1,1,107,114,0.1425564,"\int x^6 \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^6*(a + b*ArcSec[c*x]),x]","\frac{a x^7}{7}-\frac{5 b \log \left(x \left(\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right)\right)}{112 c^7}+b \sqrt{\frac{c^2 x^2-1}{c^2 x^2}} \left(-\frac{5 x^2}{112 c^5}-\frac{5 x^4}{168 c^3}-\frac{x^6}{42 c}\right)+\frac{1}{7} b x^7 \sec ^{-1}(c x)","\frac{1}{7} x^7 \left(a+b \sec ^{-1}(c x)\right)-\frac{b x^6 \sqrt{1-\frac{1}{c^2 x^2}}}{42 c}-\frac{5 b \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{112 c^7}-\frac{5 b x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{112 c^5}-\frac{5 b x^4 \sqrt{1-\frac{1}{c^2 x^2}}}{168 c^3}",1,"(a*x^7)/7 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*((-5*x^2)/(112*c^5) - (5*x^4)/(168*c^3) - x^6/(42*c)) + (b*x^7*ArcSec[c*x])/7 - (5*b*Log[x*(1 + Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])])/(112*c^7)","A",1
2,1,72,89,0.0905794,"\int x^5 \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^5*(a + b*ArcSec[c*x]),x]","\frac{a x^6}{6}+b \sqrt{\frac{c^2 x^2-1}{c^2 x^2}} \left(-\frac{4 x}{45 c^5}-\frac{2 x^3}{45 c^3}-\frac{x^5}{30 c}\right)+\frac{1}{6} b x^6 \sec ^{-1}(c x)","\frac{1}{6} x^6 \left(a+b \sec ^{-1}(c x)\right)-\frac{b x^5 \sqrt{1-\frac{1}{c^2 x^2}}}{30 c}-\frac{4 b x \sqrt{1-\frac{1}{c^2 x^2}}}{45 c^5}-\frac{2 b x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{45 c^3}",1,"(a*x^6)/6 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*((-4*x)/(45*c^5) - (2*x^3)/(45*c^3) - x^5/(30*c)) + (b*x^6*ArcSec[c*x])/6","A",1
3,1,97,89,0.0759469,"\int x^4 \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^4*(a + b*ArcSec[c*x]),x]","\frac{a x^5}{5}-\frac{3 b \log \left(x \left(\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right)\right)}{40 c^5}+b \sqrt{\frac{c^2 x^2-1}{c^2 x^2}} \left(-\frac{3 x^2}{40 c^3}-\frac{x^4}{20 c}\right)+\frac{1}{5} b x^5 \sec ^{-1}(c x)","\frac{1}{5} x^5 \left(a+b \sec ^{-1}(c x)\right)-\frac{b x^4 \sqrt{1-\frac{1}{c^2 x^2}}}{20 c}-\frac{3 b \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{40 c^5}-\frac{3 b x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{40 c^3}",1,"(a*x^5)/5 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*((-3*x^2)/(40*c^3) - x^4/(20*c)) + (b*x^5*ArcSec[c*x])/5 - (3*b*Log[x*(1 + Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])])/(40*c^5)","A",1
4,1,62,64,0.1056923,"\int x^3 \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^3*(a + b*ArcSec[c*x]),x]","\frac{a x^4}{4}+b \sqrt{\frac{c^2 x^2-1}{c^2 x^2}} \left(-\frac{x}{6 c^3}-\frac{x^3}{12 c}\right)+\frac{1}{4} b x^4 \sec ^{-1}(c x)","\frac{1}{4} x^4 \left(a+b \sec ^{-1}(c x)\right)-\frac{b x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{12 c}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}}}{6 c^3}",1,"(a*x^4)/4 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*(-1/6*x/c^3 - x^3/(12*c)) + (b*x^4*ArcSec[c*x])/4","A",1
5,1,85,64,0.054135,"\int x^2 \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^2*(a + b*ArcSec[c*x]),x]","\frac{a x^3}{3}-\frac{b x^2 \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{6 c}-\frac{b \log \left(x \left(\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right)\right)}{6 c^3}+\frac{1}{3} b x^3 \sec ^{-1}(c x)","\frac{1}{3} x^3 \left(a+b \sec ^{-1}(c x)\right)-\frac{b x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{6 c}-\frac{b \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{6 c^3}",1,"(a*x^3)/3 - (b*x^2*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(6*c) + (b*x^3*ArcSec[c*x])/3 - (b*Log[x*(1 + Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])])/(6*c^3)","A",1
6,1,50,39,0.0227672,"\int x \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x*(a + b*ArcSec[c*x]),x]","\frac{a x^2}{2}-\frac{b x \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{2 c}+\frac{1}{2} b x^2 \sec ^{-1}(c x)","\frac{1}{2} x^2 \left(a+b \sec ^{-1}(c x)\right)-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}}}{2 c}",1,"(a*x^2)/2 - (b*x*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(2*c) + (b*x^2*ArcSec[c*x])/2","A",1
7,1,59,32,0.0678565,"\int \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[a + b*ArcSec[c*x],x]","a x-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{\sqrt{c^2 x^2-1}}+b x \sec ^{-1}(c x)","a x-\frac{b \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{c}+b x \sec ^{-1}(c x)",1,"a*x + b*x*ArcSec[c*x] - (b*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[-1 + c^2*x^2]","A",1
8,1,59,64,0.0174731,"\int \frac{a+b \sec ^{-1}(c x)}{x} \, dx","Integrate[(a + b*ArcSec[c*x])/x,x]","a \log (x)+\frac{1}{2} i b \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)+\frac{1}{2} i b \sec ^{-1}(c x)^2-b \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)","\frac{i \left(a+b \sec ^{-1}(c x)\right)^2}{2 b}-\log \left(1+e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)+\frac{1}{2} i b \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)",1,"(I/2)*b*ArcSec[c*x]^2 - b*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] + a*Log[x] + (I/2)*b*PolyLog[2, -E^((2*I)*ArcSec[c*x])]","A",0
9,1,40,31,0.030987,"\int \frac{a+b \sec ^{-1}(c x)}{x^2} \, dx","Integrate[(a + b*ArcSec[c*x])/x^2,x]","-\frac{a}{x}+b c \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{b \sec ^{-1}(c x)}{x}","b c \sqrt{1-\frac{1}{c^2 x^2}}-\frac{a+b \sec ^{-1}(c x)}{x}",1,"-(a/x) + b*c*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*ArcSec[c*x])/x","A",1
10,1,66,51,0.035017,"\int \frac{a+b \sec ^{-1}(c x)}{x^3} \, dx","Integrate[(a + b*ArcSec[c*x])/x^3,x]","-\frac{a}{2 x^2}+\frac{b c \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{4 x}-\frac{1}{4} b c^2 \sin ^{-1}\left(\frac{1}{c x}\right)-\frac{b \sec ^{-1}(c x)}{2 x^2}","-\frac{a+b \sec ^{-1}(c x)}{2 x^2}+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}-\frac{1}{4} b c^2 \csc ^{-1}(c x)",1,"-1/2*a/x^2 + (b*c*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(4*x) - (b*ArcSec[c*x])/(2*x^2) - (b*c^2*ArcSin[1/(c*x)])/4","A",1
11,1,59,60,0.0480101,"\int \frac{a+b \sec ^{-1}(c x)}{x^4} \, dx","Integrate[(a + b*ArcSec[c*x])/x^4,x]","-\frac{a}{3 x^3}+b \left(\frac{2 c^3}{9}+\frac{c}{9 x^2}\right) \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{b \sec ^{-1}(c x)}{3 x^3}","-\frac{a+b \sec ^{-1}(c x)}{3 x^3}-\frac{1}{9} b c^3 \left(1-\frac{1}{c^2 x^2}\right)^{3/2}+\frac{1}{3} b c^3 \sqrt{1-\frac{1}{c^2 x^2}}",1,"-1/3*a/x^3 + b*((2*c^3)/9 + c/(9*x^2))*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*ArcSec[c*x])/(3*x^3)","A",1
12,1,78,76,0.0617215,"\int \frac{a+b \sec ^{-1}(c x)}{x^5} \, dx","Integrate[(a + b*ArcSec[c*x])/x^5,x]","-\frac{a}{4 x^4}-\frac{3}{32} b c^4 \sin ^{-1}\left(\frac{1}{c x}\right)+b \left(\frac{3 c^3}{32 x}+\frac{c}{16 x^3}\right) \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{b \sec ^{-1}(c x)}{4 x^4}","-\frac{a+b \sec ^{-1}(c x)}{4 x^4}-\frac{3}{32} b c^4 \csc ^{-1}(c x)+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{16 x^3}+\frac{3 b c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{32 x}",1,"-1/4*a/x^4 + b*(c/(16*x^3) + (3*c^3)/(32*x))*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*ArcSec[c*x])/(4*x^4) - (3*b*c^4*ArcSin[1/(c*x)])/32","A",1
13,1,69,82,0.0717149,"\int \frac{a+b \sec ^{-1}(c x)}{x^6} \, dx","Integrate[(a + b*ArcSec[c*x])/x^6,x]","-\frac{a}{5 x^5}+b \left(\frac{8 c^5}{75}+\frac{4 c^3}{75 x^2}+\frac{c}{25 x^4}\right) \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{b \sec ^{-1}(c x)}{5 x^5}","-\frac{a+b \sec ^{-1}(c x)}{5 x^5}+\frac{1}{25} b c^5 \left(1-\frac{1}{c^2 x^2}\right)^{5/2}-\frac{2}{15} b c^5 \left(1-\frac{1}{c^2 x^2}\right)^{3/2}+\frac{1}{5} b c^5 \sqrt{1-\frac{1}{c^2 x^2}}",1,"-1/5*a/x^5 + b*((8*c^5)/75 + c/(25*x^4) + (4*c^3)/(75*x^2))*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*ArcSec[c*x])/(5*x^5)","A",1
14,1,88,101,0.0786305,"\int \frac{a+b \sec ^{-1}(c x)}{x^7} \, dx","Integrate[(a + b*ArcSec[c*x])/x^7,x]","-\frac{a}{6 x^6}-\frac{5}{96} b c^6 \sin ^{-1}\left(\frac{1}{c x}\right)+b \left(\frac{5 c^5}{96 x}+\frac{5 c^3}{144 x^3}+\frac{c}{36 x^5}\right) \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{b \sec ^{-1}(c x)}{6 x^6}","-\frac{a+b \sec ^{-1}(c x)}{6 x^6}-\frac{5}{96} b c^6 \csc ^{-1}(c x)+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{36 x^5}+\frac{5 b c^5 \sqrt{1-\frac{1}{c^2 x^2}}}{96 x}+\frac{5 b c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{144 x^3}",1,"-1/6*a/x^6 + b*(c/(36*x^5) + (5*c^3)/(144*x^3) + (5*c^5)/(96*x))*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*ArcSec[c*x])/(6*x^6) - (5*b*c^6*ArcSin[1/(c*x)])/96","A",1
15,1,124,107,0.2239239,"\int x^3 \left(a+b \sec ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(a + b*ArcSec[c*x])^2,x]","\frac{c x \left(3 a^2 c^3 x^3-2 a b \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 x^2+2\right)+b^2 c x\right)-2 b c x \sec ^{-1}(c x) \left(b \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 x^2+2\right)-3 a c^3 x^3\right)+3 b^2 c^4 x^4 \sec ^{-1}(c x)^2+4 b^2 \log (x)}{12 c^4}","-\frac{b x^3 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)}{6 c}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)}{3 c^3}+\frac{1}{4} x^4 \left(a+b \sec ^{-1}(c x)\right)^2+\frac{b^2 \log (x)}{3 c^4}+\frac{b^2 x^2}{12 c^2}",1,"(c*x*(b^2*c*x + 3*a^2*c^3*x^3 - 2*a*b*Sqrt[1 - 1/(c^2*x^2)]*(2 + c^2*x^2)) - 2*b*c*x*(-3*a*c^3*x^3 + b*Sqrt[1 - 1/(c^2*x^2)]*(2 + c^2*x^2))*ArcSec[c*x] + 3*b^2*c^4*x^4*ArcSec[c*x]^2 + 4*b^2*Log[x])/(12*c^4)","A",1
16,1,225,147,1.2709834,"\int x^2 \left(a+b \sec ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(a + b*ArcSec[c*x])^2,x]","\frac{1}{3} \left(a^2 x^3+\frac{a b \left(2 x^4 \sec ^{-1}(c x)-\frac{c^3 x^3+\sqrt{c^2 x^2-1} \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)-c x}{c^4 \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{x}+\frac{b^2 \left(c^3 x^3 \sec ^{-1}(c x)^2-c^2 x^2 \sqrt{1-\frac{1}{c^2 x^2}} \sec ^{-1}(c x)-i \text{Li}_2\left(-i e^{i \sec ^{-1}(c x)}\right)+i \text{Li}_2\left(i e^{i \sec ^{-1}(c x)}\right)+c x-\sec ^{-1}(c x) \log \left(1-i e^{i \sec ^{-1}(c x)}\right)+\sec ^{-1}(c x) \log \left(1+i e^{i \sec ^{-1}(c x)}\right)\right)}{c^3}\right)","\frac{2 i b \tan ^{-1}\left(e^{i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)}{3 c^3}-\frac{b x^2 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)}{3 c}+\frac{1}{3} x^3 \left(a+b \sec ^{-1}(c x)\right)^2-\frac{i b^2 \text{Li}_2\left(-i e^{i \sec ^{-1}(c x)}\right)}{3 c^3}+\frac{i b^2 \text{Li}_2\left(i e^{i \sec ^{-1}(c x)}\right)}{3 c^3}+\frac{b^2 x}{3 c^2}",1,"(a^2*x^3 + (a*b*(2*x^4*ArcSec[c*x] - (-(c*x) + c^3*x^3 + Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(c^4*Sqrt[1 - 1/(c^2*x^2)])))/x + (b^2*(c*x - c^2*Sqrt[1 - 1/(c^2*x^2)]*x^2*ArcSec[c*x] + c^3*x^3*ArcSec[c*x]^2 - ArcSec[c*x]*Log[1 - I*E^(I*ArcSec[c*x])] + ArcSec[c*x]*Log[1 + I*E^(I*ArcSec[c*x])] - I*PolyLog[2, (-I)*E^(I*ArcSec[c*x])] + I*PolyLog[2, I*E^(I*ArcSec[c*x])]))/c^3)/3","A",0
17,1,90,56,0.151275,"\int x \left(a+b \sec ^{-1}(c x)\right)^2 \, dx","Integrate[x*(a + b*ArcSec[c*x])^2,x]","\frac{a c x \left(a c x-2 b \sqrt{1-\frac{1}{c^2 x^2}}\right)+2 b c x \sec ^{-1}(c x) \left(a c x-b \sqrt{1-\frac{1}{c^2 x^2}}\right)+b^2 c^2 x^2 \sec ^{-1}(c x)^2+2 b^2 \log (c x)}{2 c^2}","-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)}{c}+\frac{1}{2} x^2 \left(a+b \sec ^{-1}(c x)\right)^2+\frac{b^2 \log (x)}{c^2}",1,"(a*c*x*(-2*b*Sqrt[1 - 1/(c^2*x^2)] + a*c*x) + 2*b*c*x*(-(b*Sqrt[1 - 1/(c^2*x^2)]) + a*c*x)*ArcSec[c*x] + b^2*c^2*x^2*ArcSec[c*x]^2 + 2*b^2*Log[c*x])/(2*c^2)","A",1
18,1,163,92,0.1816105,"\int \left(a+b \sec ^{-1}(c x)\right)^2 \, dx","Integrate[(a + b*ArcSec[c*x])^2,x]","\frac{a^2 c x+2 a b \left(c x \sec ^{-1}(c x)+\log \left(\cos \left(\frac{1}{2} \sec ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sec ^{-1}(c x)\right)\right)-\log \left(\cos \left(\frac{1}{2} \sec ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sec ^{-1}(c x)\right)\right)\right)+b^2 \left(-2 i \text{Li}_2\left(-i e^{i \sec ^{-1}(c x)}\right)+2 i \text{Li}_2\left(i e^{i \sec ^{-1}(c x)}\right)+\sec ^{-1}(c x) \left(c x \sec ^{-1}(c x)-2 \log \left(1-i e^{i \sec ^{-1}(c x)}\right)+2 \log \left(1+i e^{i \sec ^{-1}(c x)}\right)\right)\right)}{c}","x \left(a+b \sec ^{-1}(c x)\right)^2+\frac{4 i b \tan ^{-1}\left(e^{i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)}{c}-\frac{2 i b^2 \text{Li}_2\left(-i e^{i \sec ^{-1}(c x)}\right)}{c}+\frac{2 i b^2 \text{Li}_2\left(i e^{i \sec ^{-1}(c x)}\right)}{c}",1,"(a^2*c*x + 2*a*b*(c*x*ArcSec[c*x] + Log[Cos[ArcSec[c*x]/2] - Sin[ArcSec[c*x]/2]] - Log[Cos[ArcSec[c*x]/2] + Sin[ArcSec[c*x]/2]]) + b^2*(ArcSec[c*x]*(c*x*ArcSec[c*x] - 2*Log[1 - I*E^(I*ArcSec[c*x])] + 2*Log[1 + I*E^(I*ArcSec[c*x])]) - (2*I)*PolyLog[2, (-I)*E^(I*ArcSec[c*x])] + (2*I)*PolyLog[2, I*E^(I*ArcSec[c*x])]))/c","A",1
19,1,129,93,0.1299067,"\int \frac{\left(a+b \sec ^{-1}(c x)\right)^2}{x} \, dx","Integrate[(a + b*ArcSec[c*x])^2/x,x]","a^2 \log (c x)+i b \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)+i a b \sec ^{-1}(c x)^2-2 a b \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-\frac{1}{2} b^2 \text{Li}_3\left(-e^{2 i \sec ^{-1}(c x)}\right)+\frac{1}{3} i b^2 \sec ^{-1}(c x)^3-b^2 \sec ^{-1}(c x)^2 \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)","i b \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)+\frac{i \left(a+b \sec ^{-1}(c x)\right)^3}{3 b}-\log \left(1+e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)^2-\frac{1}{2} b^2 \text{Li}_3\left(-e^{2 i \sec ^{-1}(c x)}\right)",1,"I*a*b*ArcSec[c*x]^2 + (I/3)*b^2*ArcSec[c*x]^3 - 2*a*b*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - b^2*ArcSec[c*x]^2*Log[1 + E^((2*I)*ArcSec[c*x])] + a^2*Log[c*x] + I*b*(a + b*ArcSec[c*x])*PolyLog[2, -E^((2*I)*ArcSec[c*x])] - (b^2*PolyLog[3, -E^((2*I)*ArcSec[c*x])])/2","A",0
20,1,75,50,0.1303582,"\int \frac{\left(a+b \sec ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[(a + b*ArcSec[c*x])^2/x^2,x]","\frac{-a^2+2 a b c x \sqrt{1-\frac{1}{c^2 x^2}}+2 b \sec ^{-1}(c x) \left(b c x \sqrt{1-\frac{1}{c^2 x^2}}-a\right)-b^2 \sec ^{-1}(c x)^2+2 b^2}{x}","2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)-\frac{\left(a+b \sec ^{-1}(c x)\right)^2}{x}+\frac{2 b^2}{x}",1,"(-a^2 + 2*b^2 + 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x + 2*b*(-a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x)*ArcSec[c*x] - b^2*ArcSec[c*x]^2)/x","A",1
21,1,102,94,0.1214484,"\int \frac{\left(a+b \sec ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[(a + b*ArcSec[c*x])^2/x^3,x]","\frac{-2 a^2+2 a b c x \sqrt{1-\frac{1}{c^2 x^2}}-2 a b c^2 x^2 \sin ^{-1}\left(\frac{1}{c x}\right)+2 b \sec ^{-1}(c x) \left(b c x \sqrt{1-\frac{1}{c^2 x^2}}-2 a\right)+b^2 \left(c^2 x^2-2\right) \sec ^{-1}(c x)^2+b^2}{4 x^2}","\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)}{2 x}+\frac{1}{2} \left(c^2-\frac{1}{x^2}\right) \left(a+b \sec ^{-1}(c x)\right)^2-\frac{1}{2} a b c^2 \sec ^{-1}(c x)-\frac{1}{4} b^2 c^2 \sec ^{-1}(c x)^2+\frac{b^2}{4 x^2}",1,"(-2*a^2 + b^2 + 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x + 2*b*(-2*a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x)*ArcSec[c*x] + b^2*(-2 + c^2*x^2)*ArcSec[c*x]^2 - 2*a*b*c^2*x^2*ArcSin[1/(c*x)])/(4*x^2)","A",1
22,1,108,102,0.196806,"\int \frac{\left(a+b \sec ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[(a + b*ArcSec[c*x])^2/x^4,x]","\frac{-9 a^2+6 a b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 x^2+1\right)+6 b \sec ^{-1}(c x) \left(b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 x^2+1\right)-3 a\right)+2 b^2 \left(6 c^2 x^2+1\right)-9 b^2 \sec ^{-1}(c x)^2}{27 x^3}","\frac{2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)}{9 x^2}+\frac{4}{9} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)-\frac{\left(a+b \sec ^{-1}(c x)\right)^2}{3 x^3}+\frac{4 b^2 c^2}{9 x}+\frac{2 b^2}{27 x^3}",1,"(-9*a^2 + 6*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2) + 2*b^2*(1 + 6*c^2*x^2) + 6*b*(-3*a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2))*ArcSec[c*x] - 9*b^2*ArcSec[c*x]^2)/(27*x^3)","A",1
23,1,148,134,0.1802655,"\int \frac{\left(a+b \sec ^{-1}(c x)\right)^2}{x^5} \, dx","Integrate[(a + b*ArcSec[c*x])^2/x^5,x]","\frac{-8 a^2-6 a b c^4 x^4 \sin ^{-1}\left(\frac{1}{c x}\right)+4 a b c x \sqrt{1-\frac{1}{c^2 x^2}}+2 b \sec ^{-1}(c x) \left(b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(3 c^2 x^2+2\right)-8 a\right)+6 a b c^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}+b^2 \left(3 c^4 x^4-8\right) \sec ^{-1}(c x)^2+3 b^2 c^2 x^2+b^2}{32 x^4}","\frac{3}{16} a b c^4 \sec ^{-1}(c x)+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)}{8 x^3}+\frac{3 b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)}{16 x}-\frac{\left(a+b \sec ^{-1}(c x)\right)^2}{4 x^4}+\frac{3}{32} b^2 c^4 \sec ^{-1}(c x)^2+\frac{3 b^2 c^2}{32 x^2}+\frac{b^2}{32 x^4}",1,"(-8*a^2 + b^2 + 4*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x + 3*b^2*c^2*x^2 + 6*a*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*x^3 + 2*b*(-8*a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(2 + 3*c^2*x^2))*ArcSec[c*x] + b^2*(-8 + 3*c^4*x^4)*ArcSec[c*x]^2 - 6*a*b*c^4*x^4*ArcSin[1/(c*x)])/(32*x^4)","A",1
24,1,288,207,0.8265694,"\int x^3 \left(a+b \sec ^{-1}(c x)\right)^3 \, dx","Integrate[x^3*(a + b*ArcSec[c*x])^3,x]","\frac{a^3 c^4 x^4+b \sec ^{-1}(c x) \left(c x \left(3 a^2 c^3 x^3-2 a b \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 x^2+2\right)+b^2 c x\right)-4 b^2 \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)\right)-2 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}}-a^2 b c^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}+a b^2 c^2 x^2-b^2 \sec ^{-1}(c x)^2 \left(-3 a c^4 x^4+b \left(2 c x \sqrt{1-\frac{1}{c^2 x^2}}+c^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}-2 i\right)\right)-4 a b^2 \log \left(\frac{1}{c x}\right)+b^3 c^4 x^4 \sec ^{-1}(c x)^3-b^3 c x \sqrt{1-\frac{1}{c^2 x^2}}+2 i b^3 \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)}{4 c^4}","-\frac{b^2 \log \left(1+e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)}{c^4}+\frac{b^2 x^2 \left(a+b \sec ^{-1}(c x)\right)}{4 c^2}+\frac{i b \left(a+b \sec ^{-1}(c x)\right)^2}{2 c^4}-\frac{b x^3 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)^2}{4 c}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)^2}{2 c^3}+\frac{1}{4} x^4 \left(a+b \sec ^{-1}(c x)\right)^3+\frac{i b^3 \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)}{2 c^4}-\frac{b^3 x \sqrt{1-\frac{1}{c^2 x^2}}}{4 c^3}",1,"(-2*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x - b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x + a*b^2*c^2*x^2 - a^2*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*x^3 + a^3*c^4*x^4 - b^2*(-3*a*c^4*x^4 + b*(-2*I + 2*c*Sqrt[1 - 1/(c^2*x^2)]*x + c^3*Sqrt[1 - 1/(c^2*x^2)]*x^3))*ArcSec[c*x]^2 + b^3*c^4*x^4*ArcSec[c*x]^3 + b*ArcSec[c*x]*(c*x*(b^2*c*x + 3*a^2*c^3*x^3 - 2*a*b*Sqrt[1 - 1/(c^2*x^2)]*(2 + c^2*x^2)) - 4*b^2*Log[1 + E^((2*I)*ArcSec[c*x])]) - 4*a*b^2*Log[1/(c*x)] + (2*I)*b^3*PolyLog[2, -E^((2*I)*ArcSec[c*x])])/(4*c^4)","A",0
25,1,403,236,1.4439792,"\int x^2 \left(a+b \sec ^{-1}(c x)\right)^3 \, dx","Integrate[x^2*(a + b*ArcSec[c*x])^3,x]","\frac{2 a^3 c^3 x^3+6 a^2 b c^3 x^3 \sec ^{-1}(c x)-3 a^2 b c^2 x^2 \sqrt{1-\frac{1}{c^2 x^2}}-3 a^2 b \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)+6 a b^2 c^3 x^3 \sec ^{-1}(c x)^2-6 a b^2 c^2 x^2 \sqrt{1-\frac{1}{c^2 x^2}} \sec ^{-1}(c x)-6 i b^2 \text{Li}_2\left(-i e^{i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)+6 i b^2 \text{Li}_2\left(i e^{i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)+6 a b^2 c x-6 a b^2 \sec ^{-1}(c x) \log \left(1-i e^{i \sec ^{-1}(c x)}\right)+6 a b^2 \sec ^{-1}(c x) \log \left(1+i e^{i \sec ^{-1}(c x)}\right)+2 b^3 c^3 x^3 \sec ^{-1}(c x)^3-3 b^3 c^2 x^2 \sqrt{1-\frac{1}{c^2 x^2}} \sec ^{-1}(c x)^2-6 b^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)+6 b^3 \text{Li}_3\left(-i e^{i \sec ^{-1}(c x)}\right)-6 b^3 \text{Li}_3\left(i e^{i \sec ^{-1}(c x)}\right)+6 b^3 c x \sec ^{-1}(c x)+6 i b^3 \sec ^{-1}(c x)^2 \tan ^{-1}\left(e^{i \sec ^{-1}(c x)}\right)}{6 c^3}","-\frac{i b^2 \text{Li}_2\left(-i e^{i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)}{c^3}+\frac{i b^2 \text{Li}_2\left(i e^{i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)}{c^3}+\frac{b^2 x \left(a+b \sec ^{-1}(c x)\right)}{c^2}+\frac{i b \tan ^{-1}\left(e^{i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)^2}{c^3}-\frac{b x^2 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)^2}{2 c}+\frac{1}{3} x^3 \left(a+b \sec ^{-1}(c x)\right)^3+\frac{b^3 \text{Li}_3\left(-i e^{i \sec ^{-1}(c x)}\right)}{c^3}-\frac{b^3 \text{Li}_3\left(i e^{i \sec ^{-1}(c x)}\right)}{c^3}-\frac{b^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{c^3}",1,"(6*a*b^2*c*x - 3*a^2*b*c^2*Sqrt[1 - 1/(c^2*x^2)]*x^2 + 2*a^3*c^3*x^3 + 6*b^3*c*x*ArcSec[c*x] - 6*a*b^2*c^2*Sqrt[1 - 1/(c^2*x^2)]*x^2*ArcSec[c*x] + 6*a^2*b*c^3*x^3*ArcSec[c*x] - 3*b^3*c^2*Sqrt[1 - 1/(c^2*x^2)]*x^2*ArcSec[c*x]^2 + 6*a*b^2*c^3*x^3*ArcSec[c*x]^2 + 2*b^3*c^3*x^3*ArcSec[c*x]^3 + (6*I)*b^3*ArcSec[c*x]^2*ArcTan[E^(I*ArcSec[c*x])] - 6*b^3*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]] - 6*a*b^2*ArcSec[c*x]*Log[1 - I*E^(I*ArcSec[c*x])] + 6*a*b^2*ArcSec[c*x]*Log[1 + I*E^(I*ArcSec[c*x])] - 3*a^2*b*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x] - (6*I)*b^2*(a + b*ArcSec[c*x])*PolyLog[2, (-I)*E^(I*ArcSec[c*x])] + (6*I)*b^2*(a + b*ArcSec[c*x])*PolyLog[2, I*E^(I*ArcSec[c*x])] + 6*b^3*PolyLog[3, (-I)*E^(I*ArcSec[c*x])] - 6*b^3*PolyLog[3, I*E^(I*ArcSec[c*x])])/(6*c^3)","A",0
26,1,184,126,0.495311,"\int x \left(a+b \sec ^{-1}(c x)\right)^3 \, dx","Integrate[x*(a + b*ArcSec[c*x])^3,x]","\frac{a \left(a c x \left(a c x-3 b \sqrt{1-\frac{1}{c^2 x^2}}\right)-6 b^2 \log \left(\frac{1}{c x}\right)\right)-3 b^2 \sec ^{-1}(c x)^2 \left(-a c^2 x^2+b \left(c x \sqrt{1-\frac{1}{c^2 x^2}}-i\right)\right)-3 b \sec ^{-1}(c x) \left(a c x \left(2 b \sqrt{1-\frac{1}{c^2 x^2}}-a c x\right)+2 b^2 \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)\right)+b^3 c^2 x^2 \sec ^{-1}(c x)^3+3 i b^3 \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)}{2 c^2}","-\frac{3 b^2 \log \left(1+e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)}{c^2}-\frac{3 b x \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)^2}{2 c}+\frac{3 i b \left(a+b \sec ^{-1}(c x)\right)^2}{2 c^2}+\frac{1}{2} x^2 \left(a+b \sec ^{-1}(c x)\right)^3+\frac{3 i b^3 \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)}{2 c^2}",1,"(-3*b^2*(-(a*c^2*x^2) + b*(-I + c*Sqrt[1 - 1/(c^2*x^2)]*x))*ArcSec[c*x]^2 + b^3*c^2*x^2*ArcSec[c*x]^3 - 3*b*ArcSec[c*x]*(a*c*x*(2*b*Sqrt[1 - 1/(c^2*x^2)] - a*c*x) + 2*b^2*Log[1 + E^((2*I)*ArcSec[c*x])]) + a*(a*c*x*(-3*b*Sqrt[1 - 1/(c^2*x^2)] + a*c*x) - 6*b^2*Log[1/(c*x)]) + (3*I)*b^3*PolyLog[2, -E^((2*I)*ArcSec[c*x])])/(2*c^2)","A",0
27,1,289,158,0.2457291,"\int \left(a+b \sec ^{-1}(c x)\right)^3 \, dx","Integrate[(a + b*ArcSec[c*x])^3,x]","\frac{a^3 c x-3 a^2 b \log \left(c x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)+3 a^2 b c x \sec ^{-1}(c x)-6 i b^2 \text{Li}_2\left(-i e^{i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)+6 i b^2 \text{Li}_2\left(i e^{i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)+3 a b^2 c x \sec ^{-1}(c x)^2-6 a b^2 \sec ^{-1}(c x) \log \left(1-i e^{i \sec ^{-1}(c x)}\right)+6 a b^2 \sec ^{-1}(c x) \log \left(1+i e^{i \sec ^{-1}(c x)}\right)+6 b^3 \text{Li}_3\left(-i e^{i \sec ^{-1}(c x)}\right)-6 b^3 \text{Li}_3\left(i e^{i \sec ^{-1}(c x)}\right)+b^3 c x \sec ^{-1}(c x)^3-3 b^3 \sec ^{-1}(c x)^2 \log \left(1-i e^{i \sec ^{-1}(c x)}\right)+3 b^3 \sec ^{-1}(c x)^2 \log \left(1+i e^{i \sec ^{-1}(c x)}\right)}{c}","-\frac{6 i b^2 \text{Li}_2\left(-i e^{i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)}{c}+\frac{6 i b^2 \text{Li}_2\left(i e^{i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)}{c}+x \left(a+b \sec ^{-1}(c x)\right)^3+\frac{6 i b \tan ^{-1}\left(e^{i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)^2}{c}+\frac{6 b^3 \text{Li}_3\left(-i e^{i \sec ^{-1}(c x)}\right)}{c}-\frac{6 b^3 \text{Li}_3\left(i e^{i \sec ^{-1}(c x)}\right)}{c}",1,"(a^3*c*x + 3*a^2*b*c*x*ArcSec[c*x] + 3*a*b^2*c*x*ArcSec[c*x]^2 + b^3*c*x*ArcSec[c*x]^3 - 6*a*b^2*ArcSec[c*x]*Log[1 - I*E^(I*ArcSec[c*x])] - 3*b^3*ArcSec[c*x]^2*Log[1 - I*E^(I*ArcSec[c*x])] + 6*a*b^2*ArcSec[c*x]*Log[1 + I*E^(I*ArcSec[c*x])] + 3*b^3*ArcSec[c*x]^2*Log[1 + I*E^(I*ArcSec[c*x])] - 3*a^2*b*Log[c*(1 + Sqrt[1 - 1/(c^2*x^2)])*x] - (6*I)*b^2*(a + b*ArcSec[c*x])*PolyLog[2, (-I)*E^(I*ArcSec[c*x])] + (6*I)*b^2*(a + b*ArcSec[c*x])*PolyLog[2, I*E^(I*ArcSec[c*x])] + 6*b^3*PolyLog[3, (-I)*E^(I*ArcSec[c*x])] - 6*b^3*PolyLog[3, I*E^(I*ArcSec[c*x])])/c","A",0
28,1,204,128,0.1936228,"\int \frac{\left(a+b \sec ^{-1}(c x)\right)^3}{x} \, dx","Integrate[(a + b*ArcSec[c*x])^3/x,x]","\frac{1}{4} \left(4 a^3 \log (c x)+6 i a^2 b \sec ^{-1}(c x)^2-12 a^2 b \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-6 b^2 \text{Li}_3\left(-e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)+4 i a b^2 \sec ^{-1}(c x)^3-12 a b^2 \sec ^{-1}(c x)^2 \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)+6 i b \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)^2-3 i b^3 \text{Li}_4\left(-e^{2 i \sec ^{-1}(c x)}\right)+i b^3 \sec ^{-1}(c x)^4-4 b^3 \sec ^{-1}(c x)^3 \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)\right)","-\frac{3}{2} b^2 \text{Li}_3\left(-e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)+\frac{3}{2} i b \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)^2+\frac{i \left(a+b \sec ^{-1}(c x)\right)^4}{4 b}-\log \left(1+e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)^3-\frac{3}{4} i b^3 \text{Li}_4\left(-e^{2 i \sec ^{-1}(c x)}\right)",1,"((6*I)*a^2*b*ArcSec[c*x]^2 + (4*I)*a*b^2*ArcSec[c*x]^3 + I*b^3*ArcSec[c*x]^4 - 12*a^2*b*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 12*a*b^2*ArcSec[c*x]^2*Log[1 + E^((2*I)*ArcSec[c*x])] - 4*b^3*ArcSec[c*x]^3*Log[1 + E^((2*I)*ArcSec[c*x])] + 4*a^3*Log[c*x] + (6*I)*b*(a + b*ArcSec[c*x])^2*PolyLog[2, -E^((2*I)*ArcSec[c*x])] - 6*b^2*(a + b*ArcSec[c*x])*PolyLog[3, -E^((2*I)*ArcSec[c*x])] - (3*I)*b^3*PolyLog[4, -E^((2*I)*ArcSec[c*x])])/4","A",0
