1,1,146,143,0.048099,"\int x^5 \left(a+b \sec \left(c+d x^2\right)\right) \, dx","Integrate[x^5*(a + b*Sec[c + d*x^2]),x]","\frac{a x^6}{6}-\frac{b \text{Li}_3\left(-i e^{i \left(d x^2+c\right)}\right)}{d^3}+\frac{b \text{Li}_3\left(i e^{i \left(d x^2+c\right)}\right)}{d^3}+\frac{i b x^2 \text{Li}_2\left(-i e^{i \left(d x^2+c\right)}\right)}{d^2}-\frac{i b x^2 \text{Li}_2\left(i e^{i \left(d x^2+c\right)}\right)}{d^2}-\frac{i b x^4 \tan ^{-1}\left(e^{i c+i d x^2}\right)}{d}","\frac{a x^6}{6}-\frac{b \text{Li}_3\left(-i e^{i \left(d x^2+c\right)}\right)}{d^3}+\frac{b \text{Li}_3\left(i e^{i \left(d x^2+c\right)}\right)}{d^3}+\frac{i b x^2 \text{Li}_2\left(-i e^{i \left(d x^2+c\right)}\right)}{d^2}-\frac{i b x^2 \text{Li}_2\left(i e^{i \left(d x^2+c\right)}\right)}{d^2}-\frac{i b x^4 \tan ^{-1}\left(e^{i \left(c+d x^2\right)}\right)}{d}",1,"(a*x^6)/6 - (I*b*x^4*ArcTan[E^(I*c + I*d*x^2)])/d + (I*b*x^2*PolyLog[2, (-I)*E^(I*(c + d*x^2))])/d^2 - (I*b*x^2*PolyLog[2, I*E^(I*(c + d*x^2))])/d^2 - (b*PolyLog[3, (-I)*E^(I*(c + d*x^2))])/d^3 + (b*PolyLog[3, I*E^(I*(c + d*x^2))])/d^3","A",1
2,0,0,26,0.9584215,"\int x^4 \left(a+b \sec \left(c+d x^2\right)\right) \, dx","Integrate[x^4*(a + b*Sec[c + d*x^2]),x]","\int x^4 \left(a+b \sec \left(c+d x^2\right)\right) \, dx","b \text{Int}\left(x^4 \sec \left(c+d x^2\right),x\right)+\frac{a x^5}{5}",0,"Integrate[x^4*(a + b*Sec[c + d*x^2]), x]","A",-1
3,1,95,92,0.0246259,"\int x^3 \left(a+b \sec \left(c+d x^2\right)\right) \, dx","Integrate[x^3*(a + b*Sec[c + d*x^2]),x]","\frac{a x^4}{4}+\frac{i b \text{Li}_2\left(-i e^{i \left(d x^2+c\right)}\right)}{2 d^2}-\frac{i b \text{Li}_2\left(i e^{i \left(d x^2+c\right)}\right)}{2 d^2}-\frac{i b x^2 \tan ^{-1}\left(e^{i c+i d x^2}\right)}{d}","\frac{a x^4}{4}+\frac{i b \text{Li}_2\left(-i e^{i \left(d x^2+c\right)}\right)}{2 d^2}-\frac{i b \text{Li}_2\left(i e^{i \left(d x^2+c\right)}\right)}{2 d^2}-\frac{i b x^2 \tan ^{-1}\left(e^{i \left(c+d x^2\right)}\right)}{d}",1,"(a*x^4)/4 - (I*b*x^2*ArcTan[E^(I*c + I*d*x^2)])/d + ((I/2)*b*PolyLog[2, (-I)*E^(I*(c + d*x^2))])/d^2 - ((I/2)*b*PolyLog[2, I*E^(I*(c + d*x^2))])/d^2","A",1
4,0,0,26,0.7712115,"\int x^2 \left(a+b \sec \left(c+d x^2\right)\right) \, dx","Integrate[x^2*(a + b*Sec[c + d*x^2]),x]","\int x^2 \left(a+b \sec \left(c+d x^2\right)\right) \, dx","b \text{Int}\left(x^2 \sec \left(c+d x^2\right),x\right)+\frac{a x^3}{3}",0,"Integrate[x^2*(a + b*Sec[c + d*x^2]), x]","A",-1
5,1,26,26,0.014387,"\int x \left(a+b \sec \left(c+d x^2\right)\right) \, dx","Integrate[x*(a + b*Sec[c + d*x^2]),x]","\frac{a x^2}{2}+\frac{b \tanh ^{-1}\left(\sin \left(c+d x^2\right)\right)}{2 d}","\frac{a x^2}{2}+\frac{b \tanh ^{-1}\left(\sin \left(c+d x^2\right)\right)}{2 d}",1,"(a*x^2)/2 + (b*ArcTanh[Sin[c + d*x^2]])/(2*d)","A",1
6,0,0,22,0.7659877,"\int \frac{a+b \sec \left(c+d x^2\right)}{x} \, dx","Integrate[(a + b*Sec[c + d*x^2])/x,x]","\int \frac{a+b \sec \left(c+d x^2\right)}{x} \, dx","b \text{Int}\left(\frac{\sec \left(c+d x^2\right)}{x},x\right)+a \log (x)",0,"Integrate[(a + b*Sec[c + d*x^2])/x, x]","A",-1
7,0,0,24,0.7392491,"\int \frac{a+b \sec \left(c+d x^2\right)}{x^2} \, dx","Integrate[(a + b*Sec[c + d*x^2])/x^2,x]","\int \frac{a+b \sec \left(c+d x^2\right)}{x^2} \, dx","b \text{Int}\left(\frac{\sec \left(c+d x^2\right)}{x^2},x\right)-\frac{a}{x}",0,"Integrate[(a + b*Sec[c + d*x^2])/x^2, x]","A",-1
8,1,229,242,0.6101853,"\int x^5 \left(a+b \sec \left(c+d x^2\right)\right)^2 \, dx","Integrate[x^5*(a + b*Sec[c + d*x^2])^2,x]","\frac{a^2 d^3 x^6-12 i a b d^2 x^4 \tan ^{-1}\left(e^{i \left(c+d x^2\right)}\right)+12 i a b d x^2 \text{Li}_2\left(-i e^{i \left(d x^2+c\right)}\right)-12 i a b d x^2 \text{Li}_2\left(i e^{i \left(d x^2+c\right)}\right)-12 a b \text{Li}_3\left(-i e^{i \left(d x^2+c\right)}\right)+12 a b \text{Li}_3\left(i e^{i \left(d x^2+c\right)}\right)+3 b^2 d^2 x^4 \tan \left(c+d x^2\right)-3 i b^2 \text{Li}_2\left(-e^{2 i \left(d x^2+c\right)}\right)+6 b^2 d x^2 \log \left(1+e^{2 i \left(c+d x^2\right)}\right)-3 i b^2 d^2 x^4}{6 d^3}","\frac{a^2 x^6}{6}-\frac{2 a b \text{Li}_3\left(-i e^{i \left(d x^2+c\right)}\right)}{d^3}+\frac{2 a b \text{Li}_3\left(i e^{i \left(d x^2+c\right)}\right)}{d^3}+\frac{2 i a b x^2 \text{Li}_2\left(-i e^{i \left(d x^2+c\right)}\right)}{d^2}-\frac{2 i a b x^2 \text{Li}_2\left(i e^{i \left(d x^2+c\right)}\right)}{d^2}-\frac{2 i a b x^4 \tan ^{-1}\left(e^{i \left(c+d x^2\right)}\right)}{d}-\frac{i b^2 \text{Li}_2\left(-e^{2 i \left(d x^2+c\right)}\right)}{2 d^3}+\frac{b^2 x^2 \log \left(1+e^{2 i \left(c+d x^2\right)}\right)}{d^2}+\frac{b^2 x^4 \tan \left(c+d x^2\right)}{2 d}-\frac{i b^2 x^4}{2 d}",1,"((-3*I)*b^2*d^2*x^4 + a^2*d^3*x^6 - (12*I)*a*b*d^2*x^4*ArcTan[E^(I*(c + d*x^2))] + 6*b^2*d*x^2*Log[1 + E^((2*I)*(c + d*x^2))] + (12*I)*a*b*d*x^2*PolyLog[2, (-I)*E^(I*(c + d*x^2))] - (12*I)*a*b*d*x^2*PolyLog[2, I*E^(I*(c + d*x^2))] - (3*I)*b^2*PolyLog[2, -E^((2*I)*(c + d*x^2))] - 12*a*b*PolyLog[3, (-I)*E^(I*(c + d*x^2))] + 12*a*b*PolyLog[3, I*E^(I*(c + d*x^2))] + 3*b^2*d^2*x^4*Tan[c + d*x^2])/(6*d^3)","A",1
9,0,0,21,9.6108145,"\int x^4 \left(a+b \sec \left(c+d x^2\right)\right)^2 \, dx","Integrate[x^4*(a + b*Sec[c + d*x^2])^2,x]","\int x^4 \left(a+b \sec \left(c+d x^2\right)\right)^2 \, dx","\text{Int}\left(x^4 \left(a+b \sec \left(c+d x^2\right)\right)^2,x\right)",0,"Integrate[x^4*(a + b*Sec[c + d*x^2])^2, x]","A",-1
10,1,123,133,0.4650125,"\int x^3 \left(a+b \sec \left(c+d x^2\right)\right)^2 \, dx","Integrate[x^3*(a + b*Sec[c + d*x^2])^2,x]","\frac{a^2 d^2 x^4+4 i a b \text{Li}_2\left(-i e^{i \left(d x^2+c\right)}\right)-4 i a b \text{Li}_2\left(i e^{i \left(d x^2+c\right)}\right)-8 i a b d x^2 \tan ^{-1}\left(e^{i \left(c+d x^2\right)}\right)+2 b^2 d x^2 \tan \left(c+d x^2\right)+2 b^2 \log \left(\cos \left(c+d x^2\right)\right)}{4 d^2}","\frac{a^2 x^4}{4}+\frac{i a b \text{Li}_2\left(-i e^{i \left(d x^2+c\right)}\right)}{d^2}-\frac{i a b \text{Li}_2\left(i e^{i \left(d x^2+c\right)}\right)}{d^2}-\frac{2 i a b x^2 \tan ^{-1}\left(e^{i \left(c+d x^2\right)}\right)}{d}+\frac{b^2 \log \left(\cos \left(c+d x^2\right)\right)}{2 d^2}+\frac{b^2 x^2 \tan \left(c+d x^2\right)}{2 d}",1,"(a^2*d^2*x^4 - (8*I)*a*b*d*x^2*ArcTan[E^(I*(c + d*x^2))] + 2*b^2*Log[Cos[c + d*x^2]] + (4*I)*a*b*PolyLog[2, (-I)*E^(I*(c + d*x^2))] - (4*I)*a*b*PolyLog[2, I*E^(I*(c + d*x^2))] + 2*b^2*d*x^2*Tan[c + d*x^2])/(4*d^2)","A",1
11,0,0,21,7.9073603,"\int x^2 \left(a+b \sec \left(c+d x^2\right)\right)^2 \, dx","Integrate[x^2*(a + b*Sec[c + d*x^2])^2,x]","\int x^2 \left(a+b \sec \left(c+d x^2\right)\right)^2 \, dx","\text{Int}\left(x^2 \left(a+b \sec \left(c+d x^2\right)\right)^2,x\right)",0,"Integrate[x^2*(a + b*Sec[c + d*x^2])^2, x]","A",-1
12,1,41,44,0.2130797,"\int x \left(a+b \sec \left(c+d x^2\right)\right)^2 \, dx","Integrate[x*(a + b*Sec[c + d*x^2])^2,x]","\frac{a^2 d x^2+2 a b \tanh ^{-1}\left(\sin \left(c+d x^2\right)\right)+b^2 \tan \left(c+d x^2\right)}{2 d}","\frac{a^2 x^2}{2}+\frac{a b \tanh ^{-1}\left(\sin \left(c+d x^2\right)\right)}{d}+\frac{b^2 \tan \left(c+d x^2\right)}{2 d}",1,"(a^2*d*x^2 + 2*a*b*ArcTanh[Sin[c + d*x^2]] + b^2*Tan[c + d*x^2])/(2*d)","A",1
13,0,0,21,17.8578831,"\int \frac{\left(a+b \sec \left(c+d x^2\right)\right)^2}{x} \, dx","Integrate[(a + b*Sec[c + d*x^2])^2/x,x]","\int \frac{\left(a+b \sec \left(c+d x^2\right)\right)^2}{x} \, dx","\text{Int}\left(\frac{\left(a+b \sec \left(c+d x^2\right)\right)^2}{x},x\right)",0,"Integrate[(a + b*Sec[c + d*x^2])^2/x, x]","A",-1
14,0,0,21,9.3178184,"\int \frac{\left(a+b \sec \left(c+d x^2\right)\right)^2}{x^2} \, dx","Integrate[(a + b*Sec[c + d*x^2])^2/x^2,x]","\int \frac{\left(a+b \sec \left(c+d x^2\right)\right)^2}{x^2} \, dx","\text{Int}\left(\frac{\left(a+b \sec \left(c+d x^2\right)\right)^2}{x^2},x\right)",0,"Integrate[(a + b*Sec[c + d*x^2])^2/x^2, x]","A",-1
15,1,62,90,0.2102925,"\int x \sec ^7\left(a+b x^2\right) \, dx","Integrate[x*Sec[a + b*x^2]^7,x]","\frac{15 \tanh ^{-1}\left(\sin \left(a+b x^2\right)\right)+\tan \left(a+b x^2\right) \sec \left(a+b x^2\right) \left(8 \sec ^4\left(a+b x^2\right)+10 \sec ^2\left(a+b x^2\right)+15\right)}{96 b}","\frac{5 \tanh ^{-1}\left(\sin \left(a+b x^2\right)\right)}{32 b}+\frac{\tan \left(a+b x^2\right) \sec ^5\left(a+b x^2\right)}{12 b}+\frac{5 \tan \left(a+b x^2\right) \sec ^3\left(a+b x^2\right)}{48 b}+\frac{5 \tan \left(a+b x^2\right) \sec \left(a+b x^2\right)}{32 b}",1,"(15*ArcTanh[Sin[a + b*x^2]] + Sec[a + b*x^2]*(15 + 10*Sec[a + b*x^2]^2 + 8*Sec[a + b*x^2]^4)*Tan[a + b*x^2])/(96*b)","A",1
16,1,472,382,1.1190329,"\int \frac{x^5}{a+b \sec \left(c+d x^2\right)} \, dx","Integrate[x^5/(a + b*Sec[c + d*x^2]),x]","\frac{d^3 x^6 \sqrt{e^{2 i c} \left(b^2-a^2\right)}+3 i b e^{i c} d^2 x^4 \log \left(1+\frac{a e^{i \left(2 c+d x^2\right)}}{b e^{i c}-\sqrt{e^{2 i c} \left(b^2-a^2\right)}}\right)-3 i b e^{i c} d^2 x^4 \log \left(1+\frac{a e^{i \left(2 c+d x^2\right)}}{\sqrt{e^{2 i c} \left(b^2-a^2\right)}+b e^{i c}}\right)+6 b e^{i c} d x^2 \text{Li}_2\left(-\frac{a e^{i \left(d x^2+2 c\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-6 b e^{i c} d x^2 \text{Li}_2\left(-\frac{a e^{i \left(d x^2+2 c\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+6 i b e^{i c} \text{Li}_3\left(-\frac{a e^{i \left(d x^2+2 c\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-6 i b e^{i c} \text{Li}_3\left(-\frac{a e^{i \left(d x^2+2 c\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)}{6 a d^3 \sqrt{e^{2 i c} \left(b^2-a^2\right)}}","\frac{i b \text{Li}_3\left(-\frac{a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}-\frac{i b \text{Li}_3\left(-\frac{a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}+\frac{b x^2 \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}-\frac{b x^2 \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}+\frac{i b x^4 \log \left(1+\frac{a e^{i \left(c+d x^2\right)}}{b-\sqrt{b^2-a^2}}\right)}{2 a d \sqrt{b^2-a^2}}-\frac{i b x^4 \log \left(1+\frac{a e^{i \left(c+d x^2\right)}}{\sqrt{b^2-a^2}+b}\right)}{2 a d \sqrt{b^2-a^2}}+\frac{x^6}{6 a}",1,"(d^3*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^6 + (3*I)*b*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (3*I)*b*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*b*d*E^(I*c)*x^2*PolyLog[2, -((a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 6*b*d*E^(I*c)*x^2*PolyLog[2, -((a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (6*I)*b*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (6*I)*b*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])/(6*a*d^3*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])","A",1
17,0,0,21,1.4035827,"\int \frac{x^4}{a+b \sec \left(c+d x^2\right)} \, dx","Integrate[x^4/(a + b*Sec[c + d*x^2]),x]","\int \frac{x^4}{a+b \sec \left(c+d x^2\right)} \, dx","\text{Int}\left(\frac{x^4}{a+b \sec \left(c+d x^2\right)},x\right)",0,"Integrate[x^4/(a + b*Sec[c + d*x^2]), x]","A",-1
