-c^2 d x^2}}{(m+6) \left(m^2+6 m+8\right) \sqrt{1-c^2 x^2}}-\frac{b c d^2 x^{m+2} \sqrt{d-c^2 d x^2}}{\left(m^2+8 m+12\right) \sqrt{1-c^2 x^2}}-\frac{15 b c d^2 x^{m+2} \sqrt{d-c^2 d x^2}}{(m+2)^2 (m+4) (m+6) \sqrt{1-c^2 x^2}}-\frac{b c^5 d^2 x^{m+6} \sqrt{d-c^2 d x^2}}{(m+6)^2 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d^2 x^{m+4} \sqrt{d-c^2 d x^2}}{(m+4) (m+6) \sqrt{1-c^2 x^2}}+\frac{5 b c^3 d^2 x^{m+4} \sqrt{d-c^2 d x^2}}{(m+4)^2 (m+6) \sqrt{1-c^2 x^2}}",1,"(d^2*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(-(b*c*(1 + m)*(2 + m)*(4 + m)*x*((4 + m)*(6 + m) - 2*c^2*(2 + m)*(6 + m)*x^2 + c^4*(2 + m)*(4 + m)*x^4)) + (1 + m)*(2 + m)^2*(4 + m)^2*(6 + m)*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x]) - 5*(6 + m)*(b*c*(1 + m)*(2 + m)*x*(4 + m - c^2*(2 + m)*x^2) - (1 + m)*(2 + m)^2*(4 + m)*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x]) + 3*(4 + m)*(b*c*(1 + m)*x - (1 + m)*(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - (2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] + b*c*x*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))))/((1 + m)*(2 + m)^2*(4 + m)^2*(6 + m)^2*Sqrt[1 - c^2*x^2])","A",1
150,1,237,399,0.5615909,"\int x^m \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]),x]","\frac{d x^{m+1} \sqrt{d-c^2 d x^2} \left(-\frac{3 \left(b c x \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)-(m+2) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)-(m+1) (m+2) \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+b c (m+1) x\right)}{(m+1) (m+2)^2 \sqrt{1-c^2 x^2}}+\left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{b c x \left(-c^2 (m+2) x^2+m+4\right)}{(m+2) (m+4) \sqrt{1-c^2 x^2}}\right)}{m+4}","-\frac{3 b c d x^{m+2} \sqrt{d-c^2 d x^2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{(m+1) (m+2)^2 (m+4) \sqrt{1-c^2 x^2}}+\frac{3 d x^{m+1} \sqrt{d-c^2 d x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{\left(m^3+7 m^2+14 m+8\right) \sqrt{1-c^2 x^2}}+\frac{3 d x^{m+1} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{m^2+6 m+8}+\frac{x^{m+1} \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{m+4}-\frac{b c d x^{m+2} \sqrt{d-c^2 d x^2}}{\left(m^2+6 m+8\right) \sqrt{1-c^2 x^2}}-\frac{3 b c d x^{m+2} \sqrt{d-c^2 d x^2}}{(m+2)^2 (m+4) \sqrt{1-c^2 x^2}}+\frac{b c^3 d x^{m+4} \sqrt{d-c^2 d x^2}}{(m+4)^2 \sqrt{1-c^2 x^2}}",1,"(d*x^(1 + m)*Sqrt[d - c^2*d*x^2]*(-((b*c*x*(4 + m - c^2*(2 + m)*x^2))/((2 + m)*(4 + m)*Sqrt[1 - c^2*x^2])) + (1 - c^2*x^2)*(a + b*ArcSin[c*x]) - (3*(b*c*(1 + m)*x - (1 + m)*(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]) - (2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] + b*c*x*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))/((1 + m)*(2 + m)^2*Sqrt[1 - c^2*x^2])))/(4 + m)","A",1
151,1,181,245,0.0737476,"\int x^m \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^m*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]),x]","\frac{x^{m+1} \sqrt{d-c^2 d x^2} \left(-b c x \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)+(m+2) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)+(m+1) \left(a (m+2) \sqrt{1-c^2 x^2}+b (m+2) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-b c x\right)\right)}{(m+1) (m+2)^2 \sqrt{1-c^2 x^2}}","-\frac{b c x^{m+2} \sqrt{d-c^2 d x^2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{(m+1) (m+2)^2 \sqrt{1-c^2 x^2}}+\frac{x^{m+1} \sqrt{d-c^2 d x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{\left(m^2+3 m+2\right) \sqrt{1-c^2 x^2}}+\frac{x^{m+1} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{m+2}-\frac{b c x^{m+2} \sqrt{d-c^2 d x^2}}{(m+2)^2 \sqrt{1-c^2 x^2}}",1,"(x^(1 + m)*Sqrt[d - c^2*d*x^2]*((1 + m)*(-(b*c*x) + a*(2 + m)*Sqrt[1 - c^2*x^2] + b*(2 + m)*Sqrt[1 - c^2*x^2]*ArcSin[c*x]) + (2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] - b*c*x*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))/((1 + m)*(2 + m)^2*Sqrt[1 - c^2*x^2])","A",1
152,1,129,163,0.0589731,"\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^m*(a + b*ArcSin[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{\sqrt{1-c^2 x^2} x^{m+1} \left((m+2) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)-b c x \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)\right)}{(m+1) (m+2) \sqrt{d-c^2 d x^2}}","\frac{\sqrt{1-c^2 x^2} x^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{(m+1) \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} x^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{\left(m^2+3 m+2\right) \sqrt{d-c^2 d x^2}}",1,"(x^(1 + m)*Sqrt[1 - c^2*x^2]*((2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] - b*c*x*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))/((1 + m)*(2 + m)*Sqrt[d - c^2*d*x^2])","A",1
153,1,207,272,0.2505903,"\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(3/2),x]","\frac{x^{m+1} \left(b c m x \sqrt{1-c^2 x^2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)-m (m+2) \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)+(m+1) \left((m+2) \left(a+b \sin ^{-1}(c x)\right)-b c x \sqrt{1-c^2 x^2} \, _2F_1\left(1,\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)\right)\right)}{d (m+1) (m+2) \sqrt{d-c^2 d x^2}}","\frac{b c m \sqrt{1-c^2 x^2} x^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{d \left(m^2+3 m+2\right) \sqrt{d-c^2 d x^2}}-\frac{m \sqrt{1-c^2 x^2} x^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{d (m+1) \sqrt{d-c^2 d x^2}}+\frac{x^{m+1} \left(a+b \sin ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} x^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{d (m+2) \sqrt{d-c^2 d x^2}}",1,"(x^(1 + m)*(-(m*(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2]) + (1 + m)*((2 + m)*(a + b*ArcSin[c*x]) - b*c*x*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, c^2*x^2]) + b*c*m*x*Sqrt[1 - c^2*x^2]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))/(d*(1 + m)*(2 + m)*Sqrt[d - c^2*d*x^2])","A",1
154,1,279,408,0.3890787,"\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^m*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]","\frac{x^{m+1} \left((2-m) \left(d-c^2 d x^2\right) \left(-m \sqrt{1-c^2 x^2} \left((m+2) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)-b c x \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)\right)+(m+1) (m+2) \left(a+b \sin ^{-1}(c x)\right)-b c (m+1) x \sqrt{1-c^2 x^2} \, _2F_1\left(1,\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)\right)+d (m+1) (m+2) \left(a+b \sin ^{-1}(c x)\right)-b c d (m+1) x \left(1-c^2 x^2\right)^{3/2} \, _2F_1\left(2,\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)\right)}{3 d^2 (m+1) (m+2) \left(d-c^2 d x^2\right)^{3/2}}","\frac{b c (2-m) m \sqrt{1-c^2 x^2} x^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{3 d^2 \left(m^2+3 m+2\right) \sqrt{d-c^2 d x^2}}-\frac{(2-m) m \sqrt{1-c^2 x^2} x^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 (m+1) \sqrt{d-c^2 d x^2}}+\frac{(2-m) x^{m+1} \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^{m+1} \left(a+b \sin ^{-1}(c x)\right)}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{b c (2-m) \sqrt{1-c^2 x^2} x^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{3 d^2 (m+2) \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} x^{m+2} \, _2F_1\left(2,\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{3 d^2 (m+2) \sqrt{d-c^2 d x^2}}",1,"(x^(1 + m)*(d*(1 + m)*(2 + m)*(a + b*ArcSin[c*x]) - b*c*d*(1 + m)*x*(1 - c^2*x^2)^(3/2)*Hypergeometric2F1[2, 1 + m/2, 2 + m/2, c^2*x^2] + (2 - m)*(d - c^2*d*x^2)*((1 + m)*(2 + m)*(a + b*ArcSin[c*x]) - b*c*(1 + m)*x*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, c^2*x^2] - m*Sqrt[1 - c^2*x^2]*((2 + m)*(a + b*ArcSin[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] - b*c*x*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))))/(3*d^2*(1 + m)*(2 + m)*(d - c^2*d*x^2)^(3/2))","A",1
155,1,95,100,0.0337052,"\int \frac{x^m \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^m*ArcSin[a*x])/Sqrt[1 - a^2*x^2],x]","\frac{x^{m+1} \left((m+2) \sin ^{-1}(a x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right)-a x \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;a^2 x^2\right)\right)}{(m+1) (m+2)}","\frac{x^{m+1} \sin ^{-1}(a x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right)}{m+1}-\frac{a x^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;a^2 x^2\right)}{m^2+3 m+2}",1,"(x^(1 + m)*((2 + m)*ArcSin[a*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2] - a*x*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, a^2*x^2]))/((1 + m)*(2 + m))","A",1
156,1,203,290,0.2863977,"\int x^4 \left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^4*(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2,x]","-\frac{d \left(11025 a^2 c^5 x^5 \left(5 c^2 x^2-7\right)+210 a b \sqrt{1-c^2 x^2} \left(75 c^6 x^6-57 c^4 x^4-76 c^2 x^2-152\right)+210 b \sin ^{-1}(c x) \left(105 a c^5 x^5 \left(5 c^2 x^2-7\right)+b \sqrt{1-c^2 x^2} \left(75 c^6 x^6-57 c^4 x^4-76 c^2 x^2-152\right)\right)+11025 b^2 c^5 x^5 \left(5 c^2 x^2-7\right) \sin ^{-1}(c x)^2+b^2 \left(-2250 c^7 x^7+2394 c^5 x^5+5320 c^3 x^3+31920 c x\right)\right)}{385875 c^5}","\frac{1}{7} d x^5 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{4 b d x^4 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{175 c}+\frac{2 b d \left(1-c^2 x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{49 c^5}-\frac{4 b d \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{35 c^5}+\frac{2 b d \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{21 c^5}+\frac{32 b d \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{525 c^5}+\frac{16 b d x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{525 c^3}+\frac{2}{35} d x^5 \left(a+b \sin ^{-1}(c x)\right)^2-\frac{304 b^2 d x}{3675 c^4}+\frac{2}{343} b^2 c^2 d x^7-\frac{152 b^2 d x^3}{11025 c^2}-\frac{38 b^2 d x^5}{6125}",1,"-1/385875*(d*(11025*a^2*c^5*x^5*(-7 + 5*c^2*x^2) + 210*a*b*Sqrt[1 - c^2*x^2]*(-152 - 76*c^2*x^2 - 57*c^4*x^4 + 75*c^6*x^6) + b^2*(31920*c*x + 5320*c^3*x^3 + 2394*c^5*x^5 - 2250*c^7*x^7) + 210*b*(105*a*c^5*x^5*(-7 + 5*c^2*x^2) + b*Sqrt[1 - c^2*x^2]*(-152 - 76*c^2*x^2 - 57*c^4*x^4 + 75*c^6*x^6))*ArcSin[c*x] + 11025*b^2*c^5*x^5*(-7 + 5*c^2*x^2)*ArcSin[c*x]^2))/c^5","A",1
157,1,192,202,0.167172,"\int x^3 \left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2,x]","-\frac{d \left(9 a^2 \left(4 c^6 x^6-6 c^4 x^4+1\right)+6 a b c x \sqrt{1-c^2 x^2} \left(2 c^4 x^4-2 c^2 x^2-3\right)+6 b \sin ^{-1}(c x) \left(3 a \left(4 c^6 x^6-6 c^4 x^4+1\right)+b c x \sqrt{1-c^2 x^2} \left(2 c^4 x^4-2 c^2 x^2-3\right)\right)+9 b^2 \left(4 c^6 x^6-6 c^4 x^4+1\right) \sin ^{-1}(c x)^2+b^2 c^2 x^2 \left(-2 c^4 x^4+3 c^2 x^2+9\right)\right)}{216 c^4}","-\frac{d \left(a+b \sin ^{-1}(c x)\right)^2}{24 c^4}-\frac{1}{18} b c d x^5 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{6} d x^4 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{b d x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{18 c}+\frac{b d x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{12 c^3}+\frac{1}{12} d x^4 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{108} b^2 c^2 d x^6-\frac{b^2 d x^2}{24 c^2}-\frac{1}{72} b^2 d x^4",1,"-1/216*(d*(b^2*c^2*x^2*(9 + 3*c^2*x^2 - 2*c^4*x^4) + 6*a*b*c*x*Sqrt[1 - c^2*x^2]*(-3 - 2*c^2*x^2 + 2*c^4*x^4) + 9*a^2*(1 - 6*c^4*x^4 + 4*c^6*x^6) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(-3 - 2*c^2*x^2 + 2*c^4*x^4) + 3*a*(1 - 6*c^4*x^4 + 4*c^6*x^6))*ArcSin[c*x] + 9*b^2*(1 - 6*c^4*x^4 + 4*c^6*x^6)*ArcSin[c*x]^2))/c^4","A",1
158,1,179,211,0.2780862,"\int x^2 \left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2,x]","-\frac{d \left(225 a^2 c^3 x^3 \left(3 c^2 x^2-5\right)+30 a b \sqrt{1-c^2 x^2} \left(9 c^4 x^4-13 c^2 x^2-26\right)+30 b \sin ^{-1}(c x) \left(15 a c^3 x^3 \left(3 c^2 x^2-5\right)+b \sqrt{1-c^2 x^2} \left(9 c^4 x^4-13 c^2 x^2-26\right)\right)+b^2 \left(-54 c^5 x^5+130 c^3 x^3+780 c x\right)+225 b^2 c^3 x^3 \left(3 c^2 x^2-5\right) \sin ^{-1}(c x)^2\right)}{3375 c^3}","\frac{4 b d x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{45 c}+\frac{1}{5} d x^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2 b d \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{25 c^3}+\frac{2 b d \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{15 c^3}+\frac{8 b d \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{45 c^3}+\frac{2}{15} d x^3 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{2}{125} b^2 c^2 d x^5-\frac{52 b^2 d x}{225 c^2}-\frac{26}{675} b^2 d x^3",1,"-1/3375*(d*(225*a^2*c^3*x^3*(-5 + 3*c^2*x^2) + 30*a*b*Sqrt[1 - c^2*x^2]*(-26 - 13*c^2*x^2 + 9*c^4*x^4) + b^2*(780*c*x + 130*c^3*x^3 - 54*c^5*x^5) + 30*b*(15*a*c^3*x^3*(-5 + 3*c^2*x^2) + b*Sqrt[1 - c^2*x^2]*(-26 - 13*c^2*x^2 + 9*c^4*x^4))*ArcSin[c*x] + 225*b^2*c^3*x^3*(-5 + 3*c^2*x^2)*ArcSin[c*x]^2))/c^3","A",1
159,1,157,138,0.3241648,"\int x \left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2,x]","-\frac{d \left(c x \left(8 a^2 c x \left(c^2 x^2-2\right)+2 a b \sqrt{1-c^2 x^2} \left(2 c^2 x^2-5\right)+b^2 c x \left(5-c^2 x^2\right)\right)+2 b \sin ^{-1}(c x) \left(a \left(8 c^4 x^4-16 c^2 x^2+5\right)+b c x \sqrt{1-c^2 x^2} \left(2 c^2 x^2-5\right)\right)+b^2 \left(8 c^4 x^4-16 c^2 x^2+5\right) \sin ^{-1}(c x)^2\right)}{32 c^2}","\frac{b d x \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{8 c}+\frac{3 b d x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{16 c}-\frac{d \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{4 c^2}+\frac{3 d \left(a+b \sin ^{-1}(c x)\right)^2}{32 c^2}+\frac{1}{32} b^2 c^2 d x^4-\frac{5}{32} b^2 d x^2",1,"-1/32*(d*(c*x*(b^2*c*x*(5 - c^2*x^2) + 8*a^2*c*x*(-2 + c^2*x^2) + 2*a*b*Sqrt[1 - c^2*x^2]*(-5 + 2*c^2*x^2)) + 2*b*(b*c*x*Sqrt[1 - c^2*x^2]*(-5 + 2*c^2*x^2) + a*(5 - 16*c^2*x^2 + 8*c^4*x^4))*ArcSin[c*x] + b^2*(5 - 16*c^2*x^2 + 8*c^4*x^4)*ArcSin[c*x]^2))/c^2","A",1
160,1,137,128,0.2208546,"\int \left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2,x]","-\frac{d \left(9 a^2 c x \left(c^2 x^2-3\right)+6 a b \sqrt{1-c^2 x^2} \left(c^2 x^2-7\right)+6 b \sin ^{-1}(c x) \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)\right)-2 b^2 c x \left(c^2 x^2-21\right)+9 b^2 c x \left(c^2 x^2-3\right) \sin ^{-1}(c x)^2\right)}{27 c}","\frac{1}{3} d x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{2 b d \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{9 c}+\frac{4 b d \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 c}+\frac{2}{3} d x \left(a+b \sin ^{-1}(c x)\right)^2+\frac{2}{27} b^2 c^2 d x^3-\frac{14}{9} b^2 d x",1,"-1/27*(d*(-2*b^2*c*x*(-21 + c^2*x^2) + 6*a*b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2) + 9*a^2*c*x*(-3 + c^2*x^2) + 6*b*(b*Sqrt[1 - c^2*x^2]*(-7 + c^2*x^2) + 3*a*c*x*(-3 + c^2*x^2))*ArcSin[c*x] + 9*b^2*c*x*(-3 + c^2*x^2)*ArcSin[c*x]^2))/c","A",1
161,1,236,178,0.4639205,"\int \frac{\left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2)/x,x]","\frac{1}{2} d \left(a^2 \left(-c^2\right) x^2+2 a^2 \log (x)-2 a b c^2 x^2 \sin ^{-1}(c x)+a b \left(\sin ^{-1}(c x)-c x \sqrt{1-c^2 x^2}\right)-2 i a b \left(\sin ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)\right)+4 a b \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+\frac{1}{12} b^2 \left(24 i \sin ^{-1}(c x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c x)}\right)+12 \text{Li}_3\left(e^{-2 i \sin ^{-1}(c x)}\right)+8 i \sin ^{-1}(c x)^3+24 \sin ^{-1}(c x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c x)}\right)-i \pi ^3\right)-\frac{1}{2} b^2 \sin ^{-1}(c x) \sin \left(2 \sin ^{-1}(c x)\right)+\frac{1}{4} b^2 \left(2 \sin ^{-1}(c x)^2-1\right) \cos \left(2 \sin ^{-1}(c x)\right)\right)","\frac{1}{2} d \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{1}{2} b c d x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)-i b d \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{i d \left(a+b \sin ^{-1}(c x)\right)^3}{3 b}-\frac{1}{4} d \left(a+b \sin ^{-1}(c x)\right)^2+d \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{4} b^2 c^2 d x^2+\frac{1}{2} b^2 d \text{Li}_3\left(e^{2 i \sin ^{-1}(c x)}\right)",1,"(d*(-(a^2*c^2*x^2) - 2*a*b*c^2*x^2*ArcSin[c*x] + a*b*(-(c*x*Sqrt[1 - c^2*x^2]) + ArcSin[c*x]) + (b^2*(-1 + 2*ArcSin[c*x]^2)*Cos[2*ArcSin[c*x]])/4 + 4*a*b*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 2*a^2*Log[x] - (2*I)*a*b*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c*x])]) + (b^2*((-I)*Pi^3 + (8*I)*ArcSin[c*x]^3 + 24*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] + (24*I)*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] + 12*PolyLog[3, E^((-2*I)*ArcSin[c*x])]))/12 - (b^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]])/2))/2","A",0
162,1,203,149,0.4196445,"\int \frac{\left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2)/x^2,x]","-\frac{d \left(a^2 c^2 x^2+a^2+2 a b c x \left(\sqrt{1-c^2 x^2}+c x \sin ^{-1}(c x)\right)+2 a b \left(c x \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+\sin ^{-1}(c x)\right)+b^2 c x \left(2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+c x \left(\sin ^{-1}(c x)^2-2\right)\right)-i b^2 \left(2 c x \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-2 c x \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+i \sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 c x \left(\log \left(1+e^{i \sin ^{-1}(c x)}\right)-\log \left(1-e^{i \sin ^{-1}(c x)}\right)\right)\right)\right)\right)}{x}","-2 b c d \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{d \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{x}-2 c^2 d x \left(a+b \sin ^{-1}(c x)\right)^2-4 b c d \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+2 b^2 c^2 d x+2 i b^2 c d \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-2 i b^2 c d \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)",1,"-((d*(a^2 + a^2*c^2*x^2 + 2*a*b*c*x*(Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x]) + b^2*c*x*(2*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + c*x*(-2 + ArcSin[c*x]^2)) + 2*a*b*(ArcSin[c*x] + c*x*ArcTanh[Sqrt[1 - c^2*x^2]]) - I*b^2*(I*ArcSin[c*x]*(ArcSin[c*x] + 2*c*x*(-Log[1 - E^(I*ArcSin[c*x])] + Log[1 + E^(I*ArcSin[c*x])])) + 2*c*x*PolyLog[2, -E^(I*ArcSin[c*x])] - 2*c*x*PolyLog[2, E^(I*ArcSin[c*x])])))/x)","A",0
163,1,236,193,0.3926224,"\int \frac{\left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2)/x^3,x]","\frac{1}{2} d \left(-2 a^2 c^2 \log (x)-\frac{a^2}{x^2}+2 i a b c^2 \left(\text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)\right)\right)-\frac{2 a b \left(c x \sqrt{1-c^2 x^2}+\sin ^{-1}(c x)\right)}{x^2}+\frac{1}{12} i b^2 c^2 \left(-24 \sin ^{-1}(c x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c x)}\right)+12 i \text{Li}_3\left(e^{-2 i \sin ^{-1}(c x)}\right)-8 \sin ^{-1}(c x)^3+24 i \sin ^{-1}(c x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c x)}\right)+\pi ^3\right)-\frac{b^2 \left(-2 c^2 x^2 \log (c x)+2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+\sin ^{-1}(c x)^2\right)}{x^2}\right)","i b c^2 d \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{d \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{2 x^2}-\frac{b c d \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{x}+\frac{i c^2 d \left(a+b \sin ^{-1}(c x)\right)^3}{3 b}-\frac{1}{2} c^2 d \left(a+b \sin ^{-1}(c x)\right)^2-c^2 d \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{1}{2} b^2 c^2 d \text{Li}_3\left(e^{2 i \sin ^{-1}(c x)}\right)+b^2 c^2 d \log (x)",1,"(d*(-(a^2/x^2) - (2*a*b*(c*x*Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/x^2 - 2*a^2*c^2*Log[x] - (b^2*(2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]^2 - 2*c^2*x^2*Log[c*x]))/x^2 + (2*I)*a*b*c^2*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*Log[1 - E^((2*I)*ArcSin[c*x])]) + PolyLog[2, E^((2*I)*ArcSin[c*x])]) + (I/12)*b^2*c^2*(Pi^3 - 8*ArcSin[c*x]^3 + (24*I)*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] - 24*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] + (12*I)*PolyLog[3, E^((-2*I)*ArcSin[c*x])])))/2","A",0
164,1,266,176,0.766149,"\int \frac{\left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2)/x^4,x]","\frac{d \left(3 a^2 c^2 x^2-a^2-a b c x \sqrt{1-c^2 x^2}+6 a b c^2 x^2 \sin ^{-1}(c x)+5 a b c^3 x^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-2 a b \sin ^{-1}(c x)-5 i b^2 c^3 x^3 \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+5 i b^2 c^3 x^3 \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-5 b^2 c^3 x^3 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+5 b^2 c^3 x^3 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)-b^2 c^2 x^2+3 b^2 c^2 x^2 \sin ^{-1}(c x)^2-b^2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-b^2 \sin ^{-1}(c x)^2\right)}{3 x^3}","\frac{10}{3} b c^3 d \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{b c d \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 x^2}-\frac{d \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 x^3}+\frac{2 c^2 d \left(a+b \sin ^{-1}(c x)\right)^2}{3 x}-\frac{5}{3} i b^2 c^3 d \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+\frac{5}{3} i b^2 c^3 d \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-\frac{b^2 c^2 d}{3 x}",1,"(d*(-a^2 + 3*a^2*c^2*x^2 - b^2*c^2*x^2 - a*b*c*x*Sqrt[1 - c^2*x^2] - 2*a*b*ArcSin[c*x] + 6*a*b*c^2*x^2*ArcSin[c*x] - b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - b^2*ArcSin[c*x]^2 + 3*b^2*c^2*x^2*ArcSin[c*x]^2 + 5*a*b*c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]] - 5*b^2*c^3*x^3*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 5*b^2*c^3*x^3*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (5*I)*b^2*c^3*x^3*PolyLog[2, -E^(I*ArcSin[c*x])] + (5*I)*b^2*c^3*x^3*PolyLog[2, E^(I*ArcSin[c*x])]))/(3*x^3)","A",0
165,1,253,395,0.2471134,"\int x^4 \left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^4*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2,x]","\frac{d^2 \left(99225 a^2 c^5 x^5 \left(35 c^4 x^4-90 c^2 x^2+63\right)+630 a b \sqrt{1-c^2 x^2} \left(1225 c^8 x^8-2650 c^6 x^6+789 c^4 x^4+1052 c^2 x^2+2104\right)+630 b \sin ^{-1}(c x) \left(315 a c^5 x^5 \left(35 c^4 x^4-90 c^2 x^2+63\right)+b \sqrt{1-c^2 x^2} \left(1225 c^8 x^8-2650 c^6 x^6+789 c^4 x^4+1052 c^2 x^2+2104\right)\right)+99225 b^2 c^5 x^5 \left(35 c^4 x^4-90 c^2 x^2+63\right) \sin ^{-1}(c x)^2-2 b^2 c x \left(42875 c^8 x^8-119250 c^6 x^6+49707 c^4 x^4+110460 c^2 x^2+662760\right)\right)}{31255875 c^5}","\frac{1}{9} d^2 x^5 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{4}{63} d^2 x^5 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{16 b d^2 x^4 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{1575 c}+\frac{2 b d^2 \left(1-c^2 x^2\right)^{9/2} \left(a+b \sin ^{-1}(c x)\right)}{81 c^5}-\frac{20 b d^2 \left(1-c^2 x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{441 c^5}-\frac{2 b d^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{315 c^5}+\frac{8 b d^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{189 c^5}+\frac{128 b d^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{4725 c^5}+\frac{64 b d^2 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{4725 c^3}+\frac{8}{315} d^2 x^5 \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2}{729} b^2 c^4 d^2 x^9-\frac{4208 b^2 d^2 x}{99225 c^4}+\frac{212 b^2 c^2 d^2 x^7}{27783}-\frac{2104 b^2 d^2 x^3}{297675 c^2}-\frac{526 b^2 d^2 x^5}{165375}",1,"(d^2*(99225*a^2*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4) + 630*a*b*Sqrt[1 - c^2*x^2]*(2104 + 1052*c^2*x^2 + 789*c^4*x^4 - 2650*c^6*x^6 + 1225*c^8*x^8) - 2*b^2*c*x*(662760 + 110460*c^2*x^2 + 49707*c^4*x^4 - 119250*c^6*x^6 + 42875*c^8*x^8) + 630*b*(315*a*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(2104 + 1052*c^2*x^2 + 789*c^4*x^4 - 2650*c^6*x^6 + 1225*c^8*x^8))*ArcSin[c*x] + 99225*b^2*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4)*ArcSin[c*x]^2))/(31255875*c^5)","A",1
166,1,239,302,0.2500128,"\int x^3 \left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2,x]","\frac{d^2 \left(c x \left(1152 a^2 c^3 x^3 \left(3 c^4 x^4-8 c^2 x^2+6\right)+6 a b \sqrt{1-c^2 x^2} \left(144 c^6 x^6-344 c^4 x^4+146 c^2 x^2+219\right)-b^2 c x \left(108 c^6 x^6-344 c^4 x^4+219 c^2 x^2+657\right)\right)+6 b \sin ^{-1}(c x) \left(3 a \left(384 c^8 x^8-1024 c^6 x^6+768 c^4 x^4-73\right)+b c x \sqrt{1-c^2 x^2} \left(144 c^6 x^6-344 c^4 x^4+146 c^2 x^2+219\right)\right)+9 b^2 \left(384 c^8 x^8-1024 c^6 x^6+768 c^4 x^4-73\right) \sin ^{-1}(c x)^2\right)}{27648 c^4}","-\frac{73 d^2 \left(a+b \sin ^{-1}(c x)\right)^2}{3072 c^4}-\frac{1}{32} b c d^2 x^5 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{25}{576} b c d^2 x^5 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{8} d^2 x^4 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{12} d^2 x^4 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{73 b d^2 x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{2304 c}+\frac{73 b d^2 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{1536 c^3}+\frac{1}{24} d^2 x^4 \left(a+b \sin ^{-1}(c x)\right)^2-\frac{1}{256} b^2 c^4 d^2 x^8+\frac{43 b^2 c^2 d^2 x^6}{3456}-\frac{73 b^2 d^2 x^2}{3072 c^2}-\frac{73 b^2 d^2 x^4}{9216}",1,"(d^2*(c*x*(1152*a^2*c^3*x^3*(6 - 8*c^2*x^2 + 3*c^4*x^4) - b^2*c*x*(657 + 219*c^2*x^2 - 344*c^4*x^4 + 108*c^6*x^6) + 6*a*b*Sqrt[1 - c^2*x^2]*(219 + 146*c^2*x^2 - 344*c^4*x^4 + 144*c^6*x^6)) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(219 + 146*c^2*x^2 - 344*c^4*x^4 + 144*c^6*x^6) + 3*a*(-73 + 768*c^4*x^4 - 1024*c^6*x^6 + 384*c^8*x^8))*ArcSin[c*x] + 9*b^2*(-73 + 768*c^4*x^4 - 1024*c^6*x^6 + 384*c^8*x^8)*ArcSin[c*x]^2))/(27648*c^4)","A",1
167,1,229,310,0.2075864,"\int x^2 \left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2,x]","\frac{d^2 \left(11025 a^2 c^3 x^3 \left(15 c^4 x^4-42 c^2 x^2+35\right)+210 a b \sqrt{1-c^2 x^2} \left(225 c^6 x^6-612 c^4 x^4+409 c^2 x^2+818\right)+210 b \sin ^{-1}(c x) \left(105 a c^3 x^3 \left(15 c^4 x^4-42 c^2 x^2+35\right)+b \sqrt{1-c^2 x^2} \left(225 c^6 x^6-612 c^4 x^4+409 c^2 x^2+818\right)\right)-2 b^2 c x \left(3375 c^6 x^6-12852 c^4 x^4+14315 c^2 x^2+85890\right)+11025 b^2 c^3 x^3 \left(15 c^4 x^4-42 c^2 x^2+35\right) \sin ^{-1}(c x)^2\right)}{1157625 c^3}","\frac{16 b d^2 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{315 c}+\frac{1}{7} d^2 x^3 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{4}{35} d^2 x^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2 b d^2 \left(1-c^2 x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{49 c^3}+\frac{2 b d^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{175 c^3}+\frac{8 b d^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{105 c^3}+\frac{32 b d^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{315 c^3}+\frac{8}{105} d^2 x^3 \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2}{343} b^2 c^4 d^2 x^7+\frac{136 b^2 c^2 d^2 x^5}{6125}-\frac{1636 b^2 d^2 x}{11025 c^2}-\frac{818 b^2 d^2 x^3}{33075}",1,"(d^2*(11025*a^2*c^3*x^3*(35 - 42*c^2*x^2 + 15*c^4*x^4) + 210*a*b*Sqrt[1 - c^2*x^2]*(818 + 409*c^2*x^2 - 612*c^4*x^4 + 225*c^6*x^6) - 2*b^2*c*x*(85890 + 14315*c^2*x^2 - 12852*c^4*x^4 + 3375*c^6*x^6) + 210*b*(105*a*c^3*x^3*(35 - 42*c^2*x^2 + 15*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(818 + 409*c^2*x^2 - 612*c^4*x^4 + 225*c^6*x^6))*ArcSin[c*x] + 11025*b^2*c^3*x^3*(35 - 42*c^2*x^2 + 15*c^4*x^4)*ArcSin[c*x]^2))/(1157625*c^3)","A",1
168,1,209,209,0.2854138,"\int x \left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2,x]","\frac{d^2 \left(c x \left(144 a^2 c x \left(c^4 x^4-3 c^2 x^2+3\right)+6 a b \sqrt{1-c^2 x^2} \left(8 c^4 x^4-26 c^2 x^2+33\right)+b^2 c x \left(-8 c^4 x^4+39 c^2 x^2-99\right)\right)+6 b \sin ^{-1}(c x) \left(3 a \left(16 c^6 x^6-48 c^4 x^4+48 c^2 x^2-11\right)+b c x \sqrt{1-c^2 x^2} \left(8 c^4 x^4-26 c^2 x^2+33\right)\right)+9 b^2 \left(16 c^6 x^6-48 c^4 x^4+48 c^2 x^2-11\right) \sin ^{-1}(c x)^2\right)}{864 c^2}","\frac{b d^2 x \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{18 c}+\frac{5 b d^2 x \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{72 c}+\frac{5 b d^2 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{48 c}-\frac{d^2 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{6 c^2}+\frac{5 d^2 \left(a+b \sin ^{-1}(c x)\right)^2}{96 c^2}+\frac{5}{288} b^2 c^2 d^2 x^4+\frac{b^2 d^2 \left(1-c^2 x^2\right)^3}{108 c^2}-\frac{25}{288} b^2 d^2 x^2",1,"(d^2*(c*x*(b^2*c*x*(-99 + 39*c^2*x^2 - 8*c^4*x^4) + 144*a^2*c*x*(3 - 3*c^2*x^2 + c^4*x^4) + 6*a*b*Sqrt[1 - c^2*x^2]*(33 - 26*c^2*x^2 + 8*c^4*x^4)) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(33 - 26*c^2*x^2 + 8*c^4*x^4) + 3*a*(-11 + 48*c^2*x^2 - 48*c^4*x^4 + 16*c^6*x^6))*ArcSin[c*x] + 9*b^2*(-11 + 48*c^2*x^2 - 48*c^4*x^4 + 16*c^6*x^6)*ArcSin[c*x]^2))/(864*c^2)","A",1
169,1,193,219,0.2498977,"\int \left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2,x]","\frac{d^2 \left(225 a^2 c x \left(3 c^4 x^4-10 c^2 x^2+15\right)+30 a b \sqrt{1-c^2 x^2} \left(9 c^4 x^4-38 c^2 x^2+149\right)+30 b \sin ^{-1}(c x) \left(15 a c x \left(3 c^4 x^4-10 c^2 x^2+15\right)+b \sqrt{1-c^2 x^2} \left(9 c^4 x^4-38 c^2 x^2+149\right)\right)-2 b^2 c x \left(27 c^4 x^4-190 c^2 x^2+2235\right)+225 b^2 c x \left(3 c^4 x^4-10 c^2 x^2+15\right) \sin ^{-1}(c x)^2\right)}{3375 c}","\frac{1}{5} d^2 x \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{4}{15} d^2 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{2 b d^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{25 c}+\frac{8 b d^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{45 c}+\frac{16 b d^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{15 c}+\frac{8}{15} d^2 x \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2}{125} b^2 c^4 d^2 x^5+\frac{76}{675} b^2 c^2 d^2 x^3-\frac{298}{225} b^2 d^2 x",1,"(d^2*(225*a^2*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 30*a*b*Sqrt[1 - c^2*x^2]*(149 - 38*c^2*x^2 + 9*c^4*x^4) - 2*b^2*c*x*(2235 - 190*c^2*x^2 + 27*c^4*x^4) + 30*b*(15*a*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(149 - 38*c^2*x^2 + 9*c^4*x^4))*ArcSin[c*x] + 225*b^2*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x]^2))/(3375*c)","A",1
170,1,353,271,0.4687928,"\int \frac{\left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2)/x,x]","\frac{1}{768} d^2 \left(192 a^2 c^4 x^4-768 a^2 c^2 x^2+768 a^2 \log (c x)+384 a b c^4 x^4 \sin ^{-1}(c x)-624 a b c x \sqrt{1-c^2 x^2}-1536 a b c^2 x^2 \sin ^{-1}(c x)+96 a b c^3 x^3 \sqrt{1-c^2 x^2}-768 i a b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-768 i a b \sin ^{-1}(c x)^2+624 a b \sin ^{-1}(c x)+1536 a b \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+768 i b^2 \sin ^{-1}(c x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c x)}\right)+384 b^2 \text{Li}_3\left(e^{-2 i \sin ^{-1}(c x)}\right)+256 i b^2 \sin ^{-1}(c x)^3-288 b^2 \sin ^{-1}(c x) \sin \left(2 \sin ^{-1}(c x)\right)-12 b^2 \sin ^{-1}(c x) \sin \left(4 \sin ^{-1}(c x)\right)+768 b^2 \sin ^{-1}(c x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c x)}\right)-144 b^2 \cos \left(2 \sin ^{-1}(c x)\right)+288 b^2 \sin ^{-1}(c x)^2 \cos \left(2 \sin ^{-1}(c x)\right)-3 b^2 \cos \left(4 \sin ^{-1}(c x)\right)+24 b^2 \sin ^{-1}(c x)^2 \cos \left(4 \sin ^{-1}(c x)\right)-32 i \pi ^3 b^2\right)","-\frac{1}{8} b c d^2 x \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{11}{16} b c d^2 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{4} d^2 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{2} d^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2-i b d^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{i d^2 \left(a+b \sin ^{-1}(c x)\right)^3}{3 b}-\frac{11}{32} d^2 \left(a+b \sin ^{-1}(c x)\right)^2+d^2 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{1}{32} b^2 c^4 d^2 x^4+\frac{13}{32} b^2 c^2 d^2 x^2+\frac{1}{2} b^2 d^2 \text{Li}_3\left(e^{2 i \sin ^{-1}(c x)}\right)",1,"(d^2*((-32*I)*b^2*Pi^3 - 768*a^2*c^2*x^2 + 192*a^2*c^4*x^4 - 624*a*b*c*x*Sqrt[1 - c^2*x^2] + 96*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 624*a*b*ArcSin[c*x] - 1536*a*b*c^2*x^2*ArcSin[c*x] + 384*a*b*c^4*x^4*ArcSin[c*x] - (768*I)*a*b*ArcSin[c*x]^2 + (256*I)*b^2*ArcSin[c*x]^3 - 144*b^2*Cos[2*ArcSin[c*x]] + 288*b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] - 3*b^2*Cos[4*ArcSin[c*x]] + 24*b^2*ArcSin[c*x]^2*Cos[4*ArcSin[c*x]] + 768*b^2*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] + 1536*a*b*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 768*a^2*Log[c*x] + (768*I)*b^2*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] - (768*I)*a*b*PolyLog[2, E^((2*I)*ArcSin[c*x])] + 384*b^2*PolyLog[3, E^((-2*I)*ArcSin[c*x])] - 288*b^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]] - 12*b^2*ArcSin[c*x]*Sin[4*ArcSin[c*x]]))/768","A",0
171,1,322,249,0.9730933,"\int \frac{\left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2)/x^2,x]","\frac{1}{54} d^2 \left(18 a^2 c^4 x^3-108 a^2 c^2 x-\frac{54 a^2}{x}+36 a b c^4 x^3 \sin ^{-1}(c x)+12 a b c \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right)-216 a b c \left(\sqrt{1-c^2 x^2}+c x \sin ^{-1}(c x)\right)-\frac{108 a b \left(c x \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+\sin ^{-1}(c x)\right)}{x}+2 b^2 c^2 x \left(9 c^2 x^2 \sin ^{-1}(c x)^2-2 \left(c^2 x^2+6\right)\right)-189 b^2 c \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-108 b^2 c^2 x \left(\sin ^{-1}(c x)^2-2\right)+108 i b^2 c \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-108 i b^2 c \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-\frac{54 b^2 \sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 c x \left(\log \left(1+e^{i \sin ^{-1}(c x)}\right)-\log \left(1-e^{i \sin ^{-1}(c x)}\right)\right)\right)}{x}-3 b^2 c \sin ^{-1}(c x) \cos \left(3 \sin ^{-1}(c x)\right)\right)","-\frac{4}{3} c^2 d^2 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2}{9} b c d^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{10}{3} b c d^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{d^2 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{x}-\frac{8}{3} c^2 d^2 x \left(a+b \sin ^{-1}(c x)\right)^2-4 b c d^2 \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{2}{27} b^2 c^4 d^2 x^3+\frac{32}{9} b^2 c^2 d^2 x+2 i b^2 c d^2 \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-2 i b^2 c d^2 \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)",1,"(d^2*((-54*a^2)/x - 108*a^2*c^2*x + 18*a^2*c^4*x^3 + 12*a*b*c*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + 36*a*b*c^4*x^3*ArcSin[c*x] - 189*b^2*c*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - 216*a*b*c*(Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x]) - 108*b^2*c^2*x*(-2 + ArcSin[c*x]^2) + 2*b^2*c^2*x*(-2*(6 + c^2*x^2) + 9*c^2*x^2*ArcSin[c*x]^2) - (108*a*b*(ArcSin[c*x] + c*x*ArcTanh[Sqrt[1 - c^2*x^2]]))/x - 3*b^2*c*ArcSin[c*x]*Cos[3*ArcSin[c*x]] - (54*b^2*ArcSin[c*x]*(ArcSin[c*x] + 2*c*x*(-Log[1 - E^(I*ArcSin[c*x])] + Log[1 + E^(I*ArcSin[c*x])])))/x + (108*I)*b^2*c*PolyLog[2, -E^(I*ArcSin[c*x])] - (108*I)*b^2*c*PolyLog[2, E^(I*ArcSin[c*x])]))/54","A",0
172,1,343,287,0.8645841,"\int \frac{\left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2)/x^3,x]","\frac{1}{2} d^2 \left(a^2 c^4 x^2-4 a^2 c^2 \log (x)-\frac{a^2}{x^2}+2 a b c^4 x^2 \sin ^{-1}(c x)+4 i a b c^2 \left(\sin ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)\right)+a b c^2 \left(c x \sqrt{1-c^2 x^2}-\sin ^{-1}(c x)\right)-\frac{2 a b \left(c x \sqrt{1-c^2 x^2}+\sin ^{-1}(c x)\right)}{x^2}-8 a b c^2 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+\frac{1}{6} i b^2 c^2 \left(-24 \sin ^{-1}(c x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c x)}\right)+12 i \text{Li}_3\left(e^{-2 i \sin ^{-1}(c x)}\right)-8 \sin ^{-1}(c x)^3+24 i \sin ^{-1}(c x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c x)}\right)+\pi ^3\right)-\frac{b^2 \left(-2 c^2 x^2 \log (c x)+2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+\sin ^{-1}(c x)^2\right)}{x^2}+\frac{1}{2} b^2 c^2 \sin ^{-1}(c x) \sin \left(2 \sin ^{-1}(c x)\right)-\frac{1}{4} b^2 c^2 \left(2 \sin ^{-1}(c x)^2-1\right) \cos \left(2 \sin ^{-1}(c x)\right)\right)","2 i b c^2 d^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)-c^2 d^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{b c d^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{x}-\frac{d^2 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{2 x^2}+\frac{2 i c^2 d^2 \left(a+b \sin ^{-1}(c x)\right)^3}{3 b}-\frac{1}{4} c^2 d^2 \left(a+b \sin ^{-1}(c x)\right)^2-2 c^2 d^2 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{1}{2} b c^3 d^2 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{1}{4} b^2 c^4 d^2 x^2-b^2 c^2 d^2 \text{Li}_3\left(e^{2 i \sin ^{-1}(c x)}\right)+b^2 c^2 d^2 \log (x)",1,"(d^2*(-(a^2/x^2) + a^2*c^4*x^2 + a*b*c^2*(c*x*Sqrt[1 - c^2*x^2] - ArcSin[c*x]) + 2*a*b*c^4*x^2*ArcSin[c*x] - (2*a*b*(c*x*Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/x^2 - (b^2*c^2*(-1 + 2*ArcSin[c*x]^2)*Cos[2*ArcSin[c*x]])/4 - 8*a*b*c^2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - 4*a^2*c^2*Log[x] - (b^2*(2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]^2 - 2*c^2*x^2*Log[c*x]))/x^2 + (4*I)*a*b*c^2*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c*x])]) + (I/6)*b^2*c^2*(Pi^3 - 8*ArcSin[c*x]^3 + (24*I)*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] - 24*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] + (12*I)*PolyLog[3, E^((-2*I)*ArcSin[c*x])]) + (b^2*c^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]])/2))/2","A",0
173,1,374,268,0.8432728,"\int \frac{\left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2)/x^4,x]","\frac{d^2 \left(3 a^2 c^4 x^4+6 a^2 c^2 x^2-a^2+6 a b c^4 x^4 \sin ^{-1}(c x)-a b c x \sqrt{1-c^2 x^2}+12 a b c^2 x^2 \sin ^{-1}(c x)+6 a b c^3 x^3 \sqrt{1-c^2 x^2}+11 a b c^3 x^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-2 a b \sin ^{-1}(c x)-6 b^2 c^4 x^4+3 b^2 c^4 x^4 \sin ^{-1}(c x)^2-11 i b^2 c^3 x^3 \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+11 i b^2 c^3 x^3 \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-11 b^2 c^3 x^3 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+11 b^2 c^3 x^3 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)-b^2 c^2 x^2+6 b^2 c^2 x^2 \sin ^{-1}(c x)^2-b^2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+6 b^2 c^3 x^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-b^2 \sin ^{-1}(c x)^2\right)}{3 x^3}","\frac{8}{3} c^4 d^2 x \left(a+b \sin ^{-1}(c x)\right)^2+\frac{22}{3} b c^3 d^2 \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+\frac{4 c^2 d^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 x}-\frac{b c d^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 x^2}-\frac{d^2 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{3 x^3}+\frac{5}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)-2 b^2 c^4 d^2 x-\frac{11}{3} i b^2 c^3 d^2 \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+\frac{11}{3} i b^2 c^3 d^2 \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-\frac{b^2 c^2 d^2}{3 x}",1,"(d^2*(-a^2 + 6*a^2*c^2*x^2 - b^2*c^2*x^2 + 3*a^2*c^4*x^4 - 6*b^2*c^4*x^4 - a*b*c*x*Sqrt[1 - c^2*x^2] + 6*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] - 2*a*b*ArcSin[c*x] + 12*a*b*c^2*x^2*ArcSin[c*x] + 6*a*b*c^4*x^4*ArcSin[c*x] - b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 6*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - b^2*ArcSin[c*x]^2 + 6*b^2*c^2*x^2*ArcSin[c*x]^2 + 3*b^2*c^4*x^4*ArcSin[c*x]^2 + 11*a*b*c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]] - 11*b^2*c^3*x^3*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 11*b^2*c^3*x^3*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (11*I)*b^2*c^3*x^3*PolyLog[2, -E^(I*ArcSin[c*x])] + (11*I)*b^2*c^3*x^3*PolyLog[2, E^(I*ArcSin[c*x])]))/(3*x^3)","A",0
174,1,301,476,0.4272437,"\int x^4 \left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^4*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2,x]","-\frac{d^3 \left(12006225 a^2 c^5 x^5 \left(105 c^6 x^6-385 c^4 x^4+495 c^2 x^2-231\right)+6930 a b \sqrt{1-c^2 x^2} \left(33075 c^{10} x^{10}-111475 c^8 x^8+117625 c^6 x^6-18933 c^4 x^4-25244 c^2 x^2-50488\right)+6930 b \sin ^{-1}(c x) \left(3465 a c^5 x^5 \left(105 c^6 x^6-385 c^4 x^4+495 c^2 x^2-231\right)+b \sqrt{1-c^2 x^2} \left(33075 c^{10} x^{10}-111475 c^8 x^8+117625 c^6 x^6-18933 c^4 x^4-25244 c^2 x^2-50488\right)\right)+12006225 b^2 c^5 x^5 \left(105 c^6 x^6-385 c^4 x^4+495 c^2 x^2-231\right) \sin ^{-1}(c x)^2+b^2 \left(-20837250 c^{11} x^{11}+85835750 c^9 x^9-116448750 c^7 x^7+26241138 c^5 x^5+58313640 c^3 x^3+349881840 c x\right)\right)}{13867189875 c^5}","\frac{1}{11} d^3 x^5 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{2}{33} d^3 x^5 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{8}{231} d^3 x^5 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{32 b d^3 x^4 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{5775 c}+\frac{2 b d^3 \left(1-c^2 x^2\right)^{11/2} \left(a+b \sin ^{-1}(c x)\right)}{121 c^5}-\frac{8 b d^3 \left(1-c^2 x^2\right)^{9/2} \left(a+b \sin ^{-1}(c x)\right)}{297 c^5}+\frac{2 b d^3 \left(1-c^2 x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{1617 c^5}-\frac{4 b d^3 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{1155 c^5}+\frac{16 b d^3 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{693 c^5}+\frac{256 b d^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{17325 c^5}+\frac{128 b d^3 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{17325 c^3}+\frac{16 d^3 x^5 \left(a+b \sin ^{-1}(c x)\right)^2}{1155}+\frac{2 b^2 c^6 d^3 x^{11}}{1331}-\frac{182 b^2 c^4 d^3 x^9}{29403}-\frac{100976 b^2 d^3 x}{4002075 c^4}+\frac{9410 b^2 c^2 d^3 x^7}{1120581}-\frac{50488 b^2 d^3 x^3}{12006225 c^2}-\frac{12622 b^2 d^3 x^5}{6670125}",1,"-1/13867189875*(d^3*(12006225*a^2*c^5*x^5*(-231 + 495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6) + 6930*a*b*Sqrt[1 - c^2*x^2]*(-50488 - 25244*c^2*x^2 - 18933*c^4*x^4 + 117625*c^6*x^6 - 111475*c^8*x^8 + 33075*c^10*x^10) + b^2*(349881840*c*x + 58313640*c^3*x^3 + 26241138*c^5*x^5 - 116448750*c^7*x^7 + 85835750*c^9*x^9 - 20837250*c^11*x^11) + 6930*b*(3465*a*c^5*x^5*(-231 + 495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6) + b*Sqrt[1 - c^2*x^2]*(-50488 - 25244*c^2*x^2 - 18933*c^4*x^4 + 117625*c^6*x^6 - 111475*c^8*x^8 + 33075*c^10*x^10))*ArcSin[c*x] + 12006225*b^2*c^5*x^5*(-231 + 495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6)*ArcSin[c*x]^2))/c^5","A",1
175,1,287,384,0.4535396,"\int x^3 \left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2,x]","-\frac{d^3 \left(c x \left(28800 a^2 c^3 x^3 \left(4 c^6 x^6-15 c^4 x^4+20 c^2 x^2-10\right)+30 a b \sqrt{1-c^2 x^2} \left(768 c^8 x^8-2736 c^6 x^6+3208 c^4 x^4-790 c^2 x^2-1185\right)+b^2 \left(-2304 c^9 x^9+10260 c^7 x^7-16040 c^5 x^5+5925 c^3 x^3+17775 c x\right)\right)+30 b \sin ^{-1}(c x) \left(15 a \left(512 c^{10} x^{10}-1920 c^8 x^8+2560 c^6 x^6-1280 c^4 x^4+79\right)+b c x \sqrt{1-c^2 x^2} \left(768 c^8 x^8-2736 c^6 x^6+3208 c^4 x^4-790 c^2 x^2-1185\right)\right)+225 b^2 \left(512 c^{10} x^{10}-1920 c^8 x^8+2560 c^6 x^6-1280 c^4 x^4+79\right) \sin ^{-1}(c x)^2\right)}{1152000 c^4}","-\frac{79 d^3 \left(a+b \sin ^{-1}(c x)\right)^2}{5120 c^4}-\frac{1}{50} b c d^3 x^5 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{1}{32} b c d^3 x^5 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{31}{960} b c d^3 x^5 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{10} d^3 x^4 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{3}{40} d^3 x^4 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{20} d^3 x^4 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{79 b d^3 x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{3840 c}+\frac{79 b d^3 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{2560 c^3}+\frac{1}{40} d^3 x^4 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{500} b^2 c^6 d^3 x^{10}-\frac{57 b^2 c^4 d^3 x^8}{6400}+\frac{401 b^2 c^2 d^3 x^6}{28800}-\frac{79 b^2 d^3 x^2}{5120 c^2}-\frac{79 b^2 d^3 x^4}{15360}",1,"-1/1152000*(d^3*(c*x*(28800*a^2*c^3*x^3*(-10 + 20*c^2*x^2 - 15*c^4*x^4 + 4*c^6*x^6) + 30*a*b*Sqrt[1 - c^2*x^2]*(-1185 - 790*c^2*x^2 + 3208*c^4*x^4 - 2736*c^6*x^6 + 768*c^8*x^8) + b^2*(17775*c*x + 5925*c^3*x^3 - 16040*c^5*x^5 + 10260*c^7*x^7 - 2304*c^9*x^9)) + 30*b*(b*c*x*Sqrt[1 - c^2*x^2]*(-1185 - 790*c^2*x^2 + 3208*c^4*x^4 - 2736*c^6*x^6 + 768*c^8*x^8) + 15*a*(79 - 1280*c^4*x^4 + 2560*c^6*x^6 - 1920*c^8*x^8 + 512*c^10*x^10))*ArcSin[c*x] + 225*b^2*(79 - 1280*c^4*x^4 + 2560*c^6*x^6 - 1920*c^8*x^8 + 512*c^10*x^10)*ArcSin[c*x]^2))/c^4","A",1
176,1,277,391,0.3823471,"\int x^2 \left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2,x]","-\frac{d^3 \left(99225 a^2 c^3 x^3 \left(35 c^6 x^6-135 c^4 x^4+189 c^2 x^2-105\right)+630 a b \sqrt{1-c^2 x^2} \left(1225 c^8 x^8-4675 c^6 x^6+6297 c^4 x^4-2629 c^2 x^2-5258\right)+630 b \sin ^{-1}(c x) \left(315 a c^3 x^3 \left(35 c^6 x^6-135 c^4 x^4+189 c^2 x^2-105\right)+b \sqrt{1-c^2 x^2} \left(1225 c^8 x^8-4675 c^6 x^6+6297 c^4 x^4-2629 c^2 x^2-5258\right)\right)+b^2 \left(-85750 c^9 x^9+420750 c^7 x^7-793422 c^5 x^5+552090 c^3 x^3+3312540 c x\right)+99225 b^2 c^3 x^3 \left(35 c^6 x^6-135 c^4 x^4+189 c^2 x^2-105\right) \sin ^{-1}(c x)^2\right)}{31255875 c^3}","\frac{32 b d^3 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{945 c}+\frac{1}{9} d^3 x^3 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{2}{21} d^3 x^3 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{8}{105} d^3 x^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2 b d^3 \left(1-c^2 x^2\right)^{9/2} \left(a+b \sin ^{-1}(c x)\right)}{81 c^3}+\frac{2 b d^3 \left(1-c^2 x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{441 c^3}+\frac{4 b d^3 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{525 c^3}+\frac{16 b d^3 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{315 c^3}+\frac{64 b d^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{945 c^3}+\frac{16}{315} d^3 x^3 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{2}{729} b^2 c^6 d^3 x^9-\frac{374 b^2 c^4 d^3 x^7}{27783}+\frac{4198 b^2 c^2 d^3 x^5}{165375}-\frac{10516 b^2 d^3 x}{99225 c^2}-\frac{5258 b^2 d^3 x^3}{297675}",1,"-1/31255875*(d^3*(99225*a^2*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6) + 630*a*b*Sqrt[1 - c^2*x^2]*(-5258 - 2629*c^2*x^2 + 6297*c^4*x^4 - 4675*c^6*x^6 + 1225*c^8*x^8) + b^2*(3312540*c*x + 552090*c^3*x^3 - 793422*c^5*x^5 + 420750*c^7*x^7 - 85750*c^9*x^9) + 630*b*(315*a*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6) + b*Sqrt[1 - c^2*x^2]*(-5258 - 2629*c^2*x^2 + 6297*c^4*x^4 - 4675*c^6*x^6 + 1225*c^8*x^8))*ArcSin[c*x] + 99225*b^2*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6)*ArcSin[c*x]^2))/c^3","A",1
177,1,257,268,0.344212,"\int x \left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2,x]","-\frac{d^3 \left(c x \left(1152 a^2 c x \left(c^6 x^6-4 c^4 x^4+6 c^2 x^2-4\right)+6 a b \sqrt{1-c^2 x^2} \left(48 c^6 x^6-200 c^4 x^4+326 c^2 x^2-279\right)+b^2 c x \left(-36 c^6 x^6+200 c^4 x^4-489 c^2 x^2+837\right)\right)+6 b \sin ^{-1}(c x) \left(3 a \left(128 c^8 x^8-512 c^6 x^6+768 c^4 x^4-512 c^2 x^2+93\right)+b c x \sqrt{1-c^2 x^2} \left(48 c^6 x^6-200 c^4 x^4+326 c^2 x^2-279\right)\right)+9 b^2 \left(128 c^8 x^8-512 c^6 x^6+768 c^4 x^4-512 c^2 x^2+93\right) \sin ^{-1}(c x)^2\right)}{9216 c^2}","\frac{b d^3 x \left(1-c^2 x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{32 c}+\frac{7 b d^3 x \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{192 c}+\frac{35 b d^3 x \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{768 c}+\frac{35 b d^3 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{512 c}-\frac{d^3 \left(1-c^2 x^2\right)^4 \left(a+b \sin ^{-1}(c x)\right)^2}{8 c^2}+\frac{35 d^3 \left(a+b \sin ^{-1}(c x)\right)^2}{1024 c^2}+\frac{35 b^2 c^2 d^3 x^4}{3072}+\frac{b^2 d^3 \left(1-c^2 x^2\right)^4}{256 c^2}+\frac{7 b^2 d^3 \left(1-c^2 x^2\right)^3}{1152 c^2}-\frac{175 b^2 d^3 x^2}{3072}",1,"-1/9216*(d^3*(c*x*(b^2*c*x*(837 - 489*c^2*x^2 + 200*c^4*x^4 - 36*c^6*x^6) + 1152*a^2*c*x*(-4 + 6*c^2*x^2 - 4*c^4*x^4 + c^6*x^6) + 6*a*b*Sqrt[1 - c^2*x^2]*(-279 + 326*c^2*x^2 - 200*c^4*x^4 + 48*c^6*x^6)) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(-279 + 326*c^2*x^2 - 200*c^4*x^4 + 48*c^6*x^6) + 3*a*(93 - 512*c^2*x^2 + 768*c^4*x^4 - 512*c^6*x^6 + 128*c^8*x^8))*ArcSin[c*x] + 9*b^2*(93 - 512*c^2*x^2 + 768*c^4*x^4 - 512*c^6*x^6 + 128*c^8*x^8)*ArcSin[c*x]^2))/c^2","A",1
178,1,241,298,0.4500338,"\int \left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2,x]","-\frac{d^3 \left(11025 a^2 c x \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right)+210 a b \sqrt{1-c^2 x^2} \left(75 c^6 x^6-351 c^4 x^4+757 c^2 x^2-2161\right)+210 b \sin ^{-1}(c x) \left(105 a c x \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right)+b \sqrt{1-c^2 x^2} \left(75 c^6 x^6-351 c^4 x^4+757 c^2 x^2-2161\right)\right)+2 b^2 c x \left(-1125 c^6 x^6+7371 c^4 x^4-26495 c^2 x^2+226905\right)+11025 b^2 c x \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right) \sin ^{-1}(c x)^2\right)}{385875 c}","\frac{1}{7} d^3 x \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{6}{35} d^3 x \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{8}{35} d^3 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2+\frac{2 b d^3 \left(1-c^2 x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)}{49 c}+\frac{12 b d^3 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{175 c}+\frac{16 b d^3 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{105 c}+\frac{32 b d^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{35 c}+\frac{16}{35} d^3 x \left(a+b \sin ^{-1}(c x)\right)^2+\frac{2}{343} b^2 c^6 d^3 x^7-\frac{234 b^2 c^4 d^3 x^5}{6125}+\frac{1514 b^2 c^2 d^3 x^3}{11025}-\frac{4322 b^2 d^3 x}{3675}",1,"-1/385875*(d^3*(2*b^2*c*x*(226905 - 26495*c^2*x^2 + 7371*c^4*x^4 - 1125*c^6*x^6) + 11025*a^2*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + 210*a*b*Sqrt[1 - c^2*x^2]*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6) + 210*b*(105*a*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + b*Sqrt[1 - c^2*x^2]*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6))*ArcSin[c*x] + 11025*b^2*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)*ArcSin[c*x]^2))/c","A",1
179,1,448,354,0.8141014,"\int \frac{\left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2)/x,x]","\frac{d^3 \left(-576 a^2 c^6 x^6+2592 a^2 c^4 x^4-5184 a^2 c^2 x^2+3456 a^2 \log (c x)-1152 a b c^6 x^6 \sin ^{-1}(c x)+5184 a b c^4 x^4 \sin ^{-1}(c x)-3600 a b c x \sqrt{1-c^2 x^2}-10368 a b c^2 x^2 \sin ^{-1}(c x)-192 a b c^5 x^5 \sqrt{1-c^2 x^2}+1056 a b c^3 x^3 \sqrt{1-c^2 x^2}-3456 i a b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-3456 i a b \sin ^{-1}(c x)^2+3600 a b \sin ^{-1}(c x)+6912 a b \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+3456 i b^2 \sin ^{-1}(c x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c x)}\right)+1728 b^2 \text{Li}_3\left(e^{-2 i \sin ^{-1}(c x)}\right)+1152 i b^2 \sin ^{-1}(c x)^3-1566 b^2 \sin ^{-1}(c x) \sin \left(2 \sin ^{-1}(c x)\right)-108 b^2 \sin ^{-1}(c x) \sin \left(4 \sin ^{-1}(c x)\right)-6 b^2 \sin ^{-1}(c x) \sin \left(6 \sin ^{-1}(c x)\right)+3456 b^2 \sin ^{-1}(c x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c x)}\right)-783 b^2 \cos \left(2 \sin ^{-1}(c x)\right)+1566 b^2 \sin ^{-1}(c x)^2 \cos \left(2 \sin ^{-1}(c x)\right)-27 b^2 \cos \left(4 \sin ^{-1}(c x)\right)+216 b^2 \sin ^{-1}(c x)^2 \cos \left(4 \sin ^{-1}(c x)\right)-b^2 \cos \left(6 \sin ^{-1}(c x)\right)+18 b^2 \sin ^{-1}(c x)^2 \cos \left(6 \sin ^{-1}(c x)\right)-144 i \pi ^3 b^2\right)}{3456}","-\frac{1}{18} b c d^3 x \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{7}{36} b c d^3 x \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{19}{24} b c d^3 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{6} d^3 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{4} d^3 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{2} d^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2-i b d^3 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{i d^3 \left(a+b \sin ^{-1}(c x)\right)^3}{3 b}-\frac{19}{48} d^3 \left(a+b \sin ^{-1}(c x)\right)^2+d^3 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{7}{144} b^2 c^4 d^3 x^4+\frac{71}{144} b^2 c^2 d^3 x^2-\frac{1}{108} b^2 d^3 \left(1-c^2 x^2\right)^3+\frac{1}{2} b^2 d^3 \text{Li}_3\left(e^{2 i \sin ^{-1}(c x)}\right)",1,"(d^3*((-144*I)*b^2*Pi^3 - 5184*a^2*c^2*x^2 + 2592*a^2*c^4*x^4 - 576*a^2*c^6*x^6 - 3600*a*b*c*x*Sqrt[1 - c^2*x^2] + 1056*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] - 192*a*b*c^5*x^5*Sqrt[1 - c^2*x^2] + 3600*a*b*ArcSin[c*x] - 10368*a*b*c^2*x^2*ArcSin[c*x] + 5184*a*b*c^4*x^4*ArcSin[c*x] - 1152*a*b*c^6*x^6*ArcSin[c*x] - (3456*I)*a*b*ArcSin[c*x]^2 + (1152*I)*b^2*ArcSin[c*x]^3 - 783*b^2*Cos[2*ArcSin[c*x]] + 1566*b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] - 27*b^2*Cos[4*ArcSin[c*x]] + 216*b^2*ArcSin[c*x]^2*Cos[4*ArcSin[c*x]] - b^2*Cos[6*ArcSin[c*x]] + 18*b^2*ArcSin[c*x]^2*Cos[6*ArcSin[c*x]] + 3456*b^2*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] + 6912*a*b*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 3456*a^2*Log[c*x] + (3456*I)*b^2*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] - (3456*I)*a*b*PolyLog[2, E^((2*I)*ArcSin[c*x])] + 1728*b^2*PolyLog[3, E^((-2*I)*ArcSin[c*x])] - 1566*b^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]] - 108*b^2*ArcSin[c*x]*Sin[4*ArcSin[c*x]] - 6*b^2*ArcSin[c*x]*Sin[6*ArcSin[c*x]]))/3456","A",0
180,1,483,329,1.2837242,"\int \frac{\left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2)/x^2,x]","\frac{1}{720} d^3 \left(-144 a^2 c^6 x^5+720 a^2 c^4 x^3-2160 a^2 c^2 x-\frac{720 a^2}{x}-288 a b c^6 x^5 \sin ^{-1}(c x)+1440 a b c^4 x^3 \sin ^{-1}(c x)-\frac{17568}{5} a b c \sqrt{1-c^2 x^2}-1440 a b c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-4320 a b c^2 x \sin ^{-1}(c x)-\frac{288}{5} a b c^5 x^4 \sqrt{1-c^2 x^2}+\frac{2016}{5} a b c^3 x^2 \sqrt{1-c^2 x^2}-\frac{1440 a b \sin ^{-1}(c x)}{x}-3420 b^2 c \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+3460 b^2 c^2 x-1890 b^2 c^2 x \sin ^{-1}(c x)^2-360 b^2 c^2 x \sin ^{-1}(c x)^2 \cos \left(2 \sin ^{-1}(c x)\right)+80 b^2 c^2 x \cos \left(2 \sin ^{-1}(c x)\right)+1440 i b^2 c \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-1440 i b^2 c \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-10 b^2 c \sin \left(3 \sin ^{-1}(c x)\right)+45 b^2 c \sin ^{-1}(c x)^2 \sin \left(3 \sin ^{-1}(c x)\right)+\frac{18}{25} b^2 c \sin \left(5 \sin ^{-1}(c x)\right)-9 b^2 c \sin ^{-1}(c x)^2 \sin \left(5 \sin ^{-1}(c x)\right)-\frac{720 b^2 \sin ^{-1}(c x)^2}{x}+1440 b^2 c \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-1440 b^2 c \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)-90 b^2 c \sin ^{-1}(c x) \cos \left(3 \sin ^{-1}(c x)\right)-\frac{18}{5} b^2 c \sin ^{-1}(c x) \cos \left(5 \sin ^{-1}(c x)\right)\right)","-\frac{6}{5} c^2 d^3 x \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2-\frac{8}{5} c^2 d^3 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2}{25} b c d^3 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{2}{5} b c d^3 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{22}{5} b c d^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{d^3 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{x}-\frac{16}{5} c^2 d^3 x \left(a+b \sin ^{-1}(c x)\right)^2-4 b c d^3 \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+\frac{2}{125} b^2 c^6 d^3 x^5-\frac{14}{75} b^2 c^4 d^3 x^3+\frac{122}{25} b^2 c^2 d^3 x+2 i b^2 c d^3 \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-2 i b^2 c d^3 \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)",1,"(d^3*((-720*a^2)/x - 2160*a^2*c^2*x + 3460*b^2*c^2*x + 720*a^2*c^4*x^3 - 144*a^2*c^6*x^5 - (17568*a*b*c*Sqrt[1 - c^2*x^2])/5 + (2016*a*b*c^3*x^2*Sqrt[1 - c^2*x^2])/5 - (288*a*b*c^5*x^4*Sqrt[1 - c^2*x^2])/5 - (1440*a*b*ArcSin[c*x])/x - 4320*a*b*c^2*x*ArcSin[c*x] + 1440*a*b*c^4*x^3*ArcSin[c*x] - 288*a*b*c^6*x^5*ArcSin[c*x] - 3420*b^2*c*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - (720*b^2*ArcSin[c*x]^2)/x - 1890*b^2*c^2*x*ArcSin[c*x]^2 - 1440*a*b*c*ArcTanh[Sqrt[1 - c^2*x^2]] + 80*b^2*c^2*x*Cos[2*ArcSin[c*x]] - 360*b^2*c^2*x*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] - 90*b^2*c*ArcSin[c*x]*Cos[3*ArcSin[c*x]] - (18*b^2*c*ArcSin[c*x]*Cos[5*ArcSin[c*x]])/5 + 1440*b^2*c*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - 1440*b^2*c*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + (1440*I)*b^2*c*PolyLog[2, -E^(I*ArcSin[c*x])] - (1440*I)*b^2*c*PolyLog[2, E^(I*ArcSin[c*x])] - 10*b^2*c*Sin[3*ArcSin[c*x]] + 45*b^2*c*ArcSin[c*x]^2*Sin[3*ArcSin[c*x]] + (18*b^2*c*Sin[5*ArcSin[c*x]])/25 - 9*b^2*c*ArcSin[c*x]^2*Sin[5*ArcSin[c*x]]))/720","A",0
181,1,494,371,1.3239863,"\int \frac{\left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2)/x^3,x]","\frac{1}{256} d^3 \left(-64 a^2 c^6 x^4+384 a^2 c^4 x^2-768 a^2 c^2 \log (x)-\frac{128 a^2}{x^2}-128 a b c^6 x^4 \sin ^{-1}(c x)+768 a b c^4 x^2 \sin ^{-1}(c x)+768 i a b c^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-\frac{256 a b c \sqrt{1-c^2 x^2}}{x}+768 i a b c^2 \sin ^{-1}(c x)^2-336 a b c^2 \sin ^{-1}(c x)-1536 a b c^2 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-32 a b c^5 x^3 \sqrt{1-c^2 x^2}+336 a b c^3 x \sqrt{1-c^2 x^2}-\frac{256 a b \sin ^{-1}(c x)}{x^2}-768 i b^2 c^2 \sin ^{-1}(c x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c x)}\right)-384 b^2 c^2 \text{Li}_3\left(e^{-2 i \sin ^{-1}(c x)}\right)-\frac{256 b^2 c \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{x}+256 b^2 c^2 \log (c x)-256 i b^2 c^2 \sin ^{-1}(c x)^3+160 b^2 c^2 \sin ^{-1}(c x) \sin \left(2 \sin ^{-1}(c x)\right)+4 b^2 c^2 \sin ^{-1}(c x) \sin \left(4 \sin ^{-1}(c x)\right)-768 b^2 c^2 \sin ^{-1}(c x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c x)}\right)+80 b^2 c^2 \cos \left(2 \sin ^{-1}(c x)\right)-160 b^2 c^2 \sin ^{-1}(c x)^2 \cos \left(2 \sin ^{-1}(c x)\right)+b^2 c^2 \cos \left(4 \sin ^{-1}(c x)\right)-8 b^2 c^2 \sin ^{-1}(c x)^2 \cos \left(4 \sin ^{-1}(c x)\right)+32 i \pi ^3 b^2 c^2-\frac{128 b^2 \sin ^{-1}(c x)^2}{x^2}\right)","3 i b c^2 d^3 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)-\frac{3}{4} c^2 d^3 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2-\frac{3}{2} c^2 d^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{b c d^3 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{x}-\frac{d^3 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{2 x^2}+\frac{i c^2 d^3 \left(a+b \sin ^{-1}(c x)\right)^3}{b}+\frac{3}{32} c^2 d^3 \left(a+b \sin ^{-1}(c x)\right)^2-3 c^2 d^3 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{7}{8} b c^3 d^3 x \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)+\frac{3}{16} b c^3 d^3 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{32} b^2 c^6 d^3 x^4-\frac{21}{32} b^2 c^4 d^3 x^2-\frac{3}{2} b^2 c^2 d^3 \text{Li}_3\left(e^{2 i \sin ^{-1}(c x)}\right)+b^2 c^2 d^3 \log (x)",1,"(d^3*((32*I)*b^2*c^2*Pi^3 - (128*a^2)/x^2 + 384*a^2*c^4*x^2 - 64*a^2*c^6*x^4 - (256*a*b*c*Sqrt[1 - c^2*x^2])/x + 336*a*b*c^3*x*Sqrt[1 - c^2*x^2] - 32*a*b*c^5*x^3*Sqrt[1 - c^2*x^2] - 336*a*b*c^2*ArcSin[c*x] - (256*a*b*ArcSin[c*x])/x^2 + 768*a*b*c^4*x^2*ArcSin[c*x] - 128*a*b*c^6*x^4*ArcSin[c*x] - (256*b^2*c*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/x + (768*I)*a*b*c^2*ArcSin[c*x]^2 - (128*b^2*ArcSin[c*x]^2)/x^2 - (256*I)*b^2*c^2*ArcSin[c*x]^3 + 80*b^2*c^2*Cos[2*ArcSin[c*x]] - 160*b^2*c^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] + b^2*c^2*Cos[4*ArcSin[c*x]] - 8*b^2*c^2*ArcSin[c*x]^2*Cos[4*ArcSin[c*x]] - 768*b^2*c^2*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] - 1536*a*b*c^2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - 768*a^2*c^2*Log[x] + 256*b^2*c^2*Log[c*x] - (768*I)*b^2*c^2*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] + (768*I)*a*b*c^2*PolyLog[2, E^((2*I)*ArcSin[c*x])] - 384*b^2*c^2*PolyLog[3, E^((-2*I)*ArcSin[c*x])] + 160*b^2*c^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]] + 4*b^2*c^2*ArcSin[c*x]*Sin[4*ArcSin[c*x]]))/256","A",0
182,1,480,348,1.0449674,"\int \frac{\left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2)/x^4,x]","-\frac{d^3 \left(9 a^2 c^6 x^6-81 a^2 c^4 x^4-81 a^2 c^2 x^2+9 a^2+18 a b c^6 x^6 \sin ^{-1}(c x)-162 a b c^4 x^4 \sin ^{-1}(c x)+9 a b c x \sqrt{1-c^2 x^2}-162 a b c^2 x^2 \sin ^{-1}(c x)+6 a b c^5 x^5 \sqrt{1-c^2 x^2}-150 a b c^3 x^3 \sqrt{1-c^2 x^2}-153 a b c^3 x^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+18 a b \sin ^{-1}(c x)-2 b^2 c^6 x^6+9 b^2 c^6 x^6 \sin ^{-1}(c x)^2+150 b^2 c^4 x^4-81 b^2 c^4 x^4 \sin ^{-1}(c x)^2+153 i b^2 c^3 x^3 \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-153 i b^2 c^3 x^3 \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+153 b^2 c^3 x^3 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-153 b^2 c^3 x^3 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)+9 b^2 c^2 x^2-81 b^2 c^2 x^2 \sin ^{-1}(c x)^2+9 b^2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+6 b^2 c^5 x^5 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-150 b^2 c^3 x^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+9 b^2 \sin ^{-1}(c x)^2\right)}{27 x^3}","\frac{16}{3} c^4 d^3 x \left(a+b \sin ^{-1}(c x)\right)^2+\frac{34}{3} b c^3 d^3 \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+\frac{2 c^2 d^3 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{x}-\frac{b c d^3 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{3 x^2}-\frac{d^3 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{3 x^3}+\frac{8}{3} c^4 d^3 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{1}{9} b c^3 d^3 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)+5 b c^3 d^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{2}{27} b^2 c^6 d^3 x^3-\frac{50}{9} b^2 c^4 d^3 x-\frac{17}{3} i b^2 c^3 d^3 \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+\frac{17}{3} i b^2 c^3 d^3 \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-\frac{b^2 c^2 d^3}{3 x}",1,"-1/27*(d^3*(9*a^2 - 81*a^2*c^2*x^2 + 9*b^2*c^2*x^2 - 81*a^2*c^4*x^4 + 150*b^2*c^4*x^4 + 9*a^2*c^6*x^6 - 2*b^2*c^6*x^6 + 9*a*b*c*x*Sqrt[1 - c^2*x^2] - 150*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 6*a*b*c^5*x^5*Sqrt[1 - c^2*x^2] + 18*a*b*ArcSin[c*x] - 162*a*b*c^2*x^2*ArcSin[c*x] - 162*a*b*c^4*x^4*ArcSin[c*x] + 18*a*b*c^6*x^6*ArcSin[c*x] + 9*b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - 150*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 6*b^2*c^5*x^5*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 9*b^2*ArcSin[c*x]^2 - 81*b^2*c^2*x^2*ArcSin[c*x]^2 - 81*b^2*c^4*x^4*ArcSin[c*x]^2 + 9*b^2*c^6*x^6*ArcSin[c*x]^2 - 153*a*b*c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]] + 153*b^2*c^3*x^3*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - 153*b^2*c^3*x^3*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + (153*I)*b^2*c^3*x^3*PolyLog[2, -E^(I*ArcSin[c*x])] - (153*I)*b^2*c^3*x^3*PolyLog[2, E^(I*ArcSin[c*x])]))/x^3","A",0
183,1,508,297,0.8467971,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)^2}{d-c^2 d x^2} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2),x]","-\frac{36 a^2 c^3 x^3+108 a^2 c x+54 a^2 \log (1-c x)-54 a^2 \log (c x+1)+72 a b c^3 x^3 \sin ^{-1}(c x)+24 a b c^2 x^2 \sqrt{1-c^2 x^2}+264 a b \sqrt{1-c^2 x^2}-216 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+216 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+216 a b c x \sin ^{-1}(c x)+108 i \pi  a b \sin ^{-1}(c x)-216 a b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-108 \pi  a b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+216 a b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-108 \pi  a b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+108 \pi  a b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+108 \pi  a b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+270 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+216 b^2 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)-216 b^2 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)-270 b^2 c x+135 b^2 c x \sin ^{-1}(c x)^2+2 b^2 \sin \left(3 \sin ^{-1}(c x)\right)-9 b^2 \sin ^{-1}(c x)^2 \sin \left(3 \sin ^{-1}(c x)\right)-108 b^2 \sin ^{-1}(c x)^2 \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+108 b^2 \sin ^{-1}(c x)^2 \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-6 b^2 \sin ^{-1}(c x) \cos \left(3 \sin ^{-1}(c x)\right)}{108 c^5 d}","\frac{2 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^5 d}-\frac{2 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^5 d}-\frac{2 i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^5 d}-\frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{c^4 d}-\frac{x^3 \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^2 d}-\frac{22 b \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 c^5 d}-\frac{2 b x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 c^3 d}-\frac{2 b^2 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{c^5 d}+\frac{2 b^2 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{c^5 d}+\frac{22 b^2 x}{9 c^4 d}+\frac{2 b^2 x^3}{27 c^2 d}",1,"-1/108*(108*a^2*c*x - 270*b^2*c*x + 36*a^2*c^3*x^3 + 264*a*b*Sqrt[1 - c^2*x^2] + 24*a*b*c^2*x^2*Sqrt[1 - c^2*x^2] + (108*I)*a*b*Pi*ArcSin[c*x] + 216*a*b*c*x*ArcSin[c*x] + 72*a*b*c^3*x^3*ArcSin[c*x] + 270*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 135*b^2*c*x*ArcSin[c*x]^2 - 6*b^2*ArcSin[c*x]*Cos[3*ArcSin[c*x]] - 108*a*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 216*a*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 108*b^2*ArcSin[c*x]^2*Log[1 - I*E^(I*ArcSin[c*x])] - 108*a*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 216*a*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 108*b^2*ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] + 54*a^2*Log[1 - c*x] - 54*a^2*Log[1 + c*x] + 108*a*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 108*a*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (216*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (216*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])] + 216*b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 216*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])] + 2*b^2*Sin[3*ArcSin[c*x]] - 9*b^2*ArcSin[c*x]^2*Sin[3*ArcSin[c*x]])/(c^5*d)","A",0
184,1,441,210,0.3972766,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)^2}{d-c^2 d x^2} \, dx","Integrate[(x^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2),x]","-\frac{12 a^2 c^2 x^2+12 a^2 \log \left(1-c^2 x^2\right)+12 a b c x \sqrt{1-c^2 x^2}+24 a b c^2 x^2 \sin ^{-1}(c x)-48 i a b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-48 i a b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-24 i a b \sin ^{-1}(c x)^2-12 a b \sin ^{-1}(c x)+48 i \pi  a b \sin ^{-1}(c x)+48 a b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+48 a b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+96 \pi  a b \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+24 \pi  a b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-24 \pi  a b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-24 \pi  a b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-96 \pi  a b \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+24 \pi  a b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-24 i b^2 \sin ^{-1}(c x) \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+12 b^2 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)-8 i b^2 \sin ^{-1}(c x)^3+6 b^2 \sin \left(2 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+24 b^2 \sin ^{-1}(c x)^2 \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)-6 b^2 \sin ^{-1}(c x)^2 \cos \left(2 \sin ^{-1}(c x)\right)+3 b^2 \cos \left(2 \sin ^{-1}(c x)\right)}{24 c^4 d}","\frac{i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^4 d}+\frac{i \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c^4 d}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{4 c^4 d}-\frac{\log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^4 d}-\frac{x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{2 c^2 d}-\frac{b x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 c^3 d}-\frac{b^2 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 c^4 d}+\frac{b^2 x^2}{4 c^2 d}",1,"-1/24*(12*a^2*c^2*x^2 + 12*a*b*c*x*Sqrt[1 - c^2*x^2] - 12*a*b*ArcSin[c*x] + (48*I)*a*b*Pi*ArcSin[c*x] + 24*a*b*c^2*x^2*ArcSin[c*x] - (24*I)*a*b*ArcSin[c*x]^2 - (8*I)*b^2*ArcSin[c*x]^3 + 3*b^2*Cos[2*ArcSin[c*x]] - 6*b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] + 96*a*b*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 24*a*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] + 48*a*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 24*a*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 48*a*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 24*b^2*ArcSin[c*x]^2*Log[1 + E^((2*I)*ArcSin[c*x])] + 12*a^2*Log[1 - c^2*x^2] - 96*a*b*Pi*Log[Cos[ArcSin[c*x]/2]] + 24*a*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 24*a*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (48*I)*a*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (48*I)*a*b*PolyLog[2, I*E^(I*ArcSin[c*x])] - (24*I)*b^2*ArcSin[c*x]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + 12*b^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])] + 6*b^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]])/(c^4*d)","B",0
185,1,317,218,0.3101479,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{d-c^2 d x^2} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2),x]","-\frac{2 a^2 c x+a^2 \log (1-c x)-a^2 \log (c x+1)+4 a b \sqrt{1-c^2 x^2}-4 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+4 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+4 a b c x \sin ^{-1}(c x)-4 a b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+4 a b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+4 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+4 b^2 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)-4 b^2 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)-4 b^2 c x+2 b^2 c x \sin ^{-1}(c x)^2-2 b^2 \sin ^{-1}(c x)^2 \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+2 b^2 \sin ^{-1}(c x)^2 \log \left(1+i e^{i \sin ^{-1}(c x)}\right)}{2 c^3 d}","\frac{2 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^3 d}-\frac{2 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^3 d}-\frac{2 i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^3 d}-\frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 d}-\frac{2 b \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c^3 d}-\frac{2 b^2 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{c^3 d}+\frac{2 b^2 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{c^3 d}+\frac{2 b^2 x}{c^2 d}",1,"-1/2*(2*a^2*c*x - 4*b^2*c*x + 4*a*b*Sqrt[1 - c^2*x^2] + 4*a*b*c*x*ArcSin[c*x] + 4*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 2*b^2*c*x*ArcSin[c*x]^2 - 4*a*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 2*b^2*ArcSin[c*x]^2*Log[1 - I*E^(I*ArcSin[c*x])] + 4*a*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 2*b^2*ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] + a^2*Log[1 - c*x] - a^2*Log[1 + c*x] - (4*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (4*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])] + 4*b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 4*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(c^3*d)","A",0
186,1,143,117,0.0789954,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{d-c^2 d x^2} \, dx","Integrate[(x*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2),x]","\frac{-3 a^2 \log \left(1-c^2 x^2\right)+6 i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+6 i a b \sin ^{-1}(c x)^2-12 a b \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)-3 b^2 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)+2 i b^2 \sin ^{-1}(c x)^3-6 b^2 \sin ^{-1}(c x)^2 \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)}{6 c^2 d}","\frac{i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^2 d}+\frac{i \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c^2 d}-\frac{\log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 d}-\frac{b^2 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 c^2 d}",1,"((6*I)*a*b*ArcSin[c*x]^2 + (2*I)*b^2*ArcSin[c*x]^3 - 12*a*b*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] - 6*b^2*ArcSin[c*x]^2*Log[1 + E^((2*I)*ArcSin[c*x])] - 3*a^2*Log[1 - c^2*x^2] + (6*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - 3*b^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/(6*c^2*d)","A",0
187,1,207,156,0.4951677,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{d-c^2 d x^2} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2),x]","\frac{a^2 (-\log (1-c x))+a^2 \log (c x+1)+4 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)-4 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+4 a b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-4 a b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-4 b^2 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)+4 b^2 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)-4 i b^2 \sin ^{-1}(c x)^2 \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right)}{2 c d}","\frac{2 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c d}-\frac{2 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c d}-\frac{2 i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c d}-\frac{2 b^2 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{c d}+\frac{2 b^2 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{c d}",1,"((-4*I)*b^2*ArcSin[c*x]^2*ArcTan[E^(I*ArcSin[c*x])] + 4*a*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 4*a*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - a^2*Log[1 - c*x] + a^2*Log[1 + c*x] + (4*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (4*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])] - 4*b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] + 4*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(2*c*d)","A",0
188,1,254,131,0.1889261,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x*(d - c^2*d*x^2)),x]","\frac{-12 a^2 \log \left(1-c^2 x^2\right)+24 a^2 \log (c x)+24 i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)-24 i a b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+48 a b \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-48 a b \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)+24 i b^2 \sin ^{-1}(c x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c x)}\right)+12 b^2 \text{Li}_3\left(e^{-2 i \sin ^{-1}(c x)}\right)-12 b^2 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)+16 i b^2 \sin ^{-1}(c x)^3+24 b^2 \sin ^{-1}(c x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c x)}\right)-24 b^2 \sin ^{-1}(c x)^2 \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)-i \pi ^3 b^2}{24 d}","\frac{i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d}-\frac{i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d}-\frac{2 \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d}-\frac{b^2 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 d}+\frac{b^2 \text{Li}_3\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d}",1,"((-I)*b^2*Pi^3 + (16*I)*b^2*ArcSin[c*x]^3 + 24*b^2*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] + 48*a*b*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - 48*a*b*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] - 24*b^2*ArcSin[c*x]^2*Log[1 + E^((2*I)*ArcSin[c*x])] + 24*a^2*Log[c*x] - 12*a^2*Log[1 - c^2*x^2] + (24*I)*b^2*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] + (24*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - (24*I)*a*b*PolyLog[2, E^((2*I)*ArcSin[c*x])] + 12*b^2*PolyLog[3, E^((-2*I)*ArcSin[c*x])] - 12*b^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/(24*d)","A",0
189,1,391,238,0.6845614,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^2 \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)),x]","-\frac{a^2 c \log (1-c x)-a^2 c \log (c x+1)+\frac{2 a^2}{x}+4 a b c \left(-i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+\frac{\sin ^{-1}(c x)}{c x}+\sin ^{-1}(c x) \left(-\log \left(1-i e^{i \sin ^{-1}(c x)}\right)\right)+\sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-\log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)+2 b^2 c \left(-2 i \sin ^{-1}(c x) \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+2 i \sin ^{-1}(c x) \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-2 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+2 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+2 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)-2 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)+\frac{\sin ^{-1}(c x)^2}{c x}+\sin ^{-1}(c x)^2 \left(-\log \left(1-i e^{i \sin ^{-1}(c x)}\right)\right)+\sin ^{-1}(c x)^2 \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-2 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+2 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{2 d}","\frac{2 i b c \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d}-\frac{2 i b c \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{d x}-\frac{2 i c \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d}-\frac{4 b c \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d}+\frac{2 i b^2 c \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{d}-\frac{2 i b^2 c \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{d}-\frac{2 b^2 c \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{d}+\frac{2 b^2 c \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{d}",1,"-1/2*((2*a^2)/x + a^2*c*Log[1 - c*x] - a^2*c*Log[1 + c*x] + 4*a*b*c*(ArcSin[c*x]/(c*x) - ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + Log[Cos[ArcSin[c*x]/2]] - Log[Sin[ArcSin[c*x]/2]] - I*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + I*PolyLog[2, I*E^(I*ArcSin[c*x])]) + 2*b^2*c*(ArcSin[c*x]^2/(c*x) - 2*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - ArcSin[c*x]^2*Log[1 - I*E^(I*ArcSin[c*x])] + ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] + 2*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (2*I)*PolyLog[2, -E^(I*ArcSin[c*x])] - (2*I)*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (2*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] + (2*I)*PolyLog[2, E^(I*ArcSin[c*x])] + 2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 2*PolyLog[3, I*E^(I*ArcSin[c*x])]))/d","A",0
190,1,353,210,1.1861803,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^3 \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)),x]","-\frac{a^2 c^2 \log \left(1-c^2 x^2\right)-2 a^2 c^2 \log (x)+\frac{a^2}{x^2}+2 a b c^2 \left(\frac{\sqrt{1-c^2 x^2}}{c x}+\frac{\sin ^{-1}(c x)}{c^2 x^2}-i \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+i \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-2 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+2 \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)+2 b^2 c^2 \left(\frac{\sin ^{-1}(c x)^2}{2 c^2 x^2}+\frac{\sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c x}-i \sin ^{-1}(c x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c x)}\right)-i \sin ^{-1}(c x) \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{-2 i \sin ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)-\log (c x)-\frac{2}{3} i \sin ^{-1}(c x)^3-\sin ^{-1}(c x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c x)}\right)+\sin ^{-1}(c x)^2 \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)+\frac{i \pi ^3}{24}\right)}{2 d}","\frac{i b c^2 \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d}-\frac{i b c^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d}-\frac{b c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{d x}-\frac{2 c^2 \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{2 d x^2}-\frac{b^2 c^2 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 d}+\frac{b^2 c^2 \text{Li}_3\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d}+\frac{b^2 c^2 \log (x)}{d}",1,"-1/2*(a^2/x^2 - 2*a^2*c^2*Log[x] + a^2*c^2*Log[1 - c^2*x^2] + 2*a*b*c^2*(Sqrt[1 - c^2*x^2]/(c*x) + ArcSin[c*x]/(c^2*x^2) - 2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 2*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] - I*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + I*PolyLog[2, E^((2*I)*ArcSin[c*x])]) + 2*b^2*c^2*((I/24)*Pi^3 + (Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*x) + ArcSin[c*x]^2/(2*c^2*x^2) - ((2*I)/3)*ArcSin[c*x]^3 - ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] + ArcSin[c*x]^2*Log[1 + E^((2*I)*ArcSin[c*x])] - Log[c*x] - I*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] - I*ArcSin[c*x]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - PolyLog[3, E^((-2*I)*ArcSin[c*x])]/2 + PolyLog[3, -E^((2*I)*ArcSin[c*x])]/2))/d","A",0
191,1,849,333,7.8442503,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^4 \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)),x]","-\frac{a^2 \log (1-c x) c^3}{2 d}+\frac{a^2 \log (c x+1) c^3}{2 d}-\frac{b^2 \left(\frac{1}{2} c x \sin ^{-1}(c x)^2 \csc ^4\left(\frac{1}{2} \sin ^{-1}(c x)\right)+2 \sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+\frac{8 \sin ^{-1}(c x)^2 \sin ^4\left(\frac{1}{2} \sin ^{-1}(c x)\right)}{c^3 x^3}-2 \sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+14 \sin ^{-1}(c x)^2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)+4 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)-56 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-24 \sin ^{-1}(c x)^2 \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+24 \sin ^{-1}(c x)^2 \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+56 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)-56 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-48 i \sin ^{-1}(c x) \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+48 i \sin ^{-1}(c x) \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+56 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+48 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)-48 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)+14 \sin ^{-1}(c x)^2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)+4 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) c^3}{24 d}-\frac{a^2 c^2}{d x}-\frac{2 a b \left(\frac{1}{2} \left(-\frac{i \sin ^{-1}(c x)^2}{2 c}+\frac{2 \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)}{c}+\frac{3 i \pi  \sin ^{-1}(c x)}{2 c}+\frac{2 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)}{c}-\frac{\pi  \log \left(1+i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{2 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{c}+\frac{\pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{c}-\frac{2 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c}\right) c^4-\frac{1}{2} \left(-\frac{i \sin ^{-1}(c x)^2}{2 c}+\frac{2 \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)}{c}+\frac{i \pi  \sin ^{-1}(c x)}{2 c}+\frac{2 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)}{c}+\frac{\pi  \log \left(1-i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{2 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{c}-\frac{\pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{c}-\frac{2 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c}\right) c^4-\left(-\frac{\sin ^{-1}(c x)}{x}-c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)\right) c^2+\frac{c^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right) x^3+c \sqrt{1-c^2 x^2} x+2 \sin ^{-1}(c x)}{6 x^3}\right)}{d}-\frac{a^2}{3 d x^3}","\frac{2 i b c^3 \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d}-\frac{2 i b c^3 \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d}-\frac{2 i c^3 \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d}-\frac{14 b c^3 \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 d}-\frac{b c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 d x^2}-\frac{c^2 \left(a+b \sin ^{-1}(c x)\right)^2}{d x}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{3 d x^3}+\frac{7 i b^2 c^3 \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{3 d}-\frac{7 i b^2 c^3 \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{3 d}-\frac{2 b^2 c^3 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{d}+\frac{2 b^2 c^3 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{d}-\frac{b^2 c^2}{3 d x}",1,"-1/3*a^2/(d*x^3) - (a^2*c^2)/(d*x) - (a^2*c^3*Log[1 - c*x])/(2*d) + (a^2*c^3*Log[1 + c*x])/(2*d) - (2*a*b*(-(c^2*(-(ArcSin[c*x]/x) - c*ArcTanh[Sqrt[1 - c^2*x^2]])) + (c*x*Sqrt[1 - c^2*x^2] + 2*ArcSin[c*x] + c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*x^3) + (c^4*((((3*I)/2)*Pi*ArcSin[c*x])/c - ((I/2)*ArcSin[c*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])])/c - (Pi*Log[1 + I*E^(I*ArcSin[c*x])])/c + (2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c + (Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]])/c - ((2*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/c))/2 - (c^4*(((I/2)*Pi*ArcSin[c*x])/c - ((I/2)*ArcSin[c*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])])/c + (Pi*Log[1 - I*E^(I*ArcSin[c*x])])/c + (2*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c - (Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]])/c - ((2*I)*PolyLog[2, I*E^(I*ArcSin[c*x])])/c))/2))/d - (b^2*c^3*(4*Cot[ArcSin[c*x]/2] + 14*ArcSin[c*x]^2*Cot[ArcSin[c*x]/2] + 2*ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 + (c*x*ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^4)/2 - 56*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - 24*ArcSin[c*x]^2*Log[1 - I*E^(I*ArcSin[c*x])] + 24*ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] + 56*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (56*I)*PolyLog[2, -E^(I*ArcSin[c*x])] - (48*I)*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (48*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] + (56*I)*PolyLog[2, E^(I*ArcSin[c*x])] + 48*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 48*PolyLog[3, I*E^(I*ArcSin[c*x])] - 2*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 + (8*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2]^4)/(c^3*x^3) + 4*Tan[ArcSin[c*x]/2] + 14*ArcSin[c*x]^2*Tan[ArcSin[c*x]/2]))/(24*d)","B",0
192,1,614,300,3.1382223,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^2,x]","\frac{-\frac{2 a^2 c x}{c^2 x^2-1}+4 a^2 c x+3 a^2 \log (1-c x)-3 a^2 \log (c x+1)+8 a b \sqrt{1-c^2 x^2}+\frac{2 a b \sqrt{1-c^2 x^2}}{c x-1}-\frac{2 a b \sqrt{1-c^2 x^2}}{c x+1}-12 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+12 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)+8 a b c x \sin ^{-1}(c x)-\frac{2 a b \sin ^{-1}(c x)}{c x-1}-\frac{2 a b \sin ^{-1}(c x)}{c x+1}+6 i \pi  a b \sin ^{-1}(c x)-12 a b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-6 \pi  a b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+12 a b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-6 \pi  a b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+6 \pi  a b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+6 \pi  a b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+\frac{8 b^2 c x}{c^2 x^2-1}-\frac{6 b^2 c^2 x^2 \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{6 b^2 c x \sin ^{-1}(c x)^2}{1-c^2 x^2}+2 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+\frac{2 b^2 \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{8 b^2 c^3 x^3}{1-c^2 x^2}+\frac{4 b^2 c^3 x^3 \sin ^{-1}(c x)^2}{c^2 x^2-1}+12 b^2 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)-12 b^2 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)+4 b^2 \tanh ^{-1}(c x)+12 i b^2 \sin ^{-1}(c x)^2 \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right)}{4 c^5 d^2}","-\frac{3 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^5 d^2}+\frac{3 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^5 d^2}+\frac{3 i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^5 d^2}+\frac{3 x \left(a+b \sin ^{-1}(c x)\right)^2}{2 c^4 d^2}+\frac{x^3 \left(a+b \sin ^{-1}(c x)\right)^2}{2 c^2 d^2 \left(1-c^2 x^2\right)}+\frac{2 b \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c^5 d^2}-\frac{b \left(a+b \sin ^{-1}(c x)\right)}{c^5 d^2 \sqrt{1-c^2 x^2}}+\frac{3 b^2 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{c^5 d^2}-\frac{3 b^2 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{c^5 d^2}+\frac{b^2 \tanh ^{-1}(c x)}{c^5 d^2}-\frac{2 b^2 x}{c^4 d^2}",1,"(4*a^2*c*x + (8*b^2*c^3*x^3)/(1 - c^2*x^2) + 8*a*b*Sqrt[1 - c^2*x^2] + (2*a*b*Sqrt[1 - c^2*x^2])/(-1 + c*x) - (2*a*b*Sqrt[1 - c^2*x^2])/(1 + c*x) - (2*a^2*c*x)/(-1 + c^2*x^2) + (8*b^2*c*x)/(-1 + c^2*x^2) + (6*I)*a*b*Pi*ArcSin[c*x] + 8*a*b*c*x*ArcSin[c*x] - (2*a*b*ArcSin[c*x])/(-1 + c*x) - (2*a*b*ArcSin[c*x])/(1 + c*x) + (2*b^2*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (6*b^2*c^2*x^2*ArcSin[c*x])/Sqrt[1 - c^2*x^2] + 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + (6*b^2*c*x*ArcSin[c*x]^2)/(1 - c^2*x^2) + (4*b^2*c^3*x^3*ArcSin[c*x]^2)/(-1 + c^2*x^2) + (12*I)*b^2*ArcSin[c*x]^2*ArcTan[E^(I*ArcSin[c*x])] + 4*b^2*ArcTanh[c*x] - 6*a*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 12*a*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 6*a*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 12*a*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 3*a^2*Log[1 - c*x] - 3*a^2*Log[1 + c*x] + 6*a*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 6*a*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (12*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (12*I)*b*(a + b*ArcSin[c*x])*PolyLog[2, I*E^(I*ArcSin[c*x])] + 12*b^2*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 12*b^2*PolyLog[3, I*E^(I*ArcSin[c*x])])/(4*c^5*d^2)","B",0
193,1,502,227,1.0669057,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^2,x]","\frac{-\frac{3 a^2}{c^2 x^2-1}+3 a^2 \log \left(1-c^2 x^2\right)+\frac{3 a b \sqrt{1-c^2 x^2}}{c x-1}+\frac{3 a b \sqrt{1-c^2 x^2}}{c x+1}-12 i a b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-12 i a b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-6 i a b \sin ^{-1}(c x)^2-\frac{3 a b \sin ^{-1}(c x)}{c x-1}+\frac{3 a b \sin ^{-1}(c x)}{c x+1}+12 i \pi  a b \sin ^{-1}(c x)+12 a b \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+12 a b \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+24 \pi  a b \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+6 \pi  a b \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-6 \pi  a b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-6 \pi  a b \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-24 \pi  a b \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+6 \pi  a b \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-3 b^2 \log \left(1-c^2 x^2\right)+\frac{3 b^2 \sin ^{-1}(c x)^2}{1-c^2 x^2}-\frac{6 b^2 c x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-6 i b^2 \sin ^{-1}(c x) \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+3 b^2 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)-2 i b^2 \sin ^{-1}(c x)^3+6 b^2 \sin ^{-1}(c x)^2 \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)}{6 c^4 d^2}","-\frac{i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^4 d^2}-\frac{i \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c^4 d^2}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{2 c^4 d^2}+\frac{\log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^4 d^2}+\frac{x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{b x \left(a+b \sin ^{-1}(c x)\right)}{c^3 d^2 \sqrt{1-c^2 x^2}}+\frac{b^2 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 c^4 d^2}-\frac{b^2 \log \left(1-c^2 x^2\right)}{2 c^4 d^2}",1,"((3*a*b*Sqrt[1 - c^2*x^2])/(-1 + c*x) + (3*a*b*Sqrt[1 - c^2*x^2])/(1 + c*x) - (3*a^2)/(-1 + c^2*x^2) + (12*I)*a*b*Pi*ArcSin[c*x] - (3*a*b*ArcSin[c*x])/(-1 + c*x) + (3*a*b*ArcSin[c*x])/(1 + c*x) - (6*b^2*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (6*I)*a*b*ArcSin[c*x]^2 + (3*b^2*ArcSin[c*x]^2)/(1 - c^2*x^2) - (2*I)*b^2*ArcSin[c*x]^3 + 24*a*b*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 6*a*b*Pi*Log[1 - I*E^(I*ArcSin[c*x])] + 12*a*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 6*a*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] + 12*a*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 6*b^2*ArcSin[c*x]^2*Log[1 + E^((2*I)*ArcSin[c*x])] + 3*a^2*Log[1 - c^2*x^2] - 3*b^2*Log[1 - c^2*x^2] - 24*a*b*Pi*Log[Cos[ArcSin[c*x]/2]] + 6*a*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 6*a*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (12*I)*a*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (12*I)*a*b*PolyLog[2, I*E^(I*ArcSin[c*x])] - (6*I)*b^2*ArcSin[c*x]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + 3*b^2*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/(6*c^4*d^2)","B",0
194,1,383,233,2.6447915,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^2,x]","-\frac{\frac{2 a^2 c x}{c^2 x^2-1}+a^2 (-\log (1-c x))+a^2 \log (c x+1)+\frac{2 a b \left(-2 \sqrt{1-c^2 x^2}+\cos \left(2 \sin ^{-1}(c x)\right)+\sin ^{-1}(c x) \left(2 c x-\log \left(1-i e^{i \sin ^{-1}(c x)}\right)+\log \left(1+i e^{i \sin ^{-1}(c x)}\right)+\left(\log \left(1+i e^{i \sin ^{-1}(c x)}\right)-\log \left(1-i e^{i \sin ^{-1}(c x)}\right)\right) \cos \left(2 \sin ^{-1}(c x)\right)\right)+1\right)}{c^2 x^2-1}+4 i a b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-4 i a b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+\frac{2 b^2 \sin ^{-1}(c x) \left(c x \sin ^{-1}(c x)-2 \sqrt{1-c^2 x^2}\right)}{c^2 x^2-1}-4 b^2 \left(-i \sin ^{-1}(c x) \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+i \sin ^{-1}(c x) \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+\text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)-\text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)+\tanh ^{-1}(c x)+i \sin ^{-1}(c x)^2 \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right)\right)}{4 c^3 d^2}","-\frac{i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^3 d^2}+\frac{i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c^3 d^2}+\frac{i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^3 d^2}+\frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{b \left(a+b \sin ^{-1}(c x)\right)}{c^3 d^2 \sqrt{1-c^2 x^2}}+\frac{b^2 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{c^3 d^2}-\frac{b^2 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{c^3 d^2}+\frac{b^2 \tanh ^{-1}(c x)}{c^3 d^2}",1,"-1/4*((2*a^2*c*x)/(-1 + c^2*x^2) + (2*b^2*ArcSin[c*x]*(-2*Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x]))/(-1 + c^2*x^2) + (2*a*b*(1 - 2*Sqrt[1 - c^2*x^2] + Cos[2*ArcSin[c*x]] + ArcSin[c*x]*(2*c*x - Log[1 - I*E^(I*ArcSin[c*x])] + Log[1 + I*E^(I*ArcSin[c*x])] + Cos[2*ArcSin[c*x]]*(-Log[1 - I*E^(I*ArcSin[c*x])] + Log[1 + I*E^(I*ArcSin[c*x])]))))/(-1 + c^2*x^2) - a^2*Log[1 - c*x] + a^2*Log[1 + c*x] + (4*I)*a*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (4*I)*a*b*PolyLog[2, I*E^(I*ArcSin[c*x])] - 4*b^2*(I*ArcSin[c*x]^2*ArcTan[E^(I*ArcSin[c*x])] + ArcTanh[c*x] - I*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + I*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] + PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - PolyLog[3, I*E^(I*ArcSin[c*x])]))/(c^3*d^2)","A",0
195,1,75,89,0.1910718,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^2,x]","-\frac{\frac{2 b c x \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{c^2 x^2-1}+b^2 \log \left(1-c^2 x^2\right)}{2 c^2 d^2}","-\frac{b x \left(a+b \sin ^{-1}(c x)\right)}{c d^2 \sqrt{1-c^2 x^2}}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{b^2 \log \left(1-c^2 x^2\right)}{2 c^2 d^2}",1,"-1/2*((2*b*c*x*(a + b*ArcSin[c*x]))/Sqrt[1 - c^2*x^2] + (a + b*ArcSin[c*x])^2/(-1 + c^2*x^2) + b^2*Log[1 - c^2*x^2])/(c^2*d^2)","A",1
196,1,359,230,2.6218728,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^2,x]","\frac{-\frac{2 a^2 x}{c^2 x^2-1}-\frac{a^2 \log (1-c x)}{c}+\frac{a^2 \log (c x+1)}{c}+\frac{2 a b \left(\frac{2 \left(c^2 x^2+\sqrt{1-c^2 x^2}+\sin ^{-1}(c x) \left(\left(c^2 x^2-1\right) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+\left(1-c^2 x^2\right) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-c x\right)-1\right)}{c^2 x^2-1}+2 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-2 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)\right)}{c}+\frac{4 b^2 \left(\frac{c x \sin ^{-1}(c x)^2}{2-2 c^2 x^2}-\frac{\sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+i \sin ^{-1}(c x) \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-i \sin ^{-1}(c x) \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-\text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)+\text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)+\tanh ^{-1}(c x)-i \sin ^{-1}(c x)^2 \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right)\right)}{c}}{4 d^2}","-\frac{b \left(a+b \sin ^{-1}(c x)\right)}{c d^2 \sqrt{1-c^2 x^2}}+\frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{2 d^2 \left(1-c^2 x^2\right)}+\frac{i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c d^2}-\frac{i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c d^2}-\frac{i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c d^2}-\frac{b^2 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{c d^2}+\frac{b^2 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{c d^2}+\frac{b^2 \tanh ^{-1}(c x)}{c d^2}",1,"((-2*a^2*x)/(-1 + c^2*x^2) - (a^2*Log[1 - c*x])/c + (a^2*Log[1 + c*x])/c + (2*a*b*((2*(-1 + c^2*x^2 + Sqrt[1 - c^2*x^2] + ArcSin[c*x]*(-(c*x) + (-1 + c^2*x^2)*Log[1 - I*E^(I*ArcSin[c*x])] + (1 - c^2*x^2)*Log[1 + I*E^(I*ArcSin[c*x])])))/(-1 + c^2*x^2) + (2*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (2*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]))/c + (4*b^2*(-(ArcSin[c*x]/Sqrt[1 - c^2*x^2]) + (c*x*ArcSin[c*x]^2)/(2 - 2*c^2*x^2) - I*ArcSin[c*x]^2*ArcTan[E^(I*ArcSin[c*x])] + ArcTanh[c*x] + I*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - I*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] - PolyLog[3, (-I)*E^(I*ArcSin[c*x])] + PolyLog[3, I*E^(I*ArcSin[c*x])]))/c)/(4*d^2)","A",0
197,1,365,211,1.3246381,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x*(d - c^2*d*x^2)^2),x]","\frac{\frac{a^2}{1-c^2 x^2}-a^2 \log \left(1-c^2 x^2\right)+2 a^2 \log (c x)+2 a b \left(-\frac{c x}{\sqrt{1-c^2 x^2}}+\frac{\sin ^{-1}(c x)}{1-c^2 x^2}+i \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)-i \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+2 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-2 \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)+2 b^2 \left(-\frac{1}{2} \log \left(1-c^2 x^2\right)+\frac{\sin ^{-1}(c x)^2}{2-2 c^2 x^2}-\frac{c x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+i \sin ^{-1}(c x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c x)}\right)+i \sin ^{-1}(c x) \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(e^{-2 i \sin ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)+\frac{2}{3} i \sin ^{-1}(c x)^3+\sin ^{-1}(c x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c x)}\right)-\sin ^{-1}(c x)^2 \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)-\frac{i \pi ^3}{24}\right)}{2 d^2}","-\frac{b c x \left(a+b \sin ^{-1}(c x)\right)}{d^2 \sqrt{1-c^2 x^2}}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{2 d^2 \left(1-c^2 x^2\right)}+\frac{i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}-\frac{i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}-\frac{2 \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d^2}-\frac{b^2 \log \left(1-c^2 x^2\right)}{2 d^2}-\frac{b^2 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 d^2}+\frac{b^2 \text{Li}_3\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d^2}",1,"(a^2/(1 - c^2*x^2) + 2*a^2*Log[c*x] - a^2*Log[1 - c^2*x^2] + 2*a*b*(-((c*x)/Sqrt[1 - c^2*x^2]) + ArcSin[c*x]/(1 - c^2*x^2) + 2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - 2*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] + I*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - I*PolyLog[2, E^((2*I)*ArcSin[c*x])]) + 2*b^2*((-1/24*I)*Pi^3 - (c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] + ArcSin[c*x]^2/(2 - 2*c^2*x^2) + ((2*I)/3)*ArcSin[c*x]^3 + ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] - ArcSin[c*x]^2*Log[1 + E^((2*I)*ArcSin[c*x])] - Log[1 - c^2*x^2]/2 + I*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] + I*ArcSin[c*x]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + PolyLog[3, E^((-2*I)*ArcSin[c*x])]/2 - PolyLog[3, -E^((2*I)*ArcSin[c*x])]/2))/(2*d^2)","A",0
198,1,1059,324,9.7693491,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^2 \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)^2),x]","-\frac{3 c \log (1-c x) a^2}{4 d^2}+\frac{3 c \log (c x+1) a^2}{4 d^2}-\frac{a^2}{d^2 x}-\frac{c^2 x a^2}{2 d^2 \left(c^2 x^2-1\right)}+\frac{b c \left(-2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+6 \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-6 \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+\frac{\sin ^{-1}(c x)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-\frac{\sin ^{-1}(c x)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-4 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+4 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+6 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-6 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-\frac{2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+\frac{2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}\right) a}{2 d^2}+\frac{b^2 c \left(-2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2+6 \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)^2-6 \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)^2-6 \log \left(\left(\frac{1}{2}+\frac{i}{2}\right) e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left(-i+e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)^2+6 \log \left(\frac{1}{2} e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left((1+i)+(1-i) e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)^2+6 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^2-6 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^2-2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2+\frac{\sin ^{-1}(c x)^2}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-\frac{\sin ^{-1}(c x)^2}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}+8 \log \left(1-e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+6 \pi  \log \left(\frac{1}{2} \sqrt[4]{-1} e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left(1-i e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)+6 \pi  \log \left(-\frac{1}{2} \sqrt[4]{-1} e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left(-i+e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)-8 \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-6 \pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right) \sin ^{-1}(c x)-6 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right) \sin ^{-1}(c x)+12 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-12 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-\frac{4 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+\frac{4 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}-4 \sin ^{-1}(c x)-4 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+4 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+8 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-8 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-12 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)+12 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)\right)}{4 d^2}","-\frac{b c \left(a+b \sin ^{-1}(c x)\right)}{d^2 \sqrt{1-c^2 x^2}}+\frac{3 c^2 x \left(a+b \sin ^{-1}(c x)\right)^2}{2 d^2 \left(1-c^2 x^2\right)}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{d^2 x \left(1-c^2 x^2\right)}+\frac{3 i b c \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}-\frac{3 i b c \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}-\frac{3 i c \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d^2}-\frac{4 b c \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}+\frac{2 i b^2 c \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{d^2}-\frac{2 i b^2 c \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{d^2}-\frac{3 b^2 c \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{d^2}+\frac{3 b^2 c \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{d^2}+\frac{b^2 c \tanh ^{-1}(c x)}{d^2}",1,"-(a^2/(d^2*x)) - (a^2*c^2*x)/(2*d^2*(-1 + c^2*x^2)) - (3*a^2*c*Log[1 - c*x])/(4*d^2) + (3*a^2*c*Log[1 + c*x])/(4*d^2) + (a*b*c*(-2*ArcSin[c*x]*Cot[ArcSin[c*x]/2] + 6*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 6*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - 4*Log[Cos[ArcSin[c*x]/2]] + 4*Log[Sin[ArcSin[c*x]/2]] + (6*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (6*I)*PolyLog[2, I*E^(I*ArcSin[c*x])] + ArcSin[c*x]/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^2 - (2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - ArcSin[c*x]/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 + (2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 2*ArcSin[c*x]*Tan[ArcSin[c*x]/2]))/(2*d^2) + (b^2*c*(-4*ArcSin[c*x] - 2*ArcSin[c*x]^2*Cot[ArcSin[c*x]/2] + 8*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 6*ArcSin[c*x]^2*Log[1 - I*E^(I*ArcSin[c*x])] + 6*Pi*ArcSin[c*x]*Log[((-1)^(1/4)*(1 - I*E^(I*ArcSin[c*x])))/(2*E^((I/2)*ArcSin[c*x]))] - 6*ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] - 6*ArcSin[c*x]^2*Log[((1/2 + I/2)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])] + 6*Pi*ArcSin[c*x]*Log[-1/2*((-1)^(1/4)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])] - 8*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + 6*ArcSin[c*x]^2*Log[((1 + I) + (1 - I)*E^(I*ArcSin[c*x]))/(2*E^((I/2)*ArcSin[c*x]))] - 6*Pi*ArcSin[c*x]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 4*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 6*ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 4*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 6*ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 6*Pi*ArcSin[c*x]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (8*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (12*I)*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (12*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] - (8*I)*PolyLog[2, E^(I*ArcSin[c*x])] - 12*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] + 12*PolyLog[3, I*E^(I*ArcSin[c*x])] + ArcSin[c*x]^2/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^2 - (4*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - ArcSin[c*x]^2/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 + (4*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 2*ArcSin[c*x]^2*Tan[ArcSin[c*x]/2]))/(4*d^2)","B",0
199,1,430,270,1.6233226,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^3 \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^2),x]","\frac{\frac{a^2 c^2}{1-c^2 x^2}-2 a^2 c^2 \log \left(1-c^2 x^2\right)+4 a^2 c^2 \log (x)-\frac{a^2}{x^2}+2 a b \left(2 c^2 \left(i \left(\text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)\right)+2 \sin ^{-1}(c x) \left(\log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-\log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)\right)-\frac{c \sqrt{1-c^2 x^2}}{x}+\frac{c^2 \sin ^{-1}(c x)}{1-c^2 x^2}-\frac{c^3 x}{\sqrt{1-c^2 x^2}}-\frac{\sin ^{-1}(c x)}{x^2}\right)+b^2 c^2 \left(2 \log \left(\frac{c x}{\sqrt{1-c^2 x^2}}\right)+\frac{\sin ^{-1}(c x)^2}{1-c^2 x^2}-\frac{\sin ^{-1}(c x)^2}{c^2 x^2}-\frac{2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c x}-\frac{2 c x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+4 i \sin ^{-1}(c x) \left(\text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)\right)+2 \left(\text{Li}_3\left(e^{2 i \sin ^{-1}(c x)}\right)-\text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)\right)+4 \sin ^{-1}(c x)^2 \left(\log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-\log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)\right)}{2 d^2}","\frac{2 i b c^2 \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}-\frac{2 i b c^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}+\frac{c^2 \left(a+b \sin ^{-1}(c x)\right)^2}{d^2 \left(1-c^2 x^2\right)}-\frac{b c \left(a+b \sin ^{-1}(c x)\right)}{d^2 x \sqrt{1-c^2 x^2}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{2 d^2 x^2 \left(1-c^2 x^2\right)}-\frac{4 c^2 \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d^2}-\frac{b^2 c^2 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)}{d^2}+\frac{b^2 c^2 \text{Li}_3\left(e^{2 i \sin ^{-1}(c x)}\right)}{d^2}-\frac{b^2 c^2 \log \left(1-c^2 x^2\right)}{2 d^2}+\frac{b^2 c^2 \log (x)}{d^2}",1,"(-(a^2/x^2) + (a^2*c^2)/(1 - c^2*x^2) + 4*a^2*c^2*Log[x] - 2*a^2*c^2*Log[1 - c^2*x^2] + 2*a*b*(-((c^3*x)/Sqrt[1 - c^2*x^2]) - (c*Sqrt[1 - c^2*x^2])/x - ArcSin[c*x]/x^2 + (c^2*ArcSin[c*x])/(1 - c^2*x^2) + 2*c^2*(2*ArcSin[c*x]*(Log[1 - E^((2*I)*ArcSin[c*x])] - Log[1 + E^((2*I)*ArcSin[c*x])]) + I*(PolyLog[2, -E^((2*I)*ArcSin[c*x])] - PolyLog[2, E^((2*I)*ArcSin[c*x])]))) + b^2*c^2*((-2*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (2*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*x) - ArcSin[c*x]^2/(c^2*x^2) + ArcSin[c*x]^2/(1 - c^2*x^2) + 4*ArcSin[c*x]^2*(Log[1 - E^((2*I)*ArcSin[c*x])] - Log[1 + E^((2*I)*ArcSin[c*x])]) + 2*Log[(c*x)/Sqrt[1 - c^2*x^2]] + (4*I)*ArcSin[c*x]*(PolyLog[2, -E^((2*I)*ArcSin[c*x])] - PolyLog[2, E^((2*I)*ArcSin[c*x])]) + 2*(-PolyLog[3, -E^((2*I)*ArcSin[c*x])] + PolyLog[3, E^((2*I)*ArcSin[c*x])])))/(2*d^2)","A",0