29,1,141,80,0.1739917,"\int \frac{\left(a+b \sec ^{-1}(c x)\right)^3}{x^2} \, dx","Integrate[(a + b*ArcSec[c*x])^3/x^2,x]","\frac{-a^3+3 b \sec ^{-1}(c x) \left(-a^2+2 a b c x \sqrt{1-\frac{1}{c^2 x^2}}+2 b^2\right)+3 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}}+3 b^2 \sec ^{-1}(c x)^2 \left(b c x \sqrt{1-\frac{1}{c^2 x^2}}-a\right)+6 a b^2-6 b^3 c x \sqrt{1-\frac{1}{c^2 x^2}}-b^3 \sec ^{-1}(c x)^3}{x}","\frac{6 b^2 \left(a+b \sec ^{-1}(c x)\right)}{x}+3 b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)^2-\frac{\left(a+b \sec ^{-1}(c x)\right)^3}{x}-6 b^3 c \sqrt{1-\frac{1}{c^2 x^2}}",1,"(-a^3 + 6*a*b^2 + 3*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x - 6*b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x + 3*b*(-a^2 + 2*b^2 + 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x)*ArcSec[c*x] + 3*b^2*(-a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x)*ArcSec[c*x]^2 - b^3*ArcSec[c*x]^3)/x","A",1
30,1,185,137,0.2185591,"\int \frac{\left(a+b \sec ^{-1}(c x)\right)^3}{x^3} \, dx","Integrate[(a + b*ArcSec[c*x])^3/x^3,x]","\frac{-4 a^3+3 b c^2 x^2 \left(b^2-2 a^2\right) \sin ^{-1}\left(\frac{1}{c x}\right)+6 b \sec ^{-1}(c x) \left(-2 a^2+2 a b c x \sqrt{1-\frac{1}{c^2 x^2}}+b^2\right)+6 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}}+6 b^2 \sec ^{-1}(c x)^2 \left(a \left(c^2 x^2-2\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}}\right)+6 a b^2-3 b^3 c x \sqrt{1-\frac{1}{c^2 x^2}}+2 b^3 \left(c^2 x^2-2\right) \sec ^{-1}(c x)^3}{8 x^2}","-\frac{3}{4} b^2 \left(c^2-\frac{1}{x^2}\right) \left(a+b \sec ^{-1}(c x)\right)+\frac{3 b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)^2}{4 x}+\frac{1}{2} \left(c^2-\frac{1}{x^2}\right) \left(a+b \sec ^{-1}(c x)\right)^3-\frac{1}{4} c^2 \left(a+b \sec ^{-1}(c x)\right)^3-\frac{3 b^3 c \sqrt{1-\frac{1}{c^2 x^2}}}{8 x}+\frac{3}{8} b^3 c^2 \sec ^{-1}(c x)",1,"(-4*a^3 + 6*a*b^2 + 6*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x - 3*b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x + 6*b*(-2*a^2 + b^2 + 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x)*ArcSec[c*x] + 6*b^2*(b*c*Sqrt[1 - 1/(c^2*x^2)]*x + a*(-2 + c^2*x^2))*ArcSec[c*x]^2 + 2*b^3*(-2 + c^2*x^2)*ArcSec[c*x]^3 + 3*b*(-2*a^2 + b^2)*c^2*x^2*ArcSin[1/(c*x)])/(8*x^2)","A",1
31,1,204,170,0.305017,"\int \frac{\left(a+b \sec ^{-1}(c x)\right)^3}{x^4} \, dx","Integrate[(a + b*ArcSec[c*x])^3/x^4,x]","\frac{-9 a^3+3 b \sec ^{-1}(c x) \left(-9 a^2+6 a b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 x^2+1\right)+2 b^2 \left(6 c^2 x^2+1\right)\right)+9 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 x^2+1\right)+6 a b^2 \left(6 c^2 x^2+1\right)+9 b^2 \sec ^{-1}(c x)^2 \left(b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 x^2+1\right)-3 a\right)-2 b^3 c x \sqrt{1-\frac{1}{c^2 x^2}} \left(20 c^2 x^2+1\right)-9 b^3 \sec ^{-1}(c x)^3}{27 x^3}","\frac{4 b^2 c^2 \left(a+b \sec ^{-1}(c x)\right)}{3 x}+\frac{2 b^2 \left(a+b \sec ^{-1}(c x)\right)}{9 x^3}+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)^2}{3 x^2}+\frac{2}{3} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)^2-\frac{\left(a+b \sec ^{-1}(c x)\right)^3}{3 x^3}+\frac{2}{27} b^3 c^3 \left(1-\frac{1}{c^2 x^2}\right)^{3/2}-\frac{14}{9} b^3 c^3 \sqrt{1-\frac{1}{c^2 x^2}}",1,"(-9*a^3 + 9*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2) + 6*a*b^2*(1 + 6*c^2*x^2) - 2*b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 20*c^2*x^2) + 3*b*(-9*a^2 + 6*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2) + 2*b^2*(1 + 6*c^2*x^2))*ArcSec[c*x] + 9*b^2*(-3*a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2))*ArcSec[c*x]^2 - 9*b^3*ArcSec[c*x]^3)/(27*x^3)","A",1
32,1,283,208,0.3661984,"\int \frac{\left(a+b \sec ^{-1}(c x)\right)^3}{x^5} \, dx","Integrate[(a + b*ArcSec[c*x])^3/x^5,x]","\frac{-64 a^3+9 b c^4 x^4 \left(5 b^2-8 a^2\right) \sin ^{-1}\left(\frac{1}{c x}\right)+24 b \sec ^{-1}(c x) \left(-8 a^2+2 a b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(3 c^2 x^2+2\right)+b^2 \left(3 c^2 x^2+1\right)\right)+48 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}}+72 a^2 b c^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}+72 a b^2 c^2 x^2+24 b^2 \sec ^{-1}(c x)^2 \left(a \left(3 c^4 x^4-8\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(3 c^2 x^2+2\right)\right)+24 a b^2+8 b^3 \left(3 c^4 x^4-8\right) \sec ^{-1}(c x)^3-6 b^3 c x \sqrt{1-\frac{1}{c^2 x^2}}-45 b^3 c^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{256 x^4}","\frac{9 b^2 c^2 \left(a+b \sec ^{-1}(c x)\right)}{32 x^2}+\frac{3 b^2 \left(a+b \sec ^{-1}(c x)\right)}{32 x^4}+\frac{3}{32} c^4 \left(a+b \sec ^{-1}(c x)\right)^3+\frac{3 b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)^2}{16 x^3}+\frac{9 b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \sec ^{-1}(c x)\right)^2}{32 x}-\frac{\left(a+b \sec ^{-1}(c x)\right)^3}{4 x^4}-\frac{45}{256} b^3 c^4 \sec ^{-1}(c x)-\frac{3 b^3 c \sqrt{1-\frac{1}{c^2 x^2}}}{128 x^3}-\frac{45 b^3 c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{256 x}",1,"(-64*a^3 + 24*a*b^2 + 48*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x - 6*b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x + 72*a*b^2*c^2*x^2 + 72*a^2*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*x^3 - 45*b^3*c^3*Sqrt[1 - 1/(c^2*x^2)]*x^3 + 24*b*(-8*a^2 + b^2*(1 + 3*c^2*x^2) + 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(2 + 3*c^2*x^2))*ArcSec[c*x] + 24*b^2*(b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(2 + 3*c^2*x^2) + a*(-8 + 3*c^4*x^4))*ArcSec[c*x]^2 + 8*b^3*(-8 + 3*c^4*x^4)*ArcSec[c*x]^3 + 9*b*(-8*a^2 + 5*b^2)*c^4*x^4*ArcSin[1/(c*x)])/(256*x^4)","A",1
33,0,0,15,2.5851601,"\int \frac{x}{a+b \sec ^{-1}(c x)} \, dx","Integrate[x/(a + b*ArcSec[c*x]),x]","\int \frac{x}{a+b \sec ^{-1}(c x)} \, dx","\text{Int}\left(\frac{x}{a+b \sec ^{-1}(c x)},x\right)",0,"Integrate[x/(a + b*ArcSec[c*x]), x]","A",-1
34,0,0,13,2.3142923,"\int \frac{1}{a+b \sec ^{-1}(c x)} \, dx","Integrate[(a + b*ArcSec[c*x])^(-1),x]","\int \frac{1}{a+b \sec ^{-1}(c x)} \, dx","\text{Int}\left(\frac{1}{a+b \sec ^{-1}(c x)},x\right)",0,"Integrate[(a + b*ArcSec[c*x])^(-1), x]","A",-1
35,0,0,17,0.2910225,"\int \frac{1}{x \left(a+b \sec ^{-1}(c x)\right)} \, dx","Integrate[1/(x*(a + b*ArcSec[c*x])),x]","\int \frac{1}{x \left(a+b \sec ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \sec ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x*(a + b*ArcSec[c*x])), x]","A",-1
36,1,43,46,0.0802046,"\int \frac{1}{x^2 \left(a+b \sec ^{-1}(c x)\right)} \, dx","Integrate[1/(x^2*(a + b*ArcSec[c*x])),x]","\frac{c \left(\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)-\sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)\right)}{b}","\frac{c \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)}{b}-\frac{c \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)}{b}",1,"(c*(-(CosIntegral[a/b + ArcSec[c*x]]*Sin[a/b]) + Cos[a/b]*SinIntegral[a/b + ArcSec[c*x]]))/b","A",1
37,1,56,63,0.076891,"\int \frac{1}{x^3 \left(a+b \sec ^{-1}(c x)\right)} \, dx","Integrate[1/(x^3*(a + b*ArcSec[c*x])),x]","\frac{c^2 \left(\cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 a}{b}+2 \sec ^{-1}(c x)\right)-\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 a}{b}+2 \sec ^{-1}(c x)\right)\right)}{2 b}","\frac{c^2 \cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 a}{b}+2 \sec ^{-1}(c x)\right)}{2 b}-\frac{c^2 \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 a}{b}+2 \sec ^{-1}(c x)\right)}{2 b}",1,"(c^2*(-(CosIntegral[(2*a)/b + 2*ArcSec[c*x]]*Sin[(2*a)/b]) + Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSec[c*x]]))/(2*b)","A",1
38,1,91,117,0.1786152,"\int \frac{1}{x^4 \left(a+b \sec ^{-1}(c x)\right)} \, dx","Integrate[1/(x^4*(a + b*ArcSec[c*x])),x]","\frac{c^3 \left(\sin \left(\frac{a}{b}\right) \left(-\text{Ci}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)\right)-\sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sec ^{-1}(c x)\right)\right)+\cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)+\cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sec ^{-1}(c x)\right)\right)\right)}{4 b}","-\frac{c^3 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)}{4 b}-\frac{c^3 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 a}{b}+3 \sec ^{-1}(c x)\right)}{4 b}+\frac{c^3 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)}{4 b}+\frac{c^3 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 a}{b}+3 \sec ^{-1}(c x)\right)}{4 b}",1,"(c^3*(-(CosIntegral[a/b + ArcSec[c*x]]*Sin[a/b]) - CosIntegral[3*(a/b + ArcSec[c*x])]*Sin[(3*a)/b] + Cos[a/b]*SinIntegral[a/b + ArcSec[c*x]] + Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSec[c*x])]))/(4*b)","A",1
39,0,0,15,10.2961892,"\int \frac{x}{\left(a+b \sec ^{-1}(c x)\right)^2} \, dx","Integrate[x/(a + b*ArcSec[c*x])^2,x]","\int \frac{x}{\left(a+b \sec ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x}{\left(a+b \sec ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x/(a + b*ArcSec[c*x])^2, x]","A",-1
40,0,0,13,22.7748426,"\int \frac{1}{\left(a+b \sec ^{-1}(c x)\right)^2} \, dx","Integrate[(a + b*ArcSec[c*x])^(-2),x]","\int \frac{1}{\left(a+b \sec ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(a+b \sec ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(a + b*ArcSec[c*x])^(-2), x]","A",-1
41,0,0,17,3.4666157,"\int \frac{1}{x \left(a+b \sec ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x*(a + b*ArcSec[c*x])^2),x]","\int \frac{1}{x \left(a+b \sec ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \sec ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x*(a + b*ArcSec[c*x])^2), x]","A",-1
42,1,69,75,0.2795016,"\int \frac{1}{x^2 \left(a+b \sec ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^2*(a + b*ArcSec[c*x])^2),x]","\frac{c \left(-\frac{b \sqrt{1-\frac{1}{c^2 x^2}}}{a+b \sec ^{-1}(c x)}+\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)+\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)\right)}{b^2}","\frac{c \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)}{b^2}+\frac{c \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)}{b^2}-\frac{c \sqrt{1-\frac{1}{c^2 x^2}}}{b \left(a+b \sec ^{-1}(c x)\right)}",1,"(c*(-((b*Sqrt[1 - 1/(c^2*x^2)])/(a + b*ArcSec[c*x])) + Cos[a/b]*CosIntegral[a/b + ArcSec[c*x]] + Sin[a/b]*SinIntegral[a/b + ArcSec[c*x]]))/b^2","A",1
43,1,80,84,0.3679604,"\int \frac{1}{x^3 \left(a+b \sec ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^3*(a + b*ArcSec[c*x])^2),x]","\frac{c \left(-\frac{b \sqrt{1-\frac{1}{c^2 x^2}}}{a x+b x \sec ^{-1}(c x)}+c \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sec ^{-1}(c x)\right)\right)+c \sin \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sec ^{-1}(c x)\right)\right)\right)}{b^2}","\frac{c^2 \cos \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 a}{b}+2 \sec ^{-1}(c x)\right)}{b^2}+\frac{c^2 \sin \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 a}{b}+2 \sec ^{-1}(c x)\right)}{b^2}-\frac{c^2 \sin \left(2 \sec ^{-1}(c x)\right)}{2 b \left(a+b \sec ^{-1}(c x)\right)}",1,"(c*(-((b*Sqrt[1 - 1/(c^2*x^2)])/(a*x + b*x*ArcSec[c*x])) + c*Cos[(2*a)/b]*CosIntegral[2*(a/b + ArcSec[c*x])] + c*Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcSec[c*x])]))/b^2","A",1
44,1,223,178,0.4756342,"\int \frac{1}{x^4 \left(a+b \sec ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^4*(a + b*ArcSec[c*x])^2),x]","\frac{c^3 x^2 \cos \left(\frac{a}{b}\right) \left(a+b \sec ^{-1}(c x)\right) \text{Ci}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)+3 c^3 x^2 \cos \left(\frac{3 a}{b}\right) \left(a+b \sec ^{-1}(c x)\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sec ^{-1}(c x)\right)\right)+a c^3 x^2 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)+b c^3 x^2 \sin \left(\frac{a}{b}\right) \sec ^{-1}(c x) \text{Si}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)+3 a c^3 x^2 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sec ^{-1}(c x)\right)\right)+3 b c^3 x^2 \sin \left(\frac{3 a}{b}\right) \sec ^{-1}(c x) \text{Si}\left(3 \left(\frac{a}{b}+\sec ^{-1}(c x)\right)\right)-4 b c \sqrt{1-\frac{1}{c^2 x^2}}}{4 b^2 x^2 \left(a+b \sec ^{-1}(c x)\right)}","\frac{c^3 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)}{4 b^2}+\frac{3 c^3 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 a}{b}+3 \sec ^{-1}(c x)\right)}{4 b^2}+\frac{c^3 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)}{4 b^2}+\frac{3 c^3 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 a}{b}+3 \sec ^{-1}(c x)\right)}{4 b^2}-\frac{c^3 \sin \left(3 \sec ^{-1}(c x)\right)}{4 b \left(a+b \sec ^{-1}(c x)\right)}-\frac{c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{4 b \left(a+b \sec ^{-1}(c x)\right)}",1,"(-4*b*c*Sqrt[1 - 1/(c^2*x^2)] + c^3*x^2*(a + b*ArcSec[c*x])*Cos[a/b]*CosIntegral[a/b + ArcSec[c*x]] + 3*c^3*x^2*(a + b*ArcSec[c*x])*Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcSec[c*x])] + a*c^3*x^2*Sin[a/b]*SinIntegral[a/b + ArcSec[c*x]] + b*c^3*x^2*ArcSec[c*x]*Sin[a/b]*SinIntegral[a/b + ArcSec[c*x]] + 3*a*c^3*x^2*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSec[c*x])] + 3*b*c^3*x^2*ArcSec[c*x]*Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcSec[c*x])])/(4*b^2*x^2*(a + b*ArcSec[c*x]))","A",1
45,0,0,15,3.6753692,"\int \frac{x}{\left(a+b \sec ^{-1}(c x)\right)^3} \, dx","Integrate[x/(a + b*ArcSec[c*x])^3,x]","\int \frac{x}{\left(a+b \sec ^{-1}(c x)\right)^3} \, dx","\text{Int}\left(\frac{x}{\left(a+b \sec ^{-1}(c x)\right)^3},x\right)",0,"Integrate[x/(a + b*ArcSec[c*x])^3, x]","A",-1
46,0,0,13,12.210167,"\int \frac{1}{\left(a+b \sec ^{-1}(c x)\right)^3} \, dx","Integrate[(a + b*ArcSec[c*x])^(-3),x]","\int \frac{1}{\left(a+b \sec ^{-1}(c x)\right)^3} \, dx","\text{Int}\left(\frac{1}{\left(a+b \sec ^{-1}(c x)\right)^3},x\right)",0,"Integrate[(a + b*ArcSec[c*x])^(-3), x]","A",-1
47,0,0,17,1.7933166,"\int \frac{1}{x \left(a+b \sec ^{-1}(c x)\right)^3} \, dx","Integrate[1/(x*(a + b*ArcSec[c*x])^3),x]","\int \frac{1}{x \left(a+b \sec ^{-1}(c x)\right)^3} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \sec ^{-1}(c x)\right)^3},x\right)",0,"Integrate[1/(x*(a + b*ArcSec[c*x])^3), x]","A",-1
48,1,88,103,0.3930569,"\int \frac{1}{x^2 \left(a+b \sec ^{-1}(c x)\right)^3} \, dx","Integrate[1/(x^2*(a + b*ArcSec[c*x])^3),x]","-\frac{\frac{b \left(a+b c x \sqrt{1-\frac{1}{c^2 x^2}}+b \sec ^{-1}(c x)\right)}{x \left(a+b \sec ^{-1}(c x)\right)^2}-c \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)+c \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)}{2 b^3}","\frac{c \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)}{2 b^3}-\frac{c \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)}{2 b^3}-\frac{1}{2 b^2 x \left(a+b \sec ^{-1}(c x)\right)}-\frac{c \sqrt{1-\frac{1}{c^2 x^2}}}{2 b \left(a+b \sec ^{-1}(c x)\right)^2}",1,"-1/2*((b*(a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x + b*ArcSec[c*x]))/(x*(a + b*ArcSec[c*x])^2) - c*CosIntegral[a/b + ArcSec[c*x]]*Sin[a/b] + c*Cos[a/b]*SinIntegral[a/b + ArcSec[c*x]])/b^3","A",1
49,1,114,112,0.380955,"\int \frac{1}{x^3 \left(a+b \sec ^{-1}(c x)\right)^3} \, dx","Integrate[1/(x^3*(a + b*ArcSec[c*x])^3),x]","\frac{-\frac{b^2 c \sqrt{1-\frac{1}{c^2 x^2}}}{x \left(a+b \sec ^{-1}(c x)\right)^2}+2 c^2 \left(\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(2 \left(\frac{a}{b}+\sec ^{-1}(c x)\right)\right)-\cos \left(\frac{2 a}{b}\right) \text{Si}\left(2 \left(\frac{a}{b}+\sec ^{-1}(c x)\right)\right)\right)+\frac{b \left(c^2 x^2-2\right)}{x^2 \left(a+b \sec ^{-1}(c x)\right)}}{2 b^3}","\frac{c^2 \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 a}{b}+2 \sec ^{-1}(c x)\right)}{b^3}-\frac{c^2 \cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 a}{b}+2 \sec ^{-1}(c x)\right)}{b^3}-\frac{c^2 \cos \left(2 \sec ^{-1}(c x)\right)}{2 b^2 \left(a+b \sec ^{-1}(c x)\right)}-\frac{c^2 \sin \left(2 \sec ^{-1}(c x)\right)}{4 b \left(a+b \sec ^{-1}(c x)\right)^2}",1,"(-((b^2*c*Sqrt[1 - 1/(c^2*x^2)])/(x*(a + b*ArcSec[c*x])^2)) + (b*(-2 + c^2*x^2))/(x^2*(a + b*ArcSec[c*x])) + 2*c^2*(CosIntegral[2*(a/b + ArcSec[c*x])]*Sin[(2*a)/b] - Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcSec[c*x])]))/(2*b^3)","A",1
50,1,169,228,0.464605,"\int \frac{1}{x^4 \left(a+b \sec ^{-1}(c x)\right)^3} \, dx","Integrate[1/(x^4*(a + b*ArcSec[c*x])^3),x]","\frac{-\frac{4 b^2 c \sqrt{1-\frac{1}{c^2 x^2}}}{x^2 \left(a+b \sec ^{-1}(c x)\right)^2}+c^3 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)+9 c^3 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\sec ^{-1}(c x)\right)\right)-c^3 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)-9 c^3 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\sec ^{-1}(c x)\right)\right)+\frac{8 b c^2}{a x+b x \sec ^{-1}(c x)}-\frac{12 b}{x^3 \left(a+b \sec ^{-1}(c x)\right)}}{8 b^3}","\frac{c^3 \sin \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)}{8 b^3}+\frac{9 c^3 \sin \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 a}{b}+3 \sec ^{-1}(c x)\right)}{8 b^3}-\frac{c^3 \cos \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\sec ^{-1}(c x)\right)}{8 b^3}-\frac{9 c^3 \cos \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 a}{b}+3 \sec ^{-1}(c x)\right)}{8 b^3}-\frac{3 c^3 \cos \left(3 \sec ^{-1}(c x)\right)}{8 b^2 \left(a+b \sec ^{-1}(c x)\right)}-\frac{c^2}{8 b^2 x \left(a+b \sec ^{-1}(c x)\right)}-\frac{c^3 \sin \left(3 \sec ^{-1}(c x)\right)}{8 b \left(a+b \sec ^{-1}(c x)\right)^2}-\frac{c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{8 b \left(a+b \sec ^{-1}(c x)\right)^2}",1,"((-4*b^2*c*Sqrt[1 - 1/(c^2*x^2)])/(x^2*(a + b*ArcSec[c*x])^2) - (12*b)/(x^3*(a + b*ArcSec[c*x])) + (8*b*c^2)/(a*x + b*x*ArcSec[c*x]) + c^3*CosIntegral[a/b + ArcSec[c*x]]*Sin[a/b] + 9*c^3*CosIntegral[3*(a/b + ArcSec[c*x])]*Sin[(3*a)/b] - c^3*Cos[a/b]*SinIntegral[a/b + ArcSec[c*x]] - 9*c^3*Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSec[c*x])])/(8*b^3)","A",1
51,0,0,19,6.7912463,"\int (d x)^m \left(a+b \sec ^{-1}(c x)\right)^3 \, dx","Integrate[(d*x)^m*(a + b*ArcSec[c*x])^3,x]","\int (d x)^m \left(a+b \sec ^{-1}(c x)\right)^3 \, dx","\text{Int}\left((d x)^m \left(a+b \sec ^{-1}(c x)\right)^3,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcSec[c*x])^3, x]","A",-1
52,0,0,19,4.5140048,"\int (d x)^m \left(a+b \sec ^{-1}(c x)\right)^2 \, dx","Integrate[(d*x)^m*(a + b*ArcSec[c*x])^2,x]","\int (d x)^m \left(a+b \sec ^{-1}(c x)\right)^2 \, dx","\text{Int}\left((d x)^m \left(a+b \sec ^{-1}(c x)\right)^2,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcSec[c*x])^2, x]","A",-1
53,1,82,67,0.257179,"\int (d x)^m \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[(d*x)^m*(a + b*ArcSec[c*x]),x]","\frac{x (d x)^m \left((m+1) \left(a+b \sec ^{-1}(c x)\right)+\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right)}{\sqrt{1-c^2 x^2}}\right)}{(m+1)^2}","\frac{(d x)^{m+1} \left(a+b \sec ^{-1}(c x)\right)}{d (m+1)}-\frac{b (d x)^m \, _2F_1\left(\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{c^2 x^2}\right)}{c m (m+1)}",1,"(x*(d*x)^m*((1 + m)*(a + b*ArcSec[c*x]) + (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/Sqrt[1 - c^2*x^2]))/(1 + m)^2","A",1
54,0,0,19,0.8323602,"\int \frac{(d x)^m}{a+b \sec ^{-1}(c x)} \, dx","Integrate[(d*x)^m/(a + b*ArcSec[c*x]),x]","\int \frac{(d x)^m}{a+b \sec ^{-1}(c x)} \, dx","\text{Int}\left(\frac{(d x)^m}{a+b \sec ^{-1}(c x)},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcSec[c*x]), x]","A",-1
55,0,0,19,1.7132278,"\int \frac{(d x)^m}{\left(a+b \sec ^{-1}(c x)\right)^2} \, dx","Integrate[(d*x)^m/(a + b*ArcSec[c*x])^2,x]","\int \frac{(d x)^m}{\left(a+b \sec ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{(d x)^m}{\left(a+b \sec ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcSec[c*x])^2, x]","A",-1
56,1,166,167,0.2998536,"\int (d+e x)^3 \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^3*(a + b*ArcSec[c*x]),x]","\frac{3 a c^3 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+3 b c^3 x \sec ^{-1}(c x) \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)-b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 \left(18 d^2+6 d e x+e^2 x^2\right)+2 e^2\right)-6 b d \left(2 c^2 d^2+e^2\right) \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)}{12 c^3}","\frac{(d+e x)^4 \left(a+b \sec ^{-1}(c x)\right)}{4 e}-\frac{b d e^2 x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{2 c}-\frac{b e^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{12 c}-\frac{b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(9 c^2 d^2+e^2\right)}{6 c^3}-\frac{b d \left(2 c^2 d^2+e^2\right) \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{2 c^3}+\frac{b d^4 \csc ^{-1}(c x)}{4 e}",1,"(3*a*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(2*e^2 + c^2*(18*d^2 + 6*d*e*x + e^2*x^2)) + 3*b*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcSec[c*x] - 6*b*d*(2*c^2*d^2 + e^2)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(12*c^3)","A",1
57,1,124,124,0.1898331,"\int (d+e x)^2 \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^2*(a + b*ArcSec[c*x]),x]","\frac{c^2 x \left(2 a c \left(3 d^2+3 d e x+e^2 x^2\right)-b e \sqrt{1-\frac{1}{c^2 x^2}} (6 d+e x)\right)+2 b c^3 x \sec ^{-1}(c x) \left(3 d^2+3 d e x+e^2 x^2\right)-b \left(6 c^2 d^2+e^2\right) \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)}{6 c^3}","\frac{(d+e x)^3 \left(a+b \sec ^{-1}(c x)\right)}{3 e}-\frac{b d e x \sqrt{1-\frac{1}{c^2 x^2}}}{c}-\frac{b e^2 x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{6 c}-\frac{b \left(6 c^2 d^2+e^2\right) \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{6 c^3}+\frac{b d^3 \csc ^{-1}(c x)}{3 e}",1,"(c^2*x*(-(b*e*Sqrt[1 - 1/(c^2*x^2)]*(6*d + e*x)) + 2*a*c*(3*d^2 + 3*d*e*x + e^2*x^2)) + 2*b*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcSec[c*x] - b*(6*c^2*d^2 + e^2)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(6*c^3)","A",1
58,1,114,84,0.2230036,"\int (d+e x) \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)*(a + b*ArcSec[c*x]),x]","a d x+\frac{1}{2} a e x^2-\frac{b d x \sqrt{1-\frac{1}{c^2 x^2}} \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{\sqrt{c^2 x^2-1}}-\frac{b e x \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{2 c}+b d x \sec ^{-1}(c x)+\frac{1}{2} b e x^2 \sec ^{-1}(c x)","\frac{(d+e x)^2 \left(a+b \sec ^{-1}(c x)\right)}{2 e}-\frac{b d \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{c}-\frac{b e x \sqrt{1-\frac{1}{c^2 x^2}}}{2 c}+\frac{b d^2 \csc ^{-1}(c x)}{2 e}",1,"a*d*x + (a*e*x^2)/2 - (b*e*x*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(2*c) + b*d*x*ArcSec[c*x] + (b*e*x^2*ArcSec[c*x])/2 - (b*d*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[-1 + c^2*x^2]","A",1
59,1,59,32,0.0541414,"\int \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[a + b*ArcSec[c*x],x]","a x-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{\sqrt{c^2 x^2-1}}+b x \sec ^{-1}(c x)","a x-\frac{b \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{c}+b x \sec ^{-1}(c x)",1,"a*x + b*x*ArcSec[c*x] - (b*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[-1 + c^2*x^2]","A",1
60,1,333,247,0.6215282,"\int \frac{a+b \sec ^{-1}(c x)}{d+e x} \, dx","Integrate[(a + b*ArcSec[c*x])/(d + e*x),x]","\frac{a \log (d+e x)}{e}+\frac{b \left(-i \left(\text{Li}_2\left(\frac{\left(\sqrt{e^2-c^2 d^2}-e\right) e^{i \sec ^{-1}(c x)}}{c d}\right)+\text{Li}_2\left(-\frac{\left(e+\sqrt{e^2-c^2 d^2}\right) e^{i \sec ^{-1}(c x)}}{c d}\right)\right)+\log \left(1+\frac{\left(e-\sqrt{e^2-c^2 d^2}\right) e^{i \sec ^{-1}(c x)}}{c d}\right) \left(2 \sin ^{-1}\left(\frac{\sqrt{\frac{e}{c d}+1}}{\sqrt{2}}\right)+\sec ^{-1}(c x)\right)+\log \left(1+\frac{\left(\sqrt{e^2-c^2 d^2}+e\right) e^{i \sec ^{-1}(c x)}}{c d}\right) \left(\sec ^{-1}(c x)-2 \sin ^{-1}\left(\frac{\sqrt{\frac{e}{c d}+1}}{\sqrt{2}}\right)\right)+4 i \sin ^{-1}\left(\frac{\sqrt{\frac{e}{c d}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{(e-c d) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{e^2-c^2 d^2}}\right)+\frac{1}{2} i \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)-\sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)\right)}{e}","\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{\left(e-\sqrt{e^2-c^2 d^2}\right) e^{i \sec ^{-1}(c x)}}{c d}\right)}{e}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{\left(\sqrt{e^2-c^2 d^2}+e\right) e^{i \sec ^{-1}(c x)}}{c d}\right)}{e}-\frac{\log \left(1+e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)}{e}-\frac{i b \text{Li}_2\left(-\frac{\left(e-\sqrt{e^2-c^2 d^2}\right) e^{i \sec ^{-1}(c x)}}{c d}\right)}{e}-\frac{i b \text{Li}_2\left(-\frac{\left(e+\sqrt{e^2-c^2 d^2}\right) e^{i \sec ^{-1}(c x)}}{c d}\right)}{e}+\frac{i b \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)}{2 e}",1,"(a*Log[d + e*x])/e + (b*((4*I)*ArcSin[Sqrt[1 + e/(c*d)]/Sqrt[2]]*ArcTan[((-(c*d) + e)*Tan[ArcSec[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] + (ArcSec[c*x] + 2*ArcSin[Sqrt[1 + e/(c*d)]/Sqrt[2]])*Log[1 + ((e - Sqrt[-(c^2*d^2) + e^2])*E^(I*ArcSec[c*x]))/(c*d)] + (ArcSec[c*x] - 2*ArcSin[Sqrt[1 + e/(c*d)]/Sqrt[2]])*Log[1 + ((e + Sqrt[-(c^2*d^2) + e^2])*E^(I*ArcSec[c*x]))/(c*d)] - ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - I*(PolyLog[2, ((-e + Sqrt[-(c^2*d^2) + e^2])*E^(I*ArcSec[c*x]))/(c*d)] + PolyLog[2, -(((e + Sqrt[-(c^2*d^2) + e^2])*E^(I*ArcSec[c*x]))/(c*d))]) + (I/2)*PolyLog[2, -E^((2*I)*ArcSec[c*x])]))/e","A",0