18,1,845,261,1.3967873,"\int \frac{x^3}{a+b \sec \left(c+d x^2\right)} \, dx","Integrate[x^3/(a + b*Sec[c + d*x^2]),x]","\frac{\left(b+a \cos \left(d x^2+c\right)\right) \left(x^4-\frac{2 b \left(2 \left(d x^2+c\right) \tanh ^{-1}\left(\frac{(a+b) \cot \left(\frac{1}{2} \left(d x^2+c\right)\right)}{\sqrt{a^2-b^2}}\right)-2 \left(c+\cos ^{-1}\left(-\frac{b}{a}\right)\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)}{\sqrt{a^2-b^2}}\right)+\left(\cos ^{-1}\left(-\frac{b}{a}\right)-2 i \tanh ^{-1}\left(\frac{(a+b) \cot \left(\frac{1}{2} \left(d x^2+c\right)\right)}{\sqrt{a^2-b^2}}\right)+2 i \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)}{\sqrt{a^2-b^2}}\right)\right) \log \left(\frac{\sqrt{a^2-b^2} e^{-\frac{1}{2} i \left(d x^2+c\right)}}{\sqrt{2} \sqrt{a} \sqrt{b+a \cos \left(d x^2+c\right)}}\right)+\left(\cos ^{-1}\left(-\frac{b}{a}\right)+2 i \left(\tanh ^{-1}\left(\frac{(a+b) \cot \left(\frac{1}{2} \left(d x^2+c\right)\right)}{\sqrt{a^2-b^2}}\right)-\tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)}{\sqrt{a^2-b^2}}\right)\right)\right) \log \left(\frac{\sqrt{a^2-b^2} e^{\frac{1}{2} i \left(d x^2+c\right)}}{\sqrt{2} \sqrt{a} \sqrt{b+a \cos \left(d x^2+c\right)}}\right)-\left(\cos ^{-1}\left(-\frac{b}{a}\right)-2 i \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)}{\sqrt{a^2-b^2}}\right)\right) \log \left(\frac{(a+b) \left(a-b-i \sqrt{a^2-b^2}\right) \left(i \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)+1\right)}{a \left(a+b+\sqrt{a^2-b^2} \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)\right)}\right)-\left(\cos ^{-1}\left(-\frac{b}{a}\right)+2 i \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)}{\sqrt{a^2-b^2}}\right)\right) \log \left(\frac{(a+b) \left(-i a+i b+\sqrt{a^2-b^2}\right) \left(\tan \left(\frac{1}{2} \left(d x^2+c\right)\right)+i\right)}{a \left(a+b+\sqrt{a^2-b^2} \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)\right)}\right)+i \left(\text{Li}_2\left(\frac{\left(b-i \sqrt{a^2-b^2}\right) \left(a+b-\sqrt{a^2-b^2} \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)\right)}{a \left(a+b+\sqrt{a^2-b^2} \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)\right)}\right)-\text{Li}_2\left(\frac{\left(b+i \sqrt{a^2-b^2}\right) \left(a+b-\sqrt{a^2-b^2} \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)\right)}{a \left(a+b+\sqrt{a^2-b^2} \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)\right)}\right)\right)\right)}{\sqrt{a^2-b^2} d^2}\right) \sec \left(d x^2+c\right)}{4 a \left(a+b \sec \left(d x^2+c\right)\right)}","\frac{b \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{2 a d^2 \sqrt{b^2-a^2}}-\frac{b \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{2 a d^2 \sqrt{b^2-a^2}}+\frac{i b x^2 \log \left(1+\frac{a e^{i \left(c+d x^2\right)}}{b-\sqrt{b^2-a^2}}\right)}{2 a d \sqrt{b^2-a^2}}-\frac{i b x^2 \log \left(1+\frac{a e^{i \left(c+d x^2\right)}}{\sqrt{b^2-a^2}+b}\right)}{2 a d \sqrt{b^2-a^2}}+\frac{x^4}{4 a}",1,"((b + a*Cos[c + d*x^2])*(x^4 - (2*b*(2*(c + d*x^2)*ArcTanh[((a + b)*Cot[(c + d*x^2)/2])/Sqrt[a^2 - b^2]] - 2*(c + ArcCos[-(b/a)])*ArcTanh[((a - b)*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(b/a)] - (2*I)*ArcTanh[((a + b)*Cot[(c + d*x^2)/2])/Sqrt[a^2 - b^2]] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]])*Log[Sqrt[a^2 - b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(c + d*x^2))*Sqrt[b + a*Cos[c + d*x^2]])] + (ArcCos[-(b/a)] + (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x^2)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2)*(c + d*x^2)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Cos[c + d*x^2]])] - (ArcCos[-(b/a)] - (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]])*Log[((a + b)*(a - b - I*Sqrt[a^2 - b^2])*(1 + I*Tan[(c + d*x^2)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^2)/2]))] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]])*Log[((a + b)*((-I)*a + I*b + Sqrt[a^2 - b^2])*(I + Tan[(c + d*x^2)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^2)/2]))] + I*(PolyLog[2, ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^2)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^2)/2]))] - PolyLog[2, ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^2)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^2)/2]))])))/(Sqrt[a^2 - b^2]*d^2))*Sec[c + d*x^2])/(4*a*(a + b*Sec[c + d*x^2]))","B",1
19,0,0,21,1.1690817,"\int \frac{x^2}{a+b \sec \left(c+d x^2\right)} \, dx","Integrate[x^2/(a + b*Sec[c + d*x^2]),x]","\int \frac{x^2}{a+b \sec \left(c+d x^2\right)} \, dx","\text{Int}\left(\frac{x^2}{a+b \sec \left(c+d x^2\right)},x\right)",0,"Integrate[x^2/(a + b*Sec[c + d*x^2]), x]","A",-1
20,1,67,66,0.179093,"\int \frac{x}{a+b \sec \left(c+d x^2\right)} \, dx","Integrate[x/(a + b*Sec[c + d*x^2]),x]","\frac{\frac{2 b \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} \left(c+d x^2\right)\right)}{\sqrt{a^2-b^2}}\right)}{d \sqrt{a^2-b^2}}+\frac{c}{d}+x^2}{2 a}","\frac{x^2}{2 a}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} \left(c+d x^2\right)\right)}{\sqrt{a+b}}\right)}{a d \sqrt{a-b} \sqrt{a+b}}",1,"(c/d + x^2 + (2*b*ArcTanh[((-a + b)*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*d))/(2*a)","A",1
21,0,0,21,0.9747371,"\int \frac{1}{x \left(a+b \sec \left(c+d x^2\right)\right)} \, dx","Integrate[1/(x*(a + b*Sec[c + d*x^2])),x]","\int \frac{1}{x \left(a+b \sec \left(c+d x^2\right)\right)} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \sec \left(c+d x^2\right)\right)},x\right)",0,"Integrate[1/(x*(a + b*Sec[c + d*x^2])), x]","A",-1
22,0,0,24,0.1510401,"\int \frac{a+b \sec \left(c+d x^2\right)}{x^2} \, dx","Integrate[(a + b*Sec[c + d*x^2])/x^2,x]","\int \frac{a+b \sec \left(c+d x^2\right)}{x^2} \, dx","b \text{Int}\left(\frac{\sec \left(c+d x^2\right)}{x^2},x\right)-\frac{a}{x}",0,"Integrate[(a + b*Sec[c + d*x^2])/x^2, x]","A",-1
23,1,895,1092,6.5782346,"\int \frac{x^5}{\left(a+b \sec \left(c+d x^2\right)\right)^2} \, dx","Integrate[x^5/(a + b*Sec[c + d*x^2])^2,x]","\frac{\left(b+a \cos \left(d x^2+c\right)\right) \sec ^2\left(d x^2+c\right) \left(\left(b+a \cos \left(d x^2+c\right)\right) x^6+\frac{3 b^2 \left(a \sin \left(d x^2\right)-b \sin (c)\right) x^4}{(a-b) (a+b) d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right)}-\frac{3 b \left(b+a \cos \left(d x^2+c\right)\right) \left(2 \left(1+e^{2 i c}\right) \left(-2 a^2 d e^{i c} x^2+b^2 d e^{i c} x^2+i b \sqrt{\left(b^2-a^2\right) e^{2 i c}}\right) \text{Li}_2\left(-\frac{a e^{i \left(d x^2+2 c\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+2 \left(1+e^{2 i c}\right) \left(2 a^2 d e^{i c} x^2-b^2 d e^{i c} x^2+i b \sqrt{\left(b^2-a^2\right) e^{2 i c}}\right) \text{Li}_2\left(-\frac{a e^{i \left(d x^2+2 c\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+i \left(d \left(2 b d e^{2 i c} \sqrt{\left(b^2-a^2\right) e^{2 i c}} x^2+\left(1+e^{2 i c}\right) \left(-2 a^2 d e^{i c} x^2+b^2 d e^{i c} x^2+2 i b \sqrt{\left(b^2-a^2\right) e^{2 i c}}\right) \log \left(\frac{e^{i \left(d x^2+2 c\right)} a}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right)+\left(1+e^{2 i c}\right) \left(2 a^2 d e^{i c} x^2-b^2 d e^{i c} x^2+2 i b \sqrt{\left(b^2-a^2\right) e^{2 i c}}\right) \log \left(\frac{e^{i \left(d x^2+2 c\right)} a}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right)\right) x^2-2 \left(2 a^2-b^2\right) e^{i c} \left(1+e^{2 i c}\right) \text{Li}_3\left(-\frac{a e^{i \left(d x^2+2 c\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+2 \left(2 a^2-b^2\right) e^{i c} \left(1+e^{2 i c}\right) \text{Li}_3\left(-\frac{a e^{i \left(d x^2+2 c\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)\right)\right)}{\left(a^2-b^2\right) d^3 \sqrt{\left(b^2-a^2\right) e^{2 i c}} \left(1+e^{2 i c}\right)}\right)}{6 a^2 \left(a+b \sec \left(d x^2+c\right)\right)^2}","\frac{x^6}{6 a^2}+\frac{i b \log \left(\frac{e^{i \left(d x^2+c\right)} a}{b-\sqrt{b^2-a^2}}+1\right) x^4}{a^2 \sqrt{b^2-a^2} d}-\frac{i b^3 \log \left(\frac{e^{i \left(d x^2+c\right)} a}{b-\sqrt{b^2-a^2}}+1\right) x^4}{2 a^2 \left(b^2-a^2\right)^{3/2} d}-\frac{i b \log \left(\frac{e^{i \left(d x^2+c\right)} a}{b+\sqrt{b^2-a^2}}+1\right) x^4}{a^2 \sqrt{b^2-a^2} d}+\frac{i b^3 \log \left(\frac{e^{i \left(d x^2+c\right)} a}{b+\sqrt{b^2-a^2}}+1\right) x^4}{2 a^2 \left(b^2-a^2\right)^{3/2} d}+\frac{b^2 \sin \left(d x^2+c\right) x^4}{2 a \left(a^2-b^2\right) d \left(b+a \cos \left(d x^2+c\right)\right)}-\frac{i b^2 x^4}{2 a^2 \left(a^2-b^2\right) d}+\frac{b^2 \log \left(\frac{e^{i \left(d x^2+c\right)} a}{b-i \sqrt{a^2-b^2}}+1\right) x^2}{a^2 \left(a^2-b^2\right) d^2}+\frac{b^2 \log \left(\frac{e^{i \left(d x^2+c\right)} a}{b+i \sqrt{a^2-b^2}}+1\right) x^2}{a^2 \left(a^2-b^2\right) d^2}+\frac{2 b \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^2}{a^2 \sqrt{b^2-a^2} d^2}-\frac{b^3 \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^2}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{2 b \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^2}{a^2 \sqrt{b^2-a^2} d^2}+\frac{b^3 \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^2}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{i b^2 \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{b-i \sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right) d^3}-\frac{i b^2 \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{b+i \sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right) d^3}+\frac{2 i b \text{Li}_3\left(-\frac{a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 \sqrt{b^2-a^2} d^3}-\frac{i b^3 \text{Li}_3\left(-\frac{a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 \left(b^2-a^2\right)^{3/2} d^3}-\frac{2 i b \text{Li}_3\left(-\frac{a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 \sqrt{b^2-a^2} d^3}+\frac{i b^3 \text{Li}_3\left(-\frac{a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 \left(b^2-a^2\right)^{3/2} d^3}",1,"((b + a*Cos[c + d*x^2])*Sec[c + d*x^2]^2*(x^6*(b + a*Cos[c + d*x^2]) - (3*b*(b + a*Cos[c + d*x^2])*(2*(1 + E^((2*I)*c))*(I*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*x^2 + b^2*d*E^(I*c)*x^2)*PolyLog[2, -((a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 2*(1 + E^((2*I)*c))*(I*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] + 2*a^2*d*E^(I*c)*x^2 - b^2*d*E^(I*c)*x^2)*PolyLog[2, -((a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + I*(d*x^2*(2*b*d*E^((2*I)*c)*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^2 + (1 + E^((2*I)*c))*((2*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*x^2 + b^2*d*E^(I*c)*x^2)*Log[1 + (a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (1 + E^((2*I)*c))*((2*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] + 2*a^2*d*E^(I*c)*x^2 - b^2*d*E^(I*c)*x^2)*Log[1 + (a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])]) - 2*(2*a^2 - b^2)*E^(I*c)*(1 + E^((2*I)*c))*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 2*(2*a^2 - b^2)*E^(I*c)*(1 + E^((2*I)*c))*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])))/((a^2 - b^2)*d^3*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*(1 + E^((2*I)*c))) + (3*b^2*x^4*(-(b*Sin[c]) + a*Sin[d*x^2]))/((a - b)*(a + b)*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2]))))/(6*a^2*(a + b*Sec[c + d*x^2])^2)","A",0
24,0,0,21,6.8798386,"\int \frac{x^4}{\left(a+b \sec \left(c+d x^2\right)\right)^2} \, dx","Integrate[x^4/(a + b*Sec[c + d*x^2])^2,x]","\int \frac{x^4}{\left(a+b \sec \left(c+d x^2\right)\right)^2} \, dx","\text{Int}\left(\frac{x^4}{\left(a+b \sec \left(c+d x^2\right)\right)^2},x\right)",0,"Integrate[x^4/(a + b*Sec[c + d*x^2])^2, x]","A",-1