200,1,1514,439,12.9517805,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^4 \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^2),x]","-\frac{a^2 x c^4}{2 d^2 \left(c^2 x^2-1\right)}-\frac{5 a^2 \log (1-c x) c^3}{4 d^2}+\frac{5 a^2 \log (c x+1) c^3}{4 d^2}+\frac{b^2 \left(\frac{5}{6} \sin ^{-1}(c x)^3-\frac{1}{24} \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2-\frac{1}{24} \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right) \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2+\frac{\sin ^{-1}(c x)^2}{4 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-\frac{\sin ^{-1}(c x)^2}{4 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-\frac{1}{12} \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+\frac{1}{12} \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-\frac{\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+\frac{\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+\frac{1}{12} \left(-13 \cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2-2 \cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \csc \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\frac{26}{3} \left(\frac{1}{8} i \sin ^{-1}(c x)^2-\frac{1}{2} \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+\frac{1}{2} i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)\right)+\frac{26}{3} \left(\frac{1}{2} \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-\frac{1}{2} i \left(\frac{1}{4} \sin ^{-1}(c x)^2+\text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right)\right)+\frac{1}{6} \left(-5 \sin ^{-1}(c x)^3+15 \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)^2-15 \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)^2-15 \log \left(\left(\frac{1}{2}+\frac{i}{2}\right) e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left(-i+e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)^2+15 \log \left(\frac{1}{2} e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left((1+i)+(1-i) e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)^2+15 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^2-15 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^2+15 \pi  \log \left(\frac{1}{2} \sqrt[4]{-1} e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left(1-i e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)+15 \pi  \log \left(-\frac{1}{2} \sqrt[4]{-1} e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left(-i+e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)-15 \pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right) \sin ^{-1}(c x)-15 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right) \sin ^{-1}(c x)+30 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-30 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-6 \sin ^{-1}(c x)-6 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+6 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-30 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)+30 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)\right)+\frac{1}{12} \sec \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(-13 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2-2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right) c^3}{d^2}-\frac{2 a^2 c^2}{d^2 x}+\frac{2 a b \left(-\frac{\left(\sin ^{-1}(c x)+\sqrt{1-c^2 x^2}\right) c^4}{4 \left(x c^2+c\right)}-\frac{5}{4} \left(-\frac{i \sin ^{-1}(c x)^2}{2 c}+\frac{2 \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)}{c}+\frac{3 i \pi  \sin ^{-1}(c x)}{2 c}+\frac{2 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)}{c}-\frac{\pi  \log \left(1+i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{2 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{c}+\frac{\pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{c}-\frac{2 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c}\right) c^4+\frac{5}{4} \left(-\frac{i \sin ^{-1}(c x)^2}{2 c}+\frac{2 \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)}{c}+\frac{i \pi  \sin ^{-1}(c x)}{2 c}+\frac{2 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)}{c}+\frac{\pi  \log \left(1-i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{2 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{c}-\frac{\pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{c}-\frac{2 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c}\right) c^4+\frac{\left(\sqrt{1-c^2 x^2}-\sin ^{-1}(c x)\right) c^3}{4 (c x-1)}+2 \left(-\frac{\sin ^{-1}(c x)}{x}-c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)\right) c^2-\frac{c^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right) x^3+c \sqrt{1-c^2 x^2} x+2 \sin ^{-1}(c x)}{6 x^3}\right)}{d^2}-\frac{a^2}{3 d^2 x^3}","\frac{5 i b c^3 \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}-\frac{5 i b c^3 \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}-\frac{5 i c^3 \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d^2}-\frac{26 b c^3 \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 d^2}-\frac{5 c^2 \left(a+b \sin ^{-1}(c x)\right)^2}{3 d^2 x \left(1-c^2 x^2\right)}-\frac{b c \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 x^2 \sqrt{1-c^2 x^2}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{3 d^2 x^3 \left(1-c^2 x^2\right)}+\frac{5 c^4 x \left(a+b \sin ^{-1}(c x)\right)^2}{2 d^2 \left(1-c^2 x^2\right)}-\frac{2 b c^3 \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 \sqrt{1-c^2 x^2}}+\frac{13 i b^2 c^3 \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{3 d^2}-\frac{13 i b^2 c^3 \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{3 d^2}-\frac{5 b^2 c^3 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{d^2}+\frac{5 b^2 c^3 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{d^2}+\frac{b^2 c^3 \tanh ^{-1}(c x)}{d^2}-\frac{b^2 c^2}{3 d^2 x}",1,"-1/3*a^2/(d^2*x^3) - (2*a^2*c^2)/(d^2*x) - (a^2*c^4*x)/(2*d^2*(-1 + c^2*x^2)) - (5*a^2*c^3*Log[1 - c*x])/(4*d^2) + (5*a^2*c^3*Log[1 + c*x])/(4*d^2) + (2*a*b*((c^3*(Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(4*(-1 + c*x)) - (c^4*(Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/(4*(c + c^2*x)) + 2*c^2*(-(ArcSin[c*x]/x) - c*ArcTanh[Sqrt[1 - c^2*x^2]]) - (c*x*Sqrt[1 - c^2*x^2] + 2*ArcSin[c*x] + c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*x^3) - (5*c^4*((((3*I)/2)*Pi*ArcSin[c*x])/c - ((I/2)*ArcSin[c*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])])/c - (Pi*Log[1 + I*E^(I*ArcSin[c*x])])/c + (2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c + (Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]])/c - ((2*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/c))/4 + (5*c^4*(((I/2)*Pi*ArcSin[c*x])/c - ((I/2)*ArcSin[c*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])])/c + (Pi*Log[1 - I*E^(I*ArcSin[c*x])])/c + (2*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c - (Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]])/c - ((2*I)*PolyLog[2, I*E^(I*ArcSin[c*x])])/c))/4))/d^2 + (b^2*c^3*((5*ArcSin[c*x]^3)/6 + ((-2*Cos[ArcSin[c*x]/2] - 13*ArcSin[c*x]^2*Cos[ArcSin[c*x]/2])*Csc[ArcSin[c*x]/2])/12 - (ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2)/12 - (ArcSin[c*x]^2*Cot[ArcSin[c*x]/2]*Csc[ArcSin[c*x]/2]^2)/24 + (26*((I/8)*ArcSin[c*x]^2 - (ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])])/2 + (I/2)*PolyLog[2, -E^(I*ArcSin[c*x])]))/3 + (26*((ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])])/2 - (I/2)*(ArcSin[c*x]^2/4 + PolyLog[2, E^(I*ArcSin[c*x])])))/3 + (-6*ArcSin[c*x] - 5*ArcSin[c*x]^3 + 15*ArcSin[c*x]^2*Log[1 - I*E^(I*ArcSin[c*x])] + 15*Pi*ArcSin[c*x]*Log[((-1)^(1/4)*(1 - I*E^(I*ArcSin[c*x])))/(2*E^((I/2)*ArcSin[c*x]))] - 15*ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] - 15*ArcSin[c*x]^2*Log[((1/2 + I/2)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])] + 15*Pi*ArcSin[c*x]*Log[-1/2*((-1)^(1/4)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])] + 15*ArcSin[c*x]^2*Log[((1 + I) + (1 - I)*E^(I*ArcSin[c*x]))/(2*E^((I/2)*ArcSin[c*x]))] - 15*Pi*ArcSin[c*x]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 6*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 15*ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 6*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 15*ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 15*Pi*ArcSin[c*x]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (30*I)*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (30*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] - 30*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] + 30*PolyLog[3, I*E^(I*ArcSin[c*x])])/6 + (ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2)/12 + ArcSin[c*x]^2/(4*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^2) - (ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - ArcSin[c*x]^2/(4*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + (Sec[ArcSin[c*x]/2]*(-2*Sin[ArcSin[c*x]/2] - 13*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2]))/12 - (ArcSin[c*x]^2*Sec[ArcSin[c*x]/2]^2*Tan[ArcSin[c*x]/2])/24))/d^2","B",0
201,1,667,343,6.5875603,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3,x]","\frac{\frac{60 a^2 c x}{c^2 x^2-1}+\frac{24 a^2 c x}{\left(c^2 x^2-1\right)^2}-18 a^2 \log (1-c x)+18 a^2 \log (c x+1)-\frac{60 a b \left(\sqrt{1-c^2 x^2}-\sin ^{-1}(c x)\right)}{c x-1}+\frac{60 a b \left(\sqrt{1-c^2 x^2}+\sin ^{-1}(c x)\right)}{c x+1}+\frac{4 a b \left(\sqrt{1-c^2 x^2} (c x-2)+3 \sin ^{-1}(c x)\right)}{(c x-1)^2}-\frac{4 a b \left(\sqrt{1-c^2 x^2} (c x+2)+3 \sin ^{-1}(c x)\right)}{(c x+1)^2}+18 a b \left(4 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+i \sin ^{-1}(c x)^2+\sin ^{-1}(c x) \left(-4 \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-3 i \pi \right)+2 \pi  \left(-2 \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+\log \left(1+i e^{i \sin ^{-1}(c x)}\right)+2 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-\log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)\right)\right)+18 a b \left(-4 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-i \sin ^{-1}(c x)^2+\sin ^{-1}(c x) \left(4 \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+i \pi \right)+2 \pi  \left(2 \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+\log \left(1-i e^{i \sin ^{-1}(c x)}\right)-\log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-2 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)+\frac{b^2 \left(\sin ^{-1}(c x) \left(74 \sqrt{1-c^2 x^2}+30 \cos \left(3 \sin ^{-1}(c x)\right)\right)+3 \left(3 c x-5 \sin \left(3 \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^2+2 \left(c x+\sin \left(3 \sin ^{-1}(c x)\right)\right)\right)}{\left(c^2 x^2-1\right)^2}+8 b^2 \left(9 i \sin ^{-1}(c x) \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-9 i \sin ^{-1}(c x) \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-9 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)+9 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)-14 \tanh ^{-1}(c x)-9 i \sin ^{-1}(c x)^2 \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right)\right)}{96 c^5 d^3}","\frac{3 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 c^5 d^3}-\frac{3 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 c^5 d^3}-\frac{3 i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{4 c^5 d^3}+\frac{x^3 \left(a+b \sin ^{-1}(c x)\right)^2}{4 c^2 d^3 \left(1-c^2 x^2\right)^2}+\frac{5 b \left(a+b \sin ^{-1}(c x)\right)}{4 c^5 d^3 \sqrt{1-c^2 x^2}}-\frac{b \left(a+b \sin ^{-1}(c x)\right)}{6 c^5 d^3 \left(1-c^2 x^2\right)^{3/2}}-\frac{3 x \left(a+b \sin ^{-1}(c x)\right)^2}{8 c^4 d^3 \left(1-c^2 x^2\right)}-\frac{3 b^2 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{4 c^5 d^3}+\frac{3 b^2 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{4 c^5 d^3}-\frac{7 b^2 \tanh ^{-1}(c x)}{6 c^5 d^3}+\frac{b^2 x}{12 c^4 d^3 \left(1-c^2 x^2\right)}",1,"((24*a^2*c*x)/(-1 + c^2*x^2)^2 + (60*a^2*c*x)/(-1 + c^2*x^2) - (60*a*b*(Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(-1 + c*x) + (60*a*b*(Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/(1 + c*x) + (4*a*b*((-2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[c*x]))/(-1 + c*x)^2 - (4*a*b*((2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[c*x]))/(1 + c*x)^2 - 18*a^2*Log[1 - c*x] + 18*a^2*Log[1 + c*x] + 18*a*b*(I*ArcSin[c*x]^2 + ArcSin[c*x]*((-3*I)*Pi - 4*Log[1 + I*E^(I*ArcSin[c*x])]) + 2*Pi*(-2*Log[1 + E^((-I)*ArcSin[c*x])] + Log[1 + I*E^(I*ArcSin[c*x])] + 2*Log[Cos[ArcSin[c*x]/2]] - Log[-Cos[(Pi + 2*ArcSin[c*x])/4]]) + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]) + 18*a*b*((-I)*ArcSin[c*x]^2 + ArcSin[c*x]*(I*Pi + 4*Log[1 - I*E^(I*ArcSin[c*x])]) + 2*Pi*(2*Log[1 + E^((-I)*ArcSin[c*x])] + Log[1 - I*E^(I*ArcSin[c*x])] - 2*Log[Cos[ArcSin[c*x]/2]] - Log[Sin[(Pi + 2*ArcSin[c*x])/4]]) - (4*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]) + 8*b^2*((-9*I)*ArcSin[c*x]^2*ArcTan[E^(I*ArcSin[c*x])] - 14*ArcTanh[c*x] + (9*I)*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (9*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] - 9*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] + 9*PolyLog[3, I*E^(I*ArcSin[c*x])]) + (b^2*(ArcSin[c*x]*(74*Sqrt[1 - c^2*x^2] + 30*Cos[3*ArcSin[c*x]]) + 3*ArcSin[c*x]^2*(3*c*x - 5*Sin[3*ArcSin[c*x]]) + 2*(c*x + Sin[3*ArcSin[c*x]])))/(-1 + c^2*x^2)^2)/(96*c^5*d^3)","A",0
202,1,192,172,0.1911603,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(x^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3,x]","\frac{6 a^2 c^2 x^2-3 a^2+6 a b c x \sqrt{1-c^2 x^2}+2 b \sin ^{-1}(c x) \left(a \left(6 c^2 x^2-3\right)+b c x \sqrt{1-c^2 x^2} \left(3-4 c^2 x^2\right)\right)-8 a b c^3 x^3 \sqrt{1-c^2 x^2}-b^2 c^2 x^2+4 b^2 \left(c^2 x^2-1\right)^2 \log \left(1-c^2 x^2\right)+3 b^2 \left(2 c^2 x^2-1\right) \sin ^{-1}(c x)^2+b^2}{12 c^4 d^3 \left(c^2 x^2-1\right)^2}","-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{4 c^4 d^3}+\frac{x^4 \left(a+b \sin ^{-1}(c x)\right)^2}{4 d^3 \left(1-c^2 x^2\right)^2}-\frac{b x^3 \left(a+b \sin ^{-1}(c x)\right)}{6 c d^3 \left(1-c^2 x^2\right)^{3/2}}+\frac{b x \left(a+b \sin ^{-1}(c x)\right)}{2 c^3 d^3 \sqrt{1-c^2 x^2}}+\frac{b^2}{12 c^4 d^3 \left(1-c^2 x^2\right)}+\frac{b^2 \log \left(1-c^2 x^2\right)}{3 c^4 d^3}",1,"(-3*a^2 + b^2 + 6*a^2*c^2*x^2 - b^2*c^2*x^2 + 6*a*b*c*x*Sqrt[1 - c^2*x^2] - 8*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 2*b*(b*c*x*(3 - 4*c^2*x^2)*Sqrt[1 - c^2*x^2] + a*(-3 + 6*c^2*x^2))*ArcSin[c*x] + 3*b^2*(-1 + 2*c^2*x^2)*ArcSin[c*x]^2 + 4*b^2*(-1 + c^2*x^2)^2*Log[1 - c^2*x^2])/(12*c^4*d^3*(-1 + c^2*x^2)^2)","A",1
203,1,446,341,4.6469119,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3,x]","\frac{\frac{6 a^2 c x}{c^2 x^2-1}+\frac{12 a^2 c x}{\left(c^2 x^2-1\right)^2}+3 a^2 \log (1-c x)-3 a^2 \log (c x+1)+\frac{a b \left(\sqrt{1-c^2 x^2}+12 \sin ^{-1}(c x) \left(c^3 x^3-\left(c^2 x^2-1\right)^2 \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+\left(c^2 x^2-1\right)^2 \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+c x\right)-4 \cos \left(2 \sin ^{-1}(c x)\right)+3 \cos \left(3 \sin ^{-1}(c x)\right)-\cos \left(4 \sin ^{-1}(c x)\right)-3\right)}{\left(c^2 x^2-1\right)^2}-12 i a b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+12 i a b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+\frac{b^2 \left(2 \sin ^{-1}(c x) \left(\sqrt{1-c^2 x^2}+3 \cos \left(3 \sin ^{-1}(c x)\right)\right)-3 \left(\sin \left(3 \sin ^{-1}(c x)\right)-7 c x\right) \sin ^{-1}(c x)^2+2 \left(c x+\sin \left(3 \sin ^{-1}(c x)\right)\right)\right)}{2 \left(c^2 x^2-1\right)^2}+4 b^2 \left(-3 i \sin ^{-1}(c x) \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+3 i \sin ^{-1}(c x) \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+3 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)-3 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)-2 \tanh ^{-1}(c x)+3 i \sin ^{-1}(c x)^2 \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right)\right)}{48 c^3 d^3}","-\frac{i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 c^3 d^3}+\frac{i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 c^3 d^3}+\frac{i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{4 c^3 d^3}-\frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{8 c^2 d^3 \left(1-c^2 x^2\right)}+\frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{4 c^2 d^3 \left(1-c^2 x^2\right)^2}+\frac{b \left(a+b \sin ^{-1}(c x)\right)}{4 c^3 d^3 \sqrt{1-c^2 x^2}}-\frac{b \left(a+b \sin ^{-1}(c x)\right)}{6 c^3 d^3 \left(1-c^2 x^2\right)^{3/2}}+\frac{b^2 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{4 c^3 d^3}-\frac{b^2 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{4 c^3 d^3}-\frac{b^2 \tanh ^{-1}(c x)}{6 c^3 d^3}+\frac{b^2 x}{12 c^2 d^3 \left(1-c^2 x^2\right)}",1,"((12*a^2*c*x)/(-1 + c^2*x^2)^2 + (6*a^2*c*x)/(-1 + c^2*x^2) + (a*b*(-3 + Sqrt[1 - c^2*x^2] - 4*Cos[2*ArcSin[c*x]] + 3*Cos[3*ArcSin[c*x]] - Cos[4*ArcSin[c*x]] + 12*ArcSin[c*x]*(c*x + c^3*x^3 - (-1 + c^2*x^2)^2*Log[1 - I*E^(I*ArcSin[c*x])] + (-1 + c^2*x^2)^2*Log[1 + I*E^(I*ArcSin[c*x])])))/(-1 + c^2*x^2)^2 + 3*a^2*Log[1 - c*x] - 3*a^2*Log[1 + c*x] - (12*I)*a*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (12*I)*a*b*PolyLog[2, I*E^(I*ArcSin[c*x])] + 4*b^2*((3*I)*ArcSin[c*x]^2*ArcTan[E^(I*ArcSin[c*x])] - 2*ArcTanh[c*x] - (3*I)*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (3*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] + 3*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 3*PolyLog[3, I*E^(I*ArcSin[c*x])]) + (b^2*(2*ArcSin[c*x]*(Sqrt[1 - c^2*x^2] + 3*Cos[3*ArcSin[c*x]]) - 3*ArcSin[c*x]^2*(-7*c*x + Sin[3*ArcSin[c*x]]) + 2*(c*x + Sin[3*ArcSin[c*x]])))/(2*(-1 + c^2*x^2)^2))/(48*c^3*d^3)","A",0
204,1,162,150,0.2101021,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(x*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3,x]","\frac{3 a^2-6 a b c x \sqrt{1-c^2 x^2}+2 b \sin ^{-1}(c x) \left(3 a+b c x \sqrt{1-c^2 x^2} \left(2 c^2 x^2-3\right)\right)+4 a b c^3 x^3 \sqrt{1-c^2 x^2}-b^2 c^2 x^2-2 b^2 \left(c^2 x^2-1\right)^2 \log \left(1-c^2 x^2\right)+3 b^2 \sin ^{-1}(c x)^2+b^2}{12 c^2 d^3 \left(c^2 x^2-1\right)^2}","-\frac{b x \left(a+b \sin ^{-1}(c x)\right)}{3 c d^3 \sqrt{1-c^2 x^2}}-\frac{b x \left(a+b \sin ^{-1}(c x)\right)}{6 c d^3 \left(1-c^2 x^2\right)^{3/2}}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{4 c^2 d^3 \left(1-c^2 x^2\right)^2}+\frac{b^2}{12 c^2 d^3 \left(1-c^2 x^2\right)}-\frac{b^2 \log \left(1-c^2 x^2\right)}{6 c^2 d^3}",1,"(3*a^2 + b^2 - b^2*c^2*x^2 - 6*a*b*c*x*Sqrt[1 - c^2*x^2] + 4*a*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 2*b*(3*a + b*c*x*Sqrt[1 - c^2*x^2]*(-3 + 2*c^2*x^2))*ArcSin[c*x] + 3*b^2*ArcSin[c*x]^2 - 2*b^2*(-1 + c^2*x^2)^2*Log[1 - c^2*x^2])/(12*c^2*d^3*(-1 + c^2*x^2)^2)","A",1
205,1,556,332,6.019073,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^3,x]","\frac{-\frac{36 a^2 x}{c^2 x^2-1}+\frac{24 a^2 x}{\left(c^2 x^2-1\right)^2}-\frac{18 a^2 \log (1-c x)}{c}+\frac{18 a^2 \log (c x+1)}{c}+\frac{a b \left(-72 i \left(c^2 x^2-1\right)^2 \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-70 \sqrt{1-c^2 x^2}+40 \cos \left(2 \sin ^{-1}(c x)\right)-18 \cos \left(3 \sin ^{-1}(c x)\right)+10 \cos \left(4 \sin ^{-1}(c x)\right)+3 \sin ^{-1}(c x) \left(22 c x+6 \sin \left(3 \sin ^{-1}(c x)\right)+9 \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-9 \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+12 \left(\log \left(1-i e^{i \sin ^{-1}(c x)}\right)-\log \left(1+i e^{i \sin ^{-1}(c x)}\right)\right) \cos \left(2 \sin ^{-1}(c x)\right)+3 \left(\log \left(1-i e^{i \sin ^{-1}(c x)}\right)-\log \left(1+i e^{i \sin ^{-1}(c x)}\right)\right) \cos \left(4 \sin ^{-1}(c x)\right)\right)+30\right)}{c \left(c^2 x^2-1\right)^2}+\frac{72 i a b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{4 b^2 \left(\frac{2 c x}{c^2 x^2-1}+\frac{9 c x \sin ^{-1}(c x)^2}{c^2 x^2-1}-\frac{6 c x \sin ^{-1}(c x)^2}{\left(c^2 x^2-1\right)^2}+\frac{18 \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{4 \sin ^{-1}(c x)}{\left(1-c^2 x^2\right)^{3/2}}-18 i \sin ^{-1}(c x) \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+18 i \sin ^{-1}(c x) \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+18 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)-18 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)-20 \tanh ^{-1}(c x)+18 i \sin ^{-1}(c x)^2 \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right)\right)}{c}}{96 d^3}","-\frac{3 b \left(a+b \sin ^{-1}(c x)\right)}{4 c d^3 \sqrt{1-c^2 x^2}}-\frac{b \left(a+b \sin ^{-1}(c x)\right)}{6 c d^3 \left(1-c^2 x^2\right)^{3/2}}+\frac{3 x \left(a+b \sin ^{-1}(c x)\right)^2}{8 d^3 \left(1-c^2 x^2\right)}+\frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{4 d^3 \left(1-c^2 x^2\right)^2}+\frac{3 i b \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 c d^3}-\frac{3 i b \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 c d^3}-\frac{3 i \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{4 c d^3}+\frac{b^2 x}{12 d^3 \left(1-c^2 x^2\right)}-\frac{3 b^2 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{4 c d^3}+\frac{3 b^2 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{4 c d^3}+\frac{5 b^2 \tanh ^{-1}(c x)}{6 c d^3}",1,"((24*a^2*x)/(-1 + c^2*x^2)^2 - (36*a^2*x)/(-1 + c^2*x^2) - (18*a^2*Log[1 - c*x])/c + (18*a^2*Log[1 + c*x])/c + ((72*I)*a*b*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/c - (4*b^2*((2*c*x)/(-1 + c^2*x^2) + (4*ArcSin[c*x])/(1 - c^2*x^2)^(3/2) + (18*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (6*c*x*ArcSin[c*x]^2)/(-1 + c^2*x^2)^2 + (9*c*x*ArcSin[c*x]^2)/(-1 + c^2*x^2) + (18*I)*ArcSin[c*x]^2*ArcTan[E^(I*ArcSin[c*x])] - 20*ArcTanh[c*x] - (18*I)*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (18*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] + 18*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 18*PolyLog[3, I*E^(I*ArcSin[c*x])]))/c + (a*b*(30 - 70*Sqrt[1 - c^2*x^2] + 40*Cos[2*ArcSin[c*x]] - 18*Cos[3*ArcSin[c*x]] + 10*Cos[4*ArcSin[c*x]] - (72*I)*(-1 + c^2*x^2)^2*PolyLog[2, I*E^(I*ArcSin[c*x])] + 3*ArcSin[c*x]*(22*c*x + 9*Log[1 - I*E^(I*ArcSin[c*x])] + 12*Cos[2*ArcSin[c*x]]*(Log[1 - I*E^(I*ArcSin[c*x])] - Log[1 + I*E^(I*ArcSin[c*x])]) + 3*Cos[4*ArcSin[c*x]]*(Log[1 - I*E^(I*ArcSin[c*x])] - Log[1 + I*E^(I*ArcSin[c*x])]) - 9*Log[1 + I*E^(I*ArcSin[c*x])] + 6*Sin[3*ArcSin[c*x]])))/(c*(-1 + c^2*x^2)^2))/(96*d^3)","A",0
206,1,459,296,3.8122533,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x \left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x*(d - c^2*d*x^2)^3),x]","-\frac{\frac{12 a^2}{c^2 x^2-1}-\frac{6 a^2}{\left(c^2 x^2-1\right)^2}+12 a^2 \log \left(1-c^2 x^2\right)-24 a^2 \log (c x)+4 a b \left(\frac{8 c x}{\sqrt{1-c^2 x^2}}+\frac{c x}{\left(1-c^2 x^2\right)^{3/2}}+\frac{6 \sin ^{-1}(c x)}{c^2 x^2-1}-\frac{3 \sin ^{-1}(c x)}{\left(c^2 x^2-1\right)^2}-6 i \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+6 i \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-12 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+12 \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)+b^2 \left(\frac{2}{c^2 x^2-1}+16 \log \left(1-c^2 x^2\right)+\frac{12 \sin ^{-1}(c x)^2}{c^2 x^2-1}-\frac{6 \sin ^{-1}(c x)^2}{\left(c^2 x^2-1\right)^2}+\frac{32 c x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{4 c x \sin ^{-1}(c x)}{\left(1-c^2 x^2\right)^{3/2}}-24 i \sin ^{-1}(c x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c x)}\right)-24 i \sin ^{-1}(c x) \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)-12 \text{Li}_3\left(e^{-2 i \sin ^{-1}(c x)}\right)+12 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)-16 i \sin ^{-1}(c x)^3-24 \sin ^{-1}(c x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c x)}\right)+24 \sin ^{-1}(c x)^2 \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)+i \pi ^3\right)}{24 d^3}","-\frac{4 b c x \left(a+b \sin ^{-1}(c x)\right)}{3 d^3 \sqrt{1-c^2 x^2}}-\frac{b c x \left(a+b \sin ^{-1}(c x)\right)}{6 d^3 \left(1-c^2 x^2\right)^{3/2}}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{2 d^3 \left(1-c^2 x^2\right)}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{4 d^3 \left(1-c^2 x^2\right)^2}+\frac{i b \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^3}-\frac{i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^3}-\frac{2 \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d^3}+\frac{b^2}{12 d^3 \left(1-c^2 x^2\right)}-\frac{2 b^2 \log \left(1-c^2 x^2\right)}{3 d^3}-\frac{b^2 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 d^3}+\frac{b^2 \text{Li}_3\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d^3}",1,"-1/24*((-6*a^2)/(-1 + c^2*x^2)^2 + (12*a^2)/(-1 + c^2*x^2) - 24*a^2*Log[c*x] + 12*a^2*Log[1 - c^2*x^2] + 4*a*b*((c*x)/(1 - c^2*x^2)^(3/2) + (8*c*x)/Sqrt[1 - c^2*x^2] - (3*ArcSin[c*x])/(-1 + c^2*x^2)^2 + (6*ArcSin[c*x])/(-1 + c^2*x^2) - 12*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 12*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] - (6*I)*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + (6*I)*PolyLog[2, E^((2*I)*ArcSin[c*x])]) + b^2*(I*Pi^3 + 2/(-1 + c^2*x^2) + (4*c*x*ArcSin[c*x])/(1 - c^2*x^2)^(3/2) + (32*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (6*ArcSin[c*x]^2)/(-1 + c^2*x^2)^2 + (12*ArcSin[c*x]^2)/(-1 + c^2*x^2) - (16*I)*ArcSin[c*x]^3 - 24*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] + 24*ArcSin[c*x]^2*Log[1 + E^((2*I)*ArcSin[c*x])] + 16*Log[1 - c^2*x^2] - (24*I)*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] - (24*I)*ArcSin[c*x]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - 12*PolyLog[3, E^((-2*I)*ArcSin[c*x])] + 12*PolyLog[3, -E^((2*I)*ArcSin[c*x])]))/d^3","A",0
207,1,1351,429,12.0424241,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^2 \left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)^3),x]","-\frac{15 c \log (1-c x) a^2}{16 d^3}+\frac{15 c \log (c x+1) a^2}{16 d^3}-\frac{a^2}{d^3 x}-\frac{7 c^2 x a^2}{8 d^3 \left(c^2 x^2-1\right)}+\frac{c^2 x a^2}{4 d^3 \left(c^2 x^2-1\right)^2}-\frac{b c \left(24 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-90 \left(\log \left(1-i e^{i \sin ^{-1}(c x)}\right)-\log \left(1+i e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)+24 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-\frac{3 \sin ^{-1}(c x)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4}+\frac{3 \sin ^{-1}(c x)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4}+48 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-48 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-90 i \left(\text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)\right)+\frac{44 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+\frac{2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}-\frac{44 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}-\frac{21 \sin ^{-1}(c x)-1}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}+\frac{21 \sin ^{-1}(c x)+1}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-\frac{2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}\right) a}{24 d^3}-\frac{b^2 c \left(-2 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+\frac{1}{24} \left(15 \sin ^{-1}(c x)^3-45 \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)^2+45 \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)^2+45 \log \left(\left(\frac{1}{2}+\frac{i}{2}\right) e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left(-i+e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)^2-45 \log \left(\frac{1}{2} e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left((1+i)+(1-i) e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)^2-45 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^2+45 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^2-45 \pi  \log \left(\frac{1}{2} \sqrt[4]{-1} e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left(1-i e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)-45 \pi  \log \left(-\frac{1}{2} \sqrt[4]{-1} e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left(-i+e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)+45 \pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right) \sin ^{-1}(c x)+45 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right) \sin ^{-1}(c x)-90 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+90 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+44 \sin ^{-1}(c x)+44 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-44 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+90 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)-90 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)\right)-\frac{30 c x \sin ^{-1}(c x)^3+45 \sin \left(3 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^3+15 \sin \left(5 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^3-240 \cos \left(2 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2-90 \cos \left(4 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2-54 \sin ^{-1}(c x)^2+88 c x \sin ^{-1}(c x)+96 c x \log \left(1-e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-96 c x \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-200 \sin \left(2 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+144 \log \left(1-e^{i \sin ^{-1}(c x)}\right) \sin \left(3 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-144 \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sin \left(3 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+132 \sin \left(3 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-84 \sin \left(4 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+48 \log \left(1-e^{i \sin ^{-1}(c x)}\right) \sin \left(5 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-48 \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sin \left(5 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+44 \sin \left(5 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-4 \cos \left(4 \sin ^{-1}(c x)\right)-768 i c x \left(1-c^2 x^2\right)^2 \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+4}{384 c x \left(1-c^2 x^2\right)^2}\right)}{d^3}","-\frac{7 b c \left(a+b \sin ^{-1}(c x)\right)}{4 d^3 \sqrt{1-c^2 x^2}}-\frac{b c \left(a+b \sin ^{-1}(c x)\right)}{6 d^3 \left(1-c^2 x^2\right)^{3/2}}+\frac{15 c^2 x \left(a+b \sin ^{-1}(c x)\right)^2}{8 d^3 \left(1-c^2 x^2\right)}+\frac{5 c^2 x \left(a+b \sin ^{-1}(c x)\right)^2}{4 d^3 \left(1-c^2 x^2\right)^2}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{d^3 x \left(1-c^2 x^2\right)^2}+\frac{15 i b c \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 d^3}-\frac{15 i b c \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 d^3}-\frac{15 i c \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{4 d^3}-\frac{4 b c \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^3}+\frac{b^2 c^2 x}{12 d^3 \left(1-c^2 x^2\right)}+\frac{2 i b^2 c \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{d^3}-\frac{2 i b^2 c \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{d^3}-\frac{15 b^2 c \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{4 d^3}+\frac{15 b^2 c \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{4 d^3}+\frac{11 b^2 c \tanh ^{-1}(c x)}{6 d^3}",1,"-(a^2/(d^3*x)) + (a^2*c^2*x)/(4*d^3*(-1 + c^2*x^2)^2) - (7*a^2*c^2*x)/(8*d^3*(-1 + c^2*x^2)) - (15*a^2*c*Log[1 - c*x])/(16*d^3) + (15*a^2*c*Log[1 + c*x])/(16*d^3) - (b^2*c*((-2*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (44*ArcSin[c*x] + 15*ArcSin[c*x]^3 - 45*ArcSin[c*x]^2*Log[1 - I*E^(I*ArcSin[c*x])] - 45*Pi*ArcSin[c*x]*Log[((-1)^(1/4)*(1 - I*E^(I*ArcSin[c*x])))/(2*E^((I/2)*ArcSin[c*x]))] + 45*ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] + 45*ArcSin[c*x]^2*Log[((1/2 + I/2)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])] - 45*Pi*ArcSin[c*x]*Log[-1/2*((-1)^(1/4)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])] - 45*ArcSin[c*x]^2*Log[((1 + I) + (1 - I)*E^(I*ArcSin[c*x]))/(2*E^((I/2)*ArcSin[c*x]))] + 45*Pi*ArcSin[c*x]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 44*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 45*ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 44*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + 45*ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + 45*Pi*ArcSin[c*x]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (90*I)*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (90*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] + 90*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 90*PolyLog[3, I*E^(I*ArcSin[c*x])])/24 - (4 + 88*c*x*ArcSin[c*x] - 54*ArcSin[c*x]^2 + 30*c*x*ArcSin[c*x]^3 - 240*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] - 4*Cos[4*ArcSin[c*x]] - 90*ArcSin[c*x]^2*Cos[4*ArcSin[c*x]] + 96*c*x*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - 96*c*x*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (768*I)*c*x*(1 - c^2*x^2)^2*PolyLog[2, E^(I*ArcSin[c*x])] - 200*ArcSin[c*x]*Sin[2*ArcSin[c*x]] + 132*ArcSin[c*x]*Sin[3*ArcSin[c*x]] + 45*ArcSin[c*x]^3*Sin[3*ArcSin[c*x]] + 144*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])]*Sin[3*ArcSin[c*x]] - 144*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])]*Sin[3*ArcSin[c*x]] - 84*ArcSin[c*x]*Sin[4*ArcSin[c*x]] + 44*ArcSin[c*x]*Sin[5*ArcSin[c*x]] + 15*ArcSin[c*x]^3*Sin[5*ArcSin[c*x]] + 48*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])]*Sin[5*ArcSin[c*x]] - 48*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])]*Sin[5*ArcSin[c*x]])/(384*c*x*(1 - c^2*x^2)^2)))/d^3 - (a*b*c*(24*ArcSin[c*x]*Cot[ArcSin[c*x]/2] - 90*ArcSin[c*x]*(Log[1 - I*E^(I*ArcSin[c*x])] - Log[1 + I*E^(I*ArcSin[c*x])]) + 48*Log[Cos[ArcSin[c*x]/2]] - 48*Log[Sin[ArcSin[c*x]/2]] - (90*I)*(PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - PolyLog[2, I*E^(I*ArcSin[c*x])]) - (3*ArcSin[c*x])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^4 - (-1 + 21*ArcSin[c*x])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^2 + (2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (44*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) + (3*ArcSin[c*x])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^4 - (2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 + (1 + 21*ArcSin[c*x])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (44*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + 24*ArcSin[c*x]*Tan[ArcSin[c*x]/2]))/(24*d^3)","B",0
208,1,569,403,7.9062504,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^3 \left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^3),x]","-\frac{\frac{12 a^2 c^2}{c^2 x^2-1}-\frac{3 a^2 c^2}{\left(c^2 x^2-1\right)^2}+18 a^2 c^2 \log \left(1-c^2 x^2\right)-36 a^2 c^2 \log (x)+\frac{6 a^2}{x^2}+2 a b c^2 \left(\frac{14 c x}{\sqrt{1-c^2 x^2}}+\frac{c x}{\left(1-c^2 x^2\right)^{3/2}}+\frac{6 \sqrt{1-c^2 x^2}}{c x}+\frac{12 \sin ^{-1}(c x)}{c^2 x^2-1}-\frac{3 \sin ^{-1}(c x)}{\left(c^2 x^2-1\right)^2}+\frac{6 \sin ^{-1}(c x)}{c^2 x^2}-18 i \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+18 i \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-36 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+36 \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)+12 b^2 c^2 \left(\frac{1}{24} \left(\frac{2}{c^2 x^2-1}+28 \log \left(1-c^2 x^2\right)+\frac{24 \sin ^{-1}(c x)^2}{c^2 x^2-1}+\frac{12 \sin ^{-1}(c x)^2}{c^2 x^2}-\frac{6 \sin ^{-1}(c x)^2}{\left(c^2 x^2-1\right)^2}+\frac{24 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c x}+\frac{56 c x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{4 c x \sin ^{-1}(c x)}{\left(1-c^2 x^2\right)^{3/2}}-36 \text{Li}_3\left(e^{-2 i \sin ^{-1}(c x)}\right)+36 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)-24 \log (c x)-48 i \sin ^{-1}(c x)^3-72 \sin ^{-1}(c x)^2 \log \left(1-e^{-2 i \sin ^{-1}(c x)}\right)+72 \sin ^{-1}(c x)^2 \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)+3 i \pi ^3\right)-3 i \sin ^{-1}(c x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(c x)}\right)-3 i \sin ^{-1}(c x) \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)\right)}{12 d^3}","\frac{3 i b c^2 \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^3}-\frac{3 i b c^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^3}+\frac{3 c^2 \left(a+b \sin ^{-1}(c x)\right)^2}{2 d^3 \left(1-c^2 x^2\right)}+\frac{3 c^2 \left(a+b \sin ^{-1}(c x)\right)^2}{4 d^3 \left(1-c^2 x^2\right)^2}-\frac{b c \left(a+b \sin ^{-1}(c x)\right)}{d^3 x \left(1-c^2 x^2\right)^{3/2}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{2 d^3 x^2 \left(1-c^2 x^2\right)^2}-\frac{6 c^2 \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d^3}-\frac{4 b c^3 x \left(a+b \sin ^{-1}(c x)\right)}{3 d^3 \sqrt{1-c^2 x^2}}+\frac{5 b c^3 x \left(a+b \sin ^{-1}(c x)\right)}{6 d^3 \left(1-c^2 x^2\right)^{3/2}}-\frac{3 b^2 c^2 \text{Li}_3\left(-e^{2 i \sin ^{-1}(c x)}\right)}{2 d^3}+\frac{3 b^2 c^2 \text{Li}_3\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d^3}+\frac{b^2 c^2}{12 d^3 \left(1-c^2 x^2\right)}-\frac{7 b^2 c^2 \log \left(1-c^2 x^2\right)}{6 d^3}+\frac{b^2 c^2 \log (x)}{d^3}",1,"-1/12*((6*a^2)/x^2 - (3*a^2*c^2)/(-1 + c^2*x^2)^2 + (12*a^2*c^2)/(-1 + c^2*x^2) - 36*a^2*c^2*Log[x] + 18*a^2*c^2*Log[1 - c^2*x^2] + 2*a*b*c^2*((c*x)/(1 - c^2*x^2)^(3/2) + (14*c*x)/Sqrt[1 - c^2*x^2] + (6*Sqrt[1 - c^2*x^2])/(c*x) + (6*ArcSin[c*x])/(c^2*x^2) - (3*ArcSin[c*x])/(-1 + c^2*x^2)^2 + (12*ArcSin[c*x])/(-1 + c^2*x^2) - 36*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 36*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] - (18*I)*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + (18*I)*PolyLog[2, E^((2*I)*ArcSin[c*x])]) + 12*b^2*c^2*((-3*I)*ArcSin[c*x]*PolyLog[2, E^((-2*I)*ArcSin[c*x])] - (3*I)*ArcSin[c*x]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + ((3*I)*Pi^3 + 2/(-1 + c^2*x^2) + (4*c*x*ArcSin[c*x])/(1 - c^2*x^2)^(3/2) + (56*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] + (24*Sqrt[1 - c^2*x^2]*ArcSin[c*x])/(c*x) + (12*ArcSin[c*x]^2)/(c^2*x^2) - (6*ArcSin[c*x]^2)/(-1 + c^2*x^2)^2 + (24*ArcSin[c*x]^2)/(-1 + c^2*x^2) - (48*I)*ArcSin[c*x]^3 - 72*ArcSin[c*x]^2*Log[1 - E^((-2*I)*ArcSin[c*x])] + 72*ArcSin[c*x]^2*Log[1 + E^((2*I)*ArcSin[c*x])] - 24*Log[c*x] + 28*Log[1 - c^2*x^2] - 36*PolyLog[3, E^((-2*I)*ArcSin[c*x])] + 36*PolyLog[3, -E^((2*I)*ArcSin[c*x])])/24))/d^3","A",0
209,1,1657,572,13.0285512,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^4 \left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^3),x]","-\frac{11 a^2 x c^4}{8 d^3 \left(c^2 x^2-1\right)}+\frac{a^2 x c^4}{4 d^3 \left(c^2 x^2-1\right)^2}-\frac{35 a^2 \log (1-c x) c^3}{16 d^3}+\frac{35 a^2 \log (c x+1) c^3}{16 d^3}-\frac{b^2 \left(-\frac{19}{3} i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+\frac{19}{3} i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+\frac{1}{24} \left(35 \sin ^{-1}(c x)^3-105 \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)^2+105 \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)^2+105 \log \left(\left(\frac{1}{2}+\frac{i}{2}\right) e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left(-i+e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)^2-105 \log \left(\frac{1}{2} e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left((1+i)+(1-i) e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)^2-105 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^2+105 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^2-105 \pi  \log \left(\frac{1}{2} \sqrt[4]{-1} e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left(1-i e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)-105 \pi  \log \left(-\frac{1}{2} \sqrt[4]{-1} e^{-\frac{1}{2} i \sin ^{-1}(c x)} \left(-i+e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)+105 \pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right) \sin ^{-1}(c x)+105 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right) \sin ^{-1}(c x)-210 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+210 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+68 \sin ^{-1}(c x)+68 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-68 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+210 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)-210 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)\right)+\frac{-105 c x \sin ^{-1}(c x)^3-105 \sin \left(3 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^3+35 \sin \left(5 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^3+35 \sin \left(7 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^3-210 \cos \left(6 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2+204 \sin ^{-1}(c x)^2-204 c x \sin ^{-1}(c x)-456 c x \log \left(1-e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+456 c x \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+540 \sin \left(2 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-456 \log \left(1-e^{i \sin ^{-1}(c x)}\right) \sin \left(3 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+456 \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sin \left(3 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-204 \sin \left(3 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+32 \sin \left(4 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+152 \log \left(1-e^{i \sin ^{-1}(c x)}\right) \sin \left(5 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-152 \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sin \left(5 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+68 \sin \left(5 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-116 \sin \left(6 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+152 \log \left(1-e^{i \sin ^{-1}(c x)}\right) \sin \left(7 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-152 \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sin \left(7 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+68 \sin \left(7 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+\left(658 \sin ^{-1}(c x)^2+20\right) \cos \left(2 \sin ^{-1}(c x)\right)-4 \left(35 \sin ^{-1}(c x)^2+6\right) \cos \left(4 \sin ^{-1}(c x)\right)-20 \cos \left(6 \sin ^{-1}(c x)\right)+24}{1536 c^3 x^3 \left(1-c^2 x^2\right)^2}\right) c^3}{d^3}-\frac{3 a^2 c^2}{d^3 x}-\frac{2 a b \left(\frac{11 \left(\sin ^{-1}(c x)+\sqrt{1-c^2 x^2}\right) c^4}{16 \left(x c^2+c\right)}+\frac{35}{16} \left(-\frac{i \sin ^{-1}(c x)^2}{2 c}+\frac{2 \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)}{c}+\frac{3 i \pi  \sin ^{-1}(c x)}{2 c}+\frac{2 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)}{c}-\frac{\pi  \log \left(1+i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{2 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{c}+\frac{\pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{c}-\frac{2 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c}\right) c^4-\frac{35}{16} \left(-\frac{i \sin ^{-1}(c x)^2}{2 c}+\frac{2 \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)}{c}+\frac{i \pi  \sin ^{-1}(c x)}{2 c}+\frac{2 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)}{c}+\frac{\pi  \log \left(1-i e^{i \sin ^{-1}(c x)}\right)}{c}-\frac{2 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{c}-\frac{\pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)}{c}-\frac{2 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c}\right) c^4+\frac{\left((2-c x) \sqrt{1-c^2 x^2}-3 \sin ^{-1}(c x)\right) c^3}{48 (c x-1)^2}-\frac{11 \left(\sqrt{1-c^2 x^2}-\sin ^{-1}(c x)\right) c^3}{16 (c x-1)}+\frac{\left(\sqrt{1-c^2 x^2} (c x+2)+3 \sin ^{-1}(c x)\right) c^3}{48 (c x+1)^2}-3 \left(-\frac{\sin ^{-1}(c x)}{x}-c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)\right) c^2+\frac{c^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right) x^3+c \sqrt{1-c^2 x^2} x+2 \sin ^{-1}(c x)}{6 x^3}\right)}{d^3}-\frac{a^2}{3 d^3 x^3}","\frac{35 i b c^3 \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 d^3}-\frac{35 i b c^3 \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 d^3}-\frac{35 i c^3 \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{4 d^3}-\frac{38 b c^3 \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 d^3}-\frac{7 c^2 \left(a+b \sin ^{-1}(c x)\right)^2}{3 d^3 x \left(1-c^2 x^2\right)^2}-\frac{b c \left(a+b \sin ^{-1}(c x)\right)}{3 d^3 x^2 \left(1-c^2 x^2\right)^{3/2}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{3 d^3 x^3 \left(1-c^2 x^2\right)^2}+\frac{35 c^4 x \left(a+b \sin ^{-1}(c x)\right)^2}{8 d^3 \left(1-c^2 x^2\right)}+\frac{35 c^4 x \left(a+b \sin ^{-1}(c x)\right)^2}{12 d^3 \left(1-c^2 x^2\right)^2}-\frac{29 b c^3 \left(a+b \sin ^{-1}(c x)\right)}{12 d^3 \sqrt{1-c^2 x^2}}+\frac{b c^3 \left(a+b \sin ^{-1}(c x)\right)}{6 d^3 \left(1-c^2 x^2\right)^{3/2}}+\frac{19 i b^2 c^3 \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)}{3 d^3}-\frac{19 i b^2 c^3 \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)}{3 d^3}-\frac{35 b^2 c^3 \text{Li}_3\left(-i e^{i \sin ^{-1}(c x)}\right)}{4 d^3}+\frac{35 b^2 c^3 \text{Li}_3\left(i e^{i \sin ^{-1}(c x)}\right)}{4 d^3}+\frac{17 b^2 c^3 \tanh ^{-1}(c x)}{6 d^3}+\frac{b^2 c^2}{6 d^3 x \left(1-c^2 x^2\right)}-\frac{b^2 c^2}{2 d^3 x}-\frac{b^2 c^4 x}{12 d^3 \left(1-c^2 x^2\right)}",1,"-1/3*a^2/(d^3*x^3) - (3*a^2*c^2)/(d^3*x) + (a^2*c^4*x)/(4*d^3*(-1 + c^2*x^2)^2) - (11*a^2*c^4*x)/(8*d^3*(-1 + c^2*x^2)) - (35*a^2*c^3*Log[1 - c*x])/(16*d^3) + (35*a^2*c^3*Log[1 + c*x])/(16*d^3) - (2*a*b*((c^3*((2 - c*x)*Sqrt[1 - c^2*x^2] - 3*ArcSin[c*x]))/(48*(-1 + c*x)^2) - (11*c^3*(Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(16*(-1 + c*x)) + (11*c^4*(Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/(16*(c + c^2*x)) + (c^3*((2 + c*x)*Sqrt[1 - c^2*x^2] + 3*ArcSin[c*x]))/(48*(1 + c*x)^2) - 3*c^2*(-(ArcSin[c*x]/x) - c*ArcTanh[Sqrt[1 - c^2*x^2]]) + (c*x*Sqrt[1 - c^2*x^2] + 2*ArcSin[c*x] + c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*x^3) + (35*c^4*((((3*I)/2)*Pi*ArcSin[c*x])/c - ((I/2)*ArcSin[c*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])])/c - (Pi*Log[1 + I*E^(I*ArcSin[c*x])])/c + (2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c + (Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]])/c - ((2*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/c))/16 - (35*c^4*(((I/2)*Pi*ArcSin[c*x])/c - ((I/2)*ArcSin[c*x]^2)/c + (2*Pi*Log[1 + E^((-I)*ArcSin[c*x])])/c + (Pi*Log[1 - I*E^(I*ArcSin[c*x])])/c + (2*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])])/c - (2*Pi*Log[Cos[ArcSin[c*x]/2]])/c - (Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]])/c - ((2*I)*PolyLog[2, I*E^(I*ArcSin[c*x])])/c))/16))/d^3 - (b^2*c^3*(((-19*I)/3)*PolyLog[2, -E^(I*ArcSin[c*x])] + ((19*I)/3)*PolyLog[2, E^(I*ArcSin[c*x])] + (68*ArcSin[c*x] + 35*ArcSin[c*x]^3 - 105*ArcSin[c*x]^2*Log[1 - I*E^(I*ArcSin[c*x])] - 105*Pi*ArcSin[c*x]*Log[((-1)^(1/4)*(1 - I*E^(I*ArcSin[c*x])))/(2*E^((I/2)*ArcSin[c*x]))] + 105*ArcSin[c*x]^2*Log[1 + I*E^(I*ArcSin[c*x])] + 105*ArcSin[c*x]^2*Log[((1/2 + I/2)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])] - 105*Pi*ArcSin[c*x]*Log[-1/2*((-1)^(1/4)*(-I + E^(I*ArcSin[c*x])))/E^((I/2)*ArcSin[c*x])] - 105*ArcSin[c*x]^2*Log[((1 + I) + (1 - I)*E^(I*ArcSin[c*x]))/(2*E^((I/2)*ArcSin[c*x]))] + 105*Pi*ArcSin[c*x]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 68*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 105*ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 68*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + 105*ArcSin[c*x]^2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + 105*Pi*ArcSin[c*x]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (210*I)*ArcSin[c*x]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (210*I)*ArcSin[c*x]*PolyLog[2, I*E^(I*ArcSin[c*x])] + 210*PolyLog[3, (-I)*E^(I*ArcSin[c*x])] - 210*PolyLog[3, I*E^(I*ArcSin[c*x])])/24 + (24 - 204*c*x*ArcSin[c*x] + 204*ArcSin[c*x]^2 - 105*c*x*ArcSin[c*x]^3 + (20 + 658*ArcSin[c*x]^2)*Cos[2*ArcSin[c*x]] - 4*(6 + 35*ArcSin[c*x]^2)*Cos[4*ArcSin[c*x]] - 20*Cos[6*ArcSin[c*x]] - 210*ArcSin[c*x]^2*Cos[6*ArcSin[c*x]] - 456*c*x*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 456*c*x*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + 540*ArcSin[c*x]*Sin[2*ArcSin[c*x]] - 204*ArcSin[c*x]*Sin[3*ArcSin[c*x]] - 105*ArcSin[c*x]^3*Sin[3*ArcSin[c*x]] - 456*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])]*Sin[3*ArcSin[c*x]] + 456*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])]*Sin[3*ArcSin[c*x]] + 32*ArcSin[c*x]*Sin[4*ArcSin[c*x]] + 68*ArcSin[c*x]*Sin[5*ArcSin[c*x]] + 35*ArcSin[c*x]^3*Sin[5*ArcSin[c*x]] + 152*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])]*Sin[5*ArcSin[c*x]] - 152*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])]*Sin[5*ArcSin[c*x]] - 116*ArcSin[c*x]*Sin[6*ArcSin[c*x]] + 68*ArcSin[c*x]*Sin[7*ArcSin[c*x]] + 35*ArcSin[c*x]^3*Sin[7*ArcSin[c*x]] + 152*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])]*Sin[7*ArcSin[c*x]] - 152*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])]*Sin[7*ArcSin[c*x]])/(1536*c^3*x^3*(1 - c^2*x^2)^2)))/d^3","B",0
210,1,242,374,0.3044763,"\int x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[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(225 a^2 \sqrt{1-c^2 x^2} \left(3 c^4 x^4-c^2 x^2-2\right)-30 a b c x \left(9 c^4 x^4-5 c^2 x^2-30\right)-30 b \sin ^{-1}(c x) \left(15 a \sqrt{1-c^2 x^2} \left(-3 c^4 x^4+c^2 x^2+2\right)+b c x \left(9 c^4 x^4-5 c^2 x^2-30\right)\right)-2 b^2 \sqrt{1-c^2 x^2} \left(27 c^4 x^4+11 c^2 x^2-428\right)+225 b^2 \sqrt{1-c^2 x^2} \left(3 c^4 x^4-c^2 x^2-2\right) \sin ^{-1}(c x)^2\right)}{3375 c^4 \sqrt{1-c^2 x^2}}","-\frac{x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{15 c^2}-\frac{2 b c 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{1}{5} x^4 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{2 b x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{45 c \sqrt{1-c^2 x^2}}-\frac{2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{15 c^4}+\frac{4 a b x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{1-c^2 x^2}}-\frac{2 b^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{125 c^4}+\frac{26 b^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{675 c^4}+\frac{52 b^2 \sqrt{d-c^2 d x^2}}{225 c^4}+\frac{4 b^2 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{15 c^3 \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(225*a^2*Sqrt[1 - c^2*x^2]*(-2 - c^2*x^2 + 3*c^4*x^4) - 30*a*b*c*x*(-30 - 5*c^2*x^2 + 9*c^4*x^4) - 2*b^2*Sqrt[1 - c^2*x^2]*(-428 + 11*c^2*x^2 + 27*c^4*x^4) - 30*b*(15*a*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2 - 3*c^4*x^4) + b*c*x*(-30 - 5*c^2*x^2 + 9*c^4*x^4))*ArcSin[c*x] + 225*b^2*Sqrt[1 - c^2*x^2]*(-2 - c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x]^2))/(3375*c^4*Sqrt[1 - c^2*x^2])","A",1
211,1,246,303,0.3613646,"\int x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[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(8 a^3-3 b \sin ^{-1}(c x) \left(-8 a^2+16 a b c x \left(1-2 c^2 x^2\right) \sqrt{1-c^2 x^2}+b^2 \left(8 c^4 x^4-8 c^2 x^2+1\right)\right)+24 a^2 b c x \sqrt{1-c^2 x^2} \left(2 c^2 x^2-1\right)-24 a b^2 c^2 x^2 \left(c^2 x^2-1\right)+24 b^2 \sin ^{-1}(c x)^2 \left(a+b c x \sqrt{1-c^2 x^2} \left(2 c^2 x^2-1\right)\right)+3 b^3 c x \left(1-2 c^2 x^2\right) \sqrt{1-c^2 x^2}+8 b^3 \sin ^{-1}(c x)^3\right)}{192 b c^3 \sqrt{1-c^2 x^2}}","\frac{b 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{x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{8 c^2}-\frac{b c 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} x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{\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{b^2 x \sqrt{d-c^2 d x^2}}{64 c^2}-\frac{1}{32} b^2 x^3 \sqrt{d-c^2 d x^2}-\frac{b^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]*(8*a^3 + 3*b^3*c*x*(1 - 2*c^2*x^2)*Sqrt[1 - c^2*x^2] - 24*a*b^2*c^2*x^2*(-1 + c^2*x^2) + 24*a^2*b*c*x*Sqrt[1 - c^2*x^2]*(-1 + 2*c^2*x^2) - 3*b*(-8*a^2 + 16*a*b*c*x*(1 - 2*c^2*x^2)*Sqrt[1 - c^2*x^2] + b^2*(1 - 8*c^2*x^2 + 8*c^4*x^4))*ArcSin[c*x] + 24*b^2*(a + b*c*x*Sqrt[1 - c^2*x^2]*(-1 + 2*c^2*x^2))*ArcSin[c*x]^2 + 8*b^3*ArcSin[c*x]^3))/(192*b*c^3*Sqrt[1 - c^2*x^2])","A",1
212,1,120,188,0.2967111,"\int x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2,x]","\frac{\sqrt{d-c^2 d x^2} \left(\left(c^2 x^2-1\right) \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2 b \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 \sqrt{1-c^2 x^2}}\right)}{3 c^2}","\frac{2 b 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{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^2 d}-\frac{2 b c 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{2 b^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{27 c^2}+\frac{4 b^2 \sqrt{d-c^2 d x^2}}{9 c^2}",1,"(Sqrt[d - c^2*d*x^2]*((-1 + c^2*x^2)*(a + b*ArcSin[c*x])^2 - (2*b*(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*Sqrt[1 - c^2*x^2])))/(3*c^2)","A",1
213,1,128,192,0.2349976,"\int \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2,x]","\frac{1}{6} \sqrt{d-c^2 d x^2} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^3}{b c \sqrt{1-c^2 x^2}}-\frac{3 b \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)}{2 c \sqrt{1-c^2 x^2}}+3 x \left(a+b \sin ^{-1}(c x)\right)^2\right)","\frac{\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{1}{2} x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{b c 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}{4} b^2 x \sqrt{d-c^2 d x^2}+\frac{b^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(3*x*(a + b*ArcSin[c*x])^2 + (a + b*ArcSin[c*x])^3/(b*c*Sqrt[1 - c^2*x^2]) - (3*b*(c*x*(2*a*c*x + b*Sqrt[1 - c^2*x^2]) + b*(-1 + 2*c^2*x^2)*ArcSin[c*x]))/(2*c*Sqrt[1 - c^2*x^2])))/6","A",1
214,1,391,378,1.2069581,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{x} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/x,x]","a^2 \sqrt{d-c^2 d x^2}-a^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+a^2 \sqrt{d} \log (c x)+\frac{2 a b \sqrt{d-c^2 d x^2} \left(\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-c x+\sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-\sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}+\frac{b^2 \sqrt{d-c^2 d x^2} \left(-2 \sqrt{1-c^2 x^2}+\sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2+2 i \sin ^{-1}(c x) \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-2 i \sin ^{-1}(c x) \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-2 \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)+2 \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)-2 c x \sin ^{-1}(c x)+\sin ^{-1}(c x)^2 \log \left(1-e^{i \sin ^{-1}(c x)}\right)-\sin ^{-1}(c x)^2 \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}","\frac{2 i b \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{2 i b \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{2 a b c x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}+\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}}-\frac{2 b^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}+\frac{2 b^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-2 b^2 \sqrt{d-c^2 d x^2}-\frac{2 b^2 c x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}",1,"a^2*Sqrt[d - c^2*d*x^2] + a^2*Sqrt[d]*Log[c*x] - a^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (2*a*b*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + I*PolyLog[2, -E^(I*ArcSin[c*x])] - I*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (b^2*Sqrt[d - c^2*d*x^2]*(-2*Sqrt[1 - c^2*x^2] - 2*c*x*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 + ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] - ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] + (2*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] - (2*I)*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] - 2*PolyLog[3, -E^(I*ArcSin[c*x])] + 2*PolyLog[3, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2]","A",0
215,1,257,227,1.0498546,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/x^2,x]","-\frac{a^2 \sqrt{d-c^2 d x^2}}{x}+a^2 c \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-\frac{a b \sqrt{d-c^2 d x^2} \left(2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-2 c x \log (c x)+c x \sin ^{-1}(c x)^2\right)}{x \sqrt{1-c^2 x^2}}-\frac{b^2 c \sqrt{d-c^2 d x^2} \left(\sin ^{-1}(c x) \left(\left(\frac{3 \sqrt{1-c^2 x^2}}{c x}+3 i\right) \sin ^{-1}(c x)+\sin ^{-1}(c x)^2-6 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)\right)+3 i \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)\right)}{3 \sqrt{1-c^2 x^2}}","-\frac{c \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b \sqrt{1-c^2 x^2}}-\frac{i c \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{x}+\frac{2 b c \sqrt{d-c^2 d x^2} \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{i b^2 c \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}",1,"-((a^2*Sqrt[d - c^2*d*x^2])/x) + a^2*c*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - (a*b*Sqrt[d - c^2*d*x^2]*(2*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + c*x*ArcSin[c*x]^2 - 2*c*x*Log[c*x]))/(x*Sqrt[1 - c^2*x^2]) - (b^2*c*Sqrt[d - c^2*d*x^2]*(ArcSin[c*x]*((3*I + (3*Sqrt[1 - c^2*x^2])/(c*x))*ArcSin[c*x] + ArcSin[c*x]^2 - 6*Log[1 - E^((2*I)*ArcSin[c*x])]) + (3*I)*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/(3*Sqrt[1 - c^2*x^2])","A",0
216,1,480,398,5.2846415,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/x^3,x]","\frac{1}{8} \left(-\frac{4 a^2 \sqrt{d-c^2 d x^2}}{x^2}+4 a^2 c^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-4 a^2 c^2 \sqrt{d} \log (x)+\frac{2 a b c^2 d \sqrt{1-c^2 x^2} \left(-4 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+4 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-4 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+4 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)-2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)-2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{\sqrt{d-c^2 d x^2}}+\frac{b^2 c^2 d \sqrt{1-c^2 x^2} \left(-8 i \sin ^{-1}(c x) \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+8 i \sin ^{-1}(c x) \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+8 \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)-8 \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)-4 \sin ^{-1}(c x)^2 \log \left(1-e^{i \sin ^{-1}(c x)}\right)+4 \sin ^{-1}(c x)^2 \log \left(1+e^{i \sin ^{-1}(c x)}\right)-4 \sin ^{-1}(c x) \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)-4 \sin ^{-1}(c x) \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin ^{-1}(c x)^2 \left(-\csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\sin ^{-1}(c x)^2 \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+8 \log \left(\tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)}{\sqrt{d-c^2 d x^2}}\right)","-\frac{i b c^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}+\frac{i b c^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{b c \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{x \sqrt{1-c^2 x^2}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 x^2}+\frac{c^2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}}+\frac{b^2 c^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{b^2 c^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{b^2 c^2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{\sqrt{1-c^2 x^2}}",1,"((-4*a^2*Sqrt[d - c^2*d*x^2])/x^2 - 4*a^2*c^2*Sqrt[d]*Log[x] + 4*a^2*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (2*a*b*c^2*d*Sqrt[1 - c^2*x^2]*(-2*Cot[ArcSin[c*x]/2] - ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (4*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 - 2*Tan[ArcSin[c*x]/2]))/Sqrt[d - c^2*d*x^2] + (b^2*c^2*d*Sqrt[1 - c^2*x^2]*(-4*ArcSin[c*x]*Cot[ArcSin[c*x]/2] - ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] + 8*Log[Tan[ArcSin[c*x]/2]] - (8*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] + (8*I)*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] + 8*PolyLog[3, -E^(I*ArcSin[c*x])] - 8*PolyLog[3, E^(I*ArcSin[c*x])] + ArcSin[c*x]^2*Sec[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Tan[ArcSin[c*x]/2]))/Sqrt[d - c^2*d*x^2])/8","A",0
217,1,248,314,1.2480106,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/x^4,x]","\frac{\sqrt{d-c^2 d x^2} \left(-2 \left(a^2 \left(1-c^2 x^2\right)^{3/2}+2 a b c^3 x^3 \log (c x)+a b c x+b^2 c^2 x^2 \sqrt{1-c^2 x^2}\right)-b \sin ^{-1}(c x) \left(3 a \sqrt{1-c^2 x^2}+a \cos \left(3 \sin ^{-1}(c x)\right)+4 b c^3 x^3 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+2 b c x\right)+2 i b^2 c^3 x^3 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+2 b^2 \left(i c^3 x^3+c^2 x^2 \sqrt{1-c^2 x^2}-\sqrt{1-c^2 x^2}\right) \sin ^{-1}(c x)^2\right)}{6 x^3 \sqrt{1-c^2 x^2}}","-\frac{b c \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 x^2}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 d x^3}+\frac{i c^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 \sqrt{1-c^2 x^2}}-\frac{2 b c^3 \sqrt{d-c^2 d x^2} \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 \sqrt{1-c^2 x^2}}-\frac{b^2 c^2 \sqrt{d-c^2 d x^2}}{3 x}+\frac{i b^2 c^3 \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{3 \sqrt{1-c^2 x^2}}-\frac{b^2 c^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{3 \sqrt{1-c^2 x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(2*b^2*(I*c^3*x^3 - Sqrt[1 - c^2*x^2] + c^2*x^2*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 - b*ArcSin[c*x]*(2*b*c*x + 3*a*Sqrt[1 - c^2*x^2] + a*Cos[3*ArcSin[c*x]] + 4*b*c^3*x^3*Log[1 - E^((2*I)*ArcSin[c*x])]) - 2*(a*b*c*x + b^2*c^2*x^2*Sqrt[1 - c^2*x^2] + a^2*(1 - c^2*x^2)^(3/2) + 2*a*b*c^3*x^3*Log[c*x]) + (2*I)*b^2*c^3*x^3*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/(6*x^3*Sqrt[1 - c^2*x^2])","A",0
218,1,244,503,0.3325549,"\int x^3 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[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(-11025 a^2 \left(5 c^2 x^2+2\right) \left(1-c^2 x^2\right)^{5/2}+210 a b c x \left(75 c^6 x^6-168 c^4 x^4+35 c^2 x^2+210\right)+210 b \sin ^{-1}(c x) \left(b c x \left(75 c^6 x^6-168 c^4 x^4+35 c^2 x^2+210\right)-105 a \left(1-c^2 x^2\right)^{5/2} \left(5 c^2 x^2+2\right)\right)-11025 b^2 \left(5 c^2 x^2+2\right) \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x)^2+2 b^2 \left(1125 c^6 x^6-2178 c^4 x^4-1679 c^2 x^2+18692\right) \sqrt{1-c^2 x^2}\right)}{385875 c^4 \sqrt{1-c^2 x^2}}","-\frac{d x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{35 c^2}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{175 \sqrt{1-c^2 x^2}}+\frac{1}{7} x^4 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{105 c \sqrt{1-c^2 x^2}}-\frac{2 d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{35 c^4}+\frac{4 a b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{49 \sqrt{1-c^2 x^2}}-\frac{2 b^2 d \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2}}{343 c^4}+\frac{38 b^2 d \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{6125 c^4}+\frac{304 b^2 d \sqrt{d-c^2 d x^2}}{3675 c^4}+\frac{152 b^2 d \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{11025 c^4}+\frac{4 b^2 d x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{35 c^3 \sqrt{1-c^2 x^2}}",1,"(d*Sqrt[d - c^2*d*x^2]*(-11025*a^2*(1 - c^2*x^2)^(5/2)*(2 + 5*c^2*x^2) + 210*a*b*c*x*(210 + 35*c^2*x^2 - 168*c^4*x^4 + 75*c^6*x^6) + 2*b^2*Sqrt[1 - c^2*x^2]*(18692 - 1679*c^2*x^2 - 2178*c^4*x^4 + 1125*c^6*x^6) + 210*b*(-105*a*(1 - c^2*x^2)^(5/2)*(2 + 5*c^2*x^2) + b*c*x*(210 + 35*c^2*x^2 - 168*c^4*x^4 + 75*c^6*x^6))*ArcSin[c*x] - 11025*b^2*(1 - c^2*x^2)^(5/2)*(2 + 5*c^2*x^2)*ArcSin[c*x]^2))/(385875*c^4*Sqrt[1 - c^2*x^2])","A",1
219,1,297,421,0.3336743,"\int x^2 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[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(72 a^3+3 b \sin ^{-1}(c x) \left(72 a^2-48 a b c x \sqrt{1-c^2 x^2} \left(8 c^4 x^4-14 c^2 x^2+3\right)+b^2 \left(64 c^6 x^6-168 c^4 x^4+72 c^2 x^2+7\right)\right)-72 a^2 b c x \sqrt{1-c^2 x^2} \left(8 c^4 x^4-14 c^2 x^2+3\right)+24 a b^2 c^2 x^2 \left(8 c^4 x^4-21 c^2 x^2+9\right)+72 b^2 \sin ^{-1}(c x)^2 \left(3 a+b c x \sqrt{1-c^2 x^2} \left(-8 c^4 x^4+14 c^2 x^2-3\right)\right)+b^3 c x \sqrt{1-c^2 x^2} \left(32 c^4 x^4-86 c^2 x^2-21\right)+72 b^3 \sin ^{-1}(c x)^3\right)}{3456 b c^3 \sqrt{1-c^2 x^2}}","\frac{b d x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{16 c \sqrt{1-c^2 x^2}}-\frac{d x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{16 c^2}-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{48 \sqrt{1-c^2 x^2}}+\frac{1}{6} x^3 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{d \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{b c^3 d x^6 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{18 \sqrt{1-c^2 x^2}}-\frac{7 b^2 d x \sqrt{d-c^2 d x^2}}{1152 c^2}+\frac{1}{108} b^2 c^2 d x^5 \sqrt{d-c^2 d x^2}-\frac{43 b^2 d x^3 \sqrt{d-c^2 d x^2}}{1728}+\frac{7 b^2 d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c^3 \sqrt{1-c^2 x^2}}",1,"(d*Sqrt[d - c^2*d*x^2]*(72*a^3 + 24*a*b^2*c^2*x^2*(9 - 21*c^2*x^2 + 8*c^4*x^4) - 72*a^2*b*c*x*Sqrt[1 - c^2*x^2]*(3 - 14*c^2*x^2 + 8*c^4*x^4) + b^3*c*x*Sqrt[1 - c^2*x^2]*(-21 - 86*c^2*x^2 + 32*c^4*x^4) + 3*b*(72*a^2 - 48*a*b*c*x*Sqrt[1 - c^2*x^2]*(3 - 14*c^2*x^2 + 8*c^4*x^4) + b^2*(7 + 72*c^2*x^2 - 168*c^4*x^4 + 64*c^6*x^6))*ArcSin[c*x] + 72*b^2*(3*a + b*c*x*Sqrt[1 - c^2*x^2]*(-3 + 14*c^2*x^2 - 8*c^4*x^4))*ArcSin[c*x]^2 + 72*b^3*ArcSin[c*x]^3))/(3456*b*c^3*Sqrt[1 - c^2*x^2])","A",1
220,1,159,279,0.1960336,"\int x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2,x]","\frac{2 b d \sqrt{d-c^2 d x^2} \left(15 a c x \left(3 c^4 x^4-10 c^2 x^2+15\right)+b \sqrt{1-c^2 x^2} \left(9 c^4 x^4-38 c^2 x^2+149\right)+15 b c x \left(3 c^4 x^4-10 c^2 x^2+15\right) \sin ^{-1}(c x)\right)}{1125 c^2 \sqrt{1-c^2 x^2}}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{5 c^2 d}","\frac{2 b d 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{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{5 c^2 d}-\frac{4 b c d 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 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{2 b^2 d \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{125 c^2}+\frac{16 b^2 d \sqrt{d-c^2 d x^2}}{75 c^2}+\frac{8 b^2 d \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{225 c^2}",1,"-1/5*((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/(c^2*d) + (2*b*d*Sqrt[d - c^2*d*x^2]*(15*a*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(149 - 38*c^2*x^2 + 9*c^4*x^4) + 15*b*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x]))/(1125*c^2*Sqrt[1 - c^2*x^2])","A",1
221,1,329,305,1.2074423,"\int \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(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(160 a^2 c x \sqrt{1-c^2 x^2}-64 a^2 c^3 x^3 \sqrt{1-c^2 x^2}+64 a b \cos \left(2 \sin ^{-1}(c x)\right)+4 a b \cos \left(4 \sin ^{-1}(c x)\right)-32 b^2 \sin \left(2 \sin ^{-1}(c x)\right)-b^2 \sin \left(4 \sin ^{-1}(c x)\right)\right)-96 a^2 d^{3/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+8 b d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \left(12 a+8 b \sin \left(2 \sin ^{-1}(c x)\right)+b \sin \left(4 \sin ^{-1}(c x)\right)\right)+4 b d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \left(4 a \left(8 \sin \left(2 \sin ^{-1}(c x)\right)+\sin \left(4 \sin ^{-1}(c x)\right)\right)+16 b \cos \left(2 \sin ^{-1}(c x)\right)+b \cos \left(4 \sin ^{-1}(c x)\right)\right)+32 b^2 d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^3}{256 c \sqrt{1-c^2 x^2}}","-\frac{5 b c d 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{1}{4} x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{3}{8} d x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{d \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 c^3 d 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{17}{64} b^2 d x \sqrt{d-c^2 d x^2}+\frac{17 b^2 d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}+\frac{1}{32} b^2 c^2 d x^3 \sqrt{d-c^2 d x^2}",1,"(32*b^2*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^3 - 96*a^2*d^(3/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 8*b*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*(12*a + 8*b*Sin[2*ArcSin[c*x]] + b*Sin[4*ArcSin[c*x]]) + d*Sqrt[d - c^2*d*x^2]*(160*a^2*c*x*Sqrt[1 - c^2*x^2] - 64*a^2*c^3*x^3*Sqrt[1 - c^2*x^2] + 64*a*b*Cos[2*ArcSin[c*x]] + 4*a*b*Cos[4*ArcSin[c*x]] - 32*b^2*Sin[2*ArcSin[c*x]] - b^2*Sin[4*ArcSin[c*x]]) + 4*b*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*(16*b*Cos[2*ArcSin[c*x]] + b*Cos[4*ArcSin[c*x]] + 4*a*(8*Sin[2*ArcSin[c*x]] + Sin[4*ArcSin[c*x]])))/(256*c*Sqrt[1 - c^2*x^2])","A",1
222,1,576,545,2.6474529,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/x,x]","-a^2 d^{3/2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-\frac{1}{3} a^2 d \left(c^2 x^2-4\right) \sqrt{d-c^2 d x^2}+a^2 d^{3/2} \log (c x)+\frac{2 a b d \sqrt{d-c^2 d x^2} \left(\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-c x+\sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-\sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}-\frac{a b d \sqrt{d-c^2 d x^2} \left(-3 \sin ^{-1}(c x) \left(3 \sqrt{1-c^2 x^2}+\cos \left(3 \sin ^{-1}(c x)\right)\right)+9 c x+\sin \left(3 \sin ^{-1}(c x)\right)\right)}{18 \sqrt{1-c^2 x^2}}-\frac{b^2 d \sqrt{d-c^2 d x^2} \left(2 \sqrt{1-c^2 x^2}-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2-2 i \sin ^{-1}(c x) \left(\text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right)+2 \left(\text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)\right)+2 c x \sin ^{-1}(c x)-\left(\sin ^{-1}(c x)^2 \left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)\right)\right)}{\sqrt{1-c^2 x^2}}+\frac{b^2 d \sqrt{d-c^2 d x^2} \left(27 \sqrt{1-c^2 x^2} \left(\sin ^{-1}(c x)^2-2\right)-6 \sin ^{-1}(c x) \left(9 c x+\sin \left(3 \sin ^{-1}(c x)\right)\right)+\left(9 \sin ^{-1}(c x)^2-2\right) \cos \left(3 \sin ^{-1}(c x)\right)\right)}{108 \sqrt{1-c^2 x^2}}","\frac{2 i b d \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{2 i b d \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{2 a b c d x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{2 b c d x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 \sqrt{1-c^2 x^2}}+\frac{1}{3} \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2+d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}}+\frac{2 b c^3 d 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{2 b^2 d \sqrt{d-c^2 d x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}+\frac{2 b^2 d \sqrt{d-c^2 d x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{22}{9} b^2 d \sqrt{d-c^2 d x^2}-\frac{2}{27} b^2 d \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}-\frac{2 b^2 c d x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}",1,"-1/3*(a^2*d*(-4 + c^2*x^2)*Sqrt[d - c^2*d*x^2]) + a^2*d^(3/2)*Log[c*x] - a^2*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (2*a*b*d*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + I*PolyLog[2, -E^(I*ArcSin[c*x])] - I*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] - (b^2*d*Sqrt[d - c^2*d*x^2]*(2*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 - ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) - (2*I)*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]) + 2*(PolyLog[3, -E^(I*ArcSin[c*x])] - PolyLog[3, E^(I*ArcSin[c*x])])))/Sqrt[1 - c^2*x^2] - (a*b*d*Sqrt[d - c^2*d*x^2]*(9*c*x - 3*ArcSin[c*x]*(3*Sqrt[1 - c^2*x^2] + Cos[3*ArcSin[c*x]]) + Sin[3*ArcSin[c*x]]))/(18*Sqrt[1 - c^2*x^2]) + (b^2*d*Sqrt[d - c^2*d*x^2]*(27*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]*(9*c*x + Sin[3*ArcSin[c*x]])))/(108*Sqrt[1 - c^2*x^2])","A",0
223,1,396,424,2.6669362,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/x^2,x]","\frac{36 a^2 c d^{3/2} x \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-12 a^2 d \sqrt{1-c^2 x^2} \left(c^2 x^2+2\right) \sqrt{d-c^2 d x^2}-24 a b d \sqrt{d-c^2 d x^2} \left(2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-2 c x \log (c x)+c x \sin ^{-1}(c x)^2\right)-6 a b c d x \sqrt{d-c^2 d x^2} \left(2 \sin ^{-1}(c x) \left(\sin ^{-1}(c x)+\sin \left(2 \sin ^{-1}(c x)\right)\right)+\cos \left(2 \sin ^{-1}(c x)\right)\right)-8 b^2 d \sqrt{d-c^2 d x^2} \left(\sin ^{-1}(c x) \left(3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+c x \left(\sin ^{-1}(c x)+3 i\right) \sin ^{-1}(c x)-6 c x \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)\right)+3 i c x \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)\right)-b^2 c d x \sqrt{d-c^2 d x^2} \left(4 \sin ^{-1}(c x)^3+\left(6 \sin ^{-1}(c x)^2-3\right) \sin \left(2 \sin ^{-1}(c x)\right)+6 \sin ^{-1}(c x) \cos \left(2 \sin ^{-1}(c x)\right)\right)}{24 x \sqrt{1-c^2 x^2}}","-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{c d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{2 b \sqrt{1-c^2 x^2}}-\frac{i c d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}}+b c d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{x}+\frac{2 b c d \sqrt{d-c^2 d x^2} \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}+\frac{3 b c^3 d 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{i b^2 c d \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}+\frac{1}{4} b^2 c^2 d x \sqrt{d-c^2 d x^2}-\frac{5 b^2 c d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{4 \sqrt{1-c^2 x^2}}",1,"(-12*a^2*d*Sqrt[1 - c^2*x^2]*(2 + c^2*x^2)*Sqrt[d - c^2*d*x^2] + 36*a^2*c*d^(3/2)*x*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 24*a*b*d*Sqrt[d - c^2*d*x^2]*(2*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + c*x*ArcSin[c*x]^2 - 2*c*x*Log[c*x]) - 8*b^2*d*Sqrt[d - c^2*d*x^2]*(ArcSin[c*x]*(3*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + c*x*ArcSin[c*x]*(3*I + ArcSin[c*x]) - 6*c*x*Log[1 - E^((2*I)*ArcSin[c*x])]) + (3*I)*c*x*PolyLog[2, E^((2*I)*ArcSin[c*x])]) - b^2*c*d*x*Sqrt[d - c^2*d*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]]) - 6*a*b*c*d*x*Sqrt[d - c^2*d*x^2]*(Cos[2*ArcSin[c*x]] + 2*ArcSin[c*x]*(ArcSin[c*x] + Sin[2*ArcSin[c*x]])))/(24*x*Sqrt[1 - c^2*x^2])","A",0
224,1,854,590,7.1860029,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/x^3,x]","-\frac{3}{2} a^2 d^{3/2} \log (x) c^2+\frac{3}{2} a^2 d^{3/2} \log \left(d+\sqrt{-d \left(c^2 x^2-1\right)} \sqrt{d}\right) c^2-2 a b d \sqrt{d \left(1-c^2 x^2\right)} \left(-\frac{c x}{\sqrt{1-c^2 x^2}}+\sin ^{-1}(c x)+\frac{\sin ^{-1}(c x) \left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}+\frac{i \left(\text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}\right) c^2-b^2 d \sqrt{d \left(1-c^2 x^2\right)} \left(\frac{\left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}+\sin ^{-1}(c x)^2+\frac{2 i \left(\text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{2 c x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{2 \left(\text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)-\text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}-2\right) c^2+\frac{a b d^2 \sqrt{1-c^2 x^2} \left(-\sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)-2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)-4 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+4 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)-4 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+4 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) c^2}{4 \sqrt{d \left(1-c^2 x^2\right)}}+\frac{b^2 d^2 \sqrt{1-c^2 x^2} \left(-\csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2+\sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2-4 \log \left(1-e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)^2+4 \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)^2-4 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-8 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+8 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-4 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+8 \log \left(\tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+8 \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)-8 \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)\right) c^2}{8 \sqrt{d \left(1-c^2 x^2\right)}}+\left(-c^2 d a^2-\frac{d a^2}{2 x^2}\right) \sqrt{-d \left(c^2 x^2-1\right)}","-\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}+\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{b c d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{x \sqrt{1-c^2 x^2}}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 x^2}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}}+\frac{3 a b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{b c^3 d x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}+\frac{3 b^2 c^2 d \sqrt{d-c^2 d x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{3 b^2 c^2 d \sqrt{d-c^2 d x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}+2 b^2 c^2 d \sqrt{d-c^2 d x^2}-\frac{b^2 c^2 d \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{\sqrt{1-c^2 x^2}}+\frac{3 b^2 c^3 d x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}",1,"(-(a^2*c^2*d) - (a^2*d)/(2*x^2))*Sqrt[-(d*(-1 + c^2*x^2))] - (3*a^2*c^2*d^(3/2)*Log[x])/2 + (3*a^2*c^2*d^(3/2)*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/2 - 2*a*b*c^2*d*Sqrt[d*(1 - c^2*x^2)]*(-((c*x)/Sqrt[1 - c^2*x^2]) + ArcSin[c*x] + (ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (I*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2]) - b^2*c^2*d*Sqrt[d*(1 - c^2*x^2)]*(-2 - (2*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] + ArcSin[c*x]^2 + (ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + ((2*I)*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (2*(-PolyLog[3, -E^(I*ArcSin[c*x])] + PolyLog[3, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2]) + (a*b*c^2*d^2*Sqrt[1 - c^2*x^2]*(-2*Cot[ArcSin[c*x]/2] - ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (4*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 - 2*Tan[ArcSin[c*x]/2]))/(4*Sqrt[d*(1 - c^2*x^2)]) + (b^2*c^2*d^2*Sqrt[1 - c^2*x^2]*(-4*ArcSin[c*x]*Cot[ArcSin[c*x]/2] - ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] + 8*Log[Tan[ArcSin[c*x]/2]] - (8*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] + (8*I)*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] + 8*PolyLog[3, -E^(I*ArcSin[c*x])] - 8*PolyLog[3, E^(I*ArcSin[c*x])] + ArcSin[c*x]^2*Sec[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Tan[ArcSin[c*x]/2]))/(8*Sqrt[d*(1 - c^2*x^2)])","A",0
225,1,493,400,2.1441519,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/x^4,x]","\frac{4 a^2 c^2 d x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}-a^2 d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}-3 a^2 c^3 d^{3/2} x^3 \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-a b c d x \sqrt{d-c^2 d x^2}-8 a b c^3 d x^3 \sqrt{d-c^2 d x^2} \log (c x)+b d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \left(3 a c^3 x^3+b \left(4 i c^3 x^3+4 c^2 x^2 \sqrt{1-c^2 x^2}-\sqrt{1-c^2 x^2}\right)\right)-b d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \left(2 a \left(1-4 c^2 x^2\right) \sqrt{1-c^2 x^2}+8 b c^3 x^3 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+b c x\right)-b^2 c^2 d x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}+4 i b^2 c^3 d x^3 \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+b^2 c^3 d x^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^3}{3 x^3 \sqrt{1-c^2 x^2}}","\frac{c^2 d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{x}-\frac{b c d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 x^2}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 x^3}+\frac{c^3 d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b \sqrt{1-c^2 x^2}}+\frac{4 i c^3 d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 \sqrt{1-c^2 x^2}}-\frac{8 b c^3 d \sqrt{d-c^2 d x^2} \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 \sqrt{1-c^2 x^2}}-\frac{b^2 c^2 d \sqrt{d-c^2 d x^2}}{3 x}+\frac{4 i b^2 c^3 d \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{3 \sqrt{1-c^2 x^2}}-\frac{b^2 c^3 d \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{3 \sqrt{1-c^2 x^2}}",1,"(-(a*b*c*d*x*Sqrt[d - c^2*d*x^2]) - a^2*d*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + 4*a^2*c^2*d*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] - b^2*c^2*d*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + b*d*Sqrt[d - c^2*d*x^2]*(3*a*c^3*x^3 + b*((4*I)*c^3*x^3 - Sqrt[1 - c^2*x^2] + 4*c^2*x^2*Sqrt[1 - c^2*x^2]))*ArcSin[c*x]^2 + b^2*c^3*d*x^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^3 - 3*a^2*c^3*d^(3/2)*x^3*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - b*d*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*(b*c*x + 2*a*(1 - 4*c^2*x^2)*Sqrt[1 - c^2*x^2] + 8*b*c^3*x^3*Log[1 - E^((2*I)*ArcSin[c*x])]) - 8*a*b*c^3*d*x^3*Sqrt[d - c^2*d*x^2]*Log[c*x] + (4*I)*b^2*c^3*d*x^3*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(3*x^3*Sqrt[1 - c^2*x^2])","A",0
226,1,270,651,0.4542685,"\int x^3 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[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(3969 a^2 \left(7 c^2 x^2+2\right) \left(1-c^2 x^2\right)^{7/2}+126 a b c x \left(49 c^8 x^8-171 c^6 x^6+189 c^4 x^4-21 c^2 x^2-126\right)+126 b \sin ^{-1}(c x) \left(63 a \left(7 c^2 x^2+2\right) \left(1-c^2 x^2\right)^{7/2}+b c x \left(49 c^8 x^8-171 c^6 x^6+189 c^4 x^4-21 c^2 x^2-126\right)\right)+3969 b^2 \left(7 c^2 x^2+2\right) \left(1-c^2 x^2\right)^{7/2} \sin ^{-1}(c x)^2+2 b^2 \left(343 c^8 x^8-1147 c^6 x^6+1005 c^4 x^4+899 c^2 x^2-6140\right) \sqrt{1-c^2 x^2}\right)}{250047 c^4 \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}{63 c^2}-\frac{2 b c d^2 x^5 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{21 \sqrt{1-c^2 x^2}}+\frac{1}{21} d^2 x^4 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{2 b d^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{189 c \sqrt{1-c^2 x^2}}+\frac{1}{9} x^4 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{5}{63} d x^4 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2 b c^5 d^2 x^9 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{81 \sqrt{1-c^2 x^2}}-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{63 c^4}+\frac{4 a b d^2 x \sqrt{d-c^2 d x^2}}{63 c^3 \sqrt{1-c^2 x^2}}+\frac{38 b c^3 d^2 x^7 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{441 \sqrt{1-c^2 x^2}}-\frac{2 b^2 d^2 \left(1-c^2 x^2\right)^4 \sqrt{d-c^2 d x^2}}{729 c^4}+\frac{50 b^2 d^2 \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2}}{27783 c^4}+\frac{160 b^2 d^2 \sqrt{d-c^2 d x^2}}{3969 c^4}+\frac{4 b^2 d^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{1323 c^4}+\frac{80 b^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{11907 c^4}+\frac{4 b^2 d^2 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{63 c^3 \sqrt{1-c^2 x^2}}",1,"-1/250047*(d^2*Sqrt[d - c^2*d*x^2]*(3969*a^2*(1 - c^2*x^2)^(7/2)*(2 + 7*c^2*x^2) + 126*a*b*c*x*(-126 - 21*c^2*x^2 + 189*c^4*x^4 - 171*c^6*x^6 + 49*c^8*x^8) + 2*b^2*Sqrt[1 - c^2*x^2]*(-6140 + 899*c^2*x^2 + 1005*c^4*x^4 - 1147*c^6*x^6 + 343*c^8*x^8) + 126*b*(63*a*(1 - c^2*x^2)^(7/2)*(2 + 7*c^2*x^2) + b*c*x*(-126 - 21*c^2*x^2 + 189*c^4*x^4 - 171*c^6*x^6 + 49*c^8*x^8))*ArcSin[c*x] + 3969*b^2*(1 - c^2*x^2)^(7/2)*(2 + 7*c^2*x^2)*ArcSin[c*x]^2))/(c^4*Sqrt[1 - c^2*x^2])","A",1
227,1,348,556,0.4745256,"\int x^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[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(1440 a^3+3 b \sin ^{-1}(c x) \left(1440 a^2+192 a b c x \sqrt{1-c^2 x^2} \left(48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right)+b^2 \left(-1152 c^8 x^8+4352 c^6 x^6-5664 c^4 x^4+1440 c^2 x^2+359\right)\right)+288 a^2 b c x \sqrt{1-c^2 x^2} \left(48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right)-96 a b^2 c^2 x^2 \left(36 c^6 x^6-136 c^4 x^4+177 c^2 x^2-45\right)+288 b^2 \sin ^{-1}(c x)^2 \left(15 a+b c x \sqrt{1-c^2 x^2} \left(48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right)\right)-b^3 c x \sqrt{1-c^2 x^2} \left(432 c^6 x^6-1672 c^4 x^4+2158 c^2 x^2+1077\right)+1440 b^3 \sin ^{-1}(c x)^3\right)}{110592 b c^3 \sqrt{1-c^2 x^2}}","\frac{5 b d^2 x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{128 c \sqrt{1-c^2 x^2}}-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{128 c^2}-\frac{59 b c d^2 x^4 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{384 \sqrt{1-c^2 x^2}}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{8} x^3 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{5}{48} d x^3 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{32 \sqrt{1-c^2 x^2}}+\frac{5 d^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{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{144 \sqrt{1-c^2 x^2}}-\frac{359 b^2 d^2 x \sqrt{d-c^2 d x^2}}{36864 c^2}+\frac{209 b^2 c^2 d^2 x^5 \sqrt{d-c^2 d x^2}}{13824}-\frac{1079 b^2 d^2 x^3 \sqrt{d-c^2 d x^2}}{55296}-\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{d-c^2 d x^2}+\frac{359 b^2 d^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{36864 c^3 \sqrt{1-c^2 x^2}}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(1440*a^3 - 96*a*b^2*c^2*x^2*(-45 + 177*c^2*x^2 - 136*c^4*x^4 + 36*c^6*x^6) + 288*a^2*b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6) - b^3*c*x*Sqrt[1 - c^2*x^2]*(1077 + 2158*c^2*x^2 - 1672*c^4*x^4 + 432*c^6*x^6) + 3*b*(1440*a^2 + 192*a*b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6) + b^2*(359 + 1440*c^2*x^2 - 5664*c^4*x^4 + 4352*c^6*x^6 - 1152*c^8*x^8))*ArcSin[c*x] + 288*b^2*(15*a + b*c*x*Sqrt[1 - c^2*x^2]*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6))*ArcSin[c*x]^2 + 1440*b^3*ArcSin[c*x]^3))/(110592*b*c^3*Sqrt[1 - c^2*x^2])","A",1
228,1,216,382,0.3933995,"\int x \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[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(3675 a^2 \left(1-c^2 x^2\right)^{7/2}+210 a b c x \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right)+210 b \sin ^{-1}(c x) \left(35 a \left(1-c^2 x^2\right)^{7/2}+b c x \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right)\right)+3675 b^2 \left(1-c^2 x^2\right)^{7/2} \sin ^{-1}(c x)^2+2 b^2 \left(75 c^6 x^6-351 c^4 x^4+757 c^2 x^2-2161\right) \sqrt{1-c^2 x^2}\right)}{25725 c^2 \sqrt{1-c^2 x^2}}","\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{7 c \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{7 \sqrt{1-c^2 x^2}}-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \sin ^{-1}(c x)\right)^2}{7 c^2 d}-\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{49 \sqrt{1-c^2 x^2}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{35 \sqrt{1-c^2 x^2}}+\frac{2 b^2 d^2 \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2}}{343 c^2}+\frac{32 b^2 d^2 \sqrt{d-c^2 d x^2}}{245 c^2}+\frac{12 b^2 d^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{1225 c^2}+\frac{16 b^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{735 c^2}",1,"-1/25725*(d^2*Sqrt[d - c^2*d*x^2]*(3675*a^2*(1 - c^2*x^2)^(7/2) + 210*a*b*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + 2*b^2*Sqrt[1 - c^2*x^2]*(-2161 + 757*c^2*x^2 - 351*c^4*x^4 + 75*c^6*x^6) + 210*b*(35*a*(1 - c^2*x^2)^(7/2) + b*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6))*ArcSin[c*x] + 3675*b^2*(1 - c^2*x^2)^(7/2)*ArcSin[c*x]^2))/(c^2*Sqrt[1 - c^2*x^2])","A",1
229,1,407,438,2.0854291,"\int \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]","\frac{d^2 \left(\sqrt{d-c^2 d x^2} \left(9504 a^2 c x \sqrt{1-c^2 x^2}+2304 a^2 c^5 x^5 \sqrt{1-c^2 x^2}-7488 a^2 c^3 x^3 \sqrt{1-c^2 x^2}+3240 a b \cos \left(2 \sin ^{-1}(c x)\right)+324 a b \cos \left(4 \sin ^{-1}(c x)\right)+24 a b \cos \left(6 \sin ^{-1}(c x)\right)-1620 b^2 \sin \left(2 \sin ^{-1}(c x)\right)-81 b^2 \sin \left(4 \sin ^{-1}(c x)\right)-4 b^2 \sin \left(6 \sin ^{-1}(c x)\right)\right)-4320 a^2 \sqrt{d} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+72 b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \left(60 a+45 b \sin \left(2 \sin ^{-1}(c x)\right)+9 b \sin \left(4 \sin ^{-1}(c x)\right)+b \sin \left(6 \sin ^{-1}(c x)\right)\right)+12 b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \left(540 a \sin \left(2 \sin ^{-1}(c x)\right)+108 a \sin \left(4 \sin ^{-1}(c x)\right)+12 a \sin \left(6 \sin ^{-1}(c x)\right)+270 b \cos \left(2 \sin ^{-1}(c x)\right)+27 b \cos \left(4 \sin ^{-1}(c x)\right)+2 b \cos \left(6 \sin ^{-1}(c x)\right)\right)+1440 b^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^3\right)}{13824 c \sqrt{1-c^2 x^2}}","\frac{5 d^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}{16} d^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{b d^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 \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{5 b c d^2 x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{16 \sqrt{1-c^2 x^2}}+\frac{1}{6} x \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{5}{24} d x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{1}{108} b^2 d^2 x \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}-\frac{245 b^2 d^2 x \sqrt{d-c^2 d x^2}}{1152}-\frac{65 b^2 d^2 x \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{1728}+\frac{115 b^2 d^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{1152 c \sqrt{1-c^2 x^2}}",1,"(d^2*(1440*b^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^3 - 4320*a^2*Sqrt[d]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 12*b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*(270*b*Cos[2*ArcSin[c*x]] + 27*b*Cos[4*ArcSin[c*x]] + 2*b*Cos[6*ArcSin[c*x]] + 540*a*Sin[2*ArcSin[c*x]] + 108*a*Sin[4*ArcSin[c*x]] + 12*a*Sin[6*ArcSin[c*x]]) + 72*b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*(60*a + 45*b*Sin[2*ArcSin[c*x]] + 9*b*Sin[4*ArcSin[c*x]] + b*Sin[6*ArcSin[c*x]]) + Sqrt[d - c^2*d*x^2]*(9504*a^2*c*x*Sqrt[1 - c^2*x^2] - 7488*a^2*c^3*x^3*Sqrt[1 - c^2*x^2] + 2304*a^2*c^5*x^5*Sqrt[1 - c^2*x^2] + 3240*a*b*Cos[2*ArcSin[c*x]] + 324*a*b*Cos[4*ArcSin[c*x]] + 24*a*b*Cos[6*ArcSin[c*x]] - 1620*b^2*Sin[2*ArcSin[c*x]] - 81*b^2*Sin[4*ArcSin[c*x]] - 4*b^2*Sin[6*ArcSin[c*x]])))/(13824*c*Sqrt[1 - c^2*x^2])","A",1
230,1,775,687,5.0117119,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/x,x]","\frac{d^2 \left(54000 a^2 \sqrt{d} \sqrt{1-c^2 x^2} \log (c x)-54000 a^2 \sqrt{d} \sqrt{1-c^2 x^2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+3600 a^2 \sqrt{1-c^2 x^2} \left(3 c^4 x^4-11 c^2 x^2+23\right) \sqrt{d-c^2 d x^2}-108000 a b \sqrt{d-c^2 d x^2} \left(-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)-i \left(\text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right)+c x-\sin ^{-1}(c x) \left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)\right)-6000 a b \sqrt{d-c^2 d x^2} \left(-3 \sin ^{-1}(c x) \left(3 \sqrt{1-c^2 x^2}+\cos \left(3 \sin ^{-1}(c x)\right)\right)+9 c x+\sin \left(3 \sin ^{-1}(c x)\right)\right)+30 a b \sqrt{d-c^2 d x^2} \left(-15 \sin ^{-1}(c x) \left(30 \sqrt{1-c^2 x^2}+5 \cos \left(3 \sin ^{-1}(c x)\right)-3 \cos \left(5 \sin ^{-1}(c x)\right)\right)+450 c x+25 \sin \left(3 \sin ^{-1}(c x)\right)-9 \sin \left(5 \sin ^{-1}(c x)\right)\right)-54000 b^2 \sqrt{d-c^2 d x^2} \left(2 \sqrt{1-c^2 x^2}-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2-2 i \sin ^{-1}(c x) \left(\text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right)+2 \left(\text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)\right)+2 c x \sin ^{-1}(c x)-\left(\sin ^{-1}(c x)^2 \left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)\right)\right)+1000 b^2 \sqrt{d-c^2 d x^2} \left(27 \sqrt{1-c^2 x^2} \left(\sin ^{-1}(c x)^2-2\right)-6 \sin ^{-1}(c x) \left(9 c x+\sin \left(3 \sin ^{-1}(c x)\right)\right)+\left(9 \sin ^{-1}(c x)^2-2\right) \cos \left(3 \sin ^{-1}(c x)\right)\right)-b^2 \sqrt{d-c^2 d x^2} \left(6750 \sqrt{1-c^2 x^2} \left(\sin ^{-1}(c x)^2-2\right)+30 \sin ^{-1}(c x) \left(9 \left(\sin \left(5 \sin ^{-1}(c x)\right)-50 c x\right)-25 \sin \left(3 \sin ^{-1}(c x)\right)\right)+125 \left(9 \sin ^{-1}(c x)^2-2\right) \cos \left(3 \sin ^{-1}(c x)\right)-27 \left(25 \sin ^{-1}(c x)^2-2\right) \cos \left(5 \sin ^{-1}(c x)\right)\right)\right)}{54000 \sqrt{1-c^2 x^2}}","\frac{2 i b d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{2 i b d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{2 a b c d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{16 b c d^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{15 \sqrt{1-c^2 x^2}}+d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2 d^2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}}+\frac{1}{5} \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{3} d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2 b c^5 d^2 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{22 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{45 \sqrt{1-c^2 x^2}}-\frac{2 b^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}+\frac{2 b^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{598}{225} b^2 d^2 \sqrt{d-c^2 d x^2}-\frac{2}{125} b^2 d^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}-\frac{74}{675} b^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}-\frac{2 b^2 c d^2 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}",1,"(d^2*(3600*a^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(23 - 11*c^2*x^2 + 3*c^4*x^4) + 54000*a^2*Sqrt[d]*Sqrt[1 - c^2*x^2]*Log[c*x] - 54000*a^2*Sqrt[d]*Sqrt[1 - c^2*x^2]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] - 108000*a*b*Sqrt[d - c^2*d*x^2]*(c*x - Sqrt[1 - c^2*x^2]*ArcSin[c*x] - ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) - I*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])])) - 54000*b^2*Sqrt[d - c^2*d*x^2]*(2*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 - ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) - (2*I)*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]) + 2*(PolyLog[3, -E^(I*ArcSin[c*x])] - PolyLog[3, E^(I*ArcSin[c*x])])) - 6000*a*b*Sqrt[d - c^2*d*x^2]*(9*c*x - 3*ArcSin[c*x]*(3*Sqrt[1 - c^2*x^2] + Cos[3*ArcSin[c*x]]) + Sin[3*ArcSin[c*x]]) + 1000*b^2*Sqrt[d - c^2*d*x^2]*(27*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]*(9*c*x + Sin[3*ArcSin[c*x]])) + 30*a*b*Sqrt[d - c^2*d*x^2]*(450*c*x - 15*ArcSin[c*x]*(30*Sqrt[1 - c^2*x^2] + 5*Cos[3*ArcSin[c*x]] - 3*Cos[5*ArcSin[c*x]]) + 25*Sin[3*ArcSin[c*x]] - 9*Sin[5*ArcSin[c*x]]) - b^2*Sqrt[d - c^2*d*x^2]*(6750*Sqrt[1 - c^2*x^2]*(-2 + ArcSin[c*x]^2) + 125*(-2 + 9*ArcSin[c*x]^2)*Cos[3*ArcSin[c*x]] - 27*(-2 + 25*ArcSin[c*x]^2)*Cos[5*ArcSin[c*x]] + 30*ArcSin[c*x]*(-25*Sin[3*ArcSin[c*x]] + 9*(-50*c*x + Sin[5*ArcSin[c*x]])))))/(54000*Sqrt[1 - c^2*x^2])","A",0
231,1,586,561,2.3132136,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/x^2,x]","\frac{d^2 \left(-288 a^2 c^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}-256 a^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}+480 a^2 c \sqrt{d} x \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+64 a^2 c^4 x^4 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}+512 a b c x \sqrt{d-c^2 d x^2} \log (c x)-8 b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \left(60 a c x+32 b \sqrt{1-c^2 x^2}+32 i b c x+16 b c x \sin \left(2 \sin ^{-1}(c x)\right)+b c x \sin \left(4 \sin ^{-1}(c x)\right)\right)-128 a b c x \sqrt{d-c^2 d x^2} \cos \left(2 \sin ^{-1}(c x)\right)-4 a b c x \sqrt{d-c^2 d x^2} \cos \left(4 \sin ^{-1}(c x)\right)-4 b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \left(128 a \sqrt{1-c^2 x^2}+64 a c x \sin \left(2 \sin ^{-1}(c x)\right)+4 a c x \sin \left(4 \sin ^{-1}(c x)\right)-128 b c x \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+32 b c x \cos \left(2 \sin ^{-1}(c x)\right)+b c x \cos \left(4 \sin ^{-1}(c x)\right)\right)-256 i b^2 c x \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-160 b^2 c x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^3+64 b^2 c x \sqrt{d-c^2 d x^2} \sin \left(2 \sin ^{-1}(c x)\right)+b^2 c x \sqrt{d-c^2 d x^2} \sin \left(4 \sin ^{-1}(c x)\right)\right)}{256 x \sqrt{1-c^2 x^2}}","-\frac{15}{8} c^2 d^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{5 c d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{8 b \sqrt{1-c^2 x^2}}-\frac{i c d^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}{8} b c d^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)+b c d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)+\frac{2 b c d^2 \sqrt{d-c^2 d x^2} \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{5}{4} c^2 d x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{x}+\frac{15 b c^3 d^2 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{i b^2 c d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}+\frac{31}{64} b^2 c^2 d^2 x \sqrt{d-c^2 d x^2}+\frac{1}{32} b^2 c^2 d^2 x \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}-\frac{89 b^2 c d^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{64 \sqrt{1-c^2 x^2}}",1,"(d^2*(-256*a^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] - 288*a^2*c^2*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + 64*a^2*c^4*x^4*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] - 160*b^2*c*x*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^3 + 480*a^2*c*Sqrt[d]*x*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 128*a*b*c*x*Sqrt[d - c^2*d*x^2]*Cos[2*ArcSin[c*x]] - 4*a*b*c*x*Sqrt[d - c^2*d*x^2]*Cos[4*ArcSin[c*x]] + 512*a*b*c*x*Sqrt[d - c^2*d*x^2]*Log[c*x] - (256*I)*b^2*c*x*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])] + 64*b^2*c*x*Sqrt[d - c^2*d*x^2]*Sin[2*ArcSin[c*x]] + b^2*c*x*Sqrt[d - c^2*d*x^2]*Sin[4*ArcSin[c*x]] - 4*b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*(128*a*Sqrt[1 - c^2*x^2] + 32*b*c*x*Cos[2*ArcSin[c*x]] + b*c*x*Cos[4*ArcSin[c*x]] - 128*b*c*x*Log[1 - E^((2*I)*ArcSin[c*x])] + 64*a*c*x*Sin[2*ArcSin[c*x]] + 4*a*c*x*Sin[4*ArcSin[c*x]]) - 8*b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*(60*a*c*x + (32*I)*b*c*x + 32*b*Sqrt[1 - c^2*x^2] + 16*b*c*x*Sin[2*ArcSin[c*x]] + b*c*x*Sin[4*ArcSin[c*x]])))/(256*x*Sqrt[1 - c^2*x^2])","A",0
232,1,1073,740,7.4672699,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/x^3,x]","\frac{a b c^2 \sqrt{1-c^2 x^2} \left(-\sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)-2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)-4 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+4 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)-4 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+4 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) d^3}{4 \sqrt{d \left(1-c^2 x^2\right)}}+\frac{b^2 c^2 \sqrt{1-c^2 x^2} \left(-\csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2+\sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2-4 \log \left(1-e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)^2+4 \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)^2-4 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-8 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+8 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-4 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+8 \log \left(\tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+8 \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)-8 \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)\right) d^3}{8 \sqrt{d \left(1-c^2 x^2\right)}}-\frac{5}{2} a^2 c^2 \log (x) d^{5/2}+\frac{5}{2} a^2 c^2 \log \left(d+\sqrt{-d \left(c^2 x^2-1\right)} \sqrt{d}\right) d^{5/2}-4 a b c^2 \sqrt{d \left(1-c^2 x^2\right)} \left(-\frac{c x}{\sqrt{1-c^2 x^2}}+\sin ^{-1}(c x)+\frac{\sin ^{-1}(c x) \left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}+\frac{i \left(\text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}\right) d^2-2 b^2 c^2 \sqrt{d \left(1-c^2 x^2\right)} \left(\frac{\left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}+\sin ^{-1}(c x)^2+\frac{2 i \left(\text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{2 c x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{2 \left(\text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)-\text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}-2\right) d^2-\frac{a b c^2 \sqrt{d \left(1-c^2 x^2\right)} \left(-9 c x+9 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+3 \sin ^{-1}(c x) \cos \left(3 \sin ^{-1}(c x)\right)-\sin \left(3 \sin ^{-1}(c x)\right)\right) d^2}{18 \sqrt{1-c^2 x^2}}-\frac{b^2 c^2 \sqrt{d \left(1-c^2 x^2\right)} \left(27 \sqrt{1-c^2 x^2} \left(\sin ^{-1}(c x)^2-2\right)+\left(9 \sin ^{-1}(c x)^2-2\right) \cos \left(3 \sin ^{-1}(c x)\right)-6 \sin ^{-1}(c x) \left(9 c x+\sin \left(3 \sin ^{-1}(c x)\right)\right)\right) d^2}{108 \sqrt{1-c^2 x^2}}+\sqrt{-d \left(c^2 x^2-1\right)} \left(\frac{1}{3} a^2 d^2 x^2 c^4-\frac{7}{3} a^2 d^2 c^2-\frac{a^2 d^2}{2 x^2}\right)","-\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}+\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{b c d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{x \sqrt{1-c^2 x^2}}+\frac{5 c^2 d^2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}}-\frac{5}{6} c^2 d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 x^2}-\frac{2 b c^5 d^2 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{5 a b c^3 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 \sqrt{1-c^2 x^2}}+\frac{5 b^2 c^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{5 b^2 c^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}+\frac{40}{9} b^2 c^2 d^2 \sqrt{d-c^2 d x^2}+\frac{2}{27} b^2 c^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}-\frac{b^2 c^2 d^2 \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{\sqrt{1-c^2 x^2}}+\frac{5 b^2 c^3 d^2 x \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*((-7*a^2*c^2*d^2)/3 - (a^2*d^2)/(2*x^2) + (a^2*c^4*d^2*x^2)/3) - (5*a^2*c^2*d^(5/2)*Log[x])/2 + (5*a^2*c^2*d^(5/2)*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/2 - 4*a*b*c^2*d^2*Sqrt[d*(1 - c^2*x^2)]*(-((c*x)/Sqrt[1 - c^2*x^2]) + ArcSin[c*x] + (ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (I*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2]) - 2*b^2*c^2*d^2*Sqrt[d*(1 - c^2*x^2)]*(-2 - (2*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] + ArcSin[c*x]^2 + (ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + ((2*I)*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (2*(-PolyLog[3, -E^(I*ArcSin[c*x])] + PolyLog[3, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2]) - (a*b*c^2*d^2*Sqrt[d*(1 - c^2*x^2)]*(-9*c*x + 9*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 3*ArcSin[c*x]*Cos[3*ArcSin[c*x]] - Sin[3*ArcSin[c*x]]))/(18*Sqrt[1 - c^2*x^2]) - (b^2*c^2*d^2*Sqrt[d*(1 - c^2*x^2)]*(27*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]*(9*c*x + Sin[3*ArcSin[c*x]])))/(108*Sqrt[1 - c^2*x^2]) + (a*b*c^2*d^3*Sqrt[1 - c^2*x^2]*(-2*Cot[ArcSin[c*x]/2] - ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (4*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 - 2*Tan[ArcSin[c*x]/2]))/(4*Sqrt[d*(1 - c^2*x^2)]) + (b^2*c^2*d^3*Sqrt[1 - c^2*x^2]*(-4*ArcSin[c*x]*Cot[ArcSin[c*x]/2] - ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] + 8*Log[Tan[ArcSin[c*x]/2]] - (8*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] + (8*I)*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] + 8*PolyLog[3, -E^(I*ArcSin[c*x])] - 8*PolyLog[3, E^(I*ArcSin[c*x])] + ArcSin[c*x]^2*Sec[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Tan[ArcSin[c*x]/2]))/(8*Sqrt[d*(1 - c^2*x^2)])","A",0
233,1,690,591,3.9167254,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2)/x^4,x]","\frac{d^2 \left(28 a^2 c^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}-4 a^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}+6 a^2 c^4 x^4 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}-30 a^2 c^3 \sqrt{d} x^3 \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-4 a b c x \sqrt{d-c^2 d x^2}-6 a b c^5 x^5 \sqrt{d-c^2 d x^2}+3 a b c^3 x^3 \sqrt{d-c^2 d x^2}-56 a b c^3 x^3 \sqrt{d-c^2 d x^2} \log (c x)+b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 \left(30 a c^3 x^3+3 b c^3 x^3 \sin \left(2 \sin ^{-1}(c x)\right)+4 b \left(7 i c^3 x^3+7 c^2 x^2 \sqrt{1-c^2 x^2}-\sqrt{1-c^2 x^2}\right)\right)+b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \left(6 a c^3 x^3 \sin \left(2 \sin ^{-1}(c x)\right)+48 a c^2 x^2 \sqrt{1-c^2 x^2}-6 a \sqrt{1-c^2 x^2}-2 a \cos \left(3 \sin ^{-1}(c x)\right)-56 b c^3 x^3 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+3 b c^3 x^3 \cos \left(2 \sin ^{-1}(c x)\right)-4 b c x\right)-4 b^2 c^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}-3 b^2 c^4 x^4 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}+28 i b^2 c^3 x^3 \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+10 b^2 c^3 x^3 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^3\right)}{12 x^3 \sqrt{1-c^2 x^2}}","-\frac{b c d^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)}{3 x^2}+\frac{5 c^2 d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 x}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 x^3}-\frac{5 b c^5 d^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{5}{2} c^4 d^2 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{5 c^3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{6 b \sqrt{1-c^2 x^2}}+\frac{7 i c^3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 \sqrt{1-c^2 x^2}}-\frac{7}{3} b c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)-\frac{14 b c^3 d^2 \sqrt{d-c^2 d x^2} \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 \sqrt{1-c^2 x^2}}-\frac{b^2 c^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{3 x}-\frac{7}{12} b^2 c^4 d^2 x \sqrt{d-c^2 d x^2}+\frac{7 i b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{3 \sqrt{1-c^2 x^2}}+\frac{23 b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{12 \sqrt{1-c^2 x^2}}",1,"(d^2*(-4*a*b*c*x*Sqrt[d - c^2*d*x^2] + 3*a*b*c^3*x^3*Sqrt[d - c^2*d*x^2] - 6*a*b*c^5*x^5*Sqrt[d - c^2*d*x^2] - 4*a^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + 28*a^2*c^2*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] - 4*b^2*c^2*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + 6*a^2*c^4*x^4*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] - 3*b^2*c^4*x^4*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2] + 10*b^2*c^3*x^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^3 - 30*a^2*c^3*Sqrt[d]*x^3*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 56*a*b*c^3*x^3*Sqrt[d - c^2*d*x^2]*Log[c*x] + (28*I)*b^2*c^3*x^3*Sqrt[d - c^2*d*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])] + b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*(-4*b*c*x - 6*a*Sqrt[1 - c^2*x^2] + 48*a*c^2*x^2*Sqrt[1 - c^2*x^2] + 3*b*c^3*x^3*Cos[2*ArcSin[c*x]] - 2*a*Cos[3*ArcSin[c*x]] - 56*b*c^3*x^3*Log[1 - E^((2*I)*ArcSin[c*x])] + 6*a*c^3*x^3*Sin[2*ArcSin[c*x]]) + b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2*(30*a*c^3*x^3 + 4*b*((7*I)*c^3*x^3 - Sqrt[1 - c^2*x^2] + 7*c^2*x^2*Sqrt[1 - c^2*x^2]) + 3*b*c^3*x^3*Sin[2*ArcSin[c*x]])))/(12*x^3*Sqrt[1 - c^2*x^2])","A",0
234,1,230,400,0.2004942,"\int \frac{x^5 \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^5*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\frac{225 a^2 \left(3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right)+30 a b c x \sqrt{1-c^2 x^2} \left(9 c^4 x^4+20 c^2 x^2+120\right)+30 b \sin ^{-1}(c x) \left(15 a \left(3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right)+b c x \sqrt{1-c^2 x^2} \left(9 c^4 x^4+20 c^2 x^2+120\right)\right)-2 b^2 \left(27 c^6 x^6+109 c^4 x^4+1936 c^2 x^2-2072\right)+225 b^2 \left(3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right) \sin ^{-1}(c x)^2}{3375 c^6 \sqrt{d-c^2 d x^2}}","\frac{2 b x^5 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{25 c \sqrt{d-c^2 d x^2}}-\frac{x^4 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{5 c^2 d}-\frac{8 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{15 c^6 d}+\frac{16 a b x \sqrt{1-c^2 x^2}}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{15 c^4 d}+\frac{8 b x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{45 c^3 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \left(1-c^2 x^2\right)^3}{125 c^6 \sqrt{d-c^2 d x^2}}-\frac{76 b^2 \left(1-c^2 x^2\right)^2}{675 c^6 \sqrt{d-c^2 d x^2}}+\frac{298 b^2 \left(1-c^2 x^2\right)}{225 c^6 \sqrt{d-c^2 d x^2}}+\frac{16 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{15 c^5 \sqrt{d-c^2 d x^2}}",1,"(30*a*b*c*x*Sqrt[1 - c^2*x^2]*(120 + 20*c^2*x^2 + 9*c^4*x^4) + 225*a^2*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6) - 2*b^2*(-2072 + 1936*c^2*x^2 + 109*c^4*x^4 + 27*c^6*x^6) + 30*b*(b*c*x*Sqrt[1 - c^2*x^2]*(120 + 20*c^2*x^2 + 9*c^4*x^4) + 15*a*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6))*ArcSin[c*x] + 225*b^2*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6)*ArcSin[c*x]^2)/(3375*c^6*Sqrt[d - c^2*d*x^2])","A",1
235,1,283,337,1.5537968,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\frac{32 a^2 c \sqrt{d} x \left(c^2 x^2-1\right) \left(2 c^2 x^2+3\right)-96 a^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)-4 a b \sqrt{d} \sqrt{1-c^2 x^2} \left(-4 \sin ^{-1}(c x) \left(6 \sin ^{-1}(c x)-8 \sin \left(2 \sin ^{-1}(c x)\right)+\sin \left(4 \sin ^{-1}(c x)\right)\right)+16 \cos \left(2 \sin ^{-1}(c x)\right)-\cos \left(4 \sin ^{-1}(c x)\right)\right)+b^2 \sqrt{d} \sqrt{1-c^2 x^2} \left(32 \sin ^{-1}(c x)^3+8 \left(\sin \left(4 \sin ^{-1}(c x)\right)-8 \sin \left(2 \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^2+32 \sin \left(2 \sin ^{-1}(c x)\right)-\sin \left(4 \sin ^{-1}(c x)\right)+4 \sin ^{-1}(c x) \left(\cos \left(4 \sin ^{-1}(c x)\right)-16 \cos \left(2 \sin ^{-1}(c x)\right)\right)\right)}{256 c^5 \sqrt{d} \sqrt{d-c^2 d x^2}}","\frac{b x^4 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 c \sqrt{d-c^2 d x^2}}-\frac{x^3 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{4 c^2 d}+\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{8 b c^5 \sqrt{d-c^2 d x^2}}-\frac{3 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{8 c^4 d}+\frac{3 b x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{8 c^3 \sqrt{d-c^2 d x^2}}+\frac{b^2 x^3 \left(1-c^2 x^2\right)}{32 c^2 \sqrt{d-c^2 d x^2}}-\frac{15 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{64 c^5 \sqrt{d-c^2 d x^2}}+\frac{15 b^2 x \left(1-c^2 x^2\right)}{64 c^4 \sqrt{d-c^2 d x^2}}",1,"(32*a^2*c*Sqrt[d]*x*(-1 + c^2*x^2)*(3 + 2*c^2*x^2) - 96*a^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))] + b^2*Sqrt[d]*Sqrt[1 - c^2*x^2]*(32*ArcSin[c*x]^3 + 4*ArcSin[c*x]*(-16*Cos[2*ArcSin[c*x]] + Cos[4*ArcSin[c*x]]) + 32*Sin[2*ArcSin[c*x]] - Sin[4*ArcSin[c*x]] + 8*ArcSin[c*x]^2*(-8*Sin[2*ArcSin[c*x]] + Sin[4*ArcSin[c*x]])) - 4*a*b*Sqrt[d]*Sqrt[1 - c^2*x^2]*(16*Cos[2*ArcSin[c*x]] - Cos[4*ArcSin[c*x]] - 4*ArcSin[c*x]*(6*ArcSin[c*x] - 8*Sin[2*ArcSin[c*x]] + Sin[4*ArcSin[c*x]])))/(256*c^5*Sqrt[d]*Sqrt[d - c^2*d*x^2])","A",1
236,1,176,277,0.1348273,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^3*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\frac{9 a^2 \left(c^4 x^4+c^2 x^2-2\right)+6 a b c x \sqrt{1-c^2 x^2} \left(c^2 x^2+6\right)+6 b \sin ^{-1}(c x) \left(3 a \left(c^4 x^4+c^2 x^2-2\right)+b c x \sqrt{1-c^2 x^2} \left(c^2 x^2+6\right)\right)-2 b^2 \left(c^4 x^4+19 c^2 x^2-20\right)+9 b^2 \left(c^4 x^4+c^2 x^2-2\right) \sin ^{-1}(c x)^2}{27 c^4 \sqrt{d-c^2 d x^2}}","-\frac{x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^2 d}+\frac{2 b 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 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^4 d}+\frac{4 a b x \sqrt{1-c^2 x^2}}{3 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \left(1-c^2 x^2\right)^2}{27 c^4 \sqrt{d-c^2 d x^2}}+\frac{14 b^2 \left(1-c^2 x^2\right)}{9 c^4 \sqrt{d-c^2 d x^2}}+\frac{4 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 \sqrt{d-c^2 d x^2}}",1,"(6*a*b*c*x*Sqrt[1 - c^2*x^2]*(6 + c^2*x^2) + 9*a^2*(-2 + c^2*x^2 + c^4*x^4) - 2*b^2*(-20 + 19*c^2*x^2 + c^4*x^4) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(6 + c^2*x^2) + 3*a*(-2 + c^2*x^2 + c^4*x^4))*ArcSin[c*x] + 9*b^2*(-2 + c^2*x^2 + c^4*x^4)*ArcSin[c*x]^2)/(27*c^4*Sqrt[d - c^2*d*x^2])","A",1
237,1,210,206,1.3443474,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\frac{12 a^2 c d x \left(c^2 x^2-1\right)-12 a^2 \sqrt{d} \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)-6 a b d \sqrt{1-c^2 x^2} \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)+b^2 d \sqrt{1-c^2 x^2} \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)}{24 c^3 d \sqrt{d-c^2 d x^2}}","-\frac{x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 c^2 d}+\frac{b 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{\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{b^2 x \sqrt{d-c^2 d x^2}}{4 c^2 d}-\frac{b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c^3 \sqrt{d-c^2 d x^2}}",1,"(12*a^2*c*d*x*(-1 + c^2*x^2) - 12*a^2*Sqrt[d]*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*a*b*d*Sqrt[1 - c^2*x^2]*(-2*ArcSin[c*x]^2 + Cos[2*ArcSin[c*x]] + 2*ArcSin[c*x]*Sin[2*ArcSin[c*x]]) + b^2*d*Sqrt[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]]))/(24*c^3*d*Sqrt[d - c^2*d*x^2])","A",1
238,1,86,146,0.0978067,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\frac{\left(c^2 x^2-1\right) \left(a+b \sin ^{-1}(c x)\right)^2+2 b \sqrt{1-c^2 x^2} \left(a c x+b \sqrt{1-c^2 x^2}+b c x \sin ^{-1}(c x)\right)}{c^2 \sqrt{d-c^2 d x^2}}","\frac{2 a b x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 d}+\frac{2 b^2 \left(1-c^2 x^2\right)}{c^2 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt{d-c^2 d x^2}}",1,"((-1 + c^2*x^2)*(a + b*ArcSin[c*x])^2 + 2*b*Sqrt[1 - c^2*x^2]*(a*c*x + b*Sqrt[1 - c^2*x^2] + b*c*x*ArcSin[c*x]))/(c^2*Sqrt[d - c^2*d*x^2])","A",1
239,1,64,49,0.1293654,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/Sqrt[d - c^2*d*x^2],x]","\frac{\sqrt{1-c^2 x^2} \sin ^{-1}(c x) \left(3 a^2+3 a b \sin ^{-1}(c x)+b^2 \sin ^{-1}(c x)^2\right)}{3 c \sqrt{d-c^2 d x^2}}","\frac{\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}}",1,"(Sqrt[1 - c^2*x^2]*ArcSin[c*x]*(3*a^2 + 3*a*b*ArcSin[c*x] + b^2*ArcSin[c*x]^2))/(3*c*Sqrt[d - c^2*d*x^2])","A",1
240,1,301,257,0.6883358,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x*Sqrt[d - c^2*d*x^2]),x]","-\frac{a^2 \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)}{\sqrt{d}}+\frac{a^2 \log (c x)}{\sqrt{d}}+\frac{2 a b \sqrt{1-c^2 x^2} \left(i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+\sin ^{-1}(c x) \left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)\right)}{\sqrt{d-c^2 d x^2}}+\frac{b^2 \sqrt{1-c^2 x^2} \left(2 i \sin ^{-1}(c x) \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-2 i \sin ^{-1}(c x) \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-2 \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)+2 \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)+\sin ^{-1}(c x)^2 \log \left(1-e^{i \sin ^{-1}(c x)}\right)-\sin ^{-1}(c x)^2 \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{d-c^2 d x^2}}","\frac{2 i b \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}}-\frac{2 i b \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}}-\frac{2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{d-c^2 d x^2}}+\frac{2 b^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{d-c^2 d x^2}}",1,"(a^2*Log[c*x])/Sqrt[d] - (a^2*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/Sqrt[d] + (2*a*b*Sqrt[1 - c^2*x^2]*(ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + I*PolyLog[2, -E^(I*ArcSin[c*x])] - I*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[d - c^2*d*x^2] + (b^2*Sqrt[1 - c^2*x^2]*(ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] - ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] + (2*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] - (2*I)*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] - 2*PolyLog[3, -E^(I*ArcSin[c*x])] + 2*PolyLog[3, E^(I*ArcSin[c*x])]))/Sqrt[d - c^2*d*x^2]","A",0
241,1,159,183,0.4515071,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^2 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^2*Sqrt[d - c^2*d*x^2]),x]","-\frac{\sqrt{1-c^2 x^2} \left(a \left(a \sqrt{1-c^2 x^2}-2 b c x \log (c x)\right)+2 b \sin ^{-1}(c x) \left(a \sqrt{1-c^2 x^2}-b c x \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)\right)+b^2 \left(\sqrt{1-c^2 x^2}+i c x\right) \sin ^{-1}(c x)^2+i b^2 c x \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)\right)}{x \sqrt{d-c^2 d x^2}}","-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{d x}-\frac{i c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}}+\frac{2 b c \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)}{\sqrt{d-c^2 d x^2}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{\sqrt{d-c^2 d x^2}}",1,"-((Sqrt[1 - c^2*x^2]*(b^2*(I*c*x + Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + 2*b*ArcSin[c*x]*(a*Sqrt[1 - c^2*x^2] - b*c*x*Log[1 - E^((2*I)*ArcSin[c*x])]) + a*(a*Sqrt[1 - c^2*x^2] - 2*b*c*x*Log[c*x]) + I*b^2*c*x*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/(x*Sqrt[d - c^2*d*x^2]))","A",0