61,1,142,104,0.2287483,"\int \frac{a+b \sec ^{-1}(c x)}{(d+e x)^2} \, dx","Integrate[(a + b*ArcSec[c*x])/(d + e*x)^2,x]","-\frac{a}{e (d+e x)}+\frac{b \log \left(c x \left(c d-\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d^2-e^2}\right)+e\right)}{d \sqrt{c^2 d^2-e^2}}-\frac{b \log (d+e x)}{d \sqrt{c^2 d^2-e^2}}-\frac{b \sin ^{-1}\left(\frac{1}{c x}\right)}{d e}-\frac{b \sec ^{-1}(c x)}{e (d+e x)}","-\frac{a+b \sec ^{-1}(c x)}{e (d+e x)}-\frac{b \tanh ^{-1}\left(\frac{c^2 d+\frac{e}{x}}{c \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d^2-e^2}}\right)}{d \sqrt{c^2 d^2-e^2}}-\frac{b \csc ^{-1}(c x)}{d e}",1,"-(a/(e*(d + e*x))) - (b*ArcSec[c*x])/(e*(d + e*x)) - (b*ArcSin[1/(c*x)])/(d*e) - (b*Log[d + e*x])/(d*Sqrt[c^2*d^2 - e^2]) + (b*Log[e + c*(c*d - Sqrt[c^2*d^2 - e^2]*Sqrt[1 - 1/(c^2*x^2)])*x])/(d*Sqrt[c^2*d^2 - e^2])","A",1
62,1,247,172,0.4946641,"\int \frac{a+b \sec ^{-1}(c x)}{(d+e x)^3} \, dx","Integrate[(a + b*ArcSec[c*x])/(d + e*x)^3,x]","\frac{1}{2} \left(-\frac{a}{e (d+e x)^2}+\frac{b c e x \sqrt{1-\frac{1}{c^2 x^2}}}{d \left(c^2 d^2-e^2\right) (d+e x)}+\frac{b \left(2 c^2 d^2-e^2\right) \log \left(c x \left(c d-\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d^2-e^2}\right)+e\right)}{d^2 (c d-e) (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{b \left(e^2-2 c^2 d^2\right) \log (d+e x)}{d^2 (c d-e) (c d+e) \sqrt{c^2 d^2-e^2}}-\frac{b \sin ^{-1}\left(\frac{1}{c x}\right)}{d^2 e}-\frac{b \sec ^{-1}(c x)}{e (d+e x)^2}\right)","-\frac{a+b \sec ^{-1}(c x)}{2 e (d+e x)^2}+\frac{b c e \sqrt{1-\frac{1}{c^2 x^2}}}{2 d \left(c^2 d^2-e^2\right) \left(\frac{d}{x}+e\right)}-\frac{b \left(2 c^2 d^2-e^2\right) \tanh ^{-1}\left(\frac{c^2 d+\frac{e}{x}}{c \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d^2-e^2}}\right)}{2 d^2 \left(c^2 d^2-e^2\right)^{3/2}}-\frac{b \csc ^{-1}(c x)}{2 d^2 e}",1,"(-(a/(e*(d + e*x)^2)) + (b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x)/(d*(c^2*d^2 - e^2)*(d + e*x)) - (b*ArcSec[c*x])/(e*(d + e*x)^2) - (b*ArcSin[1/(c*x)])/(d^2*e) + (b*(-2*c^2*d^2 + e^2)*Log[d + e*x])/(d^2*(c*d - e)*(c*d + e)*Sqrt[c^2*d^2 - e^2]) + (b*(2*c^2*d^2 - e^2)*Log[e + c*(c*d - Sqrt[c^2*d^2 - e^2]*Sqrt[1 - 1/(c^2*x^2)])*x])/(d^2*(c*d - e)*(c*d + e)*Sqrt[c^2*d^2 - e^2]))/2","A",1
63,1,333,372,1.479713,"\int (d+e x)^{3/2} \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^(3/2)*(a + b*ArcSec[c*x]),x]","\frac{1}{15} \left(\frac{6 a (d+e x)^{5/2}}{e}-\frac{4 b e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}{c}+\frac{4 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left(\left(9 c^2 d^2-7 c d e+e^2\right) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)-3 c^2 d^2 \Pi \left(\frac{e}{c d}+1;i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)-7 c d (c d-e) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)\right)}{c^3 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{-\frac{c}{c d+e}}}+\frac{6 b \sec ^{-1}(c x) (d+e x)^{5/2}}{e}\right)","\frac{2 (d+e x)^{5/2} \left(a+b \sec ^{-1}(c x)\right)}{5 e}+\frac{4 b d^3 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{5 c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{28 b d \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{1-c^2 x^2} \left(2 c^2 d^2+e^2\right) \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 c^4 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b e \left(1-c^2 x^2\right) \sqrt{d+e x}}{15 c^3 x \sqrt{1-\frac{1}{c^2 x^2}}}",1,"((-4*b*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])/c + (6*a*(d + e*x)^(5/2))/e + (6*b*(d + e*x)^(5/2)*ArcSec[c*x])/e + ((4*I)*b*Sqrt[(e*(1 + c*x))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*(-7*c*d*(c*d - e)*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] + (9*c^2*d^2 - 7*c*d*e + e^2)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - 3*c^2*d^2*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(c^3*e*Sqrt[-(c/(c*d + e))]*Sqrt[1 - 1/(c^2*x^2)]*x))/15","C",1
64,1,277,315,6.6010036,"\int \sqrt{d+e x} \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + e*x]*(a + b*ArcSec[c*x]),x]","\frac{2 \left(a (d+e x)^{3/2}+\frac{2 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left((2 c d-e) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)+(e-c d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)-c d \Pi \left(\frac{e}{c d}+1;i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)\right)}{c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{-\frac{c}{c d+e}}}+b \sec ^{-1}(c x) (d+e x)^{3/2}\right)}{3 e}","\frac{2 (d+e x)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{3 e}+\frac{4 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}",1,"(2*(a*(d + e*x)^(3/2) + b*(d + e*x)^(3/2)*ArcSec[c*x] + ((2*I)*b*Sqrt[(e*(1 + c*x))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*((-(c*d) + e)*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] + (2*c*d - e)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - c*d*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(c^2*Sqrt[-(c/(c*d + e))]*Sqrt[1 - 1/(c^2*x^2)]*x)))/(3*e)","C",1
65,1,212,212,2.8147549,"\int \frac{a+b \sec ^{-1}(c x)}{\sqrt{d+e x}} \, dx","Integrate[(a + b*ArcSec[c*x])/Sqrt[d + e*x],x]","\frac{2 \left(a \sqrt{d+e x}+\frac{2 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left(F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)-\Pi \left(\frac{e}{c d}+1;i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)\right)}{c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{-\frac{c}{c d+e}}}+b \sec ^{-1}(c x) \sqrt{d+e x}\right)}{e}","\frac{2 \sqrt{d+e x} \left(a+b \sec ^{-1}(c x)\right)}{e}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}",1,"(2*(a*Sqrt[d + e*x] + b*Sqrt[d + e*x]*ArcSec[c*x] + ((2*I)*b*Sqrt[(e*(1 + c*x))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*(EllipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(c*Sqrt[-(c/(c*d + e))]*Sqrt[1 - 1/(c^2*x^2)]*x)))/e","C",1
66,1,124,119,0.2898172,"\int \frac{a+b \sec ^{-1}(c x)}{(d+e x)^{3/2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(d + e*x)^(3/2),x]","-\frac{2 \left(\left(c^2 x^2-1\right) \left(a+b \sec ^{-1}(c x)\right)+2 b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)\right)}{e \left(c^2 x^2-1\right) \sqrt{d+e x}}","-\frac{2 \left(a+b \sec ^{-1}(c x)\right)}{e \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}",1,"(-2*((-1 + c^2*x^2)*(a + b*ArcSec[c*x]) + 2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]))/(e*Sqrt[d + e*x]*(-1 + c^2*x^2))","A",1
67,1,326,298,6.6359802,"\int \frac{a+b \sec ^{-1}(c x)}{(d+e x)^{5/2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(d + e*x)^(5/2),x]","\frac{2 \left(-\frac{a}{(d+e x)^{3/2}}+\frac{2 b c e^2 x \sqrt{1-\frac{1}{c^2 x^2}}}{\left(c^2 d^3-d e^2\right) \sqrt{d+e x}}-\frac{2 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left(c d F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)-c d E\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)+(c d+e) \Pi \left(\frac{e}{c d}+1;i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)\right)}{d^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left(-\frac{c}{c d+e}\right)^{3/2} (c d+e)^2}-\frac{b \sec ^{-1}(c x)}{(d+e x)^{3/2}}\right)}{3 e}","-\frac{2 \left(a+b \sec ^{-1}(c x)\right)}{3 e (d+e x)^{3/2}}-\frac{4 b e \left(1-c^2 x^2\right)}{3 c d x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{d+e x}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 d x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c d e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}",1,"(2*(-(a/(d + e*x)^(3/2)) + (2*b*c*e^2*Sqrt[1 - 1/(c^2*x^2)]*x)/((c^2*d^3 - d*e^2)*Sqrt[d + e*x]) - (b*ArcSec[c*x])/(d + e*x)^(3/2) - ((2*I)*b*Sqrt[(e*(1 + c*x))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*(-(c*d*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]) + c*d*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] + (c*d + e)*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(d^2*(-(c/(c*d + e)))^(3/2)*(c*d + e)^2*Sqrt[1 - 1/(c^2*x^2)]*x)))/(3*e)","C",1
68,1,407,540,8.9138889,"\int \frac{a+b \sec ^{-1}(c x)}{(d+e x)^{7/2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(d + e*x)^(7/2),x]","\frac{2 \left(-\frac{3 a}{(d+e x)^{5/2}}+\frac{2 b c e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2 (8 d+7 e x)-e^2 (4 d+3 e x)\right)}{\left(c^2 d^3-d e^2\right)^2 (d+e x)^{3/2}}+\frac{2 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left(-c d \left(6 c^2 d^2-c d e-3 e^2\right) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)+c d \left(7 c^2 d^2-3 e^2\right) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)-3 (c d-e) (c d+e)^2 \Pi \left(\frac{e}{c d}+1;i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)\right)}{d^3 x \sqrt{1-\frac{1}{c^2 x^2}} (c d-e) \left(-\frac{c}{c d+e}\right)^{3/2} (c d+e)^3}-\frac{3 b \sec ^{-1}(c x)}{(d+e x)^{5/2}}\right)}{15 e}","-\frac{2 \left(a+b \sec ^{-1}(c x)\right)}{5 e (d+e x)^{5/2}}-\frac{4 b e \left(1-c^2 x^2\right)}{5 c d^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{d+e x}}-\frac{16 b c e \left(1-c^2 x^2\right)}{15 x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right)^2 \sqrt{d+e x}}-\frac{4 b e \left(1-c^2 x^2\right)}{15 c d x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) (d+e x)^{3/2}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 d x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{5 c d^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b \sqrt{1-c^2 x^2} \left(7 c^2 d^2-3 e^2\right) \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^3-d e^2\right)^2 \sqrt{\frac{c (d+e x)}{c d+e}}}",1,"(2*((-3*a)/(d + e*x)^(5/2) + (2*b*c*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*(-(e^2*(4*d + 3*e*x)) + c^2*d^2*(8*d + 7*e*x)))/((c^2*d^3 - d*e^2)^2*(d + e*x)^(3/2)) - (3*b*ArcSec[c*x])/(d + e*x)^(5/2) + ((2*I)*b*Sqrt[(e*(1 + c*x))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*(c*d*(7*c^2*d^2 - 3*e^2)*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - c*d*(6*c^2*d^2 - c*d*e - 3*e^2)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - 3*(c*d - e)*(c*d + e)^2*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(d^3*(c*d - e)*(-(c/(c*d + e)))^(3/2)*(c*d + e)^3*Sqrt[1 - 1/(c^2*x^2)]*x)))/(15*e)","C",1
69,1,141,206,0.2378622,"\int x^4 \left(d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^4*(d + e*x^2)*(a + b*ArcSec[c*x]),x]","\frac{48 a c^7 x^5 \left(7 d+5 e x^2\right)+48 b c^7 x^5 \sec ^{-1}(c x) \left(7 d+5 e x^2\right)-3 b \left(42 c^2 d+25 e\right) \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)-b c^2 x^2 \sqrt{1-\frac{1}{c^2 x^2}} \left(c^4 \left(84 d x^2+40 e x^4\right)+2 c^2 \left(63 d+25 e x^2\right)+75 e\right)}{1680 c^7}","\frac{1}{5} d x^5 \left(a+b \sec ^{-1}(c x)\right)+\frac{1}{7} e x^7 \left(a+b \sec ^{-1}(c x)\right)-\frac{b e x^6 \sqrt{c^2 x^2-1}}{42 c \sqrt{c^2 x^2}}-\frac{b x \left(42 c^2 d+25 e\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{560 c^6 \sqrt{c^2 x^2}}-\frac{b x^2 \sqrt{c^2 x^2-1} \left(42 c^2 d+25 e\right)}{560 c^5 \sqrt{c^2 x^2}}-\frac{b x^4 \sqrt{c^2 x^2-1} \left(42 c^2 d+25 e\right)}{840 c^3 \sqrt{c^2 x^2}}",1,"(48*a*c^7*x^5*(7*d + 5*e*x^2) - b*c^2*Sqrt[1 - 1/(c^2*x^2)]*x^2*(75*e + 2*c^2*(63*d + 25*e*x^2) + c^4*(84*d*x^2 + 40*e*x^4)) + 48*b*c^7*x^5*(7*d + 5*e*x^2)*ArcSec[c*x] - 3*b*(42*c^2*d + 25*e)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(1680*c^7)","A",1
70,1,123,161,0.1733028,"\int x^2 \left(d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)*(a + b*ArcSec[c*x]),x]","\frac{c^2 x^2 \left(8 a c^3 x \left(5 d+3 e x^2\right)-b \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 \left(20 d+6 e x^2\right)+9 e\right)\right)+8 b c^5 x^3 \sec ^{-1}(c x) \left(5 d+3 e x^2\right)-b \left(20 c^2 d+9 e\right) \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)}{120 c^5}","\frac{1}{3} d x^3 \left(a+b \sec ^{-1}(c x)\right)+\frac{1}{5} e x^5 \left(a+b \sec ^{-1}(c x)\right)-\frac{b e x^4 \sqrt{c^2 x^2-1}}{20 c \sqrt{c^2 x^2}}-\frac{b x \left(20 c^2 d+9 e\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{120 c^4 \sqrt{c^2 x^2}}-\frac{b x^2 \sqrt{c^2 x^2-1} \left(20 c^2 d+9 e\right)}{120 c^3 \sqrt{c^2 x^2}}",1,"(c^2*x^2*(8*a*c^3*x*(5*d + 3*e*x^2) - b*Sqrt[1 - 1/(c^2*x^2)]*(9*e + c^2*(20*d + 6*e*x^2))) + 8*b*c^5*x^3*(5*d + 3*e*x^2)*ArcSec[c*x] - b*(20*c^2*d + 9*e)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(120*c^5)","A",1
71,1,150,109,0.2825384,"\int \left(d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)*(a + b*ArcSec[c*x]),x]","a d x+\frac{1}{3} a e x^3-\frac{b d x \sqrt{1-\frac{1}{c^2 x^2}} \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{\sqrt{c^2 x^2-1}}-\frac{b e x^2 \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{6 c}-\frac{b e \log \left(x \left(\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right)\right)}{6 c^3}+b d x \sec ^{-1}(c x)+\frac{1}{3} b e x^3 \sec ^{-1}(c x)","d x \left(a+b \sec ^{-1}(c x)\right)+\frac{1}{3} e x^3 \left(a+b \sec ^{-1}(c x)\right)-\frac{b x \left(6 c^2 d+e\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{6 c^2 \sqrt{c^2 x^2}}-\frac{b e x^2 \sqrt{c^2 x^2-1}}{6 c \sqrt{c^2 x^2}}",1,"a*d*x + (a*e*x^3)/3 - (b*e*x^2*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(6*c) + b*d*x*ArcSec[c*x] + (b*e*x^3*ArcSec[c*x])/3 - (b*d*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[-1 + c^2*x^2] - (b*e*Log[x*(1 + Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])])/(6*c^3)","A",1
72,1,104,87,0.1207249,"\int \frac{\left(d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSec[c*x]))/x^2,x]","-\frac{a d}{x}+a e x+b c d \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{b e x \sqrt{1-\frac{1}{c^2 x^2}} \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{\sqrt{c^2 x^2-1}}-\frac{b d \sec ^{-1}(c x)}{x}+b e x \sec ^{-1}(c x)","-\frac{d \left(a+b \sec ^{-1}(c x)\right)}{x}+e x \left(a+b \sec ^{-1}(c x)\right)+\frac{b c d \sqrt{c^2 x^2-1}}{\sqrt{c^2 x^2}}-\frac{b e x \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{\sqrt{c^2 x^2}}",1,"-((a*d)/x) + a*e*x + b*c*d*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*d*ArcSec[c*x])/x + b*e*x*ArcSec[c*x] - (b*e*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[-1 + c^2*x^2]","A",1
73,1,69,105,0.0812696,"\int \frac{\left(d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSec[c*x]))/x^4,x]","\frac{-3 a \left(d+3 e x^2\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 d x^2+d+9 e x^2\right)-3 b \sec ^{-1}(c x) \left(d+3 e x^2\right)}{9 x^3}","-\frac{d \left(a+b \sec ^{-1}(c x)\right)}{3 x^3}-\frac{e \left(a+b \sec ^{-1}(c x)\right)}{x}+\frac{b c \sqrt{c^2 x^2-1} \left(2 c^2 d+9 e\right)}{9 \sqrt{c^2 x^2}}+\frac{b c d \sqrt{c^2 x^2-1}}{9 x^2 \sqrt{c^2 x^2}}",1,"(-3*a*(d + 3*e*x^2) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 2*c^2*d*x^2 + 9*e*x^2) - 3*b*(d + 3*e*x^2)*ArcSec[c*x])/(9*x^3)","A",1
74,1,94,152,0.1297052,"\int \frac{\left(d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSec[c*x]))/x^6,x]","\frac{-15 a \left(3 d+5 e x^2\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(25 e x^2 \left(2 c^2 x^2+1\right)+3 d \left(8 c^4 x^4+4 c^2 x^2+3\right)\right)-15 b \sec ^{-1}(c x) \left(3 d+5 e x^2\right)}{225 x^5}","-\frac{d \left(a+b \sec ^{-1}(c x)\right)}{5 x^5}-\frac{e \left(a+b \sec ^{-1}(c x)\right)}{3 x^3}+\frac{b c \sqrt{c^2 x^2-1} \left(12 c^2 d+25 e\right)}{225 x^2 \sqrt{c^2 x^2}}+\frac{b c d \sqrt{c^2 x^2-1}}{25 x^4 \sqrt{c^2 x^2}}+\frac{2 b c^3 \sqrt{c^2 x^2-1} \left(12 c^2 d+25 e\right)}{225 \sqrt{c^2 x^2}}",1,"(-15*a*(3*d + 5*e*x^2) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(25*e*x^2*(1 + 2*c^2*x^2) + 3*d*(3 + 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(3*d + 5*e*x^2)*ArcSec[c*x])/(225*x^5)","A",1
75,1,110,197,0.1506485,"\int \frac{\left(d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSec[c*x]))/x^8,x]","\frac{-105 a \left(5 d+7 e x^2\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(49 e x^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)+15 d \left(16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right)\right)-105 b \sec ^{-1}(c x) \left(5 d+7 e x^2\right)}{3675 x^7}","-\frac{d \left(a+b \sec ^{-1}(c x)\right)}{7 x^7}-\frac{e \left(a+b \sec ^{-1}(c x)\right)}{5 x^5}+\frac{b c \sqrt{c^2 x^2-1} \left(30 c^2 d+49 e\right)}{1225 x^4 \sqrt{c^2 x^2}}+\frac{b c d \sqrt{c^2 x^2-1}}{49 x^6 \sqrt{c^2 x^2}}+\frac{8 b c^5 \sqrt{c^2 x^2-1} \left(30 c^2 d+49 e\right)}{3675 \sqrt{c^2 x^2}}+\frac{4 b c^3 \sqrt{c^2 x^2-1} \left(30 c^2 d+49 e\right)}{3675 x^2 \sqrt{c^2 x^2}}",1,"(-105*a*(5*d + 7*e*x^2) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(49*e*x^2*(3 + 4*c^2*x^2 + 8*c^4*x^4) + 15*d*(5 + 6*c^2*x^2 + 8*c^4*x^4 + 16*c^6*x^6)) - 105*b*(5*d + 7*e*x^2)*ArcSec[c*x])/(3675*x^7)","A",1
76,1,118,196,0.237411,"\int x^5 \left(d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^5*(d + e*x^2)*(a + b*ArcSec[c*x]),x]","\frac{1}{24} a x^6 \left(4 d+3 e x^2\right)-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^6 \left(84 d x^4+45 e x^6\right)+2 c^4 \left(56 d x^2+27 e x^4\right)+8 c^2 \left(28 d+9 e x^2\right)+144 e\right)}{2520 c^7}+\frac{1}{24} b x^6 \sec ^{-1}(c x) \left(4 d+3 e x^2\right)","\frac{1}{6} d x^6 \left(a+b \sec ^{-1}(c x)\right)+\frac{1}{8} e x^8 \left(a+b \sec ^{-1}(c x)\right)-\frac{b x \left(c^2 x^2-1\right)^{5/2} \left(4 c^2 d+9 e\right)}{120 c^7 \sqrt{c^2 x^2}}-\frac{b x \left(c^2 x^2-1\right)^{3/2} \left(8 c^2 d+9 e\right)}{72 c^7 \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(4 c^2 d+3 e\right)}{24 c^7 \sqrt{c^2 x^2}}-\frac{b e x \left(c^2 x^2-1\right)^{7/2}}{56 c^7 \sqrt{c^2 x^2}}",1,"(a*x^6*(4*d + 3*e*x^2))/24 - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(144*e + 8*c^2*(28*d + 9*e*x^2) + 2*c^4*(56*d*x^2 + 27*e*x^4) + c^6*(84*d*x^4 + 45*e*x^6)))/(2520*c^7) + (b*x^6*(4*d + 3*e*x^2)*ArcSec[c*x])/24","A",1
77,1,98,153,0.2652698,"\int x^3 \left(d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)*(a + b*ArcSec[c*x]),x]","\frac{1}{180} x \left(15 a x^3 \left(3 d+2 e x^2\right)-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} \left(3 c^4 \left(5 d x^2+2 e x^4\right)+c^2 \left(30 d+8 e x^2\right)+16 e\right)}{c^5}+15 b x^3 \sec ^{-1}(c x) \left(3 d+2 e x^2\right)\right)","\frac{1}{4} d x^4 \left(a+b \sec ^{-1}(c x)\right)+\frac{1}{6} e x^6 \left(a+b \sec ^{-1}(c x)\right)-\frac{b x \left(c^2 x^2-1\right)^{3/2} \left(3 c^2 d+4 e\right)}{36 c^5 \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(3 c^2 d+2 e\right)}{12 c^5 \sqrt{c^2 x^2}}-\frac{b e x \left(c^2 x^2-1\right)^{5/2}}{30 c^5 \sqrt{c^2 x^2}}",1,"(x*(15*a*x^3*(3*d + 2*e*x^2) - (b*Sqrt[1 - 1/(c^2*x^2)]*(16*e + c^2*(30*d + 8*e*x^2) + 3*c^4*(5*d*x^2 + 2*e*x^4)))/c^5 + 15*b*x^3*(3*d + 2*e*x^2)*ArcSec[c*x]))/180","A",1
78,1,79,138,0.1083247,"\int x \left(d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)*(a + b*ArcSec[c*x]),x]","\frac{x \left(3 a c^3 x \left(2 d+e x^2\right)+3 b c^3 x \sec ^{-1}(c x) \left(2 d+e x^2\right)-b \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 \left(6 d+e x^2\right)+2 e\right)\right)}{12 c^3}","\frac{\left(d+e x^2\right)^2 \left(a+b \sec ^{-1}(c x)\right)}{4 e}-\frac{b c d^2 x \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{4 e \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(2 c^2 d+e\right)}{4 c^3 \sqrt{c^2 x^2}}-\frac{b e x \left(c^2 x^2-1\right)^{3/2}}{12 c^3 \sqrt{c^2 x^2}}",1,"(x*(3*a*c^3*x*(2*d + e*x^2) - b*Sqrt[1 - 1/(c^2*x^2)]*(2*e + c^2*(6*d + e*x^2)) + 3*b*c^3*x*(2*d + e*x^2)*ArcSec[c*x]))/(12*c^3)","A",1
79,1,115,124,0.0902285,"\int \frac{\left(d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSec[c*x]))/x,x]","a d \log (x)+\frac{1}{2} a e x^2-\frac{b e x \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{2 c}+\frac{1}{2} i b d \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)+\frac{1}{2} i b d \sec ^{-1}(c x)^2-b d \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)+\frac{1}{2} b e x^2 \sec ^{-1}(c x)","-d \log \left(\frac{1}{x}\right) \left(a+b \sec ^{-1}(c x)\right)+\frac{1}{2} e x^2 \left(a+b \sec ^{-1}(c x)\right)-\frac{b e x \sqrt{1-\frac{1}{c^2 x^2}}}{2 c}-\frac{1}{2} i b d \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)-\frac{1}{2} i b d \csc ^{-1}(c x)^2+b d \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)-b d \log \left(\frac{1}{x}\right) \csc ^{-1}(c x)",1,"(a*e*x^2)/2 - (b*e*x*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(2*c) + (b*e*x^2*ArcSec[c*x])/2 + (I/2)*b*d*ArcSec[c*x]^2 - b*d*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] + a*d*Log[x] + (I/2)*b*d*PolyLog[2, -E^((2*I)*ArcSec[c*x])]","A",0
80,1,132,137,0.0968111,"\int \frac{\left(d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)*(a + b*ArcSec[c*x]))/x^3,x]","-\frac{a d}{2 x^2}+a e \log (x)+\frac{b c d \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{4 x}-\frac{1}{4} b c^2 d \sin ^{-1}\left(\frac{1}{c x}\right)-\frac{b d \sec ^{-1}(c x)}{2 x^2}+\frac{1}{2} i b e \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)+\frac{1}{2} i b e \sec ^{-1}(c x)^2-b e \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)","-\frac{d \left(a+b \sec ^{-1}(c x)\right)}{2 x^2}-e \log \left(\frac{1}{x}\right) \left(a+b \sec ^{-1}(c x)\right)+\frac{b c d \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}-\frac{1}{4} b c^2 d \csc ^{-1}(c x)-\frac{1}{2} i b e \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)-\frac{1}{2} i b e \csc ^{-1}(c x)^2+b e \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)-b e \log \left(\frac{1}{x}\right) \csc ^{-1}(c x)",1,"-1/2*(a*d)/x^2 + (b*c*d*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(4*x) - (b*d*ArcSec[c*x])/(2*x^2) + (I/2)*b*e*ArcSec[c*x]^2 - (b*c^2*d*ArcSin[1/(c*x)])/4 - b*e*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] + a*e*Log[x] + (I/2)*b*e*PolyLog[2, -E^((2*I)*ArcSec[c*x])]","A",0
81,1,186,252,0.3743036,"\int x^2 \left(d+e x^2\right)^2 \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)^2*(a + b*ArcSec[c*x]),x]","\frac{c^2 x^2 \left(16 a c^5 x \left(35 d^2+42 d e x^2+15 e^2 x^4\right)-b \sqrt{1-\frac{1}{c^2 x^2}} \left(8 c^4 \left(35 d^2+21 d e x^2+5 e^2 x^4\right)+2 c^2 e \left(126 d+25 e x^2\right)+75 e^2\right)\right)+16 b c^7 x^3 \sec ^{-1}(c x) \left(35 d^2+42 d e x^2+15 e^2 x^4\right)-b \left(280 c^4 d^2+252 c^2 d e+75 e^2\right) \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)}{1680 c^7}","\frac{1}{3} d^2 x^3 \left(a+b \sec ^{-1}(c x)\right)+\frac{2}{5} d e x^5 \left(a+b \sec ^{-1}(c x)\right)+\frac{1}{7} e^2 x^7 \left(a+b \sec ^{-1}(c x)\right)-\frac{b e^2 x^6 \sqrt{c^2 x^2-1}}{42 c \sqrt{c^2 x^2}}-\frac{b e x^4 \sqrt{c^2 x^2-1} \left(84 c^2 d+25 e\right)}{840 c^3 \sqrt{c^2 x^2}}-\frac{b x \left(280 c^4 d^2+252 c^2 d e+75 e^2\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{1680 c^6 \sqrt{c^2 x^2}}-\frac{b x^2 \sqrt{c^2 x^2-1} \left(280 c^4 d^2+252 c^2 d e+75 e^2\right)}{1680 c^5 \sqrt{c^2 x^2}}",1,"(c^2*x^2*(16*a*c^5*x*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) - b*Sqrt[1 - 1/(c^2*x^2)]*(75*e^2 + 2*c^2*e*(126*d + 25*e*x^2) + 8*c^4*(35*d^2 + 21*d*e*x^2 + 5*e^2*x^4))) + 16*b*c^7*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcSec[c*x] - b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(1680*c^7)","A",1
82,1,153,191,0.2408016,"\int \left(d+e x^2\right)^2 \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^2*(a + b*ArcSec[c*x]),x]","\frac{c^2 x \left(8 a c^3 \left(15 d^2+10 d e x^2+3 e^2 x^4\right)-b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 \left(40 d+6 e x^2\right)+9 e\right)\right)+8 b c^5 x \sec ^{-1}(c x) \left(15 d^2+10 d e x^2+3 e^2 x^4\right)-b \left(120 c^4 d^2+40 c^2 d e+9 e^2\right) \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)}{120 c^5}","d^2 x \left(a+b \sec ^{-1}(c x)\right)+\frac{2}{3} d e x^3 \left(a+b \sec ^{-1}(c x)\right)+\frac{1}{5} e^2 x^5 \left(a+b \sec ^{-1}(c x)\right)-\frac{b e^2 x^4 \sqrt{c^2 x^2-1}}{20 c \sqrt{c^2 x^2}}-\frac{b x \left(120 c^4 d^2+40 c^2 d e+9 e^2\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{120 c^4 \sqrt{c^2 x^2}}-\frac{b e x^2 \sqrt{c^2 x^2-1} \left(40 c^2 d+9 e\right)}{120 c^3 \sqrt{c^2 x^2}}",1,"(c^2*x*(8*a*c^3*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(9*e + c^2*(40*d + 6*e*x^2))) + 8*b*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcSec[c*x] - b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(120*c^5)","A",1
83,1,136,162,0.2050453,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \sec ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSec[c*x]))/x^2,x]","\frac{c^2 \left(2 a c \left(-3 d^2+6 d e x^2+e^2 x^4\right)+b x \sqrt{1-\frac{1}{c^2 x^2}} \left(6 c^2 d^2-e^2 x^2\right)\right)+2 b c^3 \sec ^{-1}(c x) \left(-3 d^2+6 d e x^2+e^2 x^4\right)-b e x \left(12 c^2 d+e\right) \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)}{6 c^3 x}","-\frac{d^2 \left(a+b \sec ^{-1}(c x)\right)}{x}+2 d e x \left(a+b \sec ^{-1}(c x)\right)+\frac{1}{3} e^2 x^3 \left(a+b \sec ^{-1}(c x)\right)+\frac{b c d^2 \sqrt{c^2 x^2-1}}{\sqrt{c^2 x^2}}-\frac{b e x \left(12 c^2 d+e\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{6 c^2 \sqrt{c^2 x^2}}-\frac{b e^2 x^2 \sqrt{c^2 x^2-1}}{6 c \sqrt{c^2 x^2}}",1,"(c^2*(b*Sqrt[1 - 1/(c^2*x^2)]*x*(6*c^2*d^2 - e^2*x^2) + 2*a*c*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)) + 2*b*c^3*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)*ArcSec[c*x] - b*e*(12*c^2*d + e)*x*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(6*c^3*x)","A",1