25,1,1069,596,9.0330739,"\int \frac{x^3}{\left(a+b \sec \left(c+d x^2\right)\right)^2} \, dx","Integrate[x^3/(a + b*Sec[c + d*x^2])^2,x]","\frac{\left(b+a \cos \left(d x^2+c\right)\right) \left(b^2 c \sin \left(d x^2+c\right)-b^2 \left(d x^2+c\right) \sin \left(d x^2+c\right)\right) \sec ^2\left(d x^2+c\right)}{2 a (b-a) (a+b) d^2 \left(a+b \sec \left(d x^2+c\right)\right)^2}+\frac{b \cos ^2\left(\frac{1}{2} \left(d x^2+c\right)\right) \left(b+a \cos \left(d x^2+c\right)\right) \left(-\frac{2 \left(2 a^2-b^2\right) c \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)}{\sqrt{-a-b}}\right)}{\sqrt{-a-b} \sqrt{a-b}}+b \left(\log \left(-\left(\left(b+a \cos \left(d x^2+c\right)\right) \sec ^2\left(\frac{1}{2} \left(d x^2+c\right)\right)\right)\right)-\log \left(\sec ^2\left(\frac{1}{2} \left(d x^2+c\right)\right)\right)\right)-\frac{i \left(2 a^2-b^2\right) \left(\log \left(i \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)+1\right) \log \left(\frac{i \left(\sqrt{a+b}-\sqrt{a-b} \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)\right)}{\sqrt{a-b}+i \sqrt{a+b}}\right)-\log \left(1-i \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)\right) \log \left(\frac{\sqrt{a+b}-\sqrt{a-b} \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)}{i \sqrt{a-b}+\sqrt{a+b}}\right)+\log \left(1-i \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)\right) \log \left(\frac{i \left(\sqrt{a-b} \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)+\sqrt{a+b}\right)}{\sqrt{a-b}+i \sqrt{a+b}}\right)-\log \left(i \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)+1\right) \log \left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)+\sqrt{a+b}}{i \sqrt{a-b}+\sqrt{a+b}}\right)-\text{Li}_2\left(\frac{\sqrt{a-b} \left(1-i \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)\right)}{\sqrt{a-b}-i \sqrt{a+b}}\right)+\text{Li}_2\left(\frac{\sqrt{a-b} \left(1-i \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)\right)}{\sqrt{a-b}+i \sqrt{a+b}}\right)-\text{Li}_2\left(\frac{\sqrt{a-b} \left(i \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)+1\right)}{\sqrt{a-b}-i \sqrt{a+b}}\right)+\text{Li}_2\left(\frac{\sqrt{a-b} \left(i \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)+1\right)}{\sqrt{a-b}+i \sqrt{a+b}}\right)\right)}{\sqrt{a-b} \sqrt{a+b}}\right) \left(\left(2 a^2-b^2\right) d x^2+a b \sin \left(d x^2+c\right)\right) \left(\sqrt{a+b}-\sqrt{a-b} \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)\right) \left(\sqrt{a-b} \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)+\sqrt{a+b}\right) \sec ^2\left(d x^2+c\right)}{2 a^2 \left(a^2-b^2\right) d^2 \left(a+b \sec \left(d x^2+c\right)\right)^2 \left(a b \sin \left(d x^2+c\right)-\left(2 a^2-b^2\right) \left(c-i \log \left(1-i \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)\right)+i \log \left(i \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)+1\right)\right)\right)}+\frac{\left(d x^2-c\right) \left(d x^2+c\right) \left(b+a \cos \left(d x^2+c\right)\right)^2 \sec ^2\left(d x^2+c\right)}{4 a^2 d^2 \left(a+b \sec \left(d x^2+c\right)\right)^2}","\frac{b \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 d^2 \sqrt{b^2-a^2}}-\frac{b \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 d^2 \sqrt{b^2-a^2}}+\frac{b^2 \log \left(a \cos \left(c+d x^2\right)+b\right)}{2 a^2 d^2 \left(a^2-b^2\right)}+\frac{i b x^2 \log \left(1+\frac{a e^{i \left(c+d x^2\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 d \sqrt{b^2-a^2}}-\frac{i b x^2 \log \left(1+\frac{a e^{i \left(c+d x^2\right)}}{\sqrt{b^2-a^2}+b}\right)}{a^2 d \sqrt{b^2-a^2}}+\frac{b^2 x^2 \sin \left(c+d x^2\right)}{2 a d \left(a^2-b^2\right) \left(a \cos \left(c+d x^2\right)+b\right)}-\frac{b^3 \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{2 a^2 d^2 \left(b^2-a^2\right)^{3/2}}+\frac{b^3 \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{2 a^2 d^2 \left(b^2-a^2\right)^{3/2}}-\frac{i b^3 x^2 \log \left(1+\frac{a e^{i \left(c+d x^2\right)}}{b-\sqrt{b^2-a^2}}\right)}{2 a^2 d \left(b^2-a^2\right)^{3/2}}+\frac{i b^3 x^2 \log \left(1+\frac{a e^{i \left(c+d x^2\right)}}{\sqrt{b^2-a^2}+b}\right)}{2 a^2 d \left(b^2-a^2\right)^{3/2}}+\frac{x^4}{4 a^2}",1,"((-c + d*x^2)*(c + d*x^2)*(b + a*Cos[c + d*x^2])^2*Sec[c + d*x^2]^2)/(4*a^2*d^2*(a + b*Sec[c + d*x^2])^2) + ((b + a*Cos[c + d*x^2])*Sec[c + d*x^2]^2*(b^2*c*Sin[c + d*x^2] - b^2*(c + d*x^2)*Sin[c + d*x^2]))/(2*a*(-a + b)*(a + b)*d^2*(a + b*Sec[c + d*x^2])^2) + (b*Cos[(c + d*x^2)/2]^2*(b + a*Cos[c + d*x^2])*((-2*(2*a^2 - b^2)*c*ArcTan[(Sqrt[a - b]*Tan[(c + d*x^2)/2])/Sqrt[-a - b]])/(Sqrt[-a - b]*Sqrt[a - b]) + b*(-Log[Sec[(c + d*x^2)/2]^2] + Log[-((b + a*Cos[c + d*x^2])*Sec[(c + d*x^2)/2]^2)]) - (I*(2*a^2 - b^2)*(Log[1 + I*Tan[(c + d*x^2)/2]]*Log[(I*(Sqrt[a + b] - Sqrt[a - b]*Tan[(c + d*x^2)/2]))/(Sqrt[a - b] + I*Sqrt[a + b])] - Log[1 - I*Tan[(c + d*x^2)/2]]*Log[(Sqrt[a + b] - Sqrt[a - b]*Tan[(c + d*x^2)/2])/(I*Sqrt[a - b] + Sqrt[a + b])] + Log[1 - I*Tan[(c + d*x^2)/2]]*Log[(I*(Sqrt[a + b] + Sqrt[a - b]*Tan[(c + d*x^2)/2]))/(Sqrt[a - b] + I*Sqrt[a + b])] - Log[1 + I*Tan[(c + d*x^2)/2]]*Log[(Sqrt[a + b] + Sqrt[a - b]*Tan[(c + d*x^2)/2])/(I*Sqrt[a - b] + Sqrt[a + b])] - PolyLog[2, (Sqrt[a - b]*(1 - I*Tan[(c + d*x^2)/2]))/(Sqrt[a - b] - I*Sqrt[a + b])] + PolyLog[2, (Sqrt[a - b]*(1 - I*Tan[(c + d*x^2)/2]))/(Sqrt[a - b] + I*Sqrt[a + b])] - PolyLog[2, (Sqrt[a - b]*(1 + I*Tan[(c + d*x^2)/2]))/(Sqrt[a - b] - I*Sqrt[a + b])] + PolyLog[2, (Sqrt[a - b]*(1 + I*Tan[(c + d*x^2)/2]))/(Sqrt[a - b] + I*Sqrt[a + b])]))/(Sqrt[a - b]*Sqrt[a + b]))*Sec[c + d*x^2]^2*((2*a^2 - b^2)*d*x^2 + a*b*Sin[c + d*x^2])*(Sqrt[a + b] - Sqrt[a - b]*Tan[(c + d*x^2)/2])*(Sqrt[a + b] + Sqrt[a - b]*Tan[(c + d*x^2)/2]))/(2*a^2*(a^2 - b^2)*d^2*(a + b*Sec[c + d*x^2])^2*(-((2*a^2 - b^2)*(c - I*Log[1 - I*Tan[(c + d*x^2)/2]] + I*Log[1 + I*Tan[(c + d*x^2)/2]])) + a*b*Sin[c + d*x^2]))","A",0
26,0,0,21,5.801849,"\int \frac{x^2}{\left(a+b \sec \left(c+d x^2\right)\right)^2} \, dx","Integrate[x^2/(a + b*Sec[c + d*x^2])^2,x]","\int \frac{x^2}{\left(a+b \sec \left(c+d x^2\right)\right)^2} \, dx","\text{Int}\left(\frac{x^2}{\left(a+b \sec \left(c+d x^2\right)\right)^2},x\right)",0,"Integrate[x^2/(a + b*Sec[c + d*x^2])^2, x]","A",-1
27,1,153,123,0.6886192,"\int \frac{x}{\left(a+b \sec \left(c+d x^2\right)\right)^2} \, dx","Integrate[x/(a + b*Sec[c + d*x^2])^2,x]","\frac{\frac{b \left(\left(a^2-b^2\right) \left(c+d x^2\right)+a b \sin \left(c+d x^2\right)\right)+a \left(a^2-b^2\right) \left(c+d x^2\right) \cos \left(c+d x^2\right)}{a \cos \left(c+d x^2\right)+b}-\frac{2 b \left(b^2-2 a^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} \left(c+d x^2\right)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}}{2 a^2 d (a-b) (a+b)}","-\frac{b \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} \left(c+d x^2\right)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \tan \left(c+d x^2\right)}{2 a d \left(a^2-b^2\right) \left(a+b \sec \left(c+d x^2\right)\right)}+\frac{x^2}{2 a^2}",1,"((-2*b*(-2*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (a*(a^2 - b^2)*(c + d*x^2)*Cos[c + d*x^2] + b*((a^2 - b^2)*(c + d*x^2) + a*b*Sin[c + d*x^2]))/(b + a*Cos[c + d*x^2]))/(2*a^2*(a - b)*(a + b)*d)","A",1
28,0,0,21,11.2958772,"\int \frac{1}{x \left(a+b \sec \left(c+d x^2\right)\right)^2} \, dx","Integrate[1/(x*(a + b*Sec[c + d*x^2])^2),x]","\int \frac{1}{x \left(a+b \sec \left(c+d x^2\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \sec \left(c+d x^2\right)\right)^2},x\right)",0,"Integrate[1/(x*(a + b*Sec[c + d*x^2])^2), x]","A",-1
29,0,0,21,7.3733194,"\int \frac{1}{x^2 \left(a+b \sec \left(c+d x^2\right)\right)^2} \, dx","Integrate[1/(x^2*(a + b*Sec[c + d*x^2])^2),x]","\int \frac{1}{x^2 \left(a+b \sec \left(c+d x^2\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^2 \left(a+b \sec \left(c+d x^2\right)\right)^2},x\right)",0,"Integrate[1/(x^2*(a + b*Sec[c + d*x^2])^2), x]","A",-1
30,0,0,21,7.8147026,"\int \frac{1}{x^3 \left(a+b \sec \left(c+d x^2\right)\right)^2} \, dx","Integrate[1/(x^3*(a + b*Sec[c + d*x^2])^2),x]","\int \frac{1}{x^3 \left(a+b \sec \left(c+d x^2\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^3 \left(a+b \sec \left(c+d x^2\right)\right)^2},x\right)",0,"Integrate[1/(x^3*(a + b*Sec[c + d*x^2])^2), x]","A",-1
31,1,479,476,0.1472068,"\int x^3 \left(a+b \sec \left(c+d \sqrt{x}\right)\right) \, dx","Integrate[x^3*(a + b*Sec[c + d*Sqrt[x]]),x]","\frac{a x^4}{4}-\frac{10080 i b \text{Li}_8\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^8}+\frac{10080 i b \text{Li}_8\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^8}-\frac{10080 b \sqrt{x} \text{Li}_7\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^7}+\frac{10080 b \sqrt{x} \text{Li}_7\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^7}+\frac{5040 i b x \text{Li}_6\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}-\frac{5040 i b x \text{Li}_6\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}+\frac{1680 b x^{3/2} \text{Li}_5\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{1680 b x^{3/2} \text{Li}_5\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{420 i b x^2 \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{420 i b x^2 \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{84 b x^{5/2} \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{84 b x^{5/2} \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{14 i b x^3 \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{14 i b x^3 \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 i b x^{7/2} \tan ^{-1}\left(e^{i c+i d \sqrt{x}}\right)}{d}","\frac{a x^4}{4}-\frac{10080 i b \text{Li}_8\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^8}+\frac{10080 i b \text{Li}_8\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^8}-\frac{10080 b \sqrt{x} \text{Li}_7\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^7}+\frac{10080 b \sqrt{x} \text{Li}_7\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^7}+\frac{5040 i b x \text{Li}_6\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}-\frac{5040 i b x \text{Li}_6\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}+\frac{1680 b x^{3/2} \text{Li}_5\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{1680 b x^{3/2} \text{Li}_5\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{420 i b x^2 \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{420 i b x^2 \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{84 b x^{5/2} \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{84 b x^{5/2} \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{14 i b x^3 \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{14 i b x^3 \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 i b x^{7/2} \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}",1,"(a*x^4)/4 - ((4*I)*b*x^(7/2)*ArcTan[E^(I*c + I*d*Sqrt[x])])/d + ((14*I)*b*x^3*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((14*I)*b*x^3*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))])/d^2 - (84*b*x^(5/2)*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (84*b*x^(5/2)*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))])/d^3 - ((420*I)*b*x^2*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))])/d^4 + ((420*I)*b*x^2*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))])/d^4 + (1680*b*x^(3/2)*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))])/d^5 - (1680*b*x^(3/2)*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))])/d^5 + ((5040*I)*b*x*PolyLog[6, (-I)*E^(I*(c + d*Sqrt[x]))])/d^6 - ((5040*I)*b*x*PolyLog[6, I*E^(I*(c + d*Sqrt[x]))])/d^6 - (10080*b*Sqrt[x]*PolyLog[7, (-I)*E^(I*(c + d*Sqrt[x]))])/d^7 + (10080*b*Sqrt[x]*PolyLog[7, I*E^(I*(c + d*Sqrt[x]))])/d^7 - ((10080*I)*b*PolyLog[8, (-I)*E^(I*(c + d*Sqrt[x]))])/d^8 + ((10080*I)*b*PolyLog[8, I*E^(I*(c + d*Sqrt[x]))])/d^8","A",1