242,1,487,402,6.2538738,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^3 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^3*Sqrt[d - c^2*d*x^2]),x]","\frac{-\frac{4 a^2 \sqrt{d-c^2 d x^2}}{x^2}-4 a^2 c^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+4 a^2 c^2 \sqrt{d} \log (x)+\frac{2 a b c^2 d^2 \left(1-c^2 x^2\right)^{3/2} \left(4 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-4 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+4 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-4 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)-2 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)-2 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{\left(d-c^2 d x^2\right)^{3/2}}+\frac{b^2 c^2 d^2 \left(1-c^2 x^2\right)^{3/2} \left(8 i \sin ^{-1}(c x) \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-8 i \sin ^{-1}(c x) \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-8 \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)+8 \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)+4 \sin ^{-1}(c x)^2 \log \left(1-e^{i \sin ^{-1}(c x)}\right)-4 \sin ^{-1}(c x)^2 \log \left(1+e^{i \sin ^{-1}(c x)}\right)-4 \sin ^{-1}(c x) \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)-4 \sin ^{-1}(c x) \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin ^{-1}(c x)^2 \left(-\csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\sin ^{-1}(c x)^2 \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+8 \log \left(\tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)}{\left(d-c^2 d x^2\right)^{3/2}}}{8 d}","\frac{i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}}-\frac{i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{x \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 d x^2}-\frac{c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{d-c^2 d x^2}}+\frac{b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{\sqrt{d-c^2 d x^2}}",1,"((-4*a^2*Sqrt[d - c^2*d*x^2])/x^2 + 4*a^2*c^2*Sqrt[d]*Log[x] - 4*a^2*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (2*a*b*c^2*d^2*(1 - c^2*x^2)^(3/2)*(-2*Cot[ArcSin[c*x]/2] - ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 + 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, -E^(I*ArcSin[c*x])] - (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 - 2*Tan[ArcSin[c*x]/2]))/(d - c^2*d*x^2)^(3/2) + (b^2*c^2*d^2*(1 - c^2*x^2)^(3/2)*(-4*ArcSin[c*x]*Cot[ArcSin[c*x]/2] - ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^2 + 4*ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] - 4*ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] + 8*Log[Tan[ArcSin[c*x]/2]] + (8*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] - (8*I)*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] - 8*PolyLog[3, -E^(I*ArcSin[c*x])] + 8*PolyLog[3, E^(I*ArcSin[c*x])] + ArcSin[c*x]^2*Sec[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Tan[ArcSin[c*x]/2]))/(d - c^2*d*x^2)^(3/2))/(8*d)","A",0
243,1,269,319,0.8178255,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^4 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^4*Sqrt[d - c^2*d*x^2]),x]","-\frac{\sqrt{1-c^2 x^2} \left(2 a^2 c^2 x^2 \sqrt{1-c^2 x^2}+a^2 \sqrt{1-c^2 x^2}-4 a b c^3 x^3 \log (c x)-b \sin ^{-1}(c x) \left(-2 a \sqrt{1-c^2 x^2} \left(2 c^2 x^2+1\right)+4 b c^3 x^3 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-b c x\right)+a b c x+2 i b^2 c^3 x^3 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+b^2 c^2 x^2 \sqrt{1-c^2 x^2}+b^2 \left(2 i c^3 x^3+2 c^2 x^2 \sqrt{1-c^2 x^2}+\sqrt{1-c^2 x^2}\right) \sin ^{-1}(c x)^2\right)}{3 x^3 \sqrt{d-c^2 d x^2}}","-\frac{2 c^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 d x}-\frac{b c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 x^2 \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 d x^3}-\frac{2 i c^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 \sqrt{d-c^2 d x^2}}+\frac{4 b c^3 \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 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \left(1-c^2 x^2\right)}{3 x \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 c^3 \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{3 \sqrt{d-c^2 d x^2}}",1,"-1/3*(Sqrt[1 - c^2*x^2]*(a*b*c*x + a^2*Sqrt[1 - c^2*x^2] + 2*a^2*c^2*x^2*Sqrt[1 - c^2*x^2] + b^2*c^2*x^2*Sqrt[1 - c^2*x^2] + b^2*((2*I)*c^3*x^3 + Sqrt[1 - c^2*x^2] + 2*c^2*x^2*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 - b*ArcSin[c*x]*(-(b*c*x) - 2*a*Sqrt[1 - c^2*x^2]*(1 + 2*c^2*x^2) + 4*b*c^3*x^3*Log[1 - E^((2*I)*ArcSin[c*x])]) - 4*a*b*c^3*x^3*Log[c*x] + (2*I)*b^2*c^3*x^3*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/(x^3*Sqrt[d - c^2*d*x^2])","A",0
244,1,453,549,0.8560538,"\int \frac{x^5 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^5*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","\frac{-72 a^2 c^4 x^4-288 a^2 c^2 x^2+576 a^2+432 a b \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-432 a b \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+810 a b \sin ^{-1}(c x)-372 a b \sin \left(2 \sin ^{-1}(c x)\right)+6 a b \sin \left(4 \sin ^{-1}(c x)\right)+360 a b \sin ^{-1}(c x) \cos \left(2 \sin ^{-1}(c x)\right)-18 a b \sin ^{-1}(c x) \cos \left(4 \sin ^{-1}(c x)\right)-432 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+432 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-432 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+432 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+405 b^2 \sin ^{-1}(c x)^2-372 b^2 \sin ^{-1}(c x) \sin \left(2 \sin ^{-1}(c x)\right)+6 b^2 \sin ^{-1}(c x) \sin \left(4 \sin ^{-1}(c x)\right)-376 b^2 \cos \left(2 \sin ^{-1}(c x)\right)+180 b^2 \sin ^{-1}(c x)^2 \cos \left(2 \sin ^{-1}(c x)\right)+2 b^2 \cos \left(4 \sin ^{-1}(c x)\right)-9 b^2 \sin ^{-1}(c x)^2 \cos \left(4 \sin ^{-1}(c x)\right)-378 b^2}{216 c^6 d \sqrt{d-c^2 d x^2}}","\frac{x^4 \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{8 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^6 d^2}+\frac{4 i b \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^6 d \sqrt{d-c^2 d x^2}}-\frac{16 a b x \sqrt{1-c^2 x^2}}{3 c^5 d \sqrt{d-c^2 d x^2}}+\frac{2 b x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c^5 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^4 d^2}-\frac{2 b x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c^6 d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c^6 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \left(1-c^2 x^2\right)^2}{27 c^6 d \sqrt{d-c^2 d x^2}}-\frac{32 b^2 \left(1-c^2 x^2\right)}{9 c^6 d \sqrt{d-c^2 d x^2}}-\frac{16 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d \sqrt{d-c^2 d x^2}}",1,"(576*a^2 - 378*b^2 - 288*a^2*c^2*x^2 - 72*a^2*c^4*x^4 + 810*a*b*ArcSin[c*x] + 405*b^2*ArcSin[c*x]^2 - 376*b^2*Cos[2*ArcSin[c*x]] + 360*a*b*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + 180*b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] + 2*b^2*Cos[4*ArcSin[c*x]] - 18*a*b*ArcSin[c*x]*Cos[4*ArcSin[c*x]] - 9*b^2*ArcSin[c*x]^2*Cos[4*ArcSin[c*x]] - 432*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 432*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 432*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 432*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - (432*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (432*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])] - 372*a*b*Sin[2*ArcSin[c*x]] - 372*b^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]] + 6*a*b*Sin[4*ArcSin[c*x]] + 6*b^2*ArcSin[c*x]*Sin[4*ArcSin[c*x]])/(216*c^6*d*Sqrt[d - c^2*d*x^2])","A",1
245,1,312,424,2.4442427,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","\frac{-4 a^2 c \sqrt{d} x \left(c^2 x^2-3\right)+12 a^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)+2 a b \sqrt{d} \left(\sqrt{1-c^2 x^2} \left(4 \log \left(1-c^2 x^2\right)-6 \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)+8 c x \sin ^{-1}(c x)\right)+b^2 \sqrt{d} \left(-8 i \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+\sqrt{1-c^2 x^2} \left(-4 \sin ^{-1}(c x)^3+2 \left(\sin \left(2 \sin ^{-1}(c x)\right)-4 i\right) \sin ^{-1}(c x)^2-\sin \left(2 \sin ^{-1}(c x)\right)+2 \sin ^{-1}(c x) \left(\cos \left(2 \sin ^{-1}(c x)\right)+8 \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)\right)+8 c x \sin ^{-1}(c x)^2\right)}{8 c^5 d^{3/2} \sqrt{d-c^2 d x^2}}","\frac{x^3 \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{2 b c^5 d \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^5 d \sqrt{d-c^2 d x^2}}+\frac{2 b \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^5 d \sqrt{d-c^2 d x^2}}+\frac{3 x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 c^4 d^2}-\frac{b x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 c^3 d \sqrt{d-c^2 d x^2}}-\frac{i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{c^5 d \sqrt{d-c^2 d x^2}}+\frac{b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c^5 d \sqrt{d-c^2 d x^2}}-\frac{b^2 x \left(1-c^2 x^2\right)}{4 c^4 d \sqrt{d-c^2 d x^2}}",1,"(-4*a^2*c*Sqrt[d]*x*(-3 + c^2*x^2) + 12*a^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))] + 2*a*b*Sqrt[d]*(8*c*x*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*(-6*ArcSin[c*x]^2 + Cos[2*ArcSin[c*x]] + 4*Log[1 - c^2*x^2] + 2*ArcSin[c*x]*Sin[2*ArcSin[c*x]])) + b^2*Sqrt[d]*(8*c*x*ArcSin[c*x]^2 - (8*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] + Sqrt[1 - c^2*x^2]*(-4*ArcSin[c*x]^3 + 2*ArcSin[c*x]*(Cos[2*ArcSin[c*x]] + 8*Log[1 + E^((2*I)*ArcSin[c*x])]) - Sin[2*ArcSin[c*x]] + 2*ArcSin[c*x]^2*(-4*I + Sin[2*ArcSin[c*x]]))))/(8*c^5*d^(3/2)*Sqrt[d - c^2*d*x^2])","A",0
246,1,369,412,0.597121,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","\frac{-2 a^2 c^2 x^2+4 a^2+4 a b \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-4 a b \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+6 a b \sin ^{-1}(c x)-2 a b \sin \left(2 \sin ^{-1}(c x)\right)+2 a b \sin ^{-1}(c x) \cos \left(2 \sin ^{-1}(c x)\right)-4 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+4 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-4 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+4 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+3 b^2 \sin ^{-1}(c x)^2-2 b^2 \sin ^{-1}(c x) \sin \left(2 \sin ^{-1}(c x)\right)-2 b^2 \cos \left(2 \sin ^{-1}(c x)\right)+b^2 \sin ^{-1}(c x)^2 \cos \left(2 \sin ^{-1}(c x)\right)-2 b^2}{2 c^4 d \sqrt{d-c^2 d x^2}}","\frac{x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{c^4 d^2}+\frac{4 i b \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^4 d \sqrt{d-c^2 d x^2}}-\frac{4 a b x \sqrt{1-c^2 x^2}}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 \sqrt{1-c^2 x^2} \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 \sqrt{1-c^2 x^2} \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 \left(1-c^2 x^2\right)}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{4 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}",1,"(4*a^2 - 2*b^2 - 2*a^2*c^2*x^2 + 6*a*b*ArcSin[c*x] + 3*b^2*ArcSin[c*x]^2 - 2*b^2*Cos[2*ArcSin[c*x]] + 2*a*b*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] - 4*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 4*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 4*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 4*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - (4*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (4*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])] - 2*a*b*Sin[2*ArcSin[c*x]] - 2*b^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]])/(2*c^4*d*Sqrt[d - c^2*d*x^2])","A",0
247,1,295,250,0.6119241,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","-\frac{a^2 x \sqrt{-d \left(c^2 x^2-1\right)}}{c^2 d^2 \left(c^2 x^2-1\right)}+\frac{a^2 \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^3 d^{3/2}}+\frac{a b \left(2 c x \sin ^{-1}(c x)-\sqrt{1-c^2 x^2} \left(\sin ^{-1}(c x)^2-2 \log \left(\sqrt{1-c^2 x^2}\right)\right)\right)}{c^3 d \sqrt{d \left(1-c^2 x^2\right)}}+\frac{b^2 \left(\sin ^{-1}(c x) \left(-\sqrt{1-c^2 x^2} \left(\sin ^{-1}(c x)+3 i\right) \sin ^{-1}(c x)+6 \sqrt{1-c^2 x^2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)+3 c x \sin ^{-1}(c x)\right)-3 i \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)\right)}{3 c^3 d \sqrt{d \left(1-c^2 x^2\right)}}","\frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\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{i \sqrt{1-c^2 x^2} \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} \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{i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{c^3 d \sqrt{d-c^2 d x^2}}",1,"-((a^2*x*Sqrt[-(d*(-1 + c^2*x^2))])/(c^2*d^2*(-1 + c^2*x^2))) + (a^2*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(c^3*d^(3/2)) + (a*b*(2*c*x*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*(ArcSin[c*x]^2 - 2*Log[Sqrt[1 - c^2*x^2]])))/(c^3*d*Sqrt[d*(1 - c^2*x^2)]) + (b^2*(ArcSin[c*x]*(3*c*x*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*ArcSin[c*x]*(3*I + ArcSin[c*x]) + 6*Sqrt[1 - c^2*x^2]*Log[1 + E^((2*I)*ArcSin[c*x])]) - (3*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])]))/(3*c^3*d*Sqrt[d*(1 - c^2*x^2)])","A",0
248,1,276,208,0.5769249,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","\frac{a^2+2 a b \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-2 a b \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 a b \sin ^{-1}(c x)-2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-2 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+2 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+b^2 \sin ^{-1}(c x)^2}{c^2 d \sqrt{d-c^2 d x^2}}","\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{4 i b \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{2 i b^2 \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 \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,"(a^2 + 2*a*b*ArcSin[c*x] + b^2*ArcSin[c*x]^2 - 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 2*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 2*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - (2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^2*d*Sqrt[d - c^2*d*x^2])","A",1
249,1,165,195,0.531442,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(3/2),x]","\frac{a \left(a c x+b \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)\right)+2 b \sin ^{-1}(c x) \left(a c x+b \sqrt{1-c^2 x^2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)-i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+b^2 \left(c x-i \sqrt{1-c^2 x^2}\right) \sin ^{-1}(c x)^2}{c d \sqrt{d-c^2 d x^2}}","\frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{d \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 d \sqrt{d-c^2 d x^2}}+\frac{2 b \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{i b^2 \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}}",1,"(b^2*(c*x - I*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + 2*b*ArcSin[c*x]*(a*c*x + b*Sqrt[1 - c^2*x^2]*Log[1 + E^((2*I)*ArcSin[c*x])]) + a*(a*c*x + b*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2]) - I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(c*d*Sqrt[d - c^2*d*x^2])","A",0
250,1,667,467,1.8904682,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x*(d - c^2*d*x^2)^(3/2)),x]","\frac{a^2 \sqrt{d} \sqrt{d-c^2 d x^2} \log (c x)-a^2 \sqrt{d} \sqrt{d-c^2 d x^2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+a^2 d+2 a b d \left(i \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-i \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+\sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-\sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)+\sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-\sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\sin ^{-1}(c x)\right)+b^2 d \left(2 i \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-2 i \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-2 i \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+2 i \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-2 \sqrt{1-c^2 x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)+2 \sqrt{1-c^2 x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)+\sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2 \log \left(1-e^{i \sin ^{-1}(c x)}\right)-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2 \log \left(1+e^{i \sin ^{-1}(c x)}\right)-2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+\sin ^{-1}(c x)^2\right)}{d^2 \sqrt{d-c^2 d x^2}}","\frac{2 i b \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}-\frac{2 i b \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{d \sqrt{d-c^2 d x^2}}+\frac{4 i b \sqrt{1-c^2 x^2} \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}-\frac{2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}",1,"(a^2*d + a^2*Sqrt[d]*Sqrt[d - c^2*d*x^2]*Log[c*x] - a^2*Sqrt[d]*Sqrt[d - c^2*d*x^2]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + 2*a*b*d*(ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + I*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])] - I*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(I*ArcSin[c*x])]) + b^2*d*(ArcSin[c*x]^2 + Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] - 2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] + (2*I)*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] - (2*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (2*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])] - (2*I)*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] - 2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])] + 2*Sqrt[1 - c^2*x^2]*PolyLog[3, E^(I*ArcSin[c*x])]))/(d^2*Sqrt[d - c^2*d*x^2])","A",1
251,1,322,333,0.7840881,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^2 \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)^(3/2)),x]","\frac{2 a^2 c^2 x^2-a^2+2 a b c x \sqrt{1-c^2 x^2} \log (c x)+a b c x \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)+4 a b c^2 x^2 \sin ^{-1}(c x)-2 a b \sin ^{-1}(c x)-i b^2 c x \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)-i b^2 c x \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+2 b^2 c^2 x^2 \sin ^{-1}(c x)^2-2 i b^2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2+2 b^2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+2 b^2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)-b^2 \sin ^{-1}(c x)^2}{d x \sqrt{d-c^2 d x^2}}","\frac{2 c^2 x \left(a+b \sin ^{-1}(c x)\right)^2}{d \sqrt{d-c^2 d x^2}}-\frac{2 i c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{d \sqrt{d-c^2 d x^2}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{d x \sqrt{d-c^2 d x^2}}+\frac{4 b c \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)}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}",1,"(-a^2 + 2*a^2*c^2*x^2 - 2*a*b*ArcSin[c*x] + 4*a*b*c^2*x^2*ArcSin[c*x] - b^2*ArcSin[c*x]^2 + 2*b^2*c^2*x^2*ArcSin[c*x]^2 - (2*I)*b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 + 2*b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 2*b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] + 2*a*b*c*x*Sqrt[1 - c^2*x^2]*Log[c*x] + a*b*c*x*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2] - I*b^2*c*x*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - I*b^2*c*x*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(d*x*Sqrt[d - c^2*d*x^2])","A",0
252,1,844,634,8.4545161,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^3 \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^(3/2)),x]","\frac{3 a^2 \log (x) c^2}{2 d^{3/2}}-\frac{3 a^2 \log \left(d+\sqrt{-d \left(c^2 x^2-1\right)} \sqrt{d}\right) c^2}{2 d^{3/2}}+\frac{b^2 \sqrt{1-c^2 x^2} \left(-\csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2+\sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2+\frac{8 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}-\frac{8 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+8 \sin ^{-1}(c x)^2-4 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-4 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+8 \log \left(\tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-16 \left(\sin ^{-1}(c x) \left(\log \left(1-i e^{i \sin ^{-1}(c x)}\right)-\log \left(1+i e^{i \sin ^{-1}(c x)}\right)\right)+i \left(\text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)\right)\right)+12 \left(\left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)^2+2 i \left(\text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)+2 \left(\text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)-\text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)\right)\right)\right) c^2}{8 d \sqrt{d \left(1-c^2 x^2\right)}}+\frac{a b \left(6 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \sin \left(2 \sin ^{-1}(c x)\right)-6 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \sin \left(2 \sin ^{-1}(c x)\right)-\frac{6 \cos \left(2 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+3 \cos \left(3 \sin ^{-1}(c x)\right) \log \left(1-e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-3 \cos \left(3 \sin ^{-1}(c x)\right) \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-2 \sin ^{-1}(c x)+2 \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 \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)+\sqrt{1-c^2 x^2} \left(-3 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+3 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)-2 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 \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)}{c x}\right) c}{4 d x \sqrt{d \left(1-c^2 x^2\right)}}+\sqrt{-d \left(c^2 x^2-1\right)} \left(-\frac{c^2 a^2}{d^2 \left(c^2 x^2-1\right)}-\frac{a^2}{2 d^2 x^2}\right)","\frac{3 i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}-\frac{3 i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left(a+b \sin ^{-1}(c x)\right)^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{d x \sqrt{d-c^2 d x^2}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{4 i b c^2 \sqrt{1-c^2 x^2} \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}-\frac{3 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}-\frac{3 b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}+\frac{3 b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{d \sqrt{d-c^2 d x^2}}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*(-1/2*a^2/(d^2*x^2) - (a^2*c^2)/(d^2*(-1 + c^2*x^2))) + (3*a^2*c^2*Log[x])/(2*d^(3/2)) - (3*a^2*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(3/2)) + (a*b*c*((6*I)*PolyLog[2, -E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] - (6*I)*PolyLog[2, E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] - (-2*ArcSin[c*x] + 6*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + 3*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 - E^(I*ArcSin[c*x])] - 3*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + E^(I*ArcSin[c*x])] + 2*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 2*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + Sqrt[1 - c^2*x^2]*(-3*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 3*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - 2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + 2*Sin[2*ArcSin[c*x]])/(c*x)))/(4*d*x*Sqrt[d*(1 - c^2*x^2)]) + (b^2*c^2*Sqrt[1 - c^2*x^2]*(8*ArcSin[c*x]^2 - 4*ArcSin[c*x]*Cot[ArcSin[c*x]/2] - ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^2 + 8*Log[Tan[ArcSin[c*x]/2]] - 16*(ArcSin[c*x]*(Log[1 - I*E^(I*ArcSin[c*x])] - Log[1 + I*E^(I*ArcSin[c*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - PolyLog[2, I*E^(I*ArcSin[c*x])])) + 12*(ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + (2*I)*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]) + 2*(-PolyLog[3, -E^(I*ArcSin[c*x])] + PolyLog[3, E^(I*ArcSin[c*x])])) + ArcSin[c*x]^2*Sec[ArcSin[c*x]/2]^2 + (8*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - (8*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 4*ArcSin[c*x]*Tan[ArcSin[c*x]/2]))/(8*d*Sqrt[d*(1 - c^2*x^2)])","A",0
253,1,462,483,0.904315,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^4 \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(3/2)),x]","\frac{8 a^2 c^4 x^4-4 a^2 c^2 x^2-a^2+16 a b c^4 x^4 \sin ^{-1}(c x)-a b c x \sqrt{1-c^2 x^2}-8 a b c^2 x^2 \sin ^{-1}(c x)+10 a b c^3 x^3 \sqrt{1-c^2 x^2} \log (c x)+3 a b c^3 x^3 \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)-2 a b \sin ^{-1}(c x)+b^2 c^4 x^4+8 b^2 c^4 x^4 \sin ^{-1}(c x)^2-b^2 c^2 x^2-4 b^2 c^2 x^2 \sin ^{-1}(c x)^2-b^2 c x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-3 i b^2 c^3 x^3 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)-5 i b^2 c^3 x^3 \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-8 i b^2 c^3 x^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2+10 b^2 c^3 x^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+6 b^2 c^3 x^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)-b^2 \sin ^{-1}(c x)^2}{3 d x^3 \sqrt{d-c^2 d x^2}}","-\frac{4 c^2 \left(a+b \sin ^{-1}(c x)\right)^2}{3 d x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{3 d x^3 \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left(a+b \sin ^{-1}(c x)\right)^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{8 i c^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 d \sqrt{d-c^2 d x^2}}+\frac{16 b c^3 \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 d \sqrt{d-c^2 d x^2}}-\frac{20 b c^3 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 d \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \left(1-c^2 x^2\right)}{3 d x \sqrt{d-c^2 d x^2}}-\frac{i b^2 c^3 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}-\frac{5 i b^2 c^3 \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{3 d \sqrt{d-c^2 d x^2}}",1,"(-a^2 - 4*a^2*c^2*x^2 - b^2*c^2*x^2 + 8*a^2*c^4*x^4 + b^2*c^4*x^4 - a*b*c*x*Sqrt[1 - c^2*x^2] - 2*a*b*ArcSin[c*x] - 8*a*b*c^2*x^2*ArcSin[c*x] + 16*a*b*c^4*x^4*ArcSin[c*x] - b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - b^2*ArcSin[c*x]^2 - 4*b^2*c^2*x^2*ArcSin[c*x]^2 + 8*b^2*c^4*x^4*ArcSin[c*x]^2 - (8*I)*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 + 10*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 6*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] + 10*a*b*c^3*x^3*Sqrt[1 - c^2*x^2]*Log[c*x] + 3*a*b*c^3*x^3*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2] - (3*I)*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - (5*I)*b^2*c^3*x^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(3*d*x^3*Sqrt[d - c^2*d*x^2])","A",0
254,1,594,546,1.673909,"\int \frac{x^5 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^5*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","\frac{\sqrt{d-c^2 d x^2} \left(-24 a^2 c^4 x^4+96 a^2 c^2 x^2-64 a^2-66 a b \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+66 a b \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-50 a b \sin ^{-1}(c x)+8 a b \sin \left(2 \sin ^{-1}(c x)\right)+6 a b \sin \left(4 \sin ^{-1}(c x)\right)-72 a b \sin ^{-1}(c x) \cos \left(2 \sin ^{-1}(c x)\right)-6 a b \sin ^{-1}(c x) \cos \left(4 \sin ^{-1}(c x)\right)-22 a b \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)+22 a b \cos \left(3 \sin ^{-1}(c x)\right) \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+88 i b^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-88 i b^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+66 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-66 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-25 b^2 \sin ^{-1}(c x)^2+8 b^2 \sin ^{-1}(c x) \sin \left(2 \sin ^{-1}(c x)\right)+6 b^2 \sin ^{-1}(c x) \sin \left(4 \sin ^{-1}(c x)\right)+28 b^2 \cos \left(2 \sin ^{-1}(c x)\right)-36 b^2 \sin ^{-1}(c x)^2 \cos \left(2 \sin ^{-1}(c x)\right)+6 b^2 \cos \left(4 \sin ^{-1}(c x)\right)-3 b^2 \sin ^{-1}(c x)^2 \cos \left(4 \sin ^{-1}(c x)\right)+22 b^2 \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)-22 b^2 \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)+22 b^2\right)}{24 c^6 d^3 \left(c^2 x^2-1\right)^2}","\frac{x^4 \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^2 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{8 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^6 d^3}-\frac{22 i b \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^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{16 a b x \sqrt{1-c^2 x^2}}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{11 b x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b x^3 \left(a+b \sin ^{-1}(c x)\right)}{3 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{11 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{11 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \left(1-c^2 x^2\right)}{c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{16 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(-64*a^2 + 22*b^2 + 96*a^2*c^2*x^2 - 24*a^2*c^4*x^4 - 50*a*b*ArcSin[c*x] - 25*b^2*ArcSin[c*x]^2 + 28*b^2*Cos[2*ArcSin[c*x]] - 72*a*b*ArcSin[c*x]*Cos[2*ArcSin[c*x]] - 36*b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] + 6*b^2*Cos[4*ArcSin[c*x]] - 6*a*b*ArcSin[c*x]*Cos[4*ArcSin[c*x]] - 3*b^2*ArcSin[c*x]^2*Cos[4*ArcSin[c*x]] + 66*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 22*b^2*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 - I*E^(I*ArcSin[c*x])] - 66*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - 22*b^2*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + I*E^(I*ArcSin[c*x])] - 66*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 22*a*b*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 66*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + 22*a*b*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + (88*I)*b^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (88*I)*b^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])] + 8*a*b*Sin[2*ArcSin[c*x]] + 8*b^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]] + 6*a*b*Sin[4*ArcSin[c*x]] + 6*b^2*ArcSin[c*x]*Sin[4*ArcSin[c*x]]))/(24*c^6*d^3*(-1 + c^2*x^2)^2)","A",0
255,1,374,421,1.6328454,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","\frac{a^2 c \sqrt{d} x \left(4 c^2 x^2-3\right)+3 a^2 \left(c^2 x^2-1\right) \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)-a b \sqrt{d} \left(\sqrt{1-c^2 x^2}+\left(1-c^2 x^2\right)^{3/2} \left(4 \log \left(1-c^2 x^2\right)-3 \sin ^{-1}(c x)^2\right)+2 \sin ^{-1}(c x) \sin \left(3 \sin ^{-1}(c x)\right)\right)+b^2 \sqrt{d} \left(-c^3 x^3+4 c^3 x^3 \sin ^{-1}(c x)^2+4 i \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+\left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x)^3+4 i \left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x)^2-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)-8 \left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)+c x-3 c x \sin ^{-1}(c x)^2\right)}{3 c^5 d^{5/2} \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}","\frac{x^3 \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^2 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 i \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 b \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^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b x^2 \left(a+b \sin ^{-1}(c x)\right)}{3 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{4 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 x}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}",1,"(a^2*c*Sqrt[d]*x*(-3 + 4*c^2*x^2) + 3*a^2*(-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))] + b^2*Sqrt[d]*(c*x - c^3*x^3 - Sqrt[1 - c^2*x^2]*ArcSin[c*x] - 3*c*x*ArcSin[c*x]^2 + 4*c^3*x^3*ArcSin[c*x]^2 + (4*I)*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]^2 + (1 - c^2*x^2)^(3/2)*ArcSin[c*x]^3 - 8*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] + (4*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])]) - a*b*Sqrt[d]*(Sqrt[1 - c^2*x^2] + (1 - c^2*x^2)^(3/2)*(-3*ArcSin[c*x]^2 + 4*Log[1 - c^2*x^2]) + 2*ArcSin[c*x]*Sin[3*ArcSin[c*x]]))/(3*c^5*d^(5/2)*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])","A",0
256,1,511,332,0.9292866,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","\frac{-12 a^2 c^2 x^2+8 a^2+15 a b \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-15 a b \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+4 a b \sin ^{-1}(c x)+2 a b \sin \left(2 \sin ^{-1}(c x)\right)+12 a b \sin ^{-1}(c x) \cos \left(2 \sin ^{-1}(c x)\right)+5 a b \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)-5 a b \cos \left(3 \sin ^{-1}(c x)\right) \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-20 i b^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+20 i b^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-15 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+15 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+2 b^2 \sin ^{-1}(c x)^2+2 b^2 \sin ^{-1}(c x) \sin \left(2 \sin ^{-1}(c x)\right)-2 b^2 \cos \left(2 \sin ^{-1}(c x)\right)+6 b^2 \sin ^{-1}(c x)^2 \cos \left(2 \sin ^{-1}(c x)\right)-5 b^2 \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)+5 b^2 \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)-2 b^2}{12 c^4 d^2 \left(c^2 x^2-1\right) \sqrt{d-c^2 d x^2}}","\frac{x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^2 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{2 \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{10 i b \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^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b x \left(a+b \sin ^{-1}(c x)\right)}{3 c^3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{5 i b^2 \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{5 i b^2 \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{b^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}",1,"(8*a^2 - 2*b^2 - 12*a^2*c^2*x^2 + 4*a*b*ArcSin[c*x] + 2*b^2*ArcSin[c*x]^2 - 2*b^2*Cos[2*ArcSin[c*x]] + 12*a*b*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + 6*b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] - 15*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 5*b^2*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 - I*E^(I*ArcSin[c*x])] + 15*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 5*b^2*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + I*E^(I*ArcSin[c*x])] + 15*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 5*a*b*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 15*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 5*a*b*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - (20*I)*b^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (20*I)*b^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])] + 2*a*b*Sin[2*ArcSin[c*x]] + 2*b^2*ArcSin[c*x]*Sin[2*ArcSin[c*x]])/(12*c^4*d^2*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])","A",1
257,1,303,332,0.8256061,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","\frac{-a^2 c^3 x^3+a b \sqrt{1-c^2 x^2}-a b c^2 x^2 \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)+a b \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)+b \sin ^{-1}(c x) \left(-2 a c^3 x^3+b \sqrt{1-c^2 x^2}+2 b \left(1-c^2 x^2\right)^{3/2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)+b^2 c^3 x^3-i b^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+i b^2 \left(i c^3 x^3+c^2 x^2 \sqrt{1-c^2 x^2}-\sqrt{1-c^2 x^2}\right) \sin ^{-1}(c x)^2-b^2 c x}{3 c^3 d^2 \left(c^2 x^2-1\right) \sqrt{d-c^2 d x^2}}","-\frac{b x^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{x^3 \left(a+b \sin ^{-1}(c x)\right)^2}{3 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{i \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 b \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^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 x}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{i b^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{b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}",1,"(-(b^2*c*x) - a^2*c^3*x^3 + b^2*c^3*x^3 + a*b*Sqrt[1 - c^2*x^2] + I*b^2*(I*c^3*x^3 - Sqrt[1 - c^2*x^2] + c^2*x^2*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + b*ArcSin[c*x]*(-2*a*c^3*x^3 + b*Sqrt[1 - c^2*x^2] + 2*b*(1 - c^2*x^2)^(3/2)*Log[1 + E^((2*I)*ArcSin[c*x])]) + a*b*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2] - a*b*c^2*x^2*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2] - I*b^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(3*c^3*d^2*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])","A",0
258,1,461,294,1.3087238,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","\frac{a^2 \sqrt{-d \left(c^2 x^2-1\right)}}{3 c^2 d^3 \left(c^2 x^2-1\right)^2}+\frac{a b \left(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(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)+8 \sin ^{-1}(c x)-2 \sin \left(2 \sin ^{-1}(c x)\right)+\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(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{12 c^2 d \left(d \left(1-c^2 x^2\right)\right)^{3/2}}+\frac{b^2 \left(-4 i \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+4 i \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+4 \sin ^{-1}(c x)^2-2 \sin \left(2 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+2 \cos \left(2 \sin ^{-1}(c x)\right)-\sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)+\sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)+2\right)}{12 c^2 d \left(d \left(1-c^2 x^2\right)\right)^{3/2}}","-\frac{b 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{2 i b \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{\left(a+b \sin ^{-1}(c x)\right)^2}{3 c^2 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{i b^2 \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 \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}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}",1,"(a^2*Sqrt[-(d*(-1 + c^2*x^2))])/(3*c^2*d^3*(-1 + c^2*x^2)^2) + (a*b*(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]]))/(12*c^2*d*(d*(1 - c^2*x^2))^(3/2)) + (b^2*(2 + 4*ArcSin[c*x]^2 + 2*Cos[2*ArcSin[c*x]] - 3*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 - I*E^(I*ArcSin[c*x])] + 3*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + I*E^(I*ArcSin[c*x])] - (4*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (4*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])] - 2*ArcSin[c*x]*Sin[2*ArcSin[c*x]]))/(12*c^2*d*(d*(1 - c^2*x^2))^(3/2))","A",0
259,1,320,311,1.0634869,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(d - c^2*d*x^2)^(5/2),x]","\frac{2 a^2 c^3 x^3-3 a^2 c x+a b \sqrt{1-c^2 x^2}+2 a b c^2 x^2 \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)-2 a b \sqrt{1-c^2 x^2} \log \left(1-c^2 x^2\right)+b \sin ^{-1}(c x) \left(4 a c^3 x^3-6 a c x+b \sqrt{1-c^2 x^2}-4 b \left(1-c^2 x^2\right)^{3/2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)+b^2 c^3 x^3+2 i b^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)+b^2 \left(2 c^3 x^3-2 i c^2 x^2 \sqrt{1-c^2 x^2}+2 i \sqrt{1-c^2 x^2}-3 c x\right) \sin ^{-1}(c x)^2-b^2 c x}{3 c d^2 \left(c^2 x^2-1\right) \sqrt{d-c^2 d x^2}}","-\frac{b \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 x \left(a+b \sin ^{-1}(c x)\right)^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i \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{4 b \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{x \left(a+b \sin ^{-1}(c x)\right)^2}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{2 i b^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{b^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}",1,"(-3*a^2*c*x - b^2*c*x + 2*a^2*c^3*x^3 + b^2*c^3*x^3 + a*b*Sqrt[1 - c^2*x^2] + b^2*(-3*c*x + 2*c^3*x^3 + (2*I)*Sqrt[1 - c^2*x^2] - (2*I)*c^2*x^2*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + b*ArcSin[c*x]*(-6*a*c*x + 4*a*c^3*x^3 + b*Sqrt[1 - c^2*x^2] - 4*b*(1 - c^2*x^2)^(3/2)*Log[1 + E^((2*I)*ArcSin[c*x])]) - 2*a*b*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2] + 2*a*b*c^2*x^2*Sqrt[1 - c^2*x^2]*Log[1 - c^2*x^2] + (2*I)*b^2*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/(3*c*d^2*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])","A",0
260,1,935,577,8.8939694,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x*(d - c^2*d*x^2)^(5/2)),x]","\frac{\log (c x) a^2}{d^{5/2}}-\frac{\log \left(d+\sqrt{-d \left(c^2 x^2-1\right)} \sqrt{d}\right) a^2}{d^{5/2}}+\frac{b \left(24 i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(1-c^2 x^2\right)^{3/2}-24 i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(1-c^2 x^2\right)^{3/2}+18 \sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right) \sqrt{1-c^2 x^2}-18 \sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sqrt{1-c^2 x^2}+21 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sqrt{1-c^2 x^2}-21 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sqrt{1-c^2 x^2}+20 \sin ^{-1}(c x)+12 \sin ^{-1}(c x) \cos \left(2 \sin ^{-1}(c x)\right)+6 \sin ^{-1}(c x) \cos \left(3 \sin ^{-1}(c x)\right) \log \left(1-e^{i \sin ^{-1}(c x)}\right)-6 \sin ^{-1}(c x) \cos \left(3 \sin ^{-1}(c x)\right) \log \left(1+e^{i \sin ^{-1}(c x)}\right)+7 \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)-7 \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) a}{12 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^2}{3 d^3 \left(c^2 x^2-1\right)^2}-\frac{a^2}{d^3 \left(c^2 x^2-1\right)}\right)+\frac{b^2 \left(1-c^2 x^2\right)^{3/2} \left(12 \left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)^2+\frac{2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}-\frac{2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}+14 \sin ^{-1}(c x)^2-\frac{\left(\sin ^{-1}(c x)-2\right) \sin ^{-1}(c x)}{c x-1}+24 i \left(\text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)+\frac{\left(\sin ^{-1}(c x)+2\right) \sin ^{-1}(c x)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-28 \left(\sin ^{-1}(c x) \left(\log \left(1-i e^{i \sin ^{-1}(c x)}\right)-\log \left(1+i e^{i \sin ^{-1}(c x)}\right)\right)+i \left(\text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)\right)\right)+24 \left(\text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)-\text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)\right)+\frac{2 \left(7 \sin ^{-1}(c x)^2+2\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}-\frac{2 \left(7 \sin ^{-1}(c x)^2+2\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+4\right)}{12 d \left(d \left(1-c^2 x^2\right)\right)^{3/2}}","\frac{2 i b \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i b \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b c x \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{14 i b \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 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{7 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{7 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2}{3 d^2 \sqrt{d-c^2 d x^2}}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*(a^2/(3*d^3*(-1 + c^2*x^2)^2) - a^2/(d^3*(-1 + c^2*x^2))) + (a^2*Log[c*x])/d^(5/2) - (a^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/d^(5/2) + (b^2*(1 - c^2*x^2)^(3/2)*(4 - ((-2 + ArcSin[c*x])*ArcSin[c*x])/(-1 + c*x) + 14*ArcSin[c*x]^2 + 12*ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) - 28*(ArcSin[c*x]*(Log[1 - I*E^(I*ArcSin[c*x])] - Log[1 + I*E^(I*ArcSin[c*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - PolyLog[2, I*E^(I*ArcSin[c*x])])) + (24*I)*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]) + 24*(-PolyLog[3, -E^(I*ArcSin[c*x])] + PolyLog[3, E^(I*ArcSin[c*x])]) + (2*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (2*(2 + 7*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - (2*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 + (ArcSin[c*x]*(2 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (2*(2 + 7*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(12*d*(d*(1 - c^2*x^2))^(3/2)) + (a*b*(20*ArcSin[c*x] + 12*ArcSin[c*x]*Cos[2*ArcSin[c*x]] + 18*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 6*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 - E^(I*ArcSin[c*x])] - 18*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - 6*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + E^(I*ArcSin[c*x])] + 21*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 7*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 21*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 7*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + (24*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, -E^(I*ArcSin[c*x])] - (24*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, E^(I*ArcSin[c*x])] - 2*Sin[2*ArcSin[c*x]]))/(12*d*(d*(1 - c^2*x^2))^(3/2))","A",0
261,1,352,452,2.9104082,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^2 \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^2*(d - c^2*d*x^2)^(5/2)),x]","-\frac{c \left(\frac{a^2 \left(8 c^4 x^4-12 c^2 x^2+3\right)}{c x}+a b \sqrt{1-c^2 x^2} \left(6 \left(c^2 x^2-1\right) \log (c x)+5 \left(c^2 x^2-1\right) \log \left(1-c^2 x^2\right)+1\right)+\frac{2 a b \left(8 c^4 x^4-12 c^2 x^2+3\right) \sin ^{-1}(c x)}{c x}-b^2 \left(1-c^2 x^2\right)^{3/2} \left(\frac{c x}{\sqrt{1-c^2 x^2}}-\frac{3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{c x}+\frac{5 c x \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}+\frac{c x \sin ^{-1}(c x)^2}{\left(1-c^2 x^2\right)^{3/2}}+\frac{\sin ^{-1}(c x)}{c^2 x^2-1}-5 i \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)-3 i \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-8 i \sin ^{-1}(c x)^2+6 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+10 \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)\right)}{3 d \left(d-c^2 d x^2\right)^{3/2}}","-\frac{b c \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{8 c^2 x \left(a+b \sin ^{-1}(c x)\right)^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 i c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{16 b c \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 d^2 \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{4 c^2 x \left(a+b \sin ^{-1}(c x)\right)^2}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{d x \left(d-c^2 d x^2\right)^{3/2}}-\frac{5 i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 c^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}",1,"-1/3*(c*((a^2*(3 - 12*c^2*x^2 + 8*c^4*x^4))/(c*x) + (2*a*b*(3 - 12*c^2*x^2 + 8*c^4*x^4)*ArcSin[c*x])/(c*x) + a*b*Sqrt[1 - c^2*x^2]*(1 + 6*(-1 + c^2*x^2)*Log[c*x] + 5*(-1 + c^2*x^2)*Log[1 - c^2*x^2]) - b^2*(1 - c^2*x^2)^(3/2)*((c*x)/Sqrt[1 - c^2*x^2] + ArcSin[c*x]/(-1 + c^2*x^2) - (8*I)*ArcSin[c*x]^2 + (c*x*ArcSin[c*x]^2)/(1 - c^2*x^2)^(3/2) + (5*c*x*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] - (3*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(c*x) + 6*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 10*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] - (5*I)*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - (3*I)*PolyLog[2, E^((2*I)*ArcSin[c*x])])))/(d*(d - c^2*d*x^2)^(3/2))","A",0
262,1,1090,752,11.3247287,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^3 \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^3*(d - c^2*d*x^2)^(5/2)),x]","\frac{5 a^2 \log (x) c^2}{2 d^{5/2}}-\frac{5 a^2 \log \left(d+\sqrt{-d \left(c^2 x^2-1\right)} \sqrt{d}\right) c^2}{2 d^{5/2}}+\frac{a b \sqrt{1-c^2 x^2} \left(-3 \sin ^{-1}(c x) \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)+3 \sin ^{-1}(c x) \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right)-\frac{2 \left(\sin ^{-1}(c x)-1\right)}{c x-1}+52 \sin ^{-1}(c x)-6 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)+60 \sin ^{-1}(c x) \left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)+52 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-52 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+60 i \left(\text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right)+\frac{52 \sin ^{-1}(c x) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+\frac{4 \sin ^{-1}(c x) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}-6 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\frac{52 \sin ^{-1}(c x) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+\frac{2 \left(\sin ^{-1}(c x)+1\right)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-\frac{4 \sin ^{-1}(c x) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}\right) c^2}{12 d^2 \sqrt{d \left(1-c^2 x^2\right)}}+\frac{b^2 \sqrt{1-c^2 x^2} \left(-3 \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2+3 \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2+\frac{4 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}-\frac{4 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}+52 \sin ^{-1}(c x)^2-\frac{2 \left(\sin ^{-1}(c x)-2\right) \sin ^{-1}(c x)}{c x-1}-12 \cot \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)-12 \tan \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+\frac{2 \left(\sin ^{-1}(c x)+2\right) \sin ^{-1}(c x)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}+24 \log \left(\tan \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-104 \left(\sin ^{-1}(c x) \left(\log \left(1-i e^{i \sin ^{-1}(c x)}\right)-\log \left(1+i e^{i \sin ^{-1}(c x)}\right)\right)+i \left(\text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)\right)\right)+60 \left(\left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)^2+2 i \left(\text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-\text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right) \sin ^{-1}(c x)+2 \left(\text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)-\text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)\right)\right)+\frac{4 \left(13 \sin ^{-1}(c x)^2+2\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}-\frac{4 \left(13 \sin ^{-1}(c x)^2+2\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+8\right) c^2}{24 d^2 \sqrt{d \left(1-c^2 x^2\right)}}+\sqrt{-d \left(c^2 x^2-1\right)} \left(-\frac{2 c^2 a^2}{d^3 \left(c^2 x^2-1\right)}-\frac{a^2}{2 d^3 x^2}+\frac{c^2 a^2}{3 d^3 \left(c^2 x^2-1\right)^2}\right)","\frac{5 i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left(a+b \sin ^{-1}(c x)\right)^2}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{b c \left(a+b \sin ^{-1}(c x)\right)}{d^2 x \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}+\frac{26 i b c^2 \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 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left(a+b \sin ^{-1}(c x)\right)^2}{6 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{2 d x^2 \left(d-c^2 d x^2\right)^{3/2}}+\frac{2 b c^3 x \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{13 i b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{13 i b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 b^2 c^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{d^2 \sqrt{d-c^2 d x^2}}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*(-1/2*a^2/(d^3*x^2) + (a^2*c^2)/(3*d^3*(-1 + c^2*x^2)^2) - (2*a^2*c^2)/(d^3*(-1 + c^2*x^2))) + (5*a^2*c^2*Log[x])/(2*d^(5/2)) - (5*a^2*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(5/2)) + (a*b*c^2*Sqrt[1 - c^2*x^2]*((-2*(-1 + ArcSin[c*x]))/(-1 + c*x) + 52*ArcSin[c*x] - 6*Cot[ArcSin[c*x]/2] - 3*ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 + 60*ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + 52*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 52*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + (60*I)*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]) + 3*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 + (4*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (52*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - (4*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 + (2*(1 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (52*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 6*Tan[ArcSin[c*x]/2]))/(12*d^2*Sqrt[d*(1 - c^2*x^2)]) + (b^2*c^2*Sqrt[1 - c^2*x^2]*(8 - (2*(-2 + ArcSin[c*x])*ArcSin[c*x])/(-1 + c*x) + 52*ArcSin[c*x]^2 - 12*ArcSin[c*x]*Cot[ArcSin[c*x]/2] - 3*ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^2 + 24*Log[Tan[ArcSin[c*x]/2]] - 104*(ArcSin[c*x]*(Log[1 - I*E^(I*ArcSin[c*x])] - Log[1 + I*E^(I*ArcSin[c*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - PolyLog[2, I*E^(I*ArcSin[c*x])])) + 60*(ArcSin[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + (2*I)*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLog[2, E^(I*ArcSin[c*x])]) + 2*(-PolyLog[3, -E^(I*ArcSin[c*x])] + PolyLog[3, E^(I*ArcSin[c*x])])) + 3*ArcSin[c*x]^2*Sec[ArcSin[c*x]/2]^2 + (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (4*(2 + 13*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 + (2*ArcSin[c*x]*(2 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (4*(2 + 13*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 12*ArcSin[c*x]*Tan[ArcSin[c*x]/2]))/(24*d^2*Sqrt[d*(1 - c^2*x^2)])","A",0
263,1,441,538,4.2187558,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^4 \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(5/2)),x]","\frac{-\frac{a^2 \left(16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1\right)}{x^3}-\frac{a b \left(c x \sqrt{1-c^2 x^2} \left(16 c^2 x^2 \left(c^2 x^2-1\right) \log (c x)+8 c^2 x^2 \left(c^2 x^2-1\right) \log \left(1-c^2 x^2\right)+1\right)+2 \left(16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1\right) \sin ^{-1}(c x)\right)}{x^3}+b^2 c^3 \left(1-c^2 x^2\right)^{3/2} \left(-\frac{\sqrt{1-c^2 x^2}}{c x}+\frac{c x}{\sqrt{1-c^2 x^2}}-\frac{8 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{c x}+\frac{8 c x \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}+\frac{c x \sin ^{-1}(c x)^2}{\left(1-c^2 x^2\right)^{3/2}}+\frac{\sin ^{-1}(c x)}{c^2 x^2-1}-\frac{\sin ^{-1}(c x)}{c^2 x^2}-\frac{\sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{c^3 x^3}-8 i \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)-8 i \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-16 i \sin ^{-1}(c x)^2+16 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+16 \sin ^{-1}(c x) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)\right)}{3 d \left(d-c^2 d x^2\right)^{3/2}}","-\frac{b c \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 x^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left(a+b \sin ^{-1}(c x)\right)^2}{d x \left(d-c^2 d x^2\right)^{3/2}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{3 d x^3 \left(d-c^2 d x^2\right)^{3/2}}+\frac{16 c^4 x \left(a+b \sin ^{-1}(c x)\right)^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left(a+b \sin ^{-1}(c x)\right)^2}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{16 i c^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{32 b c^3 \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 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}+\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 i b^2 c^3 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 i b^2 c^3 \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{3 d^2 \sqrt{d-c^2 d x^2}}",1,"(-((a^2*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6))/x^3) - (a*b*(2*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6)*ArcSin[c*x] + c*x*Sqrt[1 - c^2*x^2]*(1 + 16*c^2*x^2*(-1 + c^2*x^2)*Log[c*x] + 8*c^2*x^2*(-1 + c^2*x^2)*Log[1 - c^2*x^2])))/x^3 + b^2*c^3*(1 - c^2*x^2)^(3/2)*((c*x)/Sqrt[1 - c^2*x^2] - Sqrt[1 - c^2*x^2]/(c*x) - ArcSin[c*x]/(c^2*x^2) + ArcSin[c*x]/(-1 + c^2*x^2) - (16*I)*ArcSin[c*x]^2 + (c*x*ArcSin[c*x]^2)/(1 - c^2*x^2)^(3/2) + (8*c*x*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] - (Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(c^3*x^3) - (8*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(c*x) + 16*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 16*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])] - (8*I)*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - (8*I)*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/(3*d*(d - c^2*d*x^2)^(3/2))","A",0
264,1,100,157,0.0667671,"\int \frac{x^4 \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^4*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2],x]","\frac{a x \sqrt{1-a^2 x^2} \left(2 a^2 x^2+15\right)-8 a x \sqrt{1-a^2 x^2} \left(2 a^2 x^2+3\right) \sin ^{-1}(a x)^2+\left(8 a^4 x^4+24 a^2 x^2-15\right) \sin ^{-1}(a x)+8 \sin ^{-1}(a x)^3}{64 a^5}","\frac{\sin ^{-1}(a x)^3}{8 a^5}-\frac{15 \sin ^{-1}(a x)}{64 a^5}+\frac{3 x^2 \sin ^{-1}(a x)}{8 a^3}+\frac{x^3 \sqrt{1-a^2 x^2}}{32 a^2}-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{4 a^2}+\frac{15 x \sqrt{1-a^2 x^2}}{64 a^4}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{8 a^4}+\frac{x^4 \sin ^{-1}(a x)}{8 a}",1,"(a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2) + (-15 + 24*a^2*x^2 + 8*a^4*x^4)*ArcSin[a*x] - 8*a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)*ArcSin[a*x]^2 + 8*ArcSin[a*x]^3)/(64*a^5)","A",1
265,1,81,126,0.0685403,"\int \frac{x^3 \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^3*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2],x]","\frac{2 \sqrt{1-a^2 x^2} \left(a^2 x^2+20\right)-9 \sqrt{1-a^2 x^2} \left(a^2 x^2+2\right) \sin ^{-1}(a x)^2+6 a x \left(a^2 x^2+6\right) \sin ^{-1}(a x)}{27 a^4}","\frac{4 x \sin ^{-1}(a x)}{3 a^3}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^2}-\frac{2 \left(1-a^2 x^2\right)^{3/2}}{27 a^4}+\frac{14 \sqrt{1-a^2 x^2}}{9 a^4}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a^4}+\frac{2 x^3 \sin ^{-1}(a x)}{9 a}",1,"(2*Sqrt[1 - a^2*x^2]*(20 + a^2*x^2) + 6*a*x*(6 + a^2*x^2)*ArcSin[a*x] - 9*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcSin[a*x]^2)/(27*a^4)","A",1
266,1,73,89,0.0331751,"\int \frac{x^2 \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^2*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2],x]","\frac{3 a x \sqrt{1-a^2 x^2}-6 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2+\left(6 a^2 x^2-3\right) \sin ^{-1}(a x)+2 \sin ^{-1}(a x)^3}{12 a^3}","\frac{\sin ^{-1}(a x)^3}{6 a^3}-\frac{\sin ^{-1}(a x)}{4 a^3}+\frac{x \sqrt{1-a^2 x^2}}{4 a^2}-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac{x^2 \sin ^{-1}(a x)}{2 a}",1,"(3*a*x*Sqrt[1 - a^2*x^2] + (-3 + 6*a^2*x^2)*ArcSin[a*x] - 6*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2 + 2*ArcSin[a*x]^3)/(12*a^3)","A",1
267,1,51,55,0.0203903,"\int \frac{x \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2],x]","\frac{2 \sqrt{1-a^2 x^2}-\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2+2 a x \sin ^{-1}(a x)}{a^2}","\frac{2 \sqrt{1-a^2 x^2}}{a^2}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a^2}+\frac{2 x \sin ^{-1}(a x)}{a}",1,"(2*Sqrt[1 - a^2*x^2] + 2*a*x*ArcSin[a*x] - Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a^2","A",1
268,1,13,13,0.0064594,"\int \frac{\sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]^2/Sqrt[1 - a^2*x^2],x]","\frac{\sin ^{-1}(a x)^3}{3 a}","\frac{\sin ^{-1}(a x)^3}{3 a}",1,"ArcSin[a*x]^3/(3*a)","A",1
269,1,116,92,0.1281616,"\int \frac{\sin ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]^2/(x*Sqrt[1 - a^2*x^2]),x]","2 i \sin ^{-1}(a x) \text{Li}_2\left(-e^{i \sin ^{-1}(a x)}\right)-2 i \sin ^{-1}(a x) \text{Li}_2\left(e^{i \sin ^{-1}(a x)}\right)-2 \text{Li}_3\left(-e^{i \sin ^{-1}(a x)}\right)+2 \text{Li}_3\left(e^{i \sin ^{-1}(a x)}\right)+\sin ^{-1}(a x)^2 \log \left(1-e^{i \sin ^{-1}(a x)}\right)-\sin ^{-1}(a x)^2 \log \left(1+e^{i \sin ^{-1}(a x)}\right)","2 i \sin ^{-1}(a x) \text{Li}_2\left(-e^{i \sin ^{-1}(a x)}\right)-2 i \sin ^{-1}(a x) \text{Li}_2\left(e^{i \sin ^{-1}(a x)}\right)-2 \text{Li}_3\left(-e^{i \sin ^{-1}(a x)}\right)+2 \text{Li}_3\left(e^{i \sin ^{-1}(a x)}\right)-2 \sin ^{-1}(a x)^2 \tanh ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)",1,"ArcSin[a*x]^2*Log[1 - E^(I*ArcSin[a*x])] - ArcSin[a*x]^2*Log[1 + E^(I*ArcSin[a*x])] + (2*I)*ArcSin[a*x]*PolyLog[2, -E^(I*ArcSin[a*x])] - (2*I)*ArcSin[a*x]*PolyLog[2, E^(I*ArcSin[a*x])] - 2*PolyLog[3, -E^(I*ArcSin[a*x])] + 2*PolyLog[3, E^(I*ArcSin[a*x])]","A",0
270,1,72,76,0.3039191,"\int \frac{\sin ^{-1}(a x)^2}{x^2 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]^2/(x^2*Sqrt[1 - a^2*x^2]),x]","\sin ^{-1}(a x) \left(2 a \log \left(1-e^{2 i \sin ^{-1}(a x)}\right)-\frac{\left(\sqrt{1-a^2 x^2}+i a x\right) \sin ^{-1}(a x)}{x}\right)-i a \text{Li}_2\left(e^{2 i \sin ^{-1}(a x)}\right)","-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{x}-i a \text{Li}_2\left(e^{2 i \sin ^{-1}(a x)}\right)-i a \sin ^{-1}(a x)^2+2 a \sin ^{-1}(a x) \log \left(1-e^{2 i \sin ^{-1}(a x)}\right)",1,"ArcSin[a*x]*(-(((I*a*x + Sqrt[1 - a^2*x^2])*ArcSin[a*x])/x) + 2*a*Log[1 - E^((2*I)*ArcSin[a*x])]) - I*a*PolyLog[2, E^((2*I)*ArcSin[a*x])]","A",1
271,1,194,163,1.5719677,"\int \frac{\sin ^{-1}(a x)^2}{x^3 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]^2/(x^3*Sqrt[1 - a^2*x^2]),x]","\frac{1}{8} a^2 \left(8 i \sin ^{-1}(a x) \left(\text{Li}_2\left(-e^{i \sin ^{-1}(a x)}\right)-\text{Li}_2\left(e^{i \sin ^{-1}(a x)}\right)\right)+8 \left(\text{Li}_3\left(e^{i \sin ^{-1}(a x)}\right)-\text{Li}_3\left(-e^{i \sin ^{-1}(a x)}\right)\right)+4 \sin ^{-1}(a x)^2 \left(\log \left(1-e^{i \sin ^{-1}(a x)}\right)-\log \left(1+e^{i \sin ^{-1}(a x)}\right)\right)-4 \sin ^{-1}(a x) \tan \left(\frac{1}{2} \sin ^{-1}(a x)\right)-4 \sin ^{-1}(a x) \cot \left(\frac{1}{2} \sin ^{-1}(a x)\right)+\sin ^{-1}(a x)^2 \left(-\csc ^2\left(\frac{1}{2} \sin ^{-1}(a x)\right)\right)+\sin ^{-1}(a x)^2 \sec ^2\left(\frac{1}{2} \sin ^{-1}(a x)\right)+8 \log \left(\tan \left(\frac{1}{2} \sin ^{-1}(a x)\right)\right)\right)","i a^2 \sin ^{-1}(a x) \text{Li}_2\left(-e^{i \sin ^{-1}(a x)}\right)-i a^2 \sin ^{-1}(a x) \text{Li}_2\left(e^{i \sin ^{-1}(a x)}\right)-a^2 \text{Li}_3\left(-e^{i \sin ^{-1}(a x)}\right)+a^2 \text{Li}_3\left(e^{i \sin ^{-1}(a x)}\right)-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 x^2}-a^2 \tanh ^{-1}\left(\sqrt{1-a^2 x^2}\right)+a^2 \left(-\sin ^{-1}(a x)^2\right) \tanh ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)-\frac{a \sin ^{-1}(a x)}{x}",1,"(a^2*(-4*ArcSin[a*x]*Cot[ArcSin[a*x]/2] - ArcSin[a*x]^2*Csc[ArcSin[a*x]/2]^2 + 4*ArcSin[a*x]^2*(Log[1 - E^(I*ArcSin[a*x])] - Log[1 + E^(I*ArcSin[a*x])]) + 8*Log[Tan[ArcSin[a*x]/2]] + (8*I)*ArcSin[a*x]*(PolyLog[2, -E^(I*ArcSin[a*x])] - PolyLog[2, E^(I*ArcSin[a*x])]) + 8*(-PolyLog[3, -E^(I*ArcSin[a*x])] + PolyLog[3, E^(I*ArcSin[a*x])]) + ArcSin[a*x]^2*Sec[ArcSin[a*x]/2]^2 - 4*ArcSin[a*x]*Tan[ArcSin[a*x]/2]))/8","A",0
272,1,42,42,0.0702128,"\int \frac{\sin ^{-1}(a x)^2}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[ArcSin[a*x]^2/Sqrt[c - a^2*c*x^2],x]","\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a \sqrt{c-a^2 c x^2}}","\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a \sqrt{c-a^2 c x^2}}",1,"(Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(3*a*Sqrt[c - a^2*c*x^2])","A",1
273,1,108,179,0.263548,"\int \frac{\sin ^{-1}(a x)^2}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcSin[a*x]^2/(c - a^2*c*x^2)^(3/2),x]","\frac{\sin ^{-1}(a x) \left(a x \sin ^{-1}(a x)+\sqrt{1-a^2 x^2} \left(2 \log \left(1+e^{2 i \sin ^{-1}(a x)}\right)-i \sin ^{-1}(a x)\right)\right)-i \sqrt{1-a^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(a x)}\right)}{a c \sqrt{c \left(1-a^2 x^2\right)}}","-\frac{i \sqrt{1-a^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(a x)}\right)}{a c \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^2}{c \sqrt{c-a^2 c x^2}}-\frac{i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a c \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left(1+e^{2 i \sin ^{-1}(a x)}\right)}{a c \sqrt{c-a^2 c x^2}}",1,"(ArcSin[a*x]*(a*x*ArcSin[a*x] + Sqrt[1 - a^2*x^2]*((-I)*ArcSin[a*x] + 2*Log[1 + E^((2*I)*ArcSin[a*x])])) - I*Sqrt[1 - a^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[a*x])])/(a*c*Sqrt[c*(1 - a^2*x^2)])","A",1
274,1,149,283,0.7028327,"\int \frac{\sin ^{-1}(a x)^2}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcSin[a*x]^2/(c - a^2*c*x^2)^(5/2),x]","\frac{-2 i \sqrt{1-a^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(a x)}\right)+\left(a x \left(\frac{1}{1-a^2 x^2}+2\right)-2 i \sqrt{1-a^2 x^2}\right) \sin ^{-1}(a x)^2+\frac{\sin ^{-1}(a x) \left(-1+\left(4-4 a^2 x^2\right) \log \left(1+e^{2 i \sin ^{-1}(a x)}\right)\right)}{\sqrt{1-a^2 x^2}}+a x}{3 a c^2 \sqrt{c-a^2 c x^2}}","-\frac{2 i \sqrt{1-a^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(a x)}\right)}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{x}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \sin ^{-1}(a x)^2}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{3 a c^2 \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)}{3 a c^2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left(1+e^{2 i \sin ^{-1}(a x)}\right)}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^2}{3 c \left(c-a^2 c x^2\right)^{3/2}}",1,"(a*x + ((-2*I)*Sqrt[1 - a^2*x^2] + a*x*(2 + (1 - a^2*x^2)^(-1)))*ArcSin[a*x]^2 + (ArcSin[a*x]*(-1 + (4 - 4*a^2*x^2)*Log[1 + E^((2*I)*ArcSin[a*x])]))/Sqrt[1 - a^2*x^2] - (2*I)*Sqrt[1 - a^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[a*x])])/(3*a*c^2*Sqrt[c - a^2*c*x^2])","A",1
275,1,234,390,0.962923,"\int \frac{\sin ^{-1}(a x)^2}{\left(c-a^2 c x^2\right)^{7/2}} \, dx","Integrate[ArcSin[a*x]^2/(c - a^2*c*x^2)^(7/2),x]","\frac{\sqrt{1-a^2 x^2} \left(\frac{11 a x}{\sqrt{1-a^2 x^2}}+\frac{16 a x \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{8 \sin ^{-1}(a x) \left(\frac{a x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-1\right)}{1-a^2 x^2}+\frac{3 \sin ^{-1}(a x) \left(\frac{2 a x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-1\right)}{\left(1-a^2 x^2\right)^2}+\frac{a^3 x^3}{\left(1-a^2 x^2\right)^{3/2}}-16 i \text{Li}_2\left(-e^{2 i \sin ^{-1}(a x)}\right)-16 i \sin ^{-1}(a x)^2+32 \sin ^{-1}(a x) \log \left(1+e^{2 i \sin ^{-1}(a x)}\right)\right)}{30 a c^3 \sqrt{c \left(1-a^2 x^2\right)}}","-\frac{8 i \sqrt{1-a^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(a x)}\right)}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}+\frac{x}{30 c^3 \left(1-a^2 x^2\right) \sqrt{c-a^2 c x^2}}+\frac{8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{4 \sin ^{-1}(a x)}{15 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)}{10 a c^3 \left(1-a^2 x^2\right)^{3/2} \sqrt{c-a^2 c x^2}}+\frac{16 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left(1+e^{2 i \sin ^{-1}(a x)}\right)}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \sin ^{-1}(a x)^2}{15 c^2 \left(c-a^2 c x^2\right)^{3/2}}+\frac{x \sin ^{-1}(a x)^2}{5 c \left(c-a^2 c x^2\right)^{5/2}}",1,"(Sqrt[1 - a^2*x^2]*((a^3*x^3)/(1 - a^2*x^2)^(3/2) + (11*a*x)/Sqrt[1 - a^2*x^2] - (16*I)*ArcSin[a*x]^2 + (16*a*x*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2] + (8*ArcSin[a*x]*(-1 + (a*x*ArcSin[a*x])/Sqrt[1 - a^2*x^2]))/(1 - a^2*x^2) + (3*ArcSin[a*x]*(-1 + (2*a*x*ArcSin[a*x])/Sqrt[1 - a^2*x^2]))/(1 - a^2*x^2)^2 + 32*ArcSin[a*x]*Log[1 + E^((2*I)*ArcSin[a*x])] - (16*I)*PolyLog[2, -E^((2*I)*ArcSin[a*x])]))/(30*a*c^3*Sqrt[c*(1 - a^2*x^2)])","A",1
276,0,0,1312,3.6698422,"\int x^m \left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^m*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2,x]","\int x^m \left(d-c^2 d x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","\frac{d^3 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2 x^{m+1}}{m+7}+\frac{6 d^3 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 x^{m+1}}{(m+5) (m+7)}+\frac{24 d^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 x^{m+1}}{(m+7) \left(m^2+8 m+15\right)}+\frac{48 d^3 \left(a+b \sin ^{-1}(c x)\right)^2 x^{m+1}}{(m+5) (m+7) \left(m^2+4 m+3\right)}-\frac{2 b c d^3 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) x^{m+2}}{(m+7)^2}-\frac{12 b c d^3 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) x^{m+2}}{(m+5)^2 (m+7)}-\frac{10 b c d^3 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) x^{m+2}}{(m+5) (m+7)^2}-\frac{48 b c d^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) x^{m+2}}{(m+3)^2 (m+5) (m+7)}-\frac{36 b c d^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) x^{m+2}}{(m+3) (m+5)^2 (m+7)}-\frac{30 b c d^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right) x^{m+2}}{(m+7)^2 \left(m^2+8 m+15\right)}-\frac{48 b c d^3 \left(a+b \sin ^{-1}(c x)\right) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right) x^{m+2}}{(m+2) (m+3)^2 (m+5) (m+7)}-\frac{36 b c d^3 \left(a+b \sin ^{-1}(c x)\right) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right) x^{m+2}}{(m+5)^2 (m+7) \left(m^2+5 m+6\right)}-\frac{30 b c d^3 \left(a+b \sin ^{-1}(c x)\right) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right) x^{m+2}}{(m+5) (m+7)^2 \left(m^2+5 m+6\right)}-\frac{96 b c d^3 \left(a+b \sin ^{-1}(c x)\right) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right) x^{m+2}}{(m+5) (m+7) \left(m^3+6 m^2+11 m+6\right)}+\frac{48 b^2 c^2 d^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^2 x^2\right) x^{m+3}}{(m+2) (m+3)^3 (m+5) (m+7)}+\frac{36 b^2 c^2 d^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^2 x^2\right) x^{m+3}}{(m+2) (m+3)^2 (m+5)^2 (m+7)}+\frac{96 b^2 c^2 d^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^2 x^2\right) x^{m+3}}{(m+3)^2 (m+5) (m+7) \left(m^2+3 m+2\right)}+\frac{30 b^2 c^2 d^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^2 x^2\right) x^{m+3}}{(m+2) (m+3)^2 (m+5) (m+7)^2}+\frac{48 b^2 c^2 d^3 x^{m+3}}{(m+3)^3 (m+5) (m+7)}+\frac{12 b^2 c^2 d^3 x^{m+3}}{(m+3) (m+5)^2 (m+7)}+\frac{36 b^2 c^2 d^3 x^{m+3}}{(m+3)^2 (m+5)^2 (m+7)}+\frac{10 b^2 c^2 d^3 x^{m+3}}{(m+7)^2 \left(m^2+8 m+15\right)}+\frac{2 b^2 c^2 d^3 x^{m+3}}{(m+3) (m+7)^2}+\frac{30 b^2 c^2 d^3 x^{m+3}}{(m+3)^2 (m+5) (m+7)^2}-\frac{12 b^2 c^4 d^3 x^{m+5}}{(m+5)^3 (m+7)}-\frac{4 b^2 c^4 d^3 x^{m+5}}{(m+5) (m+7)^2}-\frac{10 b^2 c^4 d^3 x^{m+5}}{(m+5)^2 (m+7)^2}+\frac{2 b^2 c^6 d^3 x^{m+7}}{(m+7)^3}",1,"Integrate[x^m*(d - c^2*d*x^2)^3*(a + b*ArcSin[c*x])^2, x]","F",-1
277,0,0,756,0.154868,"\int x^m \left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^m*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2,x]","\int x^m \left(d-c^2 d x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","\frac{16 b^2 c^2 d^2 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^2 x^2\right)}{(m+3)^2 (m+5) \left(m^2+3 m+2\right)}+\frac{8 b^2 c^2 d^2 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^2 x^2\right)}{(m+2) (m+3)^3 (m+5)}+\frac{6 b^2 c^2 d^2 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^2 x^2\right)}{(m+2) (m+3)^2 (m+5)^2}-\frac{6 b c d^2 x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{(m+5)^2 \left(m^2+5 m+6\right)}-\frac{16 b c d^2 x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{(m+5) \left(m^3+6 m^2+11 m+6\right)}-\frac{8 b c d^2 x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{(m+2) (m+3)^2 (m+5)}+\frac{4 d^2 \left(1-c^2 x^2\right) x^{m+1} \left(a+b \sin ^{-1}(c x)\right)^2}{m^2+8 m+15}+\frac{d^2 \left(1-c^2 x^2\right)^2 x^{m+1} \left(a+b \sin ^{-1}(c x)\right)^2}{m+5}-\frac{2 b c d^2 \left(1-c^2 x^2\right)^{3/2} x^{m+2} \left(a+b \sin ^{-1}(c x)\right)}{(m+5)^2}-\frac{8 b c d^2 \sqrt{1-c^2 x^2} x^{m+2} \left(a+b \sin ^{-1}(c x)\right)}{(m+3)^2 (m+5)}-\frac{6 b c d^2 \sqrt{1-c^2 x^2} x^{m+2} \left(a+b \sin ^{-1}(c x)\right)}{(m+3) (m+5)^2}+\frac{8 d^2 x^{m+1} \left(a+b \sin ^{-1}(c x)\right)^2}{(m+5) \left(m^2+4 m+3\right)}-\frac{2 b^2 c^4 d^2 x^{m+5}}{(m+5)^3}+\frac{8 b^2 c^2 d^2 x^{m+3}}{(m+3)^3 (m+5)}+\frac{2 b^2 c^2 d^2 x^{m+3}}{(m+3) (m+5)^2}+\frac{6 b^2 c^2 d^2 x^{m+3}}{(m+3)^2 (m+5)^2}",1,"Integrate[x^m*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x])^2, x]","F",-1
278,0,0,371,0.1096601,"\int x^m \left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^m*(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2,x]","\int x^m \left(d-c^2 d x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","\frac{4 b^2 c^2 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^2 x^2\right)}{(m+3)^2 \left(m^2+3 m+2\right)}+\frac{2 b^2 c^2 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^2 x^2\right)}{(m+2) (m+3)^3}-\frac{4 b c d x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{m^3+6 m^2+11 m+6}-\frac{2 b c d x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{(m+2) (m+3)^2}+\frac{d \left(1-c^2 x^2\right) x^{m+1} \left(a+b \sin ^{-1}(c x)\right)^2}{m+3}-\frac{2 b c d \sqrt{1-c^2 x^2} x^{m+2} \left(a+b \sin ^{-1}(c x)\right)}{(m+3)^2}+\frac{2 d x^{m+1} \left(a+b \sin ^{-1}(c x)\right)^2}{m^2+4 m+3}+\frac{2 b^2 c^2 d x^{m+3}}{(m+3)^3}",1,"Integrate[x^m*(d - c^2*d*x^2)*(a + b*ArcSin[c*x])^2, x]","F",-1
279,0,0,30,7.4971792,"\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{d-c^2 d x^2} \, dx","Integrate[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2),x]","\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{d-c^2 d x^2} \, dx","\text{Int}\left(\frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{d-c^2 d x^2},x\right)",0,"Integrate[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2), x]","A",-1
280,0,0,280,9.1435229,"\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^2,x]","\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^2} \, dx","-\frac{b^2 c^2 (m+1) 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^2 x^2\right)}{d^2 \left(m^2+5 m+6\right)}+\frac{(1-m) \text{Int}\left(\frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{d-c^2 d x^2},x\right)}{2 d}+\frac{b c (m+1) x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2 (m+2)}+\frac{x^{m+1} \left(a+b \sin ^{-1}(c x)\right)^2}{2 d^2 \left(1-c^2 x^2\right)}-\frac{b c x^{m+2} \left(a+b \sin ^{-1}(c x)\right)}{d^2 \sqrt{1-c^2 x^2}}+\frac{b^2 c^2 x^{m+3} \, _2F_1\left(1,\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right)}{d^2 (m+3)}",0,"Integrate[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^2, x]","A",-1
281,0,0,669,10.3107934,"\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3,x]","\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^3} \, dx","-\frac{b^2 c^2 (1-m) (m+1) 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^2 x^2\right)}{6 d^3 \left(m^2+5 m+6\right)}-\frac{b^2 c^2 (3-m) (m+1) 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^2 x^2\right)}{4 d^3 \left(m^2+5 m+6\right)}+\frac{(1-m) (3-m) \text{Int}\left(\frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{d-c^2 d x^2},x\right)}{8 d^2}+\frac{b c (1-m) (m+1) x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{6 d^3 (m+2)}+\frac{b c (3-m) (m+1) x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{4 d^3 (m+2)}+\frac{(3-m) x^{m+1} \left(a+b \sin ^{-1}(c x)\right)^2}{8 d^3 \left(1-c^2 x^2\right)}+\frac{x^{m+1} \left(a+b \sin ^{-1}(c x)\right)^2}{4 d^3 \left(1-c^2 x^2\right)^2}-\frac{b c (1-m) x^{m+2} \left(a+b \sin ^{-1}(c x)\right)}{6 d^3 \sqrt{1-c^2 x^2}}-\frac{b c (3-m) x^{m+2} \left(a+b \sin ^{-1}(c x)\right)}{4 d^3 \sqrt{1-c^2 x^2}}-\frac{b c x^{m+2} \left(a+b \sin ^{-1}(c x)\right)}{6 d^3 \left(1-c^2 x^2\right)^{3/2}}+\frac{b^2 c^2 (1-m) x^{m+3} \, _2F_1\left(1,\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right)}{6 d^3 (m+3)}+\frac{b^2 c^2 (3-m) x^{m+3} \, _2F_1\left(1,\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right)}{4 d^3 (m+3)}+\frac{b^2 c^2 x^{m+3} \, _2F_1\left(2,\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right)}{6 d^3 (m+3)}",0,"Integrate[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^3, x]","A",-1
282,0,0,958,6.431251,"\int x^m \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2,x]","\int x^m \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2 x^{m+1}}{m+6}+\frac{5 d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 x^{m+1}}{(m+4) (m+6)}+\frac{15 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 x^{m+1}}{(m+6) \left(m^2+6 m+8\right)}-\frac{30 b c d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^{m+2}}{(m+2)^2 (m+4) (m+6) \sqrt{1-c^2 x^2}}-\frac{10 b c d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^{m+2}}{(m+6) \left(m^2+6 m+8\right) \sqrt{1-c^2 x^2}}-\frac{2 b c d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^{m+2}}{\left(m^2+8 m+12\right) \sqrt{1-c^2 x^2}}+\frac{10 b^2 c^2 d^2 (3 m+10) \sqrt{d-c^2 d x^2} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right) x^{m+3}}{(m+2) (m+3) (m+4)^3 (m+6) \sqrt{1-c^2 x^2}}+\frac{2 b^2 c^2 d^2 \left(15 m^2+130 m+264\right) \sqrt{d-c^2 d x^2} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right) x^{m+3}}{(m+2) (m+3) (m+4)^2 (m+6)^3 \sqrt{1-c^2 x^2}}+\frac{30 b^2 c^2 d^2 \sqrt{d-c^2 d x^2} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right) x^{m+3}}{(m+2)^2 (m+3) (m+4) (m+6) \sqrt{1-c^2 x^2}}+\frac{2 b^2 c^2 d^2 \left(m^2+15 m+52\right) \sqrt{d-c^2 d x^2} x^{m+3}}{(m+4)^2 (m+6)^3}+\frac{10 b^2 c^2 d^2 \sqrt{d-c^2 d x^2} x^{m+3}}{(m+4)^3 (m+6)}+\frac{4 b c^3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^{m+4}}{(m+4) (m+6) \sqrt{1-c^2 x^2}}+\frac{10 b c^3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^{m+4}}{(m+4)^2 (m+6) \sqrt{1-c^2 x^2}}-\frac{2 b^2 c^4 d^2 \sqrt{d-c^2 d x^2} x^{m+5}}{(m+6)^3}-\frac{2 b c^5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right) x^{m+6}}{(m+6)^2 \sqrt{1-c^2 x^2}}+\frac{15 d^3 \text{Int}\left(\frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}},x\right)}{(m+6) \left(m^2+6 m+8\right)}",0,"Integrate[x^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2, x]","A",-1
283,0,0,500,0.1638959,"\int x^m \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2,x]","\int x^m \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","\frac{3 d^2 \text{Int}\left(\frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}},x\right)}{m^2+6 m+8}+\frac{3 d x^{m+1} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{m^2+6 m+8}-\frac{2 b c d x^{m+2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{\left(m^2+6 m+8\right) \sqrt{1-c^2 x^2}}+\frac{x^{m+1} \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{m+4}-\frac{6 b c d x^{m+2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{(m+2)^2 (m+4) \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d x^{m+4} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{(m+4)^2 \sqrt{1-c^2 x^2}}+\frac{2 b^2 c^2 d (3 m+10) x^{m+3} \sqrt{d-c^2 d x^2} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right)}{(m+2) (m+3) (m+4)^3 \sqrt{1-c^2 x^2}}+\frac{6 b^2 c^2 d x^{m+3} \sqrt{d-c^2 d x^2} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right)}{(m+2)^2 (m+3) (m+4) \sqrt{1-c^2 x^2}}+\frac{2 b^2 c^2 d x^{m+3} \sqrt{d-c^2 d x^2}}{(m+4)^3}",0,"Integrate[x^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2, x]","A",-1
284,0,0,204,0.1137525,"\int x^m \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^m*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2,x]","\int x^m \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","\frac{d \text{Int}\left(\frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}},x\right)}{m+2}+\frac{x^{m+1} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{m+2}-\frac{2 b c x^{m+2} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)}{(m+2)^2 \sqrt{1-c^2 x^2}}+\frac{2 b^2 c^2 x^{m+3} \sqrt{d-c^2 d x^2} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right)}{(m+2)^2 (m+3) \sqrt{1-c^2 x^2}}",0,"Integrate[x^m*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2, x]","A",-1
285,0,0,32,3.4520973,"\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^m*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","\text{Int}\left(\frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}},x\right)",0,"Integrate[(x^m*(a + b*ArcSin[c*x])^2)/Sqrt[d - c^2*d*x^2], x]","A",-1
286,0,0,32,4.5767472,"\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}},x\right)",0,"Integrate[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2), x]","A",-1
287,0,0,32,4.7552341,"\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","\int \frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^m \left(a+b \sin ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}},x\right)",0,"Integrate[(x^m*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2), x]","A",-1
288,0,0,27,0.9448948,"\int \frac{x^m \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^m*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2],x]","\int \frac{x^m \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","\text{Int}\left(\frac{x^m \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}},x\right)",0,"Integrate[(x^m*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2], x]","A",-1
289,1,171,370,0.3996034,"\int \left(c-a^2 c x^2\right)^3 \sin ^{-1}(a x)^3 \, dx","Integrate[(c - a^2*c*x^2)^3*ArcSin[a*x]^3,x]","\frac{c^3 \left(2 \sqrt{1-a^2 x^2} \left(16875 a^6 x^6-134541 a^4 x^4+747937 a^2 x^2-22329151\right)-385875 a x \left(5 a^6 x^6-21 a^4 x^4+35 a^2 x^2-35\right) \sin ^{-1}(a x)^3-11025 \sqrt{1-a^2 x^2} \left(75 a^6 x^6-351 a^4 x^4+757 a^2 x^2-2161\right) \sin ^{-1}(a x)^2+210 a x \left(1125 a^6 x^6-7371 a^4 x^4+26495 a^2 x^2-226905\right) \sin ^{-1}(a x)\right)}{13505625 a}","\frac{6}{343} a^6 c^3 x^7 \sin ^{-1}(a x)-\frac{702 a^4 c^3 x^5 \sin ^{-1}(a x)}{6125}+\frac{1514 a^2 c^3 x^3 \sin ^{-1}(a x)}{3675}-\frac{6 c^3 \left(1-a^2 x^2\right)^{7/2}}{2401 a}-\frac{2664 c^3 \left(1-a^2 x^2\right)^{5/2}}{214375 a}-\frac{30256 c^3 \left(1-a^2 x^2\right)^{3/2}}{385875 a}-\frac{413312 c^3 \sqrt{1-a^2 x^2}}{128625 a}+\frac{1}{7} c^3 x \left(1-a^2 x^2\right)^3 \sin ^{-1}(a x)^3+\frac{6}{35} c^3 x \left(1-a^2 x^2\right)^2 \sin ^{-1}(a x)^3+\frac{8}{35} c^3 x \left(1-a^2 x^2\right) \sin ^{-1}(a x)^3+\frac{3 c^3 \left(1-a^2 x^2\right)^{7/2} \sin ^{-1}(a x)^2}{49 a}+\frac{18 c^3 \left(1-a^2 x^2\right)^{5/2} \sin ^{-1}(a x)^2}{175 a}+\frac{8 c^3 \left(1-a^2 x^2\right)^{3/2} \sin ^{-1}(a x)^2}{35 a}+\frac{48 c^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{35 a}+\frac{16}{35} c^3 x \sin ^{-1}(a x)^3-\frac{4322 c^3 x \sin ^{-1}(a x)}{1225}",1,"(c^3*(2*Sqrt[1 - a^2*x^2]*(-22329151 + 747937*a^2*x^2 - 134541*a^4*x^4 + 16875*a^6*x^6) + 210*a*x*(-226905 + 26495*a^2*x^2 - 7371*a^4*x^4 + 1125*a^6*x^6)*ArcSin[a*x] - 11025*Sqrt[1 - a^2*x^2]*(-2161 + 757*a^2*x^2 - 351*a^4*x^4 + 75*a^6*x^6)*ArcSin[a*x]^2 - 385875*a*x*(-35 + 35*a^2*x^2 - 21*a^4*x^4 + 5*a^6*x^6)*ArcSin[a*x]^3))/(13505625*a)","A",1
290,1,139,273,0.2537166,"\int \left(c-a^2 c x^2\right)^2 \sin ^{-1}(a x)^3 \, dx","Integrate[(c - a^2*c*x^2)^2*ArcSin[a*x]^3,x]","\frac{c^2 \left(-2 \sqrt{1-a^2 x^2} \left(81 a^4 x^4-842 a^2 x^2+31841\right)+1125 a x \left(3 a^4 x^4-10 a^2 x^2+15\right) \sin ^{-1}(a x)^3+225 \sqrt{1-a^2 x^2} \left(9 a^4 x^4-38 a^2 x^2+149\right) \sin ^{-1}(a x)^2-30 a x \left(27 a^4 x^4-190 a^2 x^2+2235\right) \sin ^{-1}(a x)\right)}{16875 a}","-\frac{6}{125} a^4 c^2 x^5 \sin ^{-1}(a x)+\frac{76}{225} a^2 c^2 x^3 \sin ^{-1}(a x)-\frac{6 c^2 \left(1-a^2 x^2\right)^{5/2}}{625 a}-\frac{272 c^2 \left(1-a^2 x^2\right)^{3/2}}{3375 a}-\frac{4144 c^2 \sqrt{1-a^2 x^2}}{1125 a}+\frac{1}{5} c^2 x \left(1-a^2 x^2\right)^2 \sin ^{-1}(a x)^3+\frac{4}{15} c^2 x \left(1-a^2 x^2\right) \sin ^{-1}(a x)^3+\frac{3 c^2 \left(1-a^2 x^2\right)^{5/2} \sin ^{-1}(a x)^2}{25 a}+\frac{4 c^2 \left(1-a^2 x^2\right)^{3/2} \sin ^{-1}(a x)^2}{15 a}+\frac{8 c^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{5 a}+\frac{8}{15} c^2 x \sin ^{-1}(a x)^3-\frac{298}{75} c^2 x \sin ^{-1}(a x)",1,"(c^2*(-2*Sqrt[1 - a^2*x^2]*(31841 - 842*a^2*x^2 + 81*a^4*x^4) - 30*a*x*(2235 - 190*a^2*x^2 + 27*a^4*x^4)*ArcSin[a*x] + 225*Sqrt[1 - a^2*x^2]*(149 - 38*a^2*x^2 + 9*a^4*x^4)*ArcSin[a*x]^2 + 1125*a*x*(15 - 10*a^2*x^2 + 3*a^4*x^4)*ArcSin[a*x]^3))/(16875*a)","A",1
291,1,101,158,0.1145831,"\int \left(c-a^2 c x^2\right) \sin ^{-1}(a x)^3 \, dx","Integrate[(c - a^2*c*x^2)*ArcSin[a*x]^3,x]","\frac{c \left(2 \sqrt{1-a^2 x^2} \left(a^2 x^2-61\right)-9 a x \left(a^2 x^2-3\right) \sin ^{-1}(a x)^3-9 \sqrt{1-a^2 x^2} \left(a^2 x^2-7\right) \sin ^{-1}(a x)^2+6 a x \left(a^2 x^2-21\right) \sin ^{-1}(a x)\right)}{27 a}","\frac{2}{9} a^2 c x^3 \sin ^{-1}(a x)-\frac{2 c \left(1-a^2 x^2\right)^{3/2}}{27 a}-\frac{40 c \sqrt{1-a^2 x^2}}{9 a}+\frac{1}{3} c x \left(1-a^2 x^2\right) \sin ^{-1}(a x)^3+\frac{c \left(1-a^2 x^2\right)^{3/2} \sin ^{-1}(a x)^2}{3 a}+\frac{2 c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+\frac{2}{3} c x \sin ^{-1}(a x)^3-\frac{14}{3} c x \sin ^{-1}(a x)",1,"(c*(2*Sqrt[1 - a^2*x^2]*(-61 + a^2*x^2) + 6*a*x*(-21 + a^2*x^2)*ArcSin[a*x] - 9*Sqrt[1 - a^2*x^2]*(-7 + a^2*x^2)*ArcSin[a*x]^2 - 9*a*x*(-3 + a^2*x^2)*ArcSin[a*x]^3))/(27*a)","A",1
292,1,162,200,0.2237061,"\int \frac{\sin ^{-1}(a x)^3}{c-a^2 c x^2} \, dx","Integrate[ArcSin[a*x]^3/(c - a^2*c*x^2),x]","-\frac{i \left(-3 \sin ^{-1}(a x)^2 \text{Li}_2\left(-i e^{i \sin ^{-1}(a x)}\right)+3 \sin ^{-1}(a x)^2 \text{Li}_2\left(i e^{i \sin ^{-1}(a x)}\right)-6 i \sin ^{-1}(a x) \text{Li}_3\left(-i e^{i \sin ^{-1}(a x)}\right)+6 i \sin ^{-1}(a x) \text{Li}_3\left(i e^{i \sin ^{-1}(a x)}\right)+6 \text{Li}_4\left(-i e^{i \sin ^{-1}(a x)}\right)-6 \text{Li}_4\left(i e^{i \sin ^{-1}(a x)}\right)+2 \sin ^{-1}(a x)^3 \tan ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)\right)}{a c}","\frac{3 i \sin ^{-1}(a x)^2 \text{Li}_2\left(-i e^{i \sin ^{-1}(a x)}\right)}{a c}-\frac{3 i \sin ^{-1}(a x)^2 \text{Li}_2\left(i e^{i \sin ^{-1}(a x)}\right)}{a c}-\frac{6 \sin ^{-1}(a x) \text{Li}_3\left(-i e^{i \sin ^{-1}(a x)}\right)}{a c}+\frac{6 \sin ^{-1}(a x) \text{Li}_3\left(i e^{i \sin ^{-1}(a x)}\right)}{a c}-\frac{6 i \text{Li}_4\left(-i e^{i \sin ^{-1}(a x)}\right)}{a c}+\frac{6 i \text{Li}_4\left(i e^{i \sin ^{-1}(a x)}\right)}{a c}-\frac{2 i \sin ^{-1}(a x)^3 \tan ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)}{a c}",1,"((-I)*(2*ArcSin[a*x]^3*ArcTan[E^(I*ArcSin[a*x])] - 3*ArcSin[a*x]^2*PolyLog[2, (-I)*E^(I*ArcSin[a*x])] + 3*ArcSin[a*x]^2*PolyLog[2, I*E^(I*ArcSin[a*x])] - (6*I)*ArcSin[a*x]*PolyLog[3, (-I)*E^(I*ArcSin[a*x])] + (6*I)*ArcSin[a*x]*PolyLog[3, I*E^(I*ArcSin[a*x])] + 6*PolyLog[4, (-I)*E^(I*ArcSin[a*x])] - 6*PolyLog[4, I*E^(I*ArcSin[a*x])]))/(a*c)","A",1
293,1,234,337,0.5198516,"\int \frac{\sin ^{-1}(a x)^3}{\left(c-a^2 c x^2\right)^2} \, dx","Integrate[ArcSin[a*x]^3/(c - a^2*c*x^2)^2,x]","\frac{\frac{a x \sin ^{-1}(a x)^3}{1-a^2 x^2}-\frac{3 \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-6 \sin ^{-1}(a x) \text{Li}_3\left(-i e^{i \sin ^{-1}(a x)}\right)+6 \sin ^{-1}(a x) \text{Li}_3\left(i e^{i \sin ^{-1}(a x)}\right)+3 i \left(\sin ^{-1}(a x)^2+2\right) \text{Li}_2\left(-i e^{i \sin ^{-1}(a x)}\right)-3 i \left(\sin ^{-1}(a x)^2+2\right) \text{Li}_2\left(i e^{i \sin ^{-1}(a x)}\right)-6 i \text{Li}_4\left(-i e^{i \sin ^{-1}(a x)}\right)+6 i \text{Li}_4\left(i e^{i \sin ^{-1}(a x)}\right)-2 i \sin ^{-1}(a x)^3 \tan ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)-12 i \sin ^{-1}(a x) \tan ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)}{2 a c^2}","\frac{x \sin ^{-1}(a x)^3}{2 c^2 \left(1-a^2 x^2\right)}-\frac{3 \sin ^{-1}(a x)^2}{2 a c^2 \sqrt{1-a^2 x^2}}+\frac{3 i \sin ^{-1}(a x)^2 \text{Li}_2\left(-i e^{i \sin ^{-1}(a x)}\right)}{2 a c^2}-\frac{3 i \sin ^{-1}(a x)^2 \text{Li}_2\left(i e^{i \sin ^{-1}(a x)}\right)}{2 a c^2}-\frac{3 \sin ^{-1}(a x) \text{Li}_3\left(-i e^{i \sin ^{-1}(a x)}\right)}{a c^2}+\frac{3 \sin ^{-1}(a x) \text{Li}_3\left(i e^{i \sin ^{-1}(a x)}\right)}{a c^2}+\frac{3 i \text{Li}_2\left(-i e^{i \sin ^{-1}(a x)}\right)}{a c^2}-\frac{3 i \text{Li}_2\left(i e^{i \sin ^{-1}(a x)}\right)}{a c^2}-\frac{3 i \text{Li}_4\left(-i e^{i \sin ^{-1}(a x)}\right)}{a c^2}+\frac{3 i \text{Li}_4\left(i e^{i \sin ^{-1}(a x)}\right)}{a c^2}-\frac{i \sin ^{-1}(a x)^3 \tan ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)}{a c^2}-\frac{6 i \sin ^{-1}(a x) \tan ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)}{a c^2}",1,"((-3*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2] + (a*x*ArcSin[a*x]^3)/(1 - a^2*x^2) - (12*I)*ArcSin[a*x]*ArcTan[E^(I*ArcSin[a*x])] - (2*I)*ArcSin[a*x]^3*ArcTan[E^(I*ArcSin[a*x])] + (3*I)*(2 + ArcSin[a*x]^2)*PolyLog[2, (-I)*E^(I*ArcSin[a*x])] - (3*I)*(2 + ArcSin[a*x]^2)*PolyLog[2, I*E^(I*ArcSin[a*x])] - 6*ArcSin[a*x]*PolyLog[3, (-I)*E^(I*ArcSin[a*x])] + 6*ArcSin[a*x]*PolyLog[3, I*E^(I*ArcSin[a*x])] - (6*I)*PolyLog[4, (-I)*E^(I*ArcSin[a*x])] + (6*I)*PolyLog[4, I*E^(I*ArcSin[a*x])])/(2*a*c^2)","A",1
294,1,1544,455,12.5201382,"\int \frac{\sin ^{-1}(a x)^3}{\left(c-a^2 c x^2\right)^3} \, dx","Integrate[ArcSin[a*x]^3/(c - a^2*c*x^2)^3,x]","-\frac{-\frac{\sin ^{-1}(a x)^3}{16 \left(\cos \left(\frac{1}{2} \sin ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(a x)\right)\right)^4}+\frac{\sin ^{-1}(a x)^3}{16 \left(\cos \left(\frac{1}{2} \sin ^{-1}(a x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(a x)\right)\right)^4}+\frac{\sin \left(\frac{1}{2} \sin ^{-1}(a x)\right) \sin ^{-1}(a x)^2}{8 \left(\cos \left(\frac{1}{2} \sin ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(a x)\right)\right)^3}-\frac{\sin \left(\frac{1}{2} \sin ^{-1}(a x)\right) \sin ^{-1}(a x)^2}{8 \left(\cos \left(\frac{1}{2} \sin ^{-1}(a x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(a x)\right)\right)^3}+\frac{1}{4} \left(5 \sin ^{-1}(a x)^2+1\right)-\frac{5}{2} \left(\sin ^{-1}(a x) \left(\log \left(1-i e^{i \sin ^{-1}(a x)}\right)-\log \left(1+i e^{i \sin ^{-1}(a x)}\right)\right)+i \left(\text{Li}_2\left(-i e^{i \sin ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{i \sin ^{-1}(a x)}\right)\right)\right)-\frac{3}{8} \left(\frac{1}{8} \pi ^3 \log \left(\cot \left(\frac{1}{2} \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)\right)\right)+\frac{3}{4} \pi ^2 \left(\left(\frac{\pi }{2}-\sin ^{-1}(a x)\right) \left(\log \left(1-e^{i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)}\right)-\log \left(1+e^{i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)}\right)\right)+i \left(\text{Li}_2\left(-e^{i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)}\right)-\text{Li}_2\left(e^{i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)}\right)\right)\right)-\frac{3}{2} \pi  \left(\left(\log \left(1-e^{i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)}\right)-\log \left(1+e^{i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)}\right)\right) \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)^2+2 i \left(\text{Li}_2\left(-e^{i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)}\right)-\text{Li}_2\left(e^{i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)}\right)\right) \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)+2 \left(\text{Li}_3\left(e^{i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)}\right)-\text{Li}_3\left(-e^{i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)}\right)\right)\right)+8 \left(\frac{1}{64} i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)^4-\frac{1}{8} \log \left(1+e^{i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)}\right) \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)^3+\frac{3}{8} i \text{Li}_2\left(-e^{i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)}\right) \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)^2-\frac{3}{4} \text{Li}_3\left(-e^{i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)}\right) \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)+\frac{1}{4} i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^4-\frac{1}{8} \pi ^3 \left(i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)-\log \left(1+e^{2 i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)\right)-\left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^3 \log \left(1+e^{2 i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)+\frac{3}{4} \pi ^2 \left(\frac{1}{2} i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^2-\log \left(1+e^{2 i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right) \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)+\frac{1}{2} i \text{Li}_2\left(-e^{2 i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)\right)+\frac{3}{2} i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^2 \text{Li}_2\left(-e^{2 i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)-\frac{3}{2} \pi  \left(\frac{1}{3} i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^3-\log \left(1+e^{2 i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right) \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)^2+i \text{Li}_2\left(-e^{2 i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right) \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)-\frac{1}{2} \text{Li}_3\left(-e^{2 i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)\right)-\frac{3}{2} \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right) \text{Li}_3\left(-e^{2 i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)-\frac{3}{4} i \text{Li}_4\left(-e^{i \left(\frac{\pi }{2}-\sin ^{-1}(a x)\right)}\right)-\frac{3}{4} i \text{Li}_4\left(-e^{2 i \left(\frac{1}{2} \left(\sin ^{-1}(a x)-\frac{\pi }{2}\right)+\frac{\pi }{2}\right)}\right)\right)\right)-\frac{-5 \sin \left(\frac{1}{2} \sin ^{-1}(a x)\right) \sin ^{-1}(a x)^2-\sin \left(\frac{1}{2} \sin ^{-1}(a x)\right)}{4 \left(\cos \left(\frac{1}{2} \sin ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(a x)\right)\right)}-\frac{5 \sin \left(\frac{1}{2} \sin ^{-1}(a x)\right) \sin ^{-1}(a x)^2+\sin \left(\frac{1}{2} \sin ^{-1}(a x)\right)}{4 \left(\cos \left(\frac{1}{2} \sin ^{-1}(a x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(a x)\right)\right)}-\frac{3 \sin ^{-1}(a x)^3-\sin ^{-1}(a x)^2+2 \sin ^{-1}(a x)}{16 \left(\cos \left(\frac{1}{2} \sin ^{-1}(a x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(a x)\right)\right)^2}-\frac{-3 \sin ^{-1}(a x)^3-\sin ^{-1}(a x)^2-2 \sin ^{-1}(a x)}{16 \left(\cos \left(\frac{1}{2} \sin ^{-1}(a x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(a x)\right)\right)^2}}{a c^3}","-\frac{1}{4 a c^3 \sqrt{1-a^2 x^2}}+\frac{3 x \sin ^{-1}(a x)^3}{8 c^3 \left(1-a^2 x^2\right)}+\frac{x \sin ^{-1}(a x)^3}{4 c^3 \left(1-a^2 x^2\right)^2}-\frac{9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt{1-a^2 x^2}}-\frac{\sin ^{-1}(a x)^2}{4 a c^3 \left(1-a^2 x^2\right)^{3/2}}+\frac{x \sin ^{-1}(a x)}{4 c^3 \left(1-a^2 x^2\right)}+\frac{9 i \sin ^{-1}(a x)^2 \text{Li}_2\left(-i e^{i \sin ^{-1}(a x)}\right)}{8 a c^3}-\frac{9 i \sin ^{-1}(a x)^2 \text{Li}_2\left(i e^{i \sin ^{-1}(a x)}\right)}{8 a c^3}-\frac{9 \sin ^{-1}(a x) \text{Li}_3\left(-i e^{i \sin ^{-1}(a x)}\right)}{4 a c^3}+\frac{9 \sin ^{-1}(a x) \text{Li}_3\left(i e^{i \sin ^{-1}(a x)}\right)}{4 a c^3}+\frac{5 i \text{Li}_2\left(-i e^{i \sin ^{-1}(a x)}\right)}{2 a c^3}-\frac{5 i \text{Li}_2\left(i e^{i \sin ^{-1}(a x)}\right)}{2 a c^3}-\frac{9 i \text{Li}_4\left(-i e^{i \sin ^{-1}(a x)}\right)}{4 a c^3}+\frac{9 i \text{Li}_4\left(i e^{i \sin ^{-1}(a x)}\right)}{4 a c^3}-\frac{3 i \sin ^{-1}(a x)^3 \tan ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)}{4 a c^3}-\frac{5 i \sin ^{-1}(a x) \tan ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)}{a c^3}",1,"-(((1 + 5*ArcSin[a*x]^2)/4 - (5*(ArcSin[a*x]*(Log[1 - I*E^(I*ArcSin[a*x])] - Log[1 + I*E^(I*ArcSin[a*x])]) + I*(PolyLog[2, (-I)*E^(I*ArcSin[a*x])] - PolyLog[2, I*E^(I*ArcSin[a*x])])))/2 - (3*((Pi^3*Log[Cot[(Pi/2 - ArcSin[a*x])/2]])/8 + (3*Pi^2*((Pi/2 - ArcSin[a*x])*(Log[1 - E^(I*(Pi/2 - ArcSin[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcSin[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcSin[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcSin[a*x]))])))/4 - (3*Pi*((Pi/2 - ArcSin[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcSin[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcSin[a*x]))]) + (2*I)*(Pi/2 - ArcSin[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcSin[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcSin[a*x]))]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcSin[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcSin[a*x]))])))/2 + 8*((I/64)*(Pi/2 - ArcSin[a*x])^4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)^4 - ((Pi/2 - ArcSin[a*x])^3*Log[1 + E^(I*(Pi/2 - ArcSin[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))]))/8 - (Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))] + ((3*I)/8)*(Pi/2 - ArcSin[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcSin[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)^2 - (Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))] + (I/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)^2*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))] - (3*(Pi/2 - ArcSin[a*x])*PolyLog[3, -E^(I*(Pi/2 - ArcSin[a*x]))])/4 - (3*Pi*((I/3)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)^3 - (Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)^2*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))] + I*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))] - PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))]/2))/2 - (3*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)*PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))])/2 - ((3*I)/4)*PolyLog[4, -E^(I*(Pi/2 - ArcSin[a*x]))] - ((3*I)/4)*PolyLog[4, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))])))/8 - ArcSin[a*x]^3/(16*(Cos[ArcSin[a*x]/2] - Sin[ArcSin[a*x]/2])^4) - (2*ArcSin[a*x] - ArcSin[a*x]^2 + 3*ArcSin[a*x]^3)/(16*(Cos[ArcSin[a*x]/2] - Sin[ArcSin[a*x]/2])^2) + (ArcSin[a*x]^2*Sin[ArcSin[a*x]/2])/(8*(Cos[ArcSin[a*x]/2] - Sin[ArcSin[a*x]/2])^3) + ArcSin[a*x]^3/(16*(Cos[ArcSin[a*x]/2] + Sin[ArcSin[a*x]/2])^4) - (ArcSin[a*x]^2*Sin[ArcSin[a*x]/2])/(8*(Cos[ArcSin[a*x]/2] + Sin[ArcSin[a*x]/2])^3) - (-2*ArcSin[a*x] - ArcSin[a*x]^2 - 3*ArcSin[a*x]^3)/(16*(Cos[ArcSin[a*x]/2] + Sin[ArcSin[a*x]/2])^2) - (-Sin[ArcSin[a*x]/2] - 5*ArcSin[a*x]^2*Sin[ArcSin[a*x]/2])/(4*(Cos[ArcSin[a*x]/2] - Sin[ArcSin[a*x]/2])) - (Sin[ArcSin[a*x]/2] + 5*ArcSin[a*x]^2*Sin[ArcSin[a*x]/2])/(4*(Cos[ArcSin[a*x]/2] + Sin[ArcSin[a*x]/2])))/(a*c^3))","B",0
295,1,179,533,0.8841046,"\int \left(c-a^2 c x^2\right)^{5/2} \sin ^{-1}(a x)^3 \, dx","Integrate[(c - a^2*c*x^2)^(5/2)*ArcSin[a*x]^3,x]","\frac{c^2 \sqrt{c-a^2 c x^2} \left(4320 \sin ^{-1}(a x)^4+288 \left(45 \sin \left(2 \sin ^{-1}(a x)\right)+9 \sin \left(4 \sin ^{-1}(a x)\right)+\sin \left(6 \sin ^{-1}(a x)\right)\right) \sin ^{-1}(a x)^3-12 \left(1620 \sin \left(2 \sin ^{-1}(a x)\right)+81 \sin \left(4 \sin ^{-1}(a x)\right)+4 \sin \left(6 \sin ^{-1}(a x)\right)\right) \sin ^{-1}(a x)+72 \sin ^{-1}(a x)^2 \left(270 \cos \left(2 \sin ^{-1}(a x)\right)+27 \cos \left(4 \sin ^{-1}(a x)\right)+2 \cos \left(6 \sin ^{-1}(a x)\right)\right)-9720 \cos \left(2 \sin ^{-1}(a x)\right)-243 \cos \left(4 \sin ^{-1}(a x)\right)-8 \cos \left(6 \sin ^{-1}(a x)\right)\right)}{55296 a \sqrt{1-a^2 x^2}}","\frac{865 a c^2 x^2 \sqrt{c-a^2 c x^2}}{2304 \sqrt{1-a^2 x^2}}-\frac{c^2 \left(1-a^2 x^2\right)^{5/2} \sqrt{c-a^2 c x^2}}{216 a}-\frac{15 a c^2 x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{32 \sqrt{1-a^2 x^2}}+\frac{5}{16} c^2 x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^3-\frac{245}{384} c^2 x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)-\frac{1}{36} c^2 x \left(1-a^2 x^2\right)^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)-\frac{65}{576} c^2 x \left(1-a^2 x^2\right) \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)+\frac{5 c^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^4}{64 a \sqrt{1-a^2 x^2}}+\frac{c^2 \left(1-a^2 x^2\right)^{5/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{12 a}+\frac{5 c^2 \left(1-a^2 x^2\right)^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{32 a}+\frac{115 c^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{768 a \sqrt{1-a^2 x^2}}+\frac{1}{6} x \left(c-a^2 c x^2\right)^{5/2} \sin ^{-1}(a x)^3+\frac{5}{24} c x \left(c-a^2 c x^2\right)^{3/2} \sin ^{-1}(a x)^3-\frac{65 a^3 c^2 x^4 \sqrt{c-a^2 c x^2}}{2304 \sqrt{1-a^2 x^2}}",1,"(c^2*Sqrt[c - a^2*c*x^2]*(4320*ArcSin[a*x]^4 - 9720*Cos[2*ArcSin[a*x]] - 243*Cos[4*ArcSin[a*x]] - 8*Cos[6*ArcSin[a*x]] + 72*ArcSin[a*x]^2*(270*Cos[2*ArcSin[a*x]] + 27*Cos[4*ArcSin[a*x]] + 2*Cos[6*ArcSin[a*x]]) + 288*ArcSin[a*x]^3*(45*Sin[2*ArcSin[a*x]] + 9*Sin[4*ArcSin[a*x]] + Sin[6*ArcSin[a*x]]) - 12*ArcSin[a*x]*(1620*Sin[2*ArcSin[a*x]] + 81*Sin[4*ArcSin[a*x]] + 4*Sin[6*ArcSin[a*x]])))/(55296*a*Sqrt[1 - a^2*x^2])","A",1
296,1,138,365,0.3549174,"\int \left(c-a^2 c x^2\right)^{3/2} \sin ^{-1}(a x)^3 \, dx","Integrate[(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^3,x]","\frac{c \sqrt{c-a^2 c x^2} \left(96 \sin ^{-1}(a x)^4+32 \left(8 \sin \left(2 \sin ^{-1}(a x)\right)+\sin \left(4 \sin ^{-1}(a x)\right)\right) \sin ^{-1}(a x)^3-12 \left(32 \sin \left(2 \sin ^{-1}(a x)\right)+\sin \left(4 \sin ^{-1}(a x)\right)\right) \sin ^{-1}(a x)+24 \sin ^{-1}(a x)^2 \left(16 \cos \left(2 \sin ^{-1}(a x)\right)+\cos \left(4 \sin ^{-1}(a x)\right)\right)-3 \left(64 \cos \left(2 \sin ^{-1}(a x)\right)+\cos \left(4 \sin ^{-1}(a x)\right)\right)\right)}{1024 a \sqrt{1-a^2 x^2}}","\frac{51 a c x^2 \sqrt{c-a^2 c x^2}}{128 \sqrt{1-a^2 x^2}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 \sqrt{1-a^2 x^2}}+\frac{1}{4} x \left(c-a^2 c x^2\right)^{3/2} \sin ^{-1}(a x)^3+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^3-\frac{45}{64} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)-\frac{3}{32} c x \left(1-a^2 x^2\right) \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^4}{32 a \sqrt{1-a^2 x^2}}+\frac{3 c \left(1-a^2 x^2\right)^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{16 a}+\frac{27 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{128 a \sqrt{1-a^2 x^2}}-\frac{3 a^3 c x^4 \sqrt{c-a^2 c x^2}}{128 \sqrt{1-a^2 x^2}}",1,"(c*Sqrt[c - a^2*c*x^2]*(96*ArcSin[a*x]^4 + 24*ArcSin[a*x]^2*(16*Cos[2*ArcSin[a*x]] + Cos[4*ArcSin[a*x]]) - 3*(64*Cos[2*ArcSin[a*x]] + Cos[4*ArcSin[a*x]]) + 32*ArcSin[a*x]^3*(8*Sin[2*ArcSin[a*x]] + Sin[4*ArcSin[a*x]]) - 12*ArcSin[a*x]*(32*Sin[2*ArcSin[a*x]] + Sin[4*ArcSin[a*x]])))/(1024*a*Sqrt[1 - a^2*x^2])","A",1
297,1,114,215,0.0702265,"\int \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^3 \, dx","Integrate[Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^3,x]","\frac{\sqrt{c-a^2 c x^2} \left(3 a^2 x^2+4 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3+\left(3-6 a^2 x^2\right) \sin ^{-1}(a x)^2-6 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+\sin ^{-1}(a x)^4\right)}{8 a \sqrt{1-a^2 x^2}}","\frac{3 a x^2 \sqrt{c-a^2 c x^2}}{8 \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^4}{8 a \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^3-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{4 \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^2}{8 a \sqrt{1-a^2 x^2}}-\frac{3}{4} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)",1,"(Sqrt[c - a^2*c*x^2]*(3*a^2*x^2 - 6*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + (3 - 6*a^2*x^2)*ArcSin[a*x]^2 + 4*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3 + ArcSin[a*x]^4))/(8*a*Sqrt[1 - a^2*x^2])","A",1
298,1,42,42,0.0509812,"\int \frac{\sin ^{-1}(a x)^3}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[ArcSin[a*x]^3/Sqrt[c - a^2*c*x^2],x]","\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^4}{4 a \sqrt{c-a^2 c x^2}}","\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^4}{4 a \sqrt{c-a^2 c x^2}}",1,"(Sqrt[1 - a^2*x^2]*ArcSin[a*x]^4)/(4*a*Sqrt[c - a^2*c*x^2])","A",1
299,1,157,238,0.2802972,"\int \frac{\sin ^{-1}(a x)^3}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcSin[a*x]^3/(c - a^2*c*x^2)^(3/2),x]","\frac{-6 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{Li}_2\left(-e^{2 i \sin ^{-1}(a x)}\right)+3 \sqrt{1-a^2 x^2} \text{Li}_3\left(-e^{2 i \sin ^{-1}(a x)}\right)+2 \sin ^{-1}(a x)^2 \left(\left(a x-i \sqrt{1-a^2 x^2}\right) \sin ^{-1}(a x)+3 \sqrt{1-a^2 x^2} \log \left(1+e^{2 i \sin ^{-1}(a x)}\right)\right)}{2 a c \sqrt{c-a^2 c x^2}}","-\frac{3 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{Li}_2\left(-e^{2 i \sin ^{-1}(a x)}\right)}{a c \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{1-a^2 x^2} \text{Li}_3\left(-e^{2 i \sin ^{-1}(a x)}\right)}{2 a c \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}}-\frac{i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a c \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left(1+e^{2 i \sin ^{-1}(a x)}\right)}{a c \sqrt{c-a^2 c x^2}}",1,"(2*ArcSin[a*x]^2*((a*x - I*Sqrt[1 - a^2*x^2])*ArcSin[a*x] + 3*Sqrt[1 - a^2*x^2]*Log[1 + E^((2*I)*ArcSin[a*x])]) - (6*I)*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*PolyLog[2, -E^((2*I)*ArcSin[a*x])] + 3*Sqrt[1 - a^2*x^2]*PolyLog[3, -E^((2*I)*ArcSin[a*x])])/(2*a*c*Sqrt[c - a^2*c*x^2])","A",1
300,1,211,388,0.6313334,"\int \frac{\sin ^{-1}(a x)^3}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcSin[a*x]^3/(c - a^2*c*x^2)^(5/2),x]","\frac{\left(1-a^2 x^2\right)^{3/2} \left(3 \log \left(1-a^2 x^2\right)+\frac{4 a x \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}}+\frac{2 a x \sin ^{-1}(a x)^3}{\left(1-a^2 x^2\right)^{3/2}}+\frac{3 \sin ^{-1}(a x)^2}{a^2 x^2-1}+\frac{6 a x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-12 i \sin ^{-1}(a x) \text{Li}_2\left(-e^{2 i \sin ^{-1}(a x)}\right)+6 \text{Li}_3\left(-e^{2 i \sin ^{-1}(a x)}\right)-4 i \sin ^{-1}(a x)^3+12 \sin ^{-1}(a x)^2 \log \left(1+e^{2 i \sin ^{-1}(a x)}\right)\right)}{6 a c \left(c-a^2 c x^2\right)^{3/2}}","-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{Li}_2\left(-e^{2 i \sin ^{-1}(a x)}\right)}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \text{Li}_3\left(-e^{2 i \sin ^{-1}(a x)}\right)}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left(1-a^2 x^2\right)}{2 a c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \sin ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{2 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)^2}{2 a c^2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left(1+e^{2 i \sin ^{-1}(a x)}\right)}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{3 c \left(c-a^2 c x^2\right)^{3/2}}",1,"((1 - a^2*x^2)^(3/2)*((6*a*x*ArcSin[a*x])/Sqrt[1 - a^2*x^2] + (3*ArcSin[a*x]^2)/(-1 + a^2*x^2) - (4*I)*ArcSin[a*x]^3 + (2*a*x*ArcSin[a*x]^3)/(1 - a^2*x^2)^(3/2) + (4*a*x*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2] + 12*ArcSin[a*x]^2*Log[1 + E^((2*I)*ArcSin[a*x])] + 3*Log[1 - a^2*x^2] - (12*I)*ArcSin[a*x]*PolyLog[2, -E^((2*I)*ArcSin[a*x])] + 6*PolyLog[3, -E^((2*I)*ArcSin[a*x])]))/(6*a*c*(c - a^2*c*x^2)^(3/2))","A",1
301,1,319,547,0.839608,"\int \frac{\sin ^{-1}(a x)^3}{\left(c-a^2 c x^2\right)^{7/2}} \, dx","Integrate[ArcSin[a*x]^3/(c - a^2*c*x^2)^(7/2),x]","\frac{-96 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{Li}_2\left(-e^{2 i \sin ^{-1}(a x)}\right)+48 \sqrt{1-a^2 x^2} \text{Li}_3\left(-e^{2 i \sin ^{-1}(a x)}\right)-\frac{3}{\sqrt{1-a^2 x^2}}+30 \sqrt{1-a^2 x^2} \log \left(1-a^2 x^2\right)-32 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3+\frac{16 a x \sin ^{-1}(a x)^3}{1-a^2 x^2}+\frac{12 a x \sin ^{-1}(a x)^3}{\left(a^2 x^2-1\right)^2}-\frac{24 \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{9 \sin ^{-1}(a x)^2}{\left(1-a^2 x^2\right)^{3/2}}+\frac{6 a x \sin ^{-1}(a x)}{1-a^2 x^2}+96 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left(1+e^{2 i \sin ^{-1}(a x)}\right)+32 a x \sin ^{-1}(a x)^3+60 a x \sin ^{-1}(a x)}{60 a c^3 \sqrt{c-a^2 c x^2}}","-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{Li}_2\left(-e^{2 i \sin ^{-1}(a x)}\right)}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{1-a^2 x^2} \text{Li}_3\left(-e^{2 i \sin ^{-1}(a x)}\right)}{5 a c^3 \sqrt{c-a^2 c x^2}}-\frac{1}{20 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left(1-a^2 x^2\right)}{2 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}-\frac{3 \sin ^{-1}(a x)^2}{20 a c^3 \left(1-a^2 x^2\right)^{3/2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{10 c^3 \left(1-a^2 x^2\right) \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left(1+e^{2 i \sin ^{-1}(a x)}\right)}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \sin ^{-1}(a x)^3}{15 c^2 \left(c-a^2 c x^2\right)^{3/2}}+\frac{x \sin ^{-1}(a x)^3}{5 c \left(c-a^2 c x^2\right)^{5/2}}",1,"(-3/Sqrt[1 - a^2*x^2] + 60*a*x*ArcSin[a*x] + (6*a*x*ArcSin[a*x])/(1 - a^2*x^2) - (9*ArcSin[a*x]^2)/(1 - a^2*x^2)^(3/2) - (24*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2] + 32*a*x*ArcSin[a*x]^3 + (16*a*x*ArcSin[a*x]^3)/(1 - a^2*x^2) - (32*I)*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3 + (12*a*x*ArcSin[a*x]^3)/(-1 + a^2*x^2)^2 + 96*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2*Log[1 + E^((2*I)*ArcSin[a*x])] + 30*Sqrt[1 - a^2*x^2]*Log[1 - a^2*x^2] - (96*I)*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*PolyLog[2, -E^((2*I)*ArcSin[a*x])] + 48*Sqrt[1 - a^2*x^2]*PolyLog[3, -E^((2*I)*ArcSin[a*x])])/(60*a*c^3*Sqrt[c - a^2*c*x^2])","A",1
302,0,0,27,0.9150856,"\int \frac{x^m \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^m*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2],x]","\int \frac{x^m \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","\text{Int}\left(\frac{x^m \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}},x\right)",0,"Integrate[(x^m*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2], x]","A",-1
303,1,125,191,0.0701035,"\int \frac{x^4 \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^4*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2],x]","\frac{-3 a^2 x^2 \left(a^2 x^2+15\right)-16 a x \sqrt{1-a^2 x^2} \left(2 a^2 x^2+3\right) \sin ^{-1}(a x)^3+6 a x \sqrt{1-a^2 x^2} \left(2 a^2 x^2+15\right) \sin ^{-1}(a x)+3 \left(8 a^4 x^4+24 a^2 x^2-15\right) \sin ^{-1}(a x)^2+12 \sin ^{-1}(a x)^4}{128 a^5}","\frac{3 \sin ^{-1}(a x)^4}{32 a^5}-\frac{45 \sin ^{-1}(a x)^2}{128 a^5}-\frac{45 x^2}{128 a^3}+\frac{9 x^2 \sin ^{-1}(a x)^2}{16 a^3}-\frac{x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{4 a^2}+\frac{3 x^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{32 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{8 a^4}+\frac{45 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{64 a^4}-\frac{3 x^4}{128 a}+\frac{3 x^4 \sin ^{-1}(a x)^2}{16 a}",1,"(-3*a^2*x^2*(15 + a^2*x^2) + 6*a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2)*ArcSin[a*x] + 3*(-15 + 24*a^2*x^2 + 8*a^4*x^4)*ArcSin[a*x]^2 - 16*a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)*ArcSin[a*x]^3 + 12*ArcSin[a*x]^4)/(128*a^5)","A",1
304,1,100,157,0.0687046,"\int \frac{x^3 \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^3*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2],x]","\frac{-2 a x \left(a^2 x^2+60\right)-9 \sqrt{1-a^2 x^2} \left(a^2 x^2+2\right) \sin ^{-1}(a x)^3+9 a x \left(a^2 x^2+6\right) \sin ^{-1}(a x)^2+6 \sqrt{1-a^2 x^2} \left(a^2 x^2+20\right) \sin ^{-1}(a x)}{27 a^4}","-\frac{40 x}{9 a^3}+\frac{2 x \sin ^{-1}(a x)^2}{a^3}-\frac{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^2}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^2}-\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{3 a^4}+\frac{40 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^4}-\frac{2 x^3}{27 a}+\frac{x^3 \sin ^{-1}(a x)^2}{3 a}",1,"(-2*a*x*(60 + a^2*x^2) + 6*Sqrt[1 - a^2*x^2]*(20 + a^2*x^2)*ArcSin[a*x] + 9*a*x*(6 + a^2*x^2)*ArcSin[a*x]^2 - 9*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcSin[a*x]^3)/(27*a^4)","A",1
305,1,85,107,0.0313021,"\int \frac{x^2 \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^2*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2],x]","\frac{-3 a^2 x^2-4 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3+\left(6 a^2 x^2-3\right) \sin ^{-1}(a x)^2+6 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+\sin ^{-1}(a x)^4}{8 a^3}","\frac{\sin ^{-1}(a x)^4}{8 a^3}-\frac{3 \sin ^{-1}(a x)^2}{8 a^3}-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{2 a^2}+\frac{3 x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{4 a^2}-\frac{3 x^2}{8 a}+\frac{3 x^2 \sin ^{-1}(a x)^2}{4 a}",1,"(-3*a^2*x^2 + 6*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + (-3 + 6*a^2*x^2)*ArcSin[a*x]^2 - 4*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3 + ArcSin[a*x]^4)/(8*a^3)","A",1
306,1,61,67,0.0181743,"\int \frac{x \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x*ArcSin[a*x]^3)/Sqrt[1 - a^2*x^2],x]","\frac{-\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3+6 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-6 a x+3 a x \sin ^{-1}(a x)^2}{a^2}","-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}+\frac{6 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2}-\frac{6 x}{a}+\frac{3 x \sin ^{-1}(a x)^2}{a}",1,"(-6*a*x + 6*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + 3*a*x*ArcSin[a*x]^2 - Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2","A",1
307,1,13,13,0.0042963,"\int \frac{\sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]^3/Sqrt[1 - a^2*x^2],x]","\frac{\sin ^{-1}(a x)^4}{4 a}","\frac{\sin ^{-1}(a x)^4}{4 a}",1,"ArcSin[a*x]^4/(4*a)","A",1
308,1,180,138,0.1607421,"\int \frac{\sin ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]^3/(x*Sqrt[1 - a^2*x^2]),x]","-\frac{1}{8} i \left(-24 \sin ^{-1}(a x)^2 \text{Li}_2\left(e^{-i \sin ^{-1}(a x)}\right)-24 \sin ^{-1}(a x)^2 \text{Li}_2\left(-e^{i \sin ^{-1}(a x)}\right)+48 i \sin ^{-1}(a x) \text{Li}_3\left(e^{-i \sin ^{-1}(a x)}\right)-48 i \sin ^{-1}(a x) \text{Li}_3\left(-e^{i \sin ^{-1}(a x)}\right)+48 \text{Li}_4\left(e^{-i \sin ^{-1}(a x)}\right)+48 \text{Li}_4\left(-e^{i \sin ^{-1}(a x)}\right)-2 \sin ^{-1}(a x)^4+8 i \sin ^{-1}(a x)^3 \log \left(1-e^{-i \sin ^{-1}(a x)}\right)-8 i \sin ^{-1}(a x)^3 \log \left(1+e^{i \sin ^{-1}(a x)}\right)+\pi ^4\right)","3 i \sin ^{-1}(a x)^2 \text{Li}_2\left(-e^{i \sin ^{-1}(a x)}\right)-3 i \sin ^{-1}(a x)^2 \text{Li}_2\left(e^{i \sin ^{-1}(a x)}\right)-6 \sin ^{-1}(a x) \text{Li}_3\left(-e^{i \sin ^{-1}(a x)}\right)+6 \sin ^{-1}(a x) \text{Li}_3\left(e^{i \sin ^{-1}(a x)}\right)-6 i \text{Li}_4\left(-e^{i \sin ^{-1}(a x)}\right)+6 i \text{Li}_4\left(e^{i \sin ^{-1}(a x)}\right)-2 \sin ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)",1,"(-1/8*I)*(Pi^4 - 2*ArcSin[a*x]^4 + (8*I)*ArcSin[a*x]^3*Log[1 - E^((-I)*ArcSin[a*x])] - (8*I)*ArcSin[a*x]^3*Log[1 + E^(I*ArcSin[a*x])] - 24*ArcSin[a*x]^2*PolyLog[2, E^((-I)*ArcSin[a*x])] - 24*ArcSin[a*x]^2*PolyLog[2, -E^(I*ArcSin[a*x])] + (48*I)*ArcSin[a*x]*PolyLog[3, E^((-I)*ArcSin[a*x])] - (48*I)*ArcSin[a*x]*PolyLog[3, -E^(I*ArcSin[a*x])] + 48*PolyLog[4, E^((-I)*ArcSin[a*x])] + 48*PolyLog[4, -E^(I*ArcSin[a*x])])","A",0