84,1,127,158,0.2279664,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \sec ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSec[c*x]))/x^4,x]","\frac{c \left(b c d x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 d x^2+d+18 e x^2\right)-3 a \left(d^2+6 d e x^2-3 e^2 x^4\right)\right)-9 b e^2 x^3 \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)-3 b c \sec ^{-1}(c x) \left(d^2+6 d e x^2-3 e^2 x^4\right)}{9 c x^3}","-\frac{d^2 \left(a+b \sec ^{-1}(c x)\right)}{3 x^3}-\frac{2 d e \left(a+b \sec ^{-1}(c x)\right)}{x}+e^2 x \left(a+b \sec ^{-1}(c x)\right)+\frac{b c d^2 \sqrt{c^2 x^2-1}}{9 x^2 \sqrt{c^2 x^2}}+\frac{2 b c d \sqrt{c^2 x^2-1} \left(c^2 d+9 e\right)}{9 \sqrt{c^2 x^2}}-\frac{b e^2 x \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{\sqrt{c^2 x^2}}",1,"(c*(b*c*d*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 2*c^2*d*x^2 + 18*e*x^2) - 3*a*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)) - 3*b*c*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)*ArcSec[c*x] - 9*b*e^2*x^3*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(9*c*x^3)","A",1
85,1,127,183,0.2274741,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \sec ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSec[c*x]))/x^6,x]","\frac{-15 a \left(3 d^2+10 d e x^2+15 e^2 x^4\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(50 d e x^2 \left(2 c^2 x^2+1\right)+3 d^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)+225 e^2 x^4\right)-15 b \sec ^{-1}(c x) \left(3 d^2+10 d e x^2+15 e^2 x^4\right)}{225 x^5}","-\frac{d^2 \left(a+b \sec ^{-1}(c x)\right)}{5 x^5}-\frac{2 d e \left(a+b \sec ^{-1}(c x)\right)}{3 x^3}-\frac{e^2 \left(a+b \sec ^{-1}(c x)\right)}{x}+\frac{b c d^2 \sqrt{c^2 x^2-1}}{25 x^4 \sqrt{c^2 x^2}}+\frac{2 b c d \sqrt{c^2 x^2-1} \left(6 c^2 d+25 e\right)}{225 x^2 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left(24 c^4 d^2+100 c^2 d e+225 e^2\right)}{225 \sqrt{c^2 x^2}}",1,"(-15*a*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(225*e^2*x^4 + 50*d*e*x^2*(1 + 2*c^2*x^2) + 3*d^2*(3 + 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4)*ArcSec[c*x])/(225*x^5)","A",1
86,1,153,241,0.2392291,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \sec ^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSec[c*x]))/x^8,x]","\frac{-105 a \left(15 d^2+42 d e x^2+35 e^2 x^4\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(1225 e^2 x^4 \left(2 c^2 x^2+1\right)+294 d e x^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)+45 d^2 \left(16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right)\right)-105 b \sec ^{-1}(c x) \left(15 d^2+42 d e x^2+35 e^2 x^4\right)}{11025 x^7}","-\frac{d^2 \left(a+b \sec ^{-1}(c x)\right)}{7 x^7}-\frac{2 d e \left(a+b \sec ^{-1}(c x)\right)}{5 x^5}-\frac{e^2 \left(a+b \sec ^{-1}(c x)\right)}{3 x^3}+\frac{b c d^2 \sqrt{c^2 x^2-1}}{49 x^6 \sqrt{c^2 x^2}}+\frac{2 b c d \sqrt{c^2 x^2-1} \left(15 c^2 d+49 e\right)}{1225 x^4 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left(360 c^4 d^2+1176 c^2 d e+1225 e^2\right)}{11025 x^2 \sqrt{c^2 x^2}}+\frac{2 b c^3 \sqrt{c^2 x^2-1} \left(360 c^4 d^2+1176 c^2 d e+1225 e^2\right)}{11025 \sqrt{c^2 x^2}}",1,"(-105*a*(15*d^2 + 42*d*e*x^2 + 35*e^2*x^4) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1225*e^2*x^4*(1 + 2*c^2*x^2) + 294*d*e*x^2*(3 + 4*c^2*x^2 + 8*c^4*x^4) + 45*d^2*(5 + 6*c^2*x^2 + 8*c^4*x^4 + 16*c^6*x^6)) - 105*b*(15*d^2 + 42*d*e*x^2 + 35*e^2*x^4)*ArcSec[c*x])/(11025*x^7)","A",1
87,1,162,242,0.2924335,"\int x^3 \left(d+e x^2\right)^2 \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^2*(a + b*ArcSec[c*x]),x]","\frac{1}{24} a x^4 \left(6 d^2+8 d e x^2+3 e^2 x^4\right)-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(3 c^6 \left(70 d^2 x^2+56 d e x^4+15 e^2 x^6\right)+c^4 \left(420 d^2+224 d e x^2+54 e^2 x^4\right)+8 c^2 e \left(56 d+9 e x^2\right)+144 e^2\right)}{2520 c^7}+\frac{1}{24} b x^4 \sec ^{-1}(c x) \left(6 d^2+8 d e x^2+3 e^2 x^4\right)","\frac{1}{4} d^2 x^4 \left(a+b \sec ^{-1}(c x)\right)+\frac{1}{3} d e x^6 \left(a+b \sec ^{-1}(c x)\right)+\frac{1}{8} e^2 x^8 \left(a+b \sec ^{-1}(c x)\right)-\frac{b e x \left(c^2 x^2-1\right)^{5/2} \left(8 c^2 d+9 e\right)}{120 c^7 \sqrt{c^2 x^2}}-\frac{b e^2 x \left(c^2 x^2-1\right)^{7/2}}{56 c^7 \sqrt{c^2 x^2}}-\frac{b x \left(c^2 x^2-1\right)^{3/2} \left(6 c^4 d^2+16 c^2 d e+9 e^2\right)}{72 c^7 \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(6 c^4 d^2+8 c^2 d e+3 e^2\right)}{24 c^7 \sqrt{c^2 x^2}}",1,"(a*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4))/24 - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(144*e^2 + 8*c^2*e*(56*d + 9*e*x^2) + c^4*(420*d^2 + 224*d*e*x^2 + 54*e^2*x^4) + 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)))/(2520*c^7) + (b*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcSec[c*x])/24","A",1
88,1,125,195,0.3083809,"\int x \left(d+e x^2\right)^2 \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^2*(a + b*ArcSec[c*x]),x]","\frac{1}{90} x \left(15 a x \left(3 d^2+3 d e x^2+e^2 x^4\right)-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} \left(3 c^4 \left(15 d^2+5 d e x^2+e^2 x^4\right)+2 c^2 e \left(15 d+2 e x^2\right)+8 e^2\right)}{c^5}+15 b x \sec ^{-1}(c x) \left(3 d^2+3 d e x^2+e^2 x^4\right)\right)","\frac{\left(d+e x^2\right)^3 \left(a+b \sec ^{-1}(c x)\right)}{6 e}-\frac{b c d^3 x \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{6 e \sqrt{c^2 x^2}}-\frac{b e x \left(c^2 x^2-1\right)^{3/2} \left(3 c^2 d+2 e\right)}{18 c^5 \sqrt{c^2 x^2}}-\frac{b e^2 x \left(c^2 x^2-1\right)^{5/2}}{30 c^5 \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(3 c^4 d^2+3 c^2 d e+e^2\right)}{6 c^5 \sqrt{c^2 x^2}}",1,"(x*(15*a*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) - (b*Sqrt[1 - 1/(c^2*x^2)]*(8*e^2 + 2*c^2*e*(15*d + 2*e*x^2) + 3*c^4*(15*d^2 + 5*d*e*x^2 + e^2*x^4)))/c^5 + 15*b*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4)*ArcSec[c*x]))/90","A",1
89,1,160,186,0.38785,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \sec ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSec[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 x \left(c x \sec ^{-1}(c x)-\sqrt{1-\frac{1}{c^2 x^2}}\right)}{c}-\frac{b e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 x^2+2\right)}{12 c^3}+\frac{1}{2} i b d^2 \left(\text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)+\sec ^{-1}(c x) \left(\sec ^{-1}(c x)+2 i \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)\right)\right)+\frac{1}{4} b e^2 x^4 \sec ^{-1}(c x)","-d^2 \log \left(\frac{1}{x}\right) \left(a+b \sec ^{-1}(c x)\right)+d e x^2 \left(a+b \sec ^{-1}(c x)\right)+\frac{1}{4} e^2 x^4 \left(a+b \sec ^{-1}(c x)\right)-\frac{b e^2 x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{12 c}-\frac{b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(6 c^2 d+e\right)}{6 c^3}-\frac{1}{2} i b d^2 \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)-\frac{1}{2} i b d^2 \csc ^{-1}(c x)^2+b d^2 \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)-b d^2 \log \left(\frac{1}{x}\right) \csc ^{-1}(c x)",1,"a*d*e*x^2 + (a*e^2*x^4)/4 - (b*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*(2 + c^2*x^2))/(12*c^3) + (b*e^2*x^4*ArcSec[c*x])/4 + (b*d*e*x*(-Sqrt[1 - 1/(c^2*x^2)] + c*x*ArcSec[c*x]))/c + a*d^2*Log[x] + (I/2)*b*d^2*(ArcSec[c*x]*(ArcSec[c*x] + (2*I)*Log[1 + E^((2*I)*ArcSec[c*x])]) + PolyLog[2, -E^((2*I)*ArcSec[c*x])])","A",0
90,1,187,189,0.6641762,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \sec ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcSec[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+\frac{b c d^2 \sqrt{1-\frac{1}{c^2 x^2}} \left(\frac{c^2 x^2 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{\sqrt{1-c^2 x^2}}+1\right)}{x}+\frac{2 b e^2 x \left(c x \sec ^{-1}(c x)-\sqrt{1-\frac{1}{c^2 x^2}}\right)}{c}-\frac{2 b d^2 \sec ^{-1}(c x)}{x^2}+4 i b d e \left(\text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)+\sec ^{-1}(c x) \left(\sec ^{-1}(c x)+2 i \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)\right)\right)\right)","-\frac{d^2 \left(a+b \sec ^{-1}(c x)\right)}{2 x^2}-2 d e \log \left(\frac{1}{x}\right) \left(a+b \sec ^{-1}(c x)\right)+\frac{1}{2} e^2 x^2 \left(a+b \sec ^{-1}(c x)\right)+\frac{b c d^2 \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}-\frac{1}{4} b c^2 d^2 \csc ^{-1}(c x)-\frac{b e^2 x \sqrt{1-\frac{1}{c^2 x^2}}}{2 c}-i b d e \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)-i b d e \csc ^{-1}(c x)^2+2 b d e \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)-2 b d e \log \left(\frac{1}{x}\right) \csc ^{-1}(c x)",1,"((-2*a*d^2)/x^2 + 2*a*e^2*x^2 - (2*b*d^2*ArcSec[c*x])/x^2 + (2*b*e^2*x*(-Sqrt[1 - 1/(c^2*x^2)] + c*x*ArcSec[c*x]))/c + (b*c*d^2*Sqrt[1 - 1/(c^2*x^2)]*(1 + (c^2*x^2*ArcTanh[Sqrt[1 - c^2*x^2]])/Sqrt[1 - c^2*x^2]))/x + 8*a*d*e*Log[x] + (4*I)*b*d*e*(ArcSec[c*x]*(ArcSec[c*x] + (2*I)*Log[1 + E^((2*I)*ArcSec[c*x])]) + PolyLog[2, -E^((2*I)*ArcSec[c*x])]))/4","A",0
91,1,1023,546,1.4686203,"\int \frac{x^2 \left(a+b \sec ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x^2*(a + b*ArcSec[c*x]))/(d + e*x^2),x]","\frac{a x}{e}-\frac{a \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{3/2}}+b \left(\frac{c x \sec ^{-1}(c x)+\log \left(\cos \left(\frac{1}{2} \sec ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sec ^{-1}(c x)\right)\right)-\log \left(\cos \left(\frac{1}{2} \sec ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sec ^{-1}(c x)\right)\right)}{c e}-\frac{\sqrt{d} \left(8 \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 i \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-2 \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)\right)}{4 e^{3/2}}+\frac{\sqrt{d} \left(8 \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-2 i \sec ^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+2 i \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-2 \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)\right)}{4 e^{3/2}}\right)","\frac{\sqrt{-d} \left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^{3/2}}-\frac{\sqrt{-d} \left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^{3/2}}+\frac{\sqrt{-d} \left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^{3/2}}-\frac{\sqrt{-d} \left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^{3/2}}+\frac{x \left(a+b \sec ^{-1}(c x)\right)}{e}+\frac{i b \sqrt{-d} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^{3/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^{3/2}}+\frac{i b \sqrt{-d} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^{3/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^{3/2}}-\frac{b \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{c e}",1,"(a*x)/e - (a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(3/2) + b*((c*x*ArcSec[c*x] + Log[Cos[ArcSec[c*x]/2] - Sin[ArcSec[c*x]/2]] - Log[Cos[ArcSec[c*x]/2] + Sin[ArcSec[c*x]/2]])/(c*e) - (Sqrt[d]*(8*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - (2*I)*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 2*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + PolyLog[2, -E^((2*I)*ArcSec[c*x])]))/(4*e^(3/2)) + (Sqrt[d]*(8*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - (2*I)*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 2*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + PolyLog[2, -E^((2*I)*ArcSec[c*x])]))/(4*e^(3/2)))","A",0
92,1,891,487,0.4583061,"\int \frac{x \left(a+b \sec ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x*(a + b*ArcSec[c*x]))/(d + e*x^2),x]","\frac{4 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+4 i b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+b \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+b \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+2 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+b \sec ^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+b \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 b \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)+a \log \left(e x^2+d\right)-i b \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-i b \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-i b \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-i b \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+i b \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)}{2 e}","\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e}-\frac{\log \left(1+e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)}{e}-\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e}-\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e}-\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e}-\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e}+\frac{i b \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)}{2 e}",1,"((4*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] + (4*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] + b*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 2*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + b*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 2*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + b*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + b*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*b*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] + a*Log[d + e*x^2] - I*b*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - I*b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - I*b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - I*b*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + I*b*PolyLog[2, -E^((2*I)*ArcSec[c*x])])/(2*e)","A",0
93,1,871,509,0.401289,"\int \frac{a+b \sec ^{-1}(c x)}{d+e x^2} \, dx","Integrate[(a + b*ArcSec[c*x])/(d + e*x^2),x]","\frac{2 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)-4 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+4 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-i b \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 i b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+i b \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+2 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+i b \sec ^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-i b \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 i b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+b \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-b \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-b \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+b \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)}{2 \sqrt{d} \sqrt{e}}","\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}",1,"(2*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - 4*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] + 4*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - I*b*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + I*b*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + I*b*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - I*b*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + b*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + b*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/(2*Sqrt[d]*Sqrt[e])","A",0
94,1,402,459,0.9256149,"\int \frac{a+b \sec ^{-1}(c x)}{x \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcSec[c*x])/(x*(d + e*x^2)),x]","\frac{-2 a \log \left(d+e x^2\right)+4 a \log (x)+i b \left(\text{Li}_2\left(-\frac{\left(d c^2+2 e-2 \sqrt{e \left(d c^2+e\right)}\right) e^{2 i \sec ^{-1}(c x)}}{c^2 d}\right)+\text{Li}_2\left(-\frac{\left(d c^2+2 \left(e+\sqrt{e \left(d c^2+e\right)}\right)\right) e^{2 i \sec ^{-1}(c x)}}{c^2 d}\right)-4 \sin ^{-1}\left(\sqrt{\frac{e}{c^2 d}+1}\right) \tan ^{-1}\left(\frac{c e x \sqrt{1-\frac{1}{c^2 x^2}}}{\sqrt{e \left(c^2 d+e\right)}}\right)+2 i \sec ^{-1}(c x) \log \left(1+\frac{\left(-2 \sqrt{e \left(c^2 d+e\right)}+c^2 d+2 e\right) e^{2 i \sec ^{-1}(c x)}}{c^2 d}\right)+2 i \sec ^{-1}(c x) \log \left(1+\frac{\left(2 \left(\sqrt{e \left(c^2 d+e\right)}+e\right)+c^2 d\right) e^{2 i \sec ^{-1}(c x)}}{c^2 d}\right)+2 i \sin ^{-1}\left(\sqrt{\frac{e}{c^2 d}+1}\right) \log \left(1+\frac{\left(-2 \sqrt{e \left(c^2 d+e\right)}+c^2 d+2 e\right) e^{2 i \sec ^{-1}(c x)}}{c^2 d}\right)-2 i \sin ^{-1}\left(\sqrt{\frac{e}{c^2 d}+1}\right) \log \left(1+\frac{\left(2 \left(\sqrt{e \left(c^2 d+e\right)}+e\right)+c^2 d\right) e^{2 i \sec ^{-1}(c x)}}{c^2 d}\right)+2 \sec ^{-1}(c x)^2\right)}{4 d}","-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d}+\frac{i \left(a+b \sec ^{-1}(c x)\right)^2}{2 b d}+\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d}+\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d}+\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d}+\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d}",1,"(4*a*Log[x] - 2*a*Log[d + e*x^2] + I*b*(2*ArcSec[c*x]^2 - 4*ArcSin[Sqrt[1 + e/(c^2*d)]]*ArcTan[(c*e*Sqrt[1 - 1/(c^2*x^2)]*x)/Sqrt[e*(c^2*d + e)]] + (2*I)*ArcSec[c*x]*Log[1 + ((c^2*d + 2*e - 2*Sqrt[e*(c^2*d + e)])*E^((2*I)*ArcSec[c*x]))/(c^2*d)] + (2*I)*ArcSin[Sqrt[1 + e/(c^2*d)]]*Log[1 + ((c^2*d + 2*e - 2*Sqrt[e*(c^2*d + e)])*E^((2*I)*ArcSec[c*x]))/(c^2*d)] + (2*I)*ArcSec[c*x]*Log[1 + ((c^2*d + 2*(e + Sqrt[e*(c^2*d + e)]))*E^((2*I)*ArcSec[c*x]))/(c^2*d)] - (2*I)*ArcSin[Sqrt[1 + e/(c^2*d)]]*Log[1 + ((c^2*d + 2*(e + Sqrt[e*(c^2*d + e)]))*E^((2*I)*ArcSec[c*x]))/(c^2*d)] + PolyLog[2, -(((c^2*d + 2*e - 2*Sqrt[e*(c^2*d + e)])*E^((2*I)*ArcSec[c*x]))/(c^2*d))] + PolyLog[2, -(((c^2*d + 2*(e + Sqrt[e*(c^2*d + e)]))*E^((2*I)*ArcSec[c*x]))/(c^2*d))]))/(4*d)","A",0
95,1,997,551,1.5014047,"\int \frac{a+b \sec ^{-1}(c x)}{x^2 \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcSec[c*x])/(x^2*(d + e*x^2)),x]","-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) a}{d^{3/2}}-\frac{a}{d x}+b \left(\frac{c \sqrt{1-\frac{1}{c^2 x^2}}-\frac{\sec ^{-1}(c x)}{x}}{d}-\frac{\sqrt{e} \left(8 \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 i \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-2 \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)\right)}{4 d^{3/2}}+\frac{\sqrt{e} \left(8 \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-2 i \sec ^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+2 i \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-2 \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)\right)}{4 d^{3/2}}\right)","\frac{\sqrt{e} \left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 (-d)^{3/2}}-\frac{a}{d x}+\frac{i b \sqrt{e} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}+\frac{i b \sqrt{e} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{d}-\frac{b \sec ^{-1}(c x)}{d x}",1,"-(a/(d*x)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + b*((c*Sqrt[1 - 1/(c^2*x^2)] - ArcSec[c*x]/x)/d - (Sqrt[e]*(8*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - (2*I)*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 2*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + PolyLog[2, -E^((2*I)*ArcSec[c*x])]))/(4*d^(3/2)) + (Sqrt[e]*(8*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - (2*I)*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 2*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + PolyLog[2, -E^((2*I)*ArcSec[c*x])]))/(4*d^(3/2)))","A",0
96,1,1255,608,4.2509453,"\int \frac{x^5 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^5*(a + b*ArcSec[c*x]))/(d + e*x^2)^2,x]","-\frac{\frac{2 a d^2}{e x^2+d}+4 a \log \left(e x^2+d\right) d-2 a e x^2+b \left(-2 e \sec ^{-1}(c x) x^2+\frac{2 e \sqrt{1-\frac{1}{c^2 x^2}} x}{c}+\frac{d^{3/2} \sec ^{-1}(c x)}{\sqrt{d}-i \sqrt{e} x}+\frac{d^{3/2} \sec ^{-1}(c x)}{i \sqrt{e} x+\sqrt{d}}+2 d \sin ^{-1}\left(\frac{1}{c x}\right)+16 i d \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+16 i d \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+4 d \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+8 d \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 d \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+8 d \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+4 d \sec ^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-8 d \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+4 d \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-8 d \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-8 d \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-\frac{d \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i c \sqrt{d}-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{-d c^2-e}}-\frac{d \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}-4 i d \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-4 i d \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-4 i d \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-4 i d \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i d \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)\right)}{4 e^3}","-\frac{d \left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{e^3}-\frac{d \left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{e^3}-\frac{d \left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{e^3}-\frac{d \left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{e^3}+\frac{2 d \log \left(1+e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)}{e^3}+\frac{d \left(a+b \sec ^{-1}(c x)\right)}{2 e^2 \left(\frac{d}{x^2}+e\right)}+\frac{x^2 \left(a+b \sec ^{-1}(c x)\right)}{2 e^2}+\frac{b d \tan ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{2 e^{5/2} \sqrt{c^2 d+e}}+\frac{i b d \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{e^3}+\frac{i b d \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{e^3}+\frac{i b d \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{e^3}+\frac{i b d \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{e^3}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}}}{2 c e^2}-\frac{i b d \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)}{e^3}",1,"-1/4*(-2*a*e*x^2 + (2*a*d^2)/(d + e*x^2) + 4*a*d*Log[d + e*x^2] + b*((2*e*Sqrt[1 - 1/(c^2*x^2)]*x)/c - 2*e*x^2*ArcSec[c*x] + (d^(3/2)*ArcSec[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (d^(3/2)*ArcSec[c*x])/(Sqrt[d] + I*Sqrt[e]*x) + 2*d*ArcSin[1/(c*x)] + (16*I)*d*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] + (16*I)*d*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] + 4*d*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 8*d*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 4*d*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 8*d*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 4*d*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 8*d*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 4*d*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 8*d*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 8*d*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - (d*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] - (d*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] - (4*I)*d*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*d*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*d*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*d*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*d*PolyLog[2, -E^((2*I)*ArcSec[c*x])]))/e^3","B",0
97,1,1213,570,1.3219149,"\int \frac{x^3 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^3*(a + b*ArcSec[c*x]))/(d + e*x^2)^2,x]","\frac{\frac{2 a d}{e x^2+d}+\frac{b \sec ^{-1}(c x) \sqrt{d}}{\sqrt{d}-i \sqrt{e} x}+\frac{b \sec ^{-1}(c x) \sqrt{d}}{i \sqrt{e} x+\sqrt{d}}+2 b \sin ^{-1}\left(\frac{1}{c x}\right)+8 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+8 i b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+2 b \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 b \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+4 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+2 b \sec ^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-4 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+2 b \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-4 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-4 b \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-\frac{b \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i c \sqrt{d}-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{-d c^2-e}}-\frac{b \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}+2 a \log \left(e x^2+d\right)-2 i b \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 i b \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 i b \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 i b \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+2 i b \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)}{4 e^2}","\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^2}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^2}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^2}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^2}-\frac{a+b \sec ^{-1}(c x)}{2 e \left(\frac{d}{x^2}+e\right)}-\frac{\log \left(1+e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)}{e^2}-\frac{b \tan ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{2 e^{3/2} \sqrt{c^2 d+e}}-\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^2}-\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^2}-\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^2}-\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^2}+\frac{i b \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)}{2 e^2}",1,"((2*a*d)/(d + e*x^2) + (b*Sqrt[d]*ArcSec[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (b*Sqrt[d]*ArcSec[c*x])/(Sqrt[d] + I*Sqrt[e]*x) + 2*b*ArcSin[1/(c*x)] + (8*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] + (8*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] + 2*b*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 4*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 2*b*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 4*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 2*b*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 4*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 2*b*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 4*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 4*b*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - (b*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] - (b*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] + 2*a*Log[d + e*x^2] - (2*I)*b*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*b*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*b*PolyLog[2, -E^((2*I)*ArcSec[c*x])])/(4*e^2)","B",0