32,1,351,348,0.0838272,"\int x^2 \left(a+b \sec \left(c+d \sqrt{x}\right)\right) \, dx","Integrate[x^2*(a + b*Sec[c + d*Sqrt[x]]),x]","\frac{a x^3}{3}+\frac{240 i b \text{Li}_6\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}-\frac{240 i b \text{Li}_6\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}+\frac{240 b \sqrt{x} \text{Li}_5\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{240 b \sqrt{x} \text{Li}_5\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{120 i b x \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{120 i b x \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{40 b x^{3/2} \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{40 b x^{3/2} \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{10 i b x^2 \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{10 i b x^2 \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 i b x^{5/2} \tan ^{-1}\left(e^{i c+i d \sqrt{x}}\right)}{d}","\frac{a x^3}{3}+\frac{240 i b \text{Li}_6\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}-\frac{240 i b \text{Li}_6\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}+\frac{240 b \sqrt{x} \text{Li}_5\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{240 b \sqrt{x} \text{Li}_5\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{120 i b x \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{120 i b x \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{40 b x^{3/2} \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{40 b x^{3/2} \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{10 i b x^2 \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{10 i b x^2 \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 i b x^{5/2} \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}",1,"(a*x^3)/3 - ((4*I)*b*x^(5/2)*ArcTan[E^(I*c + I*d*Sqrt[x])])/d + ((10*I)*b*x^2*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((10*I)*b*x^2*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))])/d^2 - (40*b*x^(3/2)*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (40*b*x^(3/2)*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))])/d^3 - ((120*I)*b*x*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))])/d^4 + ((120*I)*b*x*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))])/d^4 + (240*b*Sqrt[x]*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))])/d^5 - (240*b*Sqrt[x]*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))])/d^5 + ((240*I)*b*PolyLog[6, (-I)*E^(I*(c + d*Sqrt[x]))])/d^6 - ((240*I)*b*PolyLog[6, I*E^(I*(c + d*Sqrt[x]))])/d^6","A",1
33,1,223,220,0.0625497,"\int x \left(a+b \sec \left(c+d \sqrt{x}\right)\right) \, dx","Integrate[x*(a + b*Sec[c + d*Sqrt[x]]),x]","\frac{a x^2}{2}-\frac{12 i b \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{12 i b \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{12 b \sqrt{x} \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{12 b \sqrt{x} \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{6 i b x \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{6 i b x \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 i b x^{3/2} \tan ^{-1}\left(e^{i c+i d \sqrt{x}}\right)}{d}","\frac{a x^2}{2}-\frac{12 i b \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{12 i b \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{12 b \sqrt{x} \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{12 b \sqrt{x} \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{6 i b x \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{6 i b x \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 i b x^{3/2} \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}",1,"(a*x^2)/2 - ((4*I)*b*x^(3/2)*ArcTan[E^(I*c + I*d*Sqrt[x])])/d + ((6*I)*b*x*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))])/d^2 - ((6*I)*b*x*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))])/d^2 - (12*b*Sqrt[x]*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))])/d^3 + (12*b*Sqrt[x]*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))])/d^3 - ((12*I)*b*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))])/d^4 + ((12*I)*b*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))])/d^4","A",1
34,0,0,24,2.0631887,"\int \frac{a+b \sec \left(c+d \sqrt{x}\right)}{x} \, dx","Integrate[(a + b*Sec[c + d*Sqrt[x]])/x,x]","\int \frac{a+b \sec \left(c+d \sqrt{x}\right)}{x} \, dx","b \text{Int}\left(\frac{\sec \left(c+d \sqrt{x}\right)}{x},x\right)+a \log (x)",0,"Integrate[(a + b*Sec[c + d*Sqrt[x]])/x, x]","A",-1
35,0,0,26,2.1989524,"\int \frac{a+b \sec \left(c+d \sqrt{x}\right)}{x^2} \, dx","Integrate[(a + b*Sec[c + d*Sqrt[x]])/x^2,x]","\int \frac{a+b \sec \left(c+d \sqrt{x}\right)}{x^2} \, dx","b \text{Int}\left(\frac{\sec \left(c+d \sqrt{x}\right)}{x^2},x\right)-\frac{a}{x}",0,"Integrate[(a + b*Sec[c + d*Sqrt[x]])/x^2, x]","A",-1
36,1,739,749,2.1976319,"\int x^3 \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2 \, dx","Integrate[x^3*(a + b*Sec[c + d*Sqrt[x]])^2,x]","\frac{a^2 d^8 x^4-32 i a b d^7 x^{7/2} \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)+112 i a b d^6 x^3 \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)-112 i a b d^6 x^3 \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)-672 a b d^5 x^{5/2} \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)+672 a b d^5 x^{5/2} \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)-3360 i a b d^4 x^2 \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)+3360 i a b d^4 x^2 \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)+13440 a b d^3 x^{3/2} \text{Li}_5\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)-13440 a b d^3 x^{3/2} \text{Li}_5\left(i e^{i \left(c+d \sqrt{x}\right)}\right)+40320 i a b d^2 x \text{Li}_6\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)-40320 i a b d^2 x \text{Li}_6\left(i e^{i \left(c+d \sqrt{x}\right)}\right)-80640 a b d \sqrt{x} \text{Li}_7\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)+80640 a b d \sqrt{x} \text{Li}_7\left(i e^{i \left(c+d \sqrt{x}\right)}\right)-80640 i a b \text{Li}_8\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)+80640 i a b \text{Li}_8\left(i e^{i \left(c+d \sqrt{x}\right)}\right)+8 b^2 d^7 x^{7/2} \tan \left(c+d \sqrt{x}\right)+56 b^2 d^6 x^3 \log \left(1+e^{2 i \left(c+d \sqrt{x}\right)}\right)-168 i b^2 d^5 x^{5/2} \text{Li}_2\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)+420 b^2 d^4 x^2 \text{Li}_3\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)+840 i b^2 d^3 x^{3/2} \text{Li}_4\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)-1260 b^2 d^2 x \text{Li}_5\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)-1260 i b^2 d \sqrt{x} \text{Li}_6\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)+630 b^2 \text{Li}_7\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)-8 i b^2 d^7 x^{7/2}}{4 d^8}","\frac{a^2 x^4}{4}-\frac{20160 i a b \text{Li}_8\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^8}+\frac{20160 i a b \text{Li}_8\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^8}-\frac{20160 a b \sqrt{x} \text{Li}_7\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^7}+\frac{20160 a b \sqrt{x} \text{Li}_7\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^7}+\frac{10080 i a b x \text{Li}_6\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}-\frac{10080 i a b x \text{Li}_6\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}+\frac{3360 a b x^{3/2} \text{Li}_5\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{840 i a b x^2 \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{168 a b x^{5/2} \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{28 i a b x^3 \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{8 i a b x^{7/2} \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}+\frac{315 b^2 \text{Li}_7\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{2 d^8}-\frac{315 i b^2 \sqrt{x} \text{Li}_6\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^7}-\frac{315 b^2 x \text{Li}_5\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^6}+\frac{210 i b^2 x^{3/2} \text{Li}_4\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^5}+\frac{105 b^2 x^2 \text{Li}_3\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{42 i b^2 x^{5/2} \text{Li}_2\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{14 b^2 x^3 \log \left(1+e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^2}+\frac{2 b^2 x^{7/2} \tan \left(c+d \sqrt{x}\right)}{d}-\frac{2 i b^2 x^{7/2}}{d}",1,"((-8*I)*b^2*d^7*x^(7/2) + a^2*d^8*x^4 - (32*I)*a*b*d^7*x^(7/2)*ArcTan[E^(I*(c + d*Sqrt[x]))] + 56*b^2*d^6*x^3*Log[1 + E^((2*I)*(c + d*Sqrt[x]))] + (112*I)*a*b*d^6*x^3*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))] - (112*I)*a*b*d^6*x^3*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))] - (168*I)*b^2*d^5*x^(5/2)*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))] - 672*a*b*d^5*x^(5/2)*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))] + 672*a*b*d^5*x^(5/2)*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))] + 420*b^2*d^4*x^2*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))] - (3360*I)*a*b*d^4*x^2*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))] + (3360*I)*a*b*d^4*x^2*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))] + (840*I)*b^2*d^3*x^(3/2)*PolyLog[4, -E^((2*I)*(c + d*Sqrt[x]))] + 13440*a*b*d^3*x^(3/2)*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))] - 13440*a*b*d^3*x^(3/2)*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))] - 1260*b^2*d^2*x*PolyLog[5, -E^((2*I)*(c + d*Sqrt[x]))] + (40320*I)*a*b*d^2*x*PolyLog[6, (-I)*E^(I*(c + d*Sqrt[x]))] - (40320*I)*a*b*d^2*x*PolyLog[6, I*E^(I*(c + d*Sqrt[x]))] - (1260*I)*b^2*d*Sqrt[x]*PolyLog[6, -E^((2*I)*(c + d*Sqrt[x]))] - 80640*a*b*d*Sqrt[x]*PolyLog[7, (-I)*E^(I*(c + d*Sqrt[x]))] + 80640*a*b*d*Sqrt[x]*PolyLog[7, I*E^(I*(c + d*Sqrt[x]))] + 630*b^2*PolyLog[7, -E^((2*I)*(c + d*Sqrt[x]))] - (80640*I)*a*b*PolyLog[8, (-I)*E^(I*(c + d*Sqrt[x]))] + (80640*I)*a*b*PolyLog[8, I*E^(I*(c + d*Sqrt[x]))] + 8*b^2*d^7*x^(7/2)*Tan[c + d*Sqrt[x]])/(4*d^8)","A",1
37,1,543,551,1.2568989,"\int x^2 \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2 \, dx","Integrate[x^2*(a + b*Sec[c + d*Sqrt[x]])^2,x]","\frac{a^2 d^6 x^3-24 i a b d^5 x^{5/2} \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)+60 i a b d^4 x^2 \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)-60 i a b d^4 x^2 \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)-240 a b d^3 x^{3/2} \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)+240 a b d^3 x^{3/2} \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)-720 i a b d^2 x \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)+720 i a b d^2 x \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)+1440 a b d \sqrt{x} \text{Li}_5\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)-1440 a b d \sqrt{x} \text{Li}_5\left(i e^{i \left(c+d \sqrt{x}\right)}\right)+1440 i a b \text{Li}_6\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)-1440 i a b \text{Li}_6\left(i e^{i \left(c+d \sqrt{x}\right)}\right)+6 b^2 d^5 x^{5/2} \tan \left(c+d \sqrt{x}\right)+30 b^2 d^4 x^2 \log \left(1+e^{2 i \left(c+d \sqrt{x}\right)}\right)-60 i b^2 d^3 x^{3/2} \text{Li}_2\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)+90 b^2 d^2 x \text{Li}_3\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)+90 i b^2 d \sqrt{x} \text{Li}_4\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)-45 b^2 \text{Li}_5\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)-6 i b^2 d^5 x^{5/2}}{3 d^6}","\frac{a^2 x^3}{3}+\frac{480 i a b \text{Li}_6\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}-\frac{480 i a b \text{Li}_6\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}+\frac{480 a b \sqrt{x} \text{Li}_5\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{480 a b \sqrt{x} \text{Li}_5\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{240 i a b x \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{240 i a b x \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{80 a b