309,1,108,99,0.2442642,"\int \frac{\sin ^{-1}(a x)^3}{x^2 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]^3/(x^2*Sqrt[1 - a^2*x^2]),x]","\frac{1}{8} a \left(-\frac{8 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a x}+24 i \sin ^{-1}(a x) \text{Li}_2\left(e^{-2 i \sin ^{-1}(a x)}\right)+12 \text{Li}_3\left(e^{-2 i \sin ^{-1}(a x)}\right)+8 i \sin ^{-1}(a x)^3+24 \sin ^{-1}(a x)^2 \log \left(1-e^{-2 i \sin ^{-1}(a x)}\right)-i \pi ^3\right)","-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-3 i a \sin ^{-1}(a x) \text{Li}_2\left(e^{2 i \sin ^{-1}(a x)}\right)+\frac{3}{2} a \text{Li}_3\left(e^{2 i \sin ^{-1}(a x)}\right)-i a \sin ^{-1}(a x)^3+3 a \sin ^{-1}(a x)^2 \log \left(1-e^{2 i \sin ^{-1}(a x)}\right)",1,"(a*((-I)*Pi^3 + (8*I)*ArcSin[a*x]^3 - (8*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(a*x) + 24*ArcSin[a*x]^2*Log[1 - E^((-2*I)*ArcSin[a*x])] + (24*I)*ArcSin[a*x]*PolyLog[2, E^((-2*I)*ArcSin[a*x])] + 12*PolyLog[3, E^((-2*I)*ArcSin[a*x])]))/8","A",0
310,1,317,264,5.0352549,"\int \frac{\sin ^{-1}(a x)^3}{x^3 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]^3/(x^3*Sqrt[1 - a^2*x^2]),x]","\frac{1}{16} a^2 \left(24 i \sin ^{-1}(a x)^2 \text{Li}_2\left(e^{-i \sin ^{-1}(a x)}\right)+48 \sin ^{-1}(a x) \text{Li}_3\left(e^{-i \sin ^{-1}(a x)}\right)-48 \sin ^{-1}(a x) \text{Li}_3\left(-e^{i \sin ^{-1}(a x)}\right)+24 i \left(\sin ^{-1}(a x)^2+2\right) \text{Li}_2\left(-e^{i \sin ^{-1}(a x)}\right)-48 i \text{Li}_2\left(e^{i \sin ^{-1}(a x)}\right)-48 i \text{Li}_4\left(e^{-i \sin ^{-1}(a x)}\right)-48 i \text{Li}_4\left(-e^{i \sin ^{-1}(a x)}\right)+2 i \sin ^{-1}(a x)^4+8 \sin ^{-1}(a x)^3 \log \left(1-e^{-i \sin ^{-1}(a x)}\right)-8 \sin ^{-1}(a x)^3 \log \left(1+e^{i \sin ^{-1}(a x)}\right)+48 \sin ^{-1}(a x) \log \left(1-e^{i \sin ^{-1}(a x)}\right)-48 \sin ^{-1}(a x) \log \left(1+e^{i \sin ^{-1}(a x)}\right)-12 \sin ^{-1}(a x)^2 \tan \left(\frac{1}{2} \sin ^{-1}(a x)\right)-12 \sin ^{-1}(a x)^2 \cot \left(\frac{1}{2} \sin ^{-1}(a x)\right)-2 \sin ^{-1}(a x)^3 \csc ^2\left(\frac{1}{2} \sin ^{-1}(a x)\right)+2 \sin ^{-1}(a x)^3 \sec ^2\left(\frac{1}{2} \sin ^{-1}(a x)\right)-i \pi ^4\right)","\frac{3}{2} i a^2 \sin ^{-1}(a x)^2 \text{Li}_2\left(-e^{i \sin ^{-1}(a x)}\right)-\frac{3}{2} i a^2 \sin ^{-1}(a x)^2 \text{Li}_2\left(e^{i \sin ^{-1}(a x)}\right)-3 a^2 \sin ^{-1}(a x) \text{Li}_3\left(-e^{i \sin ^{-1}(a x)}\right)+3 a^2 \sin ^{-1}(a x) \text{Li}_3\left(e^{i \sin ^{-1}(a x)}\right)+3 i a^2 \text{Li}_2\left(-e^{i \sin ^{-1}(a x)}\right)-3 i a^2 \text{Li}_2\left(e^{i \sin ^{-1}(a x)}\right)-3 i a^2 \text{Li}_4\left(-e^{i \sin ^{-1}(a x)}\right)+3 i a^2 \text{Li}_4\left(e^{i \sin ^{-1}(a x)}\right)-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{2 x^2}-a^2 \sin ^{-1}(a x)^3 \tanh ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)-6 a^2 \sin ^{-1}(a x) \tanh ^{-1}\left(e^{i \sin ^{-1}(a x)}\right)-\frac{3 a \sin ^{-1}(a x)^2}{2 x}",1,"(a^2*((-I)*Pi^4 + (2*I)*ArcSin[a*x]^4 - 12*ArcSin[a*x]^2*Cot[ArcSin[a*x]/2] - 2*ArcSin[a*x]^3*Csc[ArcSin[a*x]/2]^2 + 8*ArcSin[a*x]^3*Log[1 - E^((-I)*ArcSin[a*x])] + 48*ArcSin[a*x]*Log[1 - E^(I*ArcSin[a*x])] - 48*ArcSin[a*x]*Log[1 + E^(I*ArcSin[a*x])] - 8*ArcSin[a*x]^3*Log[1 + E^(I*ArcSin[a*x])] + (24*I)*ArcSin[a*x]^2*PolyLog[2, E^((-I)*ArcSin[a*x])] + (24*I)*(2 + ArcSin[a*x]^2)*PolyLog[2, -E^(I*ArcSin[a*x])] - (48*I)*PolyLog[2, E^(I*ArcSin[a*x])] + 48*ArcSin[a*x]*PolyLog[3, E^((-I)*ArcSin[a*x])] - 48*ArcSin[a*x]*PolyLog[3, -E^(I*ArcSin[a*x])] - (48*I)*PolyLog[4, E^((-I)*ArcSin[a*x])] - (48*I)*PolyLog[4, -E^(I*ArcSin[a*x])] + 2*ArcSin[a*x]^3*Sec[ArcSin[a*x]/2]^2 - 12*ArcSin[a*x]^2*Tan[ArcSin[a*x]/2]))/16","A",0
311,1,43,67,0.1523305,"\int \frac{\left(c-a^2 c x^2\right)^3}{\sin ^{-1}(a x)} \, dx","Integrate[(c - a^2*c*x^2)^3/ArcSin[a*x],x]","\frac{c^3 \left(35 \text{Ci}\left(\sin ^{-1}(a x)\right)+21 \text{Ci}\left(3 \sin ^{-1}(a x)\right)+7 \text{Ci}\left(5 \sin ^{-1}(a x)\right)+\text{Ci}\left(7 \sin ^{-1}(a x)\right)\right)}{64 a}","\frac{35 c^3 \text{Ci}\left(\sin ^{-1}(a x)\right)}{64 a}+\frac{21 c^3 \text{Ci}\left(3 \sin ^{-1}(a x)\right)}{64 a}+\frac{7 c^3 \text{Ci}\left(5 \sin ^{-1}(a x)\right)}{64 a}+\frac{c^3 \text{Ci}\left(7 \sin ^{-1}(a x)\right)}{64 a}",1,"(c^3*(35*CosIntegral[ArcSin[a*x]] + 21*CosIntegral[3*ArcSin[a*x]] + 7*CosIntegral[5*ArcSin[a*x]] + CosIntegral[7*ArcSin[a*x]]))/(64*a)","A",1
312,1,34,50,0.0940041,"\int \frac{\left(c-a^2 c x^2\right)^2}{\sin ^{-1}(a x)} \, dx","Integrate[(c - a^2*c*x^2)^2/ArcSin[a*x],x]","\frac{c^2 \left(10 \text{Ci}\left(\sin ^{-1}(a x)\right)+5 \text{Ci}\left(3 \sin ^{-1}(a x)\right)+\text{Ci}\left(5 \sin ^{-1}(a x)\right)\right)}{16 a}","\frac{5 c^2 \text{Ci}\left(\sin ^{-1}(a x)\right)}{8 a}+\frac{5 c^2 \text{Ci}\left(3 \sin ^{-1}(a x)\right)}{16 a}+\frac{c^2 \text{Ci}\left(5 \sin ^{-1}(a x)\right)}{16 a}",1,"(c^2*(10*CosIntegral[ArcSin[a*x]] + 5*CosIntegral[3*ArcSin[a*x]] + CosIntegral[5*ArcSin[a*x]]))/(16*a)","A",1
313,1,23,29,0.0165833,"\int \frac{c-a^2 c x^2}{\sin ^{-1}(a x)} \, dx","Integrate[(c - a^2*c*x^2)/ArcSin[a*x],x]","\frac{c \left(3 \text{Ci}\left(\sin ^{-1}(a x)\right)+\text{Ci}\left(3 \sin ^{-1}(a x)\right)\right)}{4 a}","\frac{3 c \text{Ci}\left(\sin ^{-1}(a x)\right)}{4 a}+\frac{c \text{Ci}\left(3 \sin ^{-1}(a x)\right)}{4 a}",1,"(c*(3*CosIntegral[ArcSin[a*x]] + CosIntegral[3*ArcSin[a*x]]))/(4*a)","A",1
314,0,0,23,2.718131,"\int \frac{1}{\left(c-a^2 c x^2\right) \sin ^{-1}(a x)} \, dx","Integrate[1/((c - a^2*c*x^2)*ArcSin[a*x]),x]","\int \frac{1}{\left(c-a^2 c x^2\right) \sin ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{\left(c-a^2 c x^2\right) \sin ^{-1}(a x)},x\right)",0,"Integrate[1/((c - a^2*c*x^2)*ArcSin[a*x]), x]","A",-1
315,0,0,23,8.5605613,"\int \frac{1}{\left(c-a^2 c x^2\right)^2 \sin ^{-1}(a x)} \, dx","Integrate[1/((c - a^2*c*x^2)^2*ArcSin[a*x]),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^2 \sin ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{\left(c-a^2 c x^2\right)^2 \sin ^{-1}(a x)},x\right)",0,"Integrate[1/((c - a^2*c*x^2)^2*ArcSin[a*x]), x]","A",-1
316,1,152,206,0.485692,"\int \frac{x^4 \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(x^4*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]),x]","-\frac{\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 \cos \left(\frac{4 a}{b}\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\cos \left(\frac{6 a}{b}\right) \text{Ci}\left(6 \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)+2 \sin \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\sin \left(\frac{6 a}{b}\right) \text{Si}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-2 \log \left(a+b \sin ^{-1}(c x)\right)}{32 b c^5}","-\frac{\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^5}-\frac{\cos \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^5}+\frac{\cos \left(\frac{6 a}{b}\right) \text{Ci}\left(\frac{6 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^5}-\frac{\sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^5}-\frac{\sin \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^5}+\frac{\sin \left(\frac{6 a}{b}\right) \text{Si}\left(\frac{6 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^5}+\frac{\log \left(a+b \sin ^{-1}(c x)\right)}{16 b c^5}",1,"-1/32*(Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c*x])] + 2*Cos[(4*a)/b]*CosIntegral[4*(a/b + ArcSin[c*x])] - Cos[(6*a)/b]*CosIntegral[6*(a/b + ArcSin[c*x])] - 2*Log[a + b*ArcSin[c*x]] + Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] + 2*Sin[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])] - Sin[(6*a)/b]*SinIntegral[6*(a/b + ArcSin[c*x])])/(b*c^5)","A",1
317,1,135,183,0.3671039,"\int \frac{x^3 \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(x^3*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]),x]","\frac{-2 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-\sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+\cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\cos \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{16 b c^4}","-\frac{\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b c^4}-\frac{\sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^4}+\frac{\sin \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^4}+\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b c^4}+\frac{\cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^4}-\frac{\cos \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^4}",1,"(-2*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] + CosIntegral[5*(a/b + ArcSin[c*x])]*Sin[(5*a)/b] + 2*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] - Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])])/(16*b*c^4)","A",1
318,1,66,82,0.1996191,"\int \frac{x^2 \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(x^2*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]),x]","-\frac{\cos \left(\frac{4 a}{b}\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\log \left(8 \left(a+b \sin ^{-1}(c x)\right)\right)}{8 b c^3}","-\frac{\cos \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 b c^3}-\frac{\sin \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 b c^3}+\frac{\log \left(a+b \sin ^{-1}(c x)\right)}{8 b c^3}",1,"-1/8*(Cos[(4*a)/b]*CosIntegral[4*(a/b + ArcSin[c*x])] - Log[8*(a + b*ArcSin[c*x])] + Sin[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])])/(b*c^3)","A",1
319,1,91,121,0.2441986,"\int \frac{x \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(x*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]),x]","\frac{\sin \left(\frac{a}{b}\right) \left(-\text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+\cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{4 b c^2}","-\frac{\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b c^2}-\frac{\sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b c^2}+\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b c^2}+\frac{\cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b c^2}",1,"(-(CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b]) - CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] + Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])])/(4*b*c^2)","A",1
320,1,62,82,0.1836932,"\int \frac{\sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(a + b*ArcSin[c*x]),x]","\frac{\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)}{2 b c}","\frac{\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b c}+\frac{\sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b c}+\frac{\log \left(a+b \sin ^{-1}(c x)\right)}{2 b c}",1,"(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])])/(2*b*c)","A",1
321,0,0,79,3.1795342,"\int \frac{\sqrt{1-c^2 x^2}}{x \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x*(a + b*ArcSin[c*x])),x]","\int \frac{\sqrt{1-c^2 x^2}}{x \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)},x\right)+\frac{\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b}-\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b}",0,"Integrate[Sqrt[1 - c^2*x^2]/(x*(a + b*ArcSin[c*x])), x]","A",-1
322,0,0,47,0.9737434,"\int \frac{\sqrt{1-c^2 x^2}}{x^2 \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x^2*(a + b*ArcSin[c*x])),x]","\int \frac{\sqrt{1-c^2 x^2}}{x^2 \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)},x\right)-\frac{c \log \left(a+b \sin ^{-1}(c x)\right)}{b}",0,"Integrate[Sqrt[1 - c^2*x^2]/(x^2*(a + b*ArcSin[c*x])), x]","A",-1
323,0,0,31,6.0216164,"\int \frac{\sqrt{1-c^2 x^2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcSin[c*x])),x]","\int \frac{\sqrt{1-c^2 x^2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\sqrt{1-c^2 x^2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcSin[c*x])), x]","A",-1
324,0,0,31,0.8715903,"\int \frac{\sqrt{1-c^2 x^2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcSin[c*x])),x]","\int \frac{\sqrt{1-c^2 x^2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\sqrt{1-c^2 x^2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcSin[c*x])), x]","A",-1
325,1,179,245,0.8192643,"\int \frac{x^3 \left(1-c^2 x^2\right)^{3/2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(x^3*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x]),x]","\frac{-3 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-3 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{7 a}{b}\right) \text{Ci}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+3 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+3 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\cos \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\cos \left(\frac{7 a}{b}\right) \text{Si}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{64 b c^4}","-\frac{3 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{64 b c^4}-\frac{3 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b c^4}+\frac{\sin \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b c^4}+\frac{\sin \left(\frac{7 a}{b}\right) \text{Ci}\left(\frac{7 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b c^4}+\frac{3 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{64 b c^4}+\frac{3 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b c^4}-\frac{\cos \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b c^4}-\frac{\cos \left(\frac{7 a}{b}\right) \text{Si}\left(\frac{7 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b c^4}",1,"(-3*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - 3*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] + CosIntegral[5*(a/b + ArcSin[c*x])]*Sin[(5*a)/b] + CosIntegral[7*(a/b + ArcSin[c*x])]*Sin[(7*a)/b] + 3*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 3*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] - Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])] - Cos[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])])/(64*b*c^4)","A",1
326,1,165,206,0.6720529,"\int \frac{x^2 \left(1-c^2 x^2\right)^{3/2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(x^2*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x]),x]","-\frac{-\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 \cos \left(\frac{4 a}{b}\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\cos \left(\frac{6 a}{b}\right) \text{Ci}\left(6 \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)+2 \sin \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{6 a}{b}\right) \text{Si}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 \log \left(a+b \sin ^{-1}(c x)\right)-4 \log \left(8 \left(a+b \sin ^{-1}(c x)\right)\right)}{32 b c^3}","\frac{\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^3}-\frac{\cos \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^3}-\frac{\cos \left(\frac{6 a}{b}\right) \text{Ci}\left(\frac{6 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^3}+\frac{\sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^3}-\frac{\sin \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^3}-\frac{\sin \left(\frac{6 a}{b}\right) \text{Si}\left(\frac{6 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^3}+\frac{\log \left(a+b \sin ^{-1}(c x)\right)}{16 b c^3}",1,"-1/32*(-(Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c*x])]) + 2*Cos[(4*a)/b]*CosIntegral[4*(a/b + ArcSin[c*x])] + Cos[(6*a)/b]*CosIntegral[6*(a/b + ArcSin[c*x])] + 2*Log[a + b*ArcSin[c*x]] - 4*Log[8*(a + b*ArcSin[c*x])] - Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] + 2*Sin[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])] + Sin[(6*a)/b]*SinIntegral[6*(a/b + ArcSin[c*x])])/(b*c^3)","A",1
327,1,136,183,0.5724046,"\int \frac{x \left(1-c^2 x^2\right)^{3/2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(x*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x]),x]","\frac{-2 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-3 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\sin \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+3 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\cos \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{16 b c^2}","-\frac{\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b c^2}-\frac{3 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^2}-\frac{\sin \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^2}+\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b c^2}+\frac{3 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^2}+\frac{\cos \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^2}",1,"(-2*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - 3*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] - CosIntegral[5*(a/b + ArcSin[c*x])]*Sin[(5*a)/b] + 2*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 3*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])])/(16*b*c^2)","A",1
328,1,121,144,0.3653274,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(a + b*ArcSin[c*x]),x]","\frac{4 \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\cos \left(\frac{4 a}{b}\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+4 \sin \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+4 \log \left(a+b \sin ^{-1}(c x)\right)-\log \left(8 \left(a+b \sin ^{-1}(c x)\right)\right)}{8 b c}","\frac{\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b c}+\frac{\cos \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 b c}+\frac{\sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b c}+\frac{\sin \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 b c}+\frac{3 \log \left(a+b \sin ^{-1}(c x)\right)}{8 b c}",1,"(4*Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c*x])] + Cos[(4*a)/b]*CosIntegral[4*(a/b + ArcSin[c*x])] + 4*Log[a + b*ArcSin[c*x]] - Log[8*(a + b*ArcSin[c*x])] + 4*Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] + Sin[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])])/(8*b*c)","A",0
329,0,0,140,3.3416429,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcSin[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)},x\right)+\frac{5 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b}+\frac{\sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b}-\frac{5 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b}-\frac{\cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b}",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcSin[c*x])), x]","A",-1
330,0,0,107,1.3214071,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^2 \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x^2*(a + b*ArcSin[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^2 \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)},x\right)-\frac{c \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b}-\frac{c \sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b}-\frac{3 c \log \left(a+b \sin ^{-1}(c x)\right)}{2 b}",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x^2*(a + b*ArcSin[c*x])), x]","A",-1
331,0,0,31,5.7448838,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcSin[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{3/2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcSin[c*x])), x]","A",-1
332,0,0,31,0.859827,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcSin[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{3/2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcSin[c*x])), x]","A",-1
333,1,180,245,1.2679716,"\int \frac{x^3 \left(1-c^2 x^2\right)^{5/2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(x^3*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x]),x]","\frac{-6 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-8 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+3 \sin \left(\frac{7 a}{b}\right) \text{Ci}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{9 a}{b}\right) \text{Ci}\left(9 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+6 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+8 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-3 \cos \left(\frac{7 a}{b}\right) \text{Si}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\cos \left(\frac{9 a}{b}\right) \text{Si}\left(9 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{256 b c^4}","-\frac{3 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{128 b c^4}-\frac{\sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^4}+\frac{3 \sin \left(\frac{7 a}{b}\right) \text{Ci}\left(\frac{7 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{256 b c^4}+\frac{\sin \left(\frac{9 a}{b}\right) \text{Ci}\left(\frac{9 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{256 b c^4}+\frac{3 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{128 b c^4}+\frac{\cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^4}-\frac{3 \cos \left(\frac{7 a}{b}\right) \text{Si}\left(\frac{7 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{256 b c^4}-\frac{\cos \left(\frac{9 a}{b}\right) \text{Si}\left(\frac{9 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{256 b c^4}",1,"(-6*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - 8*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] + 3*CosIntegral[7*(a/b + ArcSin[c*x])]*Sin[(7*a)/b] + CosIntegral[9*(a/b + ArcSin[c*x])]*Sin[(9*a)/b] + 6*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 8*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] - 3*Cos[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])] - Cos[(9*a)/b]*SinIntegral[9*(a/b + ArcSin[c*x])])/(256*b*c^4)","A",1
334,1,209,268,1.137591,"\int \frac{x^2 \left(1-c^2 x^2\right)^{5/2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(x^2*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x]),x]","-\frac{-4 \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+4 \cos \left(\frac{4 a}{b}\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+4 \cos \left(\frac{6 a}{b}\right) \text{Ci}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\cos \left(\frac{8 a}{b}\right) \text{Ci}\left(8 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-4 \sin \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+4 \sin \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+4 \sin \left(\frac{6 a}{b}\right) \text{Si}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{8 a}{b}\right) \text{Si}\left(8 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+11 \log \left(a+b \sin ^{-1}(c x)\right)-16 \log \left(8 \left(a+b \sin ^{-1}(c x)\right)\right)}{128 b c^3}","\frac{\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^3}-\frac{\cos \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^3}-\frac{\cos \left(\frac{6 a}{b}\right) \text{Ci}\left(\frac{6 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^3}-\frac{\cos \left(\frac{8 a}{b}\right) \text{Ci}\left(\frac{8 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{128 b c^3}+\frac{\sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^3}-\frac{\sin \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^3}-\frac{\sin \left(\frac{6 a}{b}\right) \text{Si}\left(\frac{6 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c^3}-\frac{\sin \left(\frac{8 a}{b}\right) \text{Si}\left(\frac{8 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{128 b c^3}+\frac{5 \log \left(a+b \sin ^{-1}(c x)\right)}{128 b c^3}",1,"-1/128*(-4*Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c*x])] + 4*Cos[(4*a)/b]*CosIntegral[4*(a/b + ArcSin[c*x])] + 4*Cos[(6*a)/b]*CosIntegral[6*(a/b + ArcSin[c*x])] + Cos[(8*a)/b]*CosIntegral[8*(a/b + ArcSin[c*x])] + 11*Log[a + b*ArcSin[c*x]] - 16*Log[8*(a + b*ArcSin[c*x])] - 4*Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] + 4*Sin[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])] + 4*Sin[(6*a)/b]*SinIntegral[6*(a/b + ArcSin[c*x])] + Sin[(8*a)/b]*SinIntegral[8*(a/b + ArcSin[c*x])])/(b*c^3)","A",1
335,1,180,245,1.163577,"\int \frac{x \left(1-c^2 x^2\right)^{5/2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(x*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x]),x]","\frac{-5 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-9 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-5 \sin \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\sin \left(\frac{7 a}{b}\right) \text{Ci}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+5 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+9 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+5 \cos \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\cos \left(\frac{7 a}{b}\right) \text{Si}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{64 b c^2}","-\frac{5 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{64 b c^2}-\frac{9 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b c^2}-\frac{5 \sin \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b c^2}-\frac{\sin \left(\frac{7 a}{b}\right) \text{Ci}\left(\frac{7 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b c^2}+\frac{5 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{64 b c^2}+\frac{9 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b c^2}+\frac{5 \cos \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b c^2}+\frac{\cos \left(\frac{7 a}{b}\right) \text{Si}\left(\frac{7 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b c^2}",1,"(-5*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - 9*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] - 5*CosIntegral[5*(a/b + ArcSin[c*x])]*Sin[(5*a)/b] - CosIntegral[7*(a/b + ArcSin[c*x])]*Sin[(7*a)/b] + 5*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 9*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 5*Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])] + Cos[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])])/(64*b*c^2)","A",1
336,1,165,206,0.8959683,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(a + b*ArcSin[c*x]),x]","\frac{15 \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+6 \cos \left(\frac{4 a}{b}\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\cos \left(\frac{6 a}{b}\right) \text{Ci}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+15 \sin \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+6 \sin \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{6 a}{b}\right) \text{Si}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+18 \log \left(a+b \sin ^{-1}(c x)\right)-8 \log \left(8 \left(a+b \sin ^{-1}(c x)\right)\right)}{32 b c}","\frac{15 \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c}+\frac{3 \cos \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c}+\frac{\cos \left(\frac{6 a}{b}\right) \text{Ci}\left(\frac{6 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c}+\frac{15 \sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c}+\frac{3 \sin \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c}+\frac{\sin \left(\frac{6 a}{b}\right) \text{Si}\left(\frac{6 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b c}+\frac{5 \log \left(a+b \sin ^{-1}(c x)\right)}{16 b c}",1,"(15*Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c*x])] + 6*Cos[(4*a)/b]*CosIntegral[4*(a/b + ArcSin[c*x])] + Cos[(6*a)/b]*CosIntegral[6*(a/b + ArcSin[c*x])] + 18*Log[a + b*ArcSin[c*x]] - 8*Log[8*(a + b*ArcSin[c*x])] + 15*Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] + 6*Sin[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])] + Sin[(6*a)/b]*SinIntegral[6*(a/b + ArcSin[c*x])])/(32*b*c)","A",1
337,0,0,196,3.5721428,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcSin[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)},x\right)+\frac{11 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b}+\frac{7 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b}+\frac{\sin \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b}-\frac{11 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b}-\frac{7 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b}-\frac{\cos \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b}",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcSin[c*x])), x]","A",-1
338,0,0,161,1.431343,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^2 \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcSin[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^2 \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)},x\right)-\frac{c \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{b}-\frac{c \cos \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 b}-\frac{c \sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{b}-\frac{c \sin \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 b}-\frac{15 c \log \left(a+b \sin ^{-1}(c x)\right)}{8 b}",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcSin[c*x])), x]","A",-1
339,0,0,31,5.9751352,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcSin[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{5/2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcSin[c*x])), x]","A",-1
340,0,0,31,0.9796463,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcSin[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{5/2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcSin[c*x])), x]","A",-1
341,1,31,41,0.0819676,"\int \frac{x^4}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx","Integrate[x^4/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]),x]","\frac{-4 \text{Ci}\left(2 \sin ^{-1}(a x)\right)+\text{Ci}\left(4 \sin ^{-1}(a x)\right)+3 \log \left(\sin ^{-1}(a x)\right)}{8 a^5}","-\frac{\text{Ci}\left(2 \sin ^{-1}(a x)\right)}{2 a^5}+\frac{\text{Ci}\left(4 \sin ^{-1}(a x)\right)}{8 a^5}+\frac{3 \log \left(\sin ^{-1}(a x)\right)}{8 a^5}",1,"(-4*CosIntegral[2*ArcSin[a*x]] + CosIntegral[4*ArcSin[a*x]] + 3*Log[ArcSin[a*x]])/(8*a^5)","A",1
342,1,24,27,0.0740688,"\int \frac{x^3}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx","Integrate[x^3/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]),x]","\frac{3 \text{Si}\left(\sin ^{-1}(a x)\right)-\text{Si}\left(3 \sin ^{-1}(a x)\right)}{4 a^4}","\frac{3 \text{Si}\left(\sin ^{-1}(a x)\right)}{4 a^4}-\frac{\text{Si}\left(3 \sin ^{-1}(a x)\right)}{4 a^4}",1,"(3*SinIntegral[ArcSin[a*x]] - SinIntegral[3*ArcSin[a*x]])/(4*a^4)","A",1
343,1,22,27,0.0728327,"\int \frac{x^2}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx","Integrate[x^2/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]),x]","\frac{\log \left(\sin ^{-1}(a x)\right)-\text{Ci}\left(2 \sin ^{-1}(a x)\right)}{2 a^3}","\frac{\log \left(\sin ^{-1}(a x)\right)}{2 a^3}-\frac{\text{Ci}\left(2 \sin ^{-1}(a x)\right)}{2 a^3}",1,"(-CosIntegral[2*ArcSin[a*x]] + Log[ArcSin[a*x]])/(2*a^3)","A",1
344,1,22,27,0.0171374,"\int \frac{x^2}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx","Integrate[x^2/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]),x]","\frac{\log \left(\sin ^{-1}(a x)\right)-\text{Ci}\left(2 \sin ^{-1}(a x)\right)}{2 a^3}","\frac{\log \left(\sin ^{-1}(a x)\right)}{2 a^3}-\frac{\text{Ci}\left(2 \sin ^{-1}(a x)\right)}{2 a^3}",1,"(-CosIntegral[2*ArcSin[a*x]] + Log[ArcSin[a*x]])/(2*a^3)","A",1
345,1,9,9,0.0602929,"\int \frac{x}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx","Integrate[x/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]),x]","\frac{\text{Si}\left(\sin ^{-1}(a x)\right)}{a^2}","\frac{\text{Si}\left(\sin ^{-1}(a x)\right)}{a^2}",1,"SinIntegral[ArcSin[a*x]]/a^2","A",1
346,1,9,9,0.0192947,"\int \frac{1}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx","Integrate[1/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]),x]","\frac{\log \left(\sin ^{-1}(a x)\right)}{a}","\frac{\log \left(\sin ^{-1}(a x)\right)}{a}",1,"Log[ArcSin[a*x]]/a","A",1
347,0,0,27,1.2935184,"\int \frac{1}{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx","Integrate[1/(x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]),x]","\int \frac{1}{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)},x\right)",0,"Integrate[1/(x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x]","A",-1
348,0,0,27,0.1205312,"\int \frac{1}{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx","Integrate[1/(x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]),x]","\int \frac{1}{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)},x\right)",0,"Integrate[1/(x^2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x]","A",-1
349,1,136,183,0.3578259,"\int \frac{x^5}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[x^5/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])),x]","-\frac{10 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-5 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-10 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+5 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\cos \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{16 b c^6}","-\frac{5 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b c^6}+\frac{5 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^6}-\frac{\sin \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^6}+\frac{5 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b c^6}-\frac{5 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^6}+\frac{\cos \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^6}",1,"-1/16*(10*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - 5*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] + CosIntegral[5*(a/b + ArcSin[c*x])]*Sin[(5*a)/b] - 10*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 5*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] - Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])])/(b*c^6)","A",1
350,1,108,144,0.2647917,"\int \frac{x^4}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[x^4/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])),x]","\frac{-4 \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\cos \left(\frac{4 a}{b}\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-4 \sin \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+3 \log \left(a+b \sin ^{-1}(c x)\right)}{8 b c^5}","-\frac{\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b c^5}+\frac{\cos \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 b c^5}-\frac{\sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b c^5}+\frac{\sin \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 b c^5}+\frac{3 \log \left(a+b \sin ^{-1}(c x)\right)}{8 b c^5}",1,"(-4*Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSin[c*x])] + Cos[(4*a)/b]*CosIntegral[4*(a/b + ArcSin[c*x])] + 3*Log[a + b*ArcSin[c*x]] - 4*Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] + Sin[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])])/(8*b*c^5)","A",1
351,1,92,121,0.2201998,"\int \frac{x^3}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[x^3/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])),x]","-\frac{3 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-\sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-3 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+\cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{4 b c^4}","-\frac{3 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b c^4}+\frac{\sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b c^4}+\frac{3 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b c^4}-\frac{\cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b c^4}",1,"-1/4*(3*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] - CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] - 3*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])])/(b*c^4)","A",1
352,1,64,82,0.1794578,"\int \frac{x^2}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[x^2/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])),x]","-\frac{\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)}{2 b c^3}","-\frac{\cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b c^3}-\frac{\sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b c^3}+\frac{\log \left(a+b \sin ^{-1}(c x)\right)}{2 b c^3}",1,"-1/2*(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*c^3)","A",1
353,1,45,54,0.1141973,"\int \frac{x}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[x/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])),x]","\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)}{b c^2}","\frac{\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b c^2}-\frac{\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b c^2}",1,"(-(CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b]) + Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(b*c^2)","A",1
354,1,16,16,0.0491057,"\int \frac{1}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])),x]","\frac{\log \left(a+b \sin ^{-1}(c x)\right)}{b c}","\frac{\log \left(a+b \sin ^{-1}(c x)\right)}{b c}",1,"Log[a + b*ArcSin[c*x]]/(b*c)","A",1
355,0,0,31,2.8696355,"\int \frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])),x]","\int \frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]","A",-1
356,0,0,31,0.0817528,"\int \frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])),x]","\int \frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]","A",-1
357,0,0,31,3.8947506,"\int \frac{x^2}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[x^2/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])),x]","\int \frac{x^2}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x^2}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[x^2/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]","A",-1
358,0,0,29,10.676626,"\int \frac{x}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[x/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])),x]","\int \frac{x}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[x/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]","A",-1
359,0,0,28,0.1008308,"\int \frac{1}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])),x]","\int \frac{1}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]","A",-1
360,0,0,31,2.8344071,"\int \frac{1}{x \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/(x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])),x]","\int \frac{1}{x \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]","A",-1
361,0,0,31,2.0205784,"\int \frac{1}{x^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])),x]","\int \frac{1}{x^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]","A",-1
362,0,0,31,4.7291396,"\int \frac{x^2}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])),x]","\int \frac{x^2}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x^2}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x]","A",-1
363,0,0,29,26.8028224,"\int \frac{x}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[x/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])),x]","\int \frac{x}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[x/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x]","A",-1
364,0,0,28,0.1128333,"\int \frac{1}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])),x]","\int \frac{1}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x]","A",-1
365,0,0,31,6.338403,"\int \frac{1}{x \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/(x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])),x]","\int \frac{1}{x \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x]","A",-1
366,0,0,31,6.518505,"\int \frac{1}{x^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/(x^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])),x]","\int \frac{1}{x^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x]","A",-1
367,0,0,31,1.0383267,"\int \frac{x^m \left(1-c^2 x^2\right)^{5/2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(x^m*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x]),x]","\int \frac{x^m \left(1-c^2 x^2\right)^{5/2}}{a+b \sin ^{-1}(c x)} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{5/2} x^m}{a+b \sin ^{-1}(c x)},x\right)",0,"Integrate[(x^m*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x]), x]","A",-1
368,0,0,31,0.5564718,"\int \frac{x^m \left(1-c^2 x^2\right)^{3/2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x]),x]","\int \frac{x^m \left(1-c^2 x^2\right)^{3/2}}{a+b \sin ^{-1}(c x)} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{3/2} x^m}{a+b \sin ^{-1}(c x)},x\right)",0,"Integrate[(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x]), x]","A",-1
369,0,0,31,0.0860786,"\int \frac{x^m \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(x^m*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]),x]","\int \frac{x^m \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)} \, dx","\text{Int}\left(\frac{\sqrt{1-c^2 x^2} x^m}{a+b \sin ^{-1}(c x)},x\right)",0,"Integrate[(x^m*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]), x]","A",-1
370,0,0,31,0.6572971,"\int \frac{x^m}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[x^m/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])),x]","\int \frac{x^m}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x^m}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[x^m/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]","A",-1
371,0,0,31,1.2510948,"\int \frac{x^m}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[x^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])),x]","\int \frac{x^m}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x^m}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[x^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]","A",-1
372,0,0,31,1.862446,"\int \frac{x^m}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[x^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])),x]","\int \frac{x^m}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x^m}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[x^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])), x]","A",-1
373,0,0,27,0.3828291,"\int \frac{x^m}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx","Integrate[x^m/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]),x]","\int \frac{x^m}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx","\text{Int}\left(\frac{x^m}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)},x\right)",0,"Integrate[x^m/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]), x]","A",-1
374,1,83,95,0.5985806,"\int \frac{\left(c-a^2 c x^2\right)^3}{\sin ^{-1}(a x)^2} \, dx","Integrate[(c - a^2*c*x^2)^3/ArcSin[a*x]^2,x]","-\frac{c^3 \left(64 \left(1-a^2 x^2\right)^{7/2}+35 \sin ^{-1}(a x) \text{Si}\left(\sin ^{-1}(a x)\right)+63 \sin ^{-1}(a x) \text{Si}\left(3 \sin ^{-1}(a x)\right)+35 \sin ^{-1}(a x) \text{Si}\left(5 \sin ^{-1}(a x)\right)+7 \sin ^{-1}(a x) \text{Si}\left(7 \sin ^{-1}(a x)\right)\right)}{64 a \sin ^{-1}(a x)}","-\frac{c^3 \left(1-a^2 x^2\right)^{7/2}}{a \sin ^{-1}(a x)}-\frac{35 c^3 \text{Si}\left(\sin ^{-1}(a x)\right)}{64 a}-\frac{63 c^3 \text{Si}\left(3 \sin ^{-1}(a x)\right)}{64 a}-\frac{35 c^3 \text{Si}\left(5 \sin ^{-1}(a x)\right)}{64 a}-\frac{7 c^3 \text{Si}\left(7 \sin ^{-1}(a x)\right)}{64 a}",1,"-1/64*(c^3*(64*(1 - a^2*x^2)^(7/2) + 35*ArcSin[a*x]*SinIntegral[ArcSin[a*x]] + 63*ArcSin[a*x]*SinIntegral[3*ArcSin[a*x]] + 35*ArcSin[a*x]*SinIntegral[5*ArcSin[a*x]] + 7*ArcSin[a*x]*SinIntegral[7*ArcSin[a*x]]))/(a*ArcSin[a*x])","A",1
375,1,70,78,0.4761026,"\int \frac{\left(c-a^2 c x^2\right)^2}{\sin ^{-1}(a x)^2} \, dx","Integrate[(c - a^2*c*x^2)^2/ArcSin[a*x]^2,x]","-\frac{c^2 \left(16 \left(1-a^2 x^2\right)^{5/2}+10 \sin ^{-1}(a x) \text{Si}\left(\sin ^{-1}(a x)\right)+15 \sin ^{-1}(a x) \text{Si}\left(3 \sin ^{-1}(a x)\right)+5 \sin ^{-1}(a x) \text{Si}\left(5 \sin ^{-1}(a x)\right)\right)}{16 a \sin ^{-1}(a x)}","-\frac{c^2 \left(1-a^2 x^2\right)^{5/2}}{a \sin ^{-1}(a x)}-\frac{5 c^2 \text{Si}\left(\sin ^{-1}(a x)\right)}{8 a}-\frac{15 c^2 \text{Si}\left(3 \sin ^{-1}(a x)\right)}{16 a}-\frac{5 c^2 \text{Si}\left(5 \sin ^{-1}(a x)\right)}{16 a}",1,"-1/16*(c^2*(16*(1 - a^2*x^2)^(5/2) + 10*ArcSin[a*x]*SinIntegral[ArcSin[a*x]] + 15*ArcSin[a*x]*SinIntegral[3*ArcSin[a*x]] + 5*ArcSin[a*x]*SinIntegral[5*ArcSin[a*x]]))/(a*ArcSin[a*x])","A",1
376,1,55,55,0.143655,"\int \frac{c-a^2 c x^2}{\sin ^{-1}(a x)^2} \, dx","Integrate[(c - a^2*c*x^2)/ArcSin[a*x]^2,x]","-\frac{c \left(4 \left(1-a^2 x^2\right)^{3/2}+3 \sin ^{-1}(a x) \text{Si}\left(\sin ^{-1}(a x)\right)+3 \sin ^{-1}(a x) \text{Si}\left(3 \sin ^{-1}(a x)\right)\right)}{4 a \sin ^{-1}(a x)}","-\frac{c \left(1-a^2 x^2\right)^{3/2}}{a \sin ^{-1}(a x)}-\frac{3 c \text{Si}\left(\sin ^{-1}(a x)\right)}{4 a}-\frac{3 c \text{Si}\left(3 \sin ^{-1}(a x)\right)}{4 a}",1,"-1/4*(c*(4*(1 - a^2*x^2)^(3/2) + 3*ArcSin[a*x]*SinIntegral[ArcSin[a*x]] + 3*ArcSin[a*x]*SinIntegral[3*ArcSin[a*x]]))/(a*ArcSin[a*x])","A",1
377,0,0,59,4.0694035,"\int \frac{1}{\left(c-a^2 c x^2\right) \sin ^{-1}(a x)^2} \, dx","Integrate[1/((c - a^2*c*x^2)*ArcSin[a*x]^2),x]","\int \frac{1}{\left(c-a^2 c x^2\right) \sin ^{-1}(a x)^2} \, dx","\frac{a \text{Int}\left(\frac{x}{\left(1-a^2 x^2\right)^{3/2} \sin ^{-1}(a x)},x\right)}{c}-\frac{1}{a c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}",0,"Integrate[1/((c - a^2*c*x^2)*ArcSin[a*x]^2), x]","A",-1
378,0,0,60,15.5007938,"\int \frac{1}{\left(c-a^2 c x^2\right)^2 \sin ^{-1}(a x)^2} \, dx","Integrate[1/((c - a^2*c*x^2)^2*ArcSin[a*x]^2),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^2 \sin ^{-1}(a x)^2} \, dx","\frac{3 a \text{Int}\left(\frac{x}{\left(1-a^2 x^2\right)^{5/2} \sin ^{-1}(a x)},x\right)}{c^2}-\frac{1}{a c^2 \left(1-a^2 x^2\right)^{3/2} \sin ^{-1}(a x)}",0,"Integrate[1/((c - a^2*c*x^2)^2*ArcSin[a*x]^2), x]","A",-1
379,1,17,17,0.1666631,"\int \left(\frac{1}{\left(1-x^2\right) \sin ^{-1}(x)^2}-\frac{x}{\left(1-x^2\right)^{3/2} \sin ^{-1}(x)}\right) \, dx","Integrate[1/((1 - x^2)*ArcSin[x]^2) - x/((1 - x^2)^(3/2)*ArcSin[x]),x]","-\frac{1}{\sqrt{1-x^2} \sin ^{-1}(x)}","-\frac{1}{\sqrt{1-x^2} \sin ^{-1}(x)}",1,"-(1/(Sqrt[1 - x^2]*ArcSin[x]))","A",1
380,0,0,31,0.5636077,"\int \frac{x^m \sqrt{1-c^2 x^2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(x^m*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])^2,x]","\int \frac{x^m \sqrt{1-c^2 x^2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{1-c^2 x^2} x^m}{\left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(x^m*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])^2, x]","A",-1
381,1,175,214,0.5847979,"\int \frac{x^3 \sqrt{1-c^2 x^2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(x^3*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])^2,x]","\frac{\frac{16 b c^5 x^5}{a+b \sin ^{-1}(c x)}-\frac{16 b c^3 x^3}{a+b \sin ^{-1}(c x)}+2 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+3 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-5 \cos \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+3 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-5 \sin \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{16 b^2 c^4}","\frac{\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b^2 c^4}+\frac{3 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^4}-\frac{5 \cos \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^4}+\frac{\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b^2 c^4}+\frac{3 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^4}-\frac{5 \sin \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^4}-\frac{x^3 \left(1-c^2 x^2\right)}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"((-16*b*c^3*x^3)/(a + b*ArcSin[c*x]) + (16*b*c^5*x^5)/(a + b*ArcSin[c*x]) + 2*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] + 3*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] - 5*Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x])] + 2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 3*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] - 5*Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])])/(16*b^2*c^4)","A",1
382,1,82,94,0.3494963,"\int \frac{x^2 \sqrt{1-c^2 x^2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(x^2*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])^2,x]","\frac{\frac{2 b c^2 x^2 \left(c^2 x^2-1\right)}{a+b \sin ^{-1}(c x)}-\sin \left(\frac{4 a}{b}\right) \text{Ci}\left(4 \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)}{2 b^2 c^3}","-\frac{\sin \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^3}+\frac{\cos \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^3}-\frac{x^2 \left(1-c^2 x^2\right)}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"((2*b*c^2*x^2*(-1 + c^2*x^2))/(a + b*ArcSin[c*x]) - CosIntegral[4*(a/b + ArcSin[c*x])]*Sin[(4*a)/b] + Cos[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])])/(2*b^2*c^3)","A",1
383,1,125,150,0.3184011,"\int \frac{x \sqrt{1-c^2 x^2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(x*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x])^2,x]","\frac{\frac{4 b c^3 x^3}{a+b \sin ^{-1}(c x)}+\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+3 \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)+3 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\frac{4 b c x}{a+b \sin ^{-1}(c x)}}{4 b^2 c^2}","\frac{\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b^2 c^2}+\frac{3 \cos \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^2}+\frac{\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b^2 c^2}+\frac{3 \sin \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^2}-\frac{x \left(1-c^2 x^2\right)}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"((-4*b*c*x)/(a + b*ArcSin[c*x]) + (4*b*c^3*x^3)/(a + b*ArcSin[c*x]) + Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] + 3*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 3*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])])/(4*b^2*c^2)","A",1
384,1,72,86,0.2179406,"\int \frac{\sqrt{1-c^2 x^2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(a + b*ArcSin[c*x])^2,x]","\frac{\frac{b \left(c^2 x^2-1\right)}{a+b \sin ^{-1}(c x)}+\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-\cos \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{b^2 c}","\frac{\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{b^2 c}-\frac{\cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{b^2 c}-\frac{1-c^2 x^2}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"((b*(-1 + c^2*x^2))/(a + b*ArcSin[c*x]) + CosIntegral[2*(a/b + ArcSin[c*x])]*Sin[(2*a)/b] - Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])])/(b^2*c)","A",1
385,0,0,105,11.110053,"\int \frac{\sqrt{1-c^2 x^2}}{x \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x*(a + b*ArcSin[c*x])^2),x]","\int \frac{\sqrt{1-c^2 x^2}}{x \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \left(a+b \sin ^{-1}(c x)\right)},x\right)}{b c}-\frac{\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b^2}-\frac{\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b^2}-\frac{1-c^2 x^2}{b c x \left(a+b \sin ^{-1}(c x)\right)}",0,"Integrate[Sqrt[1 - c^2*x^2]/(x*(a + b*ArcSin[c*x])^2), x]","A",-1
386,0,0,57,2.4925813,"\int \frac{\sqrt{1-c^2 x^2}}{x^2 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x^2*(a + b*ArcSin[c*x])^2),x]","\int \frac{\sqrt{1-c^2 x^2}}{x^2 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^3 \left(a+b \sin ^{-1}(c x)\right)},x\right)}{b c}-\frac{1-c^2 x^2}{b c x^2 \left(a+b \sin ^{-1}(c x)\right)}",0,"Integrate[Sqrt[1 - c^2*x^2]/(x^2*(a + b*ArcSin[c*x])^2), x]","A",-1
387,0,0,31,17.458348,"\int \frac{\sqrt{1-c^2 x^2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcSin[c*x])^2),x]","\int \frac{\sqrt{1-c^2 x^2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{1-c^2 x^2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcSin[c*x])^2), x]","A",-1
388,0,0,31,3.9896544,"\int \frac{\sqrt{1-c^2 x^2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcSin[c*x])^2),x]","\int \frac{\sqrt{1-c^2 x^2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{1-c^2 x^2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcSin[c*x])^2), x]","A",-1
389,0,0,31,0.6190118,"\int \frac{x^m \left(1-c^2 x^2\right)^{3/2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x])^2,x]","\int \frac{x^m \left(1-c^2 x^2\right)^{3/2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{3/2} x^m}{\left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x])^2, x]","A",-1
390,1,399,278,1.2175771,"\int \frac{x^3 \left(1-c^2 x^2\right)^{3/2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(x^3*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x])^2,x]","-\frac{-3 \cos \left(\frac{a}{b}\right) \left(a+b \sin ^{-1}(c x)\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-9 \cos \left(\frac{3 a}{b}\right) \left(a+b \sin ^{-1}(c x)\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+5 a \cos \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+5 b \cos \left(\frac{5 a}{b}\right) \sin ^{-1}(c x) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+7 a \cos \left(\frac{7 a}{b}\right) \text{Ci}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+7 b \cos \left(\frac{7 a}{b}\right) \sin ^{-1}(c x) \text{Ci}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-3 a \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-3 b \sin \left(\frac{a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-9 a \sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-9 b \sin \left(\frac{3 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+5 a \sin \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+5 b \sin \left(\frac{5 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+7 a \sin \left(\frac{7 a}{b}\right) \text{Si}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+7 b \sin \left(\frac{7 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+64 b c^7 x^7-128 b c^5 x^5+64 b c^3 x^3}{64 b^2 c^4 \left(a+b \sin ^{-1}(c x)\right)}","\frac{3 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{64 b^2 c^4}+\frac{9 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^4}-\frac{5 \cos \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^4}-\frac{7 \cos \left(\frac{7 a}{b}\right) \text{Ci}\left(\frac{7 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^4}+\frac{3 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{64 b^2 c^4}+\frac{9 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^4}-\frac{5 \sin \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^4}-\frac{7 \sin \left(\frac{7 a}{b}\right) \text{Si}\left(\frac{7 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^4}-\frac{x^3 \left(1-c^2 x^2\right)^2}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"-1/64*(64*b*c^3*x^3 - 128*b*c^5*x^5 + 64*b*c^7*x^7 - 3*(a + b*ArcSin[c*x])*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] - 9*(a + b*ArcSin[c*x])*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + 5*a*Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x])] + 5*b*ArcSin[c*x]*Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x])] + 7*a*Cos[(7*a)/b]*CosIntegral[7*(a/b + ArcSin[c*x])] + 7*b*ArcSin[c*x]*Cos[(7*a)/b]*CosIntegral[7*(a/b + ArcSin[c*x])] - 3*a*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 3*b*ArcSin[c*x]*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 9*a*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] - 9*b*ArcSin[c*x]*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 5*a*Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])] + 5*b*ArcSin[c*x]*Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])] + 7*a*Sin[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])] + 7*b*ArcSin[c*x]*Sin[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])])/(b^2*c^4*(a + b*ArcSin[c*x]))","A",1
391,1,306,220,0.8574919,"\int \frac{x^2 \left(1-c^2 x^2\right)^{3/2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(x^2*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x])^2,x]","-\frac{-\sin \left(\frac{2 a}{b}\right) \left(a+b \sin ^{-1}(c x)\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+4 \sin \left(\frac{4 a}{b}\right) \left(a+b \sin ^{-1}(c x)\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+3 a \sin \left(\frac{6 a}{b}\right) \text{Ci}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+3 b \sin \left(\frac{6 a}{b}\right) \sin ^{-1}(c x) \text{Ci}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+a \cos \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+b \cos \left(\frac{2 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-4 a \cos \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-4 b \cos \left(\frac{4 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-3 a \cos \left(\frac{6 a}{b}\right) \text{Si}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-3 b \cos \left(\frac{6 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+16 b c^6 x^6-32 b c^4 x^4+16 b c^2 x^2}{16 b^2 c^3 \left(a+b \sin ^{-1}(c x)\right)}","\frac{\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}-\frac{\sin \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^3}-\frac{3 \sin \left(\frac{6 a}{b}\right) \text{Ci}\left(\frac{6 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}-\frac{\cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}+\frac{\cos \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^3}+\frac{3 \cos \left(\frac{6 a}{b}\right) \text{Si}\left(\frac{6 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}-\frac{x^2 \left(1-c^2 x^2\right)^2}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"-1/16*(16*b*c^2*x^2 - 32*b*c^4*x^4 + 16*b*c^6*x^6 - (a + b*ArcSin[c*x])*CosIntegral[2*(a/b + ArcSin[c*x])]*Sin[(2*a)/b] + 4*(a + b*ArcSin[c*x])*CosIntegral[4*(a/b + ArcSin[c*x])]*Sin[(4*a)/b] + 3*a*CosIntegral[6*(a/b + ArcSin[c*x])]*Sin[(6*a)/b] + 3*b*ArcSin[c*x]*CosIntegral[6*(a/b + ArcSin[c*x])]*Sin[(6*a)/b] + a*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] + b*ArcSin[c*x]*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] - 4*a*Cos[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])] - 4*b*ArcSin[c*x]*Cos[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])] - 3*a*Cos[(6*a)/b]*SinIntegral[6*(a/b + ArcSin[c*x])] - 3*b*ArcSin[c*x]*Cos[(6*a)/b]*SinIntegral[6*(a/b + ArcSin[c*x])])/(b^2*c^3*(a + b*ArcSin[c*x]))","A",1
392,1,295,214,0.5705244,"\int \frac{x \left(1-c^2 x^2\right)^{3/2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(x*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x])^2,x]","\frac{2 \cos \left(\frac{a}{b}\right) \left(a+b \sin ^{-1}(c x)\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+9 \cos \left(\frac{3 a}{b}\right) \left(a+b \sin ^{-1}(c x)\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+5 a \cos \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+5 b \cos \left(\frac{5 a}{b}\right) \sin ^{-1}(c x) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 a \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+2 b \sin \left(\frac{a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+9 a \sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+9 b \sin \left(\frac{3 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+5 a \sin \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+5 b \sin \left(\frac{5 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-16 b c^5 x^5+32 b c^3 x^3-16 b c x}{16 b^2 c^2 \left(a+b \sin ^{-1}(c x)\right)}","\frac{\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b^2 c^2}+\frac{9 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^2}+\frac{5 \cos \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^2}+\frac{\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b^2 c^2}+\frac{9 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^2}+\frac{5 \sin \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^2}-\frac{x \left(1-c^2 x^2\right)^2}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"(-16*b*c*x + 32*b*c^3*x^3 - 16*b*c^5*x^5 + 2*(a + b*ArcSin[c*x])*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] + 9*(a + b*ArcSin[c*x])*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + 5*a*Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x])] + 5*b*ArcSin[c*x]*Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x])] + 2*a*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 2*b*ArcSin[c*x]*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 9*a*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 9*b*ArcSin[c*x]*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 5*a*Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])] + 5*b*ArcSin[c*x]*Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])])/(16*b^2*c^2*(a + b*ArcSin[c*x]))","A",1
393,1,122,150,0.6439437,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(a + b*ArcSin[c*x])^2,x]","\frac{-\frac{2 b \left(c^2 x^2-1\right)^2}{a+b \sin ^{-1}(c x)}+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)}{2 b^2 c}","\frac{\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{b^2 c}+\frac{\sin \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b^2 c}-\frac{\cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{b^2 c}-\frac{\cos \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b^2 c}-\frac{\left(1-c^2 x^2\right)^2}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"((-2*b*(-1 + c^2*x^2)^2)/(a + b*ArcSin[c*x]) + 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])])/(2*b^2*c)","A",1
394,0,0,177,11.1311384,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcSin[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","-\frac{\text{Int}\left(\frac{1-c^2 x^2}{x^2 \left(a+b \sin ^{-1}(c x)\right)},x\right)}{b c}-\frac{9 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b^2}-\frac{3 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b^2}-\frac{9 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b^2}-\frac{3 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b^2}-\frac{\left(1-c^2 x^2\right)^2}{b c x \left(a+b \sin ^{-1}(c x)\right)}",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcSin[c*x])^2), x]","A",-1
395,0,0,102,4.6197402,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^2 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x^2*(a + b*ArcSin[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^2 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","-\frac{2 c \text{Int}\left(\frac{1-c^2 x^2}{x \left(a+b \sin ^{-1}(c x)\right)},x\right)}{b}-\frac{2 \text{Int}\left(\frac{1-c^2 x^2}{x^3 \left(a+b \sin ^{-1}(c x)\right)},x\right)}{b c}-\frac{\left(1-c^2 x^2\right)^2}{b c x^2 \left(a+b \sin ^{-1}(c x)\right)}",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x^2*(a + b*ArcSin[c*x])^2), x]","A",-1
396,0,0,31,17.3922491,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcSin[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{3/2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcSin[c*x])^2), x]","A",-1
397,0,0,69,2.9661644,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcSin[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","-\frac{4 \text{Int}\left(\frac{1-c^2 x^2}{x^5 \left(a+b \sin ^{-1}(c x)\right)},x\right)}{b c}-\frac{\left(1-c^2 x^2\right)^2}{b c x^4 \left(a+b \sin ^{-1}(c x)\right)}",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcSin[c*x])^2), x]","A",-1
398,0,0,31,0.6549757,"\int \frac{x^m \left(1-c^2 x^2\right)^{5/2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(x^m*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2,x]","\int \frac{x^m \left(1-c^2 x^2\right)^{5/2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{5/2} x^m}{\left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(x^m*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2, x]","A",-1
399,1,408,278,1.7311927,"\int \frac{x^3 \left(1-c^2 x^2\right)^{5/2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(x^3*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2,x]","-\frac{-6 \cos \left(\frac{a}{b}\right) \left(a+b \sin ^{-1}(c x)\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-24 \cos \left(\frac{3 a}{b}\right) \left(a+b \sin ^{-1}(c x)\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+21 a \cos \left(\frac{7 a}{b}\right) \text{Ci}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+21 b \cos \left(\frac{7 a}{b}\right) \sin ^{-1}(c x) \text{Ci}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+9 a \cos \left(\frac{9 a}{b}\right) \text{Ci}\left(9 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+9 b \cos \left(\frac{9 a}{b}\right) \sin ^{-1}(c x) \text{Ci}\left(9 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-6 a \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-6 b \sin \left(\frac{a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-24 a \sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-24 b \sin \left(\frac{3 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+21 a \sin \left(\frac{7 a}{b}\right) \text{Si}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+21 b \sin \left(\frac{7 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+9 a \sin \left(\frac{9 a}{b}\right) \text{Si}\left(9 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+9 b \sin \left(\frac{9 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(9 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-256 b c^9 x^9+768 b c^7 x^7-768 b c^5 x^5+256 b c^3 x^3}{256 b^2 c^4 \left(a+b \sin ^{-1}(c x)\right)}","\frac{3 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{128 b^2 c^4}+\frac{3 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b^2 c^4}-\frac{21 \cos \left(\frac{7 a}{b}\right) \text{Ci}\left(\frac{7 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{256 b^2 c^4}-\frac{9 \cos \left(\frac{9 a}{b}\right) \text{Ci}\left(\frac{9 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{256 b^2 c^4}+\frac{3 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{128 b^2 c^4}+\frac{3 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 b^2 c^4}-\frac{21 \sin \left(\frac{7 a}{b}\right) \text{Si}\left(\frac{7 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{256 b^2 c^4}-\frac{9 \sin \left(\frac{9 a}{b}\right) \text{Si}\left(\frac{9 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{256 b^2 c^4}-\frac{x^3 \left(1-c^2 x^2\right)^3}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"-1/256*(256*b*c^3*x^3 - 768*b*c^5*x^5 + 768*b*c^7*x^7 - 256*b*c^9*x^9 - 6*(a + b*ArcSin[c*x])*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] - 24*(a + b*ArcSin[c*x])*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + 21*a*Cos[(7*a)/b]*CosIntegral[7*(a/b + ArcSin[c*x])] + 21*b*ArcSin[c*x]*Cos[(7*a)/b]*CosIntegral[7*(a/b + ArcSin[c*x])] + 9*a*Cos[(9*a)/b]*CosIntegral[9*(a/b + ArcSin[c*x])] + 9*b*ArcSin[c*x]*Cos[(9*a)/b]*CosIntegral[9*(a/b + ArcSin[c*x])] - 6*a*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 6*b*ArcSin[c*x]*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 24*a*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] - 24*b*ArcSin[c*x]*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 21*a*Sin[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])] + 21*b*ArcSin[c*x]*Sin[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])] + 9*a*Sin[(9*a)/b]*SinIntegral[9*(a/b + ArcSin[c*x])] + 9*b*ArcSin[c*x]*Sin[(9*a)/b]*SinIntegral[9*(a/b + ArcSin[c*x])])/(b^2*c^4*(a + b*ArcSin[c*x]))","A",1
400,1,414,282,1.2200541,"\int \frac{x^2 \left(1-c^2 x^2\right)^{5/2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(x^2*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2,x]","\frac{\sin \left(\frac{2 a}{b}\right) \left(a+b \sin ^{-1}(c x)\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-2 \sin \left(\frac{4 a}{b}\right) \left(a+b \sin ^{-1}(c x)\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-3 a \sin \left(\frac{6 a}{b}\right) \text{Ci}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-3 b \sin \left(\frac{6 a}{b}\right) \sin ^{-1}(c x) \text{Ci}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-a \sin \left(\frac{8 a}{b}\right) \text{Ci}\left(8 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-b \sin \left(\frac{8 a}{b}\right) \sin ^{-1}(c x) \text{Ci}\left(8 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-a \cos \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-b \cos \left(\frac{2 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 a \cos \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 b \cos \left(\frac{4 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+3 a \cos \left(\frac{6 a}{b}\right) \text{Si}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+3 b \cos \left(\frac{6 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+a \cos \left(\frac{8 a}{b}\right) \text{Si}\left(8 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+b \cos \left(\frac{8 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(8 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+16 b c^8 x^8-48 b c^6 x^6+48 b c^4 x^4-16 b c^2 x^2}{16 b^2 c^3 \left(a+b \sin ^{-1}(c x)\right)}","\frac{\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}-\frac{\sin \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 b^2 c^3}-\frac{3 \sin \left(\frac{6 a}{b}\right) \text{Ci}\left(\frac{6 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}-\frac{\sin \left(\frac{8 a}{b}\right) \text{Ci}\left(\frac{8 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}-\frac{\cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}+\frac{\cos \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 b^2 c^3}+\frac{3 \cos \left(\frac{6 a}{b}\right) \text{Si}\left(\frac{6 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}+\frac{\cos \left(\frac{8 a}{b}\right) \text{Si}\left(\frac{8 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}-\frac{x^2 \left(1-c^2 x^2\right)^3}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"(-16*b*c^2*x^2 + 48*b*c^4*x^4 - 48*b*c^6*x^6 + 16*b*c^8*x^8 + (a + b*ArcSin[c*x])*CosIntegral[2*(a/b + ArcSin[c*x])]*Sin[(2*a)/b] - 2*(a + b*ArcSin[c*x])*CosIntegral[4*(a/b + ArcSin[c*x])]*Sin[(4*a)/b] - 3*a*CosIntegral[6*(a/b + ArcSin[c*x])]*Sin[(6*a)/b] - 3*b*ArcSin[c*x]*CosIntegral[6*(a/b + ArcSin[c*x])]*Sin[(6*a)/b] - a*CosIntegral[8*(a/b + ArcSin[c*x])]*Sin[(8*a)/b] - b*ArcSin[c*x]*CosIntegral[8*(a/b + ArcSin[c*x])]*Sin[(8*a)/b] - a*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] - b*ArcSin[c*x]*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] + 2*a*Cos[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])] + 2*b*ArcSin[c*x]*Cos[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])] + 3*a*Cos[(6*a)/b]*SinIntegral[6*(a/b + ArcSin[c*x])] + 3*b*ArcSin[c*x]*Cos[(6*a)/b]*SinIntegral[6*(a/b + ArcSin[c*x])] + a*Cos[(8*a)/b]*SinIntegral[8*(a/b + ArcSin[c*x])] + b*ArcSin[c*x]*Cos[(8*a)/b]*SinIntegral[8*(a/b + ArcSin[c*x])])/(16*b^2*c^3*(a + b*ArcSin[c*x]))","A",1
401,1,404,276,1.0907282,"\int \frac{x \left(1-c^2 x^2\right)^{5/2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(x*(1 - c^2*x^2)^(5/2))/(a + b*ArcSin[c*x])^2,x]","\frac{5 \cos \left(\frac{a}{b}\right) \left(a+b \sin ^{-1}(c x)\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+27 \cos \left(\frac{3 a}{b}\right) \left(a+b \sin ^{-1}(c x)\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+25 a \cos \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+25 b \cos \left(\frac{5 a}{b}\right) \sin ^{-1}(c x) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+7 a \cos \left(\frac{7 a}{b}\right) \text{Ci}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+7 b \cos \left(\frac{7 a}{b}\right) \sin ^{-1}(c x) \text{Ci}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+5 a \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+5 b \sin \left(\frac{a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+27 a \sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+27 b \sin \left(\frac{3 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+25 a \sin \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+25 b \sin \left(\frac{5 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+7 a \sin \left(\frac{7 a}{b}\right) \text{Si}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+7 b \sin \left(\frac{7 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(7 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+64 b c^7 x^7-192 b c^5 x^5+192 b c^3 x^3-64 b c x}{64 b^2 c^2 \left(a+b \sin ^{-1}(c x)\right)}","\frac{5 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{64 b^2 c^2}+\frac{27 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2}+\frac{25 \cos \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2}+\frac{7 \cos \left(\frac{7 a}{b}\right) \text{Ci}\left(\frac{7 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2}+\frac{5 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{64 b^2 c^2}+\frac{27 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2}+\frac{25 \sin \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2}+\frac{7 \sin \left(\frac{7 a}{b}\right) \text{Si}\left(\frac{7 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2}-\frac{x \left(1-c^2 x^2\right)^3}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"(-64*b*c*x + 192*b*c^3*x^3 - 192*b*c^5*x^5 + 64*b*c^7*x^7 + 5*(a + b*ArcSin[c*x])*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] + 27*(a + b*ArcSin[c*x])*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + 25*a*Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x])] + 25*b*ArcSin[c*x]*Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x])] + 7*a*Cos[(7*a)/b]*CosIntegral[7*(a/b + ArcSin[c*x])] + 7*b*ArcSin[c*x]*Cos[(7*a)/b]*CosIntegral[7*(a/b + ArcSin[c*x])] + 5*a*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 5*b*ArcSin[c*x]*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 27*a*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 27*b*ArcSin[c*x]*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 25*a*Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])] + 25*b*ArcSin[c*x]*Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])] + 7*a*Sin[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])] + 7*b*ArcSin[c*x]*Sin[(7*a)/b]*SinIntegral[7*(a/b + ArcSin[c*x])])/(64*b^2*c^2*(a + b*ArcSin[c*x]))","A",1
402,1,311,217,0.9187867,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(a + b*ArcSin[c*x])^2,x]","-\frac{-15 \sin \left(\frac{2 a}{b}\right) \left(a+b \sin ^{-1}(c x)\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-12 \sin \left(\frac{4 a}{b}\right) \left(a+b \sin ^{-1}(c x)\right) \text{Ci}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-3 a \sin \left(\frac{6 a}{b}\right) \text{Ci}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-3 b \sin \left(\frac{6 a}{b}\right) \sin ^{-1}(c x) \text{Ci}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+15 a \cos \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+15 b \cos \left(\frac{2 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+12 a \cos \left(\frac{4 a}{b}\right) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+12 b \cos \left(\frac{4 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(4 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+3 a \cos \left(\frac{6 a}{b}\right) \text{Si}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+3 b \cos \left(\frac{6 a}{b}\right) \sin ^{-1}(c x) \text{Si}\left(6 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-16 b c^6 x^6+48 b c^4 x^4-48 b c^2 x^2+16 b}{16 b^2 c \left(a+b \sin ^{-1}(c x)\right)}","\frac{15 \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c}+\frac{3 \sin \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b^2 c}+\frac{3 \sin \left(\frac{6 a}{b}\right) \text{Ci}\left(\frac{6 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c}-\frac{15 \cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c}-\frac{3 \cos \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b^2 c}-\frac{3 \cos \left(\frac{6 a}{b}\right) \text{Si}\left(\frac{6 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c}-\frac{\left(1-c^2 x^2\right)^3}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"-1/16*(16*b - 48*b*c^2*x^2 + 48*b*c^4*x^4 - 16*b*c^6*x^6 - 15*(a + b*ArcSin[c*x])*CosIntegral[2*(a/b + ArcSin[c*x])]*Sin[(2*a)/b] - 12*(a + b*ArcSin[c*x])*CosIntegral[4*(a/b + ArcSin[c*x])]*Sin[(4*a)/b] - 3*a*CosIntegral[6*(a/b + ArcSin[c*x])]*Sin[(6*a)/b] - 3*b*ArcSin[c*x]*CosIntegral[6*(a/b + ArcSin[c*x])]*Sin[(6*a)/b] + 15*a*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] + 15*b*ArcSin[c*x]*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])] + 12*a*Cos[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])] + 12*b*ArcSin[c*x]*Cos[(4*a)/b]*SinIntegral[4*(a/b + ArcSin[c*x])] + 3*a*Cos[(6*a)/b]*SinIntegral[6*(a/b + ArcSin[c*x])] + 3*b*ArcSin[c*x]*Cos[(6*a)/b]*SinIntegral[6*(a/b + ArcSin[c*x])])/(b^2*c*(a + b*ArcSin[c*x]))","A",1
403,0,0,235,14.1239018,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcSin[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","-\frac{\text{Int}\left(\frac{\left(1-c^2 x^2\right)^2}{x^2 \left(a+b \sin ^{-1}(c x)\right)},x\right)}{b c}-\frac{25 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b^2}-\frac{25 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2}-\frac{5 \cos \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2}-\frac{25 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b^2}-\frac{25 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2}-\frac{5 \sin \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2}-\frac{\left(1-c^2 x^2\right)^3}{b c x \left(a+b \sin ^{-1}(c x)\right)}",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcSin[c*x])^2), x]","A",-1
404,0,0,106,3.8971623,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^2 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcSin[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^2 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","-\frac{4 c \text{Int}\left(\frac{\left(1-c^2 x^2\right)^2}{x \left(a+b \sin ^{-1}(c x)\right)},x\right)}{b}-\frac{2 \text{Int}\left(\frac{\left(1-c^2 x^2\right)^2}{x^3 \left(a+b \sin ^{-1}(c x)\right)},x\right)}{b c}-\frac{\left(1-c^2 x^2\right)^3}{b c x^2 \left(a+b \sin ^{-1}(c x)\right)}",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcSin[c*x])^2), x]","A",-1
405,0,0,31,18.0213802,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcSin[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{5/2}}{x^3 \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcSin[c*x])^2), x]","A",-1
406,0,0,31,3.4450831,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcSin[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{5/2}}{x^4 \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcSin[c*x])^2), x]","A",-1
407,0,0,49,0.6923058,"\int \frac{x^m}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[x^m/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2),x]","\int \frac{x^m}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\frac{m \text{Int}\left(\frac{x^{m-1}}{a+b \sin ^{-1}(c x)},x\right)}{b c}-\frac{x^m}{b c \left(a+b \sin ^{-1}(c x)\right)}",0,"Integrate[x^m/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2), x]","A",-1
408,1,157,204,0.4133141,"\int \frac{x^5}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[x^5/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2),x]","\frac{5 \left(2 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-3 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\cos \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-3 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\sin \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)\right)}{16 b^2 c^6}-\frac{x^5}{b c \left(a+b \sin ^{-1}(c x)\right)}","\frac{5 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b^2 c^6}-\frac{15 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^6}+\frac{5 \cos \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^6}+\frac{5 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b^2 c^6}-\frac{15 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^6}+\frac{5 \sin \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^6}-\frac{x^5}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"-(x^5/(b*c*(a + b*ArcSin[c*x]))) + (5*(2*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] - 3*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x])] + 2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 3*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])]))/(16*b^2*c^6)","A",1
409,1,117,141,0.3462682,"\int \frac{x^4}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[x^4/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2),x]","\frac{-\frac{2 b c^4 x^4}{a+b \sin ^{-1}(c x)}-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)}{2 b^2 c^5}","-\frac{\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{b^2 c^5}+\frac{\sin \left(\frac{4 a}{b}\right) \text{Ci}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^5}+\frac{\cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{b^2 c^5}-\frac{\cos \left(\frac{4 a}{b}\right) \text{Si}\left(\frac{4 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^5}-\frac{x^4}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"((-2*b*c^4*x^4)/(a + b*ArcSin[c*x]) - 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])])/(2*b^2*c^5)","A",1
410,1,113,142,0.3256139,"\int \frac{x^3}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[x^3/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2),x]","\frac{3 \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^2 c^4}-\frac{x^3}{b c \left(a+b \sin ^{-1}(c x)\right)}","\frac{3 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b^2 c^4}-\frac{3 \cos \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^4}+\frac{3 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b^2 c^4}-\frac{3 \sin \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^4}-\frac{x^3}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"-(x^3/(b*c*(a + b*ArcSin[c*x]))) + (3*(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^2*c^4)","A",1
411,1,70,79,0.1840857,"\int \frac{x^2}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[x^2/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2),x]","\frac{-\frac{b c^2 x^2}{a+b \sin ^{-1}(c x)}-\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+\cos \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{b^2 c^3}","-\frac{\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{b^2 c^3}+\frac{\cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{b^2 c^3}-\frac{x^2}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"(-((b*c^2*x^2)/(a + b*ArcSin[c*x])) - CosIntegral[2*(a/b + ArcSin[c*x])]*Sin[(2*a)/b] + Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSin[c*x])])/(b^2*c^3)","A",1
412,1,59,72,0.127994,"\int \frac{x}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[x/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2),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)-\frac{b c x}{a+b \sin ^{-1}(c x)}}{b^2 c^2}","\frac{\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b^2 c^2}+\frac{\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b^2 c^2}-\frac{x}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"(-((b*c*x)/(a + b*ArcSin[c*x])) + Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] + Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]])/(b^2*c^2)","A",1
413,1,18,18,0.0095662,"\int \frac{1}{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2),x]","-\frac{1}{b c \left(a+b \sin ^{-1}(c x)\right)}","-\frac{1}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"-(1/(b*c*(a + b*ArcSin[c*x])))","A",1
414,0,0,47,8.1511338,"\int \frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2),x]","\int \frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \left(a+b \sin ^{-1}(c x)\right)},x\right)}{b c}-\frac{1}{b c x \left(a+b \sin ^{-1}(c x)\right)}",0,"Integrate[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2), x]","A",-1
415,0,0,47,1.3889337,"\int \frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2),x]","\int \frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^3 \left(a+b \sin ^{-1}(c x)\right)},x\right)}{b c}-\frac{1}{b c x^2 \left(a+b \sin ^{-1}(c x)\right)}",0,"Integrate[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2), x]","A",-1
416,0,0,31,1.2374209,"\int \frac{x^m}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[x^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{x^m}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x^m}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
417,0,0,31,66.6874545,"\int \frac{x^3}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[x^3/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{x^3}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x^3}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x^3/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
418,0,0,69,8.573629,"\int \frac{x^2}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[x^2/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{x^2}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\frac{2 \text{Int}\left(\frac{x}{\left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)},x\right)}{b c}-\frac{x^2}{b c \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}",0,"Integrate[x^2/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
419,0,0,29,68.2399039,"\int \frac{x}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[x/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{x}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
420,0,0,64,2.7598937,"\int \frac{1}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{1}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\frac{2 c \text{Int}\left(\frac{x}{\left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)},x\right)}{b}-\frac{1}{b c \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}",0,"Integrate[1/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
421,0,0,31,54.045371,"\int \frac{1}{x \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{1}{x \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
422,0,0,31,36.3780174,"\int \frac{1}{x^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{1}{x^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
423,0,0,31,2.022583,"\int \frac{x^m}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[x^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{x^m}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x^m}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
424,0,0,31,110.4988341,"\int \frac{x^3}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[x^3/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{x^3}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x^3}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x^3/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
425,0,0,31,12.8162617,"\int \frac{x^2}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{x^2}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x^2}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
426,0,0,29,113.707595,"\int \frac{x}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[x/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{x}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
427,0,0,64,4.4372135,"\int \frac{1}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{1}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\frac{4 c \text{Int}\left(\frac{x}{\left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)},x\right)}{b}-\frac{1}{b c \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}",0,"Integrate[1/((1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
428,0,0,31,86.2478825,"\int \frac{1}{x \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{1}{x \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
429,0,0,31,26.8401406,"\int \frac{1}{x^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{1}{x^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
430,1,13,13,0.0044425,"\int \frac{1}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3} \, dx","Integrate[1/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3),x]","-\frac{1}{2 a \sin ^{-1}(a x)^2}","-\frac{1}{2 a \sin ^{-1}(a x)^2}",1,"-1/2*1/(a*ArcSin[a*x]^2)","A",1
431,1,287,251,1.5691313,"\int \frac{x^3 \left(d-c^2 d x^2\right)}{\left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x^3*(d - c^2*d*x^2))/(a + b*ArcSin[c*x])^(3/2),x]","-\frac{i d e^{-\frac{6 i a}{b}} \left(-6 i e^{\frac{6 i a}{b}} \sin \left(2 \sin ^{-1}(c x)\right)+2 i e^{\frac{6 i a}{b}} \sin \left(6 \sin ^{-1}(c x)\right)+3 \sqrt{2} e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-3 \sqrt{2} e^{\frac{8 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{6} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+\sqrt{6} e^{\frac{12 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)}{32 b c^4 \sqrt{a+b \sin ^{-1}(c x)}}","-\frac{\sqrt{3 \pi } d \cos \left(\frac{6 a}{b}\right) C\left(\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{8 b^{3/2} c^4}+\frac{3 \sqrt{\pi } d \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{8 b^{3/2} c^4}+\frac{3 \sqrt{\pi } d \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{8 b^{3/2} c^4}-\frac{\sqrt{3 \pi } d \sin \left(\frac{6 a}{b}\right) S\left(\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{8 b^{3/2} c^4}-\frac{2 d x^3 \left(1-c^2 x^2\right)^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}",1,"((-1/32*I)*d*(3*Sqrt[2]*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c*x]))/b] - 3*Sqrt[2]*E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[6]*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-6*I)*(a + b*ArcSin[c*x]))/b] + Sqrt[6]*E^(((12*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((6*I)*(a + b*ArcSin[c*x]))/b] - (6*I)*E^(((6*I)*a)/b)*Sin[2*ArcSin[c*x]] + (2*I)*E^(((6*I)*a)/b)*Sin[6*ArcSin[c*x]]))/(b*c^4*E^(((6*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])","C",0
432,1,514,591,1.6730687,"\int \frac{x^2 \left(d-c^2 d x^2\right)}{\left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x^2*(d - c^2*d*x^2))/(a + b*ArcSin[c*x])^(3/2),x]","\frac{d e^{-\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}} \left(e^{\frac{5 i a}{b}+2 i \sin ^{-1}(c x)}-2 e^{\frac{5 i a}{b}+4 i \sin ^{-1}(c x)}-2 e^{\frac{5 i a}{b}+6 i \sin ^{-1}(c x)}+e^{\frac{5 i a}{b}+8 i \sin ^{-1}(c x)}+e^{\frac{5 i \left(a+2 b \sin ^{-1}(c x)\right)}{b}}+2 e^{\frac{4 i a}{b}+5 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+2 e^{\frac{6 i a}{b}+5 i \sin ^{-1}(c x)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{3} e^{\frac{2 i a}{b}+5 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{3} e^{\frac{8 i a}{b}+5 i \sin ^{-1}(c x)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{5} e^{5 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{5} e^{\frac{5 i \left(2 a+b \sin ^{-1}(c x)\right)}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+e^{\frac{5 i a}{b}}\right)}{16 b c^3 \sqrt{a+b \sin ^{-1}(c x)}}","-\frac{\sqrt{2 \pi } d \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^3}+\frac{5 \sqrt{\frac{\pi }{2}} d \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{2 b^{3/2} c^3}-\frac{\sqrt{\frac{2 \pi }{3}} d \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^3}+\frac{5 \sqrt{\frac{\pi }{6}} d \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c^3}-\frac{\sqrt{\frac{5 \pi }{2}} d \sin \left(\frac{5 a}{b}\right) C\left(\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c^3}+\frac{\sqrt{2 \pi } d \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^3}-\frac{5 \sqrt{\frac{\pi }{2}} d \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{2 b^{3/2} c^3}+\frac{\sqrt{\frac{2 \pi }{3}} d \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^3}-\frac{5 \sqrt{\frac{\pi }{6}} d \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c^3}+\frac{\sqrt{\frac{5 \pi }{2}} d \cos \left(\frac{5 a}{b}\right) S\left(\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c^3}-\frac{2 d x^2 \left(1-c^2 x^2\right)^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}",1,"(d*(E^(((5*I)*a)/b) + E^(((5*I)*a)/b + (2*I)*ArcSin[c*x]) - 2*E^(((5*I)*a)/b + (4*I)*ArcSin[c*x]) - 2*E^(((5*I)*a)/b + (6*I)*ArcSin[c*x]) + E^(((5*I)*a)/b + (8*I)*ArcSin[c*x]) + E^(((5*I)*(a + 2*b*ArcSin[c*x]))/b) + 2*E^(((4*I)*a)/b + (5*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 2*E^(((6*I)*a)/b + (5*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*E^(((2*I)*a)/b + (5*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*E^(((8*I)*a)/b + (5*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[5]*E^((5*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-5*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[5]*E^(((5*I)*(2*a + b*ArcSin[c*x]))/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((5*I)*(a + b*ArcSin[c*x]))/b]))/(16*b*c^3*E^(((5*I)*(a + b*ArcSin[c*x]))/b)*Sqrt[a + b*ArcSin[c*x]])","C",0
433,1,375,241,2.4624761,"\int \frac{x \left(d-c^2 d x^2\right)}{\left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x*(d - c^2*d*x^2))/(a + b*ArcSin[c*x])^(3/2),x]","\frac{d \left(8 \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 x)}}{\sqrt{\pi }}\right)+8 \sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi }}\right)+\frac{i e^{-\frac{4 i a}{b}} \left(2 i e^{\frac{4 i a}{b}} \sin \left(2 \sin ^{-1}(c x)\right)+i e^{\frac{4 i a}{b}} \sin \left(4 \sin ^{-1}(c x)\right)+\sqrt{2} e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{2} e^{\frac{6 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+e^{\frac{8 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)}{b \sqrt{a+b \sin ^{-1}(c x)}}\right)}{4 c^2}","\frac{\sqrt{\frac{\pi }{2}} d \cos \left(\frac{4 a}{b}\right) C\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^2}+\frac{\sqrt{\pi } d \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{b^{3/2} c^2}+\frac{\sqrt{\pi } d \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{b^{3/2} c^2}+\frac{\sqrt{\frac{\pi }{2}} d \sin \left(\frac{4 a}{b}\right) S\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^2}-\frac{2 d x \left(1-c^2 x^2\right)^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}",1,"(d*(8*(b^(-1))^(3/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[Pi]] + 8*(b^(-1))^(3/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[Pi]]*Sin[(2*a)/b] + (I*(Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[2]*E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-4*I)*(a + b*ArcSin[c*x]))/b] + E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((4*I)*(a + b*ArcSin[c*x]))/b] + (2*I)*E^(((4*I)*a)/b)*Sin[2*ArcSin[c*x]] + I*E^(((4*I)*a)/b)*Sin[4*ArcSin[c*x]]))/(b*E^(((4*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])))/(4*c^2)","C",0
434,1,348,253,1.192601,"\int \frac{d-c^2 d x^2}{\left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d - c^2*d*x^2)/(a + b*ArcSin[c*x])^(3/2),x]","\frac{d e^{-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}} \left(-3 e^{\frac{3 i a}{b}+2 i \sin ^{-1}(c x)}-3 e^{\frac{3 i a}{b}+4 i \sin ^{-1}(c x)}-e^{\frac{3 i \left(a+2 b \sin ^{-1}(c x)\right)}{b}}+3 e^{\frac{2 i a}{b}+3 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+3 e^{\frac{4 i a}{b}+3 i \sin ^{-1}(c x)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+\sqrt{3} e^{3 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+\sqrt{3} e^{3 i \left(\frac{2 a}{b}+\sin ^{-1}(c x)\right)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-e^{\frac{3 i a}{b}}\right)}{4 b c \sqrt{a+b \sin ^{-1}(c x)}}","\frac{3 \sqrt{\frac{\pi }{2}} d \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}+\frac{\sqrt{\frac{3 \pi }{2}} d \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}-\frac{3 \sqrt{\frac{\pi }{2}} d \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}-\frac{\sqrt{\frac{3 \pi }{2}} d \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}-\frac{2 d \left(1-c^2 x^2\right)^{3/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}",1,"(d*(-E^(((3*I)*a)/b) - 3*E^(((3*I)*a)/b + (2*I)*ArcSin[c*x]) - 3*E^(((3*I)*a)/b + (4*I)*ArcSin[c*x]) - E^(((3*I)*(a + 2*b*ArcSin[c*x]))/b) + 3*E^(((2*I)*a)/b + (3*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 3*E^(((4*I)*a)/b + (3*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b] + Sqrt[3]*E^((3*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + Sqrt[3]*E^((3*I)*((2*a)/b + ArcSin[c*x]))*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c*x]))/b]))/(4*b*c*E^(((3*I)*(a + b*ArcSin[c*x]))/b)*Sqrt[a + b*ArcSin[c*x]])","C",0
435,0,0,171,2.020319,"\int \frac{d-c^2 d x^2}{x \left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d - c^2*d*x^2)/(x*(a + b*ArcSin[c*x])^(3/2)),x]","\int \frac{d-c^2 d x^2}{x \left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","-\frac{2 d \text{Int}\left(\frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}},x\right)}{b c}-\frac{2 \sqrt{\pi } d \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{b^{3/2}}-\frac{2 \sqrt{\pi } d \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{b^{3/2}}-\frac{2 d \left(1-c^2 x^2\right)^{3/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}",0,"Integrate[(d - c^2*d*x^2)/(x*(a + b*ArcSin[c*x])^(3/2)), x]","A",-1
436,1,540,485,3.0433544,"\int \frac{x^3 \left(d-c^2 d x^2\right)^2}{\left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x^3*(d - c^2*d*x^2)^2)/(a + b*ArcSin[c*x])^(3/2),x]","-\frac{i d^2 e^{-\frac{8 i a}{b}} \left(-6 i e^{\frac{8 i a}{b}} \sin \left(2 \sin ^{-1}(c x)\right)-2 i e^{\frac{8 i a}{b}} \sin \left(4 \sin ^{-1}(c x)\right)+2 i e^{\frac{8 i a}{b}} \sin \left(6 \sin ^{-1}(c x)\right)+i e^{\frac{8 i a}{b}} \sin \left(8 \sin ^{-1}(c x)\right)+3 \sqrt{2} e^{\frac{6 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-3 \sqrt{2} e^{\frac{10 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+2 e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-2 e^{\frac{12 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{6} e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+\sqrt{6} e^{\frac{14 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{2} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{8 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+\sqrt{2} e^{\frac{16 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{8 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)}{64 b c^4 \sqrt{a+b \sin ^{-1}(c x)}}","\frac{\sqrt{\frac{\pi }{2}} d^2 \cos \left(\frac{4 a}{b}\right) C\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{8 b^{3/2} c^4}-\frac{\sqrt{3 \pi } d^2 \cos \left(\frac{6 a}{b}\right) C\left(\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^4}+\frac{3 \sqrt{\pi } d^2 \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{16 b^{3/2} c^4}-\frac{\sqrt{\pi } d^2 \cos \left(\frac{8 a}{b}\right) C\left(\frac{4 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{16 b^{3/2} c^4}+\frac{3 \sqrt{\pi } d^2 \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{16 b^{3/2} c^4}+\frac{\sqrt{\frac{\pi }{2}} d^2 \sin \left(\frac{4 a}{b}\right) S\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{8 b^{3/2} c^4}-\frac{\sqrt{3 \pi } d^2 \sin \left(\frac{6 a}{b}\right) S\left(\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^4}-\frac{\sqrt{\pi } d^2 \sin \left(\frac{8 a}{b}\right) S\left(\frac{4 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{16 b^{3/2} c^4}-\frac{2 d^2 x^3 \left(1-c^2 x^2\right)^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}",1,"((-1/64*I)*d^2*(3*Sqrt[2]*E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c*x]))/b] - 3*Sqrt[2]*E^(((10*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c*x]))/b] + 2*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-4*I)*(a + b*ArcSin[c*x]))/b] - 2*E^(((12*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((4*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[6]*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-6*I)*(a + b*ArcSin[c*x]))/b] + Sqrt[6]*E^(((14*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((6*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[2]*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-8*I)*(a + b*ArcSin[c*x]))/b] + Sqrt[2]*E^(((16*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((8*I)*(a + b*ArcSin[c*x]))/b] - (6*I)*E^(((8*I)*a)/b)*Sin[2*ArcSin[c*x]] - (2*I)*E^(((8*I)*a)/b)*Sin[4*ArcSin[c*x]] + (2*I)*E^(((8*I)*a)/b)*Sin[6*ArcSin[c*x]] + I*E^(((8*I)*a)/b)*Sin[8*ArcSin[c*x]]))/(b*c^4*E^(((8*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])","C",0
437,1,686,511,3.290368,"\int \frac{x^2 \left(d-c^2 d x^2\right)^2}{\left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x^2*(d - c^2*d*x^2)^2)/(a + b*ArcSin[c*x])^(3/2),x]","\frac{d^2 e^{-\frac{7 i \left(a+b \sin ^{-1}(c x)\right)}{b}} \left(3 e^{\frac{7 i a}{b}+2 i \sin ^{-1}(c x)}+e^{\frac{7 i a}{b}+4 i \sin ^{-1}(c x)}-5 e^{\frac{7 i a}{b}+6 i \sin ^{-1}(c x)}-5 e^{\frac{7 i a}{b}+8 i \sin ^{-1}(c x)}+e^{\frac{7 i a}{b}+10 i \sin ^{-1}(c x)}+3 e^{\frac{7 i a}{b}+12 i \sin ^{-1}(c x)}+e^{\frac{7 i \left(a+2 b \sin ^{-1}(c x)\right)}{b}}+5 e^{\frac{6 i a}{b}+7 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+5 e^{\frac{8 i a}{b}+7 i \sin ^{-1}(c x)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{3} e^{\frac{4 i a}{b}+7 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{3} e^{\frac{10 i a}{b}+7 i \sin ^{-1}(c x)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-3 \sqrt{5} e^{\frac{2 i a}{b}+7 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-3 \sqrt{5} e^{\frac{12 i a}{b}+7 i \sin ^{-1}(c x)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{7} e^{7 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{7 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{7} e^{\frac{7 i \left(2 a+b \sin ^{-1}(c x)\right)}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{7 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+e^{\frac{7 i a}{b}}\right)}{64 b c^3 \sqrt{a+b \sin ^{-1}(c x)}}","\frac{5 \sqrt{\frac{\pi }{2}} d^2 \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^3}-\frac{\sqrt{\frac{3 \pi }{2}} d^2 \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^3}-\frac{3 \sqrt{\frac{5 \pi }{2}} d^2 \sin \left(\frac{5 a}{b}\right) C\left(\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^3}-\frac{\sqrt{\frac{7 \pi }{2}} d^2 \sin \left(\frac{7 a}{b}\right) C\left(\frac{\sqrt{\frac{14}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^3}-\frac{5 \sqrt{\frac{\pi }{2}} d^2 \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^3}+\frac{\sqrt{\frac{3 \pi }{2}} d^2 \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^3}+\frac{3 \sqrt{\frac{5 \pi }{2}} d^2 \cos \left(\frac{5 a}{b}\right) S\left(\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^3}+\frac{\sqrt{\frac{7 \pi }{2}} d^2 \cos \left(\frac{7 a}{b}\right) S\left(\frac{\sqrt{\frac{14}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^3}-\frac{2 d^2 x^2 \left(1-c^2 x^2\right)^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}",1,"(d^2*(E^(((7*I)*a)/b) + 3*E^(((7*I)*a)/b + (2*I)*ArcSin[c*x]) + E^(((7*I)*a)/b + (4*I)*ArcSin[c*x]) - 5*E^(((7*I)*a)/b + (6*I)*ArcSin[c*x]) - 5*E^(((7*I)*a)/b + (8*I)*ArcSin[c*x]) + E^(((7*I)*a)/b + (10*I)*ArcSin[c*x]) + 3*E^(((7*I)*a)/b + (12*I)*ArcSin[c*x]) + E^(((7*I)*(a + 2*b*ArcSin[c*x]))/b) + 5*E^(((6*I)*a)/b + (7*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 5*E^(((8*I)*a)/b + (7*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*E^(((4*I)*a)/b + (7*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*E^(((10*I)*a)/b + (7*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c*x]))/b] - 3*Sqrt[5]*E^(((2*I)*a)/b + (7*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-5*I)*(a + b*ArcSin[c*x]))/b] - 3*Sqrt[5]*E^(((12*I)*a)/b + (7*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((5*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[7]*E^((7*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-7*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[7]*E^(((7*I)*(2*a + b*ArcSin[c*x]))/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((7*I)*(a + b*ArcSin[c*x]))/b]))/(64*b*c^3*E^(((7*I)*(a + b*ArcSin[c*x]))/b)*Sqrt[a + b*ArcSin[c*x]])","C",0
438,1,509,373,3.7083516,"\int \frac{x \left(d-c^2 d x^2\right)^2}{\left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x*(d - c^2*d*x^2)^2)/(a + b*ArcSin[c*x])^(3/2),x]","\frac{d^2 \left(64 \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 x)}}{\sqrt{\pi }}\right)+64 \sqrt{\pi } \left(\frac{1}{b}\right)^{3/2} \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi }}\right)+\frac{i e^{-\frac{6 i a}{b}} \left(10 i e^{\frac{6 i a}{b}} \sin \left(2 \sin ^{-1}(c x)\right)+8 i e^{\frac{6 i a}{b}} \sin \left(4 \sin ^{-1}(c x)\right)+2 i e^{\frac{6 i a}{b}} \sin \left(6 \sin ^{-1}(c x)\right)+11 \sqrt{2} e^{\frac{4 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-11 \sqrt{2} e^{\frac{8 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-8 e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+8 e^{\frac{10 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{6} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+\sqrt{6} e^{\frac{12 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)}{b \sqrt{a+b \sin ^{-1}(c x)}}\right)}{32 c^2}","\frac{\sqrt{\frac{\pi }{2}} d^2 \cos \left(\frac{4 a}{b}\right) C\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^2}+\frac{\sqrt{3 \pi } d^2 \cos \left(\frac{6 a}{b}\right) C\left(\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{8 b^{3/2} c^2}+\frac{5 \sqrt{\pi } d^2 \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{8 b^{3/2} c^2}+\frac{5 \sqrt{\pi } d^2 \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{8 b^{3/2} c^2}+\frac{\sqrt{\frac{\pi }{2}} d^2 \sin \left(\frac{4 a}{b}\right) S\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^2}+\frac{\sqrt{3 \pi } d^2 \sin \left(\frac{6 a}{b}\right) S\left(\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{8 b^{3/2} c^2}-\frac{2 d^2 x \left(1-c^2 x^2\right)^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}",1,"(d^2*(64*(b^(-1))^(3/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[Pi]] + 64*(b^(-1))^(3/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[Pi]]*Sin[(2*a)/b] + (I*(11*Sqrt[2]*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c*x]))/b] - 11*Sqrt[2]*E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c*x]))/b] - 8*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-4*I)*(a + b*ArcSin[c*x]))/b] + 8*E^(((10*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((4*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[6]*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-6*I)*(a + b*ArcSin[c*x]))/b] + Sqrt[6]*E^(((12*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((6*I)*(a + b*ArcSin[c*x]))/b] + (10*I)*E^(((6*I)*a)/b)*Sin[2*ArcSin[c*x]] + (8*I)*E^(((6*I)*a)/b)*Sin[4*ArcSin[c*x]] + (2*I)*E^(((6*I)*a)/b)*Sin[6*ArcSin[c*x]]))/(b*E^(((6*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])))/(32*c^2)","C",0
439,1,522,390,2.741542,"\int \frac{\left(d-c^2 d x^2\right)^2}{\left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d - c^2*d*x^2)^2/(a + b*ArcSin[c*x])^(3/2),x]","\frac{d^2 e^{-\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}} \left(-5 e^{\frac{5 i a}{b}+2 i \sin ^{-1}(c x)}-10 e^{\frac{5 i a}{b}+4 i \sin ^{-1}(c x)}-10 e^{\frac{5 i a}{b}+6 i \sin ^{-1}(c x)}-5 e^{\frac{5 i a}{b}+8 i \sin ^{-1}(c x)}-e^{\frac{5 i \left(a+2 b \sin ^{-1}(c x)\right)}{b}}+10 e^{\frac{4 i a}{b}+5 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+10 e^{\frac{6 i a}{b}+5 i \sin ^{-1}(c x)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+5 \sqrt{3} e^{\frac{2 i a}{b}+5 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+5 \sqrt{3} e^{\frac{8 i a}{b}+5 i \sin ^{-1}(c x)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+\sqrt{5} e^{5 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+\sqrt{5} e^{\frac{5 i \left(2 a+b \sin ^{-1}(c x)\right)}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-e^{\frac{5 i a}{b}}\right)}{16 b c \sqrt{a+b \sin ^{-1}(c x)}}","\frac{5 \sqrt{\frac{\pi }{2}} d^2 \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{2 b^{3/2} c}+\frac{5 \sqrt{\frac{3 \pi }{2}} d^2 \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c}+\frac{\sqrt{\frac{5 \pi }{2}} d^2 \sin \left(\frac{5 a}{b}\right) C\left(\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c}-\frac{5 \sqrt{\frac{\pi }{2}} d^2 \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{2 b^{3/2} c}-\frac{5 \sqrt{\frac{3 \pi }{2}} d^2 \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c}-\frac{\sqrt{\frac{5 \pi }{2}} d^2 \cos \left(\frac{5 a}{b}\right) S\left(\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c}-\frac{2 d^2 \left(1-c^2 x^2\right)^{5/2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}",1,"(d^2*(-E^(((5*I)*a)/b) - 5*E^(((5*I)*a)/b + (2*I)*ArcSin[c*x]) - 10*E^(((5*I)*a)/b + (4*I)*ArcSin[c*x]) - 10*E^(((5*I)*a)/b + (6*I)*ArcSin[c*x]) - 5*E^(((5*I)*a)/b + (8*I)*ArcSin[c*x]) - E^(((5*I)*(a + 2*b*ArcSin[c*x]))/b) + 10*E^(((4*I)*a)/b + (5*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 10*E^(((6*I)*a)/b + (5*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b] + 5*Sqrt[3]*E^(((2*I)*a)/b + (5*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + 5*Sqrt[3]*E^(((8*I)*a)/b + (5*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c*x]))/b] + Sqrt[5]*E^((5*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-5*I)*(a + b*ArcSin[c*x]))/b] + Sqrt[5]*E^(((5*I)*(2*a + b*ArcSin[c*x]))/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((5*I)*(a + b*ArcSin[c*x]))/b]))/(16*b*c*E^(((5*I)*(a + b*ArcSin[c*x]))/b)*Sqrt[a + b*ArcSin[c*x]])","C",0
440,0,0,289,4.1478773,"\int \frac{\left(d-c^2 d x^2\right)^2}{x \left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d - c^2*d*x^2)^2/(x*(a + b*ArcSin[c*x])^(3/2)),x]","\int \frac{\left(d-c^2 d x^2\right)^2}{x \left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","-\frac{2 d^2 \text{Int}\left(\frac{1}{x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}},x\right)}{b c}-\frac{\sqrt{\frac{\pi }{2}} d^2 \cos \left(\frac{4 a}{b}\right) C\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2}}-\frac{3 \sqrt{\pi } d^2 \cos \left(\frac{2 a}{b}\right) C\left(\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{b^{3/2}}-\frac{3 \sqrt{\pi } d^2 \sin \left(\frac{2 a}{b}\right) S\left(\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right)}{b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} d^2 \sin \left(\frac{4 a}{b}\right) S\left(\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2}}-\frac{2 d^2 \left(1-c^2 x^2\right)^{5/2}}{b c x \sqrt{a+b \sin ^{-1}(c x)}}",0,"Integrate[(d - c^2*d*x^2)^2/(x*(a + b*ArcSin[c*x])^(3/2)), x]","A",-1
441,0,0,42,3.8497354,"\int \left(-\frac{3 x}{8 \left(1-x^2\right) \sqrt{\sin ^{-1}(x)}}+\frac{x \sin ^{-1}(x)^{3/2}}{\left(1-x^2\right)^2}\right) \, dx","Integrate[(-3*x)/(8*(1 - x^2)*Sqrt[ArcSin[x]]) + (x*ArcSin[x]^(3/2))/(1 - x^2)^2,x]","\int \left(-\frac{3 x}{8 \left(1-x^2\right) \sqrt{\sin ^{-1}(x)}}+\frac{x \sin ^{-1}(x)^{3/2}}{\left(1-x^2\right)^2}\right) \, dx","\frac{\sin ^{-1}(x)^{3/2}}{2 \left(1-x^2\right)}-\frac{3 x \sqrt{\sin ^{-1}(x)}}{4 \sqrt{1-x^2}}",1,"Integrate[(-3*x)/(8*(1 - x^2)*Sqrt[ArcSin[x]]) + (x*ArcSin[x]^(3/2))/(1 - x^2)^2, x]","F",-1
442,1,166,227,0.2616551,"\int \left(c-a^2 c x^2\right)^{3/2} \sqrt{\sin ^{-1}(a x)} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)*Sqrt[ArcSin[a*x]],x]","\frac{c \sqrt{c-a^2 c x^2} \left(32 \sin ^{-1}(a x)^2+8 \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{3}{2},-2 i \sin ^{-1}(a x)\right)+8 \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \Gamma \left(\frac{3}{2},2 i \sin ^{-1}(a x)\right)+\sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{3}{2},-4 i \sin ^{-1}(a x)\right)+\sqrt{i \sin ^{-1}(a x)} \Gamma \left(\frac{3}{2},4 i \sin ^{-1}(a x)\right)\right)}{128 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}","-\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right)}{64 a \sqrt{1-a^2 x^2}}-\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} S\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a \sqrt{1-a^2 x^2}}+\frac{1}{4} x \left(c-a^2 c x^2\right)^{3/2} \sqrt{\sin ^{-1}(a x)}+\frac{c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{4 a \sqrt{1-a^2 x^2}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}",1,"(c*Sqrt[c - a^2*c*x^2]*(32*ArcSin[a*x]^2 + 8*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2, (-2*I)*ArcSin[a*x]] + 8*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (2*I)*ArcSin[a*x]] + Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2, (-4*I)*ArcSin[a*x]] + Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (4*I)*ArcSin[a*x]]))/(128*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])","C",0
443,1,138,130,0.0849552,"\int \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \, dx","Integrate[Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]],x]","\frac{\sqrt{c-a^2 c x^2} \left(16 \sin ^{-1}(a x) \left(3 a x \sqrt{1-a^2 x^2}+2 \sin ^{-1}(a x)\right)+3 \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \sin ^{-1}(a x)\right)+3 \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \sin ^{-1}(a x)\right)\right)}{96 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}","-\frac{\sqrt{\pi } \sqrt{c-a^2 c x^2} S\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}",1,"(Sqrt[c - a^2*c*x^2]*(16*ArcSin[a*x]*(3*a*x*Sqrt[1 - a^2*x^2] + 2*ArcSin[a*x]) + 3*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-2*I)*ArcSin[a*x]] + 3*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (2*I)*ArcSin[a*x]]))/(96*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])","C",0
444,1,44,44,0.0595576,"\int \frac{\sqrt{\sin ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[Sqrt[ArcSin[a*x]]/Sqrt[c - a^2*c*x^2],x]","\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{c-a^2 c x^2}}","\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}{3 a \sqrt{c-a^2 c x^2}}",1,"(2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))/(3*a*Sqrt[c - a^2*c*x^2])","A",1
445,0,0,91,0.7220103,"\int \frac{\sqrt{\sin ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[Sqrt[ArcSin[a*x]]/(c - a^2*c*x^2)^(3/2),x]","\int \frac{\sqrt{\sin ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","\frac{x \sqrt{\sin ^{-1}(a x)}}{c \sqrt{c-a^2 c x^2}}-\frac{a \sqrt{1-a^2 x^2} \text{Int}\left(\frac{x}{\left(1-a^2 x^2\right) \sqrt{\sin ^{-1}(a x)}},x\right)}{2 c \sqrt{c-a^2 c x^2}}",0,"Integrate[Sqrt[ArcSin[a*x]]/(c - a^2*c*x^2)^(3/2), x]","A",-1