98,1,286,131,0.6436654,"\int \frac{x \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x*(a + b*ArcSec[c*x]))/(d + e*x^2)^2,x]","\frac{-\frac{2 a}{d+e x^2}+\frac{b \sqrt{e} \log \left(\frac{4 c d \sqrt{e} x \left(c \sqrt{d}-i \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 (-d)-e}\right)+4 i d e}{b \sqrt{c^2 (-d)-e} \left(\sqrt{d}+i \sqrt{e} x\right)}\right)}{d \sqrt{c^2 (-d)-e}}+\frac{b \sqrt{e} \log \left(\frac{4 c d \sqrt{e} x \left(c \sqrt{d}+i \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 (-d)-e}\right)-4 i d e}{b \sqrt{c^2 (-d)-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{d \sqrt{c^2 (-d)-e}}-\frac{2 b \sec ^{-1}(c x)}{d+e x^2}-\frac{2 b \sin ^{-1}\left(\frac{1}{c x}\right)}{d}}{4 e}","-\frac{a+b \sec ^{-1}(c x)}{2 e \left(d+e x^2\right)}+\frac{b c x \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{2 d e \sqrt{c^2 x^2}}-\frac{b c x \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2 d+e}}\right)}{2 d \sqrt{e} \sqrt{c^2 x^2} \sqrt{c^2 d+e}}",1,"((-2*a)/(d + e*x^2) - (2*b*ArcSec[c*x])/(d + e*x^2) - (2*b*ArcSin[1/(c*x)])/d + (b*Sqrt[e]*Log[((4*I)*d*e + 4*c*d*Sqrt[e]*(c*Sqrt[d] - I*Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x)/(b*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/(d*Sqrt[-(c^2*d) - e]) + (b*Sqrt[e]*Log[((-4*I)*d*e + 4*c*d*Sqrt[e]*(c*Sqrt[d] + I*Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x)/(b*Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))])/(d*Sqrt[-(c^2*d) - e]))/(4*e)","C",1
99,0,0,546,37.0478285,"\int \frac{a+b \sec ^{-1}(c x)}{x \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcSec[c*x])/(x*(d + e*x^2)^2),x]","\int \frac{a+b \sec ^{-1}(c x)}{x \left(d+e x^2\right)^2} \, dx","-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d^2}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d^2}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d^2}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d^2}-\frac{e \left(a+b \sec ^{-1}(c x)\right)}{2 d^2 \left(\frac{d}{x^2}+e\right)}+\frac{i \left(a+b \sec ^{-1}(c x)\right)^2}{2 b d^2}+\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d^2}+\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d^2}+\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d^2}+\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d^2}-\frac{b \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{2 d^2 \sqrt{c^2 d+e}}",1,"Integrate[(a + b*ArcSec[c*x])/(x*(d + e*x^2)^2), x]","F",-1
100,1,1331,784,2.5368324,"\int \frac{x^4 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^4*(a + b*ArcSec[c*x]))/(d + e*x^2)^2,x]","\frac{4 a \sqrt{e} x+\frac{2 a d \sqrt{e} x}{e x^2+d}-6 a \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+b \left(4 \sqrt{e} x \sec ^{-1}(c x)+3 i \sqrt{d} \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \sec ^{-1}(c x)-3 i \sqrt{d} \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \sec ^{-1}(c x)-3 i \sqrt{d} \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right) \sec ^{-1}(c x)+3 i \sqrt{d} \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \sec ^{-1}(c x)+\frac{d \sec ^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}+\frac{d \sec ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}+12 \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-12 \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+6 i \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-6 i \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+6 i \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-6 i \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+\frac{i \sqrt{d} \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i c \sqrt{d}-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{-d c^2-e}}-\frac{i \sqrt{d} \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}+\frac{4 \sqrt{e} \log \left(\cos \left(\frac{1}{2} \sec ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \sec ^{-1}(c x)\right)\right)}{c}-\frac{4 \sqrt{e} \log \left(\cos \left(\frac{1}{2} \sec ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \sec ^{-1}(c x)\right)\right)}{c}-3 \sqrt{d} \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+3 \sqrt{d} \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+3 \sqrt{d} \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-3 \sqrt{d} \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)\right)}{4 e^{5/2}}","\frac{3 \sqrt{-d} \left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{4 e^{5/2}}-\frac{d \left(a+b \sec ^{-1}(c x)\right)}{4 e^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{d \left(a+b \sec ^{-1}(c x)\right)}{4 e^2 \left(\sqrt{-d} \sqrt{e}+\frac{d}{x}\right)}+\frac{x \left(a+b \sec ^{-1}(c x)\right)}{e^2}+\frac{3 i b \sqrt{-d} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 e^{5/2}}+\frac{3 i b \sqrt{-d} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 e^{5/2}}-\frac{b \sqrt{d} \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right)}{4 e^2 \sqrt{c^2 d+e}}-\frac{b \sqrt{d} \tanh ^{-1}\left(\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right)}{4 e^2 \sqrt{c^2 d+e}}-\frac{b \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{c e^2}",1,"(4*a*Sqrt[e]*x + (2*a*d*Sqrt[e]*x)/(d + e*x^2) - 6*a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*(4*Sqrt[e]*x*ArcSec[c*x] + (d*ArcSec[c*x])/((-I)*Sqrt[d] + Sqrt[e]*x) + (d*ArcSec[c*x])/(I*Sqrt[d] + Sqrt[e]*x) + 12*Sqrt[d]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - 12*Sqrt[d]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] + (3*I)*Sqrt[d]*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (6*I)*Sqrt[d]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (3*I)*Sqrt[d]*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (6*I)*Sqrt[d]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (3*I)*Sqrt[d]*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (6*I)*Sqrt[d]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (3*I)*Sqrt[d]*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (6*I)*Sqrt[d]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (I*Sqrt[d]*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] - (I*Sqrt[d]*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] + (4*Sqrt[e]*Log[Cos[ArcSec[c*x]/2] - Sin[ArcSec[c*x]/2]])/c - (4*Sqrt[e]*Log[Cos[ArcSec[c*x]/2] + Sin[ArcSec[c*x]/2]])/c - 3*Sqrt[d]*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 3*Sqrt[d]*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 3*Sqrt[d]*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 3*Sqrt[d]*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])]))/(4*e^(5/2))","A",0
101,1,1245,745,1.8031822,"\int \frac{x^2 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^2*(a + b*ArcSec[c*x]))/(d + e*x^2)^2,x]","\frac{-\frac{2 a \sqrt{e} x}{e x^2+d}+\frac{2 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d}}+b \left(-\frac{i \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \sec ^{-1}(c x)}{\sqrt{d}}+\frac{i \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \sec ^{-1}(c x)}{\sqrt{d}}+\frac{i \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right) \sec ^{-1}(c x)}{\sqrt{d}}-\frac{i \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \sec ^{-1}(c x)}{\sqrt{d}}+\frac{\sec ^{-1}(c x)}{i \sqrt{d}-\sqrt{e} x}-\frac{\sec ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-\frac{4 \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)}{\sqrt{d}}+\frac{4 \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)}{\sqrt{d}}-\frac{2 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}+\frac{2 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}-\frac{2 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}+\frac{2 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}-\frac{i \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i c \sqrt{d}-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{d} \sqrt{-d c^2-e}}+\frac{i \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{d} \sqrt{-d c^2-e}}+\frac{\text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}-\frac{\text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}-\frac{\text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}+\frac{\text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}\right)}{4 e^{3/2}}","\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{a+b \sec ^{-1}(c x)}{4 e \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{a+b \sec ^{-1}(c x)}{4 e \left(\sqrt{-d} \sqrt{e}+\frac{d}{x}\right)}+\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right)}{4 \sqrt{d} e \sqrt{c^2 d+e}}+\frac{b \tanh ^{-1}\left(\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right)}{4 \sqrt{d} e \sqrt{c^2 d+e}}",1,"((-2*a*Sqrt[e]*x)/(d + e*x^2) + (2*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + b*(ArcSec[c*x]/(I*Sqrt[d] - Sqrt[e]*x) - ArcSec[c*x]/(I*Sqrt[d] + Sqrt[e]*x) - (4*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]])/Sqrt[d] + (4*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]])/Sqrt[d] - (I*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[d] - ((2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[d] + (I*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[d] + ((2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[d] + (I*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[d] - ((2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[d] - (I*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[d] + ((2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[d] - (I*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))])/(Sqrt[d]*Sqrt[-(c^2*d) - e]) + (I*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/(Sqrt[d]*Sqrt[-(c^2*d) - e]) + PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])]/Sqrt[d] - PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])]/Sqrt[d] - PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])]/Sqrt[d] + PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])]/Sqrt[d]))/(4*e^(3/2))","A",0
102,1,1239,739,2.2829259,"\int \frac{a+b \sec ^{-1}(c x)}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcSec[c*x])/(d + e*x^2)^2,x]","\frac{1}{2} \left(\frac{a x}{d^2+e x^2 d}+\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{3/2} \sqrt{e}}+\frac{b \left(-\frac{i \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \sec ^{-1}(c x)}{\sqrt{e}}+\frac{i \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \sec ^{-1}(c x)}{\sqrt{e}}+\frac{i \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right) \sec ^{-1}(c x)}{\sqrt{e}}-\frac{i \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \sec ^{-1}(c x)}{\sqrt{e}}+\frac{\sqrt{d} \sec ^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}+\frac{\sqrt{d} \sec ^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}-\frac{4 \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)}{\sqrt{e}}+\frac{4 \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)}{\sqrt{e}}-\frac{2 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}+\frac{2 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}-\frac{2 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}+\frac{2 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}+\frac{i \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i c \sqrt{d}-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{-d c^2-e}}-\frac{i \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}+\frac{\text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}-\frac{\text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}-\frac{\text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}+\frac{\text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}\right)}{2 d^{3/2}}\right)","-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{a+b \sec ^{-1}(c x)}{4 d \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{a+b \sec ^{-1}(c x)}{4 d \left(\sqrt{-d} \sqrt{e}+\frac{d}{x}\right)}-\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right)}{4 d^{3/2} \sqrt{c^2 d+e}}-\frac{b \tanh ^{-1}\left(\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right)}{4 d^{3/2} \sqrt{c^2 d+e}}-\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}",1,"((a*x)/(d^2 + d*e*x^2) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]) + (b*((Sqrt[d]*ArcSec[c*x])/((-I)*Sqrt[d]*Sqrt[e] + e*x) + (Sqrt[d]*ArcSec[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) - (4*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]])/Sqrt[e] + (4*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]])/Sqrt[e] - (I*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[e] - ((2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[e] + (I*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[e] + ((2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[e] + (I*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[e] - ((2*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[e] - (I*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[e] + ((2*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])])/Sqrt[e] + (I*Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] - (I*Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] + PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])]/Sqrt[e] - PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])]/Sqrt[e] - PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])]/Sqrt[e] + PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])]/Sqrt[e]))/(2*d^(3/2)))/2","A",0
103,1,1291,785,2.247251,"\int \frac{a+b \sec ^{-1}(c x)}{x^2 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcSec[c*x])/(x^2*(d + e*x^2)^2),x]","\frac{-6 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) a-\frac{4 \sqrt{d} a}{x}-\frac{2 \sqrt{d} e x a}{e x^2+d}+b \left(4 \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} c-\frac{4 \sqrt{d} \sec ^{-1}(c x)}{x}-\frac{\sqrt{d} e \sec ^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}-\frac{\sqrt{d} e \sec ^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}+12 \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-12 \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)+3 i \sqrt{e} \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+6 i \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-3 i \sqrt{e} \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-6 i \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-3 i \sqrt{e} \sec ^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+6 i \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+3 i \sqrt{e} \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-6 i \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-\frac{i e \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i c \sqrt{d}-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{-d c^2-e}}+\frac{i e \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}-3 \sqrt{e} \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+3 \sqrt{e} \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+3 \sqrt{e} \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-3 \sqrt{e} \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)\right)}{4 d^{5/2}}","-\frac{3 \sqrt{e} \left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{4 (-d)^{5/2}}+\frac{e \left(a+b \sec ^{-1}(c x)\right)}{4 d^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{e \left(a+b \sec ^{-1}(c x)\right)}{4 d^2 \left(\sqrt{-d} \sqrt{e}+\frac{d}{x}\right)}-\frac{a}{d^2 x}+\frac{b e \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right)}{4 d^{5/2} \sqrt{c^2 d+e}}+\frac{b e \tanh ^{-1}\left(\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right)}{4 d^{5/2} \sqrt{c^2 d+e}}+\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{d^2}-\frac{3 i b \sqrt{e} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}+\frac{3 i b \sqrt{e} \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}-\frac{3 i b \sqrt{e} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}+\frac{3 i b \sqrt{e} \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}-\frac{b \sec ^{-1}(c x)}{d^2 x}",1,"((-4*a*Sqrt[d])/x - (2*a*Sqrt[d]*e*x)/(d + e*x^2) - 6*a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*(4*c*Sqrt[d]*Sqrt[1 - 1/(c^2*x^2)] - (4*Sqrt[d]*ArcSec[c*x])/x - (Sqrt[d]*e*ArcSec[c*x])/((-I)*Sqrt[d]*Sqrt[e] + e*x) - (Sqrt[d]*e*ArcSec[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) + 12*Sqrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - 12*Sqrt[e]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] + (3*I)*Sqrt[e]*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (6*I)*Sqrt[e]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (3*I)*Sqrt[e]*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (6*I)*Sqrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (3*I)*Sqrt[e]*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (6*I)*Sqrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (3*I)*Sqrt[e]*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (6*I)*Sqrt[e]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (I*e*Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] + (I*e*Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] - 3*Sqrt[e]*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 3*Sqrt[e]*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + 3*Sqrt[e]*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 3*Sqrt[e]*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])]))/(4*d^(5/2))","A",0
104,1,1805,707,8.0135313,"\int \frac{x^5 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^5*(a + b*ArcSec[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 i \sqrt{d} \left(\frac{i \left(\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-\frac{\log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i c \sqrt{d}-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{-d c^2-e}}\right)}{\sqrt{d}}-\frac{\sec ^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}\right)}{16 e^{5/2}}+\frac{7 i \sqrt{d} \left(-\frac{\sec ^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}-\frac{i \left(\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-\frac{\log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}\right)}{\sqrt{d}}\right)}{16 e^{5/2}}-\frac{d \left(\frac{\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-i \left(\frac{c \sqrt{d} \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}} x}{\left(d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}+\frac{\left(2 d c^2+e\right) \log \left(-\frac{4 d \sqrt{e} \sqrt{d c^2+e} \left(c \left(c \sqrt{d}-\sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+i \sqrt{e}\right)}{\left(2 d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}\right)}{\left(d c^2+e\right)^{3/2}}\right)}{d}-\frac{\sec ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x-i \sqrt{d}\right)^2}\right)}{16 e^{5/2}}-\frac{d \left(\frac{i c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}} x}{\sqrt{d} \left(d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}-\frac{\sec ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x+i \sqrt{d}\right)^2}+\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{d \sqrt{e}}-\frac{i \left(2 d c^2+e\right) \log \left(\frac{4 d \sqrt{e} \sqrt{d c^2+e} \left(c \left(\sqrt{d} c+\sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x-i \sqrt{e}\right)}{\left(2 d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{d \left(d c^2+e\right)^{3/2}}\right)}{16 e^{5/2}}+\frac{i \left(8 \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 i \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-2 \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)\right)}{4 e^3}+\frac{i \left(8 \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-2 i \sec ^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+2 i \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-2 \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)\right)}{4 e^3}\right)","\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^3}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^3}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^3}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^3}-\frac{a+b \sec ^{-1}(c x)}{2 e^2 \left(\frac{d}{x^2}+e\right)}-\frac{a+b \sec ^{-1}(c x)}{4 e \left(\frac{d}{x^2}+e\right)^2}-\frac{\log \left(1+e^{2 i \sec ^{-1}(c x)}\right) \left(a+b \sec ^{-1}(c x)\right)}{e^3}-\frac{b \left(c^2 d+2 e\right) \tan ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{8 e^{5/2} \left(c^2 d+e\right)^{3/2}}-\frac{b \tan ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{2 e^{5/2} \sqrt{c^2 d+e}}-\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^3}-\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^3}-\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^3}-\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^3}-\frac{b c d \sqrt{1-\frac{1}{c^2 x^2}}}{8 e^2 x \left(c^2 d+e\right) \left(\frac{d}{x^2}+e\right)}+\frac{i b \text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)}{2 e^3}",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*I)/16)*Sqrt[d]*(-(ArcSec[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/e^(5/2) + (((7*I)/16)*Sqrt[d]*(-(ArcSec[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/e^(5/2) - (d*(-(ArcSec[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2)) + (ArcSin[1/(c*x)]/Sqrt[e] - I*((c*Sqrt[d]*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(I*Sqrt[e] + c*(c*Sqrt[d] - Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(c^2*d + e)^(3/2)))/d))/(16*e^(5/2)) - (d*((I*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcSec[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + ArcSin[1/(c*x)]/(d*Sqrt[e]) - (I*(2*c^2*d + e)*Log[(4*d*Sqrt[e]*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(16*e^(5/2)) + ((I/4)*(8*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - (2*I)*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 2*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + PolyLog[2, -E^((2*I)*ArcSec[c*x])]))/e^3 + ((I/4)*(8*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - (2*I)*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 2*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + PolyLog[2, -E^((2*I)*ArcSec[c*x])]))/e^3)","B",0
105,1,389,157,1.4198415,"\int \frac{x^3 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^3*(a + b*ArcSec[c*x]))/(d + e*x^2)^3,x]","-\frac{\frac{8 a}{d+e x^2}-\frac{4 a d}{\left(d+e x^2\right)^2}+\frac{b \sqrt{e} \left(c^2 d+2 e\right) \log \left(-\frac{16 d e^{3/2} \sqrt{c^2 (-d)-e} \left(\sqrt{e}+c x \left(-\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 (-d)-e}+i c \sqrt{d}\right)\right)}{b \left(c^2 d+2 e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{d \left(c^2 (-d)-e\right)^{3/2}}+\frac{b \sqrt{e} \left(c^2 d+2 e\right) \log \left(\frac{16 i d e^{3/2} \sqrt{c^2 (-d)-e} \left(-\sqrt{e}+c x \left(\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 (-d)-e}+i c \sqrt{d}\right)\right)}{b \left(c^2 d+2 e\right) \left(\sqrt{d}+i \sqrt{e} x\right)}\right)}{d \left(c^2 (-d)-e\right)^{3/2}}-\frac{2 b c e x \sqrt{1-\frac{1}{c^2 x^2}}}{\left(c^2 d+e\right) \left(d+e x^2\right)}+\frac{4 b \sec ^{-1}(c x) \left(d+2 e x^2\right)}{\left(d+e x^2\right)^2}+\frac{4 b \sin ^{-1}\left(\frac{1}{c x}\right)}{d}}{16 e^2}","\frac{x^4 \left(a+b \sec ^{-1}(c x)\right)}{4 d \left(d+e x^2\right)^2}-\frac{b c x \left(c^2 d+2 e\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2 d+e}}\right)}{8 d e^{3/2} \sqrt{c^2 x^2} \left(c^2 d+e\right)^{3/2}}+\frac{b c x \sqrt{c^2 x^2-1}}{8 e \sqrt{c^2 x^2} \left(c^2 d+e\right) \left(d+e x^2\right)}",1,"-1/16*((-4*a*d)/(d + e*x^2)^2 + (8*a)/(d + e*x^2) - (2*b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x)/((c^2*d + e)*(d + e*x^2)) + (4*b*(d + 2*e*x^2)*ArcSec[c*x])/(d + e*x^2)^2 + (4*b*ArcSin[1/(c*x)])/d + (b*Sqrt[e]*(c^2*d + 2*e)*Log[(-16*d*Sqrt[-(c^2*d) - e]*e^(3/2)*(Sqrt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(c^2*d + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(-(c^2*d) - e)^(3/2)) + (b*Sqrt[e]*(c^2*d + 2*e)*Log[((16*I)*d*Sqrt[-(c^2*d) - e]*e^(3/2)*(-Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(c^2*d + 2*e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d*(-(c^2*d) - e)^(3/2)))/e^2","C",1
106,1,386,193,0.9770049,"\int \frac{x \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x*(a + b*ArcSec[c*x]))/(d + e*x^2)^3,x]","\frac{1}{16} \left(-\frac{4 a}{e \left(d+e x^2\right)^2}-\frac{b \left(3 c^2 d+2 e\right) \log \left(-\frac{16 d^2 \sqrt{e} \sqrt{c^2 (-d)-e} \left(\sqrt{e}+c x \left(-\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 (-d)-e}+i c \sqrt{d}\right)\right)}{b \left(3 c^2 d+2 e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{d^2 \sqrt{e} \left(c^2 (-d)-e\right)^{3/2}}-\frac{b \left(3 c^2 d+2 e\right) \log \left(\frac{16 i d^2 \sqrt{e} \sqrt{c^2 (-d)-e} \left(-\sqrt{e}+c x \left(\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 (-d)-e}+i c \sqrt{d}\right)\right)}{b \left(3 c^2 d+2 e\right) \left(\sqrt{d}+i \sqrt{e} x\right)}\right)}{d^2 \sqrt{e} \left(c^2 (-d)-e\right)^{3/2}}-\frac{2 b c x \sqrt{1-\frac{1}{c^2 x^2}}}{d \left(c^2 d+e\right) \left(d+e x^2\right)}-\frac{4 b \sin ^{-1}\left(\frac{1}{c x}\right)}{d^2 e}-\frac{4 b \sec ^{-1}(c x)}{e \left(d+e x^2\right)^2}\right)","-\frac{a+b \sec ^{-1}(c x)}{4 e \left(d+e x^2\right)^2}+\frac{b c x \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{4 d^2 e \sqrt{c^2 x^2}}-\frac{b c x \left(3 c^2 d+2 e\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2 d+e}}\right)}{8 d^2 \sqrt{e} \sqrt{c^2 x^2} \left(c^2 d+e\right)^{3/2}}-\frac{b c x \sqrt{c^2 x^2-1}}{8 d \sqrt{c^2 x^2} \left(c^2 d+e\right) \left(d+e x^2\right)}",1,"((-4*a)/(e*(d + e*x^2)^2) - (2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x)/(d*(c^2*d + e)*(d + e*x^2)) - (4*b*ArcSec[c*x])/(e*(d + e*x^2)^2) - (4*b*ArcSin[1/(c*x)])/(d^2*e) - (b*(3*c^2*d + 2*e)*Log[(-16*d^2*Sqrt[-(c^2*d) - e]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(3*c^2*d + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d^2*(-(c^2*d) - e)^(3/2)*Sqrt[e]) - (b*(3*c^2*d + 2*e)*Log[((16*I)*d^2*Sqrt[-(c^2*d) - e]*Sqrt[e]*(-Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(3*c^2*d + 2*e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d^2*(-(c^2*d) - e)^(3/2)*Sqrt[e]))/16","C",1