x^{3/2} \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{80 a b x^{3/2} \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{20 i a b x^2 \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{20 i a b x^2 \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{8 i a b x^{5/2} \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}-\frac{15 b^2 \text{Li}_5\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^6}+\frac{30 i b^2 \sqrt{x} \text{Li}_4\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^5}+\frac{30 b^2 x \text{Li}_3\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{20 i b^2 x^{3/2} \text{Li}_2\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{10 b^2 x^2 \log \left(1+e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^2}+\frac{2 b^2 x^{5/2} \tan \left(c+d \sqrt{x}\right)}{d}-\frac{2 i b^2 x^{5/2}}{d}",1,"((-6*I)*b^2*d^5*x^(5/2) + a^2*d^6*x^3 - (24*I)*a*b*d^5*x^(5/2)*ArcTan[E^(I*(c + d*Sqrt[x]))] + 30*b^2*d^4*x^2*Log[1 + E^((2*I)*(c + d*Sqrt[x]))] + (60*I)*a*b*d^4*x^2*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))] - (60*I)*a*b*d^4*x^2*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))] - (60*I)*b^2*d^3*x^(3/2)*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))] - 240*a*b*d^3*x^(3/2)*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))] + 240*a*b*d^3*x^(3/2)*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))] + 90*b^2*d^2*x*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))] - (720*I)*a*b*d^2*x*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))] + (720*I)*a*b*d^2*x*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))] + (90*I)*b^2*d*Sqrt[x]*PolyLog[4, -E^((2*I)*(c + d*Sqrt[x]))] + 1440*a*b*d*Sqrt[x]*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))] - 1440*a*b*d*Sqrt[x]*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))] - 45*b^2*PolyLog[5, -E^((2*I)*(c + d*Sqrt[x]))] + (1440*I)*a*b*PolyLog[6, (-I)*E^(I*(c + d*Sqrt[x]))] - (1440*I)*a*b*PolyLog[6, I*E^(I*(c + d*Sqrt[x]))] + 6*b^2*d^5*x^(5/2)*Tan[c + d*Sqrt[x]])/(3*d^6)","A",1
38,1,347,355,0.7077426,"\int x \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2 \, dx","Integrate[x*(a + b*Sec[c + d*Sqrt[x]])^2,x]","\frac{a^2 d^4 x^2-16 i a b d^3 x^{3/2} \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)+24 i a b d^2 x \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)-24 i a b d^2 x \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)-48 a b d \sqrt{x} \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)+48 a b d \sqrt{x} \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)-48 i a b \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)+48 i a b \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)+4 b^2 d^3 x^{3/2} \tan \left(c+d \sqrt{x}\right)+12 b^2 d^2 x \log \left(1+e^{2 i \left(c+d \sqrt{x}\right)}\right)-12 i b^2 d \sqrt{x} \text{Li}_2\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)+6 b^2 \text{Li}_3\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)-4 i b^2 d^3 x^{3/2}}{2 d^4}","\frac{a^2 x^2}{2}-\frac{24 i a b \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{24 i a b \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{24 a b \sqrt{x} \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{24 a b \sqrt{x} \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{12 i a b x \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{12 i a b x \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{8 i a b x^{3/2} \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}+\frac{3 b^2 \text{Li}_3\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{6 i b^2 \sqrt{x} \text{Li}_2\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{6 b^2 x \log \left(1+e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^2}+\frac{2 b^2 x^{3/2} \tan \left(c+d \sqrt{x}\right)}{d}-\frac{2 i b^2 x^{3/2}}{d}",1,"((-4*I)*b^2*d^3*x^(3/2) + a^2*d^4*x^2 - (16*I)*a*b*d^3*x^(3/2)*ArcTan[E^(I*(c + d*Sqrt[x]))] + 12*b^2*d^2*x*Log[1 + E^((2*I)*(c + d*Sqrt[x]))] + (24*I)*a*b*d^2*x*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))] - (24*I)*a*b*d^2*x*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))] - (12*I)*b^2*d*Sqrt[x]*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))] - 48*a*b*d*Sqrt[x]*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))] + 48*a*b*d*Sqrt[x]*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))] + 6*b^2*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))] - (48*I)*a*b*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))] + (48*I)*a*b*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))] + 4*b^2*d^3*x^(3/2)*Tan[c + d*Sqrt[x]])/(2*d^4)","A",1
39,0,0,23,40.0783662,"\int \frac{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}{x} \, dx","Integrate[(a + b*Sec[c + d*Sqrt[x]])^2/x,x]","\int \frac{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}{x} \, dx","\text{Int}\left(\frac{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}{x},x\right)",0,"Integrate[(a + b*Sec[c + d*Sqrt[x]])^2/x, x]","A",-1
40,0,0,23,21.0342929,"\int \frac{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}{x^2} \, dx","Integrate[(a + b*Sec[c + d*Sqrt[x]])^2/x^2,x]","\int \frac{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}{x^2} \, dx","\text{Int}\left(\frac{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}{x^2},x\right)",0,"Integrate[(a + b*Sec[c + d*Sqrt[x]])^2/x^2, x]","A",-1
41,1,1122,1041,2.6666189,"\int \frac{x^3}{a+b \sec \left(c+d \sqrt{x}\right)} \, dx","Integrate[x^3/(a + b*Sec[c + d*Sqrt[x]]),x]","\frac{\left(b+a \cos \left(c+d \sqrt{x}\right)\right) \left(x^4+\frac{8 b e^{i c} \left(7 x^3 \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^6-7 x^3 \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^6+i \left(x^{7/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^7-x^{7/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^7+42 x^{5/2} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^5-42 x^{5/2} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^5+210 i x^2 \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^4-210 i x^2 \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^4-840 x^{3/2} \text{Li}_5\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^3+840 x^{3/2} \text{Li}_5\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^3-2520 i x \text{Li}_6\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^2+2520 i x \text{Li}_6\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^2+5040 \sqrt{x} \text{Li}_7\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d-5040 \sqrt{x} \text{Li}_7\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d+5040 i \text{Li}_8\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-5040 i \text{Li}_8\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)\right)\right)}{d^8 \sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) \sec \left(c+d \sqrt{x}\right)}{4 a \left(a+b \sec \left(c+d \sqrt{x}\right)\right)}","\frac{x^4}{4 a}+\frac{2 i b \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-\sqrt{b^2-a^2}}+1\right) x^{7/2}}{a \sqrt{b^2-a^2} d}-\frac{2 i b \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+\sqrt{b^2-a^2}}+1\right) x^{7/2}}{a \sqrt{b^2-a^2} d}+\frac{14 b \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^3}{a \sqrt{b^2-a^2} d^2}-\frac{14 b \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^3}{a \sqrt{b^2-a^2} d^2}+\frac{84 i b \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{5/2}}{a \sqrt{b^2-a^2} d^3}-\frac{84 i b \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{5/2}}{a \sqrt{b^2-a^2} d^3}-\frac{420 b \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^2}{a \sqrt{b^2-a^2} d^4}+\frac{420 b \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^2}{a \sqrt{b^2-a^2} d^4}-\frac{1680 i b \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{3/2}}{a \sqrt{b^2-a^2} d^5}+\frac{1680 i b \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{3/2}}{a \sqrt{b^2-a^2} d^5}+\frac{5040 b \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x}{a \sqrt{b^2-a^2} d^6}-\frac{5040 b \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x}{a \sqrt{b^2-a^2} d^6}+\frac{10080 i b \text{Li}_7\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) \sqrt{x}}{a \sqrt{b^2-a^2} d^7}-\frac{10080 i b \text{Li}_7\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) \sqrt{x}}{a \sqrt{b^2-a^2} d^7}-\frac{10080 b \text{Li}_8\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a \sqrt{b^2-a^2} d^8}+\frac{10080 b \text{Li}_8\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a \sqrt{b^2-a^2} d^8}",1,"((b + a*Cos[c + d*Sqrt[x]])*(x^4 + (8*b*E^(I*c)*(7*d^6*x^3*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 7*d^6*x^3*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + I*(d^7*x^(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - d^7*x^(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 42*d^5*x^(5/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 42*d^5*x^(5/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (210*I)*d^4*x^2*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (210*I)*d^4*x^2*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 840*d^3*x^(3/2)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 840*d^3*x^(3/2)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (2520*I)*d^2*x*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (2520*I)*d^2*x*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 5040*d*Sqrt[x]*PolyLog[7, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 5040*d*Sqrt[x]*PolyLog[7, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (5040*I)*PolyLog[8, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (5040*I)*PolyLog[8, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])))/(d^8*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))*Sec[c + d*Sqrt[x]])/(4*a*(a + b*Sec[c + d*Sqrt[x]]))","A",1
42,1,858,781,2.1288306,"\int \frac{x^2}{a+b \sec \left(c+d \sqrt{x}\right)} \, dx","Integrate[x^2/(a + b*Sec[c + d*Sqrt[x]]),x]","\frac{\left(b+a \cos \left(c+d \sqrt{x}\right)\right) \left(x^3+\frac{6 b e^{i c} \left(5 x^2 \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^4-5 x^2 \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^4+i \left(x^{5/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^5-x^{5/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^5+20 x^{3/2} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^3-20 x^{3/2} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^3+60 i x \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^2-60 i x \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d^2-120 \sqrt{x} \text{Li}_5\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d+120 \sqrt{x} \text{Li}_5\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d-120 i \text{Li}_6\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+120 i \text{Li}_6\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)\right)\right)}{d^6 \sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) \sec \left(c+d \sqrt{x}\right)}{3 a \left(a+b \sec \left(c+d \sqrt{x}\right)\right)}","\frac{240 b \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^6 \sqrt{b^2-a^2}}-\frac{240 b \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^6 \sqrt{b^2-a^2}}-\frac{240 i b \sqrt{x} \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^5 \sqrt{b^2-a^2}}+\frac{240 i b \sqrt{x} \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^5 \sqrt{b^2-a^2}}-\frac{120 b x \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^4 \sqrt{b^2-a^2}}+\frac{120 b x \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^4 \sqrt{b^2-a^2}}+\frac{40 i b x^{3/2} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}-\frac{40 i b x^{3/2} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}+\frac{10 b x^2 \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}-\frac{10 b x^2 \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}+\frac{2 i b x^{5/2} \log \left(1+\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d \sqrt{b^2-a^2}}-\frac{2 i b x^{5/2} \log \left(1+\frac{a e^{i \left(c+d \sqrt{x}\right)}}{\sqrt{b^2-a^2}+b}\right)}{a d \sqrt{b^2-a^2}}+\frac{x^3}{3 a}",1,"((b + a*Cos[c + d*Sqrt[x]])*(x^3 + (6*b*E^(I*c)*(5*d^4*x^2*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 5*d^4*x^2*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + I*(d^5*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - d^5*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 20*d^3*x^(3/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 20*d^3*x^(3/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (60*I)*d^2*x*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (60*I)*d^2*x*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 120*d*Sqrt[x]*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 120*d*Sqrt[x]*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (120*I)*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (120*I)*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])))/(d^6*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))*Sec[c + d*Sqrt[x]])/(3*a*(a + b*Sec[c + d*Sqrt[x]]))","A",1