446,0,0,187,1.9948588,"\int \frac{\sqrt{\sin ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[Sqrt[ArcSin[a*x]]/(c - a^2*c*x^2)^(5/2),x]","\int \frac{\sqrt{\sin ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","-\frac{a \sqrt{1-a^2 x^2} \text{Int}\left(\frac{x}{\left(1-a^2 x^2\right)^2 \sqrt{\sin ^{-1}(a x)}},x\right)}{6 c^2 \sqrt{c-a^2 c x^2}}-\frac{a \sqrt{1-a^2 x^2} \text{Int}\left(\frac{x}{\left(1-a^2 x^2\right) \sqrt{\sin ^{-1}(a x)}},x\right)}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \sqrt{\sin ^{-1}(a x)}}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{x \sqrt{\sin ^{-1}(a x)}}{3 c \left(c-a^2 c x^2\right)^{3/2}}",0,"Integrate[Sqrt[ArcSin[a*x]]/(c - a^2*c*x^2)^(5/2), x]","A",-1
447,1,186,363,0.5225546,"\int \left(c-a^2 c x^2\right)^{3/2} \sin ^{-1}(a x)^{3/2} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(3/2),x]","\frac{c \sqrt{c-a^2 c x^2} \left(-240 \sqrt{\pi } \sqrt{\sin ^{-1}(a x)^2} C\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)+\sqrt{\sin ^{-1}(a x)} \left(32 \sqrt{\sin ^{-1}(a x)^2} \left(12 \sin ^{-1}(a x)^2+20 \sin \left(2 \sin ^{-1}(a x)\right) \sin ^{-1}(a x)+15 \cos \left(2 \sin ^{-1}(a x)\right)\right)+5 \sqrt{i \sin ^{-1}(a x)} \Gamma \left(\frac{5}{2},-4 i \sin ^{-1}(a x)\right)+5 \sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{5}{2},4 i \sin ^{-1}(a x)\right)\right)\right)}{2560 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)^2}}","-\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right)}{512 a \sqrt{1-a^2 x^2}}-\frac{3 \sqrt{\pi } c \sqrt{c-a^2 c x^2} C\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a \sqrt{1-a^2 x^2}}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{20 a \sqrt{1-a^2 x^2}}+\frac{1}{4} x \left(c-a^2 c x^2\right)^{3/2} \sin ^{-1}(a x)^{3/2}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}+\frac{3 c \left(1-a^2 x^2\right)^{3/2} \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 a}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{32 \sqrt{1-a^2 x^2}}+\frac{27 c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{256 a \sqrt{1-a^2 x^2}}",1,"(c*Sqrt[c - a^2*c*x^2]*(-240*Sqrt[Pi]*Sqrt[ArcSin[a*x]^2]*FresnelC[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]] + Sqrt[ArcSin[a*x]]*(5*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-4*I)*ArcSin[a*x]] + 5*Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2, (4*I)*ArcSin[a*x]] + 32*Sqrt[ArcSin[a*x]^2]*(12*ArcSin[a*x]^2 + 15*Cos[2*ArcSin[a*x]] + 20*ArcSin[a*x]*Sin[2*ArcSin[a*x]]))))/(2560*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]^2])","C",0
448,1,158,219,0.1404274,"\int \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2} \, dx","Integrate[Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2),x]","\frac{\sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \left(32 \sin ^{-1}(a x) \sqrt{\sin ^{-1}(a x)^2} \left(5 a x \sqrt{1-a^2 x^2}+2 \sin ^{-1}(a x)\right)+15 \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \Gamma \left(\frac{3}{2},-2 i \sin ^{-1}(a x)\right)+15 \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{3}{2},2 i \sin ^{-1}(a x)\right)\right)}{320 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)^2}}","-\frac{3 \sqrt{\pi } \sqrt{c-a^2 c x^2} C\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{8 \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{16 a \sqrt{1-a^2 x^2}}",1,"(Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]]*(32*ArcSin[a*x]*Sqrt[ArcSin[a*x]^2]*(5*a*x*Sqrt[1 - a^2*x^2] + 2*ArcSin[a*x]) + 15*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[3/2, (-2*I)*ArcSin[a*x]] + 15*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[3/2, (2*I)*ArcSin[a*x]]))/(320*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]^2])","C",0
449,1,44,44,0.0662352,"\int \frac{\sin ^{-1}(a x)^{3/2}}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[ArcSin[a*x]^(3/2)/Sqrt[c - a^2*c*x^2],x]","\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt{c-a^2 c x^2}}","\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{5/2}}{5 a \sqrt{c-a^2 c x^2}}",1,"(2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(5/2))/(5*a*Sqrt[c - a^2*c*x^2])","A",1
450,0,0,91,0.842994,"\int \frac{\sin ^{-1}(a x)^{3/2}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcSin[a*x]^(3/2)/(c - a^2*c*x^2)^(3/2),x]","\int \frac{\sin ^{-1}(a x)^{3/2}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","\frac{x \sin ^{-1}(a x)^{3/2}}{c \sqrt{c-a^2 c x^2}}-\frac{3 a \sqrt{1-a^2 x^2} \text{Int}\left(\frac{x \sqrt{\sin ^{-1}(a x)}}{1-a^2 x^2},x\right)}{2 c \sqrt{c-a^2 c x^2}}",0,"Integrate[ArcSin[a*x]^(3/2)/(c - a^2*c*x^2)^(3/2), x]","A",-1
451,1,180,431,0.3990814,"\int \left(c-a^2 c x^2\right)^{3/2} \sin ^{-1}(a x)^{5/2} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(5/2),x]","\frac{c \sqrt{c-a^2 c x^2} \left(1680 \sqrt{\pi } \sqrt{\sin ^{-1}(a x)} S\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)+1536 \sin ^{-1}(a x)^4+3584 \sin \left(2 \sin ^{-1}(a x)\right) \sin ^{-1}(a x)^3-3360 \sin \left(2 \sin ^{-1}(a x)\right) \sin ^{-1}(a x)+4480 \sin ^{-1}(a x)^2 \cos \left(2 \sin ^{-1}(a x)\right)-7 \sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{7}{2},-4 i \sin ^{-1}(a x)\right)-7 \sqrt{i \sin ^{-1}(a x)} \Gamma \left(\frac{7}{2},4 i \sin ^{-1}(a x)\right)\right)}{14336 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}","\frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right)}{4096 a \sqrt{1-a^2 x^2}}+\frac{15 \sqrt{\pi } c \sqrt{c-a^2 c x^2} S\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)}{128 a \sqrt{1-a^2 x^2}}+\frac{3 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{28 a \sqrt{1-a^2 x^2}}+\frac{1}{4} x \left(c-a^2 c x^2\right)^{3/2} \sin ^{-1}(a x)^{5/2}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}+\frac{5 c \left(1-a^2 x^2\right)^{3/2} \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 a}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{32 \sqrt{1-a^2 x^2}}+\frac{45 c \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{256 a \sqrt{1-a^2 x^2}}-\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}-\frac{15}{256} c x \left(1-a^2 x^2\right) \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}",1,"(c*Sqrt[c - a^2*c*x^2]*(1536*ArcSin[a*x]^4 + 4480*ArcSin[a*x]^2*Cos[2*ArcSin[a*x]] + 1680*Sqrt[Pi]*Sqrt[ArcSin[a*x]]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]] - 7*Sqrt[(-I)*ArcSin[a*x]]*Gamma[7/2, (-4*I)*ArcSin[a*x]] - 7*Sqrt[I*ArcSin[a*x]]*Gamma[7/2, (4*I)*ArcSin[a*x]] - 3360*ArcSin[a*x]*Sin[2*ArcSin[a*x]] + 3584*ArcSin[a*x]^3*Sin[2*ArcSin[a*x]]))/(14336*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])","C",0
452,1,158,247,0.1545914,"\int \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2} \, dx","Integrate[Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(5/2),x]","\frac{\sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \left(64 \left(7 a x \sqrt{1-a^2 x^2}+2 \sin ^{-1}(a x)\right) \left(\sin ^{-1}(a x)^2\right)^{3/2}+35 i \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \Gamma \left(\frac{5}{2},-2 i \sin ^{-1}(a x)\right)-35 i \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{5}{2},2 i \sin ^{-1}(a x)\right)\right)}{896 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)^2}}","\frac{15 \sqrt{\pi } \sqrt{c-a^2 c x^2} S\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)}{128 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt{1-a^2 x^2}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}-\frac{5 a x^2 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{8 \sqrt{1-a^2 x^2}}+\frac{5 \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}{16 a \sqrt{1-a^2 x^2}}-\frac{15}{32} x \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}",1,"(Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]]*(64*(ArcSin[a*x]^2)^(3/2)*(7*a*x*Sqrt[1 - a^2*x^2] + 2*ArcSin[a*x]) + (35*I)*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-2*I)*ArcSin[a*x]] - (35*I)*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2, (2*I)*ArcSin[a*x]]))/(896*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]^2])","C",0
453,1,44,44,0.0628755,"\int \frac{\sin ^{-1}(a x)^{5/2}}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[ArcSin[a*x]^(5/2)/Sqrt[c - a^2*c*x^2],x]","\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt{c-a^2 c x^2}}","\frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{7/2}}{7 a \sqrt{c-a^2 c x^2}}",1,"(2*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(7/2))/(7*a*Sqrt[c - a^2*c*x^2])","A",1
454,0,0,91,0.8045297,"\int \frac{\sin ^{-1}(a x)^{5/2}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcSin[a*x]^(5/2)/(c - a^2*c*x^2)^(3/2),x]","\int \frac{\sin ^{-1}(a x)^{5/2}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","\frac{x \sin ^{-1}(a x)^{5/2}}{c \sqrt{c-a^2 c x^2}}-\frac{5 a \sqrt{1-a^2 x^2} \text{Int}\left(\frac{x \sin ^{-1}(a x)^{3/2}}{1-a^2 x^2},x\right)}{2 c \sqrt{c-a^2 c x^2}}",0,"Integrate[ArcSin[a*x]^(5/2)/(c - a^2*c*x^2)^(3/2), x]","A",-1
455,1,183,226,0.2337349,"\int \left(a^2-x^2\right)^{3/2} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)} \, dx","Integrate[(a^2 - x^2)^(3/2)*Sqrt[ArcSin[x/a]],x]","\frac{a^3 \sqrt{a^2-x^2} \left(32 \sin ^{-1}\left(\frac{x}{a}\right)^2+8 \sqrt{2} \sqrt{-i \sin ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{3}{2},-2 i \sin ^{-1}\left(\frac{x}{a}\right)\right)+8 \sqrt{2} \sqrt{i \sin ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{3}{2},2 i \sin ^{-1}\left(\frac{x}{a}\right)\right)+\sqrt{-i \sin ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{3}{2},-4 i \sin ^{-1}\left(\frac{x}{a}\right)\right)+\sqrt{i \sin ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{3}{2},4 i \sin ^{-1}\left(\frac{x}{a}\right)\right)\right)}{128 \sqrt{1-\frac{x^2}{a^2}} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}","\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}+\frac{1}{4} x \left(a^2-x^2\right)^{3/2} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}-\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}\right)}{64 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\sqrt{\pi } a^3 \sqrt{a^2-x^2} S\left(\frac{2 \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{\sqrt{\pi }}\right)}{8 \sqrt{1-\frac{x^2}{a^2}}}+\frac{a^3 \sqrt{a^2-x^2} \sin ^{-1}\left(\frac{x}{a}\right)^{3/2}}{4 \sqrt{1-\frac{x^2}{a^2}}}",1,"(a^3*Sqrt[a^2 - x^2]*(32*ArcSin[x/a]^2 + 8*Sqrt[2]*Sqrt[(-I)*ArcSin[x/a]]*Gamma[3/2, (-2*I)*ArcSin[x/a]] + 8*Sqrt[2]*Sqrt[I*ArcSin[x/a]]*Gamma[3/2, (2*I)*ArcSin[x/a]] + Sqrt[(-I)*ArcSin[x/a]]*Gamma[3/2, (-4*I)*ArcSin[x/a]] + Sqrt[I*ArcSin[x/a]]*Gamma[3/2, (4*I)*ArcSin[x/a]]))/(128*Sqrt[1 - x^2/a^2]*Sqrt[ArcSin[x/a]])","C",0
456,1,148,126,0.0808011,"\int \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)} \, dx","Integrate[Sqrt[a^2 - x^2]*Sqrt[ArcSin[x/a]],x]","\frac{\sqrt{a^2-x^2} \left(48 x \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left(\frac{x}{a}\right)+32 a \sin ^{-1}\left(\frac{x}{a}\right)^2+3 \sqrt{2} a \sqrt{-i \sin ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{1}{2},-2 i \sin ^{-1}\left(\frac{x}{a}\right)\right)+3 \sqrt{2} a \sqrt{i \sin ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{1}{2},2 i \sin ^{-1}\left(\frac{x}{a}\right)\right)\right)}{96 \sqrt{1-\frac{x^2}{a^2}} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}","-\frac{\sqrt{\pi } a \sqrt{a^2-x^2} S\left(\frac{2 \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{\sqrt{\pi }}\right)}{8 \sqrt{1-\frac{x^2}{a^2}}}+\frac{a \sqrt{a^2-x^2} \sin ^{-1}\left(\frac{x}{a}\right)^{3/2}}{3 \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}",1,"(Sqrt[a^2 - x^2]*(48*x*Sqrt[1 - x^2/a^2]*ArcSin[x/a] + 32*a*ArcSin[x/a]^2 + 3*Sqrt[2]*a*Sqrt[(-I)*ArcSin[x/a]]*Gamma[1/2, (-2*I)*ArcSin[x/a]] + 3*Sqrt[2]*a*Sqrt[I*ArcSin[x/a]]*Gamma[1/2, (2*I)*ArcSin[x/a]]))/(96*Sqrt[1 - x^2/a^2]*Sqrt[ArcSin[x/a]])","C",0
457,1,42,42,0.0376825,"\int \frac{\sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{\sqrt{a^2-x^2}} \, dx","Integrate[Sqrt[ArcSin[x/a]]/Sqrt[a^2 - x^2],x]","\frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left(\frac{x}{a}\right)^{3/2}}{3 \sqrt{a^2-x^2}}","\frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left(\frac{x}{a}\right)^{3/2}}{3 \sqrt{a^2-x^2}}",1,"(2*a*Sqrt[1 - x^2/a^2]*ArcSin[x/a]^(3/2))/(3*Sqrt[a^2 - x^2])","A",1
458,0,0,90,0.6781665,"\int \frac{\sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{\left(a^2-x^2\right)^{3/2}} \, dx","Integrate[Sqrt[ArcSin[x/a]]/(a^2 - x^2)^(3/2),x]","\int \frac{\sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{\left(a^2-x^2\right)^{3/2}} \, dx","\frac{x \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{a^2 \sqrt{a^2-x^2}}-\frac{\sqrt{1-\frac{x^2}{a^2}} \text{Int}\left(\frac{x}{\left(1-\frac{x^2}{a^2}\right) \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}},x\right)}{2 a^3 \sqrt{a^2-x^2}}",0,"Integrate[Sqrt[ArcSin[x/a]]/(a^2 - x^2)^(3/2), x]","A",-1
459,0,0,185,2.0978388,"\int \frac{\sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{\left(a^2-x^2\right)^{5/2}} \, dx","Integrate[Sqrt[ArcSin[x/a]]/(a^2 - x^2)^(5/2),x]","\int \frac{\sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{\left(a^2-x^2\right)^{5/2}} \, dx","-\frac{\sqrt{1-\frac{x^2}{a^2}} \text{Int}\left(\frac{x}{\left(1-\frac{x^2}{a^2}\right)^2 \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}},x\right)}{6 a^5 \sqrt{a^2-x^2}}-\frac{\sqrt{1-\frac{x^2}{a^2}} \text{Int}\left(\frac{x}{\left(1-\frac{x^2}{a^2}\right) \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}},x\right)}{3 a^5 \sqrt{a^2-x^2}}+\frac{x \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{3 a^2 \left(a^2-x^2\right)^{3/2}}+\frac{2 x \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{3 a^4 \sqrt{a^2-x^2}}",0,"Integrate[Sqrt[ArcSin[x/a]]/(a^2 - x^2)^(5/2), x]","A",-1
460,1,209,359,0.5439024,"\int \left(a^2-x^2\right)^{3/2} \sin ^{-1}\left(\frac{x}{a}\right)^{3/2} \, dx","Integrate[(a^2 - x^2)^(3/2)*ArcSin[x/a]^(3/2),x]","\frac{a^3 \sqrt{a^2-x^2} \left(-240 \sqrt{\pi } \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)^2} C\left(\frac{2 \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{\sqrt{\pi }}\right)+\sqrt{\sin ^{-1}\left(\frac{x}{a}\right)} \left(32 \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)^2} \left(12 \sin ^{-1}\left(\frac{x}{a}\right)^2+20 \sin \left(2 \sin ^{-1}\left(\frac{x}{a}\right)\right) \sin ^{-1}\left(\frac{x}{a}\right)+15 \cos \left(2 \sin ^{-1}\left(\frac{x}{a}\right)\right)\right)+5 \sqrt{i \sin ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{5}{2},-4 i \sin ^{-1}\left(\frac{x}{a}\right)\right)+5 \sqrt{-i \sin ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{5}{2},4 i \sin ^{-1}\left(\frac{x}{a}\right)\right)\right)\right)}{2560 \sqrt{1-\frac{x^2}{a^2}} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)^2}}","\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sin ^{-1}\left(\frac{x}{a}\right)^{3/2}-\frac{9 a x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{4} x \left(a^2-x^2\right)^{3/2} \sin ^{-1}\left(\frac{x}{a}\right)^{3/2}+\frac{3 \left(a^2-x^2\right)^{5/2} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{32 a \sqrt{1-\frac{x^2}{a^2}}}-\frac{3 \sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}\right)}{512 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3 \sqrt{\pi } a^3 \sqrt{a^2-x^2} C\left(\frac{2 \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{\sqrt{\pi }}\right)}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3 a^3 \sqrt{a^2-x^2} \sin ^{-1}\left(\frac{x}{a}\right)^{5/2}}{20 \sqrt{1-\frac{x^2}{a^2}}}+\frac{27 a^3 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{256 \sqrt{1-\frac{x^2}{a^2}}}",1,"(a^3*Sqrt[a^2 - x^2]*(-240*Sqrt[Pi]*Sqrt[ArcSin[x/a]^2]*FresnelC[(2*Sqrt[ArcSin[x/a]])/Sqrt[Pi]] + Sqrt[ArcSin[x/a]]*(5*Sqrt[I*ArcSin[x/a]]*Gamma[5/2, (-4*I)*ArcSin[x/a]] + 5*Sqrt[(-I)*ArcSin[x/a]]*Gamma[5/2, (4*I)*ArcSin[x/a]] + 32*Sqrt[ArcSin[x/a]^2]*(12*ArcSin[x/a]^2 + 15*Cos[2*ArcSin[x/a]] + 20*ArcSin[x/a]*Sin[2*ArcSin[x/a]]))))/(2560*Sqrt[1 - x^2/a^2]*Sqrt[ArcSin[x/a]^2])","C",0
461,1,173,215,0.1544704,"\int \sqrt{a^2-x^2} \sin ^{-1}\left(\frac{x}{a}\right)^{3/2} \, dx","Integrate[Sqrt[a^2 - x^2]*ArcSin[x/a]^(3/2),x]","\frac{\sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)} \left(32 \sin ^{-1}\left(\frac{x}{a}\right) \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)^2} \left(5 x \sqrt{1-\frac{x^2}{a^2}}+2 a \sin ^{-1}\left(\frac{x}{a}\right)\right)+15 \sqrt{2} a \sqrt{i \sin ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{3}{2},-2 i \sin ^{-1}\left(\frac{x}{a}\right)\right)+15 \sqrt{2} a \sqrt{-i \sin ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{3}{2},2 i \sin ^{-1}\left(\frac{x}{a}\right)\right)\right)}{320 \sqrt{1-\frac{x^2}{a^2}} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)^2}}","-\frac{3 \sqrt{\pi } a \sqrt{a^2-x^2} C\left(\frac{2 \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{\sqrt{\pi }}\right)}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{a \sqrt{a^2-x^2} \sin ^{-1}\left(\frac{x}{a}\right)^{5/2}}{5 \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{2} x \sqrt{a^2-x^2} \sin ^{-1}\left(\frac{x}{a}\right)^{3/2}-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{8 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{3 a \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{16 \sqrt{1-\frac{x^2}{a^2}}}",1,"(Sqrt[a^2 - x^2]*Sqrt[ArcSin[x/a]]*(32*ArcSin[x/a]*Sqrt[ArcSin[x/a]^2]*(5*x*Sqrt[1 - x^2/a^2] + 2*a*ArcSin[x/a]) + 15*Sqrt[2]*a*Sqrt[I*ArcSin[x/a]]*Gamma[3/2, (-2*I)*ArcSin[x/a]] + 15*Sqrt[2]*a*Sqrt[(-I)*ArcSin[x/a]]*Gamma[3/2, (2*I)*ArcSin[x/a]]))/(320*Sqrt[1 - x^2/a^2]*Sqrt[ArcSin[x/a]^2])","C",0
462,1,42,42,0.0389128,"\int \frac{\sin ^{-1}\left(\frac{x}{a}\right)^{3/2}}{\sqrt{a^2-x^2}} \, dx","Integrate[ArcSin[x/a]^(3/2)/Sqrt[a^2 - x^2],x]","\frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left(\frac{x}{a}\right)^{5/2}}{5 \sqrt{a^2-x^2}}","\frac{2 a \sqrt{1-\frac{x^2}{a^2}} \sin ^{-1}\left(\frac{x}{a}\right)^{5/2}}{5 \sqrt{a^2-x^2}}",1,"(2*a*Sqrt[1 - x^2/a^2]*ArcSin[x/a]^(5/2))/(5*Sqrt[a^2 - x^2])","A",1
463,0,0,90,0.770557,"\int \frac{\sin ^{-1}\left(\frac{x}{a}\right)^{3/2}}{\left(a^2-x^2\right)^{3/2}} \, dx","Integrate[ArcSin[x/a]^(3/2)/(a^2 - x^2)^(3/2),x]","\int \frac{\sin ^{-1}\left(\frac{x}{a}\right)^{3/2}}{\left(a^2-x^2\right)^{3/2}} \, dx","\frac{x \sin ^{-1}\left(\frac{x}{a}\right)^{3/2}}{a^2 \sqrt{a^2-x^2}}-\frac{3 \sqrt{1-\frac{x^2}{a^2}} \text{Int}\left(\frac{x \sqrt{\sin ^{-1}\left(\frac{x}{a}\right)}}{1-\frac{x^2}{a^2}},x\right)}{2 a^3 \sqrt{a^2-x^2}}",0,"Integrate[ArcSin[x/a]^(3/2)/(a^2 - x^2)^(3/2), x]","A",-1
464,1,53,25,0.0916468,"\int \frac{x}{\sqrt{1-x^2} \sqrt{\sin ^{-1}(x)}} \, dx","Integrate[x/(Sqrt[1 - x^2]*Sqrt[ArcSin[x]]),x]","-\frac{\sqrt{-i \sin ^{-1}(x)} \Gamma \left(\frac{1}{2},-i \sin ^{-1}(x)\right)+\sqrt{i \sin ^{-1}(x)} \Gamma \left(\frac{1}{2},i \sin ^{-1}(x)\right)}{2 \sqrt{\sin ^{-1}(x)}}","\sqrt{2 \pi } S\left(\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(x)}\right)",1,"-1/2*(Sqrt[(-I)*ArcSin[x]]*Gamma[1/2, (-I)*ArcSin[x]] + Sqrt[I*ArcSin[x]]*Gamma[1/2, I*ArcSin[x]])/Sqrt[ArcSin[x]]","C",0
465,1,336,244,0.7379073,"\int \frac{\left(c-a^2 c x^2\right)^{5/2}}{\sqrt{\sin ^{-1}(a x)}} \, dx","Integrate[(c - a^2*c*x^2)^(5/2)/Sqrt[ArcSin[a*x]],x]","\frac{c^2 \sqrt{c-a^2 c x^2} \left(240 \sin ^{-1}(a x) \sqrt{\sin ^{-1}(a x)^2}-45 i \sqrt{2} \left(-i \sin ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},2 i \sin ^{-1}(a x)\right)-18 i \left(-i \sin ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},4 i \sin ^{-1}(a x)\right)-i \sqrt{6} \left(-i \sin ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},6 i \sin ^{-1}(a x)\right)+6 i \sqrt{\sin ^{-1}(a x)^2} \sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \sin ^{-1}(a x)\right)-i \sqrt{6} \sqrt{\sin ^{-1}(a x)^2} \sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},-6 i \sin ^{-1}(a x)\right)+3 i \sqrt{2} \left(16 \left(i \sin ^{-1}(a x)\right)^{3/2}+\sqrt{-i \sin ^{-1}(a x)} \sqrt{\sin ^{-1}(a x)^2}\right) \Gamma \left(\frac{1}{2},-2 i \sin ^{-1}(a x)\right)+24 i \left(i \sin ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-4 i \sin ^{-1}(a x)\right)\right)}{384 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)} \sqrt{\sin ^{-1}(a x)^2}}","\frac{3 \sqrt{\frac{\pi }{2}} c^2 \sqrt{c-a^2 c x^2} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right)}{16 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{\frac{\pi }{3}} c^2 \sqrt{c-a^2 c x^2} C\left(2 \sqrt{\frac{3}{\pi }} \sqrt{\sin ^{-1}(a x)}\right)}{32 a \sqrt{1-a^2 x^2}}+\frac{15 \sqrt{\pi } c^2 \sqrt{c-a^2 c x^2} C\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)}{32 a \sqrt{1-a^2 x^2}}+\frac{5 c^2 \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{8 a \sqrt{1-a^2 x^2}}",1,"(c^2*Sqrt[c - a^2*c*x^2]*(240*ArcSin[a*x]*Sqrt[ArcSin[a*x]^2] + (3*I)*Sqrt[2]*(16*(I*ArcSin[a*x])^(3/2) + Sqrt[(-I)*ArcSin[a*x]]*Sqrt[ArcSin[a*x]^2])*Gamma[1/2, (-2*I)*ArcSin[a*x]] - (45*I)*Sqrt[2]*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (2*I)*ArcSin[a*x]] + (24*I)*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, (-4*I)*ArcSin[a*x]] + (6*I)*Sqrt[(-I)*ArcSin[a*x]]*Sqrt[ArcSin[a*x]^2]*Gamma[1/2, (-4*I)*ArcSin[a*x]] - (18*I)*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (4*I)*ArcSin[a*x]] - I*Sqrt[6]*Sqrt[(-I)*ArcSin[a*x]]*Sqrt[ArcSin[a*x]^2]*Gamma[1/2, (-6*I)*ArcSin[a*x]] - I*Sqrt[6]*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (6*I)*ArcSin[a*x]]))/(384*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]]*Sqrt[ArcSin[a*x]^2])","C",0
466,1,182,170,0.3676222,"\int \frac{\left(c-a^2 c x^2\right)^{3/2}}{\sqrt{\sin ^{-1}(a x)}} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)/Sqrt[ArcSin[a*x]],x]","\frac{c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)} \left(24 \sqrt{\sin ^{-1}(a x)^2}-4 \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \sin ^{-1}(a x)\right)-4 \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \sin ^{-1}(a x)\right)-\sqrt{i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \sin ^{-1}(a x)\right)-\sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \sin ^{-1}(a x)\right)\right)}{32 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)^2}}","\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right)}{8 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} C\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a \sqrt{1-a^2 x^2}}+\frac{3 c \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{4 a \sqrt{1-a^2 x^2}}",1,"(c*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]]*(24*Sqrt[ArcSin[a*x]^2] - 4*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (-2*I)*ArcSin[a*x]] - 4*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (2*I)*ArcSin[a*x]] - Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (-4*I)*ArcSin[a*x]] - Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (4*I)*ArcSin[a*x]]))/(32*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]^2])","C",0
467,1,118,99,0.1598078,"\int \frac{\sqrt{c-a^2 c x^2}}{\sqrt{\sin ^{-1}(a x)}} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/Sqrt[ArcSin[a*x]],x]","\frac{\sqrt{c \left(1-a^2 x^2\right)} \left(8 \sin ^{-1}(a x)-i \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \sin ^{-1}(a x)\right)+i \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \sin ^{-1}(a x)\right)\right)}{8 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}","\frac{\sqrt{\pi } \sqrt{c-a^2 c x^2} C\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)}{2 a \sqrt{1-a^2 x^2}}+\frac{\sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}{a \sqrt{1-a^2 x^2}}",1,"(Sqrt[c*(1 - a^2*x^2)]*(8*ArcSin[a*x] - I*Sqrt[2]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-2*I)*ArcSin[a*x]] + I*Sqrt[2]*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (2*I)*ArcSin[a*x]]))/(8*a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])","C",0
468,1,42,42,0.0545131,"\int \frac{1}{\sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}} \, dx","Integrate[1/(Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]]),x]","\frac{2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{a \sqrt{c-a^2 c x^2}}","\frac{2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{a \sqrt{c-a^2 c x^2}}",1,"(2*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(a*Sqrt[c - a^2*c*x^2])","A",1
469,0,0,27,1.0031241,"\int \frac{1}{\left(c-a^2 c x^2\right)^{3/2} \sqrt{\sin ^{-1}(a x)}} \, dx","Integrate[1/((c - a^2*c*x^2)^(3/2)*Sqrt[ArcSin[a*x]]),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^{3/2} \sqrt{\sin ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{\left(c-a^2 c x^2\right)^{3/2} \sqrt{\sin ^{-1}(a x)}},x\right)",0,"Integrate[1/((c - a^2*c*x^2)^(3/2)*Sqrt[ArcSin[a*x]]), x]","A",-1
470,0,0,27,2.4152457,"\int \frac{1}{\left(c-a^2 c x^2\right)^{5/2} \sqrt{\sin ^{-1}(a x)}} \, dx","Integrate[1/((c - a^2*c*x^2)^(5/2)*Sqrt[ArcSin[a*x]]),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^{5/2} \sqrt{\sin ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{\left(c-a^2 c x^2\right)^{5/2} \sqrt{\sin ^{-1}(a x)}},x\right)",0,"Integrate[1/((c - a^2*c*x^2)^(5/2)*Sqrt[ArcSin[a*x]]), x]","A",-1
471,1,404,237,1.2648024,"\int \frac{\left(c-a^2 c x^2\right)^{5/2}}{\sin ^{-1}(a x)^{3/2}} \, dx","Integrate[(c - a^2*c*x^2)^(5/2)/ArcSin[a*x]^(3/2),x]","-\frac{c^2 \sqrt{c-a^2 c x^2} e^{-6 i \sin ^{-1}(a x)} \left(64 \sqrt{\pi } e^{6 i \sin ^{-1}(a x)} \sqrt{\sin ^{-1}(a x)} S\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)+6 e^{2 i \sin ^{-1}(a x)}+15 e^{4 i \sin ^{-1}(a x)}+20 e^{6 i \sin ^{-1}(a x)}+15 e^{8 i \sin ^{-1}(a x)}+6 e^{10 i \sin ^{-1}(a x)}+e^{12 i \sin ^{-1}(a x)}+\sqrt{2} e^{6 i \sin ^{-1}(a x)} \sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 i \sin ^{-1}(a x)\right)+\sqrt{2} e^{6 i \sin ^{-1}(a x)} \sqrt{i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 i \sin ^{-1}(a x)\right)-12 e^{6 i \sin ^{-1}(a x)} \sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \sin ^{-1}(a x)\right)-12 e^{6 i \sin ^{-1}(a x)} \sqrt{i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \sin ^{-1}(a x)\right)-\sqrt{6} e^{6 i \sin ^{-1}(a x)} \sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},-6 i \sin ^{-1}(a x)\right)-\sqrt{6} e^{6 i \sin ^{-1}(a x)} \sqrt{i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},6 i \sin ^{-1}(a x)\right)+1\right)}{32 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}","-\frac{3 \sqrt{\frac{\pi }{2}} c^2 \sqrt{c-a^2 c x^2} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right)}{2 a \sqrt{1-a^2 x^2}}-\frac{\sqrt{3 \pi } c^2 \sqrt{c-a^2 c x^2} S\left(2 \sqrt{\frac{3}{\pi }} \sqrt{\sin ^{-1}(a x)}\right)}{8 a \sqrt{1-a^2 x^2}}-\frac{15 \sqrt{\pi } c^2 \sqrt{c-a^2 c x^2} S\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)}{8 a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2} \left(c-a^2 c x^2\right)^{5/2}}{a \sqrt{\sin ^{-1}(a x)}}",1,"-1/32*(c^2*Sqrt[c - a^2*c*x^2]*(1 + 6*E^((2*I)*ArcSin[a*x]) + 15*E^((4*I)*ArcSin[a*x]) + 20*E^((6*I)*ArcSin[a*x]) + 15*E^((8*I)*ArcSin[a*x]) + 6*E^((10*I)*ArcSin[a*x]) + E^((12*I)*ArcSin[a*x]) + 64*E^((6*I)*ArcSin[a*x])*Sqrt[Pi]*Sqrt[ArcSin[a*x]]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]] + Sqrt[2]*E^((6*I)*ArcSin[a*x])*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-2*I)*ArcSin[a*x]] + Sqrt[2]*E^((6*I)*ArcSin[a*x])*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (2*I)*ArcSin[a*x]] - 12*E^((6*I)*ArcSin[a*x])*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-4*I)*ArcSin[a*x]] - 12*E^((6*I)*ArcSin[a*x])*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (4*I)*ArcSin[a*x]] - Sqrt[6]*E^((6*I)*ArcSin[a*x])*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-6*I)*ArcSin[a*x]] - Sqrt[6]*E^((6*I)*ArcSin[a*x])*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (6*I)*ArcSin[a*x]]))/(a*E^((6*I)*ArcSin[a*x])*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])","C",0
472,1,211,163,0.4424912,"\int \frac{\left(c-a^2 c x^2\right)^{3/2}}{\sin ^{-1}(a x)^{3/2}} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)/ArcSin[a*x]^(3/2),x]","-\frac{c \sqrt{c-a^2 c x^2} e^{-4 i \sin ^{-1}(a x)} \left(16 \sqrt{\pi } e^{4 i \sin ^{-1}(a x)} \sqrt{\sin ^{-1}(a x)} S\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)+6 e^{4 i \sin ^{-1}(a x)}+e^{8 i \sin ^{-1}(a x)}+8 e^{4 i \sin ^{-1}(a x)} \cos \left(2 \sin ^{-1}(a x)\right)-2 e^{4 i \sin ^{-1}(a x)} \sqrt{-i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 i \sin ^{-1}(a x)\right)-2 e^{4 i \sin ^{-1}(a x)} \sqrt{i \sin ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 i \sin ^{-1}(a x)\right)+1\right)}{8 a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}","-\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} S\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right)}{a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{\pi } c \sqrt{c-a^2 c x^2} S\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2} \left(c-a^2 c x^2\right)^{3/2}}{a \sqrt{\sin ^{-1}(a x)}}",1,"-1/8*(c*Sqrt[c - a^2*c*x^2]*(1 + 6*E^((4*I)*ArcSin[a*x]) + E^((8*I)*ArcSin[a*x]) + 8*E^((4*I)*ArcSin[a*x])*Cos[2*ArcSin[a*x]] + 16*E^((4*I)*ArcSin[a*x])*Sqrt[Pi]*Sqrt[ArcSin[a*x]]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]] - 2*E^((4*I)*ArcSin[a*x])*Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-4*I)*ArcSin[a*x]] - 2*E^((4*I)*ArcSin[a*x])*Sqrt[I*ArcSin[a*x]]*Gamma[1/2, (4*I)*ArcSin[a*x]]))/(a*E^((4*I)*ArcSin[a*x])*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])","C",0
473,1,83,98,0.1918778,"\int \frac{\sqrt{c-a^2 c x^2}}{\sin ^{-1}(a x)^{3/2}} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/ArcSin[a*x]^(3/2),x]","-\frac{\sqrt{c \left(1-a^2 x^2\right)} \left(2 \sqrt{\pi } \sqrt{\sin ^{-1}(a x)} S\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)+\cos \left(2 \sin ^{-1}(a x)\right)+1\right)}{a \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}","-\frac{2 \sqrt{\pi } \sqrt{c-a^2 c x^2} S\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)}{a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{a \sqrt{\sin ^{-1}(a x)}}",1,"-((Sqrt[c*(1 - a^2*x^2)]*(1 + Cos[2*ArcSin[a*x]] + 2*Sqrt[Pi]*Sqrt[ArcSin[a*x]]*FresnelS[(2*Sqrt[ArcSin[a*x]])/Sqrt[Pi]]))/(a*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]]))","A",1
474,1,42,42,0.0484582,"\int \frac{1}{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}} \, dx","Integrate[1/(Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2)),x]","-\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}","-\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}",1,"(-2*Sqrt[1 - a^2*x^2])/(a*Sqrt[c - a^2*c*x^2]*Sqrt[ArcSin[a*x]])","A",1
475,0,0,103,0.9361177,"\int \frac{1}{\left(c-a^2 c x^2\right)^{3/2} \sin ^{-1}(a x)^{3/2}} \, dx","Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(3/2)),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^{3/2} \sin ^{-1}(a x)^{3/2}} \, dx","\frac{4 a \sqrt{1-a^2 x^2} \text{Int}\left(\frac{x}{\left(1-a^2 x^2\right)^2 \sqrt{\sin ^{-1}(a x)}},x\right)}{c \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{1-a^2 x^2}}{a \left(c-a^2 c x^2\right)^{3/2} \sqrt{\sin ^{-1}(a x)}}",0,"Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(3/2)), x]","A",-1
476,0,0,103,2.2097495,"\int \frac{1}{\left(c-a^2 c x^2\right)^{5/2} \sin ^{-1}(a x)^{3/2}} \, dx","Integrate[1/((c - a^2*c*x^2)^(5/2)*ArcSin[a*x]^(3/2)),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^{5/2} \sin ^{-1}(a x)^{3/2}} \, dx","\frac{8 a \sqrt{1-a^2 x^2} \text{Int}\left(\frac{x}{\left(1-a^2 x^2\right)^3 \sqrt{\sin ^{-1}(a x)}},x\right)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{1-a^2 x^2}}{a \left(c-a^2 c x^2\right)^{5/2} \sqrt{\sin ^{-1}(a x)}}",0,"Integrate[1/((c - a^2*c*x^2)^(5/2)*ArcSin[a*x]^(3/2)), x]","A",-1
477,1,251,206,1.6004737,"\int \frac{\left(c-a^2 c x^2\right)^{3/2}}{\sin ^{-1}(a x)^{5/2}} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)/ArcSin[a*x]^(5/2),x]","\frac{c \sqrt{c-a^2 c x^2} \left(16 a^2 x^2+64 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-e^{-4 i \sin ^{-1}(a x)}-e^{4 i \sin ^{-1}(a x)}+8 i e^{-4 i \sin ^{-1}(a x)} \sin ^{-1}(a x)-8 i e^{4 i \sin ^{-1}(a x)} \sin ^{-1}(a x)-16 \sqrt{2} \left(-i \sin ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-2 i \sin ^{-1}(a x)\right)-16 \sqrt{2} \left(i \sin ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},2 i \sin ^{-1}(a x)\right)-16 \left(-i \sin ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-4 i \sin ^{-1}(a x)\right)-16 \left(i \sin ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},4 i \sin ^{-1}(a x)\right)-14\right)}{24 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}","-\frac{4 \sqrt{2 \pi } c \sqrt{c-a^2 c x^2} C\left(2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right)}{3 a \sqrt{1-a^2 x^2}}-\frac{8 \sqrt{\pi } c \sqrt{c-a^2 c x^2} C\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a \sqrt{1-a^2 x^2}}-\frac{2 \sqrt{1-a^2 x^2} \left(c-a^2 c x^2\right)^{3/2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{16 c x \left(1-a^2 x^2\right) \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}}",1,"(c*Sqrt[c - a^2*c*x^2]*(-14 - E^((-4*I)*ArcSin[a*x]) - E^((4*I)*ArcSin[a*x]) + 16*a^2*x^2 + ((8*I)*ArcSin[a*x])/E^((4*I)*ArcSin[a*x]) - (8*I)*E^((4*I)*ArcSin[a*x])*ArcSin[a*x] + 64*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x] - 16*Sqrt[2]*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-2*I)*ArcSin[a*x]] - 16*Sqrt[2]*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, (2*I)*ArcSin[a*x]] - 16*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-4*I)*ArcSin[a*x]] - 16*(I*ArcSin[a*x])^(3/2)*Gamma[1/2, (4*I)*ArcSin[a*x]]))/(24*a*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))","C",0
478,1,142,130,0.5244507,"\int \frac{\sqrt{c-a^2 c x^2}}{\sin ^{-1}(a x)^{5/2}} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/ArcSin[a*x]^(5/2),x]","\frac{2 \sqrt{c-a^2 c x^2} \left(a^2 x^2+4 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-\sqrt{2} \left(-i \sin ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-2 i \sin ^{-1}(a x)\right)+\frac{\sqrt{2} \sin ^{-1}(a x)^2 \Gamma \left(\frac{1}{2},2 i \sin ^{-1}(a x)\right)}{\sqrt{i \sin ^{-1}(a x)}}-1\right)}{3 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}}","-\frac{8 \sqrt{\pi } \sqrt{c-a^2 c x^2} C\left(\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right)}{3 a \sqrt{1-a^2 x^2}}+\frac{8 x \sqrt{c-a^2 c x^2}}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}{3 a \sin ^{-1}(a x)^{3/2}}",1,"(2*Sqrt[c - a^2*c*x^2]*(-1 + a^2*x^2 + 4*a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x] - Sqrt[2]*((-I)*ArcSin[a*x])^(3/2)*Gamma[1/2, (-2*I)*ArcSin[a*x]] + (Sqrt[2]*ArcSin[a*x]^2*Gamma[1/2, (2*I)*ArcSin[a*x]])/Sqrt[I*ArcSin[a*x]]))/(3*a*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^(3/2))","C",0
479,1,44,44,0.0551235,"\int \frac{1}{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{5/2}} \, dx","Integrate[1/(Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(5/2)),x]","-\frac{2 \sqrt{1-a^2 x^2}}{3 a \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}","-\frac{2 \sqrt{1-a^2 x^2}}{3 a \sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}}",1,"(-2*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[c - a^2*c*x^2]*ArcSin[a*x]^(3/2))","A",1
480,0,0,107,1.0019664,"\int \frac{1}{\left(c-a^2 c x^2\right)^{3/2} \sin ^{-1}(a x)^{5/2}} \, dx","Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(5/2)),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^{3/2} \sin ^{-1}(a x)^{5/2}} \, dx","\frac{4 a \sqrt{1-a^2 x^2} \text{Int}\left(\frac{x}{\left(1-a^2 x^2\right)^2 \sin ^{-1}(a x)^{3/2}},x\right)}{3 c \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{1-a^2 x^2}}{3 a \left(c-a^2 c x^2\right)^{3/2} \sin ^{-1}(a x)^{3/2}}",0,"Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcSin[a*x]^(5/2)), x]","A",-1
481,0,0,107,2.1120697,"\int \frac{1}{\left(c-a^2 c x^2\right)^{5/2} \sin ^{-1}(a x)^{5/2}} \, dx","Integrate[1/((c - a^2*c*x^2)^(5/2)*ArcSin[a*x]^(5/2)),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^{5/2} \sin ^{-1}(a x)^{5/2}} \, dx","\frac{8 a \sqrt{1-a^2 x^2} \text{Int}\left(\frac{x}{\left(1-a^2 x^2\right)^3 \sin ^{-1}(a x)^{3/2}},x\right)}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{1-a^2 x^2}}{3 a \left(c-a^2 c x^2\right)^{5/2} \sin ^{-1}(a x)^{3/2}}",0,"Integrate[1/((c - a^2*c*x^2)^(5/2)*ArcSin[a*x]^(5/2)), x]","A",-1
482,1,189,259,0.9030588,"\int x^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \, dx","Integrate[x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n,x]","\frac{d \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{8 \left(a+b \sin ^{-1}(c x)\right)}{b n+b}+i 4^{-n} e^{-\frac{4 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(\left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-e^{\frac{8 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)\right)}{64 c^3 \sqrt{d \left(1-c^2 x^2\right)}}","\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^{n+1}}{8 b c^3 (n+1) \sqrt{1-c^2 x^2}}+\frac{i 2^{-2 (n+3)} e^{-\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{1-c^2 x^2}}-\frac{i 2^{-2 (n+3)} e^{\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{1-c^2 x^2}}",1,"(d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*((8*(a + b*ArcSin[c*x]))/(b + b*n) + (I*(((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b] - E^(((8*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b]))/(4^n*E^(((4*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^n)))/(64*c^3*Sqrt[d*(1 - c^2*x^2)])","A",1
483,1,272,391,0.9113254,"\int x \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \, dx","Integrate[x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n,x]","\frac{d e^{-\frac{3 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(3 e^{\frac{2 i a}{b}} \left(\left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \left(-\Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)-e^{\frac{2 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)-3^{-n} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(e^{\frac{6 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+\left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)\right)}{24 c^2 \sqrt{d \left(1-c^2 x^2\right)}}","-\frac{e^{-\frac{i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{3^{-n-1} e^{-\frac{3 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{e^{\frac{i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 c^2 \sqrt{1-c^2 x^2}}-\frac{3^{-n-1} e^{\frac{3 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 c^2 \sqrt{1-c^2 x^2}}",1,"(d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*(3*E^(((2*I)*a)/b)*(-(Gamma[1 + n, ((-I)*(a + b*ArcSin[c*x]))/b]/(((-I)*(a + b*ArcSin[c*x]))/b)^n) - (E^(((2*I)*a)/b)*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b])/((I*(a + b*ArcSin[c*x]))/b)^n) - (((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-3*I)*(a + b*ArcSin[c*x]))/b] + E^(((6*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c*x]))/b])/(3^n*((a + b*ArcSin[c*x])^2/b^2)^n)))/(24*c^2*E^(((3*I)*a)/b)*Sqrt[d*(1 - c^2*x^2)])","A",1
484,1,182,259,0.8283206,"\int \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \, dx","Integrate[Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n,x]","\frac{d \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{4 a+4 b \sin ^{-1}(c x)}{b n+b}-i 2^{-n} e^{-\frac{2 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+i 2^{-n} e^{\frac{2 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)}{8 c \sqrt{d \left(1-c^2 x^2\right)}}","\frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^{n+1}}{2 b c (n+1) \sqrt{1-c^2 x^2}}-\frac{i 2^{-n-3} e^{-\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c \sqrt{1-c^2 x^2}}+\frac{i 2^{-n-3} e^{\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c \sqrt{1-c^2 x^2}}",1,"(d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*((4*a + 4*b*ArcSin[c*x])/(b + b*n) - (I*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b])/(2^n*E^(((2*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + (I*E^(((2*I)*a)/b)*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c*x]))/b])/(2^n*((I*(a + b*ArcSin[c*x]))/b)^n)))/(8*c*Sqrt[d*(1 - c^2*x^2)])","A",1
485,0,0,219,0.2336657,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n}{x} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n)/x,x]","\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n}{x} \, dx","d \text{Int}\left(\frac{\left(a+b \sin ^{-1}(c x)\right)^n}{x \sqrt{d-c^2 d x^2}},x\right)+\frac{d e^{-\frac{i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 \sqrt{d-c^2 d x^2}}+\frac{d e^{\frac{i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 \sqrt{d-c^2 d x^2}}",0,"Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n)/x, x]","A",-1
486,0,0,88,0.3148546,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n}{x^2} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n)/x^2,x]","\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n}{x^2} \, dx","d \text{Int}\left(\frac{\left(a+b \sin ^{-1}(c x)\right)^n}{x^2 \sqrt{d-c^2 d x^2}},x\right)-\frac{c d \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^{n+1}}{b (n+1) \sqrt{d-c^2 d x^2}}",0,"Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^n)/x^2, x]","A",-1
487,1,436,684,3.573906,"\int x^2 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^n \, dx","Integrate[x^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n,x]","\frac{d^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(3 i 2^{-n} e^{-\frac{2 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(e^{\frac{4 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)+3 i 4^{-n} e^{-\frac{4 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(\left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-e^{\frac{8 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)+i 6^{-n} e^{-\frac{6 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(\left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-e^{\frac{12 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)+\frac{24 \left(a+b \sin ^{-1}(c x)\right)}{b n+b}\right)}{384 c^3 \sqrt{d-c^2 d x^2}}","\frac{d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^{n+1}}{16 b c^3 (n+1) \sqrt{1-c^2 x^2}}-\frac{i d 2^{-n-7} e^{-\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{1-c^2 x^2}}+\frac{i d 2^{-2 n-7} e^{-\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{1-c^2 x^2}}+\frac{i d 2^{-n-7} 3^{-n-1} e^{-\frac{6 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{1-c^2 x^2}}+\frac{i d 2^{-n-7} e^{\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{1-c^2 x^2}}-\frac{i d 2^{-2 n-7} e^{\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{1-c^2 x^2}}-\frac{i d 2^{-n-7} 3^{-n-1} e^{\frac{6 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{1-c^2 x^2}}",1,"(d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*((24*(a + b*ArcSin[c*x]))/(b + b*n) + ((3*I)*(-(((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b]) + E^(((4*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c*x]))/b]))/(2^n*E^(((2*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^n) + ((3*I)*(((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b] - E^(((8*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b]))/(4^n*E^(((4*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^n) + (I*(((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-6*I)*(a + b*ArcSin[c*x]))/b] - E^(((12*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((6*I)*(a + b*ArcSin[c*x]))/b]))/(6^n*E^(((6*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^n)))/(384*c^3*Sqrt[d - c^2*d*x^2])","A",1
488,1,464,595,2.4123525,"\int x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^n \, dx","Integrate[x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n,x]","-\frac{d^2 15^{-n-1} e^{-\frac{5 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{-3 n} \left(\left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \left(2\ 15^{n+1} e^{\frac{6 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+3 \left(5^{n+1} e^{\frac{2 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{2 n} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+5^{n+1} e^{\frac{8 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+3^n \left(e^{\frac{10 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+\left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{3 n} \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)\right)\right)+2\ 15^{n+1} e^{\frac{4 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)}{32 c^2 \sqrt{d-c^2 d x^2}}","-\frac{d e^{-\frac{i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 c^2 \sqrt{1-c^2 x^2}}-\frac{d 3^{-n} e^{-\frac{3 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 c^2 \sqrt{1-c^2 x^2}}-\frac{d 5^{-n-1} e^{-\frac{5 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 c^2 \sqrt{1-c^2 x^2}}-\frac{d e^{\frac{i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 c^2 \sqrt{1-c^2 x^2}}-\frac{d 3^{-n} e^{\frac{3 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 c^2 \sqrt{1-c^2 x^2}}-\frac{d 5^{-n-1} e^{\frac{5 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 c^2 \sqrt{1-c^2 x^2}}",1,"-1/32*(15^(-1 - n)*d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*(2*15^(1 + n)*E^(((4*I)*a)/b)*((I*(a + b*ArcSin[c*x]))/b)^n*((a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, ((-I)*(a + b*ArcSin[c*x]))/b] + (((-I)*(a + b*ArcSin[c*x]))/b)^n*(2*15^(1 + n)*E^(((6*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b] + 3*(5^(1 + n)*E^(((2*I)*a)/b)*((I*(a + b*ArcSin[c*x]))/b)^(2*n)*((a + b*ArcSin[c*x])^2/b^2)^n*Gamma[1 + n, ((-3*I)*(a + b*ArcSin[c*x]))/b] + 5^(1 + n)*E^(((8*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c*x]))/b] + 3^n*((((-I)*(a + b*ArcSin[c*x]))/b)^n*((I*(a + b*ArcSin[c*x]))/b)^(3*n)*Gamma[1 + n, ((-5*I)*(a + b*ArcSin[c*x]))/b] + E^(((10*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, ((5*I)*(a + b*ArcSin[c*x]))/b])))))/(c^2*E^(((5*I)*a)/b)*Sqrt[d - c^2*d*x^2]*((a + b*ArcSin[c*x])^2/b^2)^(3*n))","A",1
489,1,326,466,1.9884498,"\int \left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^n \, dx","Integrate[(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n,x]","\frac{d^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(i 4^{-n} e^{-\frac{4 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(e^{\frac{8 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)-\frac{8 \left(a+b \sin ^{-1}(c x)\right)}{b n+b}+8 \left(\frac{4 a+4 b \sin ^{-1}(c x)}{b n+b}-i 2^{-n} e^{-\frac{2 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+i 2^{-n} e^{\frac{2 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)\right)}{64 c \sqrt{d-c^2 d x^2}}","\frac{3 d \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^{n+1}}{8 b c (n+1) \sqrt{1-c^2 x^2}}-\frac{i d 2^{-n-3} e^{-\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c \sqrt{1-c^2 x^2}}-\frac{i d 2^{-2 (n+3)} e^{-\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c \sqrt{1-c^2 x^2}}+\frac{i d 2^{-n-3} e^{\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c \sqrt{1-c^2 x^2}}+\frac{i d 2^{-2 (n+3)} e^{\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c \sqrt{1-c^2 x^2}}",1,"(d^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*((-8*(a + b*ArcSin[c*x]))/(b + b*n) + 8*((4*a + 4*b*ArcSin[c*x])/(b + b*n) - (I*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b])/(2^n*E^(((2*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + (I*E^(((2*I)*a)/b)*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c*x]))/b])/(2^n*((I*(a + b*ArcSin[c*x]))/b)^n)) + (I*(-(((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b]) + E^(((8*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b]))/(4^n*E^(((4*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^n)))/(64*c*Sqrt[d - c^2*d*x^2])","A",1
490,0,0,427,0.2443034,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^n}{x} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n)/x,x]","\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^n}{x} \, dx","d^2 \text{Int}\left(\frac{\left(a+b \sin ^{-1}(c x)\right)^n}{x \sqrt{d-c^2 d x^2}},x\right)+\frac{5 d^2 e^{-\frac{i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 \sqrt{d-c^2 d x^2}}+\frac{d^2 3^{-n-1} e^{-\frac{3 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 \sqrt{d-c^2 d x^2}}+\frac{5 d^2 e^{\frac{i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 \sqrt{d-c^2 d x^2}}+\frac{d^2 3^{-n-1} e^{\frac{3 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 \sqrt{d-c^2 d x^2}}",0,"Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n)/x, x]","A",-1
491,0,0,298,0.8163443,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^n}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n)/x^2,x]","\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^n}{x^2} \, dx","d^2 \text{Int}\left(\frac{\left(a+b \sin ^{-1}(c x)\right)^n}{x^2 \sqrt{d-c^2 d x^2}},x\right)-\frac{3 c d^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^{n+1}}{2 b (n+1) \sqrt{d-c^2 d x^2}}+\frac{i c d^2 2^{-n-3} e^{-\frac{2 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{\sqrt{d-c^2 d x^2}}-\frac{i c d^2 2^{-n-3} e^{\frac{2 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{\sqrt{d-c^2 d x^2}}",0,"Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^n)/x^2, x]","A",-1
492,1,989,906,4.4072375,"\int x^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^n \, dx","Integrate[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n,x]","\frac{2^{-3 n-11} 3^{-n-1} d^3 e^{-\frac{8 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(i 3^{n+1} 4^{n+2} b e^{\frac{10 i a}{b}} (n+1) \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n-i 2^{n+3} 3^{n+1} b e^{\frac{12 i a}{b}} \Gamma \left(n+1,\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n-i 2^{n+3} 3^{n+1} b e^{\frac{12 i a}{b}} n \Gamma \left(n+1,\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n-i 4^{n+2} b e^{\frac{14 i a}{b}} \Gamma \left(n+1,\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n-i 4^{n+2} b e^{\frac{14 i a}{b}} n \Gamma \left(n+1,\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n-i 3^{n+1} b e^{\frac{16 i a}{b}} \Gamma \left(n+1,\frac{8 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n-i 3^{n+1} b e^{\frac{16 i a}{b}} n \Gamma \left(n+1,\frac{8 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n+5\ 2^{3 n+4} 3^{n+1} a e^{\frac{8 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^n+5\ 2^{3 n+4} 3^{n+1} b e^{\frac{8 i a}{b}} \sin ^{-1}(c x) \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^n-i 3^{n+1} 4^{n+2} b e^{\frac{6 i a}{b}} (n+1) \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+i 2^{n+3} 3^{n+1} b e^{\frac{4 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+i 2^{n+3} 3^{n+1} b e^{\frac{4 i a}{b}} n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+i 4^{n+2} b e^{\frac{2 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+i 4^{n+2} b e^{\frac{2 i a}{b}} n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+i 3^{n+1} b \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{8 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+i 3^{n+1} b n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{8 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)}{b c^3 (n+1) \sqrt{d-c^2 d x^2}}","-\frac{i 2^{-n-7} d^2 e^{-\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n}}{c^3 \sqrt{1-c^2 x^2}}+\frac{i 2^{-2 (n+4)} d^2 e^{-\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n}}{c^3 \sqrt{1-c^2 x^2}}+\frac{i 2^{-n-7} 3^{-n-1} d^2 e^{-\frac{6 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n}}{c^3 \sqrt{1-c^2 x^2}}+\frac{i 2^{-3 n-11} d^2 e^{-\frac{8 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{8 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n}}{c^3 \sqrt{1-c^2 x^2}}+\frac{5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^{n+1}}{128 b c^3 (n+1) \sqrt{1-c^2 x^2}}+\frac{i 2^{-n-7} d^2 e^{\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{1-c^2 x^2}}-\frac{i 2^{-2 (n+4)} d^2 e^{\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{1-c^2 x^2}}-\frac{i 2^{-n-7} 3^{-n-1} d^2 e^{\frac{6 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{1-c^2 x^2}}-\frac{i 2^{-3 n-11} d^2 e^{\frac{8 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{8 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{1-c^2 x^2}}",1,"(2^(-11 - 3*n)*3^(-1 - n)*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*(5*2^(4 + 3*n)*3^(1 + n)*a*E^(((8*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^n + 5*2^(4 + 3*n)*3^(1 + n)*b*E^(((8*I)*a)/b)*ArcSin[c*x]*((a + b*ArcSin[c*x])^2/b^2)^n - I*3^(1 + n)*4^(2 + n)*b*E^(((6*I)*a)/b)*(1 + n)*((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b] + I*3^(1 + n)*4^(2 + n)*b*E^(((10*I)*a)/b)*(1 + n)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c*x]))/b] + I*2^(3 + n)*3^(1 + n)*b*E^(((4*I)*a)/b)*((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b] + I*2^(3 + n)*3^(1 + n)*b*E^(((4*I)*a)/b)*n*((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b] - I*2^(3 + n)*3^(1 + n)*b*E^(((12*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b] - I*2^(3 + n)*3^(1 + n)*b*E^(((12*I)*a)/b)*n*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b] + I*4^(2 + n)*b*E^(((2*I)*a)/b)*((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-6*I)*(a + b*ArcSin[c*x]))/b] + I*4^(2 + n)*b*E^(((2*I)*a)/b)*n*((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-6*I)*(a + b*ArcSin[c*x]))/b] - I*4^(2 + n)*b*E^(((14*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((6*I)*(a + b*ArcSin[c*x]))/b] - I*4^(2 + n)*b*E^(((14*I)*a)/b)*n*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((6*I)*(a + b*ArcSin[c*x]))/b] + I*3^(1 + n)*b*((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-8*I)*(a + b*ArcSin[c*x]))/b] + I*3^(1 + n)*b*n*((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-8*I)*(a + b*ArcSin[c*x]))/b] - I*3^(1 + n)*b*E^(((16*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((8*I)*(a + b*ArcSin[c*x]))/b] - I*3^(1 + n)*b*E^(((16*I)*a)/b)*n*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((8*I)*(a + b*ArcSin[c*x]))/b]))/(b*c^3*E^(((8*I)*a)/b)*(1 + n)*Sqrt[d - c^2*d*x^2]*((a + b*ArcSin[c*x])^2/b^2)^n)","A",1
493,1,603,815,4.3013759,"\int x \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^n \, dx","Integrate[x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n,x]","-\frac{d^3 5^{-n} 21^{-n-1} e^{-\frac{7 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{-3 n} \left(\left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \left(9\ 5^n 7^{n+1} e^{\frac{4 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{2 n} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+105^{n+1} e^{\frac{8 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+9\ 5^n 7^{n+1} e^{\frac{10 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+3^{n+1} \left(7^{n+1} e^{\frac{12 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+5^n \left(e^{\frac{14 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,\frac{7 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+\left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{3 n} \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{7 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)+7^{n+1} e^{\frac{2 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{3 n} \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,-\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)\right)+105^{n+1} e^{\frac{6 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)}{128 c^2 \sqrt{d-c^2 d x^2}}","-\frac{5 d^2 e^{-\frac{i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n}}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{3^{1-n} d^2 e^{-\frac{3 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n}}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{5^{-n} d^2 e^{-\frac{5 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n}}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{7^{-n-1} d^2 e^{-\frac{7 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{7 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n}}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{5 d^2 e^{\frac{i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{3^{1-n} d^2 e^{\frac{3 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{5^{-n} d^2 e^{\frac{5 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{1-c^2 x^2}}-\frac{7^{-n-1} d^2 e^{\frac{7 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{7 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{1-c^2 x^2}}",1,"-1/128*(21^(-1 - n)*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*(105^(1 + n)*E^(((6*I)*a)/b)*((I*(a + b*ArcSin[c*x]))/b)^n*((a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, ((-I)*(a + b*ArcSin[c*x]))/b] + (((-I)*(a + b*ArcSin[c*x]))/b)^n*(105^(1 + n)*E^(((8*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, (I*(a + b*ArcSin[c*x]))/b] + 9*5^n*7^(1 + n)*E^(((4*I)*a)/b)*((I*(a + b*ArcSin[c*x]))/b)^(2*n)*((a + b*ArcSin[c*x])^2/b^2)^n*Gamma[1 + n, ((-3*I)*(a + b*ArcSin[c*x]))/b] + 9*5^n*7^(1 + n)*E^(((10*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, ((3*I)*(a + b*ArcSin[c*x]))/b] + 3^(1 + n)*(7^(1 + n)*E^(((2*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*((I*(a + b*ArcSin[c*x]))/b)^(3*n)*Gamma[1 + n, ((-5*I)*(a + b*ArcSin[c*x]))/b] + 7^(1 + n)*E^(((12*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, ((5*I)*(a + b*ArcSin[c*x]))/b] + 5^n*((((-I)*(a + b*ArcSin[c*x]))/b)^n*((I*(a + b*ArcSin[c*x]))/b)^(3*n)*Gamma[1 + n, ((-7*I)*(a + b*ArcSin[c*x]))/b] + E^(((14*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^(2*n)*Gamma[1 + n, ((7*I)*(a + b*ArcSin[c*x]))/b])))))/(5^n*c^2*E^(((7*I)*a)/b)*Sqrt[d - c^2*d*x^2]*((a + b*ArcSin[c*x])^2/b^2)^(3*n))","A",1
494,1,477,698,4.7585418,"\int \left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^n \, dx","Integrate[(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n,x]","\frac{d^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(9 i 4^{-n} e^{\frac{4 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+i 6^{-n} e^{\frac{6 i a}{b}} \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \Gamma \left(n+1,\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-9 i 4^{-n} e^{-\frac{4 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \Gamma \left(n+1,-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-i 6^{-n} e^{-\frac{6 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^n \left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \Gamma \left(n+1,-\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-45 i 2^{-n} e^{-\frac{2 i a}{b}} \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+45 i 2^{-n} e^{\frac{2 i a}{b}} \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+\frac{120 a}{b n+b}+\frac{120 \sin ^{-1}(c x)}{n+1}\right)}{384 c \sqrt{d-c^2 d x^2}}","\frac{5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^{n+1}}{16 b c (n+1) \sqrt{1-c^2 x^2}}-\frac{15 i d^2 2^{-n-7} e^{-\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c \sqrt{1-c^2 x^2}}-\frac{3 i d^2 2^{-2 n-7} e^{-\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c \sqrt{1-c^2 x^2}}-\frac{i d^2 2^{-n-7} 3^{-n-1} e^{-\frac{6 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c \sqrt{1-c^2 x^2}}+\frac{15 i d^2 2^{-n-7} e^{\frac{2 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c \sqrt{1-c^2 x^2}}+\frac{3 i d^2 2^{-2 n-7} e^{\frac{4 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c \sqrt{1-c^2 x^2}}+\frac{i d^2 2^{-n-7} 3^{-n-1} e^{\frac{6 i a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{6 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{c \sqrt{1-c^2 x^2}}",1,"(d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^n*((120*a)/(b + b*n) + (120*ArcSin[c*x])/(1 + n) - ((45*I)*Gamma[1 + n, ((-2*I)*(a + b*ArcSin[c*x]))/b])/(2^n*E^(((2*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n) + ((45*I)*E^(((2*I)*a)/b)*Gamma[1 + n, ((2*I)*(a + b*ArcSin[c*x]))/b])/(2^n*((I*(a + b*ArcSin[c*x]))/b)^n) - ((9*I)*((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-4*I)*(a + b*ArcSin[c*x]))/b])/(4^n*E^(((4*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^n) + ((9*I)*E^(((4*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((4*I)*(a + b*ArcSin[c*x]))/b])/(4^n*((a + b*ArcSin[c*x])^2/b^2)^n) - (I*((I*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((-6*I)*(a + b*ArcSin[c*x]))/b])/(6^n*E^(((6*I)*a)/b)*((a + b*ArcSin[c*x])^2/b^2)^n) + (I*E^(((6*I)*a)/b)*(((-I)*(a + b*ArcSin[c*x]))/b)^n*Gamma[1 + n, ((6*I)*(a + b*ArcSin[c*x]))/b])/(6^n*((a + b*ArcSin[c*x])^2/b^2)^n)))/(384*c*Sqrt[d - c^2*d*x^2])","A",1
495,0,0,827,0.2486438,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^n}{x} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n)/x,x]","\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^n}{x} \, dx","\frac{11 d^3 e^{-\frac{i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n}}{16 \sqrt{d-c^2 d x^2}}-\frac{5\ 3^{-n-1} d^3 e^{-\frac{3 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n}}{32 \sqrt{d-c^2 d x^2}}+\frac{3^{-n} d^3 e^{-\frac{3 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n}}{8 \sqrt{d-c^2 d x^2}}+\frac{5^{-n-1} d^3 e^{-\frac{5 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right) \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n}}{32 \sqrt{d-c^2 d x^2}}+\frac{11 d^3 e^{\frac{i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 \sqrt{d-c^2 d x^2}}-\frac{5\ 3^{-n-1} d^3 e^{\frac{3 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 \sqrt{d-c^2 d x^2}}+\frac{3^{-n} d^3 e^{\frac{3 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{8 \sqrt{d-c^2 d x^2}}+\frac{5^{-n-1} d^3 e^{\frac{5 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{32 \sqrt{d-c^2 d x^2}}+d^3 \text{Int}\left(\frac{\left(a+b \sin ^{-1}(c x)\right)^n}{x \sqrt{d-c^2 d x^2}},x\right)",0,"Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n)/x, x]","A",-1
496,0,0,502,0.7983448,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^n}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n)/x^2,x]","\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^n}{x^2} \, dx","d^3 \text{Int}\left(\frac{\left(a+b \sin ^{-1}(c x)\right)^n}{x^2 \sqrt{d-c^2 d x^2}},x\right)-\frac{15 c d^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^{n+1}}{8 b (n+1) \sqrt{d-c^2 d x^2}}+\frac{i c d^3 2^{-n-2} e^{-\frac{2 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{\sqrt{d-c^2 d x^2}}+\frac{i c d^3 2^{-2 (n+3)} e^{-\frac{4 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{\sqrt{d-c^2 d x^2}}-\frac{i c d^3 2^{-n-2} e^{\frac{2 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{\sqrt{d-c^2 d x^2}}-\frac{i c d^3 2^{-2 (n+3)} e^{\frac{4 i a}{b}} \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^n \left(\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{\sqrt{d-c^2 d x^2}}",0,"Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^n)/x^2, x]","A",-1
497,0,0,27,0.5287428,"\int \frac{x^m \sin ^{-1}(a x)^n}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^m*ArcSin[a*x]^n)/Sqrt[1 - a^2*x^2],x]","\int \frac{x^m \sin ^{-1}(a x)^n}{\sqrt{1-a^2 x^2}} \, dx","\text{Int}\left(\frac{x^m \sin ^{-1}(a x)^n}{\sqrt{1-a^2 x^2}},x\right)",0,"Integrate[(x^m*ArcSin[a*x]^n)/Sqrt[1 - a^2*x^2], x]","A",-1
498,1,153,163,0.3423502,"\int \frac{x^3 \sin ^{-1}(a x)^n}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^3*ArcSin[a*x]^n)/Sqrt[1 - a^2*x^2],x]","-\frac{3^{-n-1} \sin ^{-1}(a x)^n \left(\sin ^{-1}(a x)^2\right)^{-2 n} \left(\left(-i \sin ^{-1}(a x)\right)^n \left(3^{n+2} \left(\sin ^{-1}(a x)^2\right)^n \Gamma \left(n+1,i \sin ^{-1}(a x)\right)-\left(\sin ^{-1}(a x)^2\right)^n \Gamma \left(n+1,3 i \sin ^{-1}(a x)\right)-\left(i \sin ^{-1}(a x)\right)^{2 n} \Gamma \left(n+1,-3 i \sin ^{-1}(a x)\right)\right)+3^{n+2} \left(i \sin ^{-1}(a x)\right)^n \left(\sin ^{-1}(a x)^2\right)^n \Gamma \left(n+1,-i \sin ^{-1}(a x)\right)\right)}{8 a^4}","-\frac{3 \sin ^{-1}(a x)^n \left(-i \sin ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-i \sin ^{-1}(a x)\right)}{8 a^4}+\frac{3^{-n-1} \sin ^{-1}(a x)^n \left(-i \sin ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-3 i \sin ^{-1}(a x)\right)}{8 a^4}-\frac{3 \left(i \sin ^{-1}(a x)\right)^{-n} \sin ^{-1}(a x)^n \Gamma \left(n+1,i \sin ^{-1}(a x)\right)}{8 a^4}+\frac{3^{-n-1} \left(i \sin ^{-1}(a x)\right)^{-n} \sin ^{-1}(a x)^n \Gamma \left(n+1,3 i \sin ^{-1}(a x)\right)}{8 a^4}",1,"-1/8*(3^(-1 - n)*ArcSin[a*x]^n*(3^(2 + n)*(I*ArcSin[a*x])^n*(ArcSin[a*x]^2)^n*Gamma[1 + n, (-I)*ArcSin[a*x]] + ((-I)*ArcSin[a*x])^n*(3^(2 + n)*(ArcSin[a*x]^2)^n*Gamma[1 + n, I*ArcSin[a*x]] - (I*ArcSin[a*x])^(2*n)*Gamma[1 + n, (-3*I)*ArcSin[a*x]] - (ArcSin[a*x]^2)^n*Gamma[1 + n, (3*I)*ArcSin[a*x]])))/(a^4*(ArcSin[a*x]^2)^(2*n))","A",1
499,1,109,109,0.2719766,"\int \frac{x^2 \sin ^{-1}(a x)^n}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^2*ArcSin[a*x]^n)/Sqrt[1 - a^2*x^2],x]","\frac{2^{-n-3} \sin ^{-1}(a x)^n \left(\sin ^{-1}(a x)^2\right)^{-n} \left(2^{n+2} \sin ^{-1}(a x) \left(\sin ^{-1}(a x)^2\right)^n-i (n+1) \left(-i \sin ^{-1}(a x)\right)^n \Gamma \left(n+1,2 i \sin ^{-1}(a x)\right)+i (n+1) \left(i \sin ^{-1}(a x)\right)^n \Gamma \left(n+1,-2 i \sin ^{-1}(a x)\right)\right)}{a^3 (n+1)}","\frac{\sin ^{-1}(a x)^{n+1}}{2 a^3 (n+1)}+\frac{i 2^{-n-3} \sin ^{-1}(a x)^n \left(-i \sin ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-2 i \sin ^{-1}(a x)\right)}{a^3}-\frac{i 2^{-n-3} \left(i \sin ^{-1}(a x)\right)^{-n} \sin ^{-1}(a x)^n \Gamma \left(n+1,2 i \sin ^{-1}(a x)\right)}{a^3}",1,"(2^(-3 - n)*ArcSin[a*x]^n*(2^(2 + n)*ArcSin[a*x]*(ArcSin[a*x]^2)^n + I*(1 + n)*(I*ArcSin[a*x])^n*Gamma[1 + n, (-2*I)*ArcSin[a*x]] - I*(1 + n)*((-I)*ArcSin[a*x])^n*Gamma[1 + n, (2*I)*ArcSin[a*x]]))/(a^3*(1 + n)*(ArcSin[a*x]^2)^n)","A",1
500,1,70,75,0.0850183,"\int \frac{x \sin ^{-1}(a x)^n}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x*ArcSin[a*x]^n)/Sqrt[1 - a^2*x^2],x]","-\frac{\sin ^{-1}(a x)^n \left(\sin ^{-1}(a x)^2\right)^{-n} \left(\left(-i \sin ^{-1}(a x)\right)^n \Gamma \left(n+1,i \sin ^{-1}(a x)\right)+\left(i \sin ^{-1}(a x)\right)^n \Gamma \left(n+1,-i \sin ^{-1}(a x)\right)\right)}{2 a^2}","-\frac{\sin ^{-1}(a x)^n \left(-i \sin ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-i \sin ^{-1}(a x)\right)}{2 a^2}-\frac{\left(i \sin ^{-1}(a x)\right)^{-n} \sin ^{-1}(a x)^n \Gamma \left(n+1,i \sin ^{-1}(a x)\right)}{2 a^2}",1,"-1/2*(ArcSin[a*x]^n*((I*ArcSin[a*x])^n*Gamma[1 + n, (-I)*ArcSin[a*x]] + ((-I)*ArcSin[a*x])^n*Gamma[1 + n, I*ArcSin[a*x]]))/(a^2*(ArcSin[a*x]^2)^n)","A",1
501,1,17,17,0.007711,"\int \frac{\sin ^{-1}(a x)^n}{\sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]^n/Sqrt[1 - a^2*x^2],x]","\frac{\sin ^{-1}(a x)^{n+1}}{a (n+1)}","\frac{\sin ^{-1}(a x)^{n+1}}{a (n+1)}",1,"ArcSin[a*x]^(1 + n)/(a*(1 + n))","A",1
502,0,0,27,3.5612906,"\int \frac{\sin ^{-1}(a x)^n}{x \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]^n/(x*Sqrt[1 - a^2*x^2]),x]","\int \frac{\sin ^{-1}(a x)^n}{x \sqrt{1-a^2 x^2}} \, dx","\text{Int}\left(\frac{\sin ^{-1}(a x)^n}{x \sqrt{1-a^2 x^2}},x\right)",0,"Integrate[ArcSin[a*x]^n/(x*Sqrt[1 - a^2*x^2]), x]","A",-1
503,0,0,27,1.0364059,"\int \frac{\sin ^{-1}(a x)^n}{x^2 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcSin[a*x]^n/(x^2*Sqrt[1 - a^2*x^2]),x]","\int \frac{\sin ^{-1}(a x)^n}{x^2 \sqrt{1-a^2 x^2}} \, dx","\text{Int}\left(\frac{\sin ^{-1}(a x)^n}{x^2 \sqrt{1-a^2 x^2}},x\right)",0,"Integrate[ArcSin[a*x]^n/(x^2*Sqrt[1 - a^2*x^2]), x]","A",-1
504,1,293,376,1.3558249,"\int (d+c d x)^{5/2} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + c*d*x)^(5/2)*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]),x]","\frac{d^2 \sqrt{c d x+d} \sqrt{f-c f x} \left(48 a \sqrt{1-c^2 x^2} \left(6 c^3 x^3+16 c^2 x^2+9 c x-16\right)-256 b c x \left(c^2 x^2-3\right)+144 b \cos \left(2 \sin ^{-1}(c x)\right)-9 b \cos \left(4 \sin ^{-1}(c x)\right)\right)-720 a d^{5/2} \sqrt{f} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+12 b d^2 \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x) \left(-64 \left(1-c^2 x^2\right)^{3/2}+24 \sin \left(2 \sin ^{-1}(c x)\right)-3 \sin \left(4 \sin ^{-1}(c x)\right)\right)+360 b d^2 \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2}{1152 c \sqrt{1-c^2 x^2}}","\frac{1}{4} c^2 d^2 x^3 \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)+\frac{5 d^2 \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)^2}{16 b c \sqrt{1-c^2 x^2}}-\frac{2 d^2 \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)}{3 c}+\frac{3}{8} d^2 x \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)-\frac{3 b c d^2 x^2 \sqrt{c d x+d} \sqrt{f-c f x}}{16 \sqrt{1-c^2 x^2}}+\frac{2 b d^2 x \sqrt{c d x+d} \sqrt{f-c f x}}{3 \sqrt{1-c^2 x^2}}-\frac{2 b c^2 d^2 x^3 \sqrt{c d x+d} \sqrt{f-c f x}}{9 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^4 \sqrt{c d x+d} \sqrt{f-c f x}}{16 \sqrt{1-c^2 x^2}}",1,"(360*b*d^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2 - 720*a*d^(5/2)*Sqrt[f]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + d^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(-256*b*c*x*(-3 + c^2*x^2) + 48*a*Sqrt[1 - c^2*x^2]*(-16 + 9*c*x + 16*c^2*x^2 + 6*c^3*x^3) + 144*b*Cos[2*ArcSin[c*x]] - 9*b*Cos[4*ArcSin[c*x]]) + 12*b*d^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]*(-64*(1 - c^2*x^2)^(3/2) + 24*Sin[2*ArcSin[c*x]] - 3*Sin[4*ArcSin[c*x]]))/(1152*c*Sqrt[1 - c^2*x^2])","A",1
505,1,260,273,1.0621504,"\int (d+c d x)^{3/2} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + c*d*x)^(3/2)*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]),x]","\frac{d \sqrt{c d x+d} \sqrt{f-c f x} \left(12 a \sqrt{1-c^2 x^2} \left(2 c^2 x^2+3 c x-2\right)-8 b c x \left(c^2 x^2-3\right)+9 b \cos \left(2 \sin ^{-1}(c x)\right)\right)-36 a d^{3/2} \sqrt{f} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+6 b d \sqrt{c d x+d} \sqrt{f-c f x} \left(3 \sin \left(2 \sin ^{-1}(c x)\right)-4 \left(1-c^2 x^2\right)^{3/2}\right) \sin ^{-1}(c x)+18 b d \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2}{72 c \sqrt{1-c^2 x^2}}","\frac{d \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)^2}{4 b c \sqrt{1-c^2 x^2}}-\frac{d \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)}{3 c}+\frac{1}{2} d x \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)-\frac{b c d x^2 \sqrt{c d x+d} \sqrt{f-c f x}}{4 \sqrt{1-c^2 x^2}}+\frac{b d x \sqrt{c d x+d} \sqrt{f-c f x}}{3 \sqrt{1-c^2 x^2}}-\frac{b c^2 d x^3 \sqrt{c d x+d} \sqrt{f-c f x}}{9 \sqrt{1-c^2 x^2}}",1,"(18*b*d*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2 - 36*a*d^(3/2)*Sqrt[f]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + d*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(-8*b*c*x*(-3 + c^2*x^2) + 12*a*Sqrt[1 - c^2*x^2]*(-2 + 3*c*x + 2*c^2*x^2) + 9*b*Cos[2*ArcSin[c*x]]) + 6*b*d*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]*(-4*(1 - c^2*x^2)^(3/2) + 3*Sin[2*ArcSin[c*x]]))/(72*c*Sqrt[1 - c^2*x^2])","A",1
506,1,207,134,0.6544541,"\int \sqrt{d+c d x} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]),x]","\frac{1}{2} a x \sqrt{d (c x+1)} \sqrt{-f (c x-1)}-\frac{a \sqrt{d} \sqrt{f} \tan ^{-1}\left(\frac{c x \sqrt{d (c x+1)} \sqrt{-f (c x-1)}}{\sqrt{d} \sqrt{f} (c x-1) (c x+1)}\right)}{2 c}+\frac{b \sqrt{c d x+d} \sqrt{f-c f x} \sqrt{-d f \left(1-c^2 x^2\right)} \left(2 \sin ^{-1}(c x) \left(\sin ^{-1}(c x)+\sin \left(2 \sin ^{-1}(c x)\right)\right)+\cos \left(2 \sin ^{-1}(c x)\right)\right)}{8 c \sqrt{1-c^2 x^2} \sqrt{(-c d x-d) (f-c f x)}}","\frac{\sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)^2}{4 b c \sqrt{1-c^2 x^2}}+\frac{1}{2} x \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)-\frac{b c x^2 \sqrt{c d x+d} \sqrt{f-c f x}}{4 \sqrt{1-c^2 x^2}}",1,"(a*x*Sqrt[-(f*(-1 + c*x))]*Sqrt[d*(1 + c*x)])/2 - (a*Sqrt[d]*Sqrt[f]*ArcTan[(c*x*Sqrt[-(f*(-1 + c*x))]*Sqrt[d*(1 + c*x)])/(Sqrt[d]*Sqrt[f]*(-1 + c*x)*(1 + c*x))])/(2*c) + (b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*Sqrt[-(d*f*(1 - c^2*x^2))]*(Cos[2*ArcSin[c*x]] + 2*ArcSin[c*x]*(ArcSin[c*x] + Sin[2*ArcSin[c*x]])))/(8*c*Sqrt[(-d - c*d*x)*(f - c*f*x)]*Sqrt[1 - c^2*x^2])","A",1
507,1,200,141,0.8336052,"\int \frac{\sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d+c d x}} \, dx","Integrate[(Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]))/Sqrt[d + c*d*x],x]","\frac{\frac{2 \sqrt{c d x+d} \sqrt{f-c f x} \left(a \sqrt{1-c^2 x^2}-b c x\right)}{\sqrt{1-c^2 x^2}}-2 a \sqrt{d} \sqrt{f} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+\frac{b \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}+2 b \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)}{2 c d}","\frac{f \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 b c \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{f \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{c \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{b f x \sqrt{1-c^2 x^2}}{\sqrt{c d x+d} \sqrt{f-c f x}}",1,"((2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(-(b*c*x) + a*Sqrt[1 - c^2*x^2]))/Sqrt[1 - c^2*x^2] + 2*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x] + (b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] - 2*a*Sqrt[d]*Sqrt[f]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))])/(2*c*d)","A",1
508,1,248,162,1.4562422,"\int \frac{\sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)}{(d+c d x)^{3/2}} \, dx","Integrate[(Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]))/(d + c*d*x)^(3/2),x]","-\frac{-2 a \sqrt{d} \sqrt{f} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+\frac{4 a \sqrt{c d x+d} \sqrt{f-c f x}}{c x+1}+\frac{b \sqrt{c d x+d} \sqrt{f-c f x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+4\right)-8 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\left(\sin ^{-1}(c x)-4\right) \sin ^{-1}(c x)-8 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{\sqrt{1-c^2 x^2} \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}}{2 c d^2}","-\frac{f^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 b c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{2 f^2 (1-c x) \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{2 b f^2 \left(1-c^2 x^2\right)^{3/2} \log (c x+1)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}",1,"-1/2*((4*a*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(1 + c*x) - 2*a*Sqrt[d]*Sqrt[f]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + (b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[ArcSin[c*x]/2]*(ArcSin[c*x]*(4 + ArcSin[c*x]) - 8*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + ((-4 + ArcSin[c*x])*ArcSin[c*x] - 8*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(c*d^2)","A",1
509,1,114,163,0.5492164,"\int \frac{\sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)}{(d+c d x)^{5/2}} \, dx","Integrate[(Sqrt[f - c*f*x]*(a + b*ArcSin[c*x]))/(d + c*d*x)^(5/2),x]","-\frac{f \sqrt{c d x+d} \left((c x-1) \left(a c x-a-b \sqrt{1-c^2 x^2}\right)+b (c x+1) \sqrt{1-c^2 x^2} \log (-f (c x+1))+b (c x-1)^2 \sin ^{-1}(c x)\right)}{3 c d^3 (c x+1)^2 \sqrt{f-c f x}}","-\frac{f^3 (1-c x)^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{2 b f^3 \left(1-c^2 x^2\right)^{5/2}}{3 c (c x+1) (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b f^3 \left(1-c^2 x^2\right)^{5/2} \log (c x+1)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}",1,"-1/3*(f*Sqrt[d + c*d*x]*((-1 + c*x)*(-a + a*c*x - b*Sqrt[1 - c^2*x^2]) + b*(-1 + c*x)^2*ArcSin[c*x] + b*(1 + c*x)*Sqrt[1 - c^2*x^2]*Log[-(f*(1 + c*x))]))/(c*d^3*(1 + c*x)^2*Sqrt[f - c*f*x])","A",1
510,1,305,414,1.7388457,"\int (d+c d x)^{5/2} (f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + c*d*x)^(5/2)*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]),x]","\frac{d^2 f \left(\sqrt{c d x+d} \sqrt{f-c f x} \left(-240 a \sqrt{1-c^2 x^2} \left(8 c^4 x^4+10 c^3 x^3-16 c^2 x^2-25 c x+8\right)+128 b c x \left(3 c^4 x^4-10 c^2 x^2+15\right)+1200 b \cos \left(2 \sin ^{-1}(c x)\right)+75 b \cos \left(4 \sin ^{-1}(c x)\right)\right)-3600 a \sqrt{d} \sqrt{f} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)-60 b \sqrt{c d x+d} \sqrt{f-c f x} \left(32 \left(1-c^2 x^2\right)^{5/2}-40 \sin \left(2 \sin ^{-1}(c x)\right)-5 \sin \left(4 \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)+1800 b \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2\right)}{9600 c \sqrt{1-c^2 x^2}}","\frac{3 d x (c d x+d)^{3/2} (f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{8 \left(1-c^2 x^2\right)}+\frac{3 d (c d x+d)^{3/2} (f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{16 b c \left(1-c^2 x^2\right)^{3/2}}-\frac{d \left(1-c^2 x^2\right) (c d x+d)^{3/2} (f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{5 c}+\frac{1}{4} d x (c d x+d)^{3/2} (f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{5 b c d x^2 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left(1-c^2 x^2\right)^{3/2}}+\frac{b d x (c d x+d)^{3/2} (f-c f x)^{3/2}}{5 \left(1-c^2 x^2\right)^{3/2}}-\frac{2 b c^2 d x^3 (c d x+d)^{3/2} (f-c f x)^{3/2}}{15 \left(1-c^2 x^2\right)^{3/2}}+\frac{b c^4 d x^5 (c d x+d)^{3/2} (f-c f x)^{3/2}}{25 \left(1-c^2 x^2\right)^{3/2}}+\frac{b c^3 d x^4 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left(1-c^2 x^2\right)^{3/2}}",1,"(d^2*f*(1800*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2 - 3600*a*Sqrt[d]*Sqrt[f]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(128*b*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) - 240*a*Sqrt[1 - c^2*x^2]*(8 - 25*c*x - 16*c^2*x^2 + 10*c^3*x^3 + 8*c^4*x^4) + 1200*b*Cos[2*ArcSin[c*x]] + 75*b*Cos[4*ArcSin[c*x]]) - 60*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]*(32*(1 - c^2*x^2)^(5/2) - 40*Sin[2*ArcSin[c*x]] - 5*Sin[4*ArcSin[c*x]])))/(9600*c*Sqrt[1 - c^2*x^2])","A",1
511,1,247,226,1.1264218,"\int (d+c d x)^{3/2} (f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]),x]","\frac{d f \sqrt{c d x+d} \sqrt{f-c f x} \left(16 a c x \sqrt{1-c^2 x^2} \left(5-2 c^2 x^2\right)+16 b \cos \left(2 \sin ^{-1}(c x)\right)+b \cos \left(4 \sin ^{-1}(c x)\right)\right)-48 a d^{3/2} f^{3/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+24 b d f \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2+4 b d f \sqrt{c d x+d} \sqrt{f-c f x} \left(8 \sin \left(2 \sin ^{-1}(c x)\right)+\sin \left(4 \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)}{128 c \sqrt{1-c^2 x^2}}","\frac{3 x (c d x+d)^{3/2} (f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{8 \left(1-c^2 x^2\right)}+\frac{3 (c d x+d)^{3/2} (f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{16 b c \left(1-c^2 x^2\right)^{3/2}}+\frac{1}{4} x (c d x+d)^{3/2} (f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{5 b c x^2 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left(1-c^2 x^2\right)^{3/2}}+\frac{b c^3 x^4 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left(1-c^2 x^2\right)^{3/2}}",1,"(24*b*d*f*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2 - 48*a*d^(3/2)*f^(3/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + d*f*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(16*a*c*x*(5 - 2*c^2*x^2)*Sqrt[1 - c^2*x^2] + 16*b*Cos[2*ArcSin[c*x]] + b*Cos[4*ArcSin[c*x]]) + 4*b*d*f*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]*(8*Sin[2*ArcSin[c*x]] + Sin[4*ArcSin[c*x]]))/(128*c*Sqrt[1 - c^2*x^2])","A",1
512,1,260,273,1.0760998,"\int \sqrt{d+c d x} (f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + c*d*x]*(f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]),x]","\frac{f \sqrt{c d x+d} \sqrt{f-c f x} \left(12 a \sqrt{1-c^2 x^2} \left(-2 c^2 x^2+3 c x+2\right)+8 b c x \left(c^2 x^2-3\right)+9 b \cos \left(2 \sin ^{-1}(c x)\right)\right)-36 a \sqrt{d} f^{3/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+6 b f \sqrt{c d x+d} \sqrt{f-c f x} \left(4 \left(1-c^2 x^2\right)^{3/2}+3 \sin \left(2 \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)+18 b f \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2}{72 c \sqrt{1-c^2 x^2}}","\frac{f \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)^2}{4 b c \sqrt{1-c^2 x^2}}+\frac{f \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)}{3 c}+\frac{1}{2} f x \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)-\frac{b c f x^2 \sqrt{c d x+d} \sqrt{f-c f x}}{4 \sqrt{1-c^2 x^2}}-\frac{b f x \sqrt{c d x+d} \sqrt{f-c f x}}{3 \sqrt{1-c^2 x^2}}+\frac{b c^2 f x^3 \sqrt{c d x+d} \sqrt{f-c f x}}{9 \sqrt{1-c^2 x^2}}",1,"(18*b*f*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2 - 36*a*Sqrt[d]*f^(3/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + f*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(12*a*(2 + 3*c*x - 2*c^2*x^2)*Sqrt[1 - c^2*x^2] + 8*b*c*x*(-3 + c^2*x^2) + 9*b*Cos[2*ArcSin[c*x]]) + 6*b*f*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]*(4*(1 - c^2*x^2)^(3/2) + 3*Sin[2*ArcSin[c*x]]))/(72*c*Sqrt[1 - c^2*x^2])","A",1
513,1,238,242,1.3161126,"\int \frac{(f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d+c d x}} \, dx","Integrate[((f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]))/Sqrt[d + c*d*x],x]","\frac{-f \sqrt{c d x+d} \sqrt{f-c f x} \left(4 a (c x-4) \sqrt{1-c^2 x^2}+16 b c x+b \cos \left(2 \sin ^{-1}(c x)\right)\right)-12 a \sqrt{d} f^{3/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)-4 b f (c x-4) \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)+6 b f \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2}{8 c d \sqrt{1-c^2 x^2}}","\frac{3 f^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{4 b c \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{f^2 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{2 f^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{c \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{b c f^2 x^2 \sqrt{1-c^2 x^2}}{4 \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{2 b f^2 x \sqrt{1-c^2 x^2}}{\sqrt{c d x+d} \sqrt{f-c f x}}",1,"(-4*b*f*(-4 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 6*b*f*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2 - 12*a*Sqrt[d]*f^(3/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] - f*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(16*b*c*x + 4*a*(-4 + c*x)*Sqrt[1 - c^2*x^2] + b*Cos[2*ArcSin[c*x]]))/(8*c*d*Sqrt[1 - c^2*x^2])","A",1
514,1,291,252,3.5532649,"\int \frac{(f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{(d+c d x)^{3/2}} \, dx","Integrate[((f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]))/(d + c*d*x)^(3/2),x]","\frac{f \left(6 a \sqrt{d} \sqrt{f} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)-\frac{\sqrt{c d x+d} \sqrt{f-c f x} \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(2 \left(a (c x+5) \left(\sqrt{1-c^2 x^2}+c x-1\right)+b c x \left(\sqrt{1-c^2 x^2}-c x-1\right)+8 b \left(\sqrt{1-c^2 x^2}-c x-1\right) \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-3 b \left(\sqrt{1-c^2 x^2}-c x-1\right) \sin ^{-1}(c x)^2+2 b (c x+5) \left(\sqrt{1-c^2 x^2}+c x-1\right) \sin ^{-1}(c x)\right)}{2 \sqrt{1-c^2 x^2} \left(\cot \left(\frac{1}{2} \sin ^{-1}(c x)\right)+1\right)}\right)}{2 c d^2}","-\frac{3 f^3 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 b c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{f^3 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{4 f^3 (1-c x) \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{b f^3 x \left(1-c^2 x^2\right)^{3/2}}{(c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{4 b f^3 \left(1-c^2 x^2\right)^{3/2} \log (c x+1)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}",1,"(f*(6*a*Sqrt[d]*Sqrt[f]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] - (Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*Csc[ArcSin[c*x]/2]^2*(2*b*(5 + c*x)*(-1 + c*x + Sqrt[1 - c^2*x^2])*ArcSin[c*x] - 3*b*(-1 - c*x + Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + 2*(b*c*x*(-1 - c*x + Sqrt[1 - c^2*x^2]) + a*(5 + c*x)*(-1 + c*x + Sqrt[1 - c^2*x^2]) + 8*b*(-1 - c*x + Sqrt[1 - c^2*x^2])*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])))/(2*Sqrt[1 - c^2*x^2]*(1 + Cot[ArcSin[c*x]/2]))))/(2*c*d^2)","A",1
515,1,599,324,5.6092846,"\int \frac{(f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{(d+c d x)^{5/2}} \, dx","Integrate[((f - c*f*x)^(3/2)*(a + b*ArcSin[c*x]))/(d + c*d*x)^(5/2),x]","\frac{f \left(-12 a \sqrt{d} \sqrt{f} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+\frac{16 a (2 c x+1) \sqrt{c d x+d} \sqrt{f-c f x}}{(c x+1)^2}-\frac{b \sqrt{c d x+d} \sqrt{f-c f x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(3 \left(\sqrt{1-c^2 x^2}+2\right) \sin ^{-1}(c x)^2+2 \left(7 \sqrt{1-c^2 x^2}+2\right) \sin ^{-1}(c x)-28 \left(\sqrt{1-c^2 x^2}+2\right) \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-4\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(9 \sin ^{-1}(c x)^2+6 \sin ^{-1}(c x)-84 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-8\right)+\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(\left(14-3 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+28 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{(c x-1) \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4}-\frac{2 b \sqrt{c d x+d} \sqrt{f-c f x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\left(\sqrt{1-c^2 x^2}+2\right) \sin ^{-1}(c x)-2 \left(\sqrt{1-c^2 x^2}+2\right) \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-2\right)+\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x)+2 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(3 \sin ^{-1}(c x)+6 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+4\right)\right)}{(c x-1) \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4}\right)}{12 c d^3}","\frac{2 f^4 (1-c x) \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{2 f^4 (1-c x)^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{f^4 \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{4 b f^4 \left(1-c^2 x^2\right)^{5/2}}{3 c (c x+1) (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{8 b f^4 \left(1-c^2 x^2\right)^{5/2} \log (c x+1)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b f^4 \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x)^2}{2 c (c d x+d)^{5/2} (f-c f x)^{5/2}}",1,"(f*((16*a*(1 + 2*c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(1 + c*x)^2 - 12*a*Sqrt[d]*Sqrt[f]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] - (b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2]*(-8 + 6*ArcSin[c*x] + 9*ArcSin[c*x]^2 - 84*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + Cos[(3*ArcSin[c*x])/2]*((14 - 3*ArcSin[c*x])*ArcSin[c*x] + 28*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + 2*(-4 + 2*(2 + 7*Sqrt[1 - c^2*x^2])*ArcSin[c*x] + 3*(2 + Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 - 28*(2 + Sqrt[1 - c^2*x^2])*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/((-1 + c*x)*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^4) - (2*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[(3*ArcSin[c*x])/2]*(ArcSin[c*x] + 2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) - Cos[ArcSin[c*x]/2]*(4 + 3*ArcSin[c*x] + 6*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + 2*(-2 + (2 + Sqrt[1 - c^2*x^2])*ArcSin[c*x] - 2*(2 + Sqrt[1 - c^2*x^2])*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/((-1 + c*x)*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^4)))/(12*c*d^3)","A",0
516,1,303,315,1.7112783,"\int (d+c d x)^{5/2} (f-c f x)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x]),x]","\frac{d^2 f^2 \left(\sqrt{c d x+d} \sqrt{f-c f x} \left(1584 a c x \sqrt{1-c^2 x^2}+384 a c^5 x^5 \sqrt{1-c^2 x^2}-1248 a c^3 x^3 \sqrt{1-c^2 x^2}+270 b \cos \left(2 \sin ^{-1}(c x)\right)+27 b \cos \left(4 \sin ^{-1}(c x)\right)+2 b \cos \left(6 \sin ^{-1}(c x)\right)\right)-720 a \sqrt{d} \sqrt{f} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+360 b \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2+12 b \sqrt{c d x+d} \sqrt{f-c f x} \left(45 \sin \left(2 \sin ^{-1}(c x)\right)+9 \sin \left(4 \sin ^{-1}(c x)\right)+\sin \left(6 \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)\right)}{2304 c \sqrt{1-c^2 x^2}}","\frac{5 x (c d x+d)^{5/2} (f-c f x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{24 \left(1-c^2 x^2\right)}+\frac{5 x (c d x+d)^{5/2} (f-c f x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{16 \left(1-c^2 x^2\right)^2}+\frac{5 (c d x+d)^{5/2} (f-c f x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{32 b c \left(1-c^2 x^2\right)^{5/2}}+\frac{1}{6} x (c d x+d)^{5/2} (f-c f x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{25 b c x^2 (c d x+d)^{5/2} (f-c f x)^{5/2}}{96 \left(1-c^2 x^2\right)^{5/2}}+\frac{b \sqrt{1-c^2 x^2} (c d x+d)^{5/2} (f-c f x)^{5/2}}{36 c}+\frac{5 b c^3 x^4 (c d x+d)^{5/2} (f-c f x)^{5/2}}{96 \left(1-c^2 x^2\right)^{5/2}}",1,"(d^2*f^2*(360*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2 - 720*a*Sqrt[d]*Sqrt[f]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(1584*a*c*x*Sqrt[1 - c^2*x^2] - 1248*a*c^3*x^3*Sqrt[1 - c^2*x^2] + 384*a*c^5*x^5*Sqrt[1 - c^2*x^2] + 270*b*Cos[2*ArcSin[c*x]] + 27*b*Cos[4*ArcSin[c*x]] + 2*b*Cos[6*ArcSin[c*x]]) + 12*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]*(45*Sin[2*ArcSin[c*x]] + 9*Sin[4*ArcSin[c*x]] + Sin[6*ArcSin[c*x]])))/(2304*c*Sqrt[1 - c^2*x^2])","A",1
517,1,305,414,1.6474036,"\int (d+c d x)^{3/2} (f-c f x)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + c*d*x)^(3/2)*(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x]),x]","\frac{d f^2 \left(\sqrt{c d x+d} \sqrt{f-c f x} \left(240 a \sqrt{1-c^2 x^2} \left(8 c^4 x^4-10 c^3 x^3-16 c^2 x^2+25 c x+8\right)-128 b c x \left(3 c^4 x^4-10 c^2 x^2+15\right)+1200 b \cos \left(2 \sin ^{-1}(c x)\right)+75 b \cos \left(4 \sin ^{-1}(c x)\right)\right)-3600 a \sqrt{d} \sqrt{f} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+60 b \sqrt{c d x+d} \sqrt{f-c f x} \left(32 \left(1-c^2 x^2\right)^{5/2}+40 \sin \left(2 \sin ^{-1}(c x)\right)+5 \sin \left(4 \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)+1800 b \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2\right)}{9600 c \sqrt{1-c^2 x^2}}","\frac{3 f x (c d x+d)^{3/2} (f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{8 \left(1-c^2 x^2\right)}+\frac{3 f (c d x+d)^{3/2} (f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{16 b c \left(1-c^2 x^2\right)^{3/2}}+\frac{f \left(1-c^2 x^2\right) (c d x+d)^{3/2} (f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{5 c}+\frac{1}{4} f x (c d x+d)^{3/2} (f-c f x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)-\frac{5 b c f x^2 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left(1-c^2 x^2\right)^{3/2}}-\frac{b f x (c d x+d)^{3/2} (f-c f x)^{3/2}}{5 \left(1-c^2 x^2\right)^{3/2}}+\frac{2 b c^2 f x^3 (c d x+d)^{3/2} (f-c f x)^{3/2}}{15 \left(1-c^2 x^2\right)^{3/2}}-\frac{b c^4 f x^5 (c d x+d)^{3/2} (f-c f x)^{3/2}}{25 \left(1-c^2 x^2\right)^{3/2}}+\frac{b c^3 f x^4 (c d x+d)^{3/2} (f-c f x)^{3/2}}{16 \left(1-c^2 x^2\right)^{3/2}}",1,"(d*f^2*(1800*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2 - 3600*a*Sqrt[d]*Sqrt[f]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(-128*b*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 240*a*Sqrt[1 - c^2*x^2]*(8 + 25*c*x - 16*c^2*x^2 - 10*c^3*x^3 + 8*c^4*x^4) + 1200*b*Cos[2*ArcSin[c*x]] + 75*b*Cos[4*ArcSin[c*x]]) + 60*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]*(32*(1 - c^2*x^2)^(5/2) + 40*Sin[2*ArcSin[c*x]] + 5*Sin[4*ArcSin[c*x]])))/(9600*c*Sqrt[1 - c^2*x^2])","A",1
518,1,293,376,1.3518525,"\int \sqrt{d+c d x} (f-c f x)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + c*d*x]*(f - c*f*x)^(5/2)*(a + b*ArcSin[c*x]),x]","\frac{f^2 \sqrt{c d x+d} \sqrt{f-c f x} \left(48 a \sqrt{1-c^2 x^2} \left(6 c^3 x^3-16 c^2 x^2+9 c x+16\right)+256 b c x \left(c^2 x^2-3\right)+144 b \cos \left(2 \sin ^{-1}(c x)\right)-9 b \cos \left(4 \sin ^{-1}(c x)\right)\right)-720 a \sqrt{d} f^{5/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)-12 b f^2 \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x) \left(-64 \left(1-c^2 x^2\right)^{3/2}-24 \sin \left(2 \sin ^{-1}(c x)\right)+3 \sin \left(4 \sin ^{-1}(c x)\right)\right)+360 b f^2 \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2}{1152 c \sqrt{1-c^2 x^2}}","\frac{1}{4} c^2 f^2 x^3 \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)+\frac{5 f^2 \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)^2}{16 b c \sqrt{1-c^2 x^2}}+\frac{2 f^2 \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)}{3 c}+\frac{3}{8} f^2 x \sqrt{c d x+d} \sqrt{f-c f x} \left(a+b \sin ^{-1}(c x)\right)-\frac{3 b c f^2 x^2 \sqrt{c d x+d} \sqrt{f-c f x}}{16 \sqrt{1-c^2 x^2}}-\frac{2 b f^2 x \sqrt{c d x+d} \sqrt{f-c f x}}{3 \sqrt{1-c^2 x^2}}+\frac{2 b c^2 f^2 x^3 \sqrt{c d x+d} \sqrt{f-c f x}}{9 \sqrt{1-c^2 x^2}}-\frac{b c^3 f^2 x^4 \sqrt{c d x+d} \sqrt{f-c f x}}{16 \sqrt{1-c^2 x^2}}",1,"(360*b*f^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2 - 720*a*Sqrt[d]*f^(5/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + f^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(256*b*c*x*(-3 + c^2*x^2) + 48*a*Sqrt[1 - c^2*x^2]*(16 + 9*c*x - 16*c^2*x^2 + 6*c^3*x^3) + 144*b*Cos[2*ArcSin[c*x]] - 9*b*Cos[4*ArcSin[c*x]]) - 12*b*f^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]*(-64*(1 - c^2*x^2)^(3/2) - 24*Sin[2*ArcSin[c*x]] + 3*Sin[4*ArcSin[c*x]]))/(1152*c*Sqrt[1 - c^2*x^2])","A",1
519,1,274,345,1.9902395,"\int \frac{(f-c f x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{d+c d x}} \, dx","Integrate[((f - c*f*x)^(5/2)*(a + b*ArcSin[c*x]))/Sqrt[d + c*d*x],x]","\frac{f^2 \sqrt{c d x+d} \sqrt{f-c f x} \left(12 a \sqrt{1-c^2 x^2} \left(2 c^2 x^2-9 c x+22\right)-270 b c x+2 b \sin \left(3 \sin ^{-1}(c x)\right)-27 b \cos \left(2 \sin ^{-1}(c x)\right)\right)-180 a \sqrt{d} f^{5/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)-6 b f^2 \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x) \left(9 \sqrt{1-c^2 x^2} (2 c x-5)+\cos \left(3 \sin ^{-1}(c x)\right)\right)+90 b f^2 \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2}{72 c d \sqrt{1-c^2 x^2}}","\frac{5 f^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{4 b c \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{c f^3 x^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{3 f^3 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{11 f^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{3 b c f^3 x^2 \sqrt{1-c^2 x^2}}{4 \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{11 b f^3 x \sqrt{1-c^2 x^2}}{3 \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{b c^2 f^3 x^3 \sqrt{1-c^2 x^2}}{9 \sqrt{c d x+d} \sqrt{f-c f x}}",1,"(90*b*f^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2 - 180*a*Sqrt[d]*f^(5/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] - 6*b*f^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]*(9*(-5 + 2*c*x)*Sqrt[1 - c^2*x^2] + Cos[3*ArcSin[c*x]]) + f^2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(-270*b*c*x + 12*a*Sqrt[1 - c^2*x^2]*(22 - 9*c*x + 2*c^2*x^2) - 27*b*Cos[2*ArcSin[c*x]] + 2*b*Sin[3*ArcSin[c*x]]))/(72*c*d*Sqrt[1 - c^2*x^2])","A",1
520,1,685,465,3.7719572,"\int \frac{(f-c f x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{(d+c d x)^{3/2}} \, dx","Integrate[((f - c*f*x)^(5/2)*(a + b*ArcSin[c*x]))/(d + c*d*x)^(3/2),x]","\frac{f^2 \left(8 a \sqrt{1-c^2 x^2} \left(c^2 x^2-7 c x-24\right) \sqrt{c d x+d} \sqrt{f-c f x} \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+120 a \sqrt{d} \sqrt{f} (c x+1) \sqrt{1-c^2 x^2} \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)-32 b (c x+1) \sqrt{c d x+d} \sqrt{f-c f x} \left(\sin ^{-1}(c x) \left(\left(\sqrt{1-c^2 x^2}-2\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\left(\sqrt{1-c^2 x^2}+2\right) \cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\sin ^{-1}(c x)^2 \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-\left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(c x+4 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)-8 b (c x+1) \sqrt{c d x+d} \sqrt{f-c f x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+4\right)-8 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\left(\sin ^{-1}(c x)-4\right) \sin ^{-1}(c x)-8 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)-b (c x+1) \sqrt{c d x+d} \sqrt{f-c f x} \left(20 \sin ^{-1}(c x)^2 \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 \sin ^{-1}(c x) \left(-24 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+7 \sin \left(\frac{3}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{5}{2} \sin ^{-1}(c x)\right)+24 \cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+7 \cos \left(\frac{3}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{5}{2} \sin ^{-1}(c x)\right)\right)-2 \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(16 c x+\cos \left(2 \sin ^{-1}(c x)\right)+32 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)\right)}{16 c d^2 (c x+1) \sqrt{1-c^2 x^2} \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}","-\frac{5 f^4 (1-c x) \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{15 f^4 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{2 f^4 (1-c x)^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{15 f^4 \left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x) \left(a+b \sin ^{-1}(c x)\right)}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{b c f^4 x^2 \left(1-c^2 x^2\right)^{3/2}}{(c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{5 b f^4 (1-c x)^2 \left(1-c^2 x^2\right)^{3/2}}{4 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{3 b f^4 x \left(1-c^2 x^2\right)^{3/2}}{2 (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{8 b f^4 \left(1-c^2 x^2\right)^{3/2} \log (c x+1)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{15 b f^4 \left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x)^2}{4 c (c d x+d)^{3/2} (f-c f x)^{3/2}}",1,"(f^2*(8*a*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*Sqrt[1 - c^2*x^2]*(-24 - 7*c*x + c^2*x^2)*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + 120*a*Sqrt[d]*Sqrt[f]*(1 + c*x)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 8*b*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[ArcSin[c*x]/2]*(ArcSin[c*x]*(4 + ArcSin[c*x]) - 8*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + ((-4 + ArcSin[c*x])*ArcSin[c*x] - 8*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]) - 32*b*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(ArcSin[c*x]^2*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - (c*x + 4*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + ArcSin[c*x]*((2 + Sqrt[1 - c^2*x^2])*Cos[ArcSin[c*x]/2] + (-2 + Sqrt[1 - c^2*x^2])*Sin[ArcSin[c*x]/2])) - b*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(20*ArcSin[c*x]^2*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 2*(16*c*x + Cos[2*ArcSin[c*x]] + 32*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + 2*ArcSin[c*x]*(24*Cos[ArcSin[c*x]/2] + 7*Cos[(3*ArcSin[c*x])/2] + Cos[(5*ArcSin[c*x])/2] - 24*Sin[ArcSin[c*x]/2] + 7*Sin[(3*ArcSin[c*x])/2] - Sin[(5*ArcSin[c*x])/2]))))/(16*c*d^2*(1 + c*x)*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]))","A",1
521,1,847,420,6.813775,"\int \frac{(f-c f x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{(d+c d x)^{5/2}} \, dx","Integrate[((f - c*f*x)^(5/2)*(a + b*ArcSin[c*x]))/(d + c*d*x)^(5/2),x]","\frac{f^2 \left(\frac{4 a \sqrt{c x d+d} \sqrt{f-c f x} \left(3 c^2 x^2+34 c x+23\right)}{(c x+1)^2}-60 a \sqrt{d} \sqrt{f} \tan ^{-1}\left(\frac{c x \sqrt{c x d+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+\frac{2 b \sqrt{c x d+d} \sqrt{f-c f x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(9 \sin ^{-1}(c x)^2+6 \sin ^{-1}(c x)-84 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-8\right)+\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(\left(14-3 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+28 \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 \left(3 \left(\sqrt{1-c^2 x^2}+2\right) \sin ^{-1}(c x)^2+2 \left(7 \sqrt{1-c^2 x^2}+2\right) \sin ^{-1}(c x)-28 \left(\sqrt{1-c^2 x^2}+2\right) \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-4\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{(1-c x) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4}+\frac{2 b \sqrt{c x d+d} \sqrt{f-c f x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x)+2 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(3 \sin ^{-1}(c x)+6 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+4\right)+2 \left(\left(\sqrt{1-c^2 x^2}+2\right) \sin ^{-1}(c x)-2 \left(\sqrt{1-c^2 x^2}+2\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\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{(1-c x) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4}+\frac{b \sqrt{c x d+d} \sqrt{f-c f x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(-18 \sin ^{-1}(c x)^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3+2 \left(6 c^2 x^2+6 c x+52 (c x+1) \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+4\right) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\sin ^{-1}(c x) \left(-24 \cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-35 \cos \left(\frac{3}{2} \sin ^{-1}(c x)\right)+3 \cos \left(\frac{5}{2} \sin ^{-1}(c x)\right)+24 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)-35 \sin \left(\frac{3}{2} \sin ^{-1}(c x)\right)-3 \sin \left(\frac{5}{2} \sin ^{-1}(c x)\right)\right)\right)}{(c x-1) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4}\right)}{12 c d^3}","\frac{5 f^5 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{10 f^5 (1-c x)^2 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{2 f^5 (1-c x)^4 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{5 f^5 \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b f^5 x \left(1-c^2 x^2\right)^{5/2}}{(c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{8 b f^5 \left(1-c^2 x^2\right)^{5/2}}{3 c (c x+1) (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{28 b f^5 \left(1-c^2 x^2\right)^{5/2} \log (c x+1)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{5 b f^5 \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x)^2}{2 c (c d x+d)^{5/2} (f-c f x)^{5/2}}",1,"(f^2*((4*a*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(23 + 34*c*x + 3*c^2*x^2))/(1 + c*x)^2 - 60*a*Sqrt[d]*Sqrt[f]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + (2*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2]*(-8 + 6*ArcSin[c*x] + 9*ArcSin[c*x]^2 - 84*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + Cos[(3*ArcSin[c*x])/2]*((14 - 3*ArcSin[c*x])*ArcSin[c*x] + 28*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + 2*(-4 + 2*(2 + 7*Sqrt[1 - c^2*x^2])*ArcSin[c*x] + 3*(2 + Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 - 28*(2 + Sqrt[1 - c^2*x^2])*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/((1 - c*x)*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^4) + (2*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[(3*ArcSin[c*x])/2]*(ArcSin[c*x] + 2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) - Cos[ArcSin[c*x]/2]*(4 + 3*ArcSin[c*x] + 6*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + 2*(-2 + (2 + Sqrt[1 - c^2*x^2])*ArcSin[c*x] - 2*(2 + Sqrt[1 - c^2*x^2])*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/((1 - c*x)*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^4) + (b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(2*(4 + 6*c*x + 6*c^2*x^2 + 52*(1 + c*x)*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 18*ArcSin[c*x]^2*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 + ArcSin[c*x]*(-24*Cos[ArcSin[c*x]/2] - 35*Cos[(3*ArcSin[c*x])/2] + 3*Cos[(5*ArcSin[c*x])/2] + 24*Sin[ArcSin[c*x]/2] - 35*Sin[(3*ArcSin[c*x])/2] - 3*Sin[(5*ArcSin[c*x])/2])))/((-1 + c*x)*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^4)))/(12*c*d^3)","B",1
522,1,270,345,2.2780275,"\int \frac{(d+c d x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{f-c f x}} \, dx","Integrate[((d + c*d*x)^(5/2)*(a + b*ArcSin[c*x]))/Sqrt[f - c*f*x],x]","-\frac{d^2 \left(\sqrt{c d x+d} \sqrt{f-c f x} \left(12 a \sqrt{1-c^2 x^2} \left(2 c^2 x^2+9 c x+22\right)-270 b c x+2 b \sin \left(3 \sin ^{-1}(c x)\right)+27 b \cos \left(2 \sin ^{-1}(c x)\right)\right)+180 a \sqrt{d} \sqrt{f} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+6 b \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x) \left(9 (2 c x+5) \sqrt{1-c^2 x^2}-\cos \left(3 \sin ^{-1}(c x)\right)\right)-90 b \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2\right)}{72 c f \sqrt{1-c^2 x^2}}","\frac{5 d^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{4 b c \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{c d^3 x^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{3 d^3 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{11 d^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{3 b c d^3 x^2 \sqrt{1-c^2 x^2}}{4 \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{11 b d^3 x \sqrt{1-c^2 x^2}}{3 \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{b c^2 d^3 x^3 \sqrt{1-c^2 x^2}}{9 \sqrt{c d x+d} \sqrt{f-c f x}}",1,"-1/72*(d^2*(-90*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2 + 180*a*Sqrt[d]*Sqrt[f]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + 6*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]*(9*(5 + 2*c*x)*Sqrt[1 - c^2*x^2] - Cos[3*ArcSin[c*x]]) + Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(-270*b*c*x + 12*a*Sqrt[1 - c^2*x^2]*(22 + 9*c*x + 2*c^2*x^2) + 27*b*Cos[2*ArcSin[c*x]] + 2*b*Sin[3*ArcSin[c*x]])))/(c*f*Sqrt[1 - c^2*x^2])","A",1
523,1,238,242,1.2365515,"\int \frac{(d+c d x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{f-c f x}} \, dx","Integrate[((d + c*d*x)^(3/2)*(a + b*ArcSin[c*x]))/Sqrt[f - c*f*x],x]","\frac{d \sqrt{c d x+d} \sqrt{f-c f x} \left(-4 a (c x+4) \sqrt{1-c^2 x^2}+16 b c x-b \cos \left(2 \sin ^{-1}(c x)\right)\right)-12 a d^{3/2} \sqrt{f} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)-4 b d (c x+4) \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)+6 b d \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2}{8 c f \sqrt{1-c^2 x^2}}","\frac{3 d^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{4 b c \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{d^2 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{2 d^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{c \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{b c d^2 x^2 \sqrt{1-c^2 x^2}}{4 \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{2 b d^2 x \sqrt{1-c^2 x^2}}{\sqrt{c d x+d} \sqrt{f-c f x}}",1,"(-4*b*d*(4 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 6*b*d*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2 - 12*a*d^(3/2)*Sqrt[f]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + d*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(16*b*c*x - 4*a*(4 + c*x)*Sqrt[1 - c^2*x^2] - b*Cos[2*ArcSin[c*x]]))/(8*c*f*Sqrt[1 - c^2*x^2])","A",1
524,1,200,141,0.8055272,"\int \frac{\sqrt{d+c d x} \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{f-c f x}} \, dx","Integrate[(Sqrt[d + c*d*x]*(a + b*ArcSin[c*x]))/Sqrt[f - c*f*x],x]","\frac{\frac{2 \sqrt{c d x+d} \sqrt{f-c f x} \left(b c x-a \sqrt{1-c^2 x^2}\right)}{\sqrt{1-c^2 x^2}}-2 a \sqrt{d} \sqrt{f} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+\frac{b \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}-2 b \sqrt{c d x+d} \sqrt{f-c f x} \sin ^{-1}(c x)}{2 c f}","\frac{d \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 b c \sqrt{c d x+d} \sqrt{f-c f x}}-\frac{d \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{c \sqrt{c d x+d} \sqrt{f-c f x}}+\frac{b d x \sqrt{1-c^2 x^2}}{\sqrt{c d x+d} \sqrt{f-c f x}}",1,"((2*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(b*c*x - a*Sqrt[1 - c^2*x^2]))/Sqrt[1 - c^2*x^2] - 2*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x] + (b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] - 2*a*Sqrt[d]*Sqrt[f]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))])/(2*c*f)","A",1
525,1,110,55,0.560102,"\int \frac{a+b \sin ^{-1}(c x)}{\sqrt{d+c d x} \sqrt{f-c f x}} \, dx","Integrate[(a + b*ArcSin[c*x])/(Sqrt[d + c*d*x]*Sqrt[f - c*f*x]),x]","\frac{\frac{b \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{\sqrt{c d x+d} \sqrt{f-c f x}}-\frac{2 a \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)}{\sqrt{d} \sqrt{f}}}{2 c}","\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 b c \sqrt{c d x+d} \sqrt{f-c f x}}",1,"((b*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(Sqrt[d + c*d*x]*Sqrt[f - c*f*x]) - (2*a*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))])/(Sqrt[d]*Sqrt[f]))/(2*c)","A",1
526,1,79,99,0.3930428,"\int \frac{a+b \sin ^{-1}(c x)}{(d+c d x)^{3/2} \sqrt{f-c f x}} \, dx","Integrate[(a + b*ArcSin[c*x])/((d + c*d*x)^(3/2)*Sqrt[f - c*f*x]),x]","\frac{\sqrt{c d x+d} \left(a (c x-1)+b \sqrt{1-c^2 x^2} \log (-f (c x+1))+b (c x-1) \sin ^{-1}(c x)\right)}{c d^2 (c x+1) \sqrt{f-c f x}}","\frac{b f \left(1-c^2 x^2\right)^{3/2} \log (c x+1)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{f (1-c x) \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}",1,"(Sqrt[d + c*d*x]*(a*(-1 + c*x) + b*(-1 + c*x)*ArcSin[c*x] + b*Sqrt[1 - c^2*x^2]*Log[-(f*(1 + c*x))]))/(c*d^2*(1 + c*x)*Sqrt[f - c*f*x])","A",1
527,1,118,265,0.5088007,"\int \frac{a+b \sin ^{-1}(c x)}{(d+c d x)^{5/2} \sqrt{f-c f x}} \, dx","Integrate[(a + b*ArcSin[c*x])/((d + c*d*x)^(5/2)*Sqrt[f - c*f*x]),x]","\frac{\sqrt{c d x+d} \left((c x+2) \left(a c x-a-b \sqrt{1-c^2 x^2}\right)+b (c x+1) \sqrt{1-c^2 x^2} \log (-f (c x+1))+b \left(c^2 x^2+c x-2\right) \sin ^{-1}(c x)\right)}{3 c d^3 (c x+1)^2 \sqrt{f-c f x}}","\frac{f^2 x \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{3 (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{2 f^2 (1-c x) \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b f^2 \left(1-c^2 x^2\right)^{5/2}}{3 c (c x+1) (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{b f^2 \left(1-c^2 x^2\right)^{5/2} \log \left(1-c^2 x^2\right)}{6 c (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{b f^2 \left(1-c^2 x^2\right)^{5/2} \tanh ^{-1}(c x)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}",1,"(Sqrt[d + c*d*x]*((2 + c*x)*(-a + a*c*x - b*Sqrt[1 - c^2*x^2]) + b*(-2 + c*x + c^2*x^2)*ArcSin[c*x] + b*(1 + c*x)*Sqrt[1 - c^2*x^2]*Log[-(f*(1 + c*x))]))/(3*c*d^3*(1 + c*x)^2*Sqrt[f - c*f*x])","A",1
528,1,768,463,4.5535152,"\int \frac{(d+c d x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{(f-c f x)^{3/2}} \, dx","Integrate[((d + c*d*x)^(5/2)*(a + b*ArcSin[c*x]))/(f - c*f*x)^(3/2),x]","\frac{d^2 \left(\frac{8 a \left(c^2 x^2+7 c x-24\right) \sqrt{c d x+d} \sqrt{f-c f x}}{c x-1}+120 a \sqrt{d} \sqrt{f} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)-\frac{8 b (c x+1) \sqrt{c d x+d} \sqrt{f-c f x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\left(\sin ^{-1}(c x)-4\right) \sin ^{-1}(c x)-8 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+4\right)-8 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{\sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-\frac{32 b (c x+1) \sqrt{c d x+d} \sqrt{f-c f x} \left(-\sin ^{-1}(c x) \left(\left(\sqrt{1-c^2 x^2}+2\right) \cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\left(\sqrt{1-c^2 x^2}-2\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\sin ^{-1}(c x)^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(c x-4 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{\sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}+\frac{b (c x+1) \sqrt{c d x+d} \sqrt{f-c f x} \left(-20 \sin ^{-1}(c x)^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 \sin ^{-1}(c x) \left(24 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)-7 \sin \left(\frac{3}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{5}{2} \sin ^{-1}(c x)\right)+24 \cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+7 \cos \left(\frac{3}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{5}{2} \sin ^{-1}(c x)\right)\right)+2 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(-16 c x+\cos \left(2 \sin ^{-1}(c x)\right)+32 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{\sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}\right)}{16 c f^2}","\frac{5 d^4 (c x+1) \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{15 d^4 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{2 d^4 (c x+1)^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{15 d^4 \left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x) \left(a+b \sin ^{-1}(c x)\right)}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{b c d^4 x^2 \left(1-c^2 x^2\right)^{3/2}}{(c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{5 b d^4 (c x+1)^2 \left(1-c^2 x^2\right)^{3/2}}{4 c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{3 b d^4 x \left(1-c^2 x^2\right)^{3/2}}{2 (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{8 b d^4 \left(1-c^2 x^2\right)^{3/2} \log (1-c x)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{15 b d^4 \left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x)^2}{4 c (c d x+d)^{3/2} (f-c f x)^{3/2}}",1,"(d^2*((8*a*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(-24 + 7*c*x + c^2*x^2))/(-1 + c*x) + 120*a*Sqrt[d]*Sqrt[f]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] - (8*b*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[ArcSin[c*x]/2]*((-4 + ArcSin[c*x])*ArcSin[c*x] - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - (ArcSin[c*x]*(4 + ArcSin[c*x]) - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) - (32*b*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(ArcSin[c*x]^2*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) + (c*x - 4*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - ArcSin[c*x]*((2 + Sqrt[1 - c^2*x^2])*Cos[ArcSin[c*x]/2] - (-2 + Sqrt[1 - c^2*x^2])*Sin[ArcSin[c*x]/2])))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (b*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(-20*ArcSin[c*x]^2*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) + 2*(-16*c*x + Cos[2*ArcSin[c*x]] + 32*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) + 2*ArcSin[c*x]*(24*Cos[ArcSin[c*x]/2] + 7*Cos[(3*ArcSin[c*x])/2] + Cos[(5*ArcSin[c*x])/2] + 24*Sin[ArcSin[c*x]/2] - 7*Sin[(3*ArcSin[c*x])/2] + Sin[(5*ArcSin[c*x])/2])))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2)))/(16*c*f^2)","A",1