107,0,0,685,52.4874813,"\int \frac{a+b \sec ^{-1}(c x)}{x \left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcSec[c*x])/(x*(d + e*x^2)^3),x]","\int \frac{a+b \sec ^{-1}(c x)}{x \left(d+e x^2\right)^3} \, dx","-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d^3}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d^3}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d^3}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1+\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d^3}+\frac{e^2 \left(a+b \sec ^{-1}(c x)\right)}{4 d^3 \left(\frac{d}{x^2}+e\right)^2}-\frac{e \left(a+b \sec ^{-1}(c x)\right)}{d^3 \left(\frac{d}{x^2}+e\right)}+\frac{i \left(a+b \sec ^{-1}(c x)\right)^2}{2 b d^3}+\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d^3}+\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d^3}+\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d^3}+\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d^3}+\frac{b \sqrt{e} \left(c^2 d+2 e\right) \tan ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{8 d^3 \left(c^2 d+e\right)^{3/2}}-\frac{b \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{d^3 \sqrt{c^2 d+e}}+\frac{b c e \sqrt{1-\frac{1}{c^2 x^2}}}{8 d^2 x \left(c^2 d+e\right) \left(\frac{d}{x^2}+e\right)}",1,"Integrate[(a + b*ArcSec[c*x])/(x*(d + e*x^2)^3), x]","F",-1
108,1,1819,1124,6.2010863,"\int \frac{x^4 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^4*(a + b*ArcSec[c*x]))/(d + e*x^2)^3,x]","-\frac{5 a x}{8 e^2 \left(e x^2+d\right)}+\frac{a d x}{4 e^2 \left(e x^2+d\right)^2}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{d} e^{5/2}}+b \left(\frac{5 \left(\frac{i \left(\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-\frac{\log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i c \sqrt{d}-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{-d c^2-e}}\right)}{\sqrt{d}}-\frac{\sec ^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}\right)}{16 e^2}+\frac{5 \left(-\frac{\sec ^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}-\frac{i \left(\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-\frac{\log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}\right)}{\sqrt{d}}\right)}{16 e^2}+\frac{i \sqrt{d} \left(\frac{\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-i \left(\frac{c \sqrt{d} \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}} x}{\left(d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}+\frac{\left(2 d c^2+e\right) \log \left(-\frac{4 d \sqrt{e} \sqrt{d c^2+e} \left(c \left(c \sqrt{d}-\sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+i \sqrt{e}\right)}{\left(2 d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}\right)}{\left(d c^2+e\right)^{3/2}}\right)}{d}-\frac{\sec ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x-i \sqrt{d}\right)^2}\right)}{16 e^2}-\frac{i \sqrt{d} \left(\frac{i c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}} x}{\sqrt{d} \left(d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}-\frac{\sec ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x+i \sqrt{d}\right)^2}+\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{d \sqrt{e}}-\frac{i \left(2 d c^2+e\right) \log \left(\frac{4 d \sqrt{e} \sqrt{d c^2+e} \left(c \left(\sqrt{d} c+\sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x-i \sqrt{e}\right)}{\left(2 d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{d \left(d c^2+e\right)^{3/2}}\right)}{16 e^2}+\frac{3 \left(8 \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 i \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-2 \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)\right)}{32 \sqrt{d} e^{5/2}}-\frac{3 \left(8 \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-2 i \sec ^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+2 i \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-2 \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)\right)}{32 \sqrt{d} e^{5/2}}\right)","\frac{b \sqrt{-d} \sqrt{1-\frac{1}{c^2 x^2}} c}{16 e^{3/2} \left(d c^2+e\right) \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{b \sqrt{-d} \sqrt{1-\frac{1}{c^2 x^2}} c}{16 e^{3/2} \left(d c^2+e\right) \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{3 \left(a+b \sec ^{-1}(c x)\right)}{16 e^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{3 \left(a+b \sec ^{-1}(c x)\right)}{16 e^2 \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{\sqrt{-d} \left(a+b \sec ^{-1}(c x)\right)}{16 e^{3/2} \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)^2}-\frac{\sqrt{-d} \left(a+b \sec ^{-1}(c x)\right)}{16 e^{3/2} \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)^2}+\frac{3 b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 \sqrt{d} e^2 \sqrt{d c^2+e}}+\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 \sqrt{d} e \left(d c^2+e\right)^{3/2}}+\frac{3 b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 \sqrt{d} e^2 \sqrt{d c^2+e}}+\frac{b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 \sqrt{d} e \left(d c^2+e\right)^{3/2}}+\frac{3 \left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left(a+b \sec ^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{i \sec ^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 \left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left(a+b \sec ^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{i \sec ^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}",1,"(a*d*x)/(4*e^2*(d + e*x^2)^2) - (5*a*x)/(8*e^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[d]*e^(5/2)) + b*((5*(-(ArcSec[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/(16*e^2) + (5*(-(ArcSec[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/(16*e^2) + ((I/16)*Sqrt[d]*(-(ArcSec[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2)) + (ArcSin[1/(c*x)]/Sqrt[e] - I*((c*Sqrt[d]*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(I*Sqrt[e] + c*(c*Sqrt[d] - Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(c^2*d + e)^(3/2)))/d))/e^2 - ((I/16)*Sqrt[d]*((I*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcSec[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + ArcSin[1/(c*x)]/(d*Sqrt[e]) - (I*(2*c^2*d + e)*Log[(4*d*Sqrt[e]*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/e^2 + (3*(8*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - (2*I)*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 2*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + PolyLog[2, -E^((2*I)*ArcSec[c*x])]))/(32*Sqrt[d]*e^(5/2)) - (3*(8*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - (2*I)*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 2*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + PolyLog[2, -E^((2*I)*ArcSec[c*x])]))/(32*Sqrt[d]*e^(5/2)))","A",0
109,1,1827,1124,6.1383543,"\int \frac{x^2 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^2*(a + b*ArcSec[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{\frac{i \left(\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-\frac{\log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i c \sqrt{d}-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{-d c^2-e}}\right)}{\sqrt{d}}-\frac{\sec ^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}}{16 d e}-\frac{-\frac{\sec ^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}-\frac{i \left(\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-\frac{\log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}\right)}{\sqrt{d}}}{16 d e}-\frac{i \left(\frac{\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-i \left(\frac{c \sqrt{d} \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}} x}{\left(d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}+\frac{\left(2 d c^2+e\right) \log \left(-\frac{4 d \sqrt{e} \sqrt{d c^2+e} \left(c \left(c \sqrt{d}-\sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+i \sqrt{e}\right)}{\left(2 d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}\right)}{\left(d c^2+e\right)^{3/2}}\right)}{d}-\frac{\sec ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x-i \sqrt{d}\right)^2}\right)}{16 \sqrt{d} e}+\frac{i \left(\frac{i c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}} x}{\sqrt{d} \left(d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}-\frac{\sec ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x+i \sqrt{d}\right)^2}+\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{d \sqrt{e}}-\frac{i \left(2 d c^2+e\right) \log \left(\frac{4 d \sqrt{e} \sqrt{d c^2+e} \left(c \left(\sqrt{d} c+\sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x-i \sqrt{e}\right)}{\left(2 d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{d \left(d c^2+e\right)^{3/2}}\right)}{16 \sqrt{d} e}+\frac{8 \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 i \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-2 \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)}{32 d^{3/2} e^{3/2}}-\frac{8 \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-2 i \sec ^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+2 i \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-2 \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)}{32 d^{3/2} e^{3/2}}\right)","\frac{b \sqrt{1-\frac{1}{c^2 x^2}} c}{16 \sqrt{-d} \sqrt{e} \left(d c^2+e\right) \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} c}{16 \sqrt{-d} \sqrt{e} \left(d c^2+e\right) \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{a+b \sec ^{-1}(c x)}{16 d e \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{a+b \sec ^{-1}(c x)}{16 d e \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{a+b \sec ^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)^2}-\frac{a+b \sec ^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)^2}+\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{3/2} e \sqrt{d c^2+e}}-\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{3/2} \left(d c^2+e\right)^{3/2}}+\frac{b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{3/2} e \sqrt{d c^2+e}}-\frac{b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{3/2} \left(d c^2+e\right)^{3/2}}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{i \sec ^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{\left(a+b \sec ^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{i \sec ^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\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*(-1/16*(-(ArcSec[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d])/(d*e) - (-(ArcSec[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d])/(16*d*e) - ((I/16)*(-(ArcSec[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2)) + (ArcSin[1/(c*x)]/Sqrt[e] - I*((c*Sqrt[d]*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(I*Sqrt[e] + c*(c*Sqrt[d] - Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(c^2*d + e)^(3/2)))/d))/(Sqrt[d]*e) + ((I/16)*((I*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcSec[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + ArcSin[1/(c*x)]/(d*Sqrt[e]) - (I*(2*c^2*d + e)*Log[(4*d*Sqrt[e]*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) + (8*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - (2*I)*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 2*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + PolyLog[2, -E^((2*I)*ArcSec[c*x])])/(32*d^(3/2)*e^(3/2)) - (8*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - (2*I)*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 2*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + PolyLog[2, -E^((2*I)*ArcSec[c*x])])/(32*d^(3/2)*e^(3/2)))","A",0
110,1,1812,1114,6.061592,"\int \frac{a+b \sec ^{-1}(c x)}{\left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcSec[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 \left(\frac{i \left(\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-\frac{\log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i c \sqrt{d}-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{-d c^2-e}}\right)}{\sqrt{d}}-\frac{\sec ^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}\right)}{16 d^2}-\frac{3 \left(-\frac{\sec ^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}-\frac{i \left(\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-\frac{\log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}\right)}{\sqrt{d}}\right)}{16 d^2}+\frac{i \left(\frac{\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-i \left(\frac{c \sqrt{d} \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}} x}{\left(d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}+\frac{\left(2 d c^2+e\right) \log \left(-\frac{4 d \sqrt{e} \sqrt{d c^2+e} \left(c \left(c \sqrt{d}-\sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+i \sqrt{e}\right)}{\left(2 d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}\right)}{\left(d c^2+e\right)^{3/2}}\right)}{d}-\frac{\sec ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x-i \sqrt{d}\right)^2}\right)}{16 d^{3/2}}-\frac{i \left(\frac{i c \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}} x}{\sqrt{d} \left(d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}-\frac{\sec ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x+i \sqrt{d}\right)^2}+\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{d \sqrt{e}}-\frac{i \left(2 d c^2+e\right) \log \left(\frac{4 d \sqrt{e} \sqrt{d c^2+e} \left(c \left(\sqrt{d} c+\sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x-i \sqrt{e}\right)}{\left(2 d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{d \left(d c^2+e\right)^{3/2}}\right)}{16 d^{3/2}}+\frac{3 \left(8 \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 i \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-2 \text{Li}_2\left(\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)\right)}{32 d^{5/2} \sqrt{e}}-\frac{3 \left(8 \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \tan \left(\frac{1}{2} \sec ^{-1}(c x)\right)}{\sqrt{d c^2+e}}\right)-2 i \sec ^{-1}(c x) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{i \sec ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-2 i \sec ^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+2 i \sec ^{-1}(c x) \log \left(1+e^{2 i \sec ^{-1}(c x)}\right)-2 \text{Li}_2\left(-\frac{i \left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)-2 \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{i \sec ^{-1}(c x)}}{c \sqrt{d}}\right)+\text{Li}_2\left(-e^{2 i \sec ^{-1}(c x)}\right)\right)}{32 d^{5/2} \sqrt{e}}\right)","\frac{b \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}} c}{16 (-d)^{3/2} \left(d c^2+e\right) \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{b \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}} c}{16 (-d)^{3/2} \left(d c^2+e\right) \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}-\frac{5 \left(a+b \sec ^{-1}(c x)\right)}{16 d^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{5 \left(a+b \sec ^{-1}(c x)\right)}{16 d^2 \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{\sqrt{e} \left(a+b \sec ^{-1}(c x)\right)}{16 (-d)^{3/2} \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)^2}-\frac{\sqrt{e} \left(a+b \sec ^{-1}(c x)\right)}{16 (-d)^{3/2} \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)^2}-\frac{5 b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{5/2} \sqrt{d c^2+e}}+\frac{b e \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{5/2} \left(d c^2+e\right)^{3/2}}-\frac{5 b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{5/2} \sqrt{d c^2+e}}+\frac{b e \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{5/2} \left(d c^2+e\right)^{3/2}}+\frac{3 \left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left(a+b \sec ^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{i \sec ^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \left(a+b \sec ^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left(a+b \sec ^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{i \sec ^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 i b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 i b \text{Li}_2\left(\frac{c \sqrt{-d} e^{i \sec ^{-1}(c x)}}{\sqrt{e}+\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*(-(ArcSec[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/(16*d^2) - (3*(-(ArcSec[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/(16*d^2) + ((I/16)*(-(ArcSec[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2)) + (ArcSin[1/(c*x)]/Sqrt[e] - I*((c*Sqrt[d]*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(I*Sqrt[e] + c*(c*Sqrt[d] - Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(c^2*d + e)^(3/2)))/d))/d^(3/2) - ((I/16)*((I*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcSec[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + ArcSin[1/(c*x)]/(d*Sqrt[e]) - (I*(2*c^2*d + e)*Log[(4*d*Sqrt[e]*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/d^(3/2) + (3*(8*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - (2*I)*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] - Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*ArcSec[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 2*PolyLog[2, (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + PolyLog[2, -E^((2*I)*ArcSec[c*x])]))/(32*d^(5/2)*Sqrt[e]) - (3*(8*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Tan[ArcSec[c*x]/2])/Sqrt[c^2*d + e]] - (2*I)*ArcSec[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - (2*I)*ArcSec[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + (2*I)*ArcSec[c*x]*Log[1 + E^((2*I)*ArcSec[c*x])] - 2*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] - 2*PolyLog[2, (I*(Sqrt[e] + Sqrt[c^2*d + e])*E^(I*ArcSec[c*x]))/(c*Sqrt[d])] + PolyLog[2, -E^((2*I)*ArcSec[c*x])]))/(32*d^(5/2)*Sqrt[e]))","A",0
111,1,366,403,0.6705389,"\int x^5 \sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^5*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]),x]","\frac{\sqrt{d+e x^2} \left(16 a c^5 \left(8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right)+16 b c^5 \sec ^{-1}(c x) \left(8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right)-b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^4 \left(-41 d^2+22 d e x^2+40 e^2 x^4\right)+2 c^2 e \left(19 d+25 e x^2\right)+75 e^2\right)\right)}{1680 c^5 e^3}-\frac{b \left(e x^4 \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^2 x^2} \left(105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right) \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)-128 c^4 d^4 \left(c^2 x^2-1\right) \sqrt{\frac{d}{e x^2}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;\frac{1}{c^2 x^2},-\frac{d}{e x^2}\right)\right)}{3360 c^5 e^3 x \left(c^2 x^2-1\right) \sqrt{d+e x^2}}","\frac{d^2 \left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{3 e^3}-\frac{2 d \left(d+e x^2\right)^{5/2} \left(a+b \sec ^{-1}(c x)\right)}{5 e^3}+\frac{\left(d+e x^2\right)^{7/2} \left(a+b \sec ^{-1}(c x)\right)}{7 e^3}+\frac{8 b c d^{7/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{105 e^3 \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{5/2}}{42 c e^2 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(29 c^2 d-25 e\right) \left(d+e x^2\right)^{3/2}}{840 c^3 e^2 \sqrt{c^2 x^2}}-\frac{b x \left(105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{1680 c^6 e^{5/2} \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(23 c^4 d^2+12 c^2 d e-75 e^2\right) \sqrt{d+e x^2}}{1680 c^5 e^2 \sqrt{c^2 x^2}}",1,"-1/3360*(b*(-128*c^4*d^4*Sqrt[1 + d/(e*x^2)]*(-1 + c^2*x^2)*AppellF1[1, 1/2, 1/2, 2, 1/(c^2*x^2), -(d/(e*x^2))] + e*(105*c^6*d^3 - 35*c^4*d^2*e + 63*c^2*d*e^2 + 75*e^3)*Sqrt[1 - 1/(c^2*x^2)]*x^4*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e*x^2)/d)]))/(c^5*e^3*x*(-1 + c^2*x^2)*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(16*a*c^5*(8*d^3 - 4*d^2*e*x^2 + 3*d*e^2*x^4 + 15*e^3*x^6) - b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(75*e^2 + 2*c^2*e*(19*d + 25*e*x^2) + c^4*(-41*d^2 + 22*d*e*x^2 + 40*e^2*x^4)) + 16*b*c^5*(8*d^3 - 4*d^2*e*x^2 + 3*d*e^2*x^4 + 15*e^3*x^6)*ArcSec[c*x]))/(1680*c^5*e^3)","C",0
112,1,327,294,0.7729747,"\int x^3 \sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^3*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]),x]","\frac{\sqrt{d+e x^2} \left(8 a c^3 \left(-2 d^2+d e x^2+3 e^2 x^4\right)+8 b c^3 \sec ^{-1}(c x) \left(-2 d^2+d e x^2+3 e^2 x^4\right)-b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 \left(7 d+6 e x^2\right)+9 e\right)\right)}{120 c^3 e^2}+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(16 c^7 d^{5/2} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)+\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \left(15 c^4 d^2-10 c^2 d e-9 e^2\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)\right)}{120 c^6 e^2 \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","-\frac{d \left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{3 e^2}+\frac{\left(d+e x^2\right)^{5/2} \left(a+b \sec ^{-1}(c x)\right)}{5 e^2}-\frac{2 b c d^{5/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{15 e^2 \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{20 c e \sqrt{c^2 x^2}}+\frac{b x \left(15 c^4 d^2-10 c^2 d e-9 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{120 c^4 e^{3/2} \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(c^2 d+9 e\right) \sqrt{d+e x^2}}{120 c^3 e \sqrt{c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(8*a*c^3*(-2*d^2 + d*e*x^2 + 3*e^2*x^4) - b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(9*e + c^2*(7*d + 6*e*x^2)) + 8*b*c^3*(-2*d^2 + d*e*x^2 + 3*e^2*x^4)*ArcSec[c*x]))/(120*c^3*e^2) + (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*(15*c^4*d^2 - 10*c^2*d*e - 9*e^2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 16*c^7*d^(5/2)*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(120*c^6*e^2*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
113,1,266,195,0.4856651,"\int x \sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]),x]","\frac{\sqrt{d+e x^2} \left(2 a c \left(d+e x^2\right)-b e x \sqrt{1-\frac{1}{c^2 x^2}}+2 b c \sec ^{-1}(c x) \left(d+e x^2\right)\right)}{6 c e}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \left(3 c^2 d+e\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)+2 c^5 d^{3/2} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)\right)}{6 c^4 e \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","\frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{3 e}+\frac{b c d^{3/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{3 e \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{6 c \sqrt{c^2 x^2}}-\frac{b x \left(3 c^2 d+e\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{6 c^2 \sqrt{e} \sqrt{c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(-(b*e*Sqrt[1 - 1/(c^2*x^2)]*x) + 2*a*c*(d + e*x^2) + 2*b*c*(d + e*x^2)*ArcSec[c*x]))/(6*c*e) - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*(3*c^2*d + e)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 2*c^5*d^(3/2)*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(6*c^4*e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
114,0,0,26,4.4718771,"\int \frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{x} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/x,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{x} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{x},x\right)",0,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/x, x]","A",-1
115,0,0,26,4.6730465,"\int \frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{x^3} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/x^3,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{x^3} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{x^3},x\right)",0,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/x^3, x]","A",-1
116,0,0,26,9.2319944,"\int x^2 \sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^2*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]),x]","\int x^2 \sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right) \, dx","\text{Int}\left(x^2 \sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right),x\right)",0,"Integrate[x^2*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]), x]","A",-1
117,0,0,23,4.3717677,"\int \sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]),x]","\int \sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right) \, dx","\text{Int}\left(\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right),x\right)",0,"Integrate[Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]), x]","A",-1
118,0,0,26,1.5029226,"\int \frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{x^2} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/x^2,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{x^2} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{x^2},x\right)",0,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/x^2, x]","A",-1