43,1,632,521,20.4419435,"\int \frac{x}{a+b \sec \left(c+d \sqrt{x}\right)} \, dx","Integrate[x/(a + b*Sec[c + d*Sqrt[x]]),x]","\frac{d^4 x^2 \sqrt{e^{2 i c} \left(b^2-a^2\right)}+4 i b e^{i c} d^3 x^{3/2} \log \left(1+\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{e^{2 i c} \left(b^2-a^2\right)}}\right)-4 i b e^{i c} d^3 x^{3/2} \log \left(1+\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{\sqrt{e^{2 i c} \left(b^2-a^2\right)}+b e^{i c}}\right)+12 b e^{i c} d^2 x \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-12 b e^{i c} d^2 x \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+24 i b e^{i c} d \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-24 i b e^{i c} d \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-24 b e^{i c} \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+24 b e^{i c} \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)}{2 a d^4 \sqrt{e^{2 i c} \left(b^2-a^2\right)}}","-\frac{12 b \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^4 \sqrt{b^2-a^2}}+\frac{12 b \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^4 \sqrt{b^2-a^2}}+\frac{12 i b \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}-\frac{12 i b \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}+\frac{6 b x \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}-\frac{6 b x \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}+\frac{2 i b x^{3/2} \log \left(1+\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d \sqrt{b^2-a^2}}-\frac{2 i b x^{3/2} \log \left(1+\frac{a e^{i \left(c+d \sqrt{x}\right)}}{\sqrt{b^2-a^2}+b}\right)}{a d \sqrt{b^2-a^2}}+\frac{x^2}{2 a}",1,"(d^4*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^2 + (4*I)*b*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (4*I)*b*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 12*b*d^2*E^(I*c)*x*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*b*d^2*E^(I*c)*x*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (24*I)*b*d*E^(I*c)*Sqrt[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (24*I)*b*d*E^(I*c)*Sqrt[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 24*b*E^(I*c)*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 24*b*E^(I*c)*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])/(2*a*d^4*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])","A",1
44,0,0,23,2.0128532,"\int \frac{1}{x \left(a+b \sec \left(c+d \sqrt{x}\right)\right)} \, dx","Integrate[1/(x*(a + b*Sec[c + d*Sqrt[x]])),x]","\int \frac{1}{x \left(a+b \sec \left(c+d \sqrt{x}\right)\right)} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \sec \left(c+d \sqrt{x}\right)\right)},x\right)",0,"Integrate[1/(x*(a + b*Sec[c + d*Sqrt[x]])), x]","A",-1
45,0,0,26,0.3896291,"\int \frac{a+b \sec \left(c+d \sqrt{x}\right)}{x^2} \, dx","Integrate[(a + b*Sec[c + d*Sqrt[x]])/x^2,x]","\int \frac{a+b \sec \left(c+d \sqrt{x}\right)}{x^2} \, dx","b \text{Int}\left(\frac{\sec \left(c+d \sqrt{x}\right)}{x^2},x\right)-\frac{a}{x}",0,"Integrate[(a + b*Sec[c + d*Sqrt[x]])/x^2, x]","A",-1
46,1,3702,3123,14.8777112,"\int \frac{x^3}{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[x^3/(a + b*Sec[c + d*Sqrt[x]])^2,x]","\text{Result too large to show}","\frac{x^4}{4 a^2}+\frac{4 i b \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-\sqrt{b^2-a^2}}+1\right) x^{7/2}}{a^2 \sqrt{b^2-a^2} d}-\frac{2 i b^3 \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-\sqrt{b^2-a^2}}+1\right) x^{7/2}}{a^2 \left(b^2-a^2\right)^{3/2} d}-\frac{4 i b \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+\sqrt{b^2-a^2}}+1\right) x^{7/2}}{a^2 \sqrt{b^2-a^2} d}+\frac{2 i b^3 \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+\sqrt{b^2-a^2}}+1\right) x^{7/2}}{a^2 \left(b^2-a^2\right)^{3/2} d}+\frac{2 b^2 \sin \left(c+d \sqrt{x}\right) x^{7/2}}{a \left(a^2-b^2\right) d \left(b+a \cos \left(c+d \sqrt{x}\right)\right)}-\frac{2 i b^2 x^{7/2}}{a^2 \left(a^2-b^2\right) d}+\frac{14 b^2 \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-i \sqrt{a^2-b^2}}+1\right) x^3}{a^2 \left(a^2-b^2\right) d^2}+\frac{14 b^2 \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+i \sqrt{a^2-b^2}}+1\right) x^3}{a^2 \left(a^2-b^2\right) d^2}+\frac{28 b \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^3}{a^2 \sqrt{b^2-a^2} d^2}-\frac{14 b^3 \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{28 b \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^3}{a^2 \sqrt{b^2-a^2} d^2}+\frac{14 b^3 \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{84 i b^2 \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) x^{5/2}}{a^2 \left(a^2-b^2\right) d^3}-\frac{84 i b^2 \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) x^{5/2}}{a^2 \left(a^2-b^2\right) d^3}+\frac{168 i b \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{5/2}}{a^2 \sqrt{b^2-a^2} d^3}-\frac{84 i b^3 \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{5/2}}{a^2 \left(b^2-a^2\right)^{3/2} d^3}-\frac{168 i b \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{5/2}}{a^2 \sqrt{b^2-a^2} d^3}+\frac{84 i b^3 \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{5/2}}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{420 b^2 \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) x^2}{a^2 \left(a^2-b^2\right) d^4}+\frac{420 b^2 \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) x^2}{a^2 \left(a^2-b^2\right) d^4}-\frac{840 b \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^2}{a^2 \sqrt{b^2-a^2} d^4}+\frac{420 b^3 \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^2}{a^2 \left(b^2-a^2\right)^{3/2} d^4}+\frac{840 b \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^2}{a^2 \sqrt{b^2-a^2} d^4}-\frac{420 b^3 \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^2}{a^2 \left(b^2-a^2\right)^{3/2} d^4}+\frac{1680 i b^2 \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) x^{3/2}}{a^2 \left(a^2-b^2\right) d^5}+\frac{1680 i b^2 \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) x^{3/2}}{a^2 \left(a^2-b^2\right) d^5}-\frac{3360 i b \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{3/2}}{a^2 \sqrt{b^2-a^2} d^5}+\frac{1680 i b^3 \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{3/2}}{a^2 \left(b^2-a^2\right)^{3/2} d^5}+\frac{3360 i b \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{3/2}}{a^2 \sqrt{b^2-a^2} d^5}-\frac{1680 i b^3 \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{3/2}}{a^2 \left(b^2-a^2\right)^{3/2} d^5}-\frac{5040 b^2 \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) x}{a^2 \left(a^2-b^2\right) d^6}-\frac{5040 b^2 \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) x}{a^2 \left(a^2-b^2\right) d^6}+\frac{10080 b \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x}{a^2 \sqrt{b^2-a^2} d^6}-\frac{5040 b^3 \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x}{a^2 \left(b^2-a^2\right)^{3/2} d^6}-\frac{10080 b \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x}{a^2 \sqrt{b^2-a^2} d^6}+\frac{5040 b^3 \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x}{a^2 \left(b^2-a^2\right)^{3/2} d^6}-\frac{10080 i b^2 \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) \sqrt{x}}{a^2 \left(a^2-b^2\right) d^7}-\frac{10080 i b^2 \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) \sqrt{x}}{a^2 \left(a^2-b^2\right) d^7}+\frac{20160 i b \text{Li}_7\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) \sqrt{x}}{a^2 \sqrt{b^2-a^2} d^7}-\frac{10080 i b^3 \text{Li}_7\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) \sqrt{x}}{a^2 \left(b^2-a^2\right)^{3/2} d^7}-\frac{20160 i b \text{Li}_7\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) \sqrt{x}}{a^2 \sqrt{b^2-a^2} d^7}+\frac{10080 i b^3 \text{Li}_7\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) \sqrt{x}}{a^2 \left(b^2-a^2\right)^{3/2} d^7}+\frac{10080 b^2 \text{Li}_7\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right) d^8}+\frac{10080 b^2 \text{Li}_7\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right) d^8}-\frac{20160 b \text{Li}_8\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 \sqrt{b^2-a^2} d^8}+\frac{10080 b^3 \text{Li}_8\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 \left(b^2-a^2\right)^{3/2} d^8}+\frac{20160 b \text{Li}_8\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 \sqrt{b^2-a^2} d^8}-\frac{10080 b^3 \text{Li}_8\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 \left(b^2-a^2\right)^{3/2} d^8}",1,"(x^4*(b + a*Cos[c + d*Sqrt[x]])^2*Sec[c + d*Sqrt[x]]^2)/(4*a^2*(a + b*Sec[c + d*Sqrt[x]])^2) + (2*b*E^(I*c)*(b + a*Cos[c + d*Sqrt[x]])^2*((-2*I)*b*E^(I*c)*x^(7/2) + ((1 + E^((2*I)*c))*(7*b*d^6*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^3*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (2*I)*a^2*d^7*E^(I*c)*x^(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - I*b^2*d^7*E^(I*c)*x^(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 7*b*d^6*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^3*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (2*I)*a^2*d^7*E^(I*c)*x^(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + I*b^2*d^7*E^(I*c)*x^(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 7*d^5*((6*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*x^(5/2)*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 7*d^5*((-6*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*x^(5/2)*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 210*b*d^4*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^2*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (84*I)*a^2*d^5*E^(I*c)*x^(5/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (42*I)*b^2*d^5*E^(I*c)*x^(5/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 210*b*d^4*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^2*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (84*I)*a^2*d^5*E^(I*c)*x^(5/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (42*I)*b^2*d^5*E^(I*c)*x^(5/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (840*I)*b*d^3*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^(3/2)*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 