529,1,514,252,3.0930151,"\int \frac{(d+c d x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{(f-c f x)^{3/2}} \, dx","Integrate[((d + c*d*x)^(3/2)*(a + b*ArcSin[c*x]))/(f - c*f*x)^(3/2),x]","\frac{d \left(6 a \sqrt{d} \sqrt{f} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+\frac{2 a (c x-5) \sqrt{c d x+d} \sqrt{f-c f x}}{c x-1}-\frac{b (c x+1) \sqrt{c d x+d} \sqrt{f-c f x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\left(\sin ^{-1}(c x)-4\right) \sin ^{-1}(c x)-8 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+4\right)-8 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{\sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-\frac{2 b (c x+1) \sqrt{c d x+d} \sqrt{f-c f x} \left(-\sin ^{-1}(c x) \left(\left(\sqrt{1-c^2 x^2}+2\right) \cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\left(\sqrt{1-c^2 x^2}-2\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\sin ^{-1}(c x)^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(c x-4 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{\sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}\right)}{2 c f^2}","-\frac{3 d^3 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 b c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{d^3 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{4 d^3 (c x+1) \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac{b d^3 x \left(1-c^2 x^2\right)^{3/2}}{(c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{4 b d^3 \left(1-c^2 x^2\right)^{3/2} \log (1-c x)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}",1,"(d*((2*a*(-5 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(-1 + c*x) + 6*a*Sqrt[d]*Sqrt[f]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] - (b*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[ArcSin[c*x]/2]*((-4 + ArcSin[c*x])*ArcSin[c*x] - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - (ArcSin[c*x]*(4 + ArcSin[c*x]) - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) - (2*b*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(ArcSin[c*x]^2*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) + (c*x - 4*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - ArcSin[c*x]*((2 + Sqrt[1 - c^2*x^2])*Cos[ArcSin[c*x]/2] - (-2 + Sqrt[1 - c^2*x^2])*Sin[ArcSin[c*x]/2])))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2)))/(2*c*f^2)","B",0
530,1,281,162,1.7152309,"\int \frac{\sqrt{d+c d x} \left(a+b \sin ^{-1}(c x)\right)}{(f-c f x)^{3/2}} \, dx","Integrate[(Sqrt[d + c*d*x]*(a + b*ArcSin[c*x]))/(f - c*f*x)^(3/2),x]","-\frac{-2 a \sqrt{d} \sqrt{f} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+\frac{4 a \sqrt{c d x+d} \sqrt{f-c f x}}{c x-1}+\frac{b (c x+1) \sqrt{c d x+d} \sqrt{f-c f x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\left(\sin ^{-1}(c x)-4\right) \sin ^{-1}(c x)-8 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+4\right)-8 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{\sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}}{2 c f^2}","-\frac{d^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{2 b c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{2 d^2 (c x+1) \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{2 b d^2 \left(1-c^2 x^2\right)^{3/2} \log (1-c x)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}",1,"-1/2*((4*a*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(-1 + c*x) - 2*a*Sqrt[d]*Sqrt[f]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + (b*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[ArcSin[c*x]/2]*((-4 + ArcSin[c*x])*ArcSin[c*x] - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - (ArcSin[c*x]*(4 + ArcSin[c*x]) - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2))/(c*f^2)","A",1
531,1,106,98,0.4754349,"\int \frac{a+b \sin ^{-1}(c x)}{\sqrt{d+c d x} (f-c f x)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(Sqrt[d + c*d*x]*(f - c*f*x)^(3/2)),x]","\frac{\sqrt{c d x+d} \sqrt{f-c f x} \left(a \left(-\sqrt{1-c^2 x^2}\right)-b \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+b (c x-1) \log (f-c f x)\right)}{c d f^2 (c x-1) \sqrt{1-c^2 x^2}}","\frac{d (c x+1) \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{b d \left(1-c^2 x^2\right)^{3/2} \log (1-c x)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}",1,"(Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(-(a*Sqrt[1 - c^2*x^2]) - b*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + b*(-1 + c*x)*Log[f - c*f*x]))/(c*d*f^2*(-1 + c*x)*Sqrt[1 - c^2*x^2])","A",1
532,1,105,96,0.5144993,"\int \frac{a+b \sin ^{-1}(c x)}{(d+c d x)^{3/2} (f-c f x)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/((d + c*d*x)^(3/2)*(f - c*f*x)^(3/2)),x]","\frac{\sqrt{c d x+d} \left(2 a c x+b \sqrt{1-c^2 x^2} \log (-f (c x+1))+b \sqrt{1-c^2 x^2} \log (f-c f x)+2 b c x \sin ^{-1}(c x)\right)}{2 c d^2 f (c x+1) \sqrt{f-c f x}}","\frac{x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{(c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac{b \left(1-c^2 x^2\right)^{3/2} \log \left(1-c^2 x^2\right)}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}",1,"(Sqrt[d + c*d*x]*(2*a*c*x + 2*b*c*x*ArcSin[c*x] + b*Sqrt[1 - c^2*x^2]*Log[-(f*(1 + c*x))] + b*Sqrt[1 - c^2*x^2]*Log[f - c*f*x]))/(2*c*d^2*f*(1 + c*x)*Sqrt[f - c*f*x])","A",1
533,1,180,255,0.6771145,"\int \frac{a+b \sin ^{-1}(c x)}{(d+c d x)^{5/2} (f-c f x)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/((d + c*d*x)^(5/2)*(f - c*f*x)^(3/2)),x]","\frac{\sqrt{c d x+d} \left(8 a c^2 x^2+8 a c x-4 a+3 b c x \sqrt{1-c^2 x^2} \log (f-c f x)+5 b (c x+1) \sqrt{1-c^2 x^2} \log (-f (c x+1))+3 b \sqrt{1-c^2 x^2} \log (f-c f x)-2 b \sqrt{1-c^2 x^2}+4 b \left(2 c^2 x^2+2 c x-1\right) \sin ^{-1}(c x)\right)}{12 c d^3 f (c x+1)^2 \sqrt{f-c f x}}","\frac{2 f x \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{3 (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{f (1-c x) \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b f \left(1-c^2 x^2\right)^{5/2}}{6 c (c x+1) (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{b f \left(1-c^2 x^2\right)^{5/2} \log \left(1-c^2 x^2\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{b f \left(1-c^2 x^2\right)^{5/2} \tanh ^{-1}(c x)}{6 c (c d x+d)^{5/2} (f-c f x)^{5/2}}",1,"(Sqrt[d + c*d*x]*(-4*a + 8*a*c*x + 8*a*c^2*x^2 - 2*b*Sqrt[1 - c^2*x^2] + 4*b*(-1 + 2*c*x + 2*c^2*x^2)*ArcSin[c*x] + 5*b*(1 + c*x)*Sqrt[1 - c^2*x^2]*Log[-(f*(1 + c*x))] + 3*b*Sqrt[1 - c^2*x^2]*Log[f - c*f*x] + 3*b*c*x*Sqrt[1 - c^2*x^2]*Log[f - c*f*x]))/(12*c*d^3*f*(1 + c*x)^2*Sqrt[f - c*f*x])","A",1
534,1,850,419,6.2115929,"\int \frac{(d+c d x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{(f-c f x)^{5/2}} \, dx","Integrate[((d + c*d*x)^(5/2)*(a + b*ArcSin[c*x]))/(f - c*f*x)^(5/2),x]","\frac{d^2 \left(-\frac{4 a \sqrt{c x d+d} \sqrt{f-c f x} \left(3 c^2 x^2-34 c x+23\right)}{(c x-1)^2}-60 a \sqrt{d} \sqrt{f} \tan ^{-1}\left(\frac{c x \sqrt{c x d+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+\frac{2 b \sqrt{c x d+d} \sqrt{f-c f x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(3 \sin ^{-1}(c x)-6 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-4\right)-\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x)-2 \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 \left(\left(\sqrt{1-c^2 x^2}+2\right) \sin ^{-1}(c x)+2 \left(\sqrt{1-c^2 x^2}+2\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\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}+\frac{2 b \sqrt{c x d+d} \sqrt{f-c f x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(9 \sin ^{-1}(c x)^2-6 \sin ^{-1}(c x)-84 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-8\right)+\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(28 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-\sin ^{-1}(c x) \left(3 \sin ^{-1}(c x)+14\right)\right)+2 \left(-3 \left(\sqrt{1-c^2 x^2}+2\right) \sin ^{-1}(c x)^2+2 \left(7 \sqrt{1-c^2 x^2}+2\right) \sin ^{-1}(c x)+28 \left(\sqrt{1-c^2 x^2}+2\right) \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+4\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}+\frac{b \sqrt{c x d+d} \sqrt{f-c f x} \left(18 \sin ^{-1}(c x)^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3+2 \left(6 c x+3 \cos \left(2 \sin ^{-1}(c x)\right)+52 (c x-1) \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-7\right) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\sin ^{-1}(c x) \left(-24 \cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-35 \cos \left(\frac{3}{2} \sin ^{-1}(c x)\right)+3 \cos \left(\frac{5}{2} \sin ^{-1}(c x)\right)-24 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+35 \sin \left(\frac{3}{2} \sin ^{-1}(c x)\right)+3 \sin \left(\frac{5}{2} \sin ^{-1}(c x)\right)\right)\right)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}\right)}{12 c f^3}","-\frac{5 d^5 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{10 d^5 (c x+1)^2 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{2 d^5 (c x+1)^4 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{5 d^5 \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{b d^5 x \left(1-c^2 x^2\right)^{5/2}}{(c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{8 b d^5 \left(1-c^2 x^2\right)^{5/2}}{3 c (1-c x) (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{28 b d^5 \left(1-c^2 x^2\right)^{5/2} \log (1-c x)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{5 b d^5 \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x)^2}{2 c (c d x+d)^{5/2} (f-c f x)^{5/2}}",1,"(d^2*((-4*a*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(23 - 34*c*x + 3*c^2*x^2))/(-1 + c*x)^2 - 60*a*Sqrt[d]*Sqrt[f]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + (2*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[ArcSin[c*x]/2]*(-4 + 3*ArcSin[c*x] - 6*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - Cos[(3*ArcSin[c*x])/2]*(ArcSin[c*x] - 2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + 2*(2 + (2 + Sqrt[1 - c^2*x^2])*ArcSin[c*x] + 2*(2 + Sqrt[1 - c^2*x^2])*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/((Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^4*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) + (2*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[ArcSin[c*x]/2]*(-8 - 6*ArcSin[c*x] + 9*ArcSin[c*x]^2 - 84*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + Cos[(3*ArcSin[c*x])/2]*(-(ArcSin[c*x]*(14 + 3*ArcSin[c*x])) + 28*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + 2*(4 + 2*(2 + 7*Sqrt[1 - c^2*x^2])*ArcSin[c*x] - 3*(2 + Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + 28*(2 + Sqrt[1 - c^2*x^2])*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/((Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^4*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) + (b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(2*(-7 + 6*c*x + 3*Cos[2*ArcSin[c*x]] + 52*(-1 + c*x)*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) + 18*ArcSin[c*x]^2*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + ArcSin[c*x]*(-24*Cos[ArcSin[c*x]/2] - 35*Cos[(3*ArcSin[c*x])/2] + 3*Cos[(5*ArcSin[c*x])/2] - 24*Sin[ArcSin[c*x]/2] + 35*Sin[(3*ArcSin[c*x])/2] + 3*Sin[(5*ArcSin[c*x])/2])))/((Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^4*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]))))/(12*c*f^3)","B",0
535,1,601,324,4.9903523,"\int \frac{(d+c d x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{(f-c f x)^{5/2}} \, dx","Integrate[((d + c*d*x)^(3/2)*(a + b*ArcSin[c*x]))/(f - c*f*x)^(5/2),x]","\frac{d \left(-12 a \sqrt{d} \sqrt{f} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{f-c f x}}{\sqrt{d} \sqrt{f} \left(c^2 x^2-1\right)}\right)+\frac{16 a (2 c x-1) \sqrt{c d x+d} \sqrt{f-c f x}}{(c x-1)^2}+\frac{2 b \sqrt{c d x+d} \sqrt{f-c f x} \left(2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\left(\sqrt{1-c^2 x^2}+2\right) \sin ^{-1}(c x)+2 \left(\sqrt{1-c^2 x^2}+2\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\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(3 \sin ^{-1}(c x)-6 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-4\right)-\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x)-2 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4 \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}+\frac{b \sqrt{c d x+d} \sqrt{f-c f x} \left(2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(-3 \left(\sqrt{1-c^2 x^2}+2\right) \sin ^{-1}(c x)^2+2 \left(7 \sqrt{1-c^2 x^2}+2\right) \sin ^{-1}(c x)+28 \left(\sqrt{1-c^2 x^2}+2\right) \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+4\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(9 \sin ^{-1}(c x)^2-6 \sin ^{-1}(c x)-84 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-8\right)+\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(28 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-\sin ^{-1}(c x) \left(3 \sin ^{-1}(c x)+14\right)\right)\right)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4 \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}\right)}{12 c f^3}","-\frac{2 d^4 (c x+1) \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{2 d^4 (c x+1)^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{d^4 \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{4 b d^4 \left(1-c^2 x^2\right)^{5/2}}{3 c (1-c x) (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{8 b d^4 \left(1-c^2 x^2\right)^{5/2} \log (1-c x)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b d^4 \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x)^2}{2 c (c d x+d)^{5/2} (f-c f x)^{5/2}}",1,"(d*((16*a*(-1 + 2*c*x)*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(-1 + c*x)^2 - 12*a*Sqrt[d]*Sqrt[f]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[f - c*f*x])/(Sqrt[d]*Sqrt[f]*(-1 + c^2*x^2))] + (2*b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[ArcSin[c*x]/2]*(-4 + 3*ArcSin[c*x] - 6*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - Cos[(3*ArcSin[c*x])/2]*(ArcSin[c*x] - 2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + 2*(2 + (2 + Sqrt[1 - c^2*x^2])*ArcSin[c*x] + 2*(2 + Sqrt[1 - c^2*x^2])*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/((Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^4*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) + (b*Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(Cos[ArcSin[c*x]/2]*(-8 - 6*ArcSin[c*x] + 9*ArcSin[c*x]^2 - 84*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + Cos[(3*ArcSin[c*x])/2]*(-(ArcSin[c*x]*(14 + 3*ArcSin[c*x])) + 28*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + 2*(4 + 2*(2 + 7*Sqrt[1 - c^2*x^2])*ArcSin[c*x] - 3*(2 + Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + 28*(2 + Sqrt[1 - c^2*x^2])*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/((Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^4*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]))))/(12*c*f^3)","A",0
536,1,126,164,0.5390738,"\int \frac{\sqrt{d+c d x} \left(a+b \sin ^{-1}(c x)\right)}{(f-c f x)^{5/2}} \, dx","Integrate[(Sqrt[d + c*d*x]*(a + b*ArcSin[c*x]))/(f - c*f*x)^(5/2),x]","\frac{\sqrt{c d x+d} \sqrt{f-c f x} \left((c x+1) \left(a \sqrt{1-c^2 x^2}+b c x-b\right)+b (c x+1) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-b (c x-1)^2 \log (f-c f x)\right)}{3 c f^3 (c x-1)^2 \sqrt{1-c^2 x^2}}","\frac{d^3 (c x+1)^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{2 b d^3 \left(1-c^2 x^2\right)^{5/2}}{3 c (1-c x) (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b d^3 \left(1-c^2 x^2\right)^{5/2} \log (1-c x)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}",1,"(Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*((1 + c*x)*(-b + b*c*x + a*Sqrt[1 - c^2*x^2]) + b*(1 + c*x)*Sqrt[1 - c^2*x^2]*ArcSin[c*x] - b*(-1 + c*x)^2*Log[f - c*f*x]))/(3*c*f^3*(-1 + c*x)^2*Sqrt[1 - c^2*x^2])","A",1
537,1,130,265,0.500732,"\int \frac{a+b \sin ^{-1}(c x)}{\sqrt{d+c d x} (f-c f x)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(Sqrt[d + c*d*x]*(f - c*f*x)^(5/2)),x]","\frac{\sqrt{c d x+d} \sqrt{f-c f x} \left(-(c x-2) \left(a \sqrt{1-c^2 x^2}+b c x-b\right)-b (c x-2) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+b (c x-1)^2 \log (f-c f x)\right)}{3 c d f^3 (c x-1)^2 \sqrt{1-c^2 x^2}}","\frac{d^2 x \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{3 (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{2 d^2 (c x+1) \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b d^2 \left(1-c^2 x^2\right)^{5/2}}{3 c (1-c x) (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{b d^2 \left(1-c^2 x^2\right)^{5/2} \log \left(1-c^2 x^2\right)}{6 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b d^2 \left(1-c^2 x^2\right)^{5/2} \tanh ^{-1}(c x)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}",1,"(Sqrt[d + c*d*x]*Sqrt[f - c*f*x]*(-((-2 + c*x)*(-b + b*c*x + a*Sqrt[1 - c^2*x^2])) - b*(-2 + c*x)*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + b*(-1 + c*x)^2*Log[f - c*f*x]))/(3*c*d*f^3*(-1 + c*x)^2*Sqrt[1 - c^2*x^2])","A",1
538,1,184,255,0.6928201,"\int \frac{a+b \sin ^{-1}(c x)}{(d+c d x)^{3/2} (f-c f x)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/((d + c*d*x)^(3/2)*(f - c*f*x)^(5/2)),x]","\frac{\sqrt{c d x+d} \left(8 a c^2 x^2-8 a c x-4 a+5 b c x \sqrt{1-c^2 x^2} \log (f-c f x)+3 b (c x-1) \sqrt{1-c^2 x^2} \log (-f (c x+1))-5 b \sqrt{1-c^2 x^2} \log (f-c f x)+2 b \sqrt{1-c^2 x^2}+4 b \left(2 c^2 x^2-2 c x-1\right) \sin ^{-1}(c x)\right)}{12 c d^2 f^2 \left(c^2 x^2-1\right) \sqrt{f-c f x}}","\frac{2 d x \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{3 (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{d (c x+1) \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b d \left(1-c^2 x^2\right)^{5/2}}{6 c (1-c x) (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{b d \left(1-c^2 x^2\right)^{5/2} \log \left(1-c^2 x^2\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b d \left(1-c^2 x^2\right)^{5/2} \tanh ^{-1}(c x)}{6 c (c d x+d)^{5/2} (f-c f x)^{5/2}}",1,"(Sqrt[d + c*d*x]*(-4*a - 8*a*c*x + 8*a*c^2*x^2 + 2*b*Sqrt[1 - c^2*x^2] + 4*b*(-1 - 2*c*x + 2*c^2*x^2)*ArcSin[c*x] + 3*b*(-1 + c*x)*Sqrt[1 - c^2*x^2]*Log[-(f*(1 + c*x))] - 5*b*Sqrt[1 - c^2*x^2]*Log[f - c*f*x] + 5*b*c*x*Sqrt[1 - c^2*x^2]*Log[f - c*f*x]))/(12*c*d^2*f^2*Sqrt[f - c*f*x]*(-1 + c^2*x^2))","A",1
539,1,178,188,0.6403067,"\int \frac{a+b \sin ^{-1}(c x)}{(d+c d x)^{5/2} (f-c f x)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/((d + c*d*x)^(5/2)*(f - c*f*x)^(5/2)),x]","\frac{\sqrt{c d x+d} \left(4 a c^3 x^3-6 a c x+2 b c^2 x^2 \sqrt{1-c^2 x^2} \log (f-c f x)-2 b \left(1-c^2 x^2\right)^{3/2} \log (-f (c x+1))-2 b \sqrt{1-c^2 x^2} \log (f-c f x)+b \sqrt{1-c^2 x^2}+2 b c x \left(2 c^2 x^2-3\right) \sin ^{-1}(c x)\right)}{6 c d^3 (c x-1) \sqrt{f-c f x} (c f x+f)^2}","\frac{2 x \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{3 (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{3 (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac{b \left(1-c^2 x^2\right)^{3/2}}{6 c (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac{b \left(1-c^2 x^2\right)^{5/2} \log \left(1-c^2 x^2\right)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}",1,"(Sqrt[d + c*d*x]*(-6*a*c*x + 4*a*c^3*x^3 + b*Sqrt[1 - c^2*x^2] + 2*b*c*x*(-3 + 2*c^2*x^2)*ArcSin[c*x] - 2*b*(1 - c^2*x^2)^(3/2)*Log[-(f*(1 + c*x))] - 2*b*Sqrt[1 - c^2*x^2]*Log[f - c*f*x] + 2*b*c^2*x^2*Sqrt[1 - c^2*x^2]*Log[f - c*f*x]))/(6*c*d^3*(-1 + c*x)*Sqrt[f - c*f*x]*(f + c*f*x)^2)","A",1
540,1,555,613,2.5980534,"\int (d+c d x)^{5/2} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c*d*x)^(5/2)*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2,x]","\frac{d^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(3 \left(1536 a^2 c^2 x^2 \sqrt{1-c^2 x^2}+864 a^2 c x \sqrt{1-c^2 x^2}-1536 a^2 \sqrt{1-c^2 x^2}+576 a^2 c^3 x^3 \sqrt{1-c^2 x^2}-1024 a b c^3 x^3+3072 a b c x-36 a b \cos \left(4 \sin ^{-1}(c x)\right)+2304 b^2 \sqrt{1-c^2 x^2}-288 b^2 \sin \left(2 \sin ^{-1}(c x)\right)+9 b^2 \sin \left(4 \sin ^{-1}(c x)\right)\right)+1728 a b \cos \left(2 \sin ^{-1}(c x)\right)+256 b^2 \cos \left(3 \sin ^{-1}(c x)\right)\right)-4320 a^2 d^{5/2} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)-72 b d^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(-60 a+48 b \sqrt{1-c^2 x^2}-24 b \sin \left(2 \sin ^{-1}(c x)\right)+3 b \sin \left(4 \sin ^{-1}(c x)\right)+16 b \cos \left(3 \sin ^{-1}(c x)\right)\right)+12 b d^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(768 a c^2 x^2 \sqrt{1-c^2 x^2}-768 a \sqrt{1-c^2 x^2}+288 a \sin \left(2 \sin ^{-1}(c x)\right)-36 a \sin \left(4 \sin ^{-1}(c x)\right)+576 b c x+64 b \sin \left(3 \sin ^{-1}(c x)\right)+144 b \cos \left(2 \sin ^{-1}(c x)\right)-9 b \cos \left(4 \sin ^{-1}(c x)\right)\right)+1440 b^2 d^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{6912 c \sqrt{1-c^2 x^2}}","\frac{1}{4} c^2 d^2 x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{3 b c d^2 x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{8 \sqrt{1-c^2 x^2}}+\frac{4 b d^2 x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{3 \sqrt{1-c^2 x^2}}+\frac{5 d^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^3}{24 b c \sqrt{1-c^2 x^2}}-\frac{2 d^2 \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c}-\frac{4 b c^2 d^2 x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{9 \sqrt{1-c^2 x^2}}-\frac{b c^3 d^2 x^4 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{8 \sqrt{1-c^2 x^2}}+\frac{3}{8} d^2 x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{1}{32} b^2 c^2 d^2 x^3 \sqrt{c d x+d} \sqrt{e-c e x}+\frac{4 b^2 d^2 \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x}}{27 c}+\frac{15 b^2 d^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}-\frac{15}{64} b^2 d^2 x \sqrt{c d x+d} \sqrt{e-c e x}+\frac{8 b^2 d^2 \sqrt{c d x+d} \sqrt{e-c e x}}{9 c}",1,"(1440*b^2*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 4320*a^2*d^(5/2)*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 12*b*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(576*b*c*x - 768*a*Sqrt[1 - c^2*x^2] + 768*a*c^2*x^2*Sqrt[1 - c^2*x^2] + 144*b*Cos[2*ArcSin[c*x]] - 9*b*Cos[4*ArcSin[c*x]] + 288*a*Sin[2*ArcSin[c*x]] + 64*b*Sin[3*ArcSin[c*x]] - 36*a*Sin[4*ArcSin[c*x]]) - 72*b*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(-60*a + 48*b*Sqrt[1 - c^2*x^2] + 16*b*Cos[3*ArcSin[c*x]] - 24*b*Sin[2*ArcSin[c*x]] + 3*b*Sin[4*ArcSin[c*x]]) + d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1728*a*b*Cos[2*ArcSin[c*x]] + 256*b^2*Cos[3*ArcSin[c*x]] + 3*(3072*a*b*c*x - 1024*a*b*c^3*x^3 - 1536*a^2*Sqrt[1 - c^2*x^2] + 2304*b^2*Sqrt[1 - c^2*x^2] + 864*a^2*c*x*Sqrt[1 - c^2*x^2] + 1536*a^2*c^2*x^2*Sqrt[1 - c^2*x^2] + 576*a^2*c^3*x^3*Sqrt[1 - c^2*x^2] - 36*a*b*Cos[4*ArcSin[c*x]] - 288*b^2*Sin[2*ArcSin[c*x]] + 9*b^2*Sin[4*ArcSin[c*x]])))/(6912*c*Sqrt[1 - c^2*x^2])","A",1
541,1,437,455,2.0123983,"\int (d+c d x)^{3/2} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c*d*x)^(3/2)*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2,x]","\frac{d \sqrt{c d x+d} \sqrt{e-c e x} \left(12 \left(3 a^2 \sqrt{1-c^2 x^2} \left(2 c^2 x^2+3 c x-2\right)-4 a b c x \left(c^2 x^2-3\right)+9 b^2 \sqrt{1-c^2 x^2}\right)+54 a b \cos \left(2 \sin ^{-1}(c x)\right)-27 b^2 \sin \left(2 \sin ^{-1}(c x)\right)+4 b^2 \cos \left(3 \sin ^{-1}(c x)\right)\right)-108 a^2 d^{3/2} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)-18 b d \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(-6 a+3 b \sqrt{1-c^2 x^2}-3 b \sin \left(2 \sin ^{-1}(c x)\right)+b \cos \left(3 \sin ^{-1}(c x)\right)\right)+6 b d \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(2 \left(12 a c^2 x^2 \sqrt{1-c^2 x^2}-12 a \sqrt{1-c^2 x^2}+9 a \sin \left(2 \sin ^{-1}(c x)\right)+9 b c x+b \sin \left(3 \sin ^{-1}(c x)\right)\right)+9 b \cos \left(2 \sin ^{-1}(c x)\right)\right)+36 b^2 d \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{216 c \sqrt{1-c^2 x^2}}","-\frac{b c d x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{1-c^2 x^2}}+\frac{2 b d x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{3 \sqrt{1-c^2 x^2}}+\frac{d \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^3}{6 b c \sqrt{1-c^2 x^2}}-\frac{d \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c}-\frac{2 b c^2 d x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{9 \sqrt{1-c^2 x^2}}+\frac{1}{2} d x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{2 b^2 d \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x}}{27 c}+\frac{b^2 d \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}-\frac{1}{4} b^2 d x \sqrt{c d x+d} \sqrt{e-c e x}+\frac{4 b^2 d \sqrt{c d x+d} \sqrt{e-c e x}}{9 c}",1,"(36*b^2*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 108*a^2*d^(3/2)*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 18*b*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(-6*a + 3*b*Sqrt[1 - c^2*x^2] + b*Cos[3*ArcSin[c*x]] - 3*b*Sin[2*ArcSin[c*x]]) + d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(12*(9*b^2*Sqrt[1 - c^2*x^2] - 4*a*b*c*x*(-3 + c^2*x^2) + 3*a^2*Sqrt[1 - c^2*x^2]*(-2 + 3*c*x + 2*c^2*x^2)) + 54*a*b*Cos[2*ArcSin[c*x]] + 4*b^2*Cos[3*ArcSin[c*x]] - 27*b^2*Sin[2*ArcSin[c*x]]) + 6*b*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(9*b*Cos[2*ArcSin[c*x]] + 2*(9*b*c*x - 12*a*Sqrt[1 - c^2*x^2] + 12*a*c^2*x^2*Sqrt[1 - c^2*x^2] + 9*a*Sin[2*ArcSin[c*x]] + b*Sin[3*ArcSin[c*x]])))/(216*c*Sqrt[1 - c^2*x^2])","A",1
542,1,288,222,1.1457759,"\int \sqrt{d+c d x} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2,x]","\frac{3 \sqrt{c d x+d} \sqrt{e-c e x} \left(4 a^2 c x \sqrt{1-c^2 x^2}+2 a b \cos \left(2 \sin ^{-1}(c x)\right)-b^2 \sin \left(2 \sin ^{-1}(c x)\right)\right)-12 a^2 \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+6 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(2 a+b \sin \left(2 \sin ^{-1}(c x)\right)\right)+6 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(2 a \sin \left(2 \sin ^{-1}(c x)\right)+b \cos \left(2 \sin ^{-1}(c x)\right)\right)+4 b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{24 c \sqrt{1-c^2 x^2}}","\frac{\sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^3}{6 b c \sqrt{1-c^2 x^2}}-\frac{b c x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{1-c^2 x^2}}+\frac{1}{2} x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}-\frac{1}{4} b^2 x \sqrt{c d x+d} \sqrt{e-c e x}",1,"(4*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 12*a^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 6*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(b*Cos[2*ArcSin[c*x]] + 2*a*Sin[2*ArcSin[c*x]]) + 6*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(2*a + b*Sin[2*ArcSin[c*x]]) + 3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(4*a^2*c*x*Sqrt[1 - c^2*x^2] + 2*a*b*Cos[2*ArcSin[c*x]] - b^2*Sin[2*ArcSin[c*x]]))/(24*c*Sqrt[1 - c^2*x^2])","A",1
543,1,296,230,1.2789322,"\int \frac{\sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d+c d x}} \, dx","Integrate[(Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/Sqrt[d + c*d*x],x]","\frac{3 \sqrt{c d x+d} \sqrt{e-c e x} \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 \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+3 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(a+b \sqrt{1-c^2 x^2}\right)-6 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(b c x-a \sqrt{1-c^2 x^2}\right)+b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{3 c d \sqrt{1-c^2 x^2}}","-\frac{2 a b e x \sqrt{1-c^2 x^2}}{\sqrt{c d x+d} \sqrt{e-c e x}}+\frac{e \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{e \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 b^2 e \left(1-c^2 x^2\right)}{c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 b^2 e x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{\sqrt{c d x+d} \sqrt{e-c e x}}",1,"(3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-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 + c*d*x]*Sqrt[e - c*e*x]*(b*c*x - a*Sqrt[1 - c^2*x^2])*ArcSin[c*x] + 3*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 3*a^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))])/(3*c*d*Sqrt[1 - c^2*x^2])","A",1
544,1,547,530,4.2656185,"\int \frac{\sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{3/2}} \, dx","Integrate[(Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(3/2),x]","\frac{3 a^2 \sqrt{d} \sqrt{e} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)-\frac{6 a^2 \sqrt{c d x+d} \sqrt{e-c e x}}{c x+1}-\frac{3 a b \sqrt{c d x+d} \sqrt{e-c e x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+4\right)-8 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\left(\sin ^{-1}(c x)-4\right) \sin ^{-1}(c x)-8 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{\sqrt{1-c^2 x^2} \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}+\frac{b^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(-24 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-\left(\sin ^{-1}(c x)^3 \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-(6+6 i) \sin ^{-1}(c x)^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+i \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+6 \sin ^{-1}(c x) \left(4 \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+i \pi \right) \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+12 \pi  \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(2 \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+\log \left(1-i e^{i \sin ^{-1}(c x)}\right)-\log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-2 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{\sqrt{1-c^2 x^2} \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}}{3 c d^2}","-\frac{e^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 i e^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 e^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{2 e^2 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 b e^2 \left(1-c^2 x^2\right)^{3/2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 i b e^2 \left(1-c^2 x^2\right)^{3/2} \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 i b^2 e^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 i b^2 e^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 i b^2 e^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}",1,"((-6*a^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(1 + c*x) + 3*a^2*Sqrt[d]*Sqrt[e]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - (3*a*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(Cos[ArcSin[c*x]/2]*(ArcSin[c*x]*(4 + ArcSin[c*x]) - 8*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + ((-4 + ArcSin[c*x])*ArcSin[c*x] - 8*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) + (b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*((-6 - 6*I)*ArcSin[c*x]^2*(Cos[ArcSin[c*x]/2] + I*Sin[ArcSin[c*x]/2]) - ArcSin[c*x]^3*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + 6*ArcSin[c*x]*(I*Pi + 4*Log[1 - I*E^(I*ArcSin[c*x])])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + 12*Pi*(2*Log[1 + E^((-I)*ArcSin[c*x])] + Log[1 - I*E^(I*ArcSin[c*x])] - 2*Log[Cos[ArcSin[c*x]/2]] - Log[Sin[(Pi + 2*ArcSin[c*x])/4]])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - (24*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(3*c*d^2)","A",1
545,1,698,486,8.2293374,"\int \frac{\sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{5/2}} \, dx","Integrate[(Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(5/2),x]","\frac{\sqrt{d (c x+1)} \sqrt{-e (c x-1)} \left(\frac{a^2}{3 d^3 (c x+1)}-\frac{2 a^2}{3 d^3 (c x+1)^2}\right)}{c}-\frac{a b \sqrt{c d x+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\sqrt{1-c^2 x^2} \sin ^{-1}(c x)-2 \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 \sin ^{-1}(c x)-4 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-2\right)+\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x)+2 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(3 \sin ^{-1}(c x)+6 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+4\right)\right)}{3 c d^3 (c x-1) \sqrt{(-c d x-d) (e-c e x)} \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4}-\frac{b^2 (c x-1) \sqrt{c d x+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(4 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+(1+i) \sin ^{-1}(c x)^2-i \pi  \sin ^{-1}(c x)-4 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)-2 \left(2 \sin ^{-1}(c x)+\pi \right) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+2 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+\frac{4 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}-\frac{2 \left(\sin ^{-1}(c x)+2\right) \sin ^{-1}(c x)}{\left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-\frac{2 \left(\sin ^{-1}(c x)^2-4\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+4 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)}{3 c d^3 \sqrt{1-c^2 x^2} \sqrt{(-c d x-d) (e-c e x)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}","\frac{i e^3 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 b e^3 \left(1-c^2 x^2\right)^{5/2} \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{e^3 \left(1-c^2 x^2\right)^{5/2} \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b e^3 \left(1-c^2 x^2\right)^{5/2} \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{e^3 \left(1-c^2 x^2\right)^{5/2} \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) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{4 i b^2 e^3 \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 b^2 e^3 \left(1-c^2 x^2\right)^{5/2} \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}",1,"(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*((-2*a^2)/(3*d^3*(1 + c*x)^2) + a^2/(3*d^3*(1 + c*x))))/c - (a*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[(3*ArcSin[c*x])/2]*(ArcSin[c*x] + 2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) - Cos[ArcSin[c*x]/2]*(4 + 3*ArcSin[c*x] + 6*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + 2*(-2 + 2*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x] - 4*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 2*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(3*c*d^3*(-1 + c*x)*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^4) - (b^2*(-1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-I)*Pi*ArcSin[c*x] + (1 + I)*ArcSin[c*x]^2 - 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 2*(Pi + 2*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])] + 4*Pi*Log[Cos[ArcSin[c*x]/2]] + 2*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, I*E^(I*ArcSin[c*x])] + (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 - (2*ArcSin[c*x]*(2 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (2*(-4 + ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(3*c*d^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^2)","A",0
546,1,574,697,4.0391457,"\int (d+c d x)^{5/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c*d*x)^(5/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2,x]","\frac{d^2 e \left(\sqrt{c d x+d} \sqrt{e-c e x} \left(-15 \left(480 a^2 \sqrt{1-c^2 x^2} \left(8 c^4 x^4+10 c^3 x^3-16 c^2 x^2-25 c x+8\right)-512 a b c x \left(3 c^4 x^4-10 c^2 x^2+15\right)-4800 b^2 \sqrt{1-c^2 x^2}+2400 b^2 \sin \left(2 \sin ^{-1}(c x)\right)+75 b^2 \sin \left(4 \sin ^{-1}(c x)\right)\right)+72000 a b \cos \left(2 \sin ^{-1}(c x)\right)+4500 a b \cos \left(4 \sin ^{-1}(c x)\right)+4000 b^2 \cos \left(3 \sin ^{-1}(c x)\right)+288 b^2 \cos \left(5 \sin ^{-1}(c x)\right)\right)-108000 a^2 \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+1800 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(5 \left(12 a-4 b \sqrt{1-c^2 x^2}+8 b \sin \left(2 \sin ^{-1}(c x)\right)+b \sin \left(4 \sin ^{-1}(c x)\right)\right)-10 b \cos \left(3 \sin ^{-1}(c x)\right)-2 b \cos \left(5 \sin ^{-1}(c x)\right)\right)-60 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(-4 \left(960 a c^2 x^2 \sqrt{1-c^2 x^2}-480 a \sqrt{1-c^2 x^2}-480 a c^4 x^4 \sqrt{1-c^2 x^2}+600 a \sin \left(2 \sin ^{-1}(c x)\right)+75 a \sin \left(4 \sin ^{-1}(c x)\right)+300 b c x+50 b \sin \left(3 \sin ^{-1}(c x)\right)+6 b \sin \left(5 \sin ^{-1}(c x)\right)\right)-1200 b \cos \left(2 \sin ^{-1}(c x)\right)-75 b \cos \left(4 \sin ^{-1}(c x)\right)\right)+36000 b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3\right)}{288000 c \sqrt{1-c^2 x^2}}","-\frac{3 b c d x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{8 \left(1-c^2 x^2\right)^{3/2}}+\frac{3 d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{8 \left(1-c^2 x^2\right)}+\frac{2 b d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{5 \left(1-c^2 x^2\right)^{3/2}}+\frac{d (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^3}{8 b c \left(1-c^2 x^2\right)^{3/2}}-\frac{d \left(1-c^2 x^2\right) (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{5 c}+\frac{b d \sqrt{1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{8 c}-\frac{4 b c^2 d x^3 (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{15 \left(1-c^2 x^2\right)^{3/2}}+\frac{2 b c^4 d x^5 (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{25 \left(1-c^2 x^2\right)^{3/2}}+\frac{1}{4} d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{15 b^2 d x (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 \left(1-c^2 x^2\right)}+\frac{2 b^2 d \left(1-c^2 x^2\right) (c d x+d)^{3/2} (e-c e x)^{3/2}}{125 c}+\frac{16 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2}}{75 c \left(1-c^2 x^2\right)}+\frac{9 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left(1-c^2 x^2\right)^{3/2}}-\frac{1}{32} b^2 d x (c d x+d)^{3/2} (e-c e x)^{3/2}+\frac{8 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2}}{225 c}",1,"(d^2*e*(36000*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 108000*a^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 1800*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(-10*b*Cos[3*ArcSin[c*x]] - 2*b*Cos[5*ArcSin[c*x]] + 5*(12*a - 4*b*Sqrt[1 - c^2*x^2] + 8*b*Sin[2*ArcSin[c*x]] + b*Sin[4*ArcSin[c*x]])) + Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(72000*a*b*Cos[2*ArcSin[c*x]] + 4000*b^2*Cos[3*ArcSin[c*x]] + 4500*a*b*Cos[4*ArcSin[c*x]] + 288*b^2*Cos[5*ArcSin[c*x]] - 15*(-4800*b^2*Sqrt[1 - c^2*x^2] - 512*a*b*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 480*a^2*Sqrt[1 - c^2*x^2]*(8 - 25*c*x - 16*c^2*x^2 + 10*c^3*x^3 + 8*c^4*x^4) + 2400*b^2*Sin[2*ArcSin[c*x]] + 75*b^2*Sin[4*ArcSin[c*x]])) - 60*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(-1200*b*Cos[2*ArcSin[c*x]] - 75*b*Cos[4*ArcSin[c*x]] - 4*(300*b*c*x - 480*a*Sqrt[1 - c^2*x^2] + 960*a*c^2*x^2*Sqrt[1 - c^2*x^2] - 480*a*c^4*x^4*Sqrt[1 - c^2*x^2] + 600*a*Sin[2*ArcSin[c*x]] + 50*b*Sin[3*ArcSin[c*x]] + 75*a*Sin[4*ArcSin[c*x]] + 6*b*Sin[5*ArcSin[c*x]]))))/(288000*c*Sqrt[1 - c^2*x^2])","A",1
547,1,373,362,2.0601131,"\int (d+c d x)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2,x]","\frac{d e \sqrt{c d x+d} \sqrt{e-c e x} \left(160 a^2 c x \sqrt{1-c^2 x^2}-64 a^2 c^3 x^3 \sqrt{1-c^2 x^2}+64 a b \cos \left(2 \sin ^{-1}(c x)\right)+4 a b \cos \left(4 \sin ^{-1}(c x)\right)-32 b^2 \sin \left(2 \sin ^{-1}(c x)\right)-b^2 \sin \left(4 \sin ^{-1}(c x)\right)\right)-96 a^2 d^{3/2} e^{3/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+8 b d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(12 a+8 b \sin \left(2 \sin ^{-1}(c x)\right)+b \sin \left(4 \sin ^{-1}(c x)\right)\right)+4 b d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(4 a \left(8 \sin \left(2 \sin ^{-1}(c x)\right)+\sin \left(4 \sin ^{-1}(c x)\right)\right)+16 b \cos \left(2 \sin ^{-1}(c x)\right)+b \cos \left(4 \sin ^{-1}(c x)\right)\right)+32 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{256 c \sqrt{1-c^2 x^2}}","\frac{(c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^3}{8 b c \left(1-c^2 x^2\right)^{3/2}}+\frac{3 x (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{8 \left(1-c^2 x^2\right)}+\frac{b \sqrt{1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{8 c}-\frac{3 b c x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{8 \left(1-c^2 x^2\right)^{3/2}}+\frac{1}{4} x (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{15 b^2 x (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 \left(1-c^2 x^2\right)}+\frac{9 b^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left(1-c^2 x^2\right)^{3/2}}-\frac{1}{32} b^2 x (c d x+d)^{3/2} (e-c e x)^{3/2}",1,"(32*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 96*a^2*d^(3/2)*e^(3/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 8*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(12*a + 8*b*Sin[2*ArcSin[c*x]] + b*Sin[4*ArcSin[c*x]]) + d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(160*a^2*c*x*Sqrt[1 - c^2*x^2] - 64*a^2*c^3*x^3*Sqrt[1 - c^2*x^2] + 64*a*b*Cos[2*ArcSin[c*x]] + 4*a*b*Cos[4*ArcSin[c*x]] - 32*b^2*Sin[2*ArcSin[c*x]] - b^2*Sin[4*ArcSin[c*x]]) + 4*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(16*b*Cos[2*ArcSin[c*x]] + b*Cos[4*ArcSin[c*x]] + 4*a*(8*Sin[2*ArcSin[c*x]] + Sin[4*ArcSin[c*x]])))/(256*c*Sqrt[1 - c^2*x^2])","A",1
548,1,440,455,2.0609218,"\int \sqrt{d+c d x} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[d + c*d*x]*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2,x]","\frac{e \sqrt{c d x+d} \sqrt{e-c e x} \left(-3 \left(4 \left(3 a^2 \sqrt{1-c^2 x^2} \left(2 c^2 x^2-3 c x-2\right)-4 a b c x \left(c^2 x^2-3\right)+9 b^2 \sqrt{1-c^2 x^2}\right)+9 b^2 \sin \left(2 \sin ^{-1}(c x)\right)\right)+54 a b \cos \left(2 \sin ^{-1}(c x)\right)-4 b^2 \cos \left(3 \sin ^{-1}(c x)\right)\right)-108 a^2 \sqrt{d} e^{3/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+18 b e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(6 a+3 b \sqrt{1-c^2 x^2}+3 b \sin \left(2 \sin ^{-1}(c x)\right)+b \cos \left(3 \sin ^{-1}(c x)\right)\right)-6 b e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(2 \left(12 a c^2 x^2 \sqrt{1-c^2 x^2}-12 a \sqrt{1-c^2 x^2}-9 a \sin \left(2 \sin ^{-1}(c x)\right)+9 b c x+b \sin \left(3 \sin ^{-1}(c x)\right)\right)-9 b \cos \left(2 \sin ^{-1}(c x)\right)\right)+36 b^2 e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{216 c \sqrt{1-c^2 x^2}}","-\frac{b c e x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{1-c^2 x^2}}-\frac{2 b e x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{3 \sqrt{1-c^2 x^2}}+\frac{e \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^3}{6 b c \sqrt{1-c^2 x^2}}+\frac{e \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c}+\frac{2 b c^2 e x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{9 \sqrt{1-c^2 x^2}}+\frac{1}{2} e x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2 b^2 e \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x}}{27 c}+\frac{b^2 e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}-\frac{1}{4} b^2 e x \sqrt{c d x+d} \sqrt{e-c e x}-\frac{4 b^2 e \sqrt{c d x+d} \sqrt{e-c e x}}{9 c}",1,"(36*b^2*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 108*a^2*Sqrt[d]*e^(3/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 18*b*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(6*a + 3*b*Sqrt[1 - c^2*x^2] + b*Cos[3*ArcSin[c*x]] + 3*b*Sin[2*ArcSin[c*x]]) + e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(54*a*b*Cos[2*ArcSin[c*x]] - 4*b^2*Cos[3*ArcSin[c*x]] - 3*(4*(9*b^2*Sqrt[1 - c^2*x^2] - 4*a*b*c*x*(-3 + c^2*x^2) + 3*a^2*Sqrt[1 - c^2*x^2]*(-2 - 3*c*x + 2*c^2*x^2)) + 9*b^2*Sin[2*ArcSin[c*x]])) - 6*b*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(-9*b*Cos[2*ArcSin[c*x]] + 2*(9*b*c*x - 12*a*Sqrt[1 - c^2*x^2] + 12*a*c^2*x^2*Sqrt[1 - c^2*x^2] - 9*a*Sin[2*ArcSin[c*x]] + b*Sin[3*ArcSin[c*x]])))/(216*c*Sqrt[1 - c^2*x^2])","A",1
549,1,358,398,2.3855557,"\int \frac{(e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d+c d x}} \, dx","Integrate[((e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/Sqrt[d + c*d*x],x]","\frac{e \sqrt{c d x+d} \sqrt{e-c e x} \left(-4 \left(a^2 (c x-4) \sqrt{1-c^2 x^2}+8 a b c x+8 b^2 \sqrt{1-c^2 x^2}\right)-2 a b \cos \left(2 \sin ^{-1}(c x)\right)+b^2 \sin \left(2 \sin ^{-1}(c x)\right)\right)-12 a^2 \sqrt{d} e^{3/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+2 b e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(6 a+8 b \sqrt{1-c^2 x^2}-b \sin \left(2 \sin ^{-1}(c x)\right)\right)-2 b e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(4 a (c x-4) \sqrt{1-c^2 x^2}+16 b c x+b \cos \left(2 \sin ^{-1}(c x)\right)\right)+4 b^2 e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{8 c d \sqrt{1-c^2 x^2}}","\frac{e^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{2 b c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 e^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{e^2 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{2 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{b c e^2 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{4 b e^2 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{c d x+d} \sqrt{e-c e x}}-\frac{4 b^2 e^2 \left(1-c^2 x^2\right)}{c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{b^2 e^2 x \left(1-c^2 x^2\right)}{4 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{b^2 e^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt{c d x+d} \sqrt{e-c e x}}",1,"(4*b^2*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 12*a^2*Sqrt[d]*e^(3/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 2*b*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(16*b*c*x + 4*a*(-4 + c*x)*Sqrt[1 - c^2*x^2] + b*Cos[2*ArcSin[c*x]]) + 2*b*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(6*a + 8*b*Sqrt[1 - c^2*x^2] - b*Sin[2*ArcSin[c*x]]) + e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-4*(8*a*b*c*x + 8*b^2*Sqrt[1 - c^2*x^2] + a^2*(-4 + c*x)*Sqrt[1 - c^2*x^2]) - 2*a*b*Cos[2*ArcSin[c*x]] + b^2*Sin[2*ArcSin[c*x]]))/(8*c*d*Sqrt[1 - c^2*x^2])","A",1
550,1,1086,714,8.5758288,"\int \frac{(e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{3/2}} \, dx","Integrate[((e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(3/2),x]","\frac{9 \sqrt{d} e^{3/2} (c x+1) \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c x d+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) a^2-3 e (c x+5) \sqrt{c x d+d} \sqrt{e-c e x} \sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) a^2-3 b e (c x+1) \sqrt{c x d+d} \sqrt{e-c e x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+4\right)-8 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)+\left(\left(\sin ^{-1}(c x)-4\right) \sin ^{-1}(c x)-8 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) a-6 b e (c x+1) \sqrt{c x d+d} \sqrt{e-c e x} \left(\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^2+\left(\left(\sqrt{1-c^2 x^2}+2\right) \cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\left(\sqrt{1-c^2 x^2}-2\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)-\left(c x+4 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right) a-b^2 e (c x+1) \sqrt{c x d+d} \sqrt{e-c e x} \left(\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^3+(6+6 i) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+i \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^2-6 i \left(\pi -4 i \log \left(1-i e^{i \sin ^{-1}(c x)}\right)\right) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)-12 \pi  \left(2 \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+\log \left(1-i e^{i \sin ^{-1}(c x)}\right)-2 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-\log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)\right) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+24 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-b^2 e (c x+1) \sqrt{c x d+d} \sqrt{e-c e x} \left(2 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^3+3 \left(\left(\sqrt{1-c^2 x^2}+(2+2 i)\right) \cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\left(\sqrt{1-c^2 x^2}-(2-2 i)\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)^2-6 i \left(-i c x-4 i \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+\pi \right) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \sin ^{-1}(c x)-6 \left(4 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+2 \pi  \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-4 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-2 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+\sqrt{1-c^2 x^2}\right) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+24 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)}{3 c d^2 (c x+1) \sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}","\frac{2 a b e^3 x \left(1-c^2 x^2\right)^{3/2}}{(c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{e^3 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^3}{b c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{e^3 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 i e^3 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 e^3 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 e^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 b e^3 \left(1-c^2 x^2\right)^{3/2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{16 i b e^3 \left(1-c^2 x^2\right)^{3/2} \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 i b^2 e^3 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 i b^2 e^3 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 i b^2 e^3 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{2 b^2 e^3 \left(1-c^2 x^2\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{2 b^2 e^3 x \left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x)}{(c d x+d)^{3/2} (e-c e x)^{3/2}}",1,"(-3*a^2*e*(5 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + 9*a^2*Sqrt[d]*e^(3/2)*(1 + c*x)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 3*a*b*e*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(Cos[ArcSin[c*x]/2]*(ArcSin[c*x]*(4 + ArcSin[c*x]) - 8*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + ((-4 + ArcSin[c*x])*ArcSin[c*x] - 8*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]) - b^2*e*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*((6 + 6*I)*ArcSin[c*x]^2*(Cos[ArcSin[c*x]/2] + I*Sin[ArcSin[c*x]/2]) + ArcSin[c*x]^3*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - (6*I)*ArcSin[c*x]*(Pi - (4*I)*Log[1 - I*E^(I*ArcSin[c*x])])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 12*Pi*(2*Log[1 + E^((-I)*ArcSin[c*x])] + Log[1 - I*E^(I*ArcSin[c*x])] - 2*Log[Cos[ArcSin[c*x]/2]] - Log[Sin[(Pi + 2*ArcSin[c*x])/4]])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + (24*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) - 6*a*b*e*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(ArcSin[c*x]^2*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - (c*x + 4*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + ArcSin[c*x]*((2 + Sqrt[1 - c^2*x^2])*Cos[ArcSin[c*x]/2] + (-2 + Sqrt[1 - c^2*x^2])*Sin[ArcSin[c*x]/2])) - b^2*e*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(2*ArcSin[c*x]^3*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - (6*I)*ArcSin[c*x]*(Pi - I*c*x - (4*I)*Log[1 - I*E^(I*ArcSin[c*x])])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) - 6*(Sqrt[1 - c^2*x^2] + 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 2*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 4*Pi*Log[Cos[ArcSin[c*x]/2]] - 2*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + (24*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + 3*ArcSin[c*x]^2*(((2 + 2*I) + Sqrt[1 - c^2*x^2])*Cos[ArcSin[c*x]/2] + ((-2 + 2*I) + Sqrt[1 - c^2*x^2])*Sin[ArcSin[c*x]/2])))/(3*c*d^2*(1 + c*x)*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]))","A",1
551,1,1438,544,10.3391472,"\int \frac{(e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{5/2}} \, dx","Integrate[((e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(5/2),x]","-\frac{e^{3/2} \tan ^{-1}\left(\frac{c x \sqrt{-e (c x-1)} \sqrt{d (c x+1)}}{\sqrt{d} \sqrt{e} (c x-1) (c x+1)}\right) a^2}{c d^{5/2}}-\frac{b e \sqrt{c x d+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(9 \sin ^{-1}(c x)^2+6 \sin ^{-1}(c x)-84 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-8\right)+\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(\left(14-3 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+28 \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 \left(6 \sin ^{-1}(c x)^2+4 \sin ^{-1}(c x)+\sqrt{1-c^2 x^2} \left(\sin ^{-1}(c x) \left(3 \sin ^{-1}(c x)+14\right)-28 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-56 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-4\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) a}{6 c d^3 (c x-1) \sqrt{(-c x d-d) (e-c e x)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4}-\frac{b e \sqrt{c x d+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x)+2 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(3 \sin ^{-1}(c x)+6 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+4\right)+2 \left(\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+2 \sin ^{-1}(c x)-2 \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-4 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-2\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) a}{3 c d^3 (c x-1) \sqrt{(-c x d-d) (e-c e x)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4}+\frac{\sqrt{-e (c x-1)} \sqrt{d (c x+1)} \left(\frac{8 a^2 e}{3 d^3 (c x+1)}-\frac{4 a^2 e}{3 d^3 (c x+1)^2}\right)}{c}-\frac{b^2 e (c x-1) \sqrt{c x d+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(\frac{4 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}+(1+i) \sin ^{-1}(c x)^2-\frac{2 \left(\sin ^{-1}(c x)+2\right) \sin ^{-1}(c x)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-i \pi  \sin ^{-1}(c x)-4 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)-2 \left(2 \sin ^{-1}(c x)+\pi \right) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+4 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+4 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-\frac{2 \left(\sin ^{-1}(c x)^2-4\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}\right)}{3 c d^3 \sqrt{(-c x d-d) (e-c e x)} \sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}+\frac{b^2 e (c x-1) \sqrt{c x d+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(-\sin ^{-1}(c x)^3-\frac{4 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}-(7+7 i) \sin ^{-1}(c x)^2+\frac{2 \left(\sin ^{-1}(c x)+2\right) \sin ^{-1}(c x)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}+7 i \pi  \sin ^{-1}(c x)+28 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+14 \left(2 \sin ^{-1}(c x)+\pi \right) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-28 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-14 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-28 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+\frac{2 \left(7 \sin ^{-1}(c x)^2-4\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}\right)}{3 c d^3 \sqrt{(-c x d-d) (e-c e x)} \sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}","\frac{e^4 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{8 i e^4 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{32 b e^4 \left(1-c^2 x^2\right)^{5/2} \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{8 e^4 \left(1-c^2 x^2\right)^{5/2} \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 b e^4 \left(1-c^2 x^2\right)^{5/2} \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 e^4 \left(1-c^2 x^2\right)^{5/2} \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) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{32 i b^2 e^4 \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{8 b^2 e^4 \left(1-c^2 x^2\right)^{5/2} \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}",1,"(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*((-4*a^2*e)/(3*d^3*(1 + c*x)^2) + (8*a^2*e)/(3*d^3*(1 + c*x))))/c - (a^2*e^(3/2)*ArcTan[(c*x*Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)])/(Sqrt[d]*Sqrt[e]*(-1 + c*x)*(1 + c*x))])/(c*d^(5/2)) - (a*b*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2]*(-8 + 6*ArcSin[c*x] + 9*ArcSin[c*x]^2 - 84*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + Cos[(3*ArcSin[c*x])/2]*((14 - 3*ArcSin[c*x])*ArcSin[c*x] + 28*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + 2*(-4 + 4*ArcSin[c*x] + 6*ArcSin[c*x]^2 + Sqrt[1 - c^2*x^2]*(ArcSin[c*x]*(14 + 3*ArcSin[c*x]) - 28*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) - 56*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(6*c*d^3*(-1 + c*x)*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^4) - (a*b*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[(3*ArcSin[c*x])/2]*(ArcSin[c*x] + 2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) - Cos[ArcSin[c*x]/2]*(4 + 3*ArcSin[c*x] + 6*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + 2*(-2 + 2*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x] - 4*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 2*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(3*c*d^3*(-1 + c*x)*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^4) - (b^2*e*(-1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-I)*Pi*ArcSin[c*x] + (1 + I)*ArcSin[c*x]^2 - 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 2*(Pi + 2*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])] + 4*Pi*Log[Cos[ArcSin[c*x]/2]] + 2*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, I*E^(I*ArcSin[c*x])] + (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 - (2*ArcSin[c*x]*(2 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (2*(-4 + ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(3*c*d^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^2) + (b^2*e*(-1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((7*I)*Pi*ArcSin[c*x] - (7 + 7*I)*ArcSin[c*x]^2 - ArcSin[c*x]^3 + 28*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 14*(Pi + 2*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])] - 28*Pi*Log[Cos[ArcSin[c*x]/2]] - 14*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (28*I)*PolyLog[2, I*E^(I*ArcSin[c*x])] - (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 + (2*ArcSin[c*x]*(2 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 + (2*(-4 + 7*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(3*c*d^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^2)","B",0
552,1,450,502,3.1262856,"\int (d+c d x)^{5/2} (e-c e x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2,x]","\frac{d^2 e^2 \left(\sqrt{c d x+d} \sqrt{e-c e x} \left(9504 a^2 c x \sqrt{1-c^2 x^2}+2304 a^2 c^5 x^5 \sqrt{1-c^2 x^2}-7488 a^2 c^3 x^3 \sqrt{1-c^2 x^2}+3240 a b \cos \left(2 \sin ^{-1}(c x)\right)+324 a b \cos \left(4 \sin ^{-1}(c x)\right)+24 a b \cos \left(6 \sin ^{-1}(c x)\right)-1620 b^2 \sin \left(2 \sin ^{-1}(c x)\right)-81 b^2 \sin \left(4 \sin ^{-1}(c x)\right)-4 b^2 \sin \left(6 \sin ^{-1}(c x)\right)\right)-4320 a^2 \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+72 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(60 a+45 b \sin \left(2 \sin ^{-1}(c x)\right)+9 b \sin \left(4 \sin ^{-1}(c x)\right)+b \sin \left(6 \sin ^{-1}(c x)\right)\right)+12 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(540 a \sin \left(2 \sin ^{-1}(c x)\right)+108 a \sin \left(4 \sin ^{-1}(c x)\right)+12 a \sin \left(6 \sin ^{-1}(c x)\right)+270 b \cos \left(2 \sin ^{-1}(c x)\right)+27 b \cos \left(4 \sin ^{-1}(c x)\right)+2 b \cos \left(6 \sin ^{-1}(c x)\right)\right)+1440 b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3\right)}{13824 c \sqrt{1-c^2 x^2}}","\frac{5 (c d x+d)^{5/2} (e-c e x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^3}{48 b c \left(1-c^2 x^2\right)^{5/2}}+\frac{5 x (c d x+d)^{5/2} (e-c e x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{24 \left(1-c^2 x^2\right)}+\frac{5 x (c d x+d)^{5/2} (e-c e x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{16 \left(1-c^2 x^2\right)^2}+\frac{b \sqrt{1-c^2 x^2} (c d x+d)^{5/2} (e-c e x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{18 c}+\frac{5 b (c d x+d)^{5/2} (e-c e x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{48 c \sqrt{1-c^2 x^2}}-\frac{5 b c x^2 (c d x+d)^{5/2} (e-c e x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)}{16 \left(1-c^2 x^2\right)^{5/2}}+\frac{1}{6} x (c d x+d)^{5/2} (e-c e x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{65 b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2}}{1728 \left(1-c^2 x^2\right)}-\frac{245 b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2}}{1152 \left(1-c^2 x^2\right)^2}+\frac{115 b^2 (c d x+d)^{5/2} (e-c e x)^{5/2} \sin ^{-1}(c x)}{1152 c \left(1-c^2 x^2\right)^{5/2}}-\frac{1}{108} b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2}",1,"(d^2*e^2*(1440*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 4320*a^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 12*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(270*b*Cos[2*ArcSin[c*x]] + 27*b*Cos[4*ArcSin[c*x]] + 2*b*Cos[6*ArcSin[c*x]] + 540*a*Sin[2*ArcSin[c*x]] + 108*a*Sin[4*ArcSin[c*x]] + 12*a*Sin[6*ArcSin[c*x]]) + 72*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(60*a + 45*b*Sin[2*ArcSin[c*x]] + 9*b*Sin[4*ArcSin[c*x]] + b*Sin[6*ArcSin[c*x]]) + Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(9504*a^2*c*x*Sqrt[1 - c^2*x^2] - 7488*a^2*c^3*x^3*Sqrt[1 - c^2*x^2] + 2304*a^2*c^5*x^5*Sqrt[1 - c^2*x^2] + 3240*a*b*Cos[2*ArcSin[c*x]] + 324*a*b*Cos[4*ArcSin[c*x]] + 24*a*b*Cos[6*ArcSin[c*x]] - 1620*b^2*Sin[2*ArcSin[c*x]] - 81*b^2*Sin[4*ArcSin[c*x]] - 4*b^2*Sin[6*ArcSin[c*x]])))/(13824*c*Sqrt[1 - c^2*x^2])","A",1
553,1,574,697,4.125124,"\int (d+c d x)^{3/2} (e-c e x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c*d*x)^(3/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2,x]","\frac{d e^2 \left(\sqrt{c d x+d} \sqrt{e-c e x} \left(-15 \left(-480 a^2 \sqrt{1-c^2 x^2} \left(8 c^4 x^4-10 c^3 x^3-16 c^2 x^2+25 c x+8\right)+512 a b c x \left(3 c^4 x^4-10 c^2 x^2+15\right)+4800 b^2 \sqrt{1-c^2 x^2}+2400 b^2 \sin \left(2 \sin ^{-1}(c x)\right)+75 b^2 \sin \left(4 \sin ^{-1}(c x)\right)\right)+72000 a b \cos \left(2 \sin ^{-1}(c x)\right)+4500 a b \cos \left(4 \sin ^{-1}(c x)\right)-4000 b^2 \cos \left(3 \sin ^{-1}(c x)\right)-288 b^2 \cos \left(5 \sin ^{-1}(c x)\right)\right)-108000 a^2 \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+1800 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(5 \left(12 a+4 b \sqrt{1-c^2 x^2}+8 b \sin \left(2 \sin ^{-1}(c x)\right)+b \sin \left(4 \sin ^{-1}(c x)\right)\right)+10 b \cos \left(3 \sin ^{-1}(c x)\right)+2 b \cos \left(5 \sin ^{-1}(c x)\right)\right)+60 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(4 \left(-960 a c^2 x^2 \sqrt{1-c^2 x^2}+480 a \sqrt{1-c^2 x^2}+480 a c^4 x^4 \sqrt{1-c^2 x^2}+600 a \sin \left(2 \sin ^{-1}(c x)\right)+75 a \sin \left(4 \sin ^{-1}(c x)\right)-300 b c x-50 b \sin \left(3 \sin ^{-1}(c x)\right)-6 b \sin \left(5 \sin ^{-1}(c x)\right)\right)+1200 b \cos \left(2 \sin ^{-1}(c x)\right)+75 b \cos \left(4 \sin ^{-1}(c x)\right)\right)+36000 b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3\right)}{288000 c \sqrt{1-c^2 x^2}}","-\frac{3 b c e x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{8 \left(1-c^2 x^2\right)^{3/2}}+\frac{3 e x (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{8 \left(1-c^2 x^2\right)}-\frac{2 b e x (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{5 \left(1-c^2 x^2\right)^{3/2}}+\frac{e (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^3}{8 b c \left(1-c^2 x^2\right)^{3/2}}+\frac{e \left(1-c^2 x^2\right) (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{5 c}+\frac{b e \sqrt{1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{8 c}+\frac{4 b c^2 e x^3 (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{15 \left(1-c^2 x^2\right)^{3/2}}-\frac{2 b c^4 e x^5 (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{25 \left(1-c^2 x^2\right)^{3/2}}+\frac{1}{4} e x (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{15 b^2 e x (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 \left(1-c^2 x^2\right)}-\frac{2 b^2 e \left(1-c^2 x^2\right) (c d x+d)^{3/2} (e-c e x)^{3/2}}{125 c}-\frac{16 b^2 e (c d x+d)^{3/2} (e-c e x)^{3/2}}{75 c \left(1-c^2 x^2\right)}+\frac{9 b^2 e (c d x+d)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left(1-c^2 x^2\right)^{3/2}}-\frac{1}{32} b^2 e x (c d x+d)^{3/2} (e-c e x)^{3/2}-\frac{8 b^2 e (c d x+d)^{3/2} (e-c e x)^{3/2}}{225 c}",1,"(d*e^2*(36000*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 108000*a^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 1800*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(10*b*Cos[3*ArcSin[c*x]] + 2*b*Cos[5*ArcSin[c*x]] + 5*(12*a + 4*b*Sqrt[1 - c^2*x^2] + 8*b*Sin[2*ArcSin[c*x]] + b*Sin[4*ArcSin[c*x]])) + Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(72000*a*b*Cos[2*ArcSin[c*x]] - 4000*b^2*Cos[3*ArcSin[c*x]] + 4500*a*b*Cos[4*ArcSin[c*x]] - 288*b^2*Cos[5*ArcSin[c*x]] - 15*(4800*b^2*Sqrt[1 - c^2*x^2] + 512*a*b*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) - 480*a^2*Sqrt[1 - c^2*x^2]*(8 + 25*c*x - 16*c^2*x^2 - 10*c^3*x^3 + 8*c^4*x^4) + 2400*b^2*Sin[2*ArcSin[c*x]] + 75*b^2*Sin[4*ArcSin[c*x]])) + 60*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(1200*b*Cos[2*ArcSin[c*x]] + 75*b*Cos[4*ArcSin[c*x]] + 4*(-300*b*c*x + 480*a*Sqrt[1 - c^2*x^2] - 960*a*c^2*x^2*Sqrt[1 - c^2*x^2] + 480*a*c^4*x^4*Sqrt[1 - c^2*x^2] + 600*a*Sin[2*ArcSin[c*x]] - 50*b*Sin[3*ArcSin[c*x]] + 75*a*Sin[4*ArcSin[c*x]] - 6*b*Sin[5*ArcSin[c*x]]))))/(288000*c*Sqrt[1 - c^2*x^2])","A",1
554,1,555,613,2.5862298,"\int \sqrt{d+c d x} (e-c e x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[d + c*d*x]*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2,x]","\frac{e^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(3 \left(-1536 a^2 c^2 x^2 \sqrt{1-c^2 x^2}+864 a^2 c x \sqrt{1-c^2 x^2}+1536 a^2 \sqrt{1-c^2 x^2}+576 a^2 c^3 x^3 \sqrt{1-c^2 x^2}+1024 a b c^3 x^3-3072 a b c x-36 a b \cos \left(4 \sin ^{-1}(c x)\right)-2304 b^2 \sqrt{1-c^2 x^2}-288 b^2 \sin \left(2 \sin ^{-1}(c x)\right)+9 b^2 \sin \left(4 \sin ^{-1}(c x)\right)\right)+1728 a b \cos \left(2 \sin ^{-1}(c x)\right)-256 b^2 \cos \left(3 \sin ^{-1}(c x)\right)\right)-4320 a^2 \sqrt{d} e^{5/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+72 b e^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(60 a+48 b \sqrt{1-c^2 x^2}+24 b \sin \left(2 \sin ^{-1}(c x)\right)-3 b \sin \left(4 \sin ^{-1}(c x)\right)+16 b \cos \left(3 \sin ^{-1}(c x)\right)\right)-12 b e^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(768 a c^2 x^2 \sqrt{1-c^2 x^2}-768 a \sqrt{1-c^2 x^2}-288 a \sin \left(2 \sin ^{-1}(c x)\right)+36 a \sin \left(4 \sin ^{-1}(c x)\right)+576 b c x+64 b \sin \left(3 \sin ^{-1}(c x)\right)-144 b \cos \left(2 \sin ^{-1}(c x)\right)+9 b \cos \left(4 \sin ^{-1}(c x)\right)\right)+1440 b^2 e^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{6912 c \sqrt{1-c^2 x^2}}","\frac{1}{4} c^2 e^2 x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{3 b c e^2 x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{8 \sqrt{1-c^2 x^2}}-\frac{4 b e^2 x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{3 \sqrt{1-c^2 x^2}}+\frac{5 e^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^3}{24 b c \sqrt{1-c^2 x^2}}+\frac{2 e^2 \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c}+\frac{4 b c^2 e^2 x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{9 \sqrt{1-c^2 x^2}}-\frac{b c^3 e^2 x^4 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{8 \sqrt{1-c^2 x^2}}+\frac{3}{8} e^2 x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{1}{32} b^2 c^2 e^2 x^3 \sqrt{c d x+d} \sqrt{e-c e x}-\frac{4 b^2 e^2 \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x}}{27 c}+\frac{15 b^2 e^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{64 c \sqrt{1-c^2 x^2}}-\frac{15}{64} b^2 e^2 x \sqrt{c d x+d} \sqrt{e-c e x}-\frac{8 b^2 e^2 \sqrt{c d x+d} \sqrt{e-c e x}}{9 c}",1,"(1440*b^2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 4320*a^2*Sqrt[d]*e^(5/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 12*b*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(576*b*c*x - 768*a*Sqrt[1 - c^2*x^2] + 768*a*c^2*x^2*Sqrt[1 - c^2*x^2] - 144*b*Cos[2*ArcSin[c*x]] + 9*b*Cos[4*ArcSin[c*x]] - 288*a*Sin[2*ArcSin[c*x]] + 64*b*Sin[3*ArcSin[c*x]] + 36*a*Sin[4*ArcSin[c*x]]) + 72*b*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(60*a + 48*b*Sqrt[1 - c^2*x^2] + 16*b*Cos[3*ArcSin[c*x]] + 24*b*Sin[2*ArcSin[c*x]] - 3*b*Sin[4*ArcSin[c*x]]) + e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1728*a*b*Cos[2*ArcSin[c*x]] - 256*b^2*Cos[3*ArcSin[c*x]] + 3*(-3072*a*b*c*x + 1024*a*b*c^3*x^3 + 1536*a^2*Sqrt[1 - c^2*x^2] - 2304*b^2*Sqrt[1 - c^2*x^2] + 864*a^2*c*x*Sqrt[1 - c^2*x^2] - 1536*a^2*c^2*x^2*Sqrt[1 - c^2*x^2] + 576*a^2*c^3*x^3*Sqrt[1 - c^2*x^2] - 36*a*b*Cos[4*ArcSin[c*x]] - 288*b^2*Sin[2*ArcSin[c*x]] + 9*b^2*Sin[4*ArcSin[c*x]])))/(6912*c*Sqrt[1 - c^2*x^2])","A",1
555,1,473,559,3.622278,"\int \frac{(e-c e x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d+c d x}} \, dx","Integrate[((e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2)/Sqrt[d + c*d*x],x]","\frac{e^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(72 a^2 c^2 x^2 \sqrt{1-c^2 x^2}-324 a^2 c x \sqrt{1-c^2 x^2}+792 a^2 \sqrt{1-c^2 x^2}-1620 a b c x+12 a b \sin \left(3 \sin ^{-1}(c x)\right)-162 a b \cos \left(2 \sin ^{-1}(c x)\right)-1620 b^2 \sqrt{1-c^2 x^2}+81 b^2 \sin \left(2 \sin ^{-1}(c x)\right)+4 b^2 \cos \left(3 \sin ^{-1}(c x)\right)\right)-540 a^2 \sqrt{d} e^{5/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+18 b e^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(30 a+45 b \sqrt{1-c^2 x^2}-9 b \sin \left(2 \sin ^{-1}(c x)\right)-b \cos \left(3 \sin ^{-1}(c x)\right)\right)-6 b e^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(108 a c x \sqrt{1-c^2 x^2}-270 a \sqrt{1-c^2 x^2}+6 a \cos \left(3 \sin ^{-1}(c x)\right)+8 b c^3 x^3+264 b c x+27 b \cos \left(2 \sin ^{-1}(c x)\right)\right)+180 b^2 e^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{216 c d \sqrt{1-c^2 x^2}}","\frac{5 e^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{6 b c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{c e^3 x^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{3 e^3 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{2 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{11 e^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{3 b c e^3 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{22 b e^3 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 b c^2 e^3 x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 b^2 e^3 \left(1-c^2 x^2\right)^2}{27 c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{3 b^2 e^3 x \left(1-c^2 x^2\right)}{4 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{68 b^2 e^3 \left(1-c^2 x^2\right)}{9 c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{3 b^2 e^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt{c d x+d} \sqrt{e-c e x}}",1,"(180*b^2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 540*a^2*Sqrt[d]*e^(5/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 6*b*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(264*b*c*x + 8*b*c^3*x^3 - 270*a*Sqrt[1 - c^2*x^2] + 108*a*c*x*Sqrt[1 - c^2*x^2] + 27*b*Cos[2*ArcSin[c*x]] + 6*a*Cos[3*ArcSin[c*x]]) + 18*b*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(30*a + 45*b*Sqrt[1 - c^2*x^2] - b*Cos[3*ArcSin[c*x]] - 9*b*Sin[2*ArcSin[c*x]]) + e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-1620*a*b*c*x + 792*a^2*Sqrt[1 - c^2*x^2] - 1620*b^2*Sqrt[1 - c^2*x^2] - 324*a^2*c*x*Sqrt[1 - c^2*x^2] + 72*a^2*c^2*x^2*Sqrt[1 - c^2*x^2] - 162*a*b*Cos[2*ArcSin[c*x]] + 4*b^2*Cos[3*ArcSin[c*x]] + 81*b^2*Sin[2*ArcSin[c*x]] + 12*a*b*Sin[3*ArcSin[c*x]]))/(216*c*d*Sqrt[1 - c^2*x^2])","A",1
556,1,2291,918,11.5367575,"\int \frac{(e-c e x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{3/2}} \, dx","Integrate[((e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(3/2),x]","\text{Result too large to show}","-\frac{5 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^3 e^4}{2 b c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{b^2 x \left(1-c^2 x^2\right)^2 e^4}{4 (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 \left(1-c^2 x^2\right)^2 e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{x \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 e^4}{2 (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 i \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 e^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 a b x \left(1-c^2 x^2\right)^{3/2} e^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 b^2 x \left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x) e^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{b^2 \left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x) e^4}{4 c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{b c x^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) e^4}{2 (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{32 i b \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{16 b \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{16 i b^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{16 i b^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 i b^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right) e^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}",1,"(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*((-4*a^2*e^2)/d^2 + (a^2*c*e^2*x)/(2*d^2) - (8*a^2*e^2)/(d^2*(1 + c*x))))/c + (15*a^2*e^(5/2)*ArcTan[(c*x*Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)])/(Sqrt[d]*Sqrt[e]*(-1 + c*x)*(1 + c*x))])/(2*c*d^(3/2)) - (a*b*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*(ArcSin[c*x]*(4 + ArcSin[c*x]) - 8*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + ((-4 + ArcSin[c*x])*ArcSin[c*x] - 8*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(c*d^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) - (4*a*b*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*(-(c*x) + 2*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]^2 - 4*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + (-(c*x) - 2*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]^2 - 4*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(c*d^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) - (b^2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*((-6*I)*Pi*ArcSin[c*x] + (6 + 6*I)*ArcSin[c*x]^2 + ArcSin[c*x]^3 - 24*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 12*(Pi + 2*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])] + 24*Pi*Log[Cos[ArcSin[c*x]/2]] + 12*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]]) + ((-6*I)*Pi*ArcSin[c*x] - (6 - 6*I)*ArcSin[c*x]^2 + ArcSin[c*x]^3 - 24*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 12*(Pi + 2*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])] + 24*Pi*Log[Cos[ArcSin[c*x]/2]] + 12*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]])*Sin[ArcSin[c*x]/2] + (24*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(3*c*d^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) - (2*b^2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*(3*Sqrt[1 - c^2*x^2]*(-2 + ArcSin[c*x]^2) + 2*((-3*I)*Pi*ArcSin[c*x] - 3*c*x*ArcSin[c*x] + (3 + 3*I)*ArcSin[c*x]^2 + ArcSin[c*x]^3 - 12*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 6*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 12*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 12*Pi*Log[Cos[ArcSin[c*x]/2]] + 6*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]])) + (3*Sqrt[1 - c^2*x^2]*(-2 + ArcSin[c*x]^2) + 2*((-3*I)*Pi*ArcSin[c*x] - 3*c*x*ArcSin[c*x] - (3 - 3*I)*ArcSin[c*x]^2 + ArcSin[c*x]^3 - 12*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 6*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 12*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 12*Pi*Log[Cos[ArcSin[c*x]/2]] + 6*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]]))*Sin[ArcSin[c*x]/2] + (24*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(3*c*d^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) - (b^2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((96*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]) + Sin[ArcSin[c*x]/2]*((-24*I)*Pi*ArcSin[c*x] - 48*c*x*ArcSin[c*x] - (24 - 24*I)*ArcSin[c*x]^2 + 10*ArcSin[c*x]^3 + 3*Sqrt[1 - c^2*x^2]*(-16 + c*x + 8*ArcSin[c*x]^2) - 3*ArcSin[c*x]*Cos[2*ArcSin[c*x]] - 96*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 48*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 96*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 96*Pi*Log[Cos[ArcSin[c*x]/2]] + 48*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - 3*ArcSin[c*x]^2*Sin[2*ArcSin[c*x]]) + Cos[ArcSin[c*x]/2]*((-24*I)*Pi*ArcSin[c*x] - 48*c*x*ArcSin[c*x] + (24 + 24*I)*ArcSin[c*x]^2 + 10*ArcSin[c*x]^3 + 3*Sqrt[1 - c^2*x^2]*(-16 + c*x + 8*ArcSin[c*x]^2) - 3*ArcSin[c*x]*Cos[2*ArcSin[c*x]] - 96*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 48*Pi*Log[1 - I*E^(I*ArcSin[c*x])] - 96*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 96*Pi*Log[Cos[ArcSin[c*x]/2]] + 48*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - 3*ArcSin[c*x]^2*Sin[2*ArcSin[c*x]])))/(12*c*d^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) - (a*b*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((15 + 14*ArcSin[c*x])*Cos[(3*ArcSin[c*x])/2] - Cos[(5*ArcSin[c*x])/2] + 2*ArcSin[c*x]*Cos[(5*ArcSin[c*x])/2] + 4*Cos[ArcSin[c*x]/2]*(-4 + 12*ArcSin[c*x] + 5*ArcSin[c*x]^2 - 16*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) - 16*Sin[ArcSin[c*x]/2] - 48*ArcSin[c*x]*Sin[ArcSin[c*x]/2] + 20*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2] - 64*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]*Sin[ArcSin[c*x]/2] - 15*Sin[(3*ArcSin[c*x])/2] + 14*ArcSin[c*x]*Sin[(3*ArcSin[c*x])/2] - Sin[(5*ArcSin[c*x])/2] - 2*ArcSin[c*x]*Sin[(5*ArcSin[c*x])/2]))/(8*c*d^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]))","B",1
557,1,2338,729,13.8036211,"\int \frac{(e-c e x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{5/2}} \, dx","Integrate[((e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2)/(d + c*d*x)^(5/2),x]","\text{Result too large to show}","-\frac{2 a b e^5 x \left(1-c^2 x^2\right)^{5/2}}{(c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{5 e^5 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{e^5 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{28 i e^5 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{112 b e^5 \left(1-c^2 x^2\right)^{5/2} \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{28 e^5 \left(1-c^2 x^2\right)^{5/2} \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{8 b e^5 \left(1-c^2 x^2\right)^{5/2} \csc ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 e^5 \left(1-c^2 x^2\right)^{5/2} \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) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{112 i b^2 e^5 \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b^2 e^5 \left(1-c^2 x^2\right)^3}{c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b^2 e^5 x \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x)}{(c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{16 b^2 e^5 \left(1-c^2 x^2\right)^{5/2} \cot \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}",1,"(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*((a^2*e^2)/d^3 - (8*a^2*e^2)/(3*d^3*(1 + c*x)^2) + (28*a^2*e^2)/(3*d^3*(1 + c*x))))/c - (5*a^2*e^(5/2)*ArcTan[(c*x*Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)])/(Sqrt[d]*Sqrt[e]*(-1 + c*x)*(1 + c*x))])/(c*d^(5/2)) - (a*b*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2]*(-8 + 6*ArcSin[c*x] + 9*ArcSin[c*x]^2 - 84*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + Cos[(3*ArcSin[c*x])/2]*((14 - 3*ArcSin[c*x])*ArcSin[c*x] + 28*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + 2*(-4 + 4*ArcSin[c*x] + 6*ArcSin[c*x]^2 + Sqrt[1 - c^2*x^2]*(ArcSin[c*x]*(14 + 3*ArcSin[c*x]) - 28*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) - 56*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(3*c*d^3*(-1 + c*x)*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^4) - (a*b*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[(3*ArcSin[c*x])/2]*(ArcSin[c*x] + 2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) - Cos[ArcSin[c*x]/2]*(4 + 3*ArcSin[c*x] + 6*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + 2*(-2 + 2*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x] - 4*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - 2*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(3*c*d^3*(-1 + c*x)*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^4) - (b^2*e^2*(-1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-6*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] + ((13 + 13*I)*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] + (3*ArcSin[c*x]^3)/Sqrt[1 - c^2*x^2] + 3*(-2 + ArcSin[c*x]^2) + (13*((-I)*Pi*ArcSin[c*x] - 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 2*(Pi + 2*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])] + 4*Pi*Log[Cos[ArcSin[c*x]/2]] + 2*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, I*E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3) - (2*ArcSin[c*x]*(2 + ArcSin[c*x]))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (2*(4 - 13*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]))))/(3*c*d^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^2) - (b^2*e^2*(-1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-I)*Pi*ArcSin[c*x] + (1 + I)*ArcSin[c*x]^2 - 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 2*(Pi + 2*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])] + 4*Pi*Log[Cos[ArcSin[c*x]/2]] + 2*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, I*E^(I*ArcSin[c*x])] + (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 - (2*ArcSin[c*x]*(2 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - (2*(-4 + ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(3*c*d^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^2) + (2*b^2*e^2*(-1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((7*I)*Pi*ArcSin[c*x] - (7 + 7*I)*ArcSin[c*x]^2 - ArcSin[c*x]^3 + 28*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 14*(Pi + 2*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])] - 28*Pi*Log[Cos[ArcSin[c*x]/2]] - 14*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (28*I)*PolyLog[2, I*E^(I*ArcSin[c*x])] - (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 + (2*ArcSin[c*x]*(2 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 + (2*(-4 + 7*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(3*c*d^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^2) - (a*b*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(3*Cos[(5*ArcSin[c*x])/2] - 3*ArcSin[c*x]*Cos[(5*ArcSin[c*x])/2] + Cos[ArcSin[c*x]/2]*(-20 + 24*ArcSin[c*x] + 27*ArcSin[c*x]^2 - 156*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + Cos[(3*ArcSin[c*x])/2]*(9 + 35*ArcSin[c*x] - 9*ArcSin[c*x]^2 + 52*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) - 20*Sin[ArcSin[c*x]/2] - 24*ArcSin[c*x]*Sin[ArcSin[c*x]/2] + 27*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2] - 156*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]*Sin[ArcSin[c*x]/2] - 9*Sin[(3*ArcSin[c*x])/2] + 35*ArcSin[c*x]*Sin[(3*ArcSin[c*x])/2] + 9*ArcSin[c*x]^2*Sin[(3*ArcSin[c*x])/2] - 52*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]*Sin[(3*ArcSin[c*x])/2] + 3*Sin[(5*ArcSin[c*x])/2] + 3*ArcSin[c*x]*Sin[(5*ArcSin[c*x])/2]))/(6*c*d^3*(-1 + c*x)*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^4)","B",0
558,1,434,559,3.8458749,"\int \frac{(d+c d x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{e-c e x}} \, dx","Integrate[((d + c*d*x)^(5/2)*(a + b*ArcSin[c*x])^2)/Sqrt[e - c*e*x],x]","-\frac{d^2 \left(\sqrt{c d x+d} \sqrt{e-c e x} \left(6 \left(6 a^2 \sqrt{1-c^2 x^2} \left(2 c^2 x^2+9 c x+22\right)-8 a b c x \left(c^2 x^2+33\right)-27 b^2 (c x+10) \sqrt{1-c^2 x^2}\right)+162 a b \cos \left(2 \sin ^{-1}(c x)\right)+4 b^2 \cos \left(3 \sin ^{-1}(c x)\right)\right)+540 a^2 \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+18 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(-30 a+9 b (2 c x+5) \sqrt{1-c^2 x^2}-b \cos \left(3 \sin ^{-1}(c x)\right)\right)-6 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(-108 a c x \sqrt{1-c^2 x^2}-270 a \sqrt{1-c^2 x^2}+6 a \cos \left(3 \sin ^{-1}(c x)\right)+8 b c^3 x^3+36 b c^2 x^2+264 b c x-9 b \cos \left(2 \sin ^{-1}(c x)\right)-18 b\right)-180 b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3\right)}{216 c e \sqrt{1-c^2 x^2}}","\frac{5 d^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{6 b c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{c d^3 x^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{3 d^3 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{2 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{11 d^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{3 b c d^3 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{22 b d^3 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 b c^2 d^3 x^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 b^2 d^3 \left(1-c^2 x^2\right)^2}{27 c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{3 b^2 d^3 x \left(1-c^2 x^2\right)}{4 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{68 b^2 d^3 \left(1-c^2 x^2\right)}{9 c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{3 b^2 d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt{c d x+d} \sqrt{e-c e x}}",1,"-1/216*(d^2*(-180*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 + 540*a^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 6*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(-18*b + 264*b*c*x + 36*b*c^2*x^2 + 8*b*c^3*x^3 - 270*a*Sqrt[1 - c^2*x^2] - 108*a*c*x*Sqrt[1 - c^2*x^2] - 9*b*Cos[2*ArcSin[c*x]] + 6*a*Cos[3*ArcSin[c*x]]) + 18*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(-30*a + 9*b*(5 + 2*c*x)*Sqrt[1 - c^2*x^2] - b*Cos[3*ArcSin[c*x]]) + Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(6*(-27*b^2*(10 + c*x)*Sqrt[1 - c^2*x^2] - 8*a*b*c*x*(33 + c^2*x^2) + 6*a^2*Sqrt[1 - c^2*x^2]*(22 + 9*c*x + 2*c^2*x^2)) + 162*a*b*Cos[2*ArcSin[c*x]] + 4*b^2*Cos[3*ArcSin[c*x]])))/(c*e*Sqrt[1 - c^2*x^2])","A",1
559,1,344,398,2.3795812,"\int \frac{(d+c d x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{e-c e x}} \, dx","Integrate[((d + c*d*x)^(3/2)*(a + b*ArcSin[c*x])^2)/Sqrt[e - c*e*x],x]","\frac{d \sqrt{c d x+d} \sqrt{e-c e x} \left(-2 a^2 (c x+4) \sqrt{1-c^2 x^2}+16 a b c x-a b \cos \left(2 \sin ^{-1}(c x)\right)+b^2 (c x+16) \sqrt{1-c^2 x^2}\right)-6 a^2 d^{3/2} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)-2 b d \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(b (c x+4) \sqrt{1-c^2 x^2}-3 a\right)+b d \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(b \left(2 c^2 x^2+16 c x-1\right)-4 a (c x+4) \sqrt{1-c^2 x^2}\right)+2 b^2 d \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{4 c e \sqrt{1-c^2 x^2}}","\frac{d^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{2 b c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 d^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{d^2 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{2 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{b c d^2 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{4 b d^2 x \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{c d x+d} \sqrt{e-c e x}}+\frac{4 b^2 d^2 \left(1-c^2 x^2\right)}{c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{b^2 d^2 x \left(1-c^2 x^2\right)}{4 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{b^2 d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt{c d x+d} \sqrt{e-c e x}}",1,"(b*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-4*a*(4 + c*x)*Sqrt[1 - c^2*x^2] + b*(-1 + 16*c*x + 2*c^2*x^2))*ArcSin[c*x] - 2*b*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-3*a + b*(4 + c*x)*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + 2*b^2*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 6*a^2*d^(3/2)*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(16*a*b*c*x - 2*a^2*(4 + c*x)*Sqrt[1 - c^2*x^2] + b^2*(16 + c*x)*Sqrt[1 - c^2*x^2] - a*b*Cos[2*ArcSin[c*x]]))/(4*c*e*Sqrt[1 - c^2*x^2])","A",1
560,1,298,231,1.2181925,"\int \frac{\sqrt{d+c d x} \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{e-c e x}} \, dx","Integrate[(Sqrt[d + c*d*x]*(a + b*ArcSin[c*x])^2)/Sqrt[e - c*e*x],x]","\frac{3 \sqrt{c d x+d} \sqrt{e-c e x} \left(a^2 \left(-\sqrt{1-c^2 x^2}\right)+2 a b c x+2 b^2 \sqrt{1-c^2 x^2}\right)-3 a^2 \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+3 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(a-b \sqrt{1-c^2 x^2}\right)+6 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(b c x-a \sqrt{1-c^2 x^2}\right)+b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{3 c e \sqrt{1-c^2 x^2}}","\frac{2 a b d x \sqrt{1-c^2 x^2}}{\sqrt{c d x+d} \sqrt{e-c e x}}+\frac{d \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{d \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 b^2 d \left(1-c^2 x^2\right)}{c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 b^2 d x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{\sqrt{c d x+d} \sqrt{e-c e x}}",1,"(3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(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 + c*d*x]*Sqrt[e - c*e*x]*(b*c*x - a*Sqrt[1 - c^2*x^2])*ArcSin[c*x] + 3*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a - b*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 3*a^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))])/(3*c*e*Sqrt[1 - c^2*x^2])","A",1
561,1,159,55,0.9803015,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d+c d x} \sqrt{e-c e x}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]),x]","\frac{-\frac{3 a^2 \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)}{\sqrt{d} \sqrt{e}}+\frac{3 a b \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{\sqrt{c d x+d} \sqrt{e-c e x}}+\frac{b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^3}{\sqrt{c d x+d} \sqrt{e-c e x}}}{3 c}","\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c \sqrt{c d x+d} \sqrt{e-c e x}}",1,"((3*a*b*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^3)/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (3*a^2*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))])/(Sqrt[d]*Sqrt[e]))/(3*c)","B",1
562,1,225,455,1.8755974,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{3/2} \sqrt{e-c e x}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/((d + c*d*x)^(3/2)*Sqrt[e - c*e*x]),x]","-\frac{\sqrt{c d x+d} \sqrt{e-c e x} \left(a \left(a c x-a+4 b \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)\right)+2 b \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \left(-a \cot \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)+2 b \log \left(1+i e^{-i \sin ^{-1}(c x)}\right)\right)+4 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{-i \sin ^{-1}(c x)}\right)+b^2 \left(-\sqrt{1-c^2 x^2}\right) \sin ^{-1}(c x)^2 \left(\cot \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)-i\right)\right)}{c d^2 e (c x-1) (c x+1)}","-\frac{i e \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{e \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{e x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{2 b e \left(1-c^2 x^2\right)^{3/2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 i b e \left(1-c^2 x^2\right)^{3/2} \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{2 i b^2 e \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 i b^2 e \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{i b^2 e \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}",1,"-((Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-(b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2*(-I + Cot[(Pi + 2*ArcSin[c*x])/4])) + 2*b*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*(-(a*Cot[(Pi + 2*ArcSin[c*x])/4]) + 2*b*Log[1 + I/E^(I*ArcSin[c*x])]) + a*(-a + a*c*x + 4*b*Sqrt[1 - c^2*x^2]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]]) + (4*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)/E^(I*ArcSin[c*x])]))/(c*d^2*e*(-1 + c*x)*(1 + c*x)))","A",0
563,1,540,896,7.5878565,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{5/2} \sqrt{e-c e x}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/((d + c*d*x)^(5/2)*Sqrt[e - c*e*x]),x]","\frac{\sqrt{d (c x+1)} \sqrt{-e (c x-1)} \left(-\frac{a^2}{3 d^3 e (c x+1)}-\frac{a^2}{3 d^3 e (c x+1)^2}\right)}{c}+\frac{a b \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{e-c e x} \left(2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\sqrt{1-c^2 x^2} \left(\sin ^{-1}(c x)-2 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-\sin ^{-1}(c x)-4 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+1\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(3 \sin ^{-1}(c x)-6 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2\right)+\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x)+2 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{3 c d^2 \sqrt{(-c d x-d) (e-c e x)} \sqrt{-d e \left(1-c^2 x^2\right)} \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}+\frac{b^2 \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{e-c e x} \left(-8 i \text{Li}_2\left(-i e^{-i \sin ^{-1}(c x)}\right)+\cot \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(2 \sin ^{-1}(c x)^2+\sin ^{-1}(c x)^2 \csc ^2\left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)+4\right)+2 \sin ^{-1}(c x) \left(-i \sin ^{-1}(c x)-4 \log \left(1+i e^{-i \sin ^{-1}(c x)}\right)+\csc ^2\left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)\right)}{6 c d^2 \sqrt{(-c d x-d) (e-c e x)} \sqrt{-d e \left(1-c^2 x^2\right)}}","\frac{c^2 e^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 x^3}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{b c e^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) x^2}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 b^2 e^2 \left(1-c^2 x^2\right)^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 e^2 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{e^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 b e^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b^2 e^2 \left(1-c^2 x^2\right)^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{i e^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 e^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{b^2 e^2 \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{b e^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 i b e^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 b e^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 i b^2 e^2 \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b^2 e^2 \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{i b^2 e^2 \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}",1,"(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*(-1/3*a^2/(d^3*e*(1 + c*x)^2) - a^2/(3*d^3*e*(1 + c*x))))/c + (b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2]*(Cot[(Pi + 2*ArcSin[c*x])/4]*(4 + 2*ArcSin[c*x]^2 + ArcSin[c*x]^2*Csc[(Pi + 2*ArcSin[c*x])/4]^2) + 2*ArcSin[c*x]*((-I)*ArcSin[c*x] + Csc[(Pi + 2*ArcSin[c*x])/4]^2 - 4*Log[1 + I/E^(I*ArcSin[c*x])]) - (8*I)*PolyLog[2, (-I)/E^(I*ArcSin[c*x])]))/(6*c*d^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[-(d*e*(1 - c^2*x^2))]) + (a*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2]*(2 + 3*ArcSin[c*x] - 6*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + Cos[(3*ArcSin[c*x])/2]*(ArcSin[c*x] + 2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) + 2*(1 - ArcSin[c*x] + Sqrt[1 - c^2*x^2]*(ArcSin[c*x] - 2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]) - 4*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(3*c*d^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3)","A",0
564,1,2041,918,14.5743954,"\int \frac{(d+c d x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{(e-c e x)^{3/2}} \, dx","Integrate[((d + c*d*x)^(5/2)*(a + b*ArcSin[c*x])^2)/(e - c*e*x)^(3/2),x]","\text{Result too large to show}","-\frac{5 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^3 d^4}{2 b c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{b^2 x \left(1-c^2 x^2\right)^2 d^4}{4 (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 b^2 \left(1-c^2 x^2\right)^2 d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{x \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 d^4}{2 (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 i \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 d^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 a b x \left(1-c^2 x^2\right)^{3/2} d^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 b^2 x \left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x) d^4}{(c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{b^2 \left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x) d^4}{4 c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{b c x^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) d^4}{2 (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{32 i b \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{16 b \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{16 i b^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}+\frac{16 i b^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 i b^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right) d^4}{c (c x d+d)^{3/2} (e-c e x)^{3/2}}",1,"(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*((4*a^2*d^2)/e^2 + (a^2*c*d^2*x)/(2*e^2) - (8*a^2*d^2)/(e^2*(-1 + c*x))))/c + (15*a^2*d^(5/2)*ArcTan[(c*x*Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)])/(Sqrt[d]*Sqrt[e]*(-1 + c*x)*(1 + c*x))])/(2*c*e^(3/2)) - (a*b*d^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*((-4 + ArcSin[c*x])*ArcSin[c*x] - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - (ArcSin[c*x]*(4 + ArcSin[c*x]) - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(c*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (4*a*b*d^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*(-(c*x) + 2*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x] - ArcSin[c*x]^2 + 4*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + (c*x + 2*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]^2 - 4*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(c*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) - (b^2*d^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-18*I)*Pi*ArcSin[c*x] - (6 - 6*I)*ArcSin[c*x]^2 + ArcSin[c*x]^3 - 24*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 12*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 24*Pi*Log[Cos[ArcSin[c*x]/2]] - 12*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (24*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (12*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])))/(3*c*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) - (b^2*d^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((96*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - ((48 - 48*I)*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] + (20*ArcSin[c*x]^3)/Sqrt[1 - c^2*x^2] - 48*(-2 + ArcSin[c*x]^2) - 6*c*x*(-1 + 2*ArcSin[c*x]^2) - (6*ArcSin[c*x]*Cos[2*ArcSin[c*x]])/Sqrt[1 - c^2*x^2] + (48*((-3*I)*Pi*ArcSin[c*x] - 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 2*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 4*Pi*Log[Cos[ArcSin[c*x]/2]] - 2*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] - (96*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]))))/(24*c*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) - (2*b^2*d^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(6 + (6*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - 3*ArcSin[c*x]^2 - ((6 - 6*I)*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] + (2*ArcSin[c*x]^3)/Sqrt[1 - c^2*x^2] + (6*((-3*I)*Pi*ArcSin[c*x] - 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 2*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 4*Pi*Log[Cos[ArcSin[c*x]/2]] - 2*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] - (12*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]))))/(3*c*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (a*b*d^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-15 + 14*ArcSin[c*x])*Cos[(3*ArcSin[c*x])/2] + Cos[(5*ArcSin[c*x])/2] + 2*ArcSin[c*x]*Cos[(5*ArcSin[c*x])/2] + Cos[ArcSin[c*x]/2]*(16 + 48*ArcSin[c*x] - 20*ArcSin[c*x]^2 + 64*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - 16*Sin[ArcSin[c*x]/2] + 48*ArcSin[c*x]*Sin[ArcSin[c*x]/2] + 20*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2] - 64*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]*Sin[ArcSin[c*x]/2] - 15*Sin[(3*ArcSin[c*x])/2] - 14*ArcSin[c*x]*Sin[(3*ArcSin[c*x])/2] - Sin[(5*ArcSin[c*x])/2] + 2*ArcSin[c*x]*Sin[(5*ArcSin[c*x])/2]))/(8*c*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2)","B",1
565,1,1255,713,10.9939492,"\int \frac{(d+c d x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{(e-c e x)^{3/2}} \, dx","Integrate[((d + c*d*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(e - c*e*x)^(3/2),x]","\frac{3 d^{3/2} \tan ^{-1}\left(\frac{c x \sqrt{-e (c x-1)} \sqrt{d (c x+1)}}{\sqrt{d} \sqrt{e} (c x-1) (c x+1)}\right) a^2}{c e^{3/2}}-\frac{b d (c x+1) \sqrt{c x d+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\left(\sin ^{-1}(c x)-4\right) \sin ^{-1}(c x)-8 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-\left(\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+4\right)-8 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) a}{c e^2 \sqrt{(-c x d-d) (e-c e x)} \sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}+\frac{2 b d (c x+1) \sqrt{c x d+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(-\sin ^{-1}(c x)^2+\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+2 \sin ^{-1}(c x)-c x+4 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)+\left(\sin ^{-1}(c x)^2-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+2 \sin ^{-1}(c x)+c x-4 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) a}{c e^2 \sqrt{(-c x d-d) (e-c e x)} \sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}+\frac{\sqrt{-e (c x-1)} \sqrt{d (c x+1)} \left(\frac{a^2 d}{e^2}-\frac{4 a^2 d}{e^2 (c x-1)}\right)}{c}-\frac{b^2 d (c x+1) \sqrt{c x d+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(\sin ^{-1}(c x)^3-\frac{12 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}-(6-6 i) \sin ^{-1}(c x)^2-18 i \pi  \sin ^{-1}(c x)-24 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+12 \left(\pi -2 \sin ^{-1}(c x)\right) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+24 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-12 \pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+24 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)\right)}{3 c e^2 \sqrt{(-c x d-d) (e-c e x)} \sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-\frac{b^2 d (c x+1) \sqrt{c x d+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(\frac{2 \sin ^{-1}(c x)^3}{\sqrt{1-c^2 x^2}}-\frac{12 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}-\frac{(6-6 i) \sin ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}-3 \sin ^{-1}(c x)^2+\frac{6 c x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}+\frac{6 \left(-3 i \pi  \sin ^{-1}(c x)-4 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+2 \left(\pi -2 \sin ^{-1}(c x)\right) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+4 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-2 \pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+4 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}+6\right)}{3 c e^2 \sqrt{(-c x d-d) (e-c e x)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}","-\frac{2 a b d^3 x \left(1-c^2 x^2\right)^{3/2}}{(c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{d^3 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^3}{b c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{d^3 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 i d^3 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 d^3 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 d^3 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 b d^3 \left(1-c^2 x^2\right)^{3/2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{16 i b d^3 \left(1-c^2 x^2\right)^{3/2} \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{8 i b^2 d^3 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 i b^2 d^3 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 i b^2 d^3 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 \left(1-c^2 x^2\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 b^2 d^3 x \left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x)}{(c d x+d)^{3/2} (e-c e x)^{3/2}}",1,"(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*((a^2*d)/e^2 - (4*a^2*d)/(e^2*(-1 + c*x))))/c + (3*a^2*d^(3/2)*ArcTan[(c*x*Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)])/(Sqrt[d]*Sqrt[e]*(-1 + c*x)*(1 + c*x))])/(c*e^(3/2)) - (a*b*d*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*((-4 + ArcSin[c*x])*ArcSin[c*x] - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - (ArcSin[c*x]*(4 + ArcSin[c*x]) - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(c*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (2*a*b*d*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*(-(c*x) + 2*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x] - ArcSin[c*x]^2 + 4*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + (c*x + 2*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]^2 - 4*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(c*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) - (b^2*d*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-18*I)*Pi*ArcSin[c*x] - (6 - 6*I)*ArcSin[c*x]^2 + ArcSin[c*x]^3 - 24*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 12*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 24*Pi*Log[Cos[ArcSin[c*x]/2]] - 12*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (24*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (12*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])))/(3*c*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) - (b^2*d*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(6 + (6*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - 3*ArcSin[c*x]^2 - ((6 - 6*I)*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] + (2*ArcSin[c*x]^3)/Sqrt[1 - c^2*x^2] + (6*((-3*I)*Pi*ArcSin[c*x] - 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 2*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 4*Pi*Log[Cos[ArcSin[c*x]/2]] - 2*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] - (12*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]))))/(3*c*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2)","A",1
566,1,513,530,6.5822869,"\int \frac{\sqrt{d+c d x} \left(a+b \sin ^{-1}(c x)\right)^2}{(e-c e x)^{3/2}} \, dx","Integrate[(Sqrt[d + c*d*x]*(a + b*ArcSin[c*x])^2)/(e - c*e*x)^(3/2),x]","-\frac{-3 a^2 \sqrt{d} \sqrt{e} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+\frac{6 a^2 \sqrt{c d x+d} \sqrt{e-c e x}}{c x-1}+\frac{3 a b (c x+1) \sqrt{c d x+d} \sqrt{e-c e x} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\left(\sin ^{-1}(c x)-4\right) \sin ^{-1}(c x)-8 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+4\right)-8 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{\sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}+\frac{b^2 (c x+1) \sqrt{c d x+d} \sqrt{e-c e x} \left(24 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+\sin ^{-1}(c x)^3-(6-6 i) \sin ^{-1}(c x)^2-18 i \pi  \sin ^{-1}(c x)-24 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+12 \left(\pi -2 \sin ^{-1}(c x)\right) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-\frac{12 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+24 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-12 \pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)\right)}{\sqrt{1-c^2 x^2} \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}}{3 c e^2}","-\frac{d^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 i d^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{2 d^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{2 d^2 x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 b d^2 \left(1-c^2 x^2\right)^{3/2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{8 i b d^2 \left(1-c^2 x^2\right)^{3/2} \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{4 i b^2 d^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 i b^2 d^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 i b^2 d^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}",1,"-1/3*((6*a^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(-1 + c*x) - 3*a^2*Sqrt[d]*Sqrt[e]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + (3*a*b*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(Cos[ArcSin[c*x]/2]*((-4 + ArcSin[c*x])*ArcSin[c*x] - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - (ArcSin[c*x]*(4 + ArcSin[c*x]) - 8*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (b^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*((-18*I)*Pi*ArcSin[c*x] - (6 - 6*I)*ArcSin[c*x]^2 + ArcSin[c*x]^3 - 24*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 12*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 24*Pi*Log[Cos[ArcSin[c*x]/2]] - 12*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (24*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (12*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2))/(c*e^2)","A",0
567,1,221,454,1.8092253,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d+c d x} (e-c e x)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(Sqrt[d + c*d*x]*(e - c*e*x)^(3/2)),x]","-\frac{\sqrt{c d x+d} \sqrt{e-c e x} \left(2 b \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \left(a \tan \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)+2 b \log \left(1+i e^{i \sin ^{-1}(c x)}\right)\right)+a \left(a c x+a+4 b \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)\right)-4 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2 \left(\tan \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)-i\right)\right)}{c d e^2 (c x-1) (c x+1)}","-\frac{i d \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{d \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{d x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{2 b d \left(1-c^2 x^2\right)^{3/2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{4 i b d \left(1-c^2 x^2\right)^{3/2} \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{2 i b^2 d \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{2 i b^2 d \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{i b^2 d \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}",1,"-((Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a*(a + a*c*x + 4*b*Sqrt[1 - c^2*x^2]*Log[Cos[(Pi + 2*ArcSin[c*x])/4]]) - (4*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2*(-I + Tan[(Pi + 2*ArcSin[c*x])/4]) + 2*b*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*(2*b*Log[1 + I*E^(I*ArcSin[c*x])] + a*Tan[(Pi + 2*ArcSin[c*x])/4])))/(c*d*e^2*(-1 + c*x)*(1 + c*x)))","A",0
568,1,550,217,1.3117484,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)),x]","\frac{a^2 c x+2 a b \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 a b \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 a b c x \sin ^{-1}(c x)-2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-i b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2+2 i \pi  b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+4 \pi  b^2 \sqrt{1-c^2 x^2} \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+2 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+\pi  b^2 \sqrt{1-c^2 x^2} \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+2 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-\pi  b^2 \sqrt{1-c^2 x^2} \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-\pi  b^2 \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-4 \pi  b^2 \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\pi  b^2 \sqrt{1-c^2 x^2} \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+b^2 c x \sin ^{-1}(c x)^2}{c d e \sqrt{c d x+d} \sqrt{e-c e x}}","-\frac{i \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{2 b \left(1-c^2 x^2\right)^{3/2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{i b^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}",1,"(a^2*c*x + 2*a*b*c*x*ArcSin[c*x] + (2*I)*b^2*Pi*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + b^2*c*x*ArcSin[c*x]^2 - I*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 + 4*b^2*Pi*Sqrt[1 - c^2*x^2]*Log[1 + E^((-I)*ArcSin[c*x])] + b^2*Pi*Sqrt[1 - c^2*x^2]*Log[1 - I*E^(I*ArcSin[c*x])] + 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - b^2*Pi*Sqrt[1 - c^2*x^2]*Log[1 + I*E^(I*ArcSin[c*x])] + 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - 4*b^2*Pi*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2]] + b^2*Pi*Sqrt[1 - c^2*x^2]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 2*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 2*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - b^2*Pi*Sqrt[1 - c^2*x^2]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])","B",1