119,1,247,328,0.6514885,"\int \frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{x^4} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/x^4,x]","\frac{\sqrt{d+e x^2} \left(-3 a \left(d+e x^2\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 d x^2+d+4 e x^2\right)-3 b \sec ^{-1}(c x) \left(d+e x^2\right)\right)}{9 d x^3}-\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(2 c^2 d \left(c^2 d+2 e\right) E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)-\left(2 c^4 d^2+5 c^2 d e+3 e^2\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{9 \sqrt{-c^2} d \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}","-\frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{3 d x^3}+\frac{2 b c \sqrt{c^2 x^2-1} \left(c^2 d+2 e\right) \sqrt{d+e x^2}}{9 d \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{9 x^2 \sqrt{c^2 x^2}}+\frac{b x \sqrt{1-c^2 x^2} \left(c^2 d+e\right) \left(2 c^2 d+3 e\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{9 d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{2 b c^2 x \sqrt{1-c^2 x^2} \left(c^2 d+2 e\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{9 d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}",1,"(Sqrt[d + e*x^2]*(-3*a*(d + e*x^2) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 2*c^2*d*x^2 + 4*e*x^2) - 3*b*(d + e*x^2)*ArcSec[c*x]))/(9*d*x^3) - ((I/9)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(2*c^2*d*(c^2*d + 2*e)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] - (2*c^4*d^2 + 5*c^2*d*e + 3*e^2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
120,1,325,453,0.7393112,"\int \frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{x^6} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/x^6,x]","\frac{\sqrt{d+e x^2} \left(-15 a \left(3 d^2+d e x^2-2 e^2 x^4\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(d e x^2 \left(19 c^2 x^2+8\right)+3 d^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)-31 e^2 x^4\right)-15 b \sec ^{-1}(c x) \left(3 d^2+d e x^2-2 e^2 x^4\right)\right)}{225 d^2 x^5}-\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d \left(24 c^4 d^2+19 c^2 d e-31 e^2\right) E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)+\left(-24 c^6 d^3-31 c^4 d^2 e+23 c^2 d e^2+30 e^3\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{225 \sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}","\frac{2 e \left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{15 d^2 x^3}-\frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{5 d x^5}+\frac{b c \sqrt{c^2 x^2-1} \left(12 c^2 d-e\right) \sqrt{d+e x^2}}{225 d x^2 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{25 d x^4 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left(24 c^4 d^2+19 c^2 d e-31 e^2\right) \sqrt{d+e x^2}}{225 d^2 \sqrt{c^2 x^2}}+\frac{b x \sqrt{1-c^2 x^2} \left(c^2 d+e\right) \left(24 c^4 d^2+7 c^2 d e-30 e^2\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{225 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \left(24 c^4 d^2+19 c^2 d e-31 e^2\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{225 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}",1,"(Sqrt[d + e*x^2]*(-15*a*(3*d^2 + d*e*x^2 - 2*e^2*x^4) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(-31*e^2*x^4 + d*e*x^2*(8 + 19*c^2*x^2) + 3*d^2*(3 + 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(3*d^2 + d*e*x^2 - 2*e^2*x^4)*ArcSec[c*x]))/(225*d^2*x^5) - ((I/225)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] + (-24*c^6*d^3 - 31*c^4*d^2*e + 23*c^2*d*e^2 + 30*e^3)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
121,1,339,374,0.6265227,"\int x^3 \left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]),x]","-\frac{\sqrt{d+e x^2} \left(48 a c^5 \left(2 d-5 e x^2\right) \left(d+e x^2\right)^2+48 b c^5 \sec ^{-1}(c x) \left(2 d-5 e x^2\right) \left(d+e x^2\right)^2+b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^4 \left(57 d^2+106 d e x^2+40 e^2 x^4\right)+2 c^2 e \left(82 d+25 e x^2\right)+75 e^2\right)\right)}{1680 c^5 e^2}-\frac{b \left(32 c^4 d^4 \left(c^2 x^2-1\right) \sqrt{\frac{d}{e x^2}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;\frac{1}{c^2 x^2},-\frac{d}{e x^2}\right)+e x^4 \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^2 x^2} \left(-35 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2+25 e^3\right) \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)}{1120 c^5 e^2 x \left(c^2 x^2-1\right) \sqrt{d+e x^2}}","-\frac{d \left(d+e x^2\right)^{5/2} \left(a+b \sec ^{-1}(c x)\right)}{5 e^2}+\frac{\left(d+e x^2\right)^{7/2} \left(a+b \sec ^{-1}(c x)\right)}{7 e^2}-\frac{2 b c d^{7/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{35 e^2 \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{5/2}}{42 c e \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(13 c^2 d+25 e\right) \left(d+e x^2\right)^{3/2}}{840 c^3 e \sqrt{c^2 x^2}}+\frac{b x \left(35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{560 c^6 e^{3/2} \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(3 c^4 d^2-38 c^2 d e-25 e^2\right) \sqrt{d+e x^2}}{560 c^5 e \sqrt{c^2 x^2}}",1,"-1/1120*(b*(32*c^4*d^4*Sqrt[1 + d/(e*x^2)]*(-1 + c^2*x^2)*AppellF1[1, 1/2, 1/2, 2, 1/(c^2*x^2), -(d/(e*x^2))] + e*(-35*c^6*d^3 + 35*c^4*d^2*e + 63*c^2*d*e^2 + 25*e^3)*Sqrt[1 - 1/(c^2*x^2)]*x^4*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e*x^2)/d)]))/(c^5*e^2*x*(-1 + c^2*x^2)*Sqrt[d + e*x^2]) - (Sqrt[d + e*x^2]*(48*a*c^5*(2*d - 5*e*x^2)*(d + e*x^2)^2 + b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(75*e^2 + 2*c^2*e*(82*d + 25*e*x^2) + c^4*(57*d^2 + 106*d*e*x^2 + 40*e^2*x^4)) + 48*b*c^5*(2*d - 5*e*x^2)*(d + e*x^2)^2*ArcSec[c*x]))/(1680*c^5*e^2)","C",0
122,1,305,262,0.7677,"\int x \left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]),x]","\frac{\sqrt{d+e x^2} \left(8 a c^3 \left(d+e x^2\right)^2+8 b c^3 \sec ^{-1}(c x) \left(d+e x^2\right)^2-b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 \left(9 d+2 e x^2\right)+3 e\right)\right)}{40 c^3 e}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(8 c^7 d^{5/2} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)+\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \left(15 c^4 d^2+10 c^2 d e+3 e^2\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)\right)}{40 c^6 e \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","\frac{\left(d+e x^2\right)^{5/2} \left(a+b \sec ^{-1}(c x)\right)}{5 e}+\frac{b c d^{5/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{5 e \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{20 c \sqrt{c^2 x^2}}-\frac{b x \left(15 c^4 d^2+10 c^2 d e+3 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{40 c^4 \sqrt{e} \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(7 c^2 d+3 e\right) \sqrt{d+e x^2}}{40 c^3 \sqrt{c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(8*a*c^3*(d + e*x^2)^2 - b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(3*e + c^2*(9*d + 2*e*x^2)) + 8*b*c^3*(d + e*x^2)^2*ArcSec[c*x]))/(40*c^3*e) - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*(15*c^4*d^2 + 10*c^2*d*e + 3*e^2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 8*c^7*d^(5/2)*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(40*c^6*e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
123,0,0,26,4.9693176,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{x} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{x},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x, x]","A",-1
124,0,0,26,5.5413914,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x^3,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{x^3} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{x^3},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x^3, x]","A",-1
125,0,0,26,9.5930031,"\int x^2 \left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]),x]","\int x^2 \left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right) \, dx","\text{Int}\left(x^2 \left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right),x\right)",0,"Integrate[x^2*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]), x]","A",-1
126,0,0,23,5.5339044,"\int \left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]),x]","\int \left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right) \, dx","\text{Int}\left(\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right),x\right)",0,"Integrate[(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]), x]","A",-1
127,0,0,26,8.8877988,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x^2,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{x^2} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{x^2},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x^2, x]","A",-1
128,0,0,26,11.4792824,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x^4,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{x^4} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{x^4},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x^4, x]","A",-1
129,1,303,416,0.6779112,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x^6,x]","\frac{\sqrt{d+e x^2} \left(-15 a \left(d+e x^2\right)^2+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(d e x^2 \left(23 c^2 x^2+11\right)+d^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)+23 e^2 x^4\right)-15 b \sec ^{-1}(c x) \left(d+e x^2\right)^2\right)}{75 d x^5}-\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d \left(8 c^4 d^2+23 c^2 d e+23 e^2\right) E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)-\left(8 c^6 d^3+27 c^4 d^2 e+34 c^2 d e^2+15 e^3\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{75 \sqrt{-c^2} d \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}","-\frac{\left(d+e x^2\right)^{5/2} \left(a+b \sec ^{-1}(c x)\right)}{5 d x^5}+\frac{4 b c \sqrt{c^2 x^2-1} \left(c^2 d+2 e\right) \sqrt{d+e x^2}}{75 x^2 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{25 x^4 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left(8 c^4 d^2+23 c^2 d e+23 e^2\right) \sqrt{d+e x^2}}{75 d \sqrt{c^2 x^2}}+\frac{b x \sqrt{1-c^2 x^2} \left(c^2 d+e\right) \left(8 c^4 d^2+19 c^2 d e+15 e^2\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{75 d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \left(8 c^4 d^2+23 c^2 d e+23 e^2\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{75 d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}",1,"(Sqrt[d + e*x^2]*(-15*a*(d + e*x^2)^2 + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(23*e^2*x^4 + d*e*x^2*(11 + 23*c^2*x^2) + d^2*(3 + 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(d + e*x^2)^2*ArcSec[c*x]))/(75*d*x^5) - ((I/75)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] - (8*c^6*d^3 + 27*c^4*d^2*e + 34*c^2*d*e^2 + 15*e^3)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
130,1,383,554,0.8558454,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]))/x^8,x]","\frac{\sqrt{d+e x^2} \left(-105 a \left(5 d-2 e x^2\right) \left(d+e x^2\right)^2+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(d e^2 x^4 \left(193 c^2 x^2+71\right)+3 d^2 e x^2 \left(176 c^4 x^4+83 c^2 x^2+61\right)+15 d^3 \left(16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right)-247 e^3 x^6\right)-105 b \sec ^{-1}(c x) \left(5 d-2 e x^2\right) \left(d+e x^2\right)^2\right)}{3675 d^2 x^7}-\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d \left(240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right) E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)-2 \left(120 c^8 d^4+324 c^6 d^3 e+221 c^4 d^2 e^2-88 c^2 d e^3-105 e^4\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{3675 \sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}","\frac{2 e \left(d+e x^2\right)^{5/2} \left(a+b \sec ^{-1}(c x)\right)}{35 d^2 x^5}-\frac{\left(d+e x^2\right)^{5/2} \left(a+b \sec ^{-1}(c x)\right)}{7 d x^7}+\frac{b c \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{5/2}}{49 d x^6 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left(30 c^2 d+11 e\right) \left(d+e x^2\right)^{3/2}}{1225 d x^4 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left(120 c^4 d^2+159 c^2 d e-37 e^2\right) \sqrt{d+e x^2}}{3675 d x^2 \sqrt{c^2 x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left(240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right) \sqrt{d+e x^2}}{3675 d^2 \sqrt{c^2 x^2}}+\frac{2 b x \sqrt{1-c^2 x^2} \left(c^2 d+e\right) \left(120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3675 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \left(240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3675 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}",1,"(Sqrt[d + e*x^2]*(-105*a*(5*d - 2*e*x^2)*(d + e*x^2)^2 + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(-247*e^3*x^6 + d*e^2*x^4*(71 + 193*c^2*x^2) + 3*d^2*e*x^2*(61 + 83*c^2*x^2 + 176*c^4*x^4) + 15*d^3*(5 + 6*c^2*x^2 + 8*c^4*x^4 + 16*c^6*x^6)) - 105*b*(5*d - 2*e*x^2)*(d + e*x^2)^2*ArcSec[c*x]))/(3675*d^2*x^7) - ((I/3675)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] - 2*(120*c^8*d^4 + 324*c^6*d^3*e + 221*c^4*d^2*e^2 - 88*c^2*d*e^3 - 105*e^4)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
131,1,328,321,0.7516459,"\int \frac{x^5 \left(a+b \sec ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x^5*(a + b*ArcSec[c*x]))/Sqrt[d + e*x^2],x]","\frac{\sqrt{d+e x^2} \left(8 a c^3 \left(8 d^2-4 d e x^2+3 e^2 x^4\right)+8 b c^3 \sec ^{-1}(c x) \left(8 d^2-4 d e x^2+3 e^2 x^4\right)+b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 \left(13 d-6 e x^2\right)-9 e\right)\right)}{120 c^3 e^3}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(64 c^7 d^{5/2} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)+\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \left(45 c^4 d^2-10 c^2 d e+9 e^2\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)\right)}{120 c^6 e^3 \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","\frac{d^2 \sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{e^3}-\frac{2 d \left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{3 e^3}+\frac{\left(d+e x^2\right)^{5/2} \left(a+b \sec ^{-1}(c x)\right)}{5 e^3}+\frac{8 b c d^{5/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{15 e^3 \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{20 c e^2 \sqrt{c^2 x^2}}-\frac{b x \left(45 c^4 d^2-10 c^2 d e+9 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{120 c^4 e^{5/2} \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(19 c^2 d-9 e\right) \sqrt{d+e x^2}}{120 c^3 e^2 \sqrt{c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(8*a*c^3*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4) + b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(-9*e + c^2*(13*d - 6*e*x^2)) + 8*b*c^3*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4)*ArcSec[c*x]))/(120*c^3*e^3) - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*(45*c^4*d^2 - 10*c^2*d*e + 9*e^2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 64*c^7*d^(5/2)*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(120*c^6*e^3*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
132,1,272,225,0.5199855,"\int \frac{x^3 \left(a+b \sec ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x^3*(a + b*ArcSec[c*x]))/Sqrt[d + e*x^2],x]","\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(\sqrt{c^2} \sqrt{e} \left(3 c^2 d-e\right) \sqrt{c^2 d+e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)+4 c^5 d^{3/2} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)\right)}{6 c^4 e^2 \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{\sqrt{d+e x^2} \left(4 a c d-2 a c e x^2+b e x \sqrt{1-\frac{1}{c^2 x^2}}+2 b c \sec ^{-1}(c x) \left(2 d-e x^2\right)\right)}{6 c e^2}","-\frac{d \sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{e^2}+\frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{3 e^2}-\frac{2 b c d^{3/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{3 e^2 \sqrt{c^2 x^2}}+\frac{b x \left(3 c^2 d-e\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{6 c^2 e^{3/2} \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{6 c e \sqrt{c^2 x^2}}",1,"-1/6*(Sqrt[d + e*x^2]*(4*a*c*d + b*e*Sqrt[1 - 1/(c^2*x^2)]*x - 2*a*c*e*x^2 + 2*b*c*(2*d - e*x^2)*ArcSec[c*x]))/(c*e^2) + (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*(3*c^2*d - e)*Sqrt[e]*Sqrt[c^2*d + e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 4*c^5*d^(3/2)*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(6*c^4*e^2*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
133,1,211,132,0.2375957,"\int \frac{x \left(a+b \sec ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x*(a + b*ArcSec[c*x]))/Sqrt[d + e*x^2],x]","\frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{e}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)+c^3 \sqrt{d} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)\right)}{c^2 e \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","\frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{e}+\frac{b c \sqrt{d} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{e \sqrt{c^2 x^2}}-\frac{b x \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{\sqrt{e} \sqrt{c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]))/e - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + c^3*Sqrt[d]*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(c^2*e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
134,0,0,26,1.4902415,"\int \frac{a+b \sec ^{-1}(c x)}{x \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(x*Sqrt[d + e*x^2]),x]","\int \frac{a+b \sec ^{-1}(c x)}{x \sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \sec ^{-1}(c x)}{x \sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcSec[c*x])/(x*Sqrt[d + e*x^2]), x]","A",-1
135,0,0,26,11.1281919,"\int \frac{a+b \sec ^{-1}(c x)}{x^3 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(x^3*Sqrt[d + e*x^2]),x]","\int \frac{a+b \sec ^{-1}(c x)}{x^3 \sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \sec ^{-1}(c x)}{x^3 \sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcSec[c*x])/(x^3*Sqrt[d + e*x^2]), x]","A",-1
136,0,0,26,9.6890142,"\int \frac{x^2 \left(a+b \sec ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x^2*(a + b*ArcSec[c*x]))/Sqrt[d + e*x^2],x]","\int \frac{x^2 \left(a+b \sec ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{x^2 \left(a+b \sec ^{-1}(c x)\right)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[(x^2*(a + b*ArcSec[c*x]))/Sqrt[d + e*x^2], x]","A",-1
137,0,0,23,0.8491172,"\int \frac{a+b \sec ^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSec[c*x])/Sqrt[d + e*x^2],x]","\int \frac{a+b \sec ^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \sec ^{-1}(c x)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcSec[c*x])/Sqrt[d + e*x^2], x]","A",-1
138,1,143,246,0.2061159,"\int \frac{a+b \sec ^{-1}(c x)}{x^2 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(x^2*Sqrt[d + e*x^2]),x]","\frac{\sqrt{d+e x^2} \left(-a+b c x \sqrt{1-\frac{1}{c^2 x^2}}-b \sec ^{-1}(c x)\right)}{d x}-\frac{b c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} E\left(\sin ^{-1}\left(\sqrt{-\frac{e}{d}} x\right)|-\frac{c^2 d}{e}\right)}{d \sqrt{1-c^2 x^2} \sqrt{-\frac{e}{d}} \sqrt{d+e x^2}}","-\frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{d x}+\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{d \sqrt{c^2 x^2}}+\frac{b x \sqrt{1-c^2 x^2} \left(c^2 d+e\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}",1,"(Sqrt[d + e*x^2]*(-a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x - b*ArcSec[c*x]))/(d*x) - (b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*EllipticE[ArcSin[Sqrt[-(e/d)]*x], -((c^2*d)/e)])/(d*Sqrt[-(e/d)]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","A",1
139,1,249,362,0.665155,"\int \frac{a+b \sec ^{-1}(c x)}{x^4 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(x^4*Sqrt[d + e*x^2]),x]","\frac{\sqrt{d+e x^2} \left(-3 a \left(d-2 e x^2\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 d x^2+d-5 e x^2\right)-3 b \sec ^{-1}(c x) \left(d-2 e x^2\right)\right)}{9 d^2 x^3}-\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d \left(2 c^2 d-5 e\right) E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)+2 \left(c^4 \left(-d^2\right)+2 c^2 d e+3 e^2\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{9 \sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}","\frac{2 e \sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{3 d^2 x}-\frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{3 d x^3}+\frac{b c \sqrt{c^2 x^2-1} \left(2 c^2 d-5 e\right) \sqrt{d+e x^2}}{9 d^2 \sqrt{c^2 x^2}}+\frac{2 b x \sqrt{1-c^2 x^2} \left(c^2 d-3 e\right) \left(c^2 d+e\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{9 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \left(2 c^2 d-5 e\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{9 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}+\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{9 d x^2 \sqrt{c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 2*c^2*d*x^2 - 5*e*x^2) - 3*a*(d - 2*e*x^2) - 3*b*(d - 2*e*x^2)*ArcSec[c*x]))/(9*d^2*x^3) - ((I/9)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(2*c^2*d - 5*e)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] + 2*(-(c^4*d^2) + 2*c^2*d*e + 3*e^2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
140,1,329,1006,0.740014,"\int \frac{a+b \sec ^{-1}(c x)}{x^6 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(x^6*Sqrt[d + e*x^2]),x]","\frac{\sqrt{d+e x^2} \left(-15 a \left(3 d^2-4 d e x^2+8 e^2 x^4\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(-d e x^2 \left(31 c^2 x^2+17\right)+3 d^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)+94 e^2 x^4\right)-15 b \sec ^{-1}(c x) \left(3 d^2-4 d e x^2+8 e^2 x^4\right)\right)}{225 d^3 x^5}-\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d \left(24 c^4 d^2-31 c^2 d e+94 e^2\right) E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)-\left(24 c^6 d^3-19 c^4 d^2 e+77 c^2 d e^2+120 e^3\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{225 \sqrt{-c^2} d^3 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}","-\frac{8 b e^2 x \sqrt{1-c^2 x^2} \sqrt{e x^2+d} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right) c^2}{15 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}+\frac{4 b e \left(2 d c^2+e\right) x \sqrt{1-c^2 x^2} \sqrt{e x^2+d} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right) c^2}{45 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}-\frac{b \left(8 d^2 c^4+3 d e c^2-2 e^2\right) x \sqrt{1-c^2 x^2} \sqrt{e x^2+d} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right) c^2}{75 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}+\frac{b \left(8 c^2 d-e\right) \left(d c^2+e\right) x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right) c^2}{75 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{e x^2+d}}-\frac{8 b e \left(d c^2+e\right) x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right) c^2}{45 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{e x^2+d}}+\frac{8 b e^2 \sqrt{c^2 x^2-1} \sqrt{e x^2+d} c}{15 d^3 \sqrt{c^2 x^2}}-\frac{4 b e \left(2 d c^2+e\right) \sqrt{c^2 x^2-1} \sqrt{e x^2+d} c}{45 d^3 \sqrt{c^2 x^2}}+\frac{b \left(8 d^2 c^4+3 d e c^2-2 e^2\right) \sqrt{c^2 x^2-1} \sqrt{e x^2+d} c}{75 d^3 \sqrt{c^2 x^2}}-\frac{4 b e \sqrt{c^2 x^2-1} \sqrt{e x^2+d} c}{45 d^2 x^2 \sqrt{c^2 x^2}}+\frac{b \left(4 d c^2+e\right) \sqrt{c^2 x^2-1} \sqrt{e x^2+d} c}{75 d^2 x^2 \sqrt{c^2 x^2}}+\frac{b \sqrt{c^2 x^2-1} \sqrt{e x^2+d} c}{25 d x^4 \sqrt{c^2 x^2}}-\frac{8 e^2 \sqrt{e x^2+d} \left(a+b \sec ^{-1}(c x)\right)}{15 d^3 x}+\frac{4 e \sqrt{e x^2+d} \left(a+b \sec ^{-1}(c x)\right)}{15 d^2 x^3}-\frac{\sqrt{e x^2+d} \left(a+b \sec ^{-1}(c x)\right)}{5 d x^5}+\frac{8 b e^2 \left(d c^2+e\right) x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{15 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{e x^2+d}}",1,"(Sqrt[d + e*x^2]*(-15*a*(3*d^2 - 4*d*e*x^2 + 8*e^2*x^4) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(94*e^2*x^4 - d*e*x^2*(17 + 31*c^2*x^2) + 3*d^2*(3 + 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(3*d^2 - 4*d*e*x^2 + 8*e^2*x^4)*ArcSec[c*x]))/(225*d^3*x^5) - ((I/225)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(24*c^4*d^2 - 31*c^2*d*e + 94*e^2)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] - (24*c^6*d^3 - 19*c^4*d^2*e + 77*c^2*d*e^2 + 120*e^3)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^3*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
141,1,303,252,0.6365399,"\int \frac{x^5 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^5*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2),x]","\frac{-2 a c \left(8 d^2+4 d e x^2-e^2 x^4\right)-b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right)-2 b c \sec ^{-1}(c x) \left(8 d^2+4 d e x^2-e^2 x^4\right)}{6 c e^3 \sqrt{d+e x^2}}+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(\sqrt{c^2} \sqrt{e} \left(9 c^2 d-e\right) \sqrt{c^2 d+e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)+16 c^5 d^{3/2} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)\right)}{6 c^4 e^3 \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","-\frac{d^2 \left(a+b \sec ^{-1}(c x)\right)}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{e^3}+\frac{\left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{3 e^3}-\frac{8 b c d^{3/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{3 e^3 \sqrt{c^2 x^2}}+\frac{b x \left(9 c^2 d-e\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{6 c^2 e^{5/2} \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{6 c e^2 \sqrt{c^2 x^2}}",1,"(-(b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2)) - 2*a*c*(8*d^2 + 4*d*e*x^2 - e^2*x^4) - 2*b*c*(8*d^2 + 4*d*e*x^2 - e^2*x^4)*ArcSec[c*x])/(6*c*e^3*Sqrt[d + e*x^2]) + (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*(9*c^2*d - e)*Sqrt[e]*Sqrt[c^2*d + e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 16*c^5*d^(3/2)*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(6*c^4*e^3*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