569,1,739,709,8.331255,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{5/2} (e-c e x)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/((d + c*d*x)^(5/2)*(e - c*e*x)^(3/2)),x]","\frac{\sqrt{d (c x+1)} \sqrt{-e (c x-1)} \left(-\frac{a^2}{4 d^3 e^2 (c x-1)}-\frac{5 a^2}{12 d^3 e^2 (c x+1)}-\frac{a^2}{6 d^3 e^2 (c x+1)^2}\right)}{c}+\frac{a b \sqrt{c d x+d} \sqrt{e-c e x} \left(2 \sin ^{-1}(c x) \left(\cos \left(2 \sin ^{-1}(c x)\right)-2 c x\right)-\sqrt{1-c^2 x^2} \left(3 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+5 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+c x \left(3 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+5 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-1\right)\right)}{3 c d^2 e \sqrt{(-c d x-d) (e-c e x)} \sqrt{-d e \left(1-c^2 x^2\right)} \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}+\frac{b^2 \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{e-c e x} \left(6 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+10 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+(1+4 i) \sin ^{-1}(c x)^2-7 i \pi  \sin ^{-1}(c x)-16 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)-5 \left(2 \sin ^{-1}(c x)+\pi \right) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+3 \left(\pi -2 \sin ^{-1}(c x)\right) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+5 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-\frac{3 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}-\frac{2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}+\frac{\left(\sin ^{-1}(c x)+2\right) \sin ^{-1}(c x)}{\left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}-\frac{\left(5 \sin ^{-1}(c x)^2+4\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+16 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-3 \pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)\right)}{6 c d^2 e \sqrt{(-c d x-d) (e-c e x)} \sqrt{-d e \left(1-c^2 x^2\right)}}","-\frac{2 i e \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 e x \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{b e \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{b e x \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{e \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{e x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{4 b e \left(1-c^2 x^2\right)^{5/2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b e \left(1-c^2 x^2\right)^{5/2} \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{i b^2 e \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{i b^2 e \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b^2 e \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{b^2 e \left(1-c^2 x^2\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{b^2 e x \left(1-c^2 x^2\right)^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}",1,"(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*(-1/4*a^2/(d^3*e^2*(-1 + c*x)) - a^2/(6*d^3*e^2*(1 + c*x)^2) - (5*a^2)/(12*d^3*e^2*(1 + c*x))))/c + (a*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(2*ArcSin[c*x]*(-2*c*x + Cos[2*ArcSin[c*x]]) - Sqrt[1 - c^2*x^2]*(-1 + 3*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 5*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + c*x*(3*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 5*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]))))/(3*c*d^2*e*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2]*((-7*I)*Pi*ArcSin[c*x] + (1 + 4*I)*ArcSin[c*x]^2 - 16*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 5*(Pi + 2*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])] + 3*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 16*Pi*Log[Cos[ArcSin[c*x]/2]] - 3*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 5*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (6*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (10*I)*PolyLog[2, I*E^(I*ArcSin[c*x])] - (3*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) - (2*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3 + (ArcSin[c*x]*(2 + ArcSin[c*x]))/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2 - ((4 + 5*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(6*c*d^2*e*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[-(d*e*(1 - c^2*x^2))])","A",0
570,1,2312,730,13.285618,"\int \frac{(d+c d x)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{(e-c e x)^{5/2}} \, dx","Integrate[((d + c*d*x)^(5/2)*(a + b*ArcSin[c*x])^2)/(e - c*e*x)^(5/2),x]","\text{Result too large to show}","\frac{2 a b d^5 x \left(1-c^2 x^2\right)^{5/2}}{(c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{5 d^5 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{d^5 \left(1-c^2 x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{28 i d^5 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{112 b d^5 \left(1-c^2 x^2\right)^{5/2} \log \left(1-i e^{-i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{28 d^5 \left(1-c^2 x^2\right)^{5/2} \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{8 b d^5 \left(1-c^2 x^2\right)^{5/2} \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{4 d^5 \left(1-c^2 x^2\right)^{5/2} \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}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{112 i b^2 d^5 \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(i e^{-i \sin ^{-1}(c x)}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 b^2 d^5 \left(1-c^2 x^2\right)^3}{c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 b^2 d^5 x \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x)}{(c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{16 b^2 d^5 \left(1-c^2 x^2\right)^{5/2} \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}",1,"(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*(-((a^2*d^2)/e^3) + (8*a^2*d^2)/(3*e^3*(-1 + c*x)^2) + (28*a^2*d^2)/(3*e^3*(-1 + c*x))))/c - (5*a^2*d^(5/2)*ArcTan[(c*x*Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)])/(Sqrt[d]*Sqrt[e]*(-1 + c*x)*(1 + c*x))])/(c*e^(5/2)) + (a*b*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*(-4 + 3*ArcSin[c*x] - 6*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - Cos[(3*ArcSin[c*x])/2]*(ArcSin[c*x] - 2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + 2*(2 + 2*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 4*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 2*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^4*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) + (a*b*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*(-8 - 6*ArcSin[c*x] + 9*ArcSin[c*x]^2 - 84*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + Cos[(3*ArcSin[c*x])/2]*(-(ArcSin[c*x]*(14 + 3*ArcSin[c*x])) + 28*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + 2*(4 + 4*ArcSin[c*x] - 6*ArcSin[c*x]^2 + 56*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + Sqrt[1 - c^2*x^2]*((14 - 3*ArcSin[c*x])*ArcSin[c*x] + 28*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]))*Sin[ArcSin[c*x]/2]))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^4*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) + (b^2*d^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-3*I)*Pi*ArcSin[c*x] + (4*ArcSin[c*x])/(-1 + c*x) - (1 - I)*ArcSin[c*x]^2 - (2*ArcSin[c*x]^2)/(-1 + c*x) - 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 2*Pi*Log[1 + I*E^(I*ArcSin[c*x])] - 4*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 4*Pi*Log[Cos[ArcSin[c*x]/2]] - 2*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (2*(4 + ArcSin[c*x]^2 + c*x*(-4 + ArcSin[c*x]^2))*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (b^2*d^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(6 + (6*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] - (2*(-2 + ArcSin[c*x])*ArcSin[c*x])/((-1 + c*x)*Sqrt[1 - c^2*x^2]) - 3*ArcSin[c*x]^2 - ((13 - 13*I)*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] + (3*ArcSin[c*x]^3)/Sqrt[1 - c^2*x^2] + (13*((-3*I)*Pi*ArcSin[c*x] - 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 2*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 4*Pi*Log[Cos[ArcSin[c*x]/2]] - 2*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3) + (2*(4 - 13*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]))))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (2*b^2*d^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-21*I)*Pi*ArcSin[c*x] - (2*(-2 + ArcSin[c*x])*ArcSin[c*x])/(-1 + c*x) - (7 - 7*I)*ArcSin[c*x]^2 + ArcSin[c*x]^3 - 28*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 14*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 28*Pi*Log[Cos[ArcSin[c*x]/2]] - 14*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (28*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (2*(4 - 7*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (a*b*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(3*Cos[(5*ArcSin[c*x])/2] + 3*ArcSin[c*x]*Cos[(5*ArcSin[c*x])/2] + Cos[ArcSin[c*x]/2]*(-20 - 24*ArcSin[c*x] + 27*ArcSin[c*x]^2 - 156*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + Cos[(3*ArcSin[c*x])/2]*(9 - 35*ArcSin[c*x] - 9*ArcSin[c*x]^2 + 52*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + 20*Sin[ArcSin[c*x]/2] - 24*ArcSin[c*x]*Sin[ArcSin[c*x]/2] - 27*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2] + 156*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]*Sin[ArcSin[c*x]/2] + 9*Sin[(3*ArcSin[c*x])/2] + 35*ArcSin[c*x]*Sin[(3*ArcSin[c*x])/2] - 9*ArcSin[c*x]^2*Sin[(3*ArcSin[c*x])/2] + 52*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]*Sin[(3*ArcSin[c*x])/2] - 3*Sin[(5*ArcSin[c*x])/2] + 3*ArcSin[c*x]*Sin[(5*ArcSin[c*x])/2]))/(6*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^4*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]))","B",0
571,1,1419,544,10.2320285,"\int \frac{(d+c d x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{(e-c e x)^{5/2}} \, dx","Integrate[((d + c*d*x)^(3/2)*(a + b*ArcSin[c*x])^2)/(e - c*e*x)^(5/2),x]","-\frac{d^{3/2} \tan ^{-1}\left(\frac{c x \sqrt{-e (c x-1)} \sqrt{d (c x+1)}}{\sqrt{d} \sqrt{e} (c x-1) (c x+1)}\right) a^2}{c e^{5/2}}+\frac{b d \sqrt{c x d+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(3 \sin ^{-1}(c x)-6 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-4\right)-\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x)-2 \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 \left(\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+2 \sin ^{-1}(c x)+2 \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+4 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) a}{3 c e^3 \sqrt{(-c x d-d) (e-c e x)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}+\frac{b d \sqrt{c x d+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(9 \sin ^{-1}(c x)^2-6 \sin ^{-1}(c x)-84 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-8\right)+\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(28 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-\sin ^{-1}(c x) \left(3 \sin ^{-1}(c x)+14\right)\right)+2 \left(-6 \sin ^{-1}(c x)^2+4 \sin ^{-1}(c x)+56 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\sqrt{1-c^2 x^2} \left(\left(14-3 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)+28 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)+4\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right) a}{6 c e^3 \sqrt{(-c x d-d) (e-c e x)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4 \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}+\frac{\sqrt{-e (c x-1)} \sqrt{d (c x+1)} \left(\frac{8 d a^2}{3 e^3 (c x-1)}+\frac{4 d a^2}{3 e^3 (c x-1)^2}\right)}{c}+\frac{b^2 d (c x+1) \sqrt{c x d+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(-\frac{2 \sin ^{-1}(c x)^2}{c x-1}-(1-i) \sin ^{-1}(c x)^2-4 \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+\frac{4 \sin ^{-1}(c x)}{c x-1}-3 i \pi  \sin ^{-1}(c x)-4 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+2 \pi  \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+4 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-2 \pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+4 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+\frac{2 \left(\sin ^{-1}(c x)^2+c x \left(\sin ^{-1}(c x)^2-4\right)+4\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}\right)}{3 c e^3 \sqrt{(-c x d-d) (e-c e x)} \sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}+\frac{b^2 d (c x+1) \sqrt{c x d+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(\sin ^{-1}(c x)^3+\frac{4 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}-(7-7 i) \sin ^{-1}(c x)^2-\frac{2 \left(\sin ^{-1}(c x)-2\right) \sin ^{-1}(c x)}{c x-1}-21 i \pi  \sin ^{-1}(c x)-28 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+14 \left(\pi -2 \sin ^{-1}(c x)\right) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+28 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-14 \pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+28 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+\frac{2 \left(4-7 \sin ^{-1}(c x)^2\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}\right)}{3 c e^3 \sqrt{(-c x d-d) (e-c e x)} \sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}","\frac{d^4 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{8 i d^4 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{32 b d^4 \left(1-c^2 x^2\right)^{5/2} \log \left(1-i e^{-i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{8 d^4 \left(1-c^2 x^2\right)^{5/2} \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 b d^4 \left(1-c^2 x^2\right)^{5/2} \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^4 \left(1-c^2 x^2\right)^{5/2} \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}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{32 i b^2 d^4 \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(i e^{-i \sin ^{-1}(c x)}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{8 b^2 d^4 \left(1-c^2 x^2\right)^{5/2} \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}",1,"(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*((4*a^2*d)/(3*e^3*(-1 + c*x)^2) + (8*a^2*d)/(3*e^3*(-1 + c*x))))/c - (a^2*d^(3/2)*ArcTan[(c*x*Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)])/(Sqrt[d]*Sqrt[e]*(-1 + c*x)*(1 + c*x))])/(c*e^(5/2)) + (a*b*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*(-4 + 3*ArcSin[c*x] - 6*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - Cos[(3*ArcSin[c*x])/2]*(ArcSin[c*x] - 2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + 2*(2 + 2*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 4*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 2*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^4*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) + (a*b*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*(-8 - 6*ArcSin[c*x] + 9*ArcSin[c*x]^2 - 84*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + Cos[(3*ArcSin[c*x])/2]*(-(ArcSin[c*x]*(14 + 3*ArcSin[c*x])) + 28*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + 2*(4 + 4*ArcSin[c*x] - 6*ArcSin[c*x]^2 + 56*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + Sqrt[1 - c^2*x^2]*((14 - 3*ArcSin[c*x])*ArcSin[c*x] + 28*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]))*Sin[ArcSin[c*x]/2]))/(6*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^4*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) + (b^2*d*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-3*I)*Pi*ArcSin[c*x] + (4*ArcSin[c*x])/(-1 + c*x) - (1 - I)*ArcSin[c*x]^2 - (2*ArcSin[c*x]^2)/(-1 + c*x) - 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 2*Pi*Log[1 + I*E^(I*ArcSin[c*x])] - 4*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 4*Pi*Log[Cos[ArcSin[c*x]/2]] - 2*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (2*(4 + ArcSin[c*x]^2 + c*x*(-4 + ArcSin[c*x]^2))*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + (b^2*d*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-21*I)*Pi*ArcSin[c*x] - (2*(-2 + ArcSin[c*x])*ArcSin[c*x])/(-1 + c*x) - (7 - 7*I)*ArcSin[c*x]^2 + ArcSin[c*x]^3 - 28*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 14*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] + 28*Pi*Log[Cos[ArcSin[c*x]/2]] - 14*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (28*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (4*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + (2*(4 - 7*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2)","B",0
572,1,687,486,8.3586328,"\int \frac{\sqrt{d+c d x} \left(a+b \sin ^{-1}(c x)\right)^2}{(e-c e x)^{5/2}} \, dx","Integrate[(Sqrt[d + c*d*x]*(a + b*ArcSin[c*x])^2)/(e - c*e*x)^(5/2),x]","\frac{\sqrt{d (c x+1)} \sqrt{-e (c x-1)} \left(\frac{a^2}{3 e^3 (c x-1)}+\frac{2 a^2}{3 e^3 (c x-1)^2}\right)}{c}+\frac{a b \sqrt{c d x+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(\sqrt{1-c^2 x^2} \sin ^{-1}(c x)+2 \sqrt{1-c^2 x^2} \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 ^{-1}(c x)+4 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(3 \sin ^{-1}(c x)-6 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-4\right)-\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x)-2 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)\right)}{3 c e^3 \sqrt{(-c d x-d) (e-c e x)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^4 \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}+\frac{b^2 (c x+1) \sqrt{c d x+d} \sqrt{e-c e x} \sqrt{-d e \left(1-c^2 x^2\right)} \left(4 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-\frac{2 \sin ^{-1}(c x)^2}{c x-1}-(1-i) \sin ^{-1}(c x)^2+\frac{4 \sin ^{-1}(c x)}{c x-1}-3 i \pi  \sin ^{-1}(c x)-4 \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-4 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+2 \pi  \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+\frac{2 \left(\sin ^{-1}(c x)^2+c x \left(\sin ^{-1}(c x)^2-4\right)+4\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}+4 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-2 \pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)\right)}{3 c e^3 \sqrt{1-c^2 x^2} \sqrt{(-c d x-d) (e-c e x)} \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^2}","-\frac{i d^3 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 b d^3 \left(1-c^2 x^2\right)^{5/2} \log \left(1-i e^{-i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{d^3 \left(1-c^2 x^2\right)^{5/2} \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b d^3 \left(1-c^2 x^2\right)^{5/2} \sec ^2\left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{d^3 \left(1-c^2 x^2\right)^{5/2} \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}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{4 i b^2 d^3 \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(i e^{-i \sin ^{-1}(c x)}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{4 b^2 d^3 \left(1-c^2 x^2\right)^{5/2} \tan \left(\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}",1,"(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*((2*a^2)/(3*e^3*(-1 + c*x)^2) + a^2/(3*e^3*(-1 + c*x))))/c + (a*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2]*(-4 + 3*ArcSin[c*x] - 6*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) - Cos[(3*ArcSin[c*x])/2]*(ArcSin[c*x] - 2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + 2*(2 + 2*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 4*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 2*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^4*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) + (b^2*(1 + c*x)*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[-(d*e*(1 - c^2*x^2))]*((-3*I)*Pi*ArcSin[c*x] + (4*ArcSin[c*x])/(-1 + c*x) - (1 - I)*ArcSin[c*x]^2 - (2*ArcSin[c*x]^2)/(-1 + c*x) - 4*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 2*Pi*Log[1 + I*E^(I*ArcSin[c*x])] - 4*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 4*Pi*Log[Cos[ArcSin[c*x]/2]] - 2*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (2*(4 + ArcSin[c*x]^2 + c*x*(-4 + ArcSin[c*x]^2))*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3))/(3*c*e^3*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2)","A",0
573,1,388,896,7.2938283,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d+c d x} (e-c e x)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(Sqrt[d + c*d*x]*(e - c*e*x)^(5/2)),x]","\frac{\sqrt{c d x+d} \sqrt{e-c e x} \left(-\frac{2 a^2 (c x-2)}{(c x-1)^2}+\frac{2 a b \left(2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(-\left(\left(\sqrt{1-c^2 x^2}-1\right) \sin ^{-1}(c x)\right)-2 \left(\sqrt{1-c^2 x^2}+2\right) \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+1\right)+\cos \left(\frac{3}{2} \sin ^{-1}(c x)\right) \left(\sin ^{-1}(c x)-2 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(3 \sin ^{-1}(c x)+6 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-2\right)\right)}{\sqrt{1-c^2 x^2} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}+\frac{b^2 \left(-8 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+4 \tan \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)+\sin ^{-1}(c x) \left(-2 \sec ^2\left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)+8 \log \left(1+i e^{i \sin ^{-1}(c x)}\right)\right)+\sin ^{-1}(c x)^2 \left(\tan \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right) \left(\sec ^2\left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)+2\right)-2 i\right)\right)}{\sqrt{1-c^2 x^2}}\right)}{6 c d e^3}","\frac{c^2 d^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 x^3}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{b c d^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) x^2}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 b^2 d^2 \left(1-c^2 x^2\right)^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{d^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 b d^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right) x}{3 (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 b^2 d^2 \left(1-c^2 x^2\right)^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{i d^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 d^2 \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{b^2 d^2 \left(1-c^2 x^2\right)^{5/2} \sin ^{-1}(c x)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{b d^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{4 i b d^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 b d^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+e^{2 i \sin ^{-1}(c x)}\right)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b^2 d^2 \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 i b^2 d^2 \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}-\frac{i b^2 d^2 \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{3 c (c x d+d)^{5/2} (e-c e x)^{5/2}}",1,"(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*((-2*a^2*(-2 + c*x))/(-1 + c*x)^2 + (2*a*b*(Cos[(3*ArcSin[c*x])/2]*(ArcSin[c*x] - 2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + Cos[ArcSin[c*x]/2]*(-2 + 3*ArcSin[c*x] + 6*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]) + 2*(1 - (-1 + Sqrt[1 - c^2*x^2])*ArcSin[c*x] - 2*(2 + Sqrt[1 - c^2*x^2])*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])*Sin[ArcSin[c*x]/2]))/(Sqrt[1 - c^2*x^2]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3) + (b^2*((-8*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + ArcSin[c*x]*(8*Log[1 + I*E^(I*ArcSin[c*x])] - 2*Sec[(Pi + 2*ArcSin[c*x])/4]^2) + 4*Tan[(Pi + 2*ArcSin[c*x])/4] + ArcSin[c*x]^2*(-2*I + (2 + Sec[(Pi + 2*ArcSin[c*x])/4]^2)*Tan[(Pi + 2*ArcSin[c*x])/4])))/Sqrt[1 - c^2*x^2]))/(6*c*d*e^3)","A",0
574,1,764,709,8.4532869,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{3/2} (e-c e x)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/((d + c*d*x)^(3/2)*(e - c*e*x)^(5/2)),x]","\frac{\sqrt{d (c x+1)} \sqrt{-e (c x-1)} \left(-\frac{5 a^2}{12 d^2 e^3 (c x-1)}-\frac{a^2}{4 d^2 e^3 (c x+1)}+\frac{a^2}{6 d^2 e^3 (c x-1)^2}\right)}{c}-\frac{a b \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{e-c e x} \left(\sqrt{1-c^2 x^2} \left(5 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+3 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-c x \left(5 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+3 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-1\right)+2 \sin ^{-1}(c x) \left(2 c x+\cos \left(2 \sin ^{-1}(c x)\right)\right)\right)}{3 c d e^2 \sqrt{(-c d x-d) (e-c e x)} \sqrt{-d e \left(1-c^2 x^2\right)} \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3 \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)}-\frac{b^2 \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{e-c e x} \left(-10 i \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-6 i \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+(1-4 i) \sin ^{-1}(c x)^2-\frac{\left(\sin ^{-1}(c x)-2\right) \sin ^{-1}(c x)}{c x-1}+9 i \pi  \sin ^{-1}(c x)+16 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+3 \left(2 \sin ^{-1}(c x)+\pi \right) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-5 \left(\pi -2 \sin ^{-1}(c x)\right) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-3 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+\frac{2 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)^3}+\frac{3 \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2}{\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)}+\frac{\left(5 \sin ^{-1}(c x)^2+4\right) \sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)}-16 \pi  \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+5 \pi  \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)\right)}{6 c d e^2 \sqrt{(-c d x-d) (e-c e x)} \sqrt{-d e \left(1-c^2 x^2\right)}}","-\frac{2 i d \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 d x \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{b d \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{b d x \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{d \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{d x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{4 b d \left(1-c^2 x^2\right)^{5/2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 i b d \left(1-c^2 x^2\right)^{5/2} \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{i b^2 d \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{i b^2 d \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b^2 d \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{b^2 d \left(1-c^2 x^2\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{b^2 d x \left(1-c^2 x^2\right)^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}",1,"(Sqrt[-(e*(-1 + c*x))]*Sqrt[d*(1 + c*x)]*(a^2/(6*d^2*e^3*(-1 + c*x)^2) - (5*a^2)/(12*d^2*e^3*(-1 + c*x)) - a^2/(4*d^2*e^3*(1 + c*x))))/c - (a*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2]*(2*ArcSin[c*x]*(2*c*x + Cos[2*ArcSin[c*x]]) + Sqrt[1 - c^2*x^2]*(-1 + 5*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 3*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - c*x*(5*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 3*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]))))/(3*c*d*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[-(d*e*(1 - c^2*x^2))]*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) - (b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2]*((9*I)*Pi*ArcSin[c*x] - ((-2 + ArcSin[c*x])*ArcSin[c*x])/(-1 + c*x) + (1 - 4*I)*ArcSin[c*x]^2 + 16*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + 3*(Pi + 2*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])] - 5*(Pi - 2*ArcSin[c*x])*Log[1 + I*E^(I*ArcSin[c*x])] - 16*Pi*Log[Cos[ArcSin[c*x]/2]] + 5*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 3*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (10*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (6*I)*PolyLog[2, I*E^(I*ArcSin[c*x])] + (2*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3 + ((4 + 5*ArcSin[c*x]^2)*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]) + (3*ArcSin[c*x]^2*Sin[ArcSin[c*x]/2])/(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])))/(6*c*d*e^2*Sqrt[(-d - c*d*x)*(e - c*e*x)]*Sqrt[-(d*e*(1 - c^2*x^2))])","A",0
575,1,722,366,9.7543969,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{5/2} (e-c e x)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/((d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)),x]","\frac{4 a^2 c x \left(3-2 c^2 x^2\right)+4 a b \left(\sqrt{1-c^2 x^2} \left(2 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 \cos \left(2 \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(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-1\right)+\sin ^{-1}(c x) \left(3 c x+\sin \left(3 \sin ^{-1}(c x)\right)\right)\right)+b^2 \left(-16 i \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-16 i \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+2 \sqrt{1-c^2 x^2} \left(-3 i \sin ^{-1}(c x)^2+\sin ^{-1}(c x) \left(6 \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+6 \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+6 i \pi -2\right)+3 \pi  \left(4 \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+\log \left(1-i e^{i \sin ^{-1}(c x)}\right)-\log \left(1+i e^{i \sin ^{-1}(c x)}\right)-\log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-4 \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)\right)\right)+c x+6 c x \sin ^{-1}(c x)^2+2 \sin \left(3 \sin ^{-1}(c x)\right) \sin ^{-1}(c x)^2+\sin \left(3 \sin ^{-1}(c x)\right)-2 i \sin ^{-1}(c x)^2 \cos \left(3 \sin ^{-1}(c x)\right)+4 i \pi  \sin ^{-1}(c x) \cos \left(3 \sin ^{-1}(c x)\right)+4 \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)+4 \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)+8 \pi  \log \left(1+e^{-i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)+2 \pi  \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)-2 \pi  \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \cos \left(3 \sin ^{-1}(c x)\right)-8 \pi  \cos \left(3 \sin ^{-1}(c x)\right) \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 \pi  \cos \left(3 \sin ^{-1}(c x)\right) \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-2 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right) \cos \left(3 \sin ^{-1}(c x)\right)\right)}{12 d^2 e^2 \left(c-c^3 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x}}","-\frac{2 i \left(1-c^2 x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{2 x \left(1-c^2 x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{b \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{4 b \left(1-c^2 x^2\right)^{5/2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}-\frac{2 i b^2 \left(1-c^2 x^2\right)^{5/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{3 c (c d x+d)^{5/2} (e-c e x)^{5/2}}+\frac{b^2 x \left(1-c^2 x^2\right)^2}{3 (c d x+d)^{5/2} (e-c e x)^{5/2}}",1,"(4*a^2*c*x*(3 - 2*c^2*x^2) + b^2*(c*x + 6*c*x*ArcSin[c*x]^2 + (4*I)*Pi*ArcSin[c*x]*Cos[3*ArcSin[c*x]] - (2*I)*ArcSin[c*x]^2*Cos[3*ArcSin[c*x]] + 8*Pi*Cos[3*ArcSin[c*x]]*Log[1 + E^((-I)*ArcSin[c*x])] + 2*Pi*Cos[3*ArcSin[c*x]]*Log[1 - I*E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 - I*E^(I*ArcSin[c*x])] - 2*Pi*Cos[3*ArcSin[c*x]]*Log[1 + I*E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]*Cos[3*ArcSin[c*x]]*Log[1 + I*E^(I*ArcSin[c*x])] - 8*Pi*Cos[3*ArcSin[c*x]]*Log[Cos[ArcSin[c*x]/2]] + 2*Pi*Cos[3*ArcSin[c*x]]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 2*Sqrt[1 - c^2*x^2]*((-3*I)*ArcSin[c*x]^2 + ArcSin[c*x]*(-2 + (6*I)*Pi + 6*Log[1 - I*E^(I*ArcSin[c*x])] + 6*Log[1 + I*E^(I*ArcSin[c*x])]) + 3*Pi*(4*Log[1 + E^((-I)*ArcSin[c*x])] + Log[1 - I*E^(I*ArcSin[c*x])] - Log[1 + I*E^(I*ArcSin[c*x])] - 4*Log[Cos[ArcSin[c*x]/2]] + Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - Log[Sin[(Pi + 2*ArcSin[c*x])/4]])) - 2*Pi*Cos[3*ArcSin[c*x]]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (16*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (16*I)*(1 - c^2*x^2)^(3/2)*PolyLog[2, I*E^(I*ArcSin[c*x])] + Sin[3*ArcSin[c*x]] + 2*ArcSin[c*x]^2*Sin[3*ArcSin[c*x]]) + 4*a*b*(Sqrt[1 - c^2*x^2]*(-1 + 2*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 2*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + 2*Cos[2*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]])) + ArcSin[c*x]*(3*c*x + Sin[3*ArcSin[c*x]])))/(12*d^2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(c - c^3*x^2))","A",0
576,1,297,351,1.1876675,"\int x^2 \sqrt{d+c d x} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2,x]","\frac{3 \sqrt{c d x+d} \sqrt{e-c e x} \left(32 a^2 c x \sqrt{1-c^2 x^2} \left(2 c^2 x^2-1\right)-4 a b \cos \left(4 \sin ^{-1}(c x)\right)+b^2 \sin \left(4 \sin ^{-1}(c x)\right)\right)-96 a^2 \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)-24 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(b \sin \left(4 \sin ^{-1}(c x)\right)-4 a\right)-12 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(4 a \sin \left(4 \sin ^{-1}(c x)\right)+b \cos \left(4 \sin ^{-1}(c x)\right)\right)+32 b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{768 c^3 \sqrt{1-c^2 x^2}}","\frac{b x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{8 c \sqrt{1-c^2 x^2}}-\frac{b c x^4 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{8 \sqrt{1-c^2 x^2}}-\frac{x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{8 c^2}+\frac{\sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^3}{24 b c^3 \sqrt{1-c^2 x^2}}+\frac{1}{4} x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{b^2 x \sqrt{c d x+d} \sqrt{e-c e x}}{64 c^2}-\frac{b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{64 c^3 \sqrt{1-c^2 x^2}}-\frac{1}{32} b^2 x^3 \sqrt{c d x+d} \sqrt{e-c e x}",1,"(32*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 96*a^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 12*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(b*Cos[4*ArcSin[c*x]] + 4*a*Sin[4*ArcSin[c*x]]) - 24*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(-4*a + b*Sin[4*ArcSin[c*x]]) + 3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(32*a^2*c*x*Sqrt[1 - c^2*x^2]*(-1 + 2*c^2*x^2) - 4*a*b*Cos[4*ArcSin[c*x]] + b^2*Sin[4*ArcSin[c*x]]))/(768*c^3*Sqrt[1 - c^2*x^2])","A",1
577,1,178,225,0.5843503,"\int x \sqrt{d+c d x} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2,x]","\frac{\sqrt{c d x+d} \sqrt{e-c e x} \left(9 a^2 \left(c^2 x^2-1\right)^2+6 a b c x \sqrt{1-c^2 x^2} \left(c^2 x^2-3\right)+6 b \sin ^{-1}(c x) \left(3 a \left(c^2 x^2-1\right)^2+b c x \sqrt{1-c^2 x^2} \left(c^2 x^2-3\right)\right)+9 b^2 \left(c^2 x^2-1\right)^2 \sin ^{-1}(c x)^2-2 b^2 \left(c^4 x^4-8 c^2 x^2+7\right)\right)}{27 c^2 \left(c^2 x^2-1\right)}","\frac{2 b x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{3 c \sqrt{1-c^2 x^2}}-\frac{\left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{3 c^2}-\frac{2 b c x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{9 \sqrt{1-c^2 x^2}}+\frac{2 b^2 \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x}}{27 c^2}+\frac{4 b^2 \sqrt{c d x+d} \sqrt{e-c e x}}{9 c^2}",1,"(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(6*a*b*c*x*Sqrt[1 - c^2*x^2]*(-3 + c^2*x^2) + 9*a^2*(-1 + c^2*x^2)^2 - 2*b^2*(7 - 8*c^2*x^2 + c^4*x^4) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(-3 + c^2*x^2) + 3*a*(-1 + c^2*x^2)^2)*ArcSin[c*x] + 9*b^2*(-1 + c^2*x^2)^2*ArcSin[c*x]^2))/(27*c^2*(-1 + c^2*x^2))","A",1
578,1,288,222,0.8321074,"\int \sqrt{d+c d x} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2,x]","\frac{3 \sqrt{c d x+d} \sqrt{e-c e x} \left(4 a^2 c x \sqrt{1-c^2 x^2}+2 a b \cos \left(2 \sin ^{-1}(c x)\right)-b^2 \sin \left(2 \sin ^{-1}(c x)\right)\right)-12 a^2 \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+6 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(2 a+b \sin \left(2 \sin ^{-1}(c x)\right)\right)+6 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(2 a \sin \left(2 \sin ^{-1}(c x)\right)+b \cos \left(2 \sin ^{-1}(c x)\right)\right)+4 b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{24 c \sqrt{1-c^2 x^2}}","\frac{\sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^3}{6 b c \sqrt{1-c^2 x^2}}-\frac{b c x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{1-c^2 x^2}}+\frac{1}{2} x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}-\frac{1}{4} b^2 x \sqrt{c d x+d} \sqrt{e-c e x}",1,"(4*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 12*a^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 6*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(b*Cos[2*ArcSin[c*x]] + 2*a*Sin[2*ArcSin[c*x]]) + 6*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(2*a + b*Sin[2*ArcSin[c*x]]) + 3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(4*a^2*c*x*Sqrt[1 - c^2*x^2] + 2*a*b*Cos[2*ArcSin[c*x]] - b^2*Sin[2*ArcSin[c*x]]))/(24*c*Sqrt[1 - c^2*x^2])","A",1
579,1,434,432,2.3059567,"\int \frac{\sqrt{d+c d x} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{x} \, dx","Integrate[(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/x,x]","a^2 \sqrt{c d x+d} \sqrt{e-c e x}+a^2 \sqrt{d} \sqrt{e} \log (c x)-a^2 \sqrt{d} \sqrt{e} \log \left(\sqrt{d} \sqrt{e} \sqrt{c d x+d} \sqrt{e-c e x}+d e\right)-\frac{2 a b \sqrt{c d x+d} \sqrt{e-c e x} \left(-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)-i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+c x-\sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+\sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}-\frac{b^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(2 \sqrt{1-c^2 x^2}-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2-2 i \sin ^{-1}(c x) \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+2 i \sin ^{-1}(c x) \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+2 \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)-2 \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)+2 c x \sin ^{-1}(c x)+\sin ^{-1}(c x)^2 \left(-\log \left(1-e^{i \sin ^{-1}(c x)}\right)\right)+\sin ^{-1}(c x)^2 \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}","\frac{2 i b \sqrt{c d x+d} \sqrt{e-c e x} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{2 i b \sqrt{c d x+d} \sqrt{e-c e x} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{2 a b c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{1-c^2 x^2}}-\frac{2 \sqrt{c d x+d} \sqrt{e-c e x} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}}+\sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2 b^2 \sqrt{c d x+d} \sqrt{e-c e x} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}+\frac{2 b^2 \sqrt{c d x+d} \sqrt{e-c e x} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{2 b^2 c x \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-2 b^2 \sqrt{c d x+d} \sqrt{e-c e x}",1,"a^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x] + a^2*Sqrt[d]*Sqrt[e]*Log[c*x] - a^2*Sqrt[d]*Sqrt[e]*Log[d*e + Sqrt[d]*Sqrt[e]*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]] - (2*a*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(c*x - Sqrt[1 - c^2*x^2]*ArcSin[c*x] - ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - I*PolyLog[2, -E^(I*ArcSin[c*x])] + I*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] - (b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(2*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 - ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] + ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] - (2*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] + (2*I)*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] + 2*PolyLog[3, -E^(I*ArcSin[c*x])] - 2*PolyLog[3, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2]","A",0
580,1,374,257,1.2527381,"\int \frac{\sqrt{d+c d x} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/x^2,x]","\frac{-3 a^2 \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{e-c e x}+3 a^2 c \sqrt{d} \sqrt{e} x \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)-3 i b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(-i a c x-i b \sqrt{1-c^2 x^2}+b c x\right)+6 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(-a \sqrt{1-c^2 x^2}+b c x \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)\right)+6 a b c x \sqrt{c d x+d} \sqrt{e-c e x} \log (c x)-3 i b^2 c x \sqrt{c d x+d} \sqrt{e-c e x} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-b^2 c x \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{3 x \sqrt{1-c^2 x^2}}","-\frac{c \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b \sqrt{1-c^2 x^2}}-\frac{i c \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}}+\frac{2 b c \sqrt{c d x+d} \sqrt{e-c e x} \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{\sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{x}-\frac{i b^2 c \sqrt{c d x+d} \sqrt{e-c e x} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}",1,"(-3*a^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2] - (3*I)*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*((-I)*a*c*x + b*c*x - I*b*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 - b^2*c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 + 3*a^2*c*Sqrt[d]*Sqrt[e]*x*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 6*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(-(a*Sqrt[1 - c^2*x^2]) + b*c*x*Log[1 - E^((2*I)*ArcSin[c*x])]) + 6*a*b*c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Log[c*x] - (3*I)*b^2*c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(3*x*Sqrt[1 - c^2*x^2])","A",0
581,1,452,509,2.1820974,"\int x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2,x]","\frac{d e \sqrt{c d x+d} \sqrt{e-c e x} \left(-864 a^2 c x \sqrt{1-c^2 x^2}-2304 a^2 c^5 x^5 \sqrt{1-c^2 x^2}+4032 a^2 c^3 x^3 \sqrt{1-c^2 x^2}+216 a b \cos \left(2 \sin ^{-1}(c x)\right)-108 a b \cos \left(4 \sin ^{-1}(c x)\right)-24 a b \cos \left(6 \sin ^{-1}(c x)\right)-108 b^2 \sin \left(2 \sin ^{-1}(c x)\right)+27 b^2 \sin \left(4 \sin ^{-1}(c x)\right)+4 b^2 \sin \left(6 \sin ^{-1}(c x)\right)\right)-864 a^2 d^{3/2} e^{3/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)-72 b d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(-12 a-3 b \sin \left(2 \sin ^{-1}(c x)\right)+3 b \sin \left(4 \sin ^{-1}(c x)\right)+b \sin \left(6 \sin ^{-1}(c x)\right)\right)-12 b d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(-36 a \sin \left(2 \sin ^{-1}(c x)\right)+36 a \sin \left(4 \sin ^{-1}(c x)\right)+12 a \sin \left(6 \sin ^{-1}(c x)\right)-18 b \cos \left(2 \sin ^{-1}(c x)\right)+9 b \cos \left(4 \sin ^{-1}(c x)\right)+2 b \cos \left(6 \sin ^{-1}(c x)\right)\right)+288 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{13824 c^3 \sqrt{1-c^2 x^2}}","\frac{b d e x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{16 c \sqrt{1-c^2 x^2}}-\frac{7 b c d e x^4 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{48 \sqrt{1-c^2 x^2}}+\frac{1}{6} d e x^3 \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{d e x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{16 c^2}+\frac{d e \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^3}{48 b c^3 \sqrt{1-c^2 x^2}}+\frac{b c^3 d e x^6 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{18 \sqrt{1-c^2 x^2}}+\frac{1}{8} d e x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{108} b^2 c^2 d e x^5 \sqrt{c d x+d} \sqrt{e-c e x}-\frac{7 b^2 d e x \sqrt{c d x+d} \sqrt{e-c e x}}{1152 c^2}+\frac{7 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{1152 c^3 \sqrt{1-c^2 x^2}}-\frac{43 b^2 d e x^3 \sqrt{c d x+d} \sqrt{e-c e x}}{1728}",1,"(288*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 864*a^2*d^(3/2)*e^(3/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 12*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(-18*b*Cos[2*ArcSin[c*x]] + 9*b*Cos[4*ArcSin[c*x]] + 2*b*Cos[6*ArcSin[c*x]] - 36*a*Sin[2*ArcSin[c*x]] + 36*a*Sin[4*ArcSin[c*x]] + 12*a*Sin[6*ArcSin[c*x]]) - 72*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(-12*a - 3*b*Sin[2*ArcSin[c*x]] + 3*b*Sin[4*ArcSin[c*x]] + b*Sin[6*ArcSin[c*x]]) + d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-864*a^2*c*x*Sqrt[1 - c^2*x^2] + 4032*a^2*c^3*x^3*Sqrt[1 - c^2*x^2] - 2304*a^2*c^5*x^5*Sqrt[1 - c^2*x^2] + 216*a*b*Cos[2*ArcSin[c*x]] - 108*a*b*Cos[4*ArcSin[c*x]] - 24*a*b*Cos[6*ArcSin[c*x]] - 108*b^2*Sin[2*ArcSin[c*x]] + 27*b^2*Sin[4*ArcSin[c*x]] + 4*b^2*Sin[6*ArcSin[c*x]]))/(13824*c^3*Sqrt[1 - c^2*x^2])","A",1
582,1,207,338,0.8084078,"\int x (d+c d x)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2,x]","-\frac{d e \sqrt{c d x+d} \sqrt{e-c e x} \left(225 a^2 \left(c^2 x^2-1\right)^3+30 a b c x \sqrt{1-c^2 x^2} \left(3 c^4 x^4-10 c^2 x^2+15\right)+30 b \sin ^{-1}(c x) \left(15 a \left(c^2 x^2-1\right)^3+b c x \sqrt{1-c^2 x^2} \left(3 c^4 x^4-10 c^2 x^2+15\right)\right)+225 b^2 \left(c^2 x^2-1\right)^3 \sin ^{-1}(c x)^2+2 b^2 \left(-9 c^6 x^6+47 c^4 x^4-187 c^2 x^2+149\right)\right)}{1125 c^2 \left(c^2 x^2-1\right)}","\frac{2 b d e x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{5 c \sqrt{1-c^2 x^2}}-\frac{d e \left(1-c^2 x^2\right)^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{5 c^2}-\frac{4 b c d e x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{15 \sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^5 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{25 \sqrt{1-c^2 x^2}}+\frac{2 b^2 d e \left(1-c^2 x^2\right)^2 \sqrt{c d x+d} \sqrt{e-c e x}}{125 c^2}+\frac{8 b^2 d e \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x}}{225 c^2}+\frac{16 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x}}{75 c^2}",1,"-1/1125*(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(225*a^2*(-1 + c^2*x^2)^3 + 30*a*b*c*x*Sqrt[1 - c^2*x^2]*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 2*b^2*(149 - 187*c^2*x^2 + 47*c^4*x^4 - 9*c^6*x^6) + 30*b*(15*a*(-1 + c^2*x^2)^3 + b*c*x*Sqrt[1 - c^2*x^2]*(15 - 10*c^2*x^2 + 3*c^4*x^4))*ArcSin[c*x] + 225*b^2*(-1 + c^2*x^2)^3*ArcSin[c*x]^2))/(c^2*(-1 + c^2*x^2))","A",1
583,1,373,362,1.2790307,"\int (d+c d x)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2,x]","\frac{d e \sqrt{c d x+d} \sqrt{e-c e x} \left(160 a^2 c x \sqrt{1-c^2 x^2}-64 a^2 c^3 x^3 \sqrt{1-c^2 x^2}+64 a b \cos \left(2 \sin ^{-1}(c x)\right)+4 a b \cos \left(4 \sin ^{-1}(c x)\right)-32 b^2 \sin \left(2 \sin ^{-1}(c x)\right)-b^2 \sin \left(4 \sin ^{-1}(c x)\right)\right)-96 a^2 d^{3/2} e^{3/2} \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+8 b d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(12 a+8 b \sin \left(2 \sin ^{-1}(c x)\right)+b \sin \left(4 \sin ^{-1}(c x)\right)\right)+4 b d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(4 a \left(8 \sin \left(2 \sin ^{-1}(c x)\right)+\sin \left(4 \sin ^{-1}(c x)\right)\right)+16 b \cos \left(2 \sin ^{-1}(c x)\right)+b \cos \left(4 \sin ^{-1}(c x)\right)\right)+32 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{256 c \sqrt{1-c^2 x^2}}","\frac{(c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^3}{8 b c \left(1-c^2 x^2\right)^{3/2}}+\frac{3 x (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{8 \left(1-c^2 x^2\right)}+\frac{b \sqrt{1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{8 c}-\frac{3 b c x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)}{8 \left(1-c^2 x^2\right)^{3/2}}+\frac{1}{4} x (c d x+d)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{15 b^2 x (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 \left(1-c^2 x^2\right)}+\frac{9 b^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left(1-c^2 x^2\right)^{3/2}}-\frac{1}{32} b^2 x (c d x+d)^{3/2} (e-c e x)^{3/2}",1,"(32*b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 96*a^2*d^(3/2)*e^(3/2)*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 8*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(12*a + 8*b*Sin[2*ArcSin[c*x]] + b*Sin[4*ArcSin[c*x]]) + d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(160*a^2*c*x*Sqrt[1 - c^2*x^2] - 64*a^2*c^3*x^3*Sqrt[1 - c^2*x^2] + 64*a*b*Cos[2*ArcSin[c*x]] + 4*a*b*Cos[4*ArcSin[c*x]] - 32*b^2*Sin[2*ArcSin[c*x]] - b^2*Sin[4*ArcSin[c*x]]) + 4*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(16*b*Cos[2*ArcSin[c*x]] + b*Cos[4*ArcSin[c*x]] + 4*a*(8*Sin[2*ArcSin[c*x]] + Sin[4*ArcSin[c*x]])))/(256*c*Sqrt[1 - c^2*x^2])","A",1
584,1,632,647,4.1229649,"\int \frac{(d+c d x)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/x,x]","-\frac{1}{3} a^2 d e \left(c^2 x^2-4\right) \sqrt{c d x+d} \sqrt{e-c e x}+a^2 d^{3/2} e^{3/2} \log (c x)-a^2 d^{3/2} e^{3/2} \log \left(\sqrt{d} \sqrt{e} \sqrt{c d x+d} \sqrt{e-c e x}+d e\right)-\frac{2 a b d e \sqrt{c d x+d} \sqrt{e-c e x} \left(-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)-i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+c x-\sin ^{-1}(c x) \log \left(1-e^{i \sin ^{-1}(c x)}\right)+\sin ^{-1}(c x) \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}+\frac{2 a b d e \sqrt{c d x+d} \sqrt{e-c e x} \left(c^3 x^3+3 \left(1-c^2 x^2\right)^{3/2} \sin ^{-1}(c x)-3 c x\right)}{9 \sqrt{1-c^2 x^2}}-\frac{b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \left(2 \sqrt{1-c^2 x^2}-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2-2 i \sin ^{-1}(c x) \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)+2 i \sin ^{-1}(c x) \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+2 \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)-2 \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)+2 c x \sin ^{-1}(c x)+\sin ^{-1}(c x)^2 \left(-\log \left(1-e^{i \sin ^{-1}(c x)}\right)\right)+\sin ^{-1}(c x)^2 \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{1-c^2 x^2}}+\frac{b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \left(27 \sqrt{1-c^2 x^2} \left(\sin ^{-1}(c x)^2-2\right)-6 \sin ^{-1}(c x) \left(9 c x+\sin \left(3 \sin ^{-1}(c x)\right)\right)+\left(9 \sin ^{-1}(c x)^2-2\right) \cos \left(3 \sin ^{-1}(c x)\right)\right)}{108 \sqrt{1-c^2 x^2}}","\frac{2 i b d e \sqrt{c d x+d} \sqrt{e-c e x} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{2 i b d e \sqrt{c d x+d} \sqrt{e-c e x} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{2 a b c d e x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{1-c^2 x^2}}-\frac{2 b c d e x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{3 \sqrt{1-c^2 x^2}}+\frac{1}{3} d e \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2 d e \sqrt{c d x+d} \sqrt{e-c e x} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}}+\frac{2 b c^3 d e x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{9 \sqrt{1-c^2 x^2}}+d e \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2-\frac{2 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}+\frac{2 b^2 d e \sqrt{c d x+d} \sqrt{e-c e x} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{2}{27} b^2 d e \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x}-\frac{2 b^2 c d e x \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}}-\frac{22}{9} b^2 d e \sqrt{c d x+d} \sqrt{e-c e x}",1,"-1/3*(a^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-4 + c^2*x^2)) + (2*a*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-3*c*x + c^3*x^3 + 3*(1 - c^2*x^2)^(3/2)*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) + a^2*d^(3/2)*e^(3/2)*Log[c*x] - a^2*d^(3/2)*e^(3/2)*Log[d*e + Sqrt[d]*Sqrt[e]*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]] - (2*a*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(c*x - Sqrt[1 - c^2*x^2]*ArcSin[c*x] - ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - I*PolyLog[2, -E^(I*ArcSin[c*x])] + I*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] - (b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(2*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x] - Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 - ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] + ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] - (2*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] + (2*I)*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] + 2*PolyLog[3, -E^(I*ArcSin[c*x])] - 2*PolyLog[3, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (b^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(27*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]*(9*c*x + Sin[3*ArcSin[c*x]])))/(108*Sqrt[1 - c^2*x^2])","A",0
585,1,538,505,2.2525628,"\int \frac{(d+c d x)^{3/2} (e-c e x)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)*(a + b*ArcSin[c*x])^2)/x^2,x]","\frac{12 a^2 c d^{3/2} e^{3/2} x \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)-4 a^2 c^2 d e x^2 \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{e-c e x}-8 a^2 d e \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{e-c e x}-2 b d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left(6 a c x+4 b \sqrt{1-c^2 x^2}+4 i b c x+b c x \sin \left(2 \sin ^{-1}(c x)\right)\right)-2 b d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left(8 a \sqrt{1-c^2 x^2}+2 a c x \sin \left(2 \sin ^{-1}(c x)\right)-8 b c x \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+b c x \cos \left(2 \sin ^{-1}(c x)\right)\right)+16 a b c d e x \sqrt{c d x+d} \sqrt{e-c e x} \log (c x)-2 a b c d e x \sqrt{c d x+d} \sqrt{e-c e x} \cos \left(2 \sin ^{-1}(c x)\right)-8 i b^2 c d e x \sqrt{c d x+d} \sqrt{e-c e x} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-4 b^2 c d e x \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3+b^2 c d e x \sqrt{c d x+d} \sqrt{e-c e x} \sin \left(2 \sin ^{-1}(c x)\right)}{8 x \sqrt{1-c^2 x^2}}","-\frac{c d e \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^3}{2 b \sqrt{1-c^2 x^2}}-\frac{i c d e \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{1-c^2 x^2}}+b c d e \sqrt{1-c^2 x^2} \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)-\frac{d e \left(1-c^2 x^2\right) \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2}{x}+\frac{2 b c d e \sqrt{c d x+d} \sqrt{e-c e x} \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}}-\frac{3}{2} c^2 d e x \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)^2+\frac{3 b c^3 d e x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left(a+b \sin ^{-1}(c x)\right)}{2 \sqrt{1-c^2 x^2}}-\frac{i b^2 c d e \sqrt{c d x+d} \sqrt{e-c e x} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{\sqrt{1-c^2 x^2}}-\frac{5 b^2 c d e \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{4 \sqrt{1-c^2 x^2}}+\frac{1}{4} b^2 c^2 d e x \sqrt{c d x+d} \sqrt{e-c e x}",1,"(-8*a^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2] - 4*a^2*c^2*d*e*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2] - 4*b^2*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 + 12*a^2*c*d^(3/2)*e^(3/2)*x*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 2*a*b*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Cos[2*ArcSin[c*x]] + 16*a*b*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Log[c*x] - (8*I)*b^2*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*PolyLog[2, E^((2*I)*ArcSin[c*x])] + b^2*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sin[2*ArcSin[c*x]] - 2*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(8*a*Sqrt[1 - c^2*x^2] + b*c*x*Cos[2*ArcSin[c*x]] - 8*b*c*x*Log[1 - E^((2*I)*ArcSin[c*x])] + 2*a*c*x*Sin[2*ArcSin[c*x]]) - 2*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(6*a*c*x + (4*I)*b*c*x + 4*b*Sqrt[1 - c^2*x^2] + b*c*x*Sin[2*ArcSin[c*x]]))/(8*x*Sqrt[1 - c^2*x^2])","A",0
586,1,326,250,1.3052014,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d+c d x} \sqrt{e-c e x}} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x])^2)/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]),x]","\frac{-3 \sqrt{d} \sqrt{e} \left(a^2 \left(4 c x-4 c^3 x^3\right)+a b \sqrt{1-c^2 x^2}+a b \cos \left(3 \sin ^{-1}(c x)\right)+2 b^2 c x \left(c^2 x^2-1\right)\right)-12 a^2 \sqrt{c d x+d} \sqrt{e-c e x} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+12 b \sqrt{d} \sqrt{e} \sin ^{-1}(c x)^2 \left(a \sqrt{1-c^2 x^2}+b c x \left(c^2 x^2-1\right)\right)-3 b \sqrt{d} \sqrt{e} \sin ^{-1}(c x) \left(2 a c x+2 a \sin \left(3 \sin ^{-1}(c x)\right)+b \sqrt{1-c^2 x^2}+b \cos \left(3 \sin ^{-1}(c x)\right)\right)+4 b^2 \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^3}{24 c^3 \sqrt{d} \sqrt{e} \sqrt{c d x+d} \sqrt{e-c e x}}","-\frac{x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{2 c^2 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{b x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{2 c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{6 b c^3 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{b^2 x \left(1-c^2 x^2\right)}{4 c^2 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c^3 \sqrt{c d x+d} \sqrt{e-c e x}}",1,"(12*b*Sqrt[d]*Sqrt[e]*(a*Sqrt[1 - c^2*x^2] + b*c*x*(-1 + c^2*x^2))*ArcSin[c*x]^2 + 4*b^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^3 - 12*a^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] - 3*Sqrt[d]*Sqrt[e]*(a*b*Sqrt[1 - c^2*x^2] + 2*b^2*c*x*(-1 + c^2*x^2) + a^2*(4*c*x - 4*c^3*x^3) + a*b*Cos[3*ArcSin[c*x]]) - 3*b*Sqrt[d]*Sqrt[e]*ArcSin[c*x]*(2*a*c*x + b*Sqrt[1 - c^2*x^2] + b*Cos[3*ArcSin[c*x]] + 2*a*Sin[3*ArcSin[c*x]]))/(24*c^3*Sqrt[d]*Sqrt[e]*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])","A",1
587,1,150,177,0.6665545,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d+c d x} \sqrt{e-c e x}} \, dx","Integrate[(x*(a + b*ArcSin[c*x])^2)/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]),x]","-\frac{\sqrt{c d x+d} \sqrt{e-c e x} \left(a^2 \left(c^2 x^2-1\right)+2 a b c x \sqrt{1-c^2 x^2}+2 b \sin ^{-1}(c x) \left(a \left(c^2 x^2-1\right)+b c x \sqrt{1-c^2 x^2}\right)-2 b^2 \left(c^2 x^2-1\right)+b^2 \left(c^2 x^2-1\right) \sin ^{-1}(c x)^2\right)}{c^2 d e (c x-1) (c x+1)}","\frac{2 a b x \sqrt{1-c^2 x^2}}{c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{\left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 b^2 \left(1-c^2 x^2\right)}{c^2 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt{c d x+d} \sqrt{e-c e x}}",1,"-((Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(2*a*b*c*x*Sqrt[1 - c^2*x^2] + a^2*(-1 + c^2*x^2) - 2*b^2*(-1 + c^2*x^2) + 2*b*(b*c*x*Sqrt[1 - c^2*x^2] + a*(-1 + c^2*x^2))*ArcSin[c*x] + b^2*(-1 + c^2*x^2)*ArcSin[c*x]^2))/(c^2*d*e*(-1 + c*x)*(1 + c*x)))","A",1
588,1,159,55,0.6757597,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d+c d x} \sqrt{e-c e x}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]),x]","\frac{-\frac{3 a^2 \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)}{\sqrt{d} \sqrt{e}}+\frac{3 a b \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{\sqrt{c d x+d} \sqrt{e-c e x}}+\frac{b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^3}{\sqrt{c d x+d} \sqrt{e-c e x}}}{3 c}","\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c \sqrt{c d x+d} \sqrt{e-c e x}}",1,"((3*a*b*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^3)/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (3*a^2*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))])/(Sqrt[d]*Sqrt[e]))/(3*c)","B",1
589,1,336,287,1.4533362,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x \sqrt{d+c d x} \sqrt{e-c e x}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]),x]","\frac{a^2 \log (c x)}{\sqrt{d} \sqrt{e}}-\frac{a^2 \log \left(\sqrt{d} \sqrt{e} \sqrt{c d x+d} \sqrt{e-c e x}+d e\right)}{\sqrt{d} \sqrt{e}}+\frac{2 a b \sqrt{1-c^2 x^2} \left(i \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-i \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)+\sin ^{-1}(c x) \left(\log \left(1-e^{i \sin ^{-1}(c x)}\right)-\log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)\right)}{\sqrt{c d x+d} \sqrt{e-c e x}}+\frac{b^2 \sqrt{1-c^2 x^2} \left(2 i \sin ^{-1}(c x) \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-2 i \sin ^{-1}(c x) \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)-2 \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)+2 \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)+\sin ^{-1}(c x)^2 \log \left(1-e^{i \sin ^{-1}(c x)}\right)-\sin ^{-1}(c x)^2 \log \left(1+e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{c d x+d} \sqrt{e-c e x}}","\frac{2 i b \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 i b \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{\sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 b^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{\sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 b^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{\sqrt{c d x+d} \sqrt{e-c e x}}",1,"(a^2*Log[c*x])/(Sqrt[d]*Sqrt[e]) - (a^2*Log[d*e + Sqrt[d]*Sqrt[e]*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]])/(Sqrt[d]*Sqrt[e]) + (2*a*b*Sqrt[1 - c^2*x^2]*(ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]) + I*PolyLog[2, -E^(I*ArcSin[c*x])] - I*PolyLog[2, E^(I*ArcSin[c*x])]))/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (b^2*Sqrt[1 - c^2*x^2]*(ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] - ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] + (2*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] - (2*I)*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] - 2*PolyLog[3, -E^(I*ArcSin[c*x])] + 2*PolyLog[3, E^(I*ArcSin[c*x])]))/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x])","A",0
590,1,189,214,1.1803422,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^2 \sqrt{d+c d x} \sqrt{e-c e x}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]),x]","\frac{a \left(a c^2 x^2-a+2 b c x \sqrt{1-c^2 x^2} \log (c x)\right)+2 b \sin ^{-1}(c x) \left(a c^2 x^2-a+b c x \sqrt{1-c^2 x^2} \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)\right)-i b^2 c x \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+b^2 \left(c^2 x^2-i c x \sqrt{1-c^2 x^2}-1\right) \sin ^{-1}(c x)^2}{x \sqrt{c d x+d} \sqrt{e-c e x}}","-\frac{\left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{x \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{i c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 b c \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)}{\sqrt{c d x+d} \sqrt{e-c e x}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{\sqrt{c d x+d} \sqrt{e-c e x}}",1,"(b^2*(-1 + c^2*x^2 - I*c*x*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 + 2*b*ArcSin[c*x]*(-a + a*c^2*x^2 + b*c*x*Sqrt[1 - c^2*x^2]*Log[1 - E^((2*I)*ArcSin[c*x])]) + a*(-a + a*c^2*x^2 + 2*b*c*x*Sqrt[1 - c^2*x^2]*Log[c*x]) - I*b^2*c*x*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/(x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])","A",0
591,1,636,295,2.5973449,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)),x]","\frac{3 a^2 \sqrt{e} \sqrt{c d x+d} \sqrt{e-c e x} \tan ^{-1}\left(\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left(c^2 x^2-1\right)}\right)+3 a^2 c \sqrt{d} e x+3 a b \sqrt{d} e \left(\sqrt{1-c^2 x^2} \left(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(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)\right)-\sin ^{-1}(c x)^2\right)+2 c x \sin ^{-1}(c x)\right)+b^2 \sqrt{d} e \left(-6 i \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-6 i \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-\sqrt{1-c^2 x^2} \sin ^{-1}(c x)^3-3 i \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2+6 i \pi  \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+6 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+6 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+12 \pi  \sqrt{1-c^2 x^2} \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+3 \pi  \sqrt{1-c^2 x^2} \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-3 \pi  \sqrt{1-c^2 x^2} \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-3 \pi  \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-12 \pi  \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+3 \pi  \sqrt{1-c^2 x^2} \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+3 c x \sin ^{-1}(c x)^2\right)}{3 c^3 d^{3/2} e^2 \sqrt{c d x+d} \sqrt{e-c e x}}","\frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^3}{3 b c^3 d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{i \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{c^3 d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 b \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^3 d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{c^3 d e \sqrt{c d x+d} \sqrt{e-c e x}}",1,"(3*a^2*c*Sqrt[d]*e*x + 3*a^2*Sqrt[e]*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 3*a*b*Sqrt[d]*e*(2*c*x*ArcSin[c*x] + Sqrt[1 - c^2*x^2]*(-ArcSin[c*x]^2 + 2*(Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]))) + b^2*Sqrt[d]*e*((6*I)*Pi*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + 3*c*x*ArcSin[c*x]^2 - (3*I)*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 - Sqrt[1 - c^2*x^2]*ArcSin[c*x]^3 + 12*Pi*Sqrt[1 - c^2*x^2]*Log[1 + E^((-I)*ArcSin[c*x])] + 3*Pi*Sqrt[1 - c^2*x^2]*Log[1 - I*E^(I*ArcSin[c*x])] + 6*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 3*Pi*Sqrt[1 - c^2*x^2]*Log[1 + I*E^(I*ArcSin[c*x])] + 6*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - 12*Pi*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2]] + 3*Pi*Sqrt[1 - c^2*x^2]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - 3*Pi*Sqrt[1 - c^2*x^2]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (6*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (6*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])]))/(3*c^3*d^(3/2)*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])","B",1
592,1,453,244,1.3462229,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx","Integrate[(x*(a + b*ArcSin[c*x])^2)/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)),x]","\frac{a^2+2 a b \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-2 a b \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 a b \sin ^{-1}(c x)-2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)+i \pi  b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-2 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-\pi  b^2 \sqrt{1-c^2 x^2} \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+2 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-\pi  b^2 \sqrt{1-c^2 x^2} \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+\pi  b^2 \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+\pi  b^2 \sqrt{1-c^2 x^2} \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+b^2 \sin ^{-1}(c x)^2}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}}","\frac{4 i b \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 e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{c^2 d e \sqrt{c d x+d} \sqrt{e-c e x}}",1,"(a^2 + 2*a*b*ArcSin[c*x] + I*b^2*Pi*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + b^2*ArcSin[c*x]^2 - b^2*Pi*Sqrt[1 - c^2*x^2]*Log[1 - I*E^(I*ArcSin[c*x])] - 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - b^2*Pi*Sqrt[1 - c^2*x^2]*Log[1 + I*E^(I*ArcSin[c*x])] + 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + b^2*Pi*Sqrt[1 - c^2*x^2]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 2*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - 2*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + b^2*Pi*Sqrt[1 - c^2*x^2]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])","A",1
593,1,550,217,0.7576315,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)),x]","\frac{a^2 c x+2 a b \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 a b \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+2 a b c x \sin ^{-1}(c x)-2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-i b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2+2 i \pi  b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)+4 \pi  b^2 \sqrt{1-c^2 x^2} \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+2 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+\pi  b^2 \sqrt{1-c^2 x^2} \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+2 b^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-\pi  b^2 \sqrt{1-c^2 x^2} \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-\pi  b^2 \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-4 \pi  b^2 \sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\pi  b^2 \sqrt{1-c^2 x^2} \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+b^2 c x \sin ^{-1}(c x)^2}{c d e \sqrt{c d x+d} \sqrt{e-c e x}}","-\frac{i \left(1-c^2 x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{x \left(1-c^2 x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}}+\frac{2 b \left(1-c^2 x^2\right)^{3/2} \log \left(1+e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}-\frac{i b^2 \left(1-c^2 x^2\right)^{3/2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{c (c d x+d)^{3/2} (e-c e x)^{3/2}}",1,"(a^2*c*x + 2*a*b*c*x*ArcSin[c*x] + (2*I)*b^2*Pi*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + b^2*c*x*ArcSin[c*x]^2 - I*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 + 4*b^2*Pi*Sqrt[1 - c^2*x^2]*Log[1 + E^((-I)*ArcSin[c*x])] + b^2*Pi*Sqrt[1 - c^2*x^2]*Log[1 - I*E^(I*ArcSin[c*x])] + 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - b^2*Pi*Sqrt[1 - c^2*x^2]*Log[1 + I*E^(I*ArcSin[c*x])] + 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - 4*b^2*Pi*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2]] + b^2*Pi*Sqrt[1 - c^2*x^2]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 2*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 2*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - b^2*Pi*Sqrt[1 - c^2*x^2]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] - (2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])])/(c*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])","B",1
594,1,877,548,5.7611534,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)),x]","\frac{\sqrt{d} \sqrt{e} \log (c x) a^2-\sqrt{d} \sqrt{e} \log \left(d e+\sqrt{d} \sqrt{c x d+d} \sqrt{e-c e x} \sqrt{e}\right) a^2-\frac{\sqrt{c x d+d} \sqrt{e-c e x} a^2}{c^2 x^2-1}+\frac{2 b d e \left(\sqrt{1-c^2 x^2} \log \left(1-e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-\sqrt{1-c^2 x^2} \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+\sin ^{-1}(c x)+\sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-\sqrt{1-c^2 x^2} \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+i \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right)-i \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right)\right) a}{\sqrt{c x d+d} \sqrt{e-c e x}}+\frac{b^2 d e \left(\sqrt{1-c^2 x^2} \log \left(1-e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)^2-\sqrt{1-c^2 x^2} \log \left(1+e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)^2+\sin ^{-1}(c x)^2-2 \sqrt{1-c^2 x^2} \log \left(1-i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+2 \sqrt{1-c^2 x^2} \log \left(1+i e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+2 i \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)-2 i \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \sin ^{-1}(c x)+i \pi  \sqrt{1-c^2 x^2} \sin ^{-1}(c x)-\pi  \sqrt{1-c^2 x^2} \log \left(1-i e^{i \sin ^{-1}(c x)}\right)-\pi  \sqrt{1-c^2 x^2} \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+\pi  \sqrt{1-c^2 x^2} \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)+\pi  \sqrt{1-c^2 x^2} \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-2 i \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)+2 i \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-2 \sqrt{1-c^2 x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)+2 \sqrt{1-c^2 x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)\right)}{\sqrt{c x d+d} \sqrt{e-c e x}}}{d^2 e^2}","\frac{2 i b \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 i b \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{4 i b \sqrt{1-c^2 x^2} \tan ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)^2}{d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)}{d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 i b^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 b^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(-e^{i \sin ^{-1}(c x)}\right)}{d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 b^2 \sqrt{1-c^2 x^2} \text{Li}_3\left(e^{i \sin ^{-1}(c x)}\right)}{d e \sqrt{c d x+d} \sqrt{e-c e x}}",1,"(-((a^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(-1 + c^2*x^2)) + a^2*Sqrt[d]*Sqrt[e]*Log[c*x] - a^2*Sqrt[d]*Sqrt[e]*Log[d*e + Sqrt[d]*Sqrt[e]*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]] + (2*a*b*d*e*(ArcSin[c*x] + Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] + Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] - Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] + I*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])] - I*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(I*ArcSin[c*x])]))/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (b^2*d*e*(I*Pi*Sqrt[1 - c^2*x^2]*ArcSin[c*x] + ArcSin[c*x]^2 + Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2*Log[1 - E^(I*ArcSin[c*x])] - Pi*Sqrt[1 - c^2*x^2]*Log[1 - I*E^(I*ArcSin[c*x])] - 2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - Pi*Sqrt[1 - c^2*x^2]*Log[1 + I*E^(I*ArcSin[c*x])] + 2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x])] + Pi*Sqrt[1 - c^2*x^2]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + Pi*Sqrt[1 - c^2*x^2]*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (2*I)*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSin[c*x])] - (2*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I*ArcSin[c*x])] + (2*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x])] - (2*I)*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] - 2*Sqrt[1 - c^2*x^2]*PolyLog[3, -E^(I*ArcSin[c*x])] + 2*Sqrt[1 - c^2*x^2]*PolyLog[3, E^(I*ArcSin[c*x])]))/(Sqrt[d + c*d*x]*Sqrt[e - c*e*x]))/(d^2*e^2)","A",1
595,1,564,396,2.5879339,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{x^2 (d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(x^2*(d + c*d*x)^(3/2)*(e - c*e*x)^(3/2)),x]","\frac{c \csc \left(\frac{1}{2} \sin ^{-1}(c x)\right) \sec \left(\frac{1}{2} \sin ^{-1}(c x)\right) \left(4 a^2 c^2 x^2-2 a^2+2 a b \log (c x) \sin \left(2 \sin ^{-1}(c x)\right)-4 a b \sin ^{-1}(c x) \cos \left(2 \sin ^{-1}(c x)\right)+2 a b \sin \left(2 \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 a b \sin \left(2 \sin ^{-1}(c x)\right) \log \left(\sin \left(\frac{1}{2} \sin ^{-1}(c x)\right)+\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)-2 i b^2 \sin \left(2 \sin ^{-1}(c x)\right) \text{Li}_2\left(-i e^{i \sin ^{-1}(c x)}\right)-2 i b^2 \sin \left(2 \sin ^{-1}(c x)\right) \text{Li}_2\left(i e^{i \sin ^{-1}(c x)}\right)-i b^2 \sin \left(2 \sin ^{-1}(c x)\right) \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-2 i b^2 \sin ^{-1}(c x)^2 \sin \left(2 \sin ^{-1}(c x)\right)+2 i \pi  b^2 \sin ^{-1}(c x) \sin \left(2 \sin ^{-1}(c x)\right)+4 \pi  b^2 \sin \left(2 \sin ^{-1}(c x)\right) \log \left(1+e^{-i \sin ^{-1}(c x)}\right)+2 b^2 \sin ^{-1}(c x) \sin \left(2 \sin ^{-1}(c x)\right) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+\pi  b^2 \sin \left(2 \sin ^{-1}(c x)\right) \log \left(1-i e^{i \sin ^{-1}(c x)}\right)+2 b^2 \sin ^{-1}(c x) \sin \left(2 \sin ^{-1}(c x)\right) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)-\pi  b^2 \sin \left(2 \sin ^{-1}(c x)\right) \log \left(1+i e^{i \sin ^{-1}(c x)}\right)+2 b^2 \sin ^{-1}(c x) \sin \left(2 \sin ^{-1}(c x)\right) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-\pi  b^2 \sin \left(2 \sin ^{-1}(c x)\right) \log \left(\sin \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)-2 b^2 \sin ^{-1}(c x)^2 \cos \left(2 \sin ^{-1}(c x)\right)-4 \pi  b^2 \sin \left(2 \sin ^{-1}(c x)\right) \log \left(\cos \left(\frac{1}{2} \sin ^{-1}(c x)\right)\right)+\pi  b^2 \sin \left(2 \sin ^{-1}(c x)\right) \log \left(-\cos \left(\frac{1}{4} \left(2 \sin ^{-1}(c x)+\pi \right)\right)\right)\right)}{4 d e \sqrt{c d x+d} \sqrt{e-c e x}}","-\frac{2 i c \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)^2}{d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{4 b c \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)}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{4 b c \sqrt{1-c^2 x^2} \tanh ^{-1}\left(e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d e \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 c^2 x \left(a+b \sin ^{-1}(c x)\right)^2}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{d e x \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left(-e^{2 i \sin ^{-1}(c x)}\right)}{d e \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{i b^2 c \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{d e \sqrt{c d x+d} \sqrt{e-c e x}}",1,"(c*Csc[ArcSin[c*x]/2]*Sec[ArcSin[c*x]/2]*(-2*a^2 + 4*a^2*c^2*x^2 - 4*a*b*ArcSin[c*x]*Cos[2*ArcSin[c*x]] - 2*b^2*ArcSin[c*x]^2*Cos[2*ArcSin[c*x]] + (2*I)*b^2*Pi*ArcSin[c*x]*Sin[2*ArcSin[c*x]] - (2*I)*b^2*ArcSin[c*x]^2*Sin[2*ArcSin[c*x]] + 4*b^2*Pi*Log[1 + E^((-I)*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] + b^2*Pi*Log[1 - I*E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] + 2*b^2*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] - b^2*Pi*Log[1 + I*E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] + 2*b^2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] + 2*b^2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] + 2*a*b*Log[c*x]*Sin[2*ArcSin[c*x]] - 4*b^2*Pi*Log[Cos[ArcSin[c*x]/2]]*Sin[2*ArcSin[c*x]] + b^2*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]]*Sin[2*ArcSin[c*x]] + 2*a*b*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]]*Sin[2*ArcSin[c*x]] + 2*a*b*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]]*Sin[2*ArcSin[c*x]] - b^2*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]]*Sin[2*ArcSin[c*x]] - (2*I)*b^2*PolyLog[2, (-I)*E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] - (2*I)*b^2*PolyLog[2, I*E^(I*ArcSin[c*x])]*Sin[2*ArcSin[c*x]] - I*b^2*PolyLog[2, E^((2*I)*ArcSin[c*x])]*Sin[2*ArcSin[c*x]]))/(4*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])","A",0
596,1,115,152,0.1351588,"\int x^4 \left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^4*(d + e*x^2)*(a + b*ArcSin[c*x]),x]","\frac{105 a x^5 \left(7 d+5 e x^2\right)+\frac{b \sqrt{1-c^2 x^2} \left(3 c^6 \left(49 d x^4+25 e x^6\right)+2 c^4 \left(98 d x^2+45 e x^4\right)+8 c^2 \left(49 d+15 e x^2\right)+240 e\right)}{c^7}+105 b x^5 \sin ^{-1}(c x) \left(7 d+5 e x^2\right)}{3675}","\frac{1}{5} d x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{7} e x^7 \left(a+b \sin ^{-1}(c x)\right)+\frac{b \left(1-c^2 x^2\right)^{5/2} \left(7 c^2 d+15 e\right)}{175 c^7}-\frac{b \left(1-c^2 x^2\right)^{3/2} \left(14 c^2 d+15 e\right)}{105 c^7}+\frac{b \sqrt{1-c^2 x^2} \left(7 c^2 d+5 e\right)}{35 c^7}-\frac{b e \left(1-c^2 x^2\right)^{7/2}}{49 c^7}",1,"(105*a*x^5*(7*d + 5*e*x^2) + (b*Sqrt[1 - c^2*x^2]*(240*e + 8*c^2*(49*d + 15*e*x^2) + 2*c^4*(98*d*x^2 + 45*e*x^4) + 3*c^6*(49*d*x^4 + 25*e*x^6)))/c^7 + 105*b*x^5*(7*d + 5*e*x^2)*ArcSin[c*x])/3675","A",1
597,1,116,149,0.0912838,"\int x^3 \left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)*(a + b*ArcSin[c*x]),x]","\frac{24 a c^6 x^4 \left(3 d+2 e x^2\right)+3 b \sin ^{-1}(c x) \left(8 c^6 \left(3 d x^4+2 e x^6\right)-9 c^2 d-5 e\right)+b c x \sqrt{1-c^2 x^2} \left(2 c^4 \left(9 d x^2+4 e x^4\right)+c^2 \left(27 d+10 e x^2\right)+15 e\right)}{288 c^6}","\frac{1}{4} d x^4 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{6} e x^6 \left(a+b \sin ^{-1}(c x)\right)+\frac{b e x^5 \sqrt{1-c^2 x^2}}{36 c}-\frac{b \left(9 c^2 d+5 e\right) \sin ^{-1}(c x)}{96 c^6}+\frac{b x \sqrt{1-c^2 x^2} \left(9 c^2 d+5 e\right)}{96 c^5}+\frac{b x^3 \sqrt{1-c^2 x^2} \left(9 c^2 d+5 e\right)}{144 c^3}",1,"(24*a*c^6*x^4*(3*d + 2*e*x^2) + b*c*x*Sqrt[1 - c^2*x^2]*(15*e + c^2*(27*d + 10*e*x^2) + 2*c^4*(9*d*x^2 + 4*e*x^4)) + 3*b*(-9*c^2*d - 5*e + 8*c^6*(3*d*x^4 + 2*e*x^6))*ArcSin[c*x])/(288*c^6)","A",1
598,1,96,120,0.1049027,"\int x^2 \left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)*(a + b*ArcSin[c*x]),x]","\frac{1}{225} \left(15 a x^3 \left(5 d+3 e x^2\right)+\frac{b \sqrt{1-c^2 x^2} \left(c^4 \left(25 d x^2+9 e x^4\right)+2 c^2 \left(25 d+6 e x^2\right)+24 e\right)}{c^5}+15 b x^3 \sin ^{-1}(c x) \left(5 d+3 e x^2\right)\right)","\frac{1}{3} d x^3 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{5} e x^5 \left(a+b \sin ^{-1}(c x)\right)-\frac{b \left(1-c^2 x^2\right)^{3/2} \left(5 c^2 d+6 e\right)}{45 c^5}+\frac{b \sqrt{1-c^2 x^2} \left(5 c^2 d+3 e\right)}{15 c^5}+\frac{b e \left(1-c^2 x^2\right)^{5/2}}{25 c^5}",1,"(15*a*x^3*(5*d + 3*e*x^2) + (b*Sqrt[1 - c^2*x^2]*(24*e + 2*c^2*(25*d + 6*e*x^2) + c^4*(25*d*x^2 + 9*e*x^4)))/c^5 + 15*b*x^3*(5*d + 3*e*x^2)*ArcSin[c*x])/225","A",1
599,1,95,122,0.0727273,"\int x \left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)*(a + b*ArcSin[c*x]),x]","\frac{c x \left(8 a c^3 x \left(2 d+e x^2\right)+b \sqrt{1-c^2 x^2} \left(2 c^2 \left(4 d+e x^2\right)+3 e\right)\right)+b \sin ^{-1}(c x) \left(8 c^4 \left(2 d x^2+e x^4\right)-8 c^2 d-3 e\right)}{32 c^4}","\frac{\left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{4 e}+\frac{b x \sqrt{1-c^2 x^2} \left(d+e x^2\right)}{16 c}-\frac{b \left(8 c^4 d^2+8 c^2 d e+3 e^2\right) \sin ^{-1}(c x)}{32 c^4 e}+\frac{3 b x \sqrt{1-c^2 x^2} \left(2 c^2 d+e\right)}{32 c^3}",1,"(c*x*(8*a*c^3*x*(2*d + e*x^2) + b*Sqrt[1 - c^2*x^2]*(3*e + 2*c^2*(4*d + e*x^2))) + b*(-8*c^2*d - 3*e + 8*c^4*(2*d*x^2 + e*x^4))*ArcSin[c*x])/(32*c^4)","A",1
600,1,71,81,0.0771723,"\int \left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)*(a + b*ArcSin[c*x]),x]","\frac{1}{9} \left(3 a x \left(3 d+e x^2\right)+\frac{b \sqrt{1-c^2 x^2} \left(c^2 \left(9 d+e x^2\right)+2 e\right)}{c^3}+3 b x \sin ^{-1}(c x) \left(3 d+e x^2\right)\right)","d x \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} e x^3 \left(a+b \sin ^{-1}(c x)\right)+\frac{b \sqrt{1-c^2 x^2} \left(3 c^2 d+e\right)}{3 c^3}-\frac{b e \left(1-c^2 x^2\right)^{3/2}}{9 c^3}",1,"(3*a*x*(3*d + e*x^2) + (b*Sqrt[1 - c^2*x^2]*(2*e + c^2*(9*d + e*x^2)))/c^3 + 3*b*x*(3*d + e*x^2)*ArcSin[c*x])/9","A",1
601,1,108,132,0.2060453,"\int \frac{\left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSin[c*x]))/x,x]","\frac{1}{2} \left(2 a d \log (x)+a e x^2+\frac{b e \left(c x \sqrt{1-c^2 x^2}-\sin ^{-1}(c x)\right)}{2 c^2}-i b d \left(\sin ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)\right)+2 b d \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+b e x^2 \sin ^{-1}(c x)\right)","d \log (x) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{2} e x^2 \left(a+b \sin ^{-1}(c x)\right)+\frac{b e x \sqrt{1-c^2 x^2}}{4 c}-\frac{b e \sin ^{-1}(c x)}{4 c^2}-\frac{1}{2} i b d \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-\frac{1}{2} i b d \sin ^{-1}(c x)^2+b d \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-b d \log (x) \sin ^{-1}(c x)",1,"(a*e*x^2 + (b*e*(c*x*Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(2*c^2) + b*e*x^2*ArcSin[c*x] + 2*b*d*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 2*a*d*Log[x] - I*b*d*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c*x])]))/2","A",1
602,1,71,66,0.0631636,"\int \frac{\left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSin[c*x]))/x^2,x]","-\frac{a d}{x}+a e x-b c d \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+\frac{b e \sqrt{1-c^2 x^2}}{c}-\frac{b d \sin ^{-1}(c x)}{x}+b e x \sin ^{-1}(c x)","-\frac{d \left(a+b \sin ^{-1}(c x)\right)}{x}+e x \left(a+b \sin ^{-1}(c x)\right)-b c d \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+\frac{b e \sqrt{1-c^2 x^2}}{c}",1,"-((a*d)/x) + a*e*x + (b*e*Sqrt[1 - c^2*x^2])/c - (b*d*ArcSin[c*x])/x + b*e*x*ArcSin[c*x] - b*c*d*ArcTanh[Sqrt[1 - c^2*x^2]]","A",1
603,1,104,119,0.1286937,"\int \frac{\left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSin[c*x]))/x^3,x]","-\frac{a d-2 a e x^2 \log (x)+b c d x \sqrt{1-c^2 x^2}+b \sin ^{-1}(c x) \left(d-2 e x^2 \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)\right)+i b e x^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+i b e x^2 \sin ^{-1}(c x)^2}{2 x^2}","-\frac{d \left(a+b \sin ^{-1}(c x)\right)}{2 x^2}+e \log (x) \left(a+b \sin ^{-1}(c x)\right)-\frac{b c d \sqrt{1-c^2 x^2}}{2 x}-\frac{1}{2} i b e \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-\frac{1}{2} i b e \sin ^{-1}(c x)^2+b e \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-b e \log (x) \sin ^{-1}(c x)",1,"-1/2*(a*d + b*c*d*x*Sqrt[1 - c^2*x^2] + I*b*e*x^2*ArcSin[c*x]^2 + b*ArcSin[c*x]*(d - 2*e*x^2*Log[1 - E^((2*I)*ArcSin[c*x])]) - 2*a*e*x^2*Log[x] + I*b*e*x^2*PolyLog[2, E^((2*I)*ArcSin[c*x])])/x^2","A",1
604,1,109,85,0.0500182,"\int \frac{\left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSin[c*x]))/x^4,x]","-\frac{a d}{3 x^3}-\frac{a e}{x}-\frac{b c d \sqrt{1-c^2 x^2}}{6 x^2}-b c e \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-\frac{1}{6} b c^3 d \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-\frac{b d \sin ^{-1}(c x)}{3 x^3}-\frac{b e \sin ^{-1}(c x)}{x}","-\frac{d \left(a+b \sin ^{-1}(c x)\right)}{3 x^3}-\frac{e \left(a+b \sin ^{-1}(c x)\right)}{x}-\frac{1}{6} b c \left(c^2 d+6 e\right) \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-\frac{b c d \sqrt{1-c^2 x^2}}{6 x^2}",1,"-1/3*(a*d)/x^3 - (a*e)/x - (b*c*d*Sqrt[1 - c^2*x^2])/(6*x^2) - (b*d*ArcSin[c*x])/(3*x^3) - (b*e*ArcSin[c*x])/x - (b*c^3*d*ArcTanh[Sqrt[1 - c^2*x^2]])/6 - b*c*e*ArcTanh[Sqrt[1 - c^2*x^2]]","A",1
605,1,187,241,0.2126786,"\int x^4 \left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^4*(d + e*x^2)^2*(a + b*ArcSin[c*x]),x]","\frac{315 a x^5 \left(63 d^2+90 d e x^2+35 e^2 x^4\right)+\frac{b \sqrt{1-c^2 x^2} \left(c^8 \left(3969 d^2 x^4+4050 d e x^6+1225 e^2 x^8\right)+4 c^6 \left(1323 d^2 x^2+1215 d e x^4+350 e^2 x^6\right)+24 c^4 \left(441 d^2+270 d e x^2+70 e^2 x^4\right)+160 c^2 e \left(81 d+14 e x^2\right)+4480 e^2\right)}{c^9}+315 b x^5 \sin ^{-1}(c x) \left(63 d^2+90 d e x^2+35 e^2 x^4\right)}{99225}","\frac{1}{5} d^2 x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{2}{7} d e x^7 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{9} e^2 x^9 \left(a+b \sin ^{-1}(c x)\right)-\frac{2 b e \left(1-c^2 x^2\right)^{7/2} \left(9 c^2 d+14 e\right)}{441 c^9}+\frac{b e^2 \left(1-c^2 x^2\right)^{9/2}}{81 c^9}+\frac{b \left(1-c^2 x^2\right)^{5/2} \left(21 c^4 d^2+90 c^2 d e+70 e^2\right)}{525 c^9}-\frac{2 b \left(1-c^2 x^2\right)^{3/2} \left(63 c^4 d^2+135 c^2 d e+70 e^2\right)}{945 c^9}+\frac{b \sqrt{1-c^2 x^2} \left(63 c^4 d^2+90 c^2 d e+35 e^2\right)}{315 c^9}",1,"(315*a*x^5*(63*d^2 + 90*d*e*x^2 + 35*e^2*x^4) + (b*Sqrt[1 - c^2*x^2]*(4480*e^2 + 160*c^2*e*(81*d + 14*e*x^2) + 24*c^4*(441*d^2 + 270*d*e*x^2 + 70*e^2*x^4) + 4*c^6*(1323*d^2*x^2 + 1215*d*e*x^4 + 350*e^2*x^6) + c^8*(3969*d^2*x^4 + 4050*d*e*x^6 + 1225*e^2*x^8)))/c^9 + 315*b*x^5*(63*d^2 + 90*d*e*x^2 + 35*e^2*x^4)*ArcSin[c*x])/99225","A",1
606,1,190,241,0.1778584,"\int x^3 \left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^2*(a + b*ArcSin[c*x]),x]","\frac{384 a c^8 x^4 \left(6 d^2+8 d e x^2+3 e^2 x^4\right)+3 b \sin ^{-1}(c x) \left(128 c^8 \left(6 d^2 x^4+8 d e x^6+3 e^2 x^8\right)-288 c^4 d^2-320 c^2 d e-105 e^2\right)+b c x \sqrt{1-c^2 x^2} \left(16 c^6 \left(36 d^2 x^2+32 d e x^4+9 e^2 x^6\right)+8 c^4 \left(108 d^2+80 d e x^2+21 e^2 x^4\right)+30 c^2 e \left(32 d+7 e x^2\right)+315 e^2\right)}{9216 c^8}","\frac{1}{4} d^2 x^4 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} d e x^6 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{8} e^2 x^8 \left(a+b \sin ^{-1}(c x)\right)+\frac{b e^2 x^7 \sqrt{1-c^2 x^2}}{64 c}+\frac{b e x^5 \sqrt{1-c^2 x^2} \left(64 c^2 d+21 e\right)}{1152 c^3}-\frac{b \left(288 c^4 d^2+320 c^2 d e+105 e^2\right) \sin ^{-1}(c x)}{3072 c^8}+\frac{b x \sqrt{1-c^2 x^2} \left(288 c^4 d^2+320 c^2 d e+105 e^2\right)}{3072 c^7}+\frac{b x^3 \sqrt{1-c^2 x^2} \left(288 c^4 d^2+320 c^2 d e+105 e^2\right)}{4608 c^5}",1,"(384*a*c^8*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + b*c*x*Sqrt[1 - c^2*x^2]*(315*e^2 + 30*c^2*e*(32*d + 7*e*x^2) + 8*c^4*(108*d^2 + 80*d*e*x^2 + 21*e^2*x^4) + 16*c^6*(36*d^2*x^2 + 32*d*e*x^4 + 9*e^2*x^6)) + 3*b*(-288*c^4*d^2 - 320*c^2*d*e - 105*e^2 + 128*c^8*(6*d^2*x^4 + 8*d*e*x^6 + 3*e^2*x^8))*ArcSin[c*x])/(9216*c^8)","A",1
607,1,158,198,0.2025851,"\int x^2 \left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)^2*(a + b*ArcSin[c*x]),x]","\frac{105 a x^3 \left(35 d^2+42 d e x^2+15 e^2 x^4\right)+\frac{b \sqrt{1-c^2 x^2} \left(c^6 \left(1225 d^2 x^2+882 d e x^4+225 e^2 x^6\right)+2 c^4 \left(1225 d^2+588 d e x^2+135 e^2 x^4\right)+24 c^2 e \left(98 d+15 e x^2\right)+720 e^2\right)}{c^7}+105 b x^3 \sin ^{-1}(c x) \left(35 d^2+42 d e x^2+15 e^2 x^4\right)}{11025}","\frac{1}{3} d^2 x^3 \left(a+b \sin ^{-1}(c x)\right)+\frac{2}{5} d e x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{7} e^2 x^7 \left(a+b \sin ^{-1}(c x)\right)+\frac{b e \left(1-c^2 x^2\right)^{5/2} \left(14 c^2 d+15 e\right)}{175 c^7}-\frac{b e^2 \left(1-c^2 x^2\right)^{7/2}}{49 c^7}-\frac{b \left(1-c^2 x^2\right)^{3/2} \left(35 c^4 d^2+84 c^2 d e+45 e^2\right)}{315 c^7}+\frac{b \sqrt{1-c^2 x^2} \left(35 c^4 d^2+42 c^2 d e+15 e^2\right)}{105 c^7}",1,"(105*a*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) + (b*Sqrt[1 - c^2*x^2]*(720*e^2 + 24*c^2*e*(98*d + 15*e*x^2) + 2*c^4*(1225*d^2 + 588*d*e*x^2 + 135*e^2*x^4) + c^6*(1225*d^2*x^2 + 882*d*e*x^4 + 225*e^2*x^6)))/c^7 + 105*b*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcSin[c*x])/11025","A",1
608,1,159,183,0.1621377,"\int x \left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^2*(a + b*ArcSin[c*x]),x]","\frac{c x \left(48 a c^5 x \left(3 d^2+3 d e x^2+e^2 x^4\right)+b \sqrt{1-c^2 x^2} \left(4 c^4 \left(18 d^2+9 d e x^2+2 e^2 x^4\right)+2 c^2 e \left(27 d+5 e x^2\right)+15 e^2\right)\right)+3 b \sin ^{-1}(c x) \left(16 c^6 \left(3 d^2 x^2+3 d e x^4+e^2 x^6\right)-24 c^4 d^2-18 c^2 d e-5 e^2\right)}{288 c^6}","\frac{\left(d+e x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)}{6 e}+\frac{b x \sqrt{1-c^2 x^2} \left(d+e x^2\right)^2}{36 c}+\frac{5 b x \sqrt{1-c^2 x^2} \left(2 c^2 d+e\right) \left(d+e x^2\right)}{144 c^3}-\frac{b \left(2 c^2 d+e\right) \left(8 c^4 d^2+8 c^2 d e+5 e^2\right) \sin ^{-1}(c x)}{96 c^6 e}+\frac{b x \sqrt{1-c^2 x^2} \left(44 c^4 d^2+44 c^2 d e+15 e^2\right)}{288 c^5}",1,"(c*x*(48*a*c^5*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) + b*Sqrt[1 - c^2*x^2]*(15*e^2 + 2*c^2*e*(27*d + 5*e*x^2) + 4*c^4*(18*d^2 + 9*d*e*x^2 + 2*e^2*x^4))) + 3*b*(-24*c^4*d^2 - 18*c^2*d*e - 5*e^2 + 16*c^6*(3*d^2*x^2 + 3*d*e*x^4 + e^2*x^6))*ArcSin[c*x])/(288*c^6)","A",1
609,1,125,150,0.1692435,"\int \left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^2*(a + b*ArcSin[c*x]),x]","\frac{1}{225} \left(15 a x \left(15 d^2+10 d e x^2+3 e^2 x^4\right)+\frac{b \sqrt{1-c^2 x^2} \left(c^4 \left(225 d^2+50 d e x^2+9 e^2 x^4\right)+4 c^2 e \left(25 d+3 e x^2\right)+24 e^2\right)}{c^5}+15 b x \sin ^{-1}(c x) \left(15 d^2+10 d e x^2+3 e^2 x^4\right)\right)","d^2 x \left(a+b \sin ^{-1}(c x)\right)+\frac{2}{3} d e x^3 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{5} e^2 x^5 \left(a+b \sin ^{-1}(c x)\right)-\frac{2 b e \left(1-c^2 x^2\right)^{3/2} \left(5 c^2 d+3 e\right)}{45 c^5}+\frac{b e^2 \left(1-c^2 x^2\right)^{5/2}}{25 c^5}+\frac{b \sqrt{1-c^2 x^2} \left(15 c^4 d^2+10 c^2 d e+3 e^2\right)}{15 c^5}",1,"(15*a*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + (b*Sqrt[1 - c^2*x^2]*(24*e^2 + 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)))/c^5 + 15*b*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcSin[c*x])/225","A",1
610,1,184,229,0.3753462,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSin[c*x]))/x,x]","a d^2 \log (x)+a d e x^2+\frac{1}{4} a e^2 x^4+\frac{b d e \left(c x \sqrt{1-c^2 x^2}-\sin ^{-1}(c x)\right)}{2 c^2}+\frac{b e^2 \left(c x \sqrt{1-c^2 x^2} \left(2 c^2 x^2+3\right)-3 \sin ^{-1}(c x)\right)}{32 c^4}-\frac{1}{2} i b d^2 \left(\sin ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)\right)+b d^2 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+b d e x^2 \sin ^{-1}(c x)+\frac{1}{4} b e^2 x^4 \sin ^{-1}(c x)","d^2 \log (x) \left(a+b \sin ^{-1}(c x)\right)+d e x^2 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{4} e^2 x^4 \left(a+b \sin ^{-1}(c x)\right)-\frac{3 b e^2 \sin ^{-1}(c x)}{32 c^4}+\frac{b d e x \sqrt{1-c^2 x^2}}{2 c}-\frac{b d e \sin ^{-1}(c x)}{2 c^2}+\frac{b e^2 x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{3 b e^2 x \sqrt{1-c^2 x^2}}{32 c^3}-\frac{1}{2} i b d^2 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-\frac{1}{2} i b d^2 \sin ^{-1}(c x)^2+b d^2 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-b d^2 \log (x) \sin ^{-1}(c x)",1,"a*d*e*x^2 + (a*e^2*x^4)/4 + (b*e^2*(c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2) - 3*ArcSin[c*x]))/(32*c^4) + (b*d*e*(c*x*Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(2*c^2) + b*d*e*x^2*ArcSin[c*x] + (b*e^2*x^4*ArcSin[c*x])/4 + b*d^2*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + a*d^2*Log[x] - (I/2)*b*d^2*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c*x])])","A",1
611,1,129,126,0.1531367,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSin[c*x]))/x^2,x]","\frac{1}{9} \left(-\frac{9 a d^2}{x}+18 a d e x+3 a e^2 x^3-9 b c d^2 \log \left(\sqrt{1-c^2 x^2}+1\right)+\frac{b e \sqrt{1-c^2 x^2} \left(c^2 \left(18 d+e x^2\right)+2 e\right)}{c^3}+\frac{3 b \sin ^{-1}(c x) \left(-3 d^2+6 d e x^2+e^2 x^4\right)}{x}+9 b c d^2 \log (x)\right)","-\frac{d^2 \left(a+b \sin ^{-1}(c x)\right)}{x}+2 d e x \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} e^2 x^3 \left(a+b \sin ^{-1}(c x)\right)-b c d^2 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+\frac{b e \sqrt{1-c^2 x^2} \left(6 c^2 d+e\right)}{3 c^3}-\frac{b e^2 \left(1-c^2 x^2\right)^{3/2}}{9 c^3}",1,"((-9*a*d^2)/x + 18*a*d*e*x + 3*a*e^2*x^3 + (b*e*Sqrt[1 - c^2*x^2]*(2*e + c^2*(18*d + e*x^2)))/c^3 + (3*b*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)*ArcSin[c*x])/x + 9*b*c*d^2*Log[x] - 9*b*c*d^2*Log[1 + Sqrt[1 - c^2*x^2]])/9","A",1
612,1,159,185,0.3555447,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSin[c*x]))/x^3,x]","\frac{1}{4} \left(-\frac{2 a d^2}{x^2}+8 a d e \log (x)+2 a e^2 x^2+b \sin ^{-1}(c x) \left(-\frac{e^2}{c^2}+8 d e \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-\frac{2 d^2}{x^2}+2 e^2 x^2\right)-\frac{2 b c d^2 \sqrt{1-c^2 x^2}}{x}+\frac{b e^2 x \sqrt{1-c^2 x^2}}{c}-4 i b d e \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-4 i b d e \sin ^{-1}(c x)^2\right)","-\frac{d^2 \left(a+b \sin ^{-1}(c x)\right)}{2 x^2}+2 d e \log (x) \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{2} e^2 x^2 \left(a+b \sin ^{-1}(c x)\right)-\frac{b c d^2 \sqrt{1-c^2 x^2}}{2 x}+\frac{b e^2 x \sqrt{1-c^2 x^2}}{4 c}-\frac{b e^2 \sin ^{-1}(c x)}{4 c^2}-i b d e \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-i b d e \sin ^{-1}(c x)^2+2 b d e \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-2 b d e \log (x) \sin ^{-1}(c x)",1,"((-2*a*d^2)/x^2 + 2*a*e^2*x^2 - (2*b*c*d^2*Sqrt[1 - c^2*x^2])/x + (b*e^2*x*Sqrt[1 - c^2*x^2])/c - (4*I)*b*d*e*ArcSin[c*x]^2 + b*ArcSin[c*x]*(-(e^2/c^2) - (2*d^2)/x^2 + 2*e^2*x^2 + 8*d*e*Log[1 - E^((2*I)*ArcSin[c*x])]) + 8*a*d*e*Log[x] - (4*I)*b*d*e*PolyLog[2, E^((2*I)*ArcSin[c*x])])/4","A",1
613,1,140,126,0.1744211,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSin[c*x]))/x^4,x]","\frac{1}{6} \left(-\frac{2 a d^2}{x^3}-\frac{12 a d e}{x}+6 a e^2 x+6 b \sqrt{1-c^2 x^2} \left(\frac{e^2}{c}-\frac{c d^2}{6 x^2}\right)-b c d \left(c^2 d+12 e\right) \log \left(\sqrt{1-c^2 x^2}+1\right)+b c d \log (x) \left(c^2 d+12 e\right)-\frac{2 b \sin ^{-1}(c x) \left(d^2+6 d e x^2-3 e^2 x^4\right)}{x^3}\right)","-\frac{d^2 \left(a+b \sin ^{-1}(c x)\right)}{3 x^3}-\frac{2 d e \left(a+b \sin ^{-1}(c x)\right)}{x}+e^2 x \left(a+b \sin ^{-1}(c x)\right)-\frac{b c d^2 \sqrt{1-c^2 x^2}}{6 x^2}-\frac{1}{6} b c d \left(c^2 d+12 e\right) \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+\frac{b e^2 \sqrt{1-c^2 x^2}}{c}",1,"((-2*a*d^2)/x^3 - (12*a*d*e)/x + 6*a*e^2*x + 6*b*(e^2/c - (c*d^2)/(6*x^2))*Sqrt[1 - c^2*x^2] - (2*b*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)*ArcSin[c*x])/x^3 + b*c*d*(c^2*d + 12*e)*Log[x] - b*c*d*(c^2*d + 12*e)*Log[1 + Sqrt[1 - c^2*x^2]])/6","A",1
614,1,271,341,0.2862255,"\int x^4 \left(d+e x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^4*(d + e*x^2)^3*(a + b*ArcSin[c*x]),x]","\frac{3465 a x^5 \left(231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right)+\frac{b \sqrt{1-c^2 x^2} \left(c^{10} x^4 \left(160083 d^3+245025 d^2 e x^2+148225 d e^2 x^4+33075 e^3 x^6\right)+2 c^8 \left(106722 d^3 x^2+147015 d^2 e x^4+84700 d e^2 x^6+18375 e^3 x^8\right)+24 c^6 \left(17787 d^3+16335 d^2 e x^2+8470 d e^2 x^4+1750 e^3 x^6\right)+80 c^4 e \left(9801 d^2+3388 d e x^2+630 e^2 x^4\right)+4480 c^2 e^2 \left(121 d+15 e x^2\right)+134400 e^3\right)}{c^{11}}+3465 b x^5 \sin ^{-1}(c x) \left(231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right)}{4002075}","\frac{1}{5} d^3 x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{3}{7} d^2 e x^7 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} d e^2 x^9 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{11} e^3 x^{11} \left(a+b \sin ^{-1}(c x)\right)+\frac{b e^2 \left(1-c^2 x^2\right)^{9/2} \left(11 c^2 d+15 e\right)}{297 c^{11}}-\frac{b e^3 \left(1-c^2 x^2\right)^{11/2}}{121 c^{11}}-\frac{b e \left(1-c^2 x^2\right)^{7/2} \left(99 c^4 d^2+308 c^2 d e+210 e^2\right)}{1617 c^{11}}+\frac{b \left(1-c^2 x^2\right)^{5/2} \left(77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right)}{1925 c^{11}}-\frac{b \left(1-c^2 x^2\right)^{3/2} \left(462 c^6 d^3+1485 c^4 d^2 e+1540 c^2 d e^2+525 e^3\right)}{3465 c^{11}}+\frac{b \sqrt{1-c^2 x^2} \left(231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3\right)}{1155 c^{11}}",1,"(3465*a*x^5*(231*d^3 + 495*d^2*e*x^2 + 385*d*e^2*x^4 + 105*e^3*x^6) + (b*Sqrt[1 - c^2*x^2]*(134400*e^3 + 4480*c^2*e^2*(121*d + 15*e*x^2) + 80*c^4*e*(9801*d^2 + 3388*d*e*x^2 + 630*e^2*x^4) + 24*c^6*(17787*d^3 + 16335*d^2*e*x^2 + 8470*d*e^2*x^4 + 1750*e^3*x^6) + c^10*x^4*(160083*d^3 + 245025*d^2*e*x^2 + 148225*d*e^2*x^4 + 33075*e^3*x^6) + 2*c^8*(106722*d^3*x^2 + 147015*d^2*e*x^4 + 84700*d*e^2*x^6 + 18375*e^3*x^8)))/c^11 + 3465*b*x^5*(231*d^3 + 495*d^2*e*x^2 + 385*d*e^2*x^4 + 105*e^3*x^6)*ArcSin[c*x])/4002075","A",1
615,1,276,380,0.2675687,"\int x^3 \left(d+e x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^3*(a + b*ArcSin[c*x]),x]","\frac{c x \left(1920 a c^9 x^3 \left(10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right)+b \sqrt{1-c^2 x^2} \left(16 c^8 \left(300 d^3 x^2+400 d^2 e x^4+225 d e^2 x^6+48 e^3 x^8\right)+8 c^6 \left(900 d^3+1000 d^2 e x^2+525 d e^2 x^4+108 e^3 x^6\right)+6 c^4 e \left(2000 d^2+875 d e x^2+168 e^2 x^4\right)+315 c^2 e^2 \left(25 d+4 e x^2\right)+1890 e^3\right)\right)+15 b \sin ^{-1}(c x) \left(128 c^{10} x^4 \left(10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right)-480 c^6 d^3-800 c^4 d^2 e-525 c^2 d e^2-126 e^3\right)}{76800 c^{10}}","\frac{\left(d+e x^2\right)^5 \left(a+b \sin ^{-1}(c x)\right)}{10 e^2}-\frac{d \left(d+e x^2\right)^4 \left(a+b \sin ^{-1}(c x)\right)}{8 e^2}+\frac{b x \sqrt{1-c^2 x^2} \left(d+e x^2\right)^4}{100 c e}+\frac{b x \sqrt{1-c^2 x^2} \left(11 c^2 d+18 e\right) \left(d+e x^2\right)^3}{1600 c^3 e}+\frac{b x \sqrt{1-c^2 x^2} \left(26 c^4 d^2+201 c^2 d e+126 e^2\right) \left(d+e x^2\right)^2}{9600 c^5 e}+\frac{b \left(128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right) \sin ^{-1}(c x)}{5120 c^{10} e^2}-\frac{b x \sqrt{1-c^2 x^2} \left(136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right) \left(d+e x^2\right)}{38400 c^7 e}-\frac{b x \sqrt{1-c^2 x^2} \left(1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right)}{76800 c^9 e}",1,"(c*x*(1920*a*c^9*x^3*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6) + b*Sqrt[1 - c^2*x^2]*(1890*e^3 + 315*c^2*e^2*(25*d + 4*e*x^2) + 6*c^4*e*(2000*d^2 + 875*d*e*x^2 + 168*e^2*x^4) + 8*c^6*(900*d^3 + 1000*d^2*e*x^2 + 525*d*e^2*x^4 + 108*e^3*x^6) + 16*c^8*(300*d^3*x^2 + 400*d^2*e*x^4 + 225*d*e^2*x^6 + 48*e^3*x^8))) + 15*b*(-480*c^6*d^3 - 800*c^4*d^2*e - 525*c^2*d*e^2 - 126*e^3 + 128*c^10*x^4*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6))*ArcSin[c*x])/(76800*c^10)","A",1
616,1,231,287,0.2612614,"\int x^2 \left(d+e x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)^3*(a + b*ArcSin[c*x]),x]","\frac{315 a x^3 \left(105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right)+\frac{b \sqrt{1-c^2 x^2} \left(c^8 \left(11025 d^3 x^2+11907 d^2 e x^4+6075 d e^2 x^6+1225 e^3 x^8\right)+2 c^6 \left(11025 d^3+7938 d^2 e x^2+3645 d e^2 x^4+700 e^3 x^6\right)+24 c^4 e \left(1323 d^2+405 d e x^2+70 e^2 x^4\right)+80 c^2 e^2 \left(243 d+28 e x^2\right)+4480 e^3\right)}{c^9}+315 b x^3 \sin ^{-1}(c x) \left(105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right)}{99225}","\frac{1}{3} d^3 x^3 \left(a+b \sin ^{-1}(c x)\right)+\frac{3}{5} d^2 e x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{3}{7} d e^2 x^7 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{9} e^3 x^9 \left(a+b \sin ^{-1}(c x)\right)-\frac{b e^2 \left(1-c^2 x^2\right)^{7/2} \left(27 c^2 d+28 e\right)}{441 c^9}+\frac{b e^3 \left(1-c^2 x^2\right)^{9/2}}{81 c^9}+\frac{b e \left(1-c^2 x^2\right)^{5/2} \left(63 c^4 d^2+135 c^2 d e+70 e^2\right)}{525 c^9}-\frac{b \left(1-c^2 x^2\right)^{3/2} \left(105 c^6 d^3+378 c^4 d^2 e+405 c^2 d e^2+140 e^3\right)}{945 c^9}+\frac{b \sqrt{1-c^2 x^2} \left(105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right)}{315 c^9}",1,"(315*a*x^3*(105*d^3 + 189*d^2*e*x^2 + 135*d*e^2*x^4 + 35*e^3*x^6) + (b*Sqrt[1 - c^2*x^2]*(4480*e^3 + 80*c^2*e^2*(243*d + 28*e*x^2) + 24*c^4*e*(1323*d^2 + 405*d*e*x^2 + 70*e^2*x^4) + 2*c^6*(11025*d^3 + 7938*d^2*e*x^2 + 3645*d*e^2*x^4 + 700*e^3*x^6) + c^8*(11025*d^3*x^2 + 11907*d^2*e*x^4 + 6075*d*e^2*x^6 + 1225*e^3*x^8)))/c^9 + 315*b*x^3*(105*d^3 + 189*d^2*e*x^2 + 135*d*e^2*x^4 + 35*e^3*x^6)*ArcSin[c*x])/99225","A",1
617,1,232,258,0.2238203,"\int x \left(d+e x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^3*(a + b*ArcSin[c*x]),x]","\frac{c x \left(384 a c^7 x \left(4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right)+b \sqrt{1-c^2 x^2} \left(16 c^6 \left(48 d^3+36 d^2 e x^2+16 d e^2 x^4+3 e^3 x^6\right)+8 c^4 e \left(108 d^2+40 d e x^2+7 e^2 x^4\right)+10 c^2 e^2 \left(48 d+7 e x^2\right)+105 e^3\right)\right)+3 b \sin ^{-1}(c x) \left(128 c^8 \left(4 d^3 x^2+6 d^2 e x^4+4 d e^2 x^6+e^3 x^8\right)-256 c^6 d^3-288 c^4 d^2 e-160 c^2 d e^2-35 e^3\right)}{3072 c^8}","\frac{\left(d+e x^2\right)^4 \left(a+b \sin ^{-1}(c x)\right)}{8 e}+\frac{b x \sqrt{1-c^2 x^2} \left(d+e x^2\right)^3}{64 c}+\frac{7 b x \sqrt{1-c^2 x^2} \left(2 c^2 d+e\right) \left(d+e x^2\right)^2}{384 c^3}+\frac{5 b x \sqrt{1-c^2 x^2} \left(2 c^2 d+e\right) \left(40 c^4 d^2+40 c^2 d e+21 e^2\right)}{3072 c^7}+\frac{b x \sqrt{1-c^2 x^2} \left(104 c^4 d^2+104 c^2 d e+35 e^2\right) \left(d+e x^2\right)}{1536 c^5}-\frac{b \left(128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right) \sin ^{-1}(c x)}{1024 c^8 e}",1,"(c*x*(384*a*c^7*x*(4*d^3 + 6*d^2*e*x^2 + 4*d*e^2*x^4 + e^3*x^6) + b*Sqrt[1 - c^2*x^2]*(105*e^3 + 10*c^2*e^2*(48*d + 7*e*x^2) + 8*c^4*e*(108*d^2 + 40*d*e*x^2 + 7*e^2*x^4) + 16*c^6*(48*d^3 + 36*d^2*e*x^2 + 16*d*e^2*x^4 + 3*e^3*x^6))) + 3*b*(-256*c^6*d^3 - 288*c^4*d^2*e - 160*c^2*d*e^2 - 35*e^3 + 128*c^8*(4*d^3*x^2 + 6*d^2*e*x^4 + 4*d*e^2*x^6 + e^3*x^8))*ArcSin[c*x])/(3072*c^8)","A",1
618,1,187,225,0.2364811,"\int \left(d+e x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^3*(a + b*ArcSin[c*x]),x]","\frac{105 a x \left(35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right)+\frac{b \sqrt{1-c^2 x^2} \left(c^6 \left(3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right)+2 c^4 e \left(1225 d^2+294 d e x^2+45 e^2 x^4\right)+24 c^2 e^2 \left(49 d+5 e x^2\right)+240 e^3\right)}{c^7}+105 b x \sin ^{-1}(c x) \left(35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right)}{3675}","d^3 x \left(a+b \sin ^{-1}(c x)\right)+d^2 e x^3 \left(a+b \sin ^{-1}(c x)\right)+\frac{3}{5} d e^2 x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{7} e^3 x^7 \left(a+b \sin ^{-1}(c x)\right)+\frac{3 b e^2 \left(1-c^2 x^2\right)^{5/2} \left(7 c^2 d+5 e\right)}{175 c^7}-\frac{b e^3 \left(1-c^2 x^2\right)^{7/2}}{49 c^7}-\frac{b e \left(1-c^2 x^2\right)^{3/2} \left(35 c^4 d^2+42 c^2 d e+15 e^2\right)}{105 c^7}+\frac{b \sqrt{1-c^2 x^2} \left(35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right)}{35 c^7}",1,"(105*a*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6) + (b*Sqrt[1 - c^2*x^2]*(240*e^3 + 24*c^2*e^2*(49*d + 5*e*x^2) + 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e^2*x^4 + 75*e^3*x^6)))/c^7 + 105*b*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6)*ArcSin[c*x])/3675","A",1
619,1,278,357,0.3792287,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcSin[c*x]))/x,x]","a d^3 \log (x)+\frac{3}{2} a d^2 e x^2+\frac{3}{4} a d e^2 x^4+\frac{1}{6} a e^3 x^6+\frac{3 b d^2 e \left(c x \sqrt{1-c^2 x^2}-\sin ^{-1}(c x)\right)}{4 c^2}+\frac{3 b d e^2 \left(c x \sqrt{1-c^2 x^2} \left(2 c^2 x^2+3\right)-3 \sin ^{-1}(c x)\right)}{32 c^4}+\frac{b e^3 \left(c x \sqrt{1-c^2 x^2} \left(8 c^4 x^4+10 c^2 x^2+15\right)-15 \sin ^{-1}(c x)\right)}{288 c^6}-\frac{1}{2} i b d^3 \left(\sin ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)\right)+b d^3 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)+\frac{3}{2} b d^2 e x^2 \sin ^{-1}(c x)+\frac{3}{4} b d e^2 x^4 \sin ^{-1}(c x)+\frac{1}{6} b e^3 x^6 \sin ^{-1}(c x)","d^3 \log (x) \left(a+b \sin ^{-1}(c x)\right)+\frac{3}{2} d^2 e x^2 \left(a+b \sin ^{-1}(c x)\right)+\frac{3}{4} d e^2 x^4 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{6} e^3 x^6 \left(a+b \sin ^{-1}(c x)\right)-\frac{5 b e^3 \sin ^{-1}(c x)}{96 c^6}-\frac{9 b d e^2 \sin ^{-1}(c x)}{32 c^4}+\frac{3 b d^2 e x \sqrt{1-c^2 x^2}}{4 c}-\frac{3 b d^2 e \sin ^{-1}(c x)}{4 c^2}+\frac{3 b d e^2 x^3 \sqrt{1-c^2 x^2}}{16 c}+\frac{b e^3 x^5 \sqrt{1-c^2 x^2}}{36 c}+\frac{5 b e^3 x \sqrt{1-c^2 x^2}}{96 c^5}+\frac{9 b d e^2 x \sqrt{1-c^2 x^2}}{32 c^3}+\frac{5 b e^3 x^3 \sqrt{1-c^2 x^2}}{144 c^3}-\frac{1}{2} i b d^3 \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-\frac{1}{2} i b d^3 \sin ^{-1}(c x)^2+b d^3 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-b d^3 \log (x) \sin ^{-1}(c x)",1,"(3*a*d^2*e*x^2)/2 + (3*a*d*e^2*x^4)/4 + (a*e^3*x^6)/6 + (b*e^3*(c*x*Sqrt[1 - c^2*x^2]*(15 + 10*c^2*x^2 + 8*c^4*x^4) - 15*ArcSin[c*x]))/(288*c^6) + (3*b*d*e^2*(c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2) - 3*ArcSin[c*x]))/(32*c^4) + (3*b*d^2*e*(c*x*Sqrt[1 - c^2*x^2] - ArcSin[c*x]))/(4*c^2) + (3*b*d^2*e*x^2*ArcSin[c*x])/2 + (3*b*d*e^2*x^4*ArcSin[c*x])/4 + (b*e^3*x^6*ArcSin[c*x])/6 + b*d^3*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + a*d^3*Log[x] - (I/2)*b*d^3*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c*x])])","A",1
620,1,183,190,0.2162866,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcSin[c*x]))/x^2,x]","-\frac{a d^3}{x}+3 a d^2 e x+a d e^2 x^3+\frac{1}{5} a e^3 x^5-b c d^3 \log \left(\sqrt{1-c^2 x^2}+1\right)+\frac{b e \sqrt{1-c^2 x^2} \left(c^4 \left(225 d^2+25 d e x^2+3 e^2 x^4\right)+2 c^2 e \left(25 d+2 e x^2\right)+8 e^2\right)}{75 c^5}+b c d^3 \log (x)+\frac{b \sin ^{-1}(c x) \left(-5 d^3+15 d^2 e x^2+5 d e^2 x^4+e^3 x^6\right)}{5 x}","-\frac{d^3 \left(a+b \sin ^{-1}(c x)\right)}{x}+3 d^2 e x \left(a+b \sin ^{-1}(c x)\right)+d e^2 x^3 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{5} e^3 x^5 \left(a+b \sin ^{-1}(c x)\right)-b c d^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-\frac{b e^2 \left(1-c^2 x^2\right)^{3/2} \left(5 c^2 d+2 e\right)}{15 c^5}+\frac{b e^3 \left(1-c^2 x^2\right)^{5/2}}{25 c^5}+\frac{b e \sqrt{1-c^2 x^2} \left(15 c^4 d^2+5 c^2 d e+e^2\right)}{5 c^5}",1,"-((a*d^3)/x) + 3*a*d^2*e*x + a*d*e^2*x^3 + (a*e^3*x^5)/5 + (b*e*Sqrt[1 - c^2*x^2]*(8*e^2 + 2*c^2*e*(25*d + 2*e*x^2) + c^4*(225*d^2 + 25*d*e*x^2 + 3*e^2*x^4)))/(75*c^5) + (b*(-5*d^3 + 15*d^2*e*x^2 + 5*d*e^2*x^4 + e^3*x^6)*ArcSin[c*x])/(5*x) + b*c*d^3*Log[x] - b*c*d^3*Log[1 + Sqrt[1 - c^2*x^2]]","A",1
621,1,220,262,0.4653489,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcSin[c*x]))/x^3,x]","\frac{1}{32} \left(-\frac{16 a d^3}{x^2}+96 a d^2 e \log (x)+48 a d e^2 x^2+8 a e^3 x^4-\frac{16 b d^3 \left(c x \sqrt{1-c^2 x^2}+\sin ^{-1}(c x)\right)}{x^2}+\frac{24 b d e^2 \left(c x \sqrt{1-c^2 x^2}+\left(2 c^2 x^2-1\right) \sin ^{-1}(c x)\right)}{c^2}+\frac{b e^3 \left(\left(8 c^4 x^4-3\right) \sin ^{-1}(c x)+c x \sqrt{1-c^2 x^2} \left(2 c^2 x^2+3\right)\right)}{c^4}+96 b d^2 e \left(\sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-\frac{1}{2} i \left(\sin ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)\right)\right)\right)","-\frac{d^3 \left(a+b \sin ^{-1}(c x)\right)}{2 x^2}+3 d^2 e \log (x) \left(a+b \sin ^{-1}(c x)\right)+\frac{3}{2} d e^2 x^2 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{4} e^3 x^4 \left(a+b \sin ^{-1}(c x)\right)-\frac{b c d^3 \sqrt{1-c^2 x^2}}{2 x}+\frac{b e^3 x^3 \sqrt{1-c^2 x^2}}{16 c}-\frac{3 b e^2 \left(8 c^2 d+e\right) \sin ^{-1}(c x)}{32 c^4}+\frac{3 b e^2 x \sqrt{1-c^2 x^2} \left(8 c^2 d+e\right)}{32 c^3}-\frac{3}{2} i b d^2 e \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-\frac{3}{2} i b d^2 e \sin ^{-1}(c x)^2+3 b d^2 e \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)-3 b d^2 e \log (x) \sin ^{-1}(c x)",1,"((-16*a*d^3)/x^2 + 48*a*d*e^2*x^2 + 8*a*e^3*x^4 - (16*b*d^3*(c*x*Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/x^2 + (24*b*d*e^2*(c*x*Sqrt[1 - c^2*x^2] + (-1 + 2*c^2*x^2)*ArcSin[c*x]))/c^2 + (b*e^3*(c*x*Sqrt[1 - c^2*x^2]*(3 + 2*c^2*x^2) + (-3 + 8*c^4*x^4)*ArcSin[c*x]))/c^4 + 96*a*d^2*e*Log[x] + 96*b*d^2*e*(ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - (I/2)*(ArcSin[c*x]^2 + PolyLog[2, E^((2*I)*ArcSin[c*x])])))/32","A",1
622,1,194,186,0.2985655,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcSin[c*x]))/x^4,x]","\frac{1}{6} \left(-\frac{2 a d^3}{x^3}-\frac{18 a d^2 e}{x}+18 a d e^2 x+2 a e^3 x^3-b c d^2 \left(c^2 d+18 e\right) \log \left(\sqrt{1-c^2 x^2}+1\right)+b c d^2 \log (x) \left(c^2 d+18 e\right)+\frac{b \sqrt{1-c^2 x^2} \left(-3 c^4 d^3+2 c^2 e^2 x^2 \left(27 d+e x^2\right)+4 e^3 x^2\right)}{3 c^3 x^2}+\frac{2 b \sin ^{-1}(c x) \left(-d^3-9 d^2 e x^2+9 d e^2 x^4+e^3 x^6\right)}{x^3}\right)","-\frac{d^3 \left(a+b \sin ^{-1}(c x)\right)}{3 x^3}-\frac{3 d^2 e \left(a+b \sin ^{-1}(c x)\right)}{x}+3 d e^2 x \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{3} e^3 x^3 \left(a+b \sin ^{-1}(c x)\right)-\frac{b c d^3 \sqrt{1-c^2 x^2}}{6 x^2}-\frac{1}{6} b c d^2 \left(c^2 d+18 e\right) \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+\frac{b e^2 \sqrt{1-c^2 x^2} \left(9 c^2 d+e\right)}{3 c^3}-\frac{b e^3 \left(1-c^2 x^2\right)^{3/2}}{9 c^3}",1,"((-2*a*d^3)/x^3 - (18*a*d^2*e)/x + 18*a*d*e^2*x + 2*a*e^3*x^3 + (b*Sqrt[1 - c^2*x^2]*(-3*c^4*d^3 + 4*e^3*x^2 + 2*c^2*e^2*x^2*(27*d + e*x^2)))/(3*c^3*x^2) + (2*b*(-d^3 - 9*d^2*e*x^2 + 9*d*e^2*x^4 + e^3*x^6)*ArcSin[c*x])/x^3 + b*c*d^2*(c^2*d + 18*e)*Log[x] - b*c*d^2*(c^2*d + 18*e)*Log[1 + Sqrt[1 - c^2*x^2]])/6","A",1
623,1,260,317,0.3381433,"\int \left(d+e x^2\right)^4 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^4*(a + b*ArcSin[c*x]),x]","\frac{315 a x \left(315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right)+\frac{b \sqrt{1-c^2 x^2} \left(c^8 \left(99225 d^4+44100 d^3 e x^2+23814 d^2 e^2 x^4+8100 d e^3 x^6+1225 e^4 x^8\right)+8 c^6 e \left(11025 d^3+3969 d^2 e x^2+1215 d e^2 x^4+175 e^3 x^6\right)+48 c^4 e^2 \left(1323 d^2+270 d e x^2+35 e^2 x^4\right)+320 c^2 e^3 \left(81 d+7 e x^2\right)+4480 e^4\right)}{c^9}+315 b x \sin ^{-1}(c x) \left(315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right)}{99225}","d^4 x \left(a+b \sin ^{-1}(c x)\right)+\frac{4}{3} d^3 e x^3 \left(a+b \sin ^{-1}(c x)\right)+\frac{6}{5} d^2 e^2 x^5 \left(a+b \sin ^{-1}(c x)\right)+\frac{4}{7} d e^3 x^7 \left(a+b \sin ^{-1}(c x)\right)+\frac{1}{9} e^4 x^9 \left(a+b \sin ^{-1}(c x)\right)-\frac{4 b e^3 \left(1-c^2 x^2\right)^{7/2} \left(9 c^2 d+7 e\right)}{441 c^9}+\frac{b e^4 \left(1-c^2 x^2\right)^{9/2}}{81 c^9}+\frac{2 b e^2 \left(1-c^2 x^2\right)^{5/2} \left(63 c^4 d^2+90 c^2 d e+35 e^2\right)}{525 c^9}-\frac{4 b e \left(1-c^2 x^2\right)^{3/2} \left(105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right)}{945 c^9}+\frac{b \sqrt{1-c^2 x^2} \left(315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4\right)}{315 c^9}",1,"(315*a*x*(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 35*e^4*x^8) + (b*Sqrt[1 - c^2*x^2]*(4480*e^4 + 320*c^2*e^3*(81*d + 7*e*x^2) + 48*c^4*e^2*(1323*d^2 + 270*d*e*x^2 + 35*e^2*x^4) + 8*c^6*e*(11025*d^3 + 3969*d^2*e*x^2 + 1215*d*e^2*x^4 + 175*e^3*x^6) + c^8*(99225*d^4 + 44100*d^3*e*x^2 + 23814*d^2*e^2*x^4 + 8100*d*e^3*x^6 + 1225*e^4*x^8)))/c^9 + 315*b*x*(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 35*e^4*x^8)*ArcSin[c*x])/99225","A",1
624,1,515,653,0.9741153,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x]))/(d + e*x^2),x]","\frac{a d^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{5/2}}-\frac{a d x}{e^2}+\frac{a x^3}{3 e}+\frac{b \left(d^{3/2} \left(-2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)-2 \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)-\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right)+\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)+d^{3/2} \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)-\frac{4 d \sqrt{e} \left(\sqrt{1-c^2 x^2}+c x \sin ^{-1}(c x)\right)}{c}+\frac{4 e^{3/2} \left(3 c^3 x^3 \sin ^{-1}(c x)+\sqrt{1-c^2 x^2} \left(c^2 x^2+2\right)\right)}{9 c^3}\right)}{4 e^{5/2}}","\frac{(-d)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^{5/2}}-\frac{(-d)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^{5/2}}+\frac{(-d)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^{5/2}}-\frac{(-d)^{3/2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^{5/2}}+\frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{3 e}-\frac{a d x}{e^2}+\frac{i b (-d)^{3/2} \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 e^{5/2}}-\frac{i b (-d)^{3/2} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 e^{5/2}}+\frac{i b (-d)^{3/2} \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 e^{5/2}}-\frac{i b (-d)^{3/2} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 e^{5/2}}-\frac{b d \sqrt{1-c^2 x^2}}{c e^2}-\frac{b \left(1-c^2 x^2\right)^{3/2}}{9 c^3 e}+\frac{b \sqrt{1-c^2 x^2}}{3 c^3 e}-\frac{b d x \sin ^{-1}(c x)}{e^2}",1,"-((a*d*x)/e^2) + (a*x^3)/(3*e) + (a*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(5/2) + (b*((-4*d*Sqrt[e]*(Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x]))/c + (4*e^(3/2)*(Sqrt[1 - c^2*x^2]*(2 + c^2*x^2) + 3*c^3*x^3*ArcSin[c*x]))/(9*c^3) + d^(3/2)*(-(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))) - 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] - 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]) + d^(3/2)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])))/(4*e^(5/2))","A",0
625,1,454,559,0.3398034,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x^3*(a + b*ArcSin[c*x]))/(d + e*x^2),x]","\frac{-2 a c^2 d \log \left(d+e x^2\right)+2 a c^2 e x^2+b \left(i c^2 d \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right)+\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)+i c^2 d \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)+e \left(c x \sqrt{1-c^2 x^2}+\left(2 c^2 x^2-1\right) \sin ^{-1}(c x)\right)\right)}{4 c^2 e^2}","-\frac{d \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^2}-\frac{d \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^2}-\frac{d \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^2}-\frac{d \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^2}+\frac{i d \left(a+b \sin ^{-1}(c x)\right)^2}{2 b e^2}+\frac{x^2 \left(a+b \sin ^{-1}(c x)\right)}{2 e}+\frac{i b d \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 e^2}+\frac{i b d \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 e^2}+\frac{i b d \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 e^2}+\frac{i b d \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 e^2}+\frac{b x \sqrt{1-c^2 x^2}}{4 c e}-\frac{b \sin ^{-1}(c x)}{4 c^2 e}",1,"(2*a*c^2*e*x^2 - 2*a*c^2*d*Log[d + e*x^2] + b*(e*(c*x*Sqrt[1 - c^2*x^2] + (-1 + 2*c^2*x^2)*ArcSin[c*x]) + I*c^2*d*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]) + I*c^2*d*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])))/(4*c^2*e^2)","A",0
626,1,456,579,0.3280285,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x]))/(d + e*x^2),x]","\frac{-4 a c \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+4 a c \sqrt{e} x+b \left(c \sqrt{d} \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right)+\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)-c \sqrt{d} \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)+4 \sqrt{e} \left(\sqrt{1-c^2 x^2}+c x \sin ^{-1}(c x)\right)\right)}{4 c e^{3/2}}","\frac{\sqrt{-d} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^{3/2}}-\frac{\sqrt{-d} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^{3/2}}+\frac{\sqrt{-d} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^{3/2}}-\frac{\sqrt{-d} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^{3/2}}+\frac{a x}{e}+\frac{i b \sqrt{-d} \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 e^{3/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 e^{3/2}}+\frac{i b \sqrt{-d} \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 e^{3/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 e^{3/2}}+\frac{b \sqrt{1-c^2 x^2}}{c e}+\frac{b x \sin ^{-1}(c x)}{e}",1,"(4*a*c*Sqrt[e]*x - 4*a*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*(4*Sqrt[e]*(Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x]) + c*Sqrt[d]*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]) - c*Sqrt[d]*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])))/(4*c*e^(3/2))","A",0
627,1,399,491,0.0935428,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x*(a + b*ArcSin[c*x]))/(d + e*x^2),x]","-\frac{i \left(i a \log \left(d+e x^2\right)+b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)+b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)+b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+i b \sin ^{-1}(c x) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right)+i b \sin ^{-1}(c x) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right)+i b \sin ^{-1}(c x) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)+i b \sin ^{-1}(c x) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)+b \sin ^{-1}(c x)^2\right)}{2 e}","\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e}-\frac{i \left(a+b \sin ^{-1}(c x)\right)^2}{2 b e}-\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 e}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 e}-\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 e}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 e}",1,"((-1/2*I)*(b*ArcSin[c*x]^2 + I*b*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + I*b*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + I*b*ArcSin[c*x]*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])] + I*b*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])] + I*a*Log[d + e*x^2] + b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))] + b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/e","A",0
628,1,490,541,0.4366136,"\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{2 a \sqrt{-d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)-i b \sqrt{d} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)+i b \sqrt{d} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-i c \sqrt{-d}}\right)+i b \sqrt{d} \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)-i b \sqrt{d} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)-b \sqrt{d} \sin ^{-1}(c x) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)+b \sqrt{d} \sin ^{-1}(c x) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-i c \sqrt{-d}}\right)+b \sqrt{d} \sin ^{-1}(c x) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)-b \sqrt{d} \sin ^{-1}(c x) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 \sqrt{-d^2} \sqrt{e}}","\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}",1,"(2*a*Sqrt[-d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - b*Sqrt[d]*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])] + b*Sqrt[d]*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/((-I)*c*Sqrt[-d] + Sqrt[c^2*d + e])] + b*Sqrt[d]*ArcSin[c*x]*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])] - b*Sqrt[d]*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])] - I*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])] + I*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/((-I)*c*Sqrt[-d] + Sqrt[c^2*d + e])] + I*b*Sqrt[d]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))] - I*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*Sqrt[-d^2]*Sqrt[e])","A",0
629,1,441,518,0.7590673,"\int \frac{a+b \sin ^{-1}(c x)}{x \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcSin[c*x])/(x*(d + e*x^2)),x]","-\frac{a \log \left(d+e x^2\right)}{2 d}+\frac{a \log (x)}{d}+\frac{b \left(i \text{Li}_2\left(\frac{\left(2 d c^2+e-2 \sqrt{c^2 d \left(d c^2+e\right)}\right) e^{2 i \sin ^{-1}(c x)}}{e}\right)+i \text{Li}_2\left(\frac{\left(2 d c^2+e+2 \sqrt{c^2 d \left(d c^2+e\right)}\right) e^{2 i \sin ^{-1}(c x)}}{e}\right)-4 i \sin ^{-1}\left(\sqrt{-\frac{c^2 d}{e}}\right) \tan ^{-1}\left(\frac{c x \left(c^2 d+e\right)}{\sqrt{1-c^2 x^2} \sqrt{c^2 d \left(c^2 d+e\right)}}\right)-2 \sin ^{-1}\left(\sqrt{-\frac{c^2 d}{e}}\right) \log \left(1-\frac{\left(-2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{2 i \sin ^{-1}(c x)}}{e}\right)-2 \sin ^{-1}(c x) \log \left(1-\frac{\left(-2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{2 i \sin ^{-1}(c x)}}{e}\right)+2 \sin ^{-1}\left(\sqrt{-\frac{c^2 d}{e}}\right) \log \left(1-\frac{\left(2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{2 i \sin ^{-1}(c x)}}{e}\right)-2 \sin ^{-1}(c x) \log \left(1-\frac{\left(2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{2 i \sin ^{-1}(c x)}}{e}\right)-2 i \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)+4 \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)\right)}{4 d}","-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d}+\frac{\log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d}+\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 d}+\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 d}+\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 d}+\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 d}-\frac{i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d}",1,"(a*Log[x])/d - (a*Log[d + e*x^2])/(2*d) + (b*((-4*I)*ArcSin[Sqrt[-((c^2*d)/e)]]*ArcTan[(c*(c^2*d + e)*x)/(Sqrt[c^2*d*(c^2*d + e)]*Sqrt[1 - c^2*x^2])] + 4*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] - 2*ArcSin[Sqrt[-((c^2*d)/e)]]*Log[1 - ((2*c^2*d + e - 2*Sqrt[c^2*d*(c^2*d + e)])*E^((2*I)*ArcSin[c*x]))/e] - 2*ArcSin[c*x]*Log[1 - ((2*c^2*d + e - 2*Sqrt[c^2*d*(c^2*d + e)])*E^((2*I)*ArcSin[c*x]))/e] + 2*ArcSin[Sqrt[-((c^2*d)/e)]]*Log[1 - ((2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])*E^((2*I)*ArcSin[c*x]))/e] - 2*ArcSin[c*x]*Log[1 - ((2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])*E^((2*I)*ArcSin[c*x]))/e] - (2*I)*PolyLog[2, E^((2*I)*ArcSin[c*x])] + I*PolyLog[2, ((2*c^2*d + e - 2*Sqrt[c^2*d*(c^2*d + e)])*E^((2*I)*ArcSin[c*x]))/e] + I*PolyLog[2, ((2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])*E^((2*I)*ArcSin[c*x]))/e]))/(4*d)","A",0
630,1,455,579,0.3543846,"\int \frac{a+b \sin ^{-1}(c x)}{x^2 \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^2*(d + e*x^2)),x]","\frac{-4 a \sqrt{e} x \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)-4 a \sqrt{d}+b \sqrt{e} x \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right)+\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)-b \sqrt{e} x \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)-4 b \sqrt{d} \left(c x \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+\sin ^{-1}(c x)\right)}{4 d^{3/2} x}","\frac{\sqrt{e} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 (-d)^{3/2}}-\frac{a+b \sin ^{-1}(c x)}{d x}+\frac{i b \sqrt{e} \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}+\frac{i b \sqrt{e} \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}-\frac{b c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{d}",1,"(-4*a*Sqrt[d] - 4*a*Sqrt[e]*x*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - 4*b*Sqrt[d]*(ArcSin[c*x] + c*x*ArcTanh[Sqrt[1 - c^2*x^2]]) + b*Sqrt[e]*x*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]) - b*Sqrt[e]*x*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/(4*d^(3/2)*x)","A",0
631,1,483,573,2.3645221,"\int \frac{a+b \sin ^{-1}(c x)}{x^3 \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^3*(d + e*x^2)),x]","\frac{4 a e \log \left(d+e x^2\right)-\frac{4 a d}{x^2}-8 a e \log (x)+2 b \left(-i e \text{Li}_2\left(\frac{\left(2 d c^2+e-2 \sqrt{c^2 d \left(d c^2+e\right)}\right) e^{2 i \sin ^{-1}(c x)}}{e}\right)-i e \text{Li}_2\left(\frac{\left(2 d c^2+e+2 \sqrt{c^2 d \left(d c^2+e\right)}\right) e^{2 i \sin ^{-1}(c x)}}{e}\right)+4 i e \sin ^{-1}\left(\sqrt{-\frac{c^2 d}{e}}\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d \left(c^2 d+e\right)}}{c d \sqrt{1-c^2 x^2}}\right)+2 e \sin ^{-1}\left(\sqrt{-\frac{c^2 d}{e}}\right) \log \left(1-\frac{\left(-2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{2 i \sin ^{-1}(c x)}}{e}\right)+2 e \sin ^{-1}(c x) \log \left(1-\frac{\left(-2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{2 i \sin ^{-1}(c x)}}{e}\right)-2 e \sin ^{-1}\left(\sqrt{-\frac{c^2 d}{e}}\right) \log \left(1-\frac{\left(2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{2 i \sin ^{-1}(c x)}}{e}\right)+2 e \sin ^{-1}(c x) \log \left(1-\frac{\left(2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{2 i \sin ^{-1}(c x)}}{e}\right)-\frac{2 c d \sqrt{1-c^2 x^2}}{x}-\frac{2 d \sin ^{-1}(c x)}{x^2}+2 i e \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)-4 e \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)\right)}{8 d^2}","\frac{e \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^2}+\frac{e \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^2}+\frac{e \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^2}+\frac{e \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^2}-\frac{e \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}-\frac{a+b \sin ^{-1}(c x)}{2 d x^2}-\frac{i b e \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 d^2}-\frac{i b e \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 d^2}-\frac{i b e \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 d^2}-\frac{i b e \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 d^2}-\frac{b c \sqrt{1-c^2 x^2}}{2 d x}+\frac{i b e \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d^2}",1,"((-4*a*d)/x^2 - 8*a*e*Log[x] + 4*a*e*Log[d + e*x^2] + 2*b*((-2*c*d*Sqrt[1 - c^2*x^2])/x - (2*d*ArcSin[c*x])/x^2 + (4*I)*e*ArcSin[Sqrt[-((c^2*d)/e)]]*ArcTan[(Sqrt[c^2*d*(c^2*d + e)]*x)/(c*d*Sqrt[1 - c^2*x^2])] - 4*e*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 2*e*ArcSin[Sqrt[-((c^2*d)/e)]]*Log[1 - ((2*c^2*d + e - 2*Sqrt[c^2*d*(c^2*d + e)])*E^((2*I)*ArcSin[c*x]))/e] + 2*e*ArcSin[c*x]*Log[1 - ((2*c^2*d + e - 2*Sqrt[c^2*d*(c^2*d + e)])*E^((2*I)*ArcSin[c*x]))/e] - 2*e*ArcSin[Sqrt[-((c^2*d)/e)]]*Log[1 - ((2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])*E^((2*I)*ArcSin[c*x]))/e] + 2*e*ArcSin[c*x]*Log[1 - ((2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])*E^((2*I)*ArcSin[c*x]))/e] + (2*I)*e*PolyLog[2, E^((2*I)*ArcSin[c*x])] - I*e*PolyLog[2, ((2*c^2*d + e - 2*Sqrt[c^2*d*(c^2*d + e)])*E^((2*I)*ArcSin[c*x]))/e] - I*e*PolyLog[2, ((2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])*E^((2*I)*ArcSin[c*x]))/e]))/(8*d^2)","A",0
632,1,531,649,0.438626,"\int \frac{a+b \sin ^{-1}(c x)}{x^4 \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^4*(d + e*x^2)),x]","\frac{a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{5/2}}+\frac{a e}{d^2 x}-\frac{a}{3 d x^3}+b \left(-\frac{e^{3/2} \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right)+\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)}{4 d^{5/2}}+\frac{e^{3/2} \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)}{4 d^{5/2}}-\frac{e \left(-c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)-\frac{\sin ^{-1}(c x)}{x}\right)}{d^2}-\frac{c x \sqrt{1-c^2 x^2}+c^3 x^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+2 \sin ^{-1}(c x)}{6 d x^3}\right)","\frac{e^{3/2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 (-d)^{5/2}}-\frac{e^{3/2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 (-d)^{5/2}}+\frac{e^{3/2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 (-d)^{5/2}}-\frac{e^{3/2} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 (-d)^{5/2}}+\frac{e \left(a+b \sin ^{-1}(c x)\right)}{d^2 x}-\frac{a+b \sin ^{-1}(c x)}{3 d x^3}+\frac{b c e \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{d^2}+\frac{i b e^{3/2} \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 (-d)^{5/2}}-\frac{i b e^{3/2} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 (-d)^{5/2}}+\frac{i b e^{3/2} \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 (-d)^{5/2}}-\frac{i b e^{3/2} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 (-d)^{5/2}}-\frac{b c \sqrt{1-c^2 x^2}}{6 d x^2}-\frac{b c^3 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{6 d}",1,"-1/3*a/(d*x^3) + (a*e)/(d^2*x) + (a*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(5/2) + b*(-((e*(-(ArcSin[c*x]/x) - c*ArcTanh[Sqrt[1 - c^2*x^2]]))/d^2) - (c*x*Sqrt[1 - c^2*x^2] + 2*ArcSin[c*x] + c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*d*x^3) - (e^(3/2)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]))/(4*d^(5/2)) + (e^(3/2)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/(4*d^(5/2)))","A",0
633,1,593,574,1.0693963,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^3*(a + b*ArcSin[c*x]))/(d + e*x^2)^2,x]","\frac{\frac{2 a d}{d+e x^2}+2 a \log \left(d+e x^2\right)+b \left(-i \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right)+\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)-i \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)+\sqrt{d} \left(\frac{\sin ^{-1}(c x)}{\sqrt{d}+i \sqrt{e} x}-\frac{c \tan ^{-1}\left(\frac{c^2 \sqrt{d} x+i \sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{\sqrt{c^2 d+e}}\right)-i \sqrt{d} \left(-\frac{c \tanh ^{-1}\left(\frac{\sqrt{e}+i c^2 \sqrt{d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{\sqrt{c^2 d+e}}-\frac{\sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right)\right)}{4 e^2}","\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^2}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^2}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^2}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^2}+\frac{d \left(a+b \sin ^{-1}(c x)\right)}{2 e^2 \left(d+e x^2\right)}-\frac{i \left(a+b \sin ^{-1}(c x)\right)^2}{2 b e^2}-\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 e^2}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 e^2}-\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 e^2}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 e^2}-\frac{b c \sqrt{d} \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{2 e^2 \sqrt{c^2 d+e}}",1,"((2*a*d)/(d + e*x^2) + 2*a*Log[d + e*x^2] + b*(Sqrt[d]*(ArcSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqrt[e] + c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) - I*Sqrt[d]*(-(ArcSin[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) - I*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]) - I*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])))/(4*e^2)","A",0
634,1,87,86,0.1495173,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x*(a + b*ArcSin[c*x]))/(d + e*x^2)^2,x]","-\frac{\frac{a}{d+e x^2}-\frac{b c \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{\sqrt{d} \sqrt{c^2 d+e}}+\frac{b \sin ^{-1}(c x)}{d+e x^2}}{2 e}","\frac{-a-b \sin ^{-1}(c x)}{2 e \left(d+e x^2\right)}+\frac{b c \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{2 \sqrt{d} e \sqrt{c^2 d+e}}",1,"-1/2*(a/(d + e*x^2) + (b*ArcSin[c*x])/(d + e*x^2) - (b*c*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(Sqrt[d]*Sqrt[c^2*d + e]))/e","A",1
635,0,0,597,3.918498,"\int \frac{a+b \sin ^{-1}(c x)}{x \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])/(x*(d + e*x^2)^2),x]","\int \frac{a+b \sin ^{-1}(c x)}{x \left(d+e x^2\right)^2} \, dx","-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^2}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^2}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^2}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^2}+\frac{\log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^2}+\frac{a+b \sin ^{-1}(c x)}{2 d \left(d+e x^2\right)}-\frac{b c \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{2 d^{3/2} \sqrt{c^2 d+e}}+\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 d^2}+\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 d^2}+\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 d^2}+\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 d^2}-\frac{i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d^2}",1,"Integrate[(a + b*ArcSin[c*x])/(x*(d + e*x^2)^2), x]","F",-1
636,0,0,632,6.4439053,"\int \frac{a+b \sin ^{-1}(c x)}{x^3 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^3*(d + e*x^2)^2),x]","\int \frac{a+b \sin ^{-1}(c x)}{x^3 \left(d+e x^2\right)^2} \, dx","\frac{e \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{d^3}+\frac{e \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{d^3}+\frac{e \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{d^3}+\frac{e \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{d^3}-\frac{2 e \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^3}-\frac{e \left(a+b \sin ^{-1}(c x)\right)}{2 d^2 \left(d+e x^2\right)}-\frac{a+b \sin ^{-1}(c x)}{2 d^2 x^2}+\frac{b c e \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{2 d^{5/2} \sqrt{c^2 d+e}}-\frac{i b e \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{d^3}-\frac{i b e \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{d^3}-\frac{i b e \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{d^3}-\frac{i b e \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{d^3}-\frac{b c \sqrt{1-c^2 x^2}}{2 d^2 x}+\frac{i b e \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{d^3}",1,"Integrate[(a + b*ArcSin[c*x])/(x^3*(d + e*x^2)^2), x]","F",-1
637,1,649,787,1.5601689,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x]))/(d + e*x^2)^2,x]","\frac{\frac{4 a d \sqrt{e} x}{d+e x^2}-12 a \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+8 a \sqrt{e} x+b \left(3 \sqrt{d} \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right)+\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)-3 \sqrt{d} \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)+2 i d \left(\frac{\sin ^{-1}(c x)}{\sqrt{d}+i \sqrt{e} x}-\frac{c \tan ^{-1}\left(\frac{c^2 \sqrt{d} x+i \sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{\sqrt{c^2 d+e}}\right)+2 d \left(\frac{c \tanh ^{-1}\left(\frac{\sqrt{e}+i c^2 \sqrt{d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{\sqrt{c^2 d+e}}+\frac{\sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right)+\frac{8 \sqrt{e} \left(\sqrt{1-c^2 x^2}+c x \sin ^{-1}(c x)\right)}{c}\right)}{8 e^{5/2}}","\frac{3 \sqrt{-d} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{4 e^{5/2}}-\frac{d \left(a+b \sin ^{-1}(c x)\right)}{4 e^{5/2} \left(\sqrt{-d}-\sqrt{e} x\right)}+\frac{d \left(a+b \sin ^{-1}(c x)\right)}{4 e^{5/2} \left(\sqrt{-d}+\sqrt{e} x\right)}+\frac{a x}{e^2}+\frac{3 i b \sqrt{-d} \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{4 e^{5/2}}+\frac{3 i b \sqrt{-d} \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{4 e^{5/2}}+\frac{b c d \tanh ^{-1}\left(\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{b c d \tanh ^{-1}\left(\frac{c^2 \sqrt{-d} x+\sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{4 e^{5/2} \sqrt{c^2 d+e}}+\frac{b \sqrt{1-c^2 x^2}}{c e^2}+\frac{b x \sin ^{-1}(c x)}{e^2}",1,"(8*a*Sqrt[e]*x + (4*a*d*Sqrt[e]*x)/(d + e*x^2) - 12*a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*((8*Sqrt[e]*(Sqrt[1 - c^2*x^2] + c*x*ArcSin[c*x]))/c + (2*I)*d*(ArcSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqrt[e] + c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) + 2*d*(ArcSin[c*x]/(I*Sqrt[d] + Sqrt[e]*x) + (c*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) + 3*Sqrt[d]*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]) - 3*Sqrt[d]*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])))/(8*e^(5/2))","A",0
638,1,603,745,1.1879757,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x]))/(d + e*x^2)^2,x]","\frac{-\frac{4 a \sqrt{e} x}{d+e x^2}+\frac{4 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d}}+b \left(-\frac{2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right)+\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)}{\sqrt{d}}+\frac{2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)}{\sqrt{d}}-\frac{2 c \tanh ^{-1}\left(\frac{\sqrt{e}+i c^2 \sqrt{d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{\sqrt{c^2 d+e}}-2 i \left(\frac{\sin ^{-1}(c x)}{\sqrt{d}+i \sqrt{e} x}-\frac{c \tan ^{-1}\left(\frac{c^2 \sqrt{d} x+i \sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{\sqrt{c^2 d+e}}\right)-\frac{2 \sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right)}{8 e^{3/2}}","\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{a+b \sin ^{-1}(c x)}{4 e^{3/2} \left(\sqrt{-d}-\sqrt{e} x\right)}-\frac{a+b \sin ^{-1}(c x)}{4 e^{3/2} \left(\sqrt{-d}+\sqrt{e} x\right)}+\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{b c \tanh ^{-1}\left(\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{4 e^{3/2} \sqrt{c^2 d+e}}-\frac{b c \tanh ^{-1}\left(\frac{c^2 \sqrt{-d} x+\sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{4 e^{3/2} \sqrt{c^2 d+e}}",1,"((-4*a*Sqrt[e]*x)/(d + e*x^2) + (4*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + b*((-2*ArcSin[c*x])/(I*Sqrt[d] + Sqrt[e]*x) - (2*I)*(ArcSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqrt[e] + c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) - (2*c*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e] - (ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))])/Sqrt[d] + (ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])/Sqrt[d]))/(8*e^(3/2))","A",0
639,1,591,757,1.7923287,"\int \frac{a+b \sin ^{-1}(c x)}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])/(d + e*x^2)^2,x]","\frac{1}{2} \left(\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{3/2} \sqrt{e}}+\frac{a x}{d^2+d e x^2}+\frac{b \left(\text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)-\text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)-\text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+i \sqrt{d} \left(\frac{\sin ^{-1}(c x)}{\sqrt{d}+i \sqrt{e} x}-\frac{c \tan ^{-1}\left(\frac{c^2 \sqrt{d} x+i \sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{\sqrt{c^2 d+e}}\right)+\sqrt{d} \left(\frac{c \tanh ^{-1}\left(\frac{\sqrt{e}+i c^2 \sqrt{d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{\sqrt{c^2 d+e}}+\frac{\sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right)+i \sin ^{-1}(c x) \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)-i \sin ^{-1}(c x) \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right)+\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)}{2 d^{3/2} \sqrt{e}}\right)","-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left(\sqrt{-d}-\sqrt{e} x\right)}+\frac{a+b \sin ^{-1}(c x)}{4 d \sqrt{e} \left(\sqrt{-d}+\sqrt{e} x\right)}-\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{b c \tanh ^{-1}\left(\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{4 d \sqrt{e} \sqrt{c^2 d+e}}+\frac{b c \tanh ^{-1}\left(\frac{c^2 \sqrt{-d} x+\sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{4 d \sqrt{e} \sqrt{c^2 d+e}}",1,"((a*x)/(d^2 + d*e*x^2) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]) + (b*(I*Sqrt[d]*(ArcSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqrt[e] + c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) + Sqrt[d]*(ArcSin[c*x]/(I*Sqrt[d] + Sqrt[e]*x) + (c*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) + I*ArcSin[c*x]*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]) - I*ArcSin[c*x]*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]) + PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] - PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] - PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))] + PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/(2*d^(3/2)*Sqrt[e]))/2","A",0
640,1,672,795,1.4941082,"\int \frac{a+b \sin ^{-1}(c x)}{x^2 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^2*(d + e*x^2)^2),x]","\frac{-\frac{4 a \sqrt{d} e x}{d+e x^2}-12 a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)-\frac{8 a \sqrt{d}}{x}+b \left(3 \sqrt{e} \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right)+\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)-3 \sqrt{e} \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)-2 i \sqrt{d} \sqrt{e} \left(\frac{\sin ^{-1}(c x)}{\sqrt{d}+i \sqrt{e} x}-\frac{c \tan ^{-1}\left(\frac{c^2 \sqrt{d} x+i \sqrt{e}}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{\sqrt{c^2 d+e}}\right)+2 \sqrt{d} \sqrt{e} \left(-\frac{c \tanh ^{-1}\left(\frac{\sqrt{e}+i c^2 \sqrt{d} x}{\sqrt{1-c^2 x^2} \sqrt{c^2 d+e}}\right)}{\sqrt{c^2 d+e}}-\frac{\sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right)-\frac{8 \sqrt{d} \left(c x \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)+\sin ^{-1}(c x)\right)}{x}\right)}{8 d^{5/2}}","-\frac{3 \sqrt{e} \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right) \left(a+b \sin ^{-1}(c x)\right)}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right) \left(a+b \sin ^{-1}(c x)\right)}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right) \left(a+b \sin ^{-1}(c x)\right)}{4 (-d)^{5/2}}-\frac{a+b \sin ^{-1}(c x)}{d^2 x}+\frac{\sqrt{e} \left(a+b \sin ^{-1}(c x)\right)}{4 d^2 \left(\sqrt{-d}-\sqrt{e} x\right)}-\frac{\sqrt{e} \left(a+b \sin ^{-1}(c x)\right)}{4 d^2 \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{b c \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right)}{4 d^2 \sqrt{d c^2+e}}-\frac{b c \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{-d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right)}{4 d^2 \sqrt{d c^2+e}}-\frac{b c \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{d^2}-\frac{3 i b \sqrt{e} \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}+\frac{3 i b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}-\frac{3 i b \sqrt{e} \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}+\frac{3 i b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}",1,"((-8*a*Sqrt[d])/x - (4*a*Sqrt[d]*e*x)/(d + e*x^2) - 12*a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*((-2*I)*Sqrt[d]*Sqrt[e]*(ArcSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqrt[e] + c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) + 2*Sqrt[d]*Sqrt[e]*(-(ArcSin[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) - (8*Sqrt[d]*(ArcSin[c*x] + c*x*ArcTanh[Sqrt[1 - c^2*x^2]]))/x + 3*Sqrt[e]*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]) - 3*Sqrt[e]*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])))/(8*d^(5/2))","A",0
641,1,1030,705,6.9215964,"\int \frac{x^5 \left(a+b \sin ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^5*(a + b*ArcSin[c*x]))/(d + e*x^2)^3,x]","-\frac{a d^2}{4 e^3 \left(e x^2+d\right)^2}+\frac{a d}{e^3 \left(e x^2+d\right)}+\frac{a \log \left(e x^2+d\right)}{2 e^3}+b \left(\frac{7 \sqrt{d} \left(\frac{\sin ^{-1}(c x)}{i \sqrt{e} x+\sqrt{d}}-\frac{c \tan ^{-1}\left(\frac{\sqrt{d} x c^2+i \sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right)}{\sqrt{d c^2+e}}\right)}{16 e^3}-\frac{7 i \sqrt{d} \left(-\frac{\sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-\frac{c \tanh ^{-1}\left(\frac{i \sqrt{d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right)}{\sqrt{d c^2+e}}\right)}{16 e^3}-\frac{d \left(-\frac{i \sqrt{d} \left(\log \left(\frac{e \sqrt{d c^2+e} \left(-i \sqrt{d} x c^2+\sqrt{e}+\sqrt{d c^2+e} \sqrt{1-c^2 x^2}\right)}{c^3 \left(d+i \sqrt{e} x \sqrt{d}\right)}\right)+\log (4)\right) c^3}{\sqrt{e} \left(d c^2+e\right)^{3/2}}-\frac{\sqrt{1-c^2 x^2} c}{\left(d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}-\frac{\sin ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x-i \sqrt{d}\right)^2}\right)}{16 e^{5/2}}-\frac{d \left(\frac{i \sqrt{d} \left(\log \left(\frac{e \sqrt{d c^2+e} \left(i \sqrt{d} x c^2+\sqrt{e}+\sqrt{d c^2+e} \sqrt{1-c^2 x^2}\right)}{c^3 \left(d-i \sqrt{d} \sqrt{e} x\right)}\right)+\log (4)\right) c^3}{\sqrt{e} \left(d c^2+e\right)^{3/2}}-\frac{\sqrt{1-c^2 x^2} c}{\left(d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}-\frac{\sin ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x+i \sqrt{d}\right)^2}\right)}{16 e^{5/2}}-\frac{i \left(\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{c \sqrt{d}-\sqrt{d c^2+e}}+1\right)+\log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d} c+\sqrt{d c^2+e}}+1\right)\right)\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)\right)}{4 e^3}-\frac{i \left(\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d c^2+e}-c \sqrt{d}}+1\right)+\log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)\right)\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)\right)}{4 e^3}\right)","\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^3}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^3}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^3}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 e^3}-\frac{d^2 \left(a+b \sin ^{-1}(c x)\right)}{4 e^3 \left(d+e x^2\right)^2}+\frac{d \left(a+b \sin ^{-1}(c x)\right)}{e^3 \left(d+e x^2\right)}-\frac{i \left(a+b \sin ^{-1}(c x)\right)^2}{2 b e^3}-\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 e^3}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 e^3}-\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 e^3}-\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 e^3}+\frac{b c \sqrt{d} \left(2 c^2 d+e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{8 e^3 \left(c^2 d+e\right)^{3/2}}-\frac{b c \sqrt{d} \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{e^3 \sqrt{c^2 d+e}}+\frac{b c d x \sqrt{1-c^2 x^2}}{8 e^2 \left(c^2 d+e\right) \left(d+e x^2\right)}",1,"-1/4*(a*d^2)/(e^3*(d + e*x^2)^2) + (a*d)/(e^3*(d + e*x^2)) + (a*Log[d + e*x^2])/(2*e^3) + b*((7*Sqrt[d]*(ArcSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqrt[e] + c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]))/(16*e^3) - (((7*I)/16)*Sqrt[d]*(-(ArcSin[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]))/e^3 - (d*(-((c*Sqrt[1 - c^2*x^2])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))) - ArcSin[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - (I*c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(16*e^(5/2)) - (d*(-((c*Sqrt[1 - c^2*x^2])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))) - ArcSin[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + (I*c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(16*e^(5/2)) - ((I/4)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]))/e^3 - ((I/4)*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/e^3)","A",0
642,1,152,153,0.5436456,"\int \frac{x^3 \left(a+b \sin ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^3*(a + b*ArcSin[c*x]))/(d + e*x^2)^3,x]","\frac{-\frac{2 a \left(d+2 e x^2\right)+\frac{b c e x \sqrt{1-c^2 x^2} \left(d+e x^2\right)}{c^2 d+e}}{\left(d+e x^2\right)^2}+\frac{b c \left(2 c^2 d+3 e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{\sqrt{d} \left(c^2 d+e\right)^{3/2}}-\frac{2 b \sin ^{-1}(c x) \left(d+2 e x^2\right)}{\left(d+e x^2\right)^2}}{8 e^2}","\frac{x^4 \left(a+b \sin ^{-1}(c x)\right)}{4 d \left(d+e x^2\right)^2}+\frac{b c \left(2 c^2 d+3 e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{8 \sqrt{d} e^2 \left(c^2 d+e\right)^{3/2}}-\frac{b c x \sqrt{1-c^2 x^2}}{8 e \left(c^2 d+e\right) \left(d+e x^2\right)}-\frac{b \sin ^{-1}(c x)}{4 d e^2}",1,"(-(((b*c*e*x*Sqrt[1 - c^2*x^2]*(d + e*x^2))/(c^2*d + e) + 2*a*(d + 2*e*x^2))/(d + e*x^2)^2) - (2*b*(d + 2*e*x^2)*ArcSin[c*x])/(d + e*x^2)^2 + (b*c*(2*c^2*d + 3*e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(Sqrt[d]*(c^2*d + e)^(3/2)))/(8*e^2)","A",1
643,1,141,133,0.5912799,"\int \frac{x \left(a+b \sin ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x*(a + b*ArcSin[c*x]))/(d + e*x^2)^3,x]","\frac{1}{8} \left(\frac{\frac{b c x \sqrt{1-c^2 x^2} \left(d+e x^2\right)}{d \left(c^2 d+e\right)}-\frac{2 a}{e}}{\left(d+e x^2\right)^2}+\frac{b c \left(2 c^2 d+e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{d^{3/2} e \left(c^2 d+e\right)^{3/2}}-\frac{2 b \sin ^{-1}(c x)}{e \left(d+e x^2\right)^2}\right)","-\frac{a+b \sin ^{-1}(c x)}{4 e \left(d+e x^2\right)^2}+\frac{b c \left(2 c^2 d+e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{8 d^{3/2} e \left(c^2 d+e\right)^{3/2}}+\frac{b c x \sqrt{1-c^2 x^2}}{8 d \left(c^2 d+e\right) \left(d+e x^2\right)}",1,"(((-2*a)/e + (b*c*x*Sqrt[1 - c^2*x^2]*(d + e*x^2))/(d*(c^2*d + e)))/(d + e*x^2)^2 - (2*b*ArcSin[c*x])/(e*(d + e*x^2)^2) + (b*c*(2*c^2*d + e)*ArcTan[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(d^(3/2)*e*(c^2*d + e)^(3/2)))/8","A",1
644,0,0,727,7.4635089,"\int \frac{a+b \sin ^{-1}(c x)}{x \left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcSin[c*x])/(x*(d + e*x^2)^3),x]","\int \frac{a+b \sin ^{-1}(c x)}{x \left(d+e x^2\right)^3} \, dx","-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^3}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^3}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^3}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^3}+\frac{\log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^3}+\frac{a+b \sin ^{-1}(c x)}{2 d^2 \left(d+e x^2\right)}+\frac{a+b \sin ^{-1}(c x)}{4 d \left(d+e x^2\right)^2}-\frac{b c \left(2 c^2 d+e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{8 d^{5/2} \left(c^2 d+e\right)^{3/2}}-\frac{b c \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{2 d^{5/2} \sqrt{c^2 d+e}}+\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 d^3}+\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 d^3}+\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 d^3}+\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 d^3}-\frac{b c e x \sqrt{1-c^2 x^2}}{8 d^2 \left(c^2 d+e\right) \left(d+e x^2\right)}-\frac{i b \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d^3}",1,"Integrate[(a + b*ArcSin[c*x])/(x*(d + e*x^2)^3), x]","F",-1
645,0,0,783,10.0306003,"\int \frac{a+b \sin ^{-1}(c x)}{x^3 \left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcSin[c*x])/(x^3*(d + e*x^2)^3),x]","\int \frac{a+b \sin ^{-1}(c x)}{x^3 \left(d+e x^2\right)^3} \, dx","\frac{3 e \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^4}+\frac{3 e \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{-\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^4}+\frac{3 e \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^4}+\frac{3 e \left(a+b \sin ^{-1}(c x)\right) \log \left(1+\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 d+e}+i c \sqrt{-d}}\right)}{2 d^4}-\frac{3 e \log \left(1-e^{2 i \sin ^{-1}(c x)}\right) \left(a+b \sin ^{-1}(c x)\right)}{d^4}-\frac{e \left(a+b \sin ^{-1}(c x)\right)}{d^3 \left(d+e x^2\right)}-\frac{a+b \sin ^{-1}(c x)}{2 d^3 x^2}-\frac{e \left(a+b \sin ^{-1}(c x)\right)}{4 d^2 \left(d+e x^2\right)^2}+\frac{b c e \left(2 c^2 d+e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{8 d^{7/2} \left(c^2 d+e\right)^{3/2}}+\frac{b c e \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{d^{7/2} \sqrt{c^2 d+e}}-\frac{3 i b e \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 d^4}-\frac{3 i b e \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 d^4}-\frac{3 i b e \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 d^4}-\frac{3 i b e \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 d^4}+\frac{b c e^2 x \sqrt{1-c^2 x^2}}{8 d^3 \left(c^2 d+e\right) \left(d+e x^2\right)}-\frac{b c \sqrt{1-c^2 x^2}}{2 d^3 x}+\frac{3 i b e \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 d^4}",1,"Integrate[(a + b*ArcSin[c*x])/(x^3*(d + e*x^2)^3), x]","F",-1
646,1,1014,1082,6.0652683,"\int \frac{x^4 \left(a+b \sin ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^4*(a + b*ArcSin[c*x]))/(d + e*x^2)^3,x]","\frac{\frac{b d \left(\log \left(\frac{e \sqrt{d c^2+e} \left(-i \sqrt{d} x c^2+\sqrt{e}+\sqrt{d c^2+e} \sqrt{1-c^2 x^2}\right)}{c^3 \left(d+i \sqrt{e} x \sqrt{d}\right)}\right)+\log (4)\right) c^3}{\left(d c^2+e\right)^{3/2}}+\frac{b d \left(\log \left(\frac{e \sqrt{d c^2+e} \left(i \sqrt{d} x c^2+\sqrt{e}+\sqrt{d c^2+e} \sqrt{1-c^2 x^2}\right)}{c^3 \left(d-i \sqrt{d} \sqrt{e} x\right)}\right)+\log (4)\right) c^3}{\left(d c^2+e\right)^{3/2}}-\frac{5 b \tanh ^{-1}\left(\frac{i \sqrt{d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right) c}{\sqrt{d c^2+e}}-\frac{i b \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} c}{\left(d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}+\frac{i b \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} c}{\left(d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}-\frac{5 b \sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}+\frac{i b \sqrt{d} \sin ^{-1}(c x)}{\left(i \sqrt{e} x+\sqrt{d}\right)^2}+\frac{i b \sqrt{d} \sin ^{-1}(c x)}{\left(\sqrt{e} x+i \sqrt{d}\right)^2}+\frac{6 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d}}-5 i b \left(\frac{\sin ^{-1}(c x)}{i \sqrt{e} x+\sqrt{d}}-\frac{c \tan ^{-1}\left(\frac{\sqrt{d} x c^2+i \sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right)}{\sqrt{d c^2+e}}\right)+\frac{3 i b \sin ^{-1}(c x) \left(\log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d c^2+e}-c \sqrt{d}}+1\right)+\log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)\right)}{\sqrt{d}}-\frac{3 i b \sin ^{-1}(c x) \left(\log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{c \sqrt{d}-\sqrt{d c^2+e}}+1\right)+\log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d} c+\sqrt{d c^2+e}}+1\right)\right)}{\sqrt{d}}+\frac{3 b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)}{\sqrt{d}}-\frac{3 b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)}{\sqrt{d}}-\frac{3 b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)}{\sqrt{d}}+\frac{3 b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)}{\sqrt{d}}-\frac{10 a \sqrt{e} x}{e x^2+d}+\frac{4 a d \sqrt{e} x}{\left(e x^2+d\right)^2}}{16 e^{5/2}}","\frac{b d \tanh ^{-1}\left(\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right) c^3}{16 e^{5/2} \left(d c^2+e\right)^{3/2}}+\frac{b d \tanh ^{-1}\left(\frac{\sqrt{-d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right) c^3}{16 e^{5/2} \left(d c^2+e\right)^{3/2}}-\frac{5 b \tanh ^{-1}\left(\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right) c}{16 e^{5/2} \sqrt{d c^2+e}}-\frac{5 b \tanh ^{-1}\left(\frac{\sqrt{-d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right) c}{16 e^{5/2} \sqrt{d c^2+e}}+\frac{b \sqrt{-d} \sqrt{1-c^2 x^2} c}{16 e^2 \left(d c^2+e\right) \left(\sqrt{-d}-\sqrt{e} x\right)}+\frac{b \sqrt{-d} \sqrt{1-c^2 x^2} c}{16 e^2 \left(d c^2+e\right) \left(\sqrt{e} x+\sqrt{-d}\right)}+\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{16 e^{5/2} \left(\sqrt{-d}-\sqrt{e} x\right)}-\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{16 e^{5/2} \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{\sqrt{-d} \left(a+b \sin ^{-1}(c x)\right)}{16 e^{5/2} \left(\sqrt{-d}-\sqrt{e} x\right)^2}+\frac{\sqrt{-d} \left(a+b \sin ^{-1}(c x)\right)}{16 e^{5/2} \left(\sqrt{e} x+\sqrt{-d}\right)^2}+\frac{3 \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left(a+b \sin ^{-1}(c x)\right) \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left(a+b \sin ^{-1}(c x)\right) \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}",1,"(((-I)*b*c*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) + (I*b*c*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) + (4*a*d*Sqrt[e]*x)/(d + e*x^2)^2 - (10*a*Sqrt[e]*x)/(d + e*x^2) + (I*b*Sqrt[d]*ArcSin[c*x])/(Sqrt[d] + I*Sqrt[e]*x)^2 + (I*b*Sqrt[d]*ArcSin[c*x])/(I*Sqrt[d] + Sqrt[e]*x)^2 - (5*b*ArcSin[c*x])/(I*Sqrt[d] + Sqrt[e]*x) + (6*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] - (5*I)*b*(ArcSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqrt[e] + c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) - (5*b*c*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e] + ((3*I)*b*ArcSin[c*x]*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/Sqrt[d] - ((3*I)*b*ArcSin[c*x]*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/Sqrt[d] + (b*c^3*d*(Log[4] + Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(c^2*d + e)^(3/2) + (b*c^3*d*(Log[4] + Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))]))/(c^2*d + e)^(3/2) + (3*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])])/Sqrt[d] - (3*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])])/Sqrt[d] - (3*b*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))])/Sqrt[d] + (3*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])/Sqrt[d])/(16*e^(5/2))","A",0
647,1,1064,1092,6.0774448,"\int \frac{x^2 \left(a+b \sin ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^2*(a + b*ArcSin[c*x]))/(d + e*x^2)^3,x]","\frac{a x}{8 d e \left(e x^2+d\right)}-\frac{a x}{4 e \left(e x^2+d\right)^2}+\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{3/2} e^{3/2}}+b \left(\frac{i \left(\frac{\sin ^{-1}(c x)}{i \sqrt{e} x+\sqrt{d}}-\frac{c \tan ^{-1}\left(\frac{\sqrt{d} x c^2+i \sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right)}{\sqrt{d c^2+e}}\right)}{16 d e^{3/2}}-\frac{-\frac{\sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-\frac{c \tanh ^{-1}\left(\frac{i \sqrt{d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right)}{\sqrt{d c^2+e}}}{16 d e^{3/2}}-\frac{i \left(-\frac{i \sqrt{d} \left(\log \left(\frac{e \sqrt{d c^2+e} \left(-i \sqrt{d} x c^2+\sqrt{e}+\sqrt{d c^2+e} \sqrt{1-c^2 x^2}\right)}{c^3 \left(d+i \sqrt{e} x \sqrt{d}\right)}\right)+\log (4)\right) c^3}{\sqrt{e} \left(d c^2+e\right)^{3/2}}-\frac{\sqrt{1-c^2 x^2} c}{\left(d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}-\frac{\sin ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x-i \sqrt{d}\right)^2}\right)}{16 \sqrt{d} e}+\frac{i \left(\frac{i \sqrt{d} \left(\log \left(\frac{e \sqrt{d c^2+e} \left(i \sqrt{d} x c^2+\sqrt{e}+\sqrt{d c^2+e} \sqrt{1-c^2 x^2}\right)}{c^3 \left(d-i \sqrt{d} \sqrt{e} x\right)}\right)+\log (4)\right) c^3}{\sqrt{e} \left(d c^2+e\right)^{3/2}}-\frac{\sqrt{1-c^2 x^2} c}{\left(d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}-\frac{\sin ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x+i \sqrt{d}\right)^2}\right)}{16 \sqrt{d} e}-\frac{\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{c \sqrt{d}-\sqrt{d c^2+e}}+1\right)+\log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d} c+\sqrt{d c^2+e}}+1\right)\right)\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)}{32 d^{3/2} e^{3/2}}+\frac{\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d c^2+e}-c \sqrt{d}}+1\right)+\log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)\right)\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)}{32 d^{3/2} e^{3/2}}\right)","-\frac{b \tanh ^{-1}\left(\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right) c^3}{16 e^{3/2} \left(d c^2+e\right)^{3/2}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{-d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right) c^3}{16 e^{3/2} \left(d c^2+e\right)^{3/2}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right) c}{16 d e^{3/2} \sqrt{d c^2+e}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{-d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right) c}{16 d e^{3/2} \sqrt{d c^2+e}}+\frac{b \sqrt{1-c^2 x^2} c}{16 \sqrt{-d} e \left(d c^2+e\right) \left(\sqrt{-d}-\sqrt{e} x\right)}+\frac{b \sqrt{1-c^2 x^2} c}{16 \sqrt{-d} e \left(d c^2+e\right) \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left(\sqrt{-d}-\sqrt{e} x\right)}+\frac{a+b \sin ^{-1}(c x)}{16 d e^{3/2} \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left(\sqrt{-d}-\sqrt{e} x\right)^2}+\frac{a+b \sin ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left(\sqrt{e} x+\sqrt{-d}\right)^2}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{\left(a+b \sin ^{-1}(c x)\right) \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}",1,"-1/4*(a*x)/(e*(d + e*x^2)^2) + (a*x)/(8*d*e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) + b*(((I/16)*(ArcSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqrt[e] + c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]))/(d*e^(3/2)) - (-(ArcSin[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e])/(16*d*e^(3/2)) - ((I/16)*(-((c*Sqrt[1 - c^2*x^2])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))) - ArcSin[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - (I*c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) + ((I/16)*(-((c*Sqrt[1 - c^2*x^2])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))) - ArcSin[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + (I*c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) - (ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))])/(32*d^(3/2)*e^(3/2)) + (ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])/(32*d^(3/2)*e^(3/2)))","A",0
648,1,1055,1092,6.1042741,"\int \frac{a+b \sin ^{-1}(c x)}{\left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcSin[c*x])/(d + e*x^2)^3,x]","\frac{3 a x}{8 d^2 \left(e x^2+d\right)}+\frac{a x}{4 d \left(e x^2+d\right)^2}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{5/2} \sqrt{e}}+b \left(\frac{3 i \left(\frac{\sin ^{-1}(c x)}{i \sqrt{e} x+\sqrt{d}}-\frac{c \tan ^{-1}\left(\frac{\sqrt{d} x c^2+i \sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right)}{\sqrt{d c^2+e}}\right)}{16 d^2 \sqrt{e}}-\frac{3 \left(-\frac{\sin ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-\frac{c \tanh ^{-1}\left(\frac{i \sqrt{d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right)}{\sqrt{d c^2+e}}\right)}{16 d^2 \sqrt{e}}+\frac{i \left(-\frac{i \sqrt{d} \left(\log \left(\frac{e \sqrt{d c^2+e} \left(-i \sqrt{d} x c^2+\sqrt{e}+\sqrt{d c^2+e} \sqrt{1-c^2 x^2}\right)}{c^3 \left(d+i \sqrt{e} x \sqrt{d}\right)}\right)+\log (4)\right) c^3}{\sqrt{e} \left(d c^2+e\right)^{3/2}}-\frac{\sqrt{1-c^2 x^2} c}{\left(d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}-\frac{\sin ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x-i \sqrt{d}\right)^2}\right)}{16 d^{3/2}}-\frac{i \left(\frac{i \sqrt{d} \left(\log \left(\frac{e \sqrt{d c^2+e} \left(i \sqrt{d} x c^2+\sqrt{e}+\sqrt{d c^2+e} \sqrt{1-c^2 x^2}\right)}{c^3 \left(d-i \sqrt{d} \sqrt{e} x\right)}\right)+\log (4)\right) c^3}{\sqrt{e} \left(d c^2+e\right)^{3/2}}-\frac{\sqrt{1-c^2 x^2} c}{\left(d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}-\frac{\sin ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x+i \sqrt{d}\right)^2}\right)}{16 d^{3/2}}-\frac{3 \left(\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{c \sqrt{d}-\sqrt{d c^2+e}}+1\right)+\log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d} c+\sqrt{d c^2+e}}+1\right)\right)\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)\right)}{32 d^{5/2} \sqrt{e}}+\frac{3 \left(\sin ^{-1}(c x) \left(\sin ^{-1}(c x)+2 i \left(\log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d c^2+e}-c \sqrt{d}}+1\right)+\log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)\right)\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)\right)}{32 d^{5/2} \sqrt{e}}\right)","\frac{b \tanh ^{-1}\left(\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right) c^3}{16 d \sqrt{e} \left(d c^2+e\right)^{3/2}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{-d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right) c^3}{16 d \sqrt{e} \left(d c^2+e\right)^{3/2}}+\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right) c}{16 d^2 \sqrt{e} \sqrt{d c^2+e}}+\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{-d} x c^2+\sqrt{e}}{\sqrt{d c^2+e} \sqrt{1-c^2 x^2}}\right) c}{16 d^2 \sqrt{e} \sqrt{d c^2+e}}+\frac{b \sqrt{1-c^2 x^2} c}{16 (-d)^{3/2} \left(d c^2+e\right) \left(\sqrt{-d}-\sqrt{e} x\right)}+\frac{b \sqrt{1-c^2 x^2} c}{16 (-d)^{3/2} \left(d c^2+e\right) \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{16 d^2 \sqrt{e} \left(\sqrt{-d}-\sqrt{e} x\right)}+\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{16 d^2 \sqrt{e} \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left(\sqrt{-d}-\sqrt{e} x\right)^2}+\frac{a+b \sin ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left(\sqrt{e} x+\sqrt{-d}\right)^2}+\frac{3 \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left(a+b \sin ^{-1}(c x)\right) \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \left(a+b \sin ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left(a+b \sin ^{-1}(c x)\right) \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 i b \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 i b \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}",1,"(a*x)/(4*d*(d + e*x^2)^2) + (3*a*x)/(8*d^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]) + b*((((3*I)/16)*(ArcSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqrt[e] + c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]))/(d^2*Sqrt[e]) - (3*(-(ArcSin[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]))/(16*d^2*Sqrt[e]) + ((I/16)*(-((c*Sqrt[1 - c^2*x^2])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))) - ArcSin[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - (I*c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/d^(3/2) - ((I/16)*(-((c*Sqrt[1 - c^2*x^2])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))) - ArcSin[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + (I*c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/d^(3/2) - (3*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]))/(32*d^(5/2)*Sqrt[e]) + (3*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]))/(32*d^(5/2)*Sqrt[e]))","A",0
649,0,0,23,6.3453807,"\int \sqrt{d+e x^2} \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + e*x^2]*(a + b*ArcSin[c*x]),x]","\int \sqrt{d+e x^2} \left(a+b \sin ^{-1}(c x)\right) \, dx","\text{Int}\left(\sqrt{d+e x^2} \left(a+b \sin ^{-1}(c x)\right),x\right)",0,"Integrate[Sqrt[d + e*x^2]*(a + b*ArcSin[c*x]), x]","A",-1
650,0,0,23,4.4620402,"\int \frac{a+b \sin ^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])/Sqrt[d + e*x^2],x]","\int \frac{a+b \sin ^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \sin ^{-1}(c x)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcSin[c*x])/Sqrt[d + e*x^2], x]","A",-1
651,1,74,70,0.1731659,"\int \frac{a+b \sin ^{-1}(c x)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(d + e*x^2)^(3/2),x]","\frac{x \left(2 \left(a+b \sin ^{-1}(c x)\right)-b c x \sqrt{\frac{e x^2}{d}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;c^2 x^2,-\frac{e x^2}{d}\right)\right)}{2 d \sqrt{d+e x^2}}","\frac{x \left(a+b \sin ^{-1}(c x)\right)}{d \sqrt{d+e x^2}}+\frac{b \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{d \sqrt{e}}",1,"(x*(-(b*c*x*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e*x^2)/d)]) + 2*(a + b*ArcSin[c*x])))/(2*d*Sqrt[d + e*x^2])","C",0
652,1,190,146,0.2850285,"\int \frac{a+b \sin ^{-1}(c x)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(d + e*x^2)^(5/2),x]","\sqrt{d+e x^2} \left(\frac{2 a x}{3 d^2 \left(d+e x^2\right)}+\frac{a x}{3 d \left(d+e x^2\right)^2}\right)-\frac{b c x^2 \sqrt{\frac{d+e x^2}{d}} F_1\left(1;\frac{1}{2},\frac{1}{2};2;c^2 x^2,-\frac{e x^2}{d}\right)}{3 d^2 \sqrt{d+e x^2}}+\frac{b c \sqrt{1-c^2 x^2}}{3 d \left(c^2 d+e\right) \sqrt{d+e x^2}}+\frac{b x \sin ^{-1}(c x) \left(3 d+2 e x^2\right)}{3 d^2 \left(d+e x^2\right)^{3/2}}","\frac{2 x \left(a+b \sin ^{-1}(c x)\right)}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left(a+b \sin ^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}+\frac{2 b \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{3 d^2 \sqrt{e}}+\frac{b c \sqrt{1-c^2 x^2}}{3 d \left(c^2 d+e\right) \sqrt{d+e x^2}}",1,"(b*c*Sqrt[1 - c^2*x^2])/(3*d*(c^2*d + e)*Sqrt[d + e*x^2]) + Sqrt[d + e*x^2]*((a*x)/(3*d*(d + e*x^2)^2) + (2*a*x)/(3*d^2*(d + e*x^2))) - (b*c*x^2*Sqrt[(d + e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e*x^2)/d)])/(3*d^2*Sqrt[d + e*x^2]) + (b*x*(3*d + 2*e*x^2)*ArcSin[c*x])/(3*d^2*(d + e*x^2)^(3/2))","C",0
653,1,188,226,0.4747889,"\int \frac{a+b \sin ^{-1}(c x)}{\left(d+e x^2\right)^{7/2}} \, dx","Integrate[(a + b*ArcSin[c*x])/(d + e*x^2)^(7/2),x]","\frac{a x \left(15 d^2+20 d e x^2+8 e^2 x^4\right)-4 b c x^2 \sqrt{\frac{e x^2}{d}+1} \left(d+e x^2\right)^2 F_1\left(1;\frac{1}{2},\frac{1}{2};2;c^2 x^2,-\frac{e x^2}{d}\right)+\frac{b c d \sqrt{1-c^2 x^2} \left(d+e x^2\right) \left(c^2 d \left(7 d+6 e x^2\right)+e \left(5 d+4 e x^2\right)\right)}{\left(c^2 d+e\right)^2}+b x \sin ^{-1}(c x) \left(15 d^2+20 d e x^2+8 e^2 x^4\right)}{15 d^3 \left(d+e x^2\right)^{5/2}}","\frac{8 x \left(a+b \sin ^{-1}(c x)\right)}{15 d^3 \sqrt{d+e x^2}}+\frac{4 x \left(a+b \sin ^{-1}(c x)\right)}{15 d^2 \left(d+e x^2\right)^{3/2}}+\frac{x \left(a+b \sin ^{-1}(c x)\right)}{5 d \left(d+e x^2\right)^{5/2}}+\frac{8 b \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{15 d^3 \sqrt{e}}+\frac{2 b c \sqrt{1-c^2 x^2} \left(3 c^2 d+2 e\right)}{15 d^2 \left(c^2 d+e\right)^2 \sqrt{d+e x^2}}+\frac{b c \sqrt{1-c^2 x^2}}{15 d \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}",1,"(a*x*(15*d^2 + 20*d*e*x^2 + 8*e^2*x^4) + (b*c*d*Sqrt[1 - c^2*x^2]*(d + e*x^2)*(e*(5*d + 4*e*x^2) + c^2*d*(7*d + 6*e*x^2)))/(c^2*d + e)^2 - 4*b*c*x^2*(d + e*x^2)^2*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e*x^2)/d)] + b*x*(15*d^2 + 20*d*e*x^2 + 8*e^2*x^4)*ArcSin[c*x])/(15*d^3*(d + e*x^2)^(5/2))","C",0
654,0,0,484,5.3910542,"\int (f x)^m \left(d+e x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcSin[c*x]),x]","\int (f x)^m \left(d+e x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right) \, dx","\frac{d^3 (f x)^{m+1} \left(a+b \sin ^{-1}(c x)\right)}{f (m+1)}+\frac{3 d^2 e (f x)^{m+3} \left(a+b \sin ^{-1}(c x)\right)}{f^3 (m+3)}+\frac{3 d e^2 (f x)^{m+5} \left(a+b \sin ^{-1}(c x)\right)}{f^5 (m+5)}+\frac{e^3 (f x)^{m+7} \left(a+b \sin ^{-1}(c x)\right)}{f^7 (m+7)}+\frac{b e^3 \sqrt{1-c^2 x^2} (f x)^{m+6}}{c f^6 (m+7)^2}+\frac{b e^2 \sqrt{1-c^2 x^2} (f x)^{m+4} \left(3 c^2 d (m+7)^2+e \left(m^2+11 m+30\right)\right)}{c^3 f^4 (m+5)^2 (m+7)^2}+\frac{b e \sqrt{1-c^2 x^2} (f x)^{m+2} \left(3 c^4 d^2 \left(m^2+12 m+35\right)^2+3 c^2 d e (m+7)^2 \left(m^2+7 m+12\right)+e^2 \left(m^4+18 m^3+119 m^2+342 m+360\right)\right)}{c^5 f^2 (m+3)^2 (m+5)^2 (m+7)^2}-\frac{b (f x)^{m+2} \left(\frac{c^6 d^3 (m+3) (m+5) (m+7)}{m+1}+\frac{e (m+2) \left(3 c^4 d^2 \left(m^2+12 m+35\right)^2+3 c^2 d e (m+7)^2 \left(m^2+7 m+12\right)+e^2 \left(m^4+18 m^3+119 m^2+342 m+360\right)\right)}{(m+3) (m+5) (m+7)}\right) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{c^5 f^2 (m+2) (m+3) (m+5) (m+7)}",1,"Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcSin[c*x]), x]","F",-1
655,1,2792,293,7.0478257,"\int (f x)^m \left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^2*(a + b*ArcSin[c*x]),x]","\text{Result too large to show}","\frac{d^2 (f x)^{m+1} \left(a+b \sin ^{-1}(c x)\right)}{f (m+1)}+\frac{2 d e (f x)^{m+3} \left(a+b \sin ^{-1}(c x)\right)}{f^3 (m+3)}+\frac{e^2 (f x)^{m+5} \left(a+b \sin ^{-1}(c x)\right)}{f^5 (m+5)}+\frac{b e^2 \sqrt{1-c^2 x^2} (f x)^{m+4}}{c f^4 (m+5)^2}+\frac{b e \sqrt{1-c^2 x^2} (f x)^{m+2} \left(2 c^2 d (m+5)^2+e \left(m^2+7 m+12\right)\right)}{c^3 f^2 (m+3)^2 (m+5)^2}-\frac{b (f x)^{m+2} \left(\frac{c^4 d^2 (m+3) (m+5)}{m+1}+\frac{e (m+2) \left(2 c^2 d (m+5)^2+e \left(m^2+7 m+12\right)\right)}{(m+3) (m+5)}\right) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{c^3 f^2 (m+2) (m+3) (m+5)}",1,"(x*(f*x)^m*(-2*(d + e*x^2)^2*(-((2 + m)*(a + b*ArcSin[c*x])) + b*c*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, c^2*x^2]) + 8*e*x^2*(d + e*x^2)*(-a - b*ArcSin[c*x] + (b*c*x*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(3 + m) + b*c*x*Gamma[2 + m/2]*HypergeometricPFQRegularized[{1/2, 1 + m/2}, {2 + m/2}, c^2*x^2] - (b*c*x*Gamma[2 + m/2]*Gamma[(3 + m)/2]*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(5 + m)/2}, c^2*x^2])/Gamma[1 + m/2]) - (4*e*x^2*(d + 3*e*x^2)*((-2*Gamma[1 + m/2]*((4 + m)*(a + b*ArcSin[c*x]) - b*c*x*Hypergeometric2F1[1/2, 2 + m/2, 3 + m/2, c^2*x^2]))/(4 + m) + b*c*(3 + m)*x*Gamma[1 + m/2]*Gamma[2 + m/2]*HypergeometricPFQRegularized[{1/2, 1 + m/2}, {3 + m/2}, c^2*x^2] + (2*b*c*x*Gamma[2 + m/2]*((3 + m)*Gamma[2 + m/2]*Gamma[(3 + m)/2] - Gamma[1 + m/2]*Gamma[(5 + m)/2])*HypergeometricPFQRegularized[{1/2, 2 + m/2}, {3 + m/2}, c^2*x^2])/Gamma[(3 + m)/2] - 2*b*c*x*((3 + m)*Gamma[2 + m/2]*Gamma[(3 + m)/2] - Gamma[1 + m/2]*Gamma[(5 + m)/2])*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(5 + m)/2}, c^2*x^2]))/((3 + m)*Gamma[1 + m/2]) - (8*e^2*x^4*(30*a*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2] + 6*a*m*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2] + 30*b*ArcSin[c*x]*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2] + 6*b*m*ArcSin[c*x]*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2] - 6*b*c*x*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2]*Hypergeometric2F1[1/2, (5 + m)/2, (7 + m)/2, c^2*x^2] - b*c*(60 + 47*m + 12*m^2 + m^3)*x*Gamma[1 + m/2]*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]*HypergeometricPFQRegularized[{1/2, 1 + m/2}, {3 + m/2}, c^2*x^2] + b*c*(5 + m)*x*Gamma[2 + m/2]*(-6*(12 + 7*m + m^2)*Gamma[2 + m/2]^2*Gamma[(3 + m)/2] + Gamma[1 + m/2]*(-6*Gamma[3 + m/2]*Gamma[(3 + m)/2] + (4 + m)*Gamma[2 + m/2]*((6 + 5*m + m^2)*Gamma[(3 + m)/2] + 6*Gamma[(5 + m)/2])))*HypergeometricPFQRegularized[{1/2, 2 + m/2}, {3 + m/2}, c^2*x^2] + 180*b*c*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]^2*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(7 + m)/2}, c^2*x^2] + 141*b*c*m*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]^2*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(7 + m)/2}, c^2*x^2] + 36*b*c*m^2*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]^2*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(7 + m)/2}, c^2*x^2] + 3*b*c*m^3*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]^2*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(7 + m)/2}, c^2*x^2] - 60*b*c*x*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(7 + m)/2}, c^2*x^2] - 27*b*c*m*x*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(7 + m)/2}, c^2*x^2] - 3*b*c*m^2*x*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(7 + m)/2}, c^2*x^2] + 240*b*c*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2] + 188*b*c*m*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2] + 48*b*c*m^2*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2] + 4*b*c*m^3*x*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2] + 30*b*c*x*Gamma[1 + m/2]*Gamma[3 + m/2]*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2] + 6*b*c*m*x*Gamma[1 + m/2]*Gamma[3 + m/2]*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2] - 120*b*c*x*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(5 + m)/2]^2*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2] - 54*b*c*m*x*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(5 + m)/2]^2*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2] - 6*b*c*m^2*x*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(5 + m)/2]^2*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2]))/((3 + m)*(4 + m)*(5 + m)*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2]) + (4*e^2*x^4*(-(b*c*(360 + 342*m + 119*m^2 + 18*m^3 + m^4)*x*Gamma[1 + m/2]*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]*Gamma[(5 + m)/2]*HypergeometricPFQRegularized[{1/2, 1 + m/2}, {4 + m/2}, c^2*x^2]) + b*c*(30 + 11*m + m^2)*x*Gamma[2 + m/2]*Gamma[(5 + m)/2]*(-6*(12 + 7*m + m^2)*Gamma[2 + m/2]^2*Gamma[(3 + m)/2] + Gamma[1 + m/2]*(-6*Gamma[3 + m/2]*Gamma[(3 + m)/2] + (4 + m)*Gamma[2 + m/2]*((6 + 5*m + m^2)*Gamma[(3 + m)/2] + 6*Gamma[(5 + m)/2])))*HypergeometricPFQRegularized[{1/2, 2 + m/2}, {4 + m/2}, c^2*x^2] - 4*b*c*(6 + m)*x*Gamma[3 + m/2]*((60 + 47*m + 12*m^2 + m^3)*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]*Gamma[(5 + m)/2] + 3*(5 + m)*Gamma[1 + m/2]*Gamma[3 + m/2]*Gamma[(3 + m)/2]*Gamma[(5 + m)/2] - Gamma[1 + m/2]*Gamma[2 + m/2]*(2*(20 + 9*m + m^2)*Gamma[(5 + m)/2]^2 + 3*Gamma[(3 + m)/2]*Gamma[(7 + m)/2]))*HypergeometricPFQRegularized[{1/2, 3 + m/2}, {4 + m/2}, c^2*x^2] + 2*Gamma[(5 + m)/2]*(6*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2]*((6 + m)*(a + b*ArcSin[c*x]) - b*c*x*Hypergeometric2F1[1/2, 3 + m/2, 4 + m/2, c^2*x^2]) + b*c*(120 + 74*m + 15*m^2 + m^3)*x*Gamma[2 + m/2]*Gamma[(3 + m)/2]*((3 + m)*Gamma[2 + m/2]*Gamma[(3 + m)/2] - Gamma[1 + m/2]*Gamma[(5 + m)/2])*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(7 + m)/2}, c^2*x^2] - b*c*(6 + m)*x*((60 + 47*m + 12*m^2 + m^3)*Gamma[2 + m/2]^2*Gamma[(3 + m)/2]*((3 + m)*Gamma[(3 + m)/2] - 4*Gamma[(5 + m)/2]) - 6*(5 + m)*Gamma[1 + m/2]*Gamma[3 + m/2]*Gamma[(3 + m)/2]*Gamma[(5 + m)/2] + Gamma[1 + m/2]*Gamma[2 + m/2]*(6*(20 + 9*m + m^2)*Gamma[(5 + m)/2]^2 - Gamma[(3 + m)/2]*((60 + 47*m + 12*m^2 + m^3)*Gamma[(5 + m)/2] - 6*Gamma[(7 + m)/2])))*HypergeometricPFQRegularized[{1/2, (5 + m)/2}, {(7 + m)/2}, c^2*x^2])))/((3 + m)*(4 + m)*(5 + m)*(6 + m)*Gamma[1 + m/2]*Gamma[2 + m/2]*Gamma[(3 + m)/2]*Gamma[(5 + m)/2])))/(2*(1 + m)*(2 + m))","C",0
656,1,508,161,2.217151,"\int (f x)^m \left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)*(a + b*ArcSin[c*x]),x]","\frac{x (f x)^m \left(2 e x^2 \left(b c x \Gamma \left(\frac{m}{2}+2\right) \, _2\tilde{F}_1\left(\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)-\frac{b c x \Gamma \left(\frac{m}{2}+2\right) \Gamma \left(\frac{m+3}{2}\right) \, _2\tilde{F}_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right)}{\Gamma \left(\frac{m}{2}+1\right)}-a+\frac{b c x \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right)}{m+3}-b \sin ^{-1}(c x)\right)-\frac{e x^2 \left(b c (m+3) x \Gamma \left(\frac{m}{2}+1\right) \Gamma \left(\frac{m}{2}+2\right) \, _2\tilde{F}_1\left(\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+3;c^2 x^2\right)+\frac{2 b c x \Gamma \left(\frac{m}{2}+2\right) \left((m+3) \Gamma \left(\frac{m}{2}+2\right) \Gamma \left(\frac{m+3}{2}\right)-\Gamma \left(\frac{m}{2}+1\right) \Gamma \left(\frac{m+5}{2}\right)\right) \, _2\tilde{F}_1\left(\frac{1}{2},\frac{m}{2}+2;\frac{m}{2}+3;c^2 x^2\right)}{\Gamma \left(\frac{m+3}{2}\right)}-2 b c x \left((m+3) \Gamma \left(\frac{m}{2}+2\right) \Gamma \left(\frac{m+3}{2}\right)-\Gamma \left(\frac{m}{2}+1\right) \Gamma \left(\frac{m+5}{2}\right)\right) \, _2\tilde{F}_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right)-\frac{2 \Gamma \left(\frac{m}{2}+1\right) \left((m+4) \left(a+b \sin ^{-1}(c x)\right)-b c x \, _2F_1\left(\frac{1}{2},\frac{m}{2}+2;\frac{m}{2}+3;c^2 x^2\right)\right)}{m+4}\right)}{(m+3) \Gamma \left(\frac{m}{2}+1\right)}-\left(d+e x^2\right) \left(b c x \, _2F_1\left(\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)-(m+2) \left(a+b \sin ^{-1}(c x)\right)\right)\right)}{(m+1) (m+2)}","\frac{d (f x)^{m+1} \left(a+b \sin ^{-1}(c x)\right)}{f (m+1)}+\frac{e (f x)^{m+3} \left(a+b \sin ^{-1}(c x)\right)}{f^3 (m+3)}-\frac{b (f x)^{m+2} \left(c^2 d (m+3)^2+e (m+1) (m+2)\right) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{c f^2 (m+1) (m+2) (m+3)^2}+\frac{b e \sqrt{1-c^2 x^2} (f x)^{m+2}}{c f^2 (m+3)^2}",1,"(x*(f*x)^m*(-((d + e*x^2)*(-((2 + m)*(a + b*ArcSin[c*x])) + b*c*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, c^2*x^2])) + 2*e*x^2*(-a - b*ArcSin[c*x] + (b*c*x*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, c^2*x^2])/(3 + m) + b*c*x*Gamma[2 + m/2]*HypergeometricPFQRegularized[{1/2, 1 + m/2}, {2 + m/2}, c^2*x^2] - (b*c*x*Gamma[2 + m/2]*Gamma[(3 + m)/2]*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(5 + m)/2}, c^2*x^2])/Gamma[1 + m/2]) - (e*x^2*((-2*Gamma[1 + m/2]*((4 + m)*(a + b*ArcSin[c*x]) - b*c*x*Hypergeometric2F1[1/2, 2 + m/2, 3 + m/2, c^2*x^2]))/(4 + m) + b*c*(3 + m)*x*Gamma[1 + m/2]*Gamma[2 + m/2]*HypergeometricPFQRegularized[{1/2, 1 + m/2}, {3 + m/2}, c^2*x^2] + (2*b*c*x*Gamma[2 + m/2]*((3 + m)*Gamma[2 + m/2]*Gamma[(3 + m)/2] - Gamma[1 + m/2]*Gamma[(5 + m)/2])*HypergeometricPFQRegularized[{1/2, 2 + m/2}, {3 + m/2}, c^2*x^2])/Gamma[(3 + m)/2] - 2*b*c*x*((3 + m)*Gamma[2 + m/2]*Gamma[(3 + m)/2] - Gamma[1 + m/2]*Gamma[(5 + m)/2])*HypergeometricPFQRegularized[{1/2, (3 + m)/2}, {(5 + m)/2}, c^2*x^2]))/((3 + m)*Gamma[1 + m/2])))/((1 + m)*(2 + m))","C",0
657,0,0,26,8.9213672,"\int \frac{(f x)^m \left(a+b \sin ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[((f*x)^m*(a + b*ArcSin[c*x]))/(d + e*x^2),x]","\int \frac{(f x)^m \left(a+b \sin ^{-1}(c x)\right)}{d+e x^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \sin ^{-1}(c x)\right)}{d+e x^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcSin[c*x]))/(d + e*x^2), x]","A",-1
658,0,0,26,10.997352,"\int \frac{(f x)^m \left(a+b \sin ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[((f*x)^m*(a + b*ArcSin[c*x]))/(d + e*x^2)^2,x]","\int \frac{(f x)^m \left(a+b \sin ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \sin ^{-1}(c x)\right)}{\left(d+e x^2\right)^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcSin[c*x]))/(d + e*x^2)^2, x]","A",-1
659,1,435,569,0.5635712,"\int \left(d+e x^2\right)^3 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x^2)^3*(a + b*ArcSin[c*x])^2,x]","-2 b d^3 \left(b x-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}\right)-\frac{2 b d^2 e \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)}{9 c^3}-\frac{2 b d e^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)}{375 c^5}-\frac{2 b e^3 \left(-105 a \sqrt{1-c^2 x^2} \left(5 c^6 x^6+6 c^4 x^4+8 c^2 x^2+16\right)+b c x \left(75 c^6 x^6+126 c^4 x^4+280 c^2 x^2+1680\right)-105 b \sqrt{1-c^2 x^2} \left(5 c^6 x^6+6 c^4 x^4+8 c^2 x^2+16\right) \sin ^{-1}(c x)\right)}{25725 c^7}+d^3 x \left(a+b \sin ^{-1}(c x)\right)^2+d^2 e x^3 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{3}{5} d e^2 x^5 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{7} e^3 x^7 \left(a+b \sin ^{-1}(c x)\right)^2","\frac{2 b d^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{2 b d^2 e x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 c}+\frac{6 b d e^2 x^4 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{25 c}+\frac{2 b e^3 x^6 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{49 c}+\frac{32 b e^3 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{245 c^7}+\frac{16 b d e^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{25 c^5}+\frac{16 b e^3 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{245 c^5}+\frac{4 b d^2 e \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{3 c^3}+\frac{8 b d e^2 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{25 c^3}+\frac{12 b e^3 x^4 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{245 c^3}+d^3 x \left(a+b \sin ^{-1}(c x)\right)^2+d^2 e x^3 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{3}{5} d e^2 x^5 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{7} e^3 x^7 \left(a+b \sin ^{-1}(c x)\right)^2-\frac{32 b^2 e^3 x}{245 c^6}-\frac{16 b^2 d e^2 x}{25 c^4}-\frac{16 b^2 e^3 x^3}{735 c^4}-\frac{4 b^2 d^2 e x}{3 c^2}-\frac{8 b^2 d e^2 x^3}{75 c^2}-\frac{12 b^2 e^3 x^5}{1225 c^2}-2 b^2 d^3 x-\frac{2}{9} b^2 d^2 e x^3-\frac{6}{125} b^2 d e^2 x^5-\frac{2}{343} b^2 e^3 x^7",1,"d^3*x*(a + b*ArcSin[c*x])^2 + d^2*e*x^3*(a + b*ArcSin[c*x])^2 + (3*d*e^2*x^5*(a + b*ArcSin[c*x])^2)/5 + (e^3*x^7*(a + b*ArcSin[c*x])^2)/7 - (2*b*d^2*e*(-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]))/(9*c^3) - (2*b*d*e^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]))/(375*c^5) - (2*b*e^3*(-105*a*Sqrt[1 - c^2*x^2]*(16 + 8*c^2*x^2 + 6*c^4*x^4 + 5*c^6*x^6) + b*c*x*(1680 + 280*c^2*x^2 + 126*c^4*x^4 + 75*c^6*x^6) - 105*b*Sqrt[1 - c^2*x^2]*(16 + 8*c^2*x^2 + 6*c^4*x^4 + 5*c^6*x^6)*ArcSin[c*x]))/(25725*c^7) - 2*b*d^3*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c)","A",1
660,1,291,335,0.384263,"\int \left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x^2)^2*(a + b*ArcSin[c*x])^2,x]","\frac{225 a^2 c^5 x \left(15 d^2+10 d e x^2+3 e^2 x^4\right)+30 a b \sqrt{1-c^2 x^2} \left(c^4 \left(225 d^2+50 d e x^2+9 e^2 x^4\right)+4 c^2 e \left(25 d+3 e x^2\right)+24 e^2\right)+30 b \sin ^{-1}(c x) \left(15 a c^5 x \left(15 d^2+10 d e x^2+3 e^2 x^4\right)+b \sqrt{1-c^2 x^2} \left(c^4 \left(225 d^2+50 d e x^2+9 e^2 x^4\right)+4 c^2 e \left(25 d+3 e x^2\right)+24 e^2\right)\right)+225 b^2 c^5 x \sin ^{-1}(c x)^2 \left(15 d^2+10 d e x^2+3 e^2 x^4\right)-2 b^2 c x \left(c^4 \left(3375 d^2+250 d e x^2+27 e^2 x^4\right)+60 c^2 e \left(25 d+e x^2\right)+360 e^2\right)}{3375 c^5}","\frac{2 b d^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{4 b d e x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 c}+\frac{2 b e^2 x^4 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{25 c}+\frac{16 b e^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{75 c^5}+\frac{8 b d e \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 c^3}+\frac{8 b e^2 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{75 c^3}+d^2 x \left(a+b \sin ^{-1}(c x)\right)^2+\frac{2}{3} d e x^3 \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{5} e^2 x^5 \left(a+b \sin ^{-1}(c x)\right)^2-\frac{16 b^2 e^2 x}{75 c^4}-\frac{8 b^2 d e x}{9 c^2}-\frac{8 b^2 e^2 x^3}{225 c^2}-2 b^2 d^2 x-\frac{4}{27} b^2 d e x^3-\frac{2}{125} b^2 e^2 x^5",1,"(225*a^2*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + 30*a*b*Sqrt[1 - c^2*x^2]*(24*e^2 + 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)) - 2*b^2*c*x*(360*e^2 + 60*c^2*e*(25*d + e*x^2) + c^4*(3375*d^2 + 250*d*e*x^2 + 27*e^2*x^4)) + 30*b*(15*a*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + b*Sqrt[1 - c^2*x^2]*(24*e^2 + 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)))*ArcSin[c*x] + 225*b^2*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcSin[c*x]^2)/(3375*c^5)","A",1
661,1,148,156,0.2777,"\int \left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x^2)*(a + b*ArcSin[c*x])^2,x]","-2 b d \left(b x-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}\right)-\frac{2}{27} b e \left(-\frac{3 x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{6 \left(\frac{b x}{c}-\frac{\sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c^2}\right)}{c}+b x^3\right)+d x \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{3} e x^3 \left(a+b \sin ^{-1}(c x)\right)^2","\frac{2 b d \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{c}+\frac{2 b e x^2 \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 c}+\frac{4 b e \sqrt{1-c^2 x^2} \left(a+b \sin ^{-1}(c x)\right)}{9 c^3}+d x \left(a+b \sin ^{-1}(c x)\right)^2+\frac{1}{3} e x^3 \left(a+b \sin ^{-1}(c x)\right)^2-\frac{4 b^2 e x}{9 c^2}-2 b^2 d x-\frac{2}{27} b^2 e x^3",1,"d*x*(a + b*ArcSin[c*x])^2 + (e*x^3*(a + b*ArcSin[c*x])^2)/3 - 2*b*d*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c) - (2*b*e*(b*x^3 - (3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (6*((b*x)/c - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^2))/c))/27","A",1
662,1,47,47,0.0921968,"\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
663,1,1101,821,0.8651454,"\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{2 \sqrt{-d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) a^2-2 b \sqrt{d} \sin ^{-1}(c x) \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right) a+2 b \sqrt{d} \sin ^{-1}(c x) \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d c^2+e}-i c \sqrt{-d}}+1\right) a+2 b \sqrt{d} \sin ^{-1}(c x) \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right) a-2 b \sqrt{d} \sin ^{-1}(c x) \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right) a+2 i b \sqrt{d} \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right) a-2 i b \sqrt{d} \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right) a-b^2 \sqrt{d} \sin ^{-1}(c x)^2 \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right)+b^2 \sqrt{d} \sin ^{-1}(c x)^2 \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{\sqrt{d c^2+e}-i c \sqrt{-d}}+1\right)+b^2 \sqrt{d} \sin ^{-1}(c x)^2 \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)-b^2 \sqrt{d} \sin ^{-1}(c x)^2 \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right)-2 i b \sqrt{d} \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)+2 i b \sqrt{d} \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-i c \sqrt{-d}}\right)+2 i b^2 \sqrt{d} \sin ^{-1}(c x) \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)-2 i b^2 \sqrt{d} \sin ^{-1}(c x) \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)+2 b^2 \sqrt{d} \text{Li}_3\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)-2 b^2 \sqrt{d} \text{Li}_3\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{\sqrt{d c^2+e}-i c \sqrt{-d}}\right)-2 b^2 \sqrt{d} \text{Li}_3\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)+2 b^2 \sqrt{d} \text{Li}_3\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 \sqrt{-d^2} \sqrt{e}}","-\frac{\text{Li}_3\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right) b^2}{\sqrt{-d} \sqrt{e}}+\frac{\text{Li}_3\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right) b^2}{\sqrt{-d} \sqrt{e}}-\frac{\text{Li}_3\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right) b^2}{\sqrt{-d} \sqrt{e}}+\frac{\text{Li}_3\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right) b^2}{\sqrt{-d} \sqrt{e}}+\frac{i \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right) b}{\sqrt{-d} \sqrt{e}}-\frac{i \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right) b}{\sqrt{-d} \sqrt{e}}+\frac{i \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right) b}{\sqrt{-d} \sqrt{e}}-\frac{i \left(a+b \sin ^{-1}(c x)\right) \text{Li}_2\left(\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right) b}{\sqrt{-d} \sqrt{e}}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i c \sqrt{-d}-\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i c \sqrt{-d}-\sqrt{d c^2+e}}+1\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(1-\frac{\sqrt{e} e^{i \sin ^{-1}(c x)}}{i \sqrt{-d} c+\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \sin ^{-1}(c x)\right)^2 \log \left(\frac{e^{i \sin ^{-1}(c x)} \sqrt{e}}{i \sqrt{-d} c+\sqrt{d c^2+e}}+1\right)}{2 \sqrt{-d} \sqrt{e}}",1,"(2*a^2*Sqrt[-d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - 2*a*b*Sqrt[d]*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])] - b^2*Sqrt[d]*ArcSin[c*x]^2*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])] + 2*a*b*Sqrt[d]*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/((-I)*c*Sqrt[-d] + Sqrt[c^2*d + e])] + b^2*Sqrt[d]*ArcSin[c*x]^2*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/((-I)*c*Sqrt[-d] + Sqrt[c^2*d + e])] + 2*a*b*Sqrt[d]*ArcSin[c*x]*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])] + b^2*Sqrt[d]*ArcSin[c*x]^2*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])] - 2*a*b*Sqrt[d]*ArcSin[c*x]*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])] - b^2*Sqrt[d]*ArcSin[c*x]^2*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])] - (2*I)*b*Sqrt[d]*(a + b*ArcSin[c*x])*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])] + (2*I)*b*Sqrt[d]*(a + b*ArcSin[c*x])*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/((-I)*c*Sqrt[-d] + Sqrt[c^2*d + e])] + (2*I)*a*b*Sqrt[d]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))] + (2*I)*b^2*Sqrt[d]*ArcSin[c*x]*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))] - (2*I)*a*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])] - (2*I)*b^2*Sqrt[d]*ArcSin[c*x]*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])] + 2*b^2*Sqrt[d]*PolyLog[3, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])] - 2*b^2*Sqrt[d]*PolyLog[3, (Sqrt[e]*E^(I*ArcSin[c*x]))/((-I)*c*Sqrt[-d] + Sqrt[c^2*d + e])] - 2*b^2*Sqrt[d]*PolyLog[3, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))] + 2*b^2*Sqrt[d]*PolyLog[3, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*Sqrt[-d^2]*Sqrt[e])","A",0
664,0,0,25,18.0859575,"\int \sqrt{d+e x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^2,x]","\int \sqrt{d+e x^2} \left(a+b \sin ^{-1}(c x)\right)^2 \, dx","\text{Int}\left(\sqrt{d+e x^2} \left(a+b \sin ^{-1}(c x)\right)^2,x\right)",0,"Integrate[Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^2, x]","A",-1
665,0,0,25,13.0582286,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/Sqrt[d + e*x^2],x]","\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcSin[c*x])^2/Sqrt[d + e*x^2], x]","A",-1
666,0,0,25,4.3320096,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(d + e*x^2)^(3/2),x]","\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(a + b*ArcSin[c*x])^2/(d + e*x^2)^(3/2), x]","A",-1
667,0,0,25,8.7640017,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^2/(d + e*x^2)^(5/2),x]","\int \frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{\left(a+b \sin ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(a + b*ArcSin[c*x])^2/(d + e*x^2)^(5/2), x]","A",-1
668,1,253,387,0.7249701,"\int \frac{\left(d+e x^2\right)^2}{a+b \sin ^{-1}(c x)} \, dx","Integrate[(d + e*x^2)^2/(a + b*ArcSin[c*x]),x]","\frac{16 c^4 d^2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-e \cos \left(\frac{3 a}{b}\right) \left(8 c^2 d+3 e\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+8 c^2 d e \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-8 c^2 d e \sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 \cos \left(\frac{a}{b}\right) \left(8 c^4 d^2+4 c^2 d e+e^2\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+e^2 \cos \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 e^2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-3 e^2 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+e^2 \sin \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{16 b c^5}","\frac{e^2 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b c^5}-\frac{3 e^2 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^5}+\frac{e^2 \cos \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^5}+\frac{e^2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b c^5}-\frac{3 e^2 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^5}+\frac{e^2 \sin \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b c^5}+\frac{d e \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{2 b c^3}-\frac{d e \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b c^3}+\frac{d e \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{2 b c^3}-\frac{d e \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b c^3}+\frac{d^2 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b c}+\frac{d^2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b c}",1,"(2*(8*c^4*d^2 + 4*c^2*d*e + e^2)*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] - e*(8*c^2*d + 3*e)*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + e^2*Cos[(5*a)/b]*CosIntegral[5*(a/b + ArcSin[c*x])] + 16*c^4*d^2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 8*c^2*d*e*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 2*e^2*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 8*c^2*d*e*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] - 3*e^2*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + e^2*Sin[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])])/(16*b*c^5)","A",1
669,1,125,179,0.3249041,"\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+e\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+4 c^2 d \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-e \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+e \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-e \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 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b c^3}-\frac{e \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b c^3}+\frac{e \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{4 b c^3}-\frac{e \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{4 b c^3}+\frac{d \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b c}+\frac{d \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{b c}",1,"((4*c^2*d + e)*Cos[a/b]*CosIntegral[a/b + ArcSin[c*x]] - e*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSin[c*x])] + 4*c^2*d*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] + e*Sin[a/b]*SinIntegral[a/b + ArcSin[c*x]] - e*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])])/(4*b*c^3)","A",1
670,1,44,53,0.0653254,"\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
671,0,0,23,0.8094921,"\int \frac{1}{\left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x^2)*(a + b*ArcSin[c*x])),x]","\int \frac{1}{\left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right) \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
672,0,0,23,3.8532343,"\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x^2)^2*(a + b*ArcSin[c*x])),x]","\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x^2)^2*(a + b*ArcSin[c*x])), x]","A",-1
673,0,0,25,1.3369894,"\int \frac{\sqrt{d+e x^2}}{a+b \sin ^{-1}(c x)} \, dx","Integrate[Sqrt[d + e*x^2]/(a + b*ArcSin[c*x]),x]","\int \frac{\sqrt{d+e x^2}}{a+b \sin ^{-1}(c x)} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2}}{a+b \sin ^{-1}(c x)},x\right)",0,"Integrate[Sqrt[d + e*x^2]/(a + b*ArcSin[c*x]), x]","A",-1
674,0,0,25,1.2113762,"\int \frac{1}{\sqrt{d+e x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/(Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])),x]","\int \frac{1}{\sqrt{d+e x^2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\sqrt{d+e x^2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])), x]","A",-1
675,0,0,25,1.6980822,"\int \frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcSin[c*x])),x]","\int \frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcSin[c*x])), x]","A",-1
676,0,0,25,4.0752522,"\int \frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcSin[c*x])),x]","\int \frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcSin[c*x])), x]","A",-1
677,1,359,498,2.2494804,"\int \frac{\left(d+e x^2\right)^2}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[(d + e*x^2)^2/(a + b*ArcSin[c*x])^2,x]","-\frac{16 c^4 d^2 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+3 e \sin \left(\frac{3 a}{b}\right) \left(8 c^2 d+3 e\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+8 c^2 d e \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-24 c^2 d e \cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)-2 \sin \left(\frac{a}{b}\right) \left(8 c^4 d^2+4 c^2 d e+e^2\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+\frac{16 b c^4 d^2 \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+\frac{32 b c^4 d e x^2 \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+\frac{16 b c^4 e^2 x^4 \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}-5 e^2 \sin \left(\frac{5 a}{b}\right) \text{Ci}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+2 e^2 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-9 e^2 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+5 e^2 \cos \left(\frac{5 a}{b}\right) \text{Si}\left(5 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)}{16 b^2 c^5}","\frac{e^2 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b^2 c^5}-\frac{9 e^2 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^5}+\frac{5 e^2 \sin \left(\frac{5 a}{b}\right) \text{Ci}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^5}-\frac{e^2 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{8 b^2 c^5}+\frac{9 e^2 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^5}-\frac{5 e^2 \cos \left(\frac{5 a}{b}\right) \text{Si}\left(\frac{5 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^5}+\frac{d e \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{2 b^2 c^3}-\frac{3 d e \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^3}-\frac{d e \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a+b \sin ^{-1}(c x)}{b}\right)}{2 b^2 c^3}+\frac{3 d e \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 \left(a+b \sin ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^3}+\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^2 \sqrt{1-c^2 x^2}}{b c \left(a+b \sin ^{-1}(c x)\right)}-\frac{e^2 x^4 \sqrt{1-c^2 x^2}}{b c \left(a+b \sin ^{-1}(c x)\right)}",1,"-1/16*((16*b*c^4*d^2*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + (32*b*c^4*d*e*x^2*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + (16*b*c^4*e^2*x^4*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) - 2*(8*c^4*d^2 + 4*c^2*d*e + e^2)*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] + 3*e*(8*c^2*d + 3*e)*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] - 5*e^2*CosIntegral[5*(a/b + ArcSin[c*x])]*Sin[(5*a)/b] + 16*c^4*d^2*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 8*c^2*d*e*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + 2*e^2*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 24*c^2*d*e*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] - 9*e^2*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])] + 5*e^2*Cos[(5*a)/b]*SinIntegral[5*(a/b + ArcSin[c*x])])/(b^2*c^5)","A",1
678,1,191,249,1.0398803,"\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+e\right) \text{Ci}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+4 c^2 d \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)+\frac{4 b c^2 d \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+\frac{4 b c^2 e x^2 \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+3 e \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sin ^{-1}(c x)\right)\right)+e \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sin ^{-1}(c x)\right)-3 e \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 \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 \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 \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 \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{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^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*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) + (4*b*c^2*e*x^2*Sqrt[1 - c^2*x^2])/(a + b*ArcSin[c*x]) - (4*c^2*d + e)*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] + 3*e*CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] + 4*c^2*d*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] + e*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] - 3*e*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])])/(b^2*c^3)","A",1
679,1,72,86,0.2185923,"\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
680,0,0,23,22.7710347,"\int \frac{1}{\left(d+e x^2\right) \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}{\left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right) \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
681,0,0,23,56.358272,"\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d + e*x^2)^2*(a + b*ArcSin[c*x])^2),x]","\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/((d + e*x^2)^2*(a + b*ArcSin[c*x])^2), x]","A",-1
682,0,0,25,7.7934012,"\int \frac{\sqrt{d+e x^2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[d + e*x^2]/(a + b*ArcSin[c*x])^2,x]","\int \frac{\sqrt{d+e x^2}}{\left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2}}{\left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[Sqrt[d + e*x^2]/(a + b*ArcSin[c*x])^2, x]","A",-1
683,0,0,25,12.4015742,"\int \frac{1}{\sqrt{d+e x^2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/(Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^2),x]","\int \frac{1}{\sqrt{d+e x^2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\sqrt{d+e x^2} \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^2), x]","A",-1
684,0,0,25,27.4057018,"\int \frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
685,0,0,25,51.9907269,"\int \frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcSin[c*x])^2),x]","\int \frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \sin ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcSin[c*x])^2), x]","A",-1
686,1,400,754,1.6808203,"\int \left(d+e x^2\right)^2 \sqrt{a+b \sin ^{-1}(c x)} \, dx","Integrate[(d + e*x^2)^2*Sqrt[a + b*ArcSin[c*x]],x]","\frac{b e^{-\frac{5 i a}{b}} \left(-e \left(25 \sqrt{3} e^{\frac{2 i a}{b}} \left(8 c^2 d+3 e\right) \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+25 \sqrt{3} e^{\frac{8 i a}{b}} \left(8 c^2 d+3 e\right) \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-9 \sqrt{5} e \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+e^{\frac{10 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)\right)+450 e^{\frac{4 i a}{b}} \left(8 c^4 d^2+4 c^2 d e+e^2\right) \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+450 e^{\frac{6 i a}{b}} \left(8 c^4 d^2+4 c^2 d e+e^2\right) \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)}{7200 c^5 \sqrt{a+b \sin ^{-1}(c x)}}","\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 x)}}{\sqrt{b}}\right)}{8 c^5}-\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 x)}}{\sqrt{b}}\right)}{16 c^5}+\frac{\sqrt{\frac{\pi }{10}} \sqrt{b} e^2 \sin \left(\frac{5 a}{b}\right) C\left(\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{80 c^5}-\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 x)}}{\sqrt{b}}\right)}{8 c^5}+\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 x)}}{\sqrt{b}}\right)}{16 c^5}-\frac{\sqrt{\frac{\pi }{10}} \sqrt{b} e^2 \cos \left(\frac{5 a}{b}\right) S\left(\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{80 c^5}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} d e \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{2 c^3}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} d e \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{6 c^3}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} d e \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{2 c^3}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} d e \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{6 c^3}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} d^2 \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{c}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} d^2 \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{c}+d^2 x \sqrt{a+b \sin ^{-1}(c x)}+\frac{2}{3} d e x^3 \sqrt{a+b \sin ^{-1}(c x)}+\frac{1}{5} e^2 x^5 \sqrt{a+b \sin ^{-1}(c x)}",1,"(b*(450*(8*c^4*d^2 + 4*c^2*d*e + e^2)*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 450*(8*c^4*d^2 + 4*c^2*d*e + e^2)*E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b] - e*(25*Sqrt[3]*(8*c^2*d + 3*e)*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + 25*Sqrt[3]*(8*c^2*d + 3*e)*E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcSin[c*x]))/b] - 9*Sqrt[5]*e*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-5*I)*(a + b*ArcSin[c*x]))/b] + E^(((10*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((5*I)*(a + b*ArcSin[c*x]))/b]))))/(7200*c^5*E^(((5*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])","C",0
687,1,244,369,0.6733452,"\int \left(d+e x^2\right) \sqrt{a+b \sin ^{-1}(c x)} \, dx","Integrate[(d + e*x^2)*Sqrt[a + b*ArcSin[c*x]],x]","\frac{b e^{-\frac{3 i a}{b}} \left(9 e^{\frac{2 i a}{b}} \left(4 c^2 d+e\right) \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+9 e^{\frac{4 i a}{b}} \left(4 c^2 d+e\right) \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{3} e \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+e^{\frac{6 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)\right)}{72 c^3 \sqrt{a+b \sin ^{-1}(c x)}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c^3}-\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} e \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{12 c^3}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c^3}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{b} e \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{12 c^3}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} d \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{c}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} d \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{c}+d x \sqrt{a+b \sin ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sin ^{-1}(c x)}",1,"(b*(9*(4*c^2*d + e)*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 9*(4*c^2*d + e)*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*e*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcSin[c*x]))/b])))/(72*c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])","C",0
688,1,119,120,0.1388373,"\int \sqrt{a+b \sin ^{-1}(c x)} \, dx","Integrate[Sqrt[a + b*ArcSin[c*x]],x]","\frac{b e^{-\frac{i a}{b}} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)}{2 c \sqrt{a+b \sin ^{-1}(c 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 x)}}{\sqrt{b}}\right)}{c}-\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 x)}}{\sqrt{b}}\right)}{c}+x \sqrt{a+b \sin ^{-1}(c x)}",1,"(b*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b]))/(2*c*E^((I*a)/b)*Sqrt[a + b*ArcSin[c*x]])","C",0
689,0,0,25,10.9265792,"\int \frac{\sqrt{a+b \sin ^{-1}(c x)}}{d+e x^2} \, dx","Integrate[Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2),x]","\int \frac{\sqrt{a+b \sin ^{-1}(c x)}}{d+e x^2} \, dx","\text{Int}\left(\frac{\sqrt{a+b \sin ^{-1}(c x)}}{d+e x^2},x\right)",0,"Integrate[Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2), x]","A",-1
690,0,0,25,23.4923754,"\int \frac{\sqrt{a+b \sin ^{-1}(c x)}}{\left(d+e x^2\right)^2} \, dx","Integrate[Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2)^2,x]","\int \frac{\sqrt{a+b \sin ^{-1}(c x)}}{\left(d+e x^2\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{a+b \sin ^{-1}(c x)}}{\left(d+e x^2\right)^2},x\right)",0,"Integrate[Sqrt[a + b*ArcSin[c*x]]/(d + e*x^2)^2, x]","A",-1
691,1,873,482,10.2766624,"\int \left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right)^{3/2} \, dx","Integrate[(d + e*x^2)*(a + b*ArcSin[c*x])^(3/2),x]","\frac{a b d e^{-\frac{i a}{b}} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)}{2 c \sqrt{a+b \sin ^{-1}(c x)}}+\frac{a b e e^{-\frac{3 i a}{b}} \left(9 e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+9 e^{\frac{4 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{3} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+e^{\frac{6 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)\right)}{72 c^3 \sqrt{a+b \sin ^{-1}(c x)}}+\frac{b d \left(2 \sqrt{a+b \sin ^{-1}(c x)} \left(2 c x \sin ^{-1}(c x)+3 \sqrt{1-c^2 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 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 x)}\right) \left(2 a \cos \left(\frac{a}{b}\right)-3 b \sin \left(\frac{a}{b}\right)\right)\right)}{4 c}+\frac{b e \left(18 \sqrt{a+b \sin ^{-1}(c x)} \left(2 c x \sin ^{-1}(c x)+3 \sqrt{1-c^2 x^2}\right)-9 \sqrt{\frac{1}{b}} \sqrt{2 \pi } C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}\right) \left(3 b \cos \left(\frac{a}{b}\right)+2 a \sin \left(\frac{a}{b}\right)\right)+9 \sqrt{\frac{1}{b}} \sqrt{2 \pi } S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}\right) \left(2 a \cos \left(\frac{a}{b}\right)-3 b \sin \left(\frac{a}{b}\right)\right)+\sqrt{\frac{1}{b}} \sqrt{6 \pi } C\left(\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}\right) \left(b \cos \left(\frac{3 a}{b}\right)+2 a \sin \left(\frac{3 a}{b}\right)\right)+\sqrt{\frac{1}{b}} \sqrt{6 \pi } S\left(\sqrt{\frac{1}{b}} \sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}\right) \left(b \sin \left(\frac{3 a}{b}\right)-2 a \cos \left(\frac{3 a}{b}\right)\right)-6 \sqrt{a+b \sin ^{-1}(c x)} \left(\cos \left(3 \sin ^{-1}(c x)\right)+2 \sin ^{-1}(c x) \sin \left(3 \sin ^{-1}(c x)\right)\right)\right)}{144 c^3}","-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{8 c^3}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} e \cos \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{24 c^3}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{8 c^3}+\frac{\sqrt{\frac{\pi }{6}} b^{3/2} e \sin \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{24 c^3}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} d \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{2 c}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} d \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{2 c}+\frac{3 b d \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{2 c}+\frac{b e x^2 \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{6 c}+\frac{b e \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{3 c^3}+d x \left(a+b \sin ^{-1}(c x)\right)^{3/2}+\frac{1}{3} e x^3 \left(a+b \sin ^{-1}(c x)\right)^{3/2}",1,"(a*b*d*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b]))/(2*c*E^((I*a)/b)*Sqrt[a + b*ArcSin[c*x]]) + (a*b*e*(9*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 9*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] + E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((3*I)*(a + b*ArcSin[c*x]))/b])))/(72*c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]]) + (b*d*(2*Sqrt[a + b*ArcSin[c*x]]*(3*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x]) - Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*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*x]]]*(2*a*Cos[a/b] - 3*b*Sin[a/b])))/(4*c) + (b*e*(18*Sqrt[a + b*ArcSin[c*x]]*(3*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x]) - 9*Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) + 9*Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]]]*(2*a*Cos[a/b] - 3*b*Sin[a/b]) + Sqrt[b^(-1)]*Sqrt[6*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]]]*(b*Cos[(3*a)/b] + 2*a*Sin[(3*a)/b]) + Sqrt[b^(-1)]*Sqrt[6*Pi]*FresnelS[Sqrt[b^(-1)]*Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c*x]]]*(-2*a*Cos[(3*a)/b] + b*Sin[(3*a)/b]) - 6*Sqrt[a + b*ArcSin[c*x]]*(Cos[3*ArcSin[c*x]] + 2*ArcSin[c*x]*Sin[3*ArcSin[c*x]])))/(144*c^3)","C",0
692,1,291,159,2.9196367,"\int \left(a+b \sin ^{-1}(c x)\right)^{3/2} \, dx","Integrate[(a + b*ArcSin[c*x])^(3/2),x]","\frac{b \left(2 \left(3 \sqrt{1-c^2 x^2}+2 c x \sin ^{-1}(c x)\right) \sqrt{a+b \sin ^{-1}(c x)}-\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 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 x)}\right)+\frac{2 a e^{-\frac{i a}{b}} \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{3}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)}{\sqrt{a+b \sin ^{-1}(c x)}}\right)}{4 c}","-\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 x)}}{\sqrt{b}}\right)}{2 c}-\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 x)}}{\sqrt{b}}\right)}{2 c}+\frac{3 b \sqrt{1-c^2 x^2} \sqrt{a+b \sin ^{-1}(c x)}}{2 c}+x \left(a+b \sin ^{-1}(c x)\right)^{3/2}",1,"(b*(2*Sqrt[a + b*ArcSin[c*x]]*(3*Sqrt[1 - c^2*x^2] + 2*c*x*ArcSin[c*x]) + (2*a*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b]))/(E^((I*a)/b)*Sqrt[a + b*ArcSin[c*x]]) - Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sqrt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*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*x]]]*(2*a*Cos[a/b] - 3*b*Sin[a/b])))/(4*c)","C",0
693,0,0,25,3.8726888,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^{3/2}}{d+e x^2} \, dx","Integrate[(a + b*ArcSin[c*x])^(3/2)/(d + e*x^2),x]","\int \frac{\left(a+b \sin ^{-1}(c x)\right)^{3/2}}{d+e x^2} \, dx","\text{Int}\left(\frac{\left(a+b \sin ^{-1}(c x)\right)^{3/2}}{d+e x^2},x\right)",0,"Integrate[(a + b*ArcSin[c*x])^(3/2)/(d + e*x^2), x]","A",-1
694,0,0,25,12.7739245,"\int \frac{\left(a+b \sin ^{-1}(c x)\right)^{3/2}}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcSin[c*x])^(3/2)/(d + e*x^2)^2,x]","\int \frac{\left(a+b \sin ^{-1}(c x)\right)^{3/2}}{\left(d+e x^2\right)^2} \, dx","\text{Int}\left(\frac{\left(a+b \sin ^{-1}(c x)\right)^{3/2}}{\left(d+e x^2\right)^2},x\right)",0,"Integrate[(a + b*ArcSin[c*x])^(3/2)/(d + e*x^2)^2, x]","A",-1
695,1,401,679,1.7547082,"\int \frac{\left(d+e x^2\right)^2}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx","Integrate[(d + e*x^2)^2/Sqrt[a + b*ArcSin[c*x]],x]","\frac{i e^{-\frac{5 i a}{b}} \left(e \left(5 \sqrt{3} e^{\frac{2 i a}{b}} \left(8 c^2 d+3 e\right) \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-5 \sqrt{3} e^{\frac{8 i a}{b}} \left(8 c^2 d+3 e\right) \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-3 \sqrt{5} e \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-e^{\frac{10 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{5 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)\right)-30 e^{\frac{4 i a}{b}} \left(8 c^4 d^2+4 c^2 d e+e^2\right) \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+30 e^{\frac{6 i a}{b}} \left(8 c^4 d^2+4 c^2 d e+e^2\right) \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)}{480 c^5 \sqrt{a+b \sin ^{-1}(c 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 x)}}{\sqrt{b}}\right)}{4 \sqrt{b} c^5}-\frac{\sqrt{\frac{3 \pi }{2}} e^2 \cos \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{8 \sqrt{b} c^5}+\frac{\sqrt{\frac{\pi }{10}} e^2 \cos \left(\frac{5 a}{b}\right) C\left(\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{8 \sqrt{b} c^5}+\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 x)}}{\sqrt{b}}\right)}{4 \sqrt{b} c^5}-\frac{\sqrt{\frac{3 \pi }{2}} e^2 \sin \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{8 \sqrt{b} c^5}+\frac{\sqrt{\frac{\pi }{10}} e^2 \sin \left(\frac{5 a}{b}\right) S\left(\frac{\sqrt{\frac{10}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{8 \sqrt{b} c^5}+\frac{\sqrt{\frac{\pi }{2}} d e \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{\sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{6}} d e \cos \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{\sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{2}} d e \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{\sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{6}} d e \sin \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{\sqrt{b} c^3}+\frac{\sqrt{2 \pi } d^2 \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{\sqrt{b} c}+\frac{\sqrt{2 \pi } d^2 \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{\sqrt{b} c}",1,"((I/480)*(-30*(8*c^4*d^2 + 4*c^2*d*e + e^2)*E^(((4*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b] + 30*(8*c^4*d^2 + 4*c^2*d*e + e^2)*E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b] + e*(5*Sqrt[3]*(8*c^2*d + 3*e)*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] - 5*Sqrt[3]*(8*c^2*d + 3*e)*E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c*x]))/b] - 3*Sqrt[5]*e*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-5*I)*(a + b*ArcSin[c*x]))/b] - E^(((10*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((5*I)*(a + b*ArcSin[c*x]))/b]))))/(c^5*E^(((5*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])","C",0
696,1,246,329,0.6822209,"\int \frac{d+e x^2}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx","Integrate[(d + e*x^2)/Sqrt[a + b*ArcSin[c*x]],x]","-\frac{i e^{-\frac{3 i a}{b}} \left(3 e^{\frac{2 i a}{b}} \left(4 c^2 d+e\right) \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-3 e^{\frac{4 i a}{b}} \left(4 c^2 d+e\right) \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{3} e \left(\sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-e^{\frac{6 i a}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)\right)}{24 c^3 \sqrt{a+b \sin ^{-1}(c x)}}","\frac{\sqrt{\frac{\pi }{2}} e \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{6}} e \cos \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{2}} e \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{6}} e \sin \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c^3}+\frac{\sqrt{2 \pi } d \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{\sqrt{b} c}+\frac{\sqrt{2 \pi } d \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{\sqrt{b} c}",1,"((-1/24*I)*(3*(4*c^2*d + e)*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b] - 3*(4*c^2*d + e)*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*e*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] - E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c*x]))/b])))/(c^3*E^(((3*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])","C",0
697,1,121,101,0.1497999,"\int \frac{1}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx","Integrate[1/Sqrt[a + b*ArcSin[c*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 x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)\right)}{2 c \sqrt{a+b \sin ^{-1}(c x)}}","\frac{\sqrt{2 \pi } \cos \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{\sqrt{b} c}+\frac{\sqrt{2 \pi } \sin \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{\sqrt{b} c}",1,"((I/2)*(-(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b]) + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b]))/(c*E^((I*a)/b)*Sqrt[a + b*ArcSin[c*x]])","C",0
698,0,0,25,0.195489,"\int \frac{1}{\left(d+e x^2\right) \sqrt{a+b \sin ^{-1}(c x)}} \, dx","Integrate[1/((d + e*x^2)*Sqrt[a + b*ArcSin[c*x]]),x]","\int \frac{1}{\left(d+e x^2\right) \sqrt{a+b \sin ^{-1}(c x)}} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right) \sqrt{a+b \sin ^{-1}(c x)}},x\right)",0,"Integrate[1/((d + e*x^2)*Sqrt[a + b*ArcSin[c*x]]), x]","A",-1
699,0,0,25,0.3571948,"\int \frac{1}{\left(d+e x^2\right)^2 \sqrt{a+b \sin ^{-1}(c x)}} \, dx","Integrate[1/((d + e*x^2)^2*Sqrt[a + b*ArcSin[c*x]]),x]","\int \frac{1}{\left(d+e x^2\right)^2 \sqrt{a+b \sin ^{-1}(c x)}} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^2 \sqrt{a+b \sin ^{-1}(c x)}},x\right)",0,"Integrate[1/((d + e*x^2)^2*Sqrt[a + b*ArcSin[c*x]]), x]","A",-1
700,1,417,394,1.3205329,"\int \frac{d+e x^2}{\left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d + e*x^2)/(a + b*ArcSin[c*x])^(3/2),x]","\frac{e^{-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}} \left(\left(4 c^2 d+e\right) e^{\frac{2 i a}{b}+3 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+\left(4 c^2 d+e\right) e^{\frac{4 i a}{b}+3 i \sin ^{-1}(c x)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-4 c^2 d e^{\frac{3 i a}{b}+2 i \sin ^{-1}(c x)}-4 c^2 d e^{\frac{3 i a}{b}+4 i \sin ^{-1}(c x)}-e e^{\frac{3 i a}{b}+2 i \sin ^{-1}(c x)}-e e^{\frac{3 i a}{b}+4 i \sin ^{-1}(c x)}+e e^{\frac{3 i \left(a+2 b \sin ^{-1}(c x)\right)}{b}}-\sqrt{3} e e^{3 i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-\sqrt{3} e e^{3 i \left(\frac{2 a}{b}+\sin ^{-1}(c x)\right)} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{3 i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+e e^{\frac{3 i a}{b}}\right)}{4 b c^3 \sqrt{a+b \sin ^{-1}(c x)}}","\frac{\sqrt{\frac{\pi }{2}} e \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^3}-\frac{\sqrt{\frac{3 \pi }{2}} e \sin \left(\frac{3 a}{b}\right) C\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^3}-\frac{\sqrt{\frac{\pi }{2}} e \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^3}+\frac{\sqrt{\frac{3 \pi }{2}} e \cos \left(\frac{3 a}{b}\right) S\left(\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c^3}+\frac{2 \sqrt{2 \pi } d \sin \left(\frac{a}{b}\right) C\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}-\frac{2 \sqrt{2 \pi } d \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}-\frac{2 d \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}-\frac{2 e x^2 \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}",1,"(e*E^(((3*I)*a)/b) - 4*c^2*d*E^(((3*I)*a)/b + (2*I)*ArcSin[c*x]) - e*E^(((3*I)*a)/b + (2*I)*ArcSin[c*x]) - 4*c^2*d*E^(((3*I)*a)/b + (4*I)*ArcSin[c*x]) - e*E^(((3*I)*a)/b + (4*I)*ArcSin[c*x]) + e*E^(((3*I)*(a + 2*b*ArcSin[c*x]))/b) + (4*c^2*d + e)*E^(((2*I)*a)/b + (3*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b] + (4*c^2*d + e)*E^(((4*I)*a)/b + (3*I)*ArcSin[c*x])*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*e*E^((3*I)*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-3*I)*(a + b*ArcSin[c*x]))/b] - Sqrt[3]*e*E^((3*I)*((2*a)/b + ArcSin[c*x]))*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((3*I)*(a + b*ArcSin[c*x]))/b])/(4*b*c^3*E^(((3*I)*(a + b*ArcSin[c*x]))/b)*Sqrt[a + b*ArcSin[c*x]])","C",0
701,1,167,137,0.3666852,"\int \frac{1}{\left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(a + b*ArcSin[c*x])^(-3/2),x]","\frac{e^{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \left(e^{i \sin ^{-1}(c x)} \sqrt{-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},-\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)+e^{\frac{i a}{b}} \left(e^{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \sqrt{\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}} \Gamma \left(\frac{1}{2},\frac{i \left(a+b \sin ^{-1}(c x)\right)}{b}\right)-e^{2 i \sin ^{-1}(c x)}-1\right)\right)}{b c \sqrt{a+b \sin ^{-1}(c 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 x)}}{\sqrt{b}}\right)}{b^{3/2} c}-\frac{2 \sqrt{2 \pi } \cos \left(\frac{a}{b}\right) S\left(\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}-\frac{2 \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \sin ^{-1}(c x)}}",1,"(E^(I*ArcSin[c*x])*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-I)*(a + b*ArcSin[c*x]))/b] + E^((I*a)/b)*(-1 - E^((2*I)*ArcSin[c*x]) + E^((I*(a + b*ArcSin[c*x]))/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, (I*(a + b*ArcSin[c*x]))/b]))/(b*c*E^((I*(a + b*ArcSin[c*x]))/b)*Sqrt[a + b*ArcSin[c*x]])","C",0
702,0,0,25,0.2016021,"\int \frac{1}{\left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[1/((d + e*x^2)*(a + b*ArcSin[c*x])^(3/2)),x]","\int \frac{1}{\left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right) \left(a+b \sin ^{-1}(c x)\right)^{3/2}},x\right)",0,"Integrate[1/((d + e*x^2)*(a + b*ArcSin[c*x])^(3/2)), x]","A",-1
703,0,0,25,0.3433559,"\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[1/((d + e*x^2)^2*(a + b*ArcSin[c*x])^(3/2)),x]","\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^{3/2}} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^2 \left(a+b \sin ^{-1}(c x)\right)^{3/2}},x\right)",0,"Integrate[1/((d + e*x^2)^2*(a + b*ArcSin[c*x])^(3/2)), x]","A",-1