142,1,221,157,0.3626177,"\int \frac{x^3 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^3*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2),x]","\frac{\left(2 d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right)}{e^2 \sqrt{d+e x^2}}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)+2 c^3 \sqrt{d} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)\right)}{c^2 e^2 \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","\frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{e^2}+\frac{d \left(a+b \sec ^{-1}(c x)\right)}{e^2 \sqrt{d+e x^2}}-\frac{b x \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{e^{3/2} \sqrt{c^2 x^2}}+\frac{2 b c \sqrt{d} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{e^2 \sqrt{c^2 x^2}}",1,"((2*d + e*x^2)*(a + b*ArcSec[c*x]))/(e^2*Sqrt[d + e*x^2]) - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 2*c^3*Sqrt[d]*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(c^2*e^2*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
143,1,95,80,0.1517574,"\int \frac{x \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2),x]","\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)}{\sqrt{d} e \sqrt{c^2 x^2-1}}-\frac{a+b \sec ^{-1}(c x)}{e \sqrt{d+e x^2}}","-\frac{a+b \sec ^{-1}(c x)}{e \sqrt{d+e x^2}}-\frac{b c x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{\sqrt{d} e \sqrt{c^2 x^2}}",1,"-((a + b*ArcSec[c*x])/(e*Sqrt[d + e*x^2])) + (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]])/(Sqrt[d]*e*Sqrt[-1 + c^2*x^2])","A",1
144,0,0,26,15.674621,"\int \frac{a+b \sec ^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(x*(d + e*x^2)^(3/2)),x]","\int \frac{a+b \sec ^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{a+b \sec ^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(a + b*ArcSec[c*x])/(x*(d + e*x^2)^(3/2)), x]","A",-1
145,0,0,26,22.1721702,"\int \frac{a+b \sec ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(x^3*(d + e*x^2)^(3/2)),x]","\int \frac{a+b \sec ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{a+b \sec ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(a + b*ArcSec[c*x])/(x^3*(d + e*x^2)^(3/2)), x]","A",-1
146,0,0,26,8.1938437,"\int \frac{x^4 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^4*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2),x]","\int \frac{x^4 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^4 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(x^4*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2), x]","A",-1
147,0,0,26,3.8701764,"\int \frac{x^2 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2),x]","\int \frac{x^2 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^2 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(x^2*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2), x]","A",-1
148,1,113,109,0.1991051,"\int \frac{a+b \sec ^{-1}(c x)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(d + e*x^2)^(3/2),x]","\frac{x \left(a+b \sec ^{-1}(c x)\right)}{d \sqrt{d+e x^2}}-\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d \left(c^3 x^2-c\right) \sqrt{d+e x^2}}","\frac{x \left(a+b \sec ^{-1}(c x)\right)}{d \sqrt{d+e x^2}}-\frac{b x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}",1,"(x*(a + b*ArcSec[c*x]))/(d*Sqrt[d + e*x^2]) - (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(d*(-c + c^3*x^2)*Sqrt[d + e*x^2])","A",1
149,1,212,274,0.4848225,"\int \frac{a+b \sec ^{-1}(c x)}{x^2 \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(x^2*(d + e*x^2)^(3/2)),x]","\frac{-a \left(d+2 e x^2\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right)-b \sec ^{-1}(c x) \left(d+2 e x^2\right)}{d^2 x \sqrt{d+e x^2}}-\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)-\left(c^2 d+2 e\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{\sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}","-\frac{2 e x \left(a+b \sec ^{-1}(c x)\right)}{d^2 \sqrt{d+e x^2}}-\frac{a+b \sec ^{-1}(c x)}{d x \sqrt{d+e x^2}}+\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{d^2 \sqrt{c^2 x^2}}+\frac{b x \sqrt{1-c^2 x^2} \left(c^2 d+2 e\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}",1,"(b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2) - a*(d + 2*e*x^2) - b*(d + 2*e*x^2)*ArcSec[c*x])/(d^2*x*Sqrt[d + e*x^2]) - (I*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] - (c^2*d + 2*e)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
150,1,292,701,0.7391677,"\int \frac{a+b \sec ^{-1}(c x)}{x^4 \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(x^4*(d + e*x^2)^(3/2)),x]","\frac{-3 a \left(d^2-4 d e x^2-8 e^2 x^4\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(d^2 \left(2 c^2 x^2+1\right)+d e x^2 \left(2 c^2 x^2-13\right)-14 e^2 x^4\right)-3 b \sec ^{-1}(c x) \left(d^2-4 d e x^2-8 e^2 x^4\right)}{9 d^3 x^3 \sqrt{d+e x^2}}-\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(2 c^2 d \left(c^2 d-7 e\right) E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)+\left(-2 c^4 d^2+13 c^2 d e+24 e^2\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{9 \sqrt{-c^2} d^3 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}","\frac{8 e^2 x \left(a+b \sec ^{-1}(c x)\right)}{3 d^3 \sqrt{d+e x^2}}+\frac{4 e \left(a+b \sec ^{-1}(c x)\right)}{3 d^2 x \sqrt{d+e x^2}}-\frac{a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt{d+e x^2}}-\frac{8 b e^2 x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{2 b c \sqrt{c^2 x^2-1} \left(c^2 d-e\right) \sqrt{d+e x^2}}{9 d^3 \sqrt{c^2 x^2}}-\frac{4 b c e \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{3 d^3 \sqrt{c^2 x^2}}-\frac{2 b c^2 x \sqrt{1-c^2 x^2} \left(c^2 d-e\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{9 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}+\frac{4 b c^2 e x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}+\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{9 d^2 x^2 \sqrt{c^2 x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \left(2 c^2 d-e\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{9 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{4 b c^2 e x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}",1,"(-3*a*(d^2 - 4*d*e*x^2 - 8*e^2*x^4) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(-14*e^2*x^4 + d*e*x^2*(-13 + 2*c^2*x^2) + d^2*(1 + 2*c^2*x^2)) - 3*b*(d^2 - 4*d*e*x^2 - 8*e^2*x^4)*ArcSec[c*x])/(9*d^3*x^3*Sqrt[d + e*x^2]) - ((I/9)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(2*c^2*d*(c^2*d - 7*e)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] + (-2*c^4*d^2 + 13*c^2*d*e + 24*e^2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^3*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
151,1,312,244,0.5388944,"\int \frac{x^5 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^5*(a + b*ArcSec[c*x]))/(d + e*x^2)^(5/2),x]","\frac{a \left(c^2 d+e\right) \left(8 d^2+12 d e x^2+3 e^2 x^4\right)+b \left(c^2 d+e\right) \sec ^{-1}(c x) \left(8 d^2+12 d e x^2+3 e^2 x^4\right)-b c d e x \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right)}{3 e^3 \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(3 \sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)+8 c^3 \sqrt{d} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)\right)}{3 c^2 e^3 \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","-\frac{d^2 \left(a+b \sec ^{-1}(c x)\right)}{3 e^3 \left(d+e x^2\right)^{3/2}}+\frac{2 d \left(a+b \sec ^{-1}(c x)\right)}{e^3 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right)}{e^3}-\frac{b x \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{e^{5/2} \sqrt{c^2 x^2}}+\frac{8 b c \sqrt{d} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{3 e^3 \sqrt{c^2 x^2}}-\frac{b c d x \sqrt{c^2 x^2-1}}{3 e^2 \sqrt{c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}",1,"(-(b*c*d*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2)) + a*(c^2*d + e)*(8*d^2 + 12*d*e*x^2 + 3*e^2*x^4) + b*(c^2*d + e)*(8*d^2 + 12*d*e*x^2 + 3*e^2*x^4)*ArcSec[c*x])/(3*e^3*(c^2*d + e)*(d + e*x^2)^(3/2)) - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(3*Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 8*c^3*Sqrt[d]*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(3*c^2*e^3*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
152,1,172,163,0.2727987,"\int \frac{x^3 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^3*(a + b*ArcSec[c*x]))/(d + e*x^2)^(5/2),x]","\frac{-a \left(c^2 d+e\right) \left(2 d+3 e x^2\right)+b c e x \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right)-b \left(c^2 d+e\right) \sec ^{-1}(c x) \left(2 d+3 e x^2\right)}{3 e^2 \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}+\frac{2 b c x \sqrt{1-\frac{1}{c^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)}{3 \sqrt{d} e^2 \sqrt{c^2 x^2-1}}","-\frac{a+b \sec ^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{d \left(a+b \sec ^{-1}(c x)\right)}{3 e^2 \left(d+e x^2\right)^{3/2}}-\frac{2 b c x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{3 \sqrt{d} e^2 \sqrt{c^2 x^2}}+\frac{b c x \sqrt{c^2 x^2-1}}{3 e \sqrt{c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}",1,"(b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2) - a*(c^2*d + e)*(2*d + 3*e*x^2) - b*(c^2*d + e)*(2*d + 3*e*x^2)*ArcSec[c*x])/(3*e^2*(c^2*d + e)*(d + e*x^2)^(3/2)) + (2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]])/(3*Sqrt[d]*e^2*Sqrt[-1 + c^2*x^2])","A",1
153,1,158,138,0.2097521,"\int \frac{x \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x*(a + b*ArcSec[c*x]))/(d + e*x^2)^(5/2),x]","\frac{-a d \left(c^2 d+e\right)-b c e x \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right)-b d \left(c^2 d+e\right) \sec ^{-1}(c x)}{3 d e \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}+\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)}{3 d^{3/2} e \sqrt{c^2 x^2-1}}","-\frac{a+b \sec ^{-1}(c x)}{3 e \left(d+e x^2\right)^{3/2}}-\frac{b c x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{3 d^{3/2} e \sqrt{c^2 x^2}}-\frac{b c x \sqrt{c^2 x^2-1}}{3 d \sqrt{c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}",1,"(-(a*d*(c^2*d + e)) - b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2) - b*d*(c^2*d + e)*ArcSec[c*x])/(3*d*e*(c^2*d + e)*(d + e*x^2)^(3/2)) + (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]])/(3*d^(3/2)*e*Sqrt[-1 + c^2*x^2])","A",1
154,0,0,26,26.7998485,"\int \frac{a+b \sec ^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(x*(d + e*x^2)^(5/2)),x]","\int \frac{a+b \sec ^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{a+b \sec ^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(a + b*ArcSec[c*x])/(x*(d + e*x^2)^(5/2)), x]","A",-1
155,0,0,26,33.1228818,"\int \frac{a+b \sec ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(x^3*(d + e*x^2)^(5/2)),x]","\int \frac{a+b \sec ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{a+b \sec ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(a + b*ArcSec[c*x])/(x^3*(d + e*x^2)^(5/2)), x]","A",-1
156,0,0,26,10.3222226,"\int \frac{x^6 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^6*(a + b*ArcSec[c*x]))/(d + e*x^2)^(5/2),x]","\int \frac{x^6 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^6 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(x^6*(a + b*ArcSec[c*x]))/(d + e*x^2)^(5/2), x]","A",-1
157,0,0,26,9.6736844,"\int \frac{x^4 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^4*(a + b*ArcSec[c*x]))/(d + e*x^2)^(5/2),x]","\int \frac{x^4 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^4 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(x^4*(a + b*ArcSec[c*x]))/(d + e*x^2)^(5/2), x]","A",-1
158,1,186,276,0.2924805,"\int \frac{x^2 \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^2*(a + b*ArcSec[c*x]))/(d + e*x^2)^(5/2),x]","\frac{x^2 \left(a x \left(c^2 d+e\right)-b c \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right)+b x \left(c^2 d+e\right) \sec ^{-1}(c x)\right)}{3 d \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}+\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} E\left(\sin ^{-1}\left(\sqrt{-\frac{e}{d}} x\right)|-\frac{c^2 d}{e}\right)}{3 d \sqrt{1-c^2 x^2} \sqrt{-\frac{e}{d}} \left(c^2 d+e\right) \sqrt{d+e x^2}}","\frac{x^3 \left(a+b \sec ^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}-\frac{b c x^2 \sqrt{c^2 x^2-1}}{3 d \sqrt{c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}-\frac{b x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d e \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d e \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \left(c^2 d+e\right) \sqrt{\frac{e x^2}{d}+1}}",1,"(x^2*(a*(c^2*d + e)*x - b*c*Sqrt[1 - 1/(c^2*x^2)]*(d + e*x^2) + b*(c^2*d + e)*x*ArcSec[c*x]))/(3*d*(c^2*d + e)*(d + e*x^2)^(3/2)) + (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*EllipticE[ArcSin[Sqrt[-(e/d)]*x], -((c^2*d)/e)])/(3*d*Sqrt[-(e/d)]*(c^2*d + e)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","A",1
159,1,248,296,0.5535263,"\int \frac{a+b \sec ^{-1}(c x)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(d + e*x^2)^(5/2),x]","\frac{x \left(a \left(c^2 d+e\right) \left(3 d+2 e x^2\right)+b c e x \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right)+b \left(c^2 d+e\right) \sec ^{-1}(c x) \left(3 d+2 e x^2\right)\right)}{3 d^2 \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}-\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(2 \left(c^2 d+e\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)+c^2 d E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{3 \sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}","\frac{2 x \left(a+b \sec ^{-1}(c x)\right)}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left(a+b \sec ^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}+\frac{b c e x^2 \sqrt{c^2 x^2-1}}{3 d^2 \sqrt{c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}-\frac{2 b x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \left(c^2 d+e\right) \sqrt{\frac{e x^2}{d}+1}}",1,"(x*(b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2) + a*(c^2*d + e)*(3*d + 2*e*x^2) + b*(c^2*d + e)*(3*d + 2*e*x^2)*ArcSec[c*x]))/(3*d^2*(c^2*d + e)*(d + e*x^2)^(3/2)) - ((I/3)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] + 2*(c^2*d + e)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^2*(c^2*d + e)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
160,1,323,631,0.7825044,"\int \frac{a+b \sec ^{-1}(c x)}{x^2 \left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSec[c*x])/(x^2*(d + e*x^2)^(5/2)),x]","\frac{-a \left(c^2 d+e\right) \left(3 d^2+12 d e x^2+8 e^2 x^4\right)-b \left(c^2 d+e\right) \sec ^{-1}(c x) \left(3 d^2+12 d e x^2+8 e^2 x^4\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right) \left(3 c^2 d \left(d+e x^2\right)+e \left(3 d+2 e x^2\right)\right)}{3 d^3 x \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}-\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d \left(3 c^2 d+2 e\right) E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)-\left(3 c^4 d^2+11 c^2 d e+8 e^2\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{3 \sqrt{-c^2} d^3 \sqrt{1-c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}","-\frac{8 e x \left(a+b \sec ^{-1}(c x)\right)}{3 d^3 \sqrt{d+e x^2}}-\frac{4 e x \left(a+b \sec ^{-1}(c x)\right)}{3 d^2 \left(d+e x^2\right)^{3/2}}-\frac{a+b \sec ^{-1}(c x)}{d x \left(d+e x^2\right)^{3/2}}-\frac{4 b c e^2 x^2 \sqrt{c^2 x^2-1}}{3 d^3 \sqrt{c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}+\frac{b c \sqrt{c^2 x^2-1} \left(c^2 d+2 e\right) \sqrt{d+e x^2}}{d^3 \sqrt{c^2 x^2} \left(c^2 d+e\right)}+\frac{8 b e x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \left(c^2 d+2 e\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \left(c^2 d+e\right) \sqrt{\frac{e x^2}{d}+1}}+\frac{4 b c^2 e x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \left(c^2 d+e\right) \sqrt{\frac{e x^2}{d}+1}}-\frac{b c e \sqrt{c^2 x^2-1}}{d^2 \sqrt{c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}",1,"(-(a*(c^2*d + e)*(3*d^2 + 12*d*e*x^2 + 8*e^2*x^4)) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2)*(3*c^2*d*(d + e*x^2) + e*(3*d + 2*e*x^2)) - b*(c^2*d + e)*(3*d^2 + 12*d*e*x^2 + 8*e^2*x^4)*ArcSec[c*x])/(3*d^3*(c^2*d + e)*x*(d + e*x^2)^(3/2)) - ((I/3)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(3*c^2*d + 2*e)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] - (3*c^4*d^2 + 11*c^2*d*e + 8*e^2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^3*(c^2*d + e)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
161,0,0,589,0.2222566,"\int (f x)^m \left(d+e x^2\right)^3 \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcSec[c*x]),x]","\int (f x)^m \left(d+e x^2\right)^3 \left(a+b \sec ^{-1}(c x)\right) \, dx","\frac{d^3 (f x)^{m+1} \left(a+b \sec ^{-1}(c x)\right)}{f (m+1)}+\frac{3 d^2 e (f x)^{m+3} \left(a+b \sec ^{-1}(c x)\right)}{f^3 (m+3)}+\frac{3 d e^2 (f x)^{m+5} \left(a+b \sec ^{-1}(c x)\right)}{f^5 (m+5)}+\frac{e^3 (f x)^{m+7} \left(a+b \sec ^{-1}(c x)\right)}{f^7 (m+7)}-\frac{b e^3 x \sqrt{c^2 x^2-1} (f x)^{m+5}}{c f^5 (m+6) (m+7) \sqrt{c^2 x^2}}-\frac{b e^2 x \sqrt{c^2 x^2-1} (f x)^{m+3} \left(3 c^2 d \left(m^2+13 m+42\right)+e (m+5)^2\right)}{c^3 f^3 (m+4) (m+5) (m+6) (m+7) \sqrt{c^2 x^2}}-\frac{b e x \sqrt{c^2 x^2-1} (f x)^{m+1} \left(3 c^4 d^2 \left(m^4+22 m^3+179 m^2+638 m+840\right)+3 c^2 d e (m+3)^2 \left(m^2+13 m+42\right)+e^2 \left(m^2+8 m+15\right)^2\right)}{c^5 f (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) \sqrt{c^2 x^2}}-\frac{b x \sqrt{1-c^2 x^2} (f x)^{m+1} \left(\frac{c^6 d^3 (m+2) (m+4) (m+6)}{m+1}+\frac{e (m+1) \left(3 c^4 d^2 \left(m^4+22 m^3+179 m^2+638 m+840\right)+3 c^2 d e (m+3)^2 \left(m^2+13 m+42\right)+e^2 \left(m^2+8 m+15\right)^2\right)}{(m+3) (m+5) (m+7)}\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right)}{c^5 f (m+1) (m+2) (m+4) (m+6) \sqrt{c^2 x^2} \sqrt{c^2 x^2-1}}",1,"Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcSec[c*x]), x]","F",-1
162,0,0,374,0.1461039,"\int (f x)^m \left(d+e x^2\right)^2 \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^2*(a + b*ArcSec[c*x]),x]","\int (f x)^m \left(d+e x^2\right)^2 \left(a+b \sec ^{-1}(c x)\right) \, dx","\frac{d^2 (f x)^{m+1} \left(a+b \sec ^{-1}(c x)\right)}{f (m+1)}+\frac{2 d e (f x)^{m+3} \left(a+b \sec ^{-1}(c x)\right)}{f^3 (m+3)}+\frac{e^2 (f x)^{m+5} \left(a+b \sec ^{-1}(c x)\right)}{f^5 (m+5)}-\frac{b e^2 x \sqrt{c^2 x^2-1} (f x)^{m+3}}{c f^3 (m+4) (m+5) \sqrt{c^2 x^2}}-\frac{b e x \sqrt{c^2 x^2-1} (f x)^{m+1} \left(2 c^2 d \left(m^2+9 m+20\right)+e (m+3)^2\right)}{c^3 f (m+2) (m+3) (m+4) (m+5) \sqrt{c^2 x^2}}-\frac{b x \sqrt{1-c^2 x^2} (f x)^{m+1} \left(c^4 d^2 (m+2) (m+3) (m+4) (m+5)+e (m+1)^2 \left(2 c^2 d \left(m^2+9 m+20\right)+e (m+3)^2\right)\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right)}{c^3 f (m+1)^2 (m+2) (m+3) (m+4) (m+5) \sqrt{c^2 x^2} \sqrt{c^2 x^2-1}}",1,"Integrate[(f*x)^m*(d + e*x^2)^2*(a + b*ArcSec[c*x]), x]","F",-1
163,0,0,178,0.1095675,"\int (f x)^m \left(d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)*(a + b*ArcSec[c*x]),x]","\int (f x)^m \left(d+e x^2\right) \left(a+b \sec ^{-1}(c x)\right) \, dx","\frac{d x^{m+1} \left(a+b \sec ^{-1}(c x)\right)}{m+1}+\frac{e x^{m+3} \left(a+b \sec ^{-1}(c x)\right)}{m+3}+\frac{b \sqrt{c^2 x^2-1} x^{m+2} \left(c^2 d (m+2) (m+3)+e (m+1)^2\right) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+3}{2};c^2 x^2\right)}{c (m+1)^2 (m+2) (m+3) \sqrt{c^2 x^2}}-\frac{b e \sqrt{c^2 x^2-1} x^{m+2}}{c \left(m^2+5 m+6\right) \sqrt{c^2 x^2}}",1,"Integrate[(f*x)^m*(d + e*x^2)*(a + b*ArcSec[c*x]), x]","F",-1
164,0,0,26,2.0402454,"\int \frac{(f x)^m \left(a+b \sec ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[((f*x)^m*(a + b*ArcSec[c*x]))/(d + e*x^2),x]","\int \frac{(f x)^m \left(a+b \sec ^{-1}(c x)\right)}{d+e x^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \sec ^{-1}(c x)\right)}{d+e x^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcSec[c*x]))/(d + e*x^2), x]","A",-1
165,0,0,26,5.3874525,"\int \frac{(f x)^m \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[((f*x)^m*(a + b*ArcSec[c*x]))/(d + e*x^2)^2,x]","\int \frac{(f x)^m \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcSec[c*x]))/(d + e*x^2)^2, x]","A",-1
166,0,0,28,0.8977317,"\int (f x)^m \left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]),x]","\int (f x)^m \left(d+e x^2\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right) \, dx","\text{Int}\left(\left(d+e x^2\right)^{3/2} (f x)^m \left(a+b \sec ^{-1}(c x)\right),x\right)",0,"Integrate[(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcSec[c*x]), x]","A",-1
167,0,0,28,0.1127642,"\int (f x)^m \sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]),x]","\int (f x)^m \sqrt{d+e x^2} \left(a+b \sec ^{-1}(c x)\right) \, dx","\text{Int}\left(\sqrt{d+e x^2} (f x)^m \left(a+b \sec ^{-1}(c x)\right),x\right)",0,"Integrate[(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcSec[c*x]), x]","A",-1
168,0,0,28,1.1466461,"\int \frac{(f x)^m \left(a+b \sec ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[((f*x)^m*(a + b*ArcSec[c*x]))/Sqrt[d + e*x^2],x]","\int \frac{(f x)^m \left(a+b \sec ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \sec ^{-1}(c x)\right)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcSec[c*x]))/Sqrt[d + e*x^2], x]","A",-1
169,0,0,28,1.4170525,"\int \frac{(f x)^m \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[((f*x)^m*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2),x]","\int \frac{(f x)^m \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \sec ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcSec[c*x]))/(d + e*x^2)^(3/2), x]","A",-1
170,1,194,401,0.250996,"\int \frac{x^{11} \left(a+b \sec ^{-1}(c x)\right)}{\sqrt{1-c^4 x^4}} \, dx","Integrate[(x^11*(a + b*ArcSec[c*x]))/Sqrt[1 - c^4*x^4],x]","\frac{-105 a \sqrt{1-c^4 x^4} \left(3 c^8 x^8+4 c^4 x^4+8\right)-105 b \sqrt{1-c^4 x^4} \left(3 c^8 x^8+4 c^4 x^4+8\right) \sec ^{-1}(c x)+840 b \tan ^{-1}\left(\frac{c x \sqrt{1-\frac{1}{c^2 x^2}}}{\sqrt{1-c^4 x^4}}\right)+\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^4 x^4} \left(35 c^8 x^8+5 c^6 x^6+78 c^4 x^4+36 c^2 x^2+768\right)}{c^2 x^2-1}}{3150 c^{12}}","-\frac{\left(1-c^4 x^4\right)^{5/2} \left(a+b \sec ^{-1}(c x)\right)}{10 c^{12}}+\frac{\left(1-c^4 x^4\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{3 c^{12}}-\frac{\sqrt{1-c^4 x^4} \left(a+b \sec ^{-1}(c x)\right)}{2 c^{12}}+\frac{b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{9/2}}{90 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{3 b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{7/2}}{70 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{13 b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{5/2}}{150 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{7 b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{3/2}}{90 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{15 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{4 b \sqrt{1-c^2 x^2} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{15 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}",1,"(-105*a*Sqrt[1 - c^4*x^4]*(8 + 4*c^4*x^4 + 3*c^8*x^8) + (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 - c^4*x^4]*(768 + 36*c^2*x^2 + 78*c^4*x^4 + 5*c^6*x^6 + 35*c^8*x^8))/(-1 + c^2*x^2) - 105*b*Sqrt[1 - c^4*x^4]*(8 + 4*c^4*x^4 + 3*c^8*x^8)*ArcSec[c*x] + 840*b*ArcTan[(c*Sqrt[1 - 1/(c^2*x^2)]*x)/Sqrt[1 - c^4*x^4]])/(3150*c^12)","A",1
171,1,159,268,0.3254887,"\int \frac{x^7 \left(a+b \sec ^{-1}(c x)\right)}{\sqrt{1-c^4 x^4}} \, dx","Integrate[(x^7*(a + b*ArcSec[c*x]))/Sqrt[1 - c^4*x^4],x]","\frac{-15 a \sqrt{1-c^4 x^4} \left(c^4 x^4+2\right)-15 b \sqrt{1-c^4 x^4} \left(c^4 x^4+2\right) \sec ^{-1}(c x)+\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^4 x^4} \left(3 c^4 x^4+c^2 x^2+28\right)}{c^2 x^2-1}+30 b \tan ^{-1}\left(\frac{c x \sqrt{1-\frac{1}{c^2 x^2}}}{\sqrt{1-c^4 x^4}}\right)}{90 c^8}","\frac{\left(1-c^4 x^4\right)^{3/2} \left(a+b \sec ^{-1}(c x)\right)}{6 c^8}-\frac{\sqrt{1-c^4 x^4} \left(a+b \sec ^{-1}(c x)\right)}{2 c^8}+\frac{b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{5/2}}{30 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{3/2}}{18 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{3 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{b \sqrt{1-c^2 x^2} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{3 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}}",1,"(-15*a*Sqrt[1 - c^4*x^4]*(2 + c^4*x^4) + (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 - c^4*x^4]*(28 + c^2*x^2 + 3*c^4*x^4))/(-1 + c^2*x^2) - 15*b*Sqrt[1 - c^4*x^4]*(2 + c^4*x^4)*ArcSec[c*x] + 30*b*ArcTan[(c*Sqrt[1 - 1/(c^2*x^2)]*x)/Sqrt[1 - c^4*x^4]])/(90*c^8)","A",1
172,1,118,126,0.2202815,"\int \frac{x^3 \left(a+b \sec ^{-1}(c x)\right)}{\sqrt{1-c^4 x^4}} \, dx","Integrate[(x^3*(a + b*ArcSec[c*x]))/Sqrt[1 - c^4*x^4],x]","\frac{\frac{\sqrt{1-c^4 x^4} \left(-a c^2 x^2+a+b c x \sqrt{1-\frac{1}{c^2 x^2}}\right)}{c^2 x^2-1}-b \sqrt{1-c^4 x^4} \sec ^{-1}(c x)+b \tan ^{-1}\left(\frac{c x \sqrt{1-\frac{1}{c^2 x^2}}}{\sqrt{1-c^4 x^4}}\right)}{2 c^4}","-\frac{\sqrt{1-c^4 x^4} \left(a+b \sec ^{-1}(c x)\right)}{2 c^4}+\frac{b x \sqrt{1-c^4 x^4}}{2 c^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1}}-\frac{b x \tan ^{-1}\left(\frac{\sqrt{1-c^4 x^4}}{\sqrt{c^2 x^2-1}}\right)}{2 c^3 \sqrt{c^2 x^2}}",1,"(((a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x - a*c^2*x^2)*Sqrt[1 - c^4*x^4])/(-1 + c^2*x^2) - b*Sqrt[1 - c^4*x^4]*ArcSec[c*x] + b*ArcTan[(c*Sqrt[1 - 1/(c^2*x^2)]*x)/Sqrt[1 - c^4*x^4]])/(2*c^4)","A",1
173,0,0,29,0.35975,"\int \frac{a+b \sec ^{-1}(c x)}{x \sqrt{1-c^4 x^4}} \, dx","Integrate[(a + b*ArcSec[c*x])/(x*Sqrt[1 - c^4*x^4]),x]","\int \frac{a+b \sec ^{-1}(c x)}{x \sqrt{1-c^4 x^4}} \, dx","\text{Int}\left(\frac{a+b \sec ^{-1}(c x)}{x \sqrt{1-c^4 x^4}},x\right)",0,"Integrate[(a + b*ArcSec[c*x])/(x*Sqrt[1 - c^4*x^4]), x]","A",-1
174,0,0,29,6.5478861,"\int \frac{a+b \sec ^{-1}(c x)}{x^5 \sqrt{1-c^4 x^4}} \, dx","Integrate[(a + b*ArcSec[c*x])/(x^5*Sqrt[1 - c^4*x^4]),x]","\int \frac{a+b \sec ^{-1}(c x)}{x^5 \sqrt{1-c^4 x^4}} \, dx","\text{Int}\left(\frac{a+b \sec ^{-1}(c x)}{x^5 \sqrt{1-c^4 x^4}},x\right)",0,"Integrate[(a + b*ArcSec[c*x])/(x^5*Sqrt[1 - c^4*x^4]), x]","A",-1