420*a^2*d^4*E^(I*c)*x^2*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 210*b^2*d^4*E^(I*c)*x^2*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (840*I)*b*d^3*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^(3/2)*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 420*a^2*d^4*E^(I*c)*x^2*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 210*b^2*d^4*E^(I*c)*x^2*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 2520*b*d^2*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (1680*I)*a^2*d^3*E^(I*c)*x^(3/2)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (840*I)*b^2*d^3*E^(I*c)*x^(3/2)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 2520*b*d^2*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (1680*I)*a^2*d^3*E^(I*c)*x^(3/2)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (840*I)*b^2*d^3*E^(I*c)*x^(3/2)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (5040*I)*b*d*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 5040*a^2*d^2*E^(I*c)*x*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 2520*b^2*d^2*E^(I*c)*x*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (5040*I)*b*d*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 5040*a^2*d^2*E^(I*c)*x*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 2520*b^2*d^2*E^(I*c)*x*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 5040*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*PolyLog[7, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (10080*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[7, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (5040*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[7, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 5040*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*PolyLog[7, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (10080*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[7, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (5040*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[7, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 10080*a^2*E^(I*c)*PolyLog[8, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 5040*b^2*E^(I*c)*PolyLog[8, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 10080*a^2*E^(I*c)*PolyLog[8, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 5040*b^2*E^(I*c)*PolyLog[8, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/(d^7*E^(I*c)*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))*Sec[c + d*Sqrt[x]]^2)/(a^2*(a^2 - b^2)*d*(1 + E^((2*I)*c))*(a + b*Sec[c + d*Sqrt[x]])^2) + (2*(b + a*Cos[c + d*Sqrt[x]])*Sec[c + d*Sqrt[x]]^2*(b^3*x^(7/2)*Sin[c] - a*b^2*x^(7/2)*Sin[d*Sqrt[x]]))/(a^2*(-a + b)*(a + b)*d*(a + b*Sec[c + d*Sqrt[x]])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2]))","A",0
47,1,2777,2323,13.2615494,"\int \frac{x^2}{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[x^2/(a + b*Sec[c + d*Sqrt[x]])^2,x]","\text{Result too large to show}","-\frac{2 i x^{5/2} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-\sqrt{b^2-a^2}}+1\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}+\frac{2 i x^{5/2} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+\sqrt{b^2-a^2}}+1\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}-\frac{10 x^2 \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}+\frac{10 x^2 \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{40 i x^{3/2} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{40 i x^{3/2} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{120 x \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^4}-\frac{120 x \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^4}+\frac{240 i \sqrt{x} \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^5}-\frac{240 i \sqrt{x} \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^5}-\frac{240 \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^6}+\frac{240 \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^6}-\frac{2 i x^{5/2} b^2}{a^2 \left(a^2-b^2\right) d}+\frac{10 x^2 \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-i \sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}+\frac{10 x^2 \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+i \sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}-\frac{40 i x^{3/2} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}-\frac{40 i x^{3/2} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}+\frac{120 x \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^4}+\frac{120 x \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^4}+\frac{240 i \sqrt{x} \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^5}+\frac{240 i \sqrt{x} \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^5}-\frac{240 \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^6}-\frac{240 \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^6}+\frac{2 x^{5/2} \sin \left(c+d \sqrt{x}\right) b^2}{a \left(a^2-b^2\right) d \left(b+a \cos \left(c+d \sqrt{x}\right)\right)}+\frac{4 i x^{5/2} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-\sqrt{b^2-a^2}}+1\right) b}{a^2 \sqrt{b^2-a^2} d}-\frac{4 i x^{5/2} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+\sqrt{b^2-a^2}}+1\right) b}{a^2 \sqrt{b^2-a^2} d}+\frac{20 x^2 \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}-\frac{20 x^2 \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}+\frac{80 i x^{3/2} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{80 i x^{3/2} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{240 x \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^4}+\frac{240 x \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^4}-\frac{480 i \sqrt{x} \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^5}+\frac{480 i \sqrt{x} \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^5}+\frac{480 \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^6}-\frac{480 \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^6}+\frac{x^3}{3 a^2}",1,"((-4*I)*b^2*E^((2*I)*c)*x^(5/2)*(b + a*Cos[c + d*Sqrt[x]])^2*Sec[c + d*Sqrt[x]]^2)/(a^2*(a^2 - b^2)*d*(1 + E^((2*I)*c))*(a + b*Sec[c + d*Sqrt[x]])^2) + (x^3*(b + a*Cos[c + d*Sqrt[x]])^2*Sec[c + d*Sqrt[x]]^2)/(3*a^2*(a + b*Sec[c + d*Sqrt[x]])^2) + (2*b*(b + a*Cos[c + d*Sqrt[x]])^2*(5*b*d^4*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (2*I)*a^2*d^5*E^(I*c)*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - I*b^2*d^5*E^(I*c)*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 5*b*d^4*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (2*I)*a^2*d^5*E^(I*c)*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + I*b^2*d^5*E^(I*c)*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 5*d^3*((4*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*x^(3/2)*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 5*d^3*((-4*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*x^(3/2)*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 60*b*d^2*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (40*I)*a^2*d^3*E^(I*c)*x^(3/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (20*I)*b^2*d^3*E^(I*c)*x^(3/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 60*b*d^2*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (40*I)*a^2*d^3*E^(I*c)*x^(3/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (20*I)*b^2*d^3*E^(I*c)*x^(3/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (120*I)*b*d*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 120*a^2*d^2*E^(I*c)*x*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 60*b^2*d^2*E^(I*c)*x*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (120*I)*b*d*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 120*a^2*d^2*E^(I*c)*x*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 60*b^2*d^2*E^(I*c)*x*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 120*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (240*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (120*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 120*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (240*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (120*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 240*a^2*E^(I*c)*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 120*b^2*E^(I*c)*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 240*a^2*E^(I*c)*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 120*b^2*E^(I*c)*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])*Sec[c + d*Sqrt[x]]^2)/(a^2*(a^2 - b^2)*d^6*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*(a + b*Sec[c + d*Sqrt[x]])^2) + (2*(b + a*Cos[c + d*Sqrt[x]])*Sec[c + d*Sqrt[x]]^2*(b^3*x^(5/2)*Sin[c] - a*b^2*x^(5/2)*Sin[d*Sqrt[x]]))/(a^2*(-a + b)*(a + b)*d*(a + b*Sec[c + d*Sqrt[x]])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2]))","A",0
48,1,1767,1523,15.9381038,"\int \frac{x}{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[x/(a + b*Sec[c + d*Sqrt[x]])^2,x]","\frac{2 \left(b+a \cos \left(c+d \sqrt{x}\right)\right) \left(b^3 x^{3/2} \sin (c)-a b^2 x^{3/2} \sin \left(d \sqrt{x}\right)\right) \sec ^2\left(c+d \sqrt{x}\right)}{a^2 (b-a) (a+b) d \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2 \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right)}+\frac{x^2 \left(b+a \cos \left(c+d \sqrt{x}\right)\right)^2 \sec ^2\left(c+d \sqrt{x}\right)}{2 a^2 \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}+\frac{2 b \left(b+a \cos \left(c+d \sqrt{x}\right)\right)^2 \left(\frac{2 i a^2 e^{i c} x^{3/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^3-i b^2 e^{i c} x^{3/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^3-2 i a^2 e^{i c} x^{3/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^3+i b^2 e^{i c} x^{3/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^3+3 b \sqrt{\left(b^2-a^2\right) e^{2 i c}} x \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2+3 b \sqrt{\left(b^2-a^2\right) e^{2 i c}} x \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) d^2-3 \left(-2 d e^{i c} \sqrt{x} a^2+2 i b \sqrt{\left(b^2-a^2\right) e^{2 i c}}+b^2 d e^{i c} \sqrt{x}\right) \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d+3 \left(-2 d e^{i c} \sqrt{x} a^2-2 i b \sqrt{\left(b^2-a^2\right) e^{2 i c}}+b^2 d e^{i c} \sqrt{x}\right) \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d+12 i a^2 e^{i c} \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d-6 i b^2 e^{i c} \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d-12 i a^2 e^{i c} \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d+6 i b^2 e^{i c} \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) d+6 b \sqrt{\left(b^2-a^2\right) e^{2 i c}} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+6 b \sqrt{\left(b^2-a^2\right) e^{2 i c}} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-12 a^2 e^{i c} \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+6 b^2 e^{i c} \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+12 a^2 e^{i c} \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-6 b^2 e^{i c} \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)}{\sqrt{\left(b^2-a^2\right) e^{2 i c}}}-\frac{2 i b d^3 e^{2 i c} x^{3/2}}{1+e^{2 i c}}\right) \sec ^2\left(c+d \sqrt{x}\right)}{a^2 \left(a^2-b^2\right) d^4 \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}","-\frac{2 i x^{3/2} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-\sqrt{b^2-a^2}}+1\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}+\frac{2 i x^{3/2} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+\sqrt{b^2-a^2}}+1\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}-\frac{6 x \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}+\frac{6 x \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{12 i \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{12 i \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{12 \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^4}-\frac{12 \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^4}-\frac{2 i x^{3/2} b^2}{a^2 \left(a^2-b^2\right) d}+\frac{6 x \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-i \sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}+\frac{6 x \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+i \sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}-\frac{12 i \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}-\frac{12 i \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}+\frac{12 \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^4}+\frac{12 \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^4}+\frac{2 x^{3/2} \sin \left(c+d \sqrt{x}\right) b^2}{a \left(a^2-b^2\right) d \left(b+a \cos \left(c+d \sqrt{x}\right)\right)}+\frac{4 i x^{3/2} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-\sqrt{b^2-a^2}}+1\right) b}{a^2 \sqrt{b^2-a^2} d}-\frac{4 i x^{3/2} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+\sqrt{b^2-a^2}}+1\right) b}{a^2 \sqrt{b^2-a^2} d}+\frac{12 x \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}-\frac{12 x \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}+\frac{24 i \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{24 i \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{24 \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^4}+\frac{24 \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^4}+\frac{x^2}{2 a^2}",1,"(x^2*(b + a*Cos[c + d*Sqrt[x]])^2*Sec[c + d*Sqrt[x]]^2)/(2*a^2*(a + b*Sec[c + d*Sqrt[x]])^2) + (2*b*(b + a*Cos[c + d*Sqrt[x]])^2*(((-2*I)*b*d^3*E^((2*I)*c)*x^(3/2))/(1 + E^((2*I)*c)) + (3*b*d^2*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (2*I)*a^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - I*b^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 3*b*d^2*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (2*I)*a^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + I*b^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 3*d*((2*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*Sqrt[x]*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 3*d*((-2*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*Sqrt[x]*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 6*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (12*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (6*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 6*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (12*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (6*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*a^2*E^(I*c)*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 6*b^2*E^(I*c)*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*a^2*E^(I*c)*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 6*b^2*E^(I*c)*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)])*Sec[c + d*Sqrt[x]]^2)/(a^2*(a^2 - b^2)*d^4*(a + b*Sec[c + d*Sqrt[x]])^2) + (2*(b + a*Cos[c + d*Sqrt[x]])*Sec[c + d*Sqrt[x]]^2*(b^3*x^(3/2)*Sin[c] - a*b^2*x^(3/2)*Sin[d*Sqrt[x]]))/(a^2*(-a + b)*(a + b)*d*(a + b*Sec[c + d*Sqrt[x]])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2]))","A",0
49,0,0,23,44.416654,"\int \frac{1}{x \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[1/(x*(a + b*Sec[c + d*Sqrt[x]])^2),x]","\int \frac{1}{x \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2},x\right)",0,"Integrate[1/(x*(a + b*Sec[c + d*Sqrt[x]])^2), x]","A",-1
50,0,0,23,31.5650995,"\int \frac{1}{x^2 \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[1/(x^2*(a + b*Sec[c + d*Sqrt[x]])^2),x]","\int \frac{1}{x^2 \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^2 \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2},x\right)",0,"Integrate[1/(x^2*(a + b*Sec[c + d*Sqrt[x]])^2), x]","A",-1
51,1,281,284,0.2537444,"\int x^{3/2} \left(a+b \sec \left(c+d \sqrt{x}\right)\right) \, dx","Integrate[x^(3/2)*(a + b*Sec[c + d*Sqrt[x]]),x]","\frac{2 \left(a d^5 x^{5/2}-10 i b d^4 x^2 \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)+20 i b d^3 x^{3/2} \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)-20 i b d^3 x^{3/2} \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)-60 b d^2 x \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)+60 b d^2 x \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)-120 i b d \sqrt{x} \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)+120 i b d \sqrt{x} \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)+120 b \text{Li}_5\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)-120 b \text{Li}_5\left(i e^{i \left(c+d \sqrt{x}\right)}\right)\right)}{5 d^5}","\frac{2}{5} a x^{5/2}+\frac{48 b \text{Li}_5\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{48 b \text{Li}_5\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{48 i b \sqrt{x} \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{48 i b \sqrt{x} \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{24 b x \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{24 b x \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{8 i b x^{3/2} \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{8 i b x^{3/2} \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 i b x^2 \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}",1,"(2*(a*d^5*x^(5/2) - (10*I)*b*d^4*x^2*ArcTan[E^(I*(c + d*Sqrt[x]))] + (20*I)*b*d^3*x^(3/2)*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))] - (20*I)*b*d^3*x^(3/2)*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))] - 60*b*d^2*x*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))] + 60*b*d^2*x*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))] - (120*I)*b*d*Sqrt[x]*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))] + (120*I)*b*d*Sqrt[x]*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))] + 120*b*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))] - 120*b*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))]))/(5*d^5)","A",1
52,1,155,158,0.1354721,"\int \sqrt{x} \left(a+b \sec \left(c+d \sqrt{x}\right)\right) \, dx","Integrate[Sqrt[x]*(a + b*Sec[c + d*Sqrt[x]]),x]","\frac{2 \left(a d^3 x^{3/2}-6 i b d^2 x \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)+6 i b d \sqrt{x} \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)-6 i b d \sqrt{x} \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)-6 b \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)+6 b \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)\right)}{3 d^3}","\frac{2}{3} a x^{3/2}-\frac{4 b \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{4 b \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{4 i b \sqrt{x} \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 i b \sqrt{x} \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 i b x \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}",1,"(2*(a*d^3*x^(3/2) - (6*I)*b*d^2*x*ArcTan[E^(I*(c + d*Sqrt[x]))] + (6*I)*b*d*Sqrt[x]*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))] - (6*I)*b*d*Sqrt[x]*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))] - 6*b*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))] + 6*b*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))]))/(3*d^3)","A",1
53,1,26,26,0.082696,"\int \frac{a+b \sec \left(c+d \sqrt{x}\right)}{\sqrt{x}} \, dx","Integrate[(a + b*Sec[c + d*Sqrt[x]])/Sqrt[x],x]","2 a \sqrt{x}+\frac{2 b \tanh ^{-1}\left(\sin \left(c+d \sqrt{x}\right)\right)}{d}","2 a \sqrt{x}+\frac{2 b \tanh ^{-1}\left(\sin \left(c+d \sqrt{x}\right)\right)}{d}",1,"2*a*Sqrt[x] + (2*b*ArcTanh[Sin[c + d*Sqrt[x]]])/d","A",1
54,0,0,30,2.8986013,"\int \frac{a+b \sec \left(c+d \sqrt{x}\right)}{x^{3/2}} \, dx","Integrate[(a + b*Sec[c + d*Sqrt[x]])/x^(3/2),x]","\int \frac{a+b \sec \left(c+d \sqrt{x}\right)}{x^{3/2}} \, dx","b \text{Int}\left(\frac{\sec \left(c+d \sqrt{x}\right)}{x^{3/2}},x\right)-\frac{2 a}{\sqrt{x}}",0,"Integrate[(a + b*Sec[c + d*Sqrt[x]])/x^(3/2), x]","A",-1
55,0,0,32,2.8334074,"\int \frac{a+b \sec \left(c+d \sqrt{x}\right)}{x^{5/2}} \, dx","Integrate[(a + b*Sec[c + d*Sqrt[x]])/x^(5/2),x]","\int \frac{a+b \sec \left(c+d \sqrt{x}\right)}{x^{5/2}} \, dx","b \text{Int}\left(\frac{\sec \left(c+d \sqrt{x}\right)}{x^{5/2}},x\right)-\frac{2 a}{3 x^{3/2}}",0,"Integrate[(a + b*Sec[c + d*Sqrt[x]])/x^(5/2), x]","A",-1
56,1,443,451,1.583115,"\int x^{3/2} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2 \, dx","Integrate[x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2,x]","\frac{2 \left(a^2 d^5 x^{5/2}-20 i a b d^4 x^2 \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)+40 i a b d^3 x^{3/2} \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)-40 i a b d^3 x^{3/2} \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)-120 a b d^2 x \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)+120 a b d^2 x \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)-240 i a b d \sqrt{x} \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)+240 i a b d \sqrt{x} \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)+240 a b \text{Li}_5\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)-240 a b \text{Li}_5\left(i e^{i \left(c+d \sqrt{x}\right)}\right)+5 b^2 d^4 x^2 \tan \left(c+d \sqrt{x}\right)+20 b^2 d^3 x^{3/2} \log \left(1+e^{2 i \left(c+d \sqrt{x}\right)}\right)-30 i b^2 d^2 x \text{Li}_2\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)+30 b^2 d \sqrt{x} \text{Li}_3\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)+15 i b^2 \text{Li}_4\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)-5 i b^2 d^4 x^2\right)}{5 d^5}","\frac{2}{5} a^2 x^{5/2}+\frac{96 a b \text{Li}_5\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{96 a b \text{Li}_5\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{96 i a b \sqrt{x} \text{Li}_4\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{96 i a b \sqrt{x} \text{Li}_4\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{48 a b x \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{48 a b x \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{16 i a b x^{3/2} \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{16 i a b x^{3/2} \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{8 i a b x^2 \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}+\frac{6 i b^2 \text{Li}_4\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^5}+\frac{12 b^2 \sqrt{x} \text{Li}_3\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{12 i b^2 x \text{Li}_2\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{8 b^2 x^{3/2} \log \left(1+e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^2}+\frac{2 b^2 x^2 \tan \left(c+d \sqrt{x}\right)}{d}-\frac{2 i b^2 x^2}{d}",1,"(2*((-5*I)*b^2*d^4*x^2 + a^2*d^5*x^(5/2) - (20*I)*a*b*d^4*x^2*ArcTan[E^(I*(c + d*Sqrt[x]))] + 20*b^2*d^3*x^(3/2)*Log[1 + E^((2*I)*(c + d*Sqrt[x]))] + (40*I)*a*b*d^3*x^(3/2)*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))] - (40*I)*a*b*d^3*x^(3/2)*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))] - (30*I)*b^2*d^2*x*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))] - 120*a*b*d^2*x*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))] + 120*a*b*d^2*x*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))] + 30*b^2*d*Sqrt[x]*PolyLog[3, -E^((2*I)*(c + d*Sqrt[x]))] - (240*I)*a*b*d*Sqrt[x]*PolyLog[4, (-I)*E^(I*(c + d*Sqrt[x]))] + (240*I)*a*b*d*Sqrt[x]*PolyLog[4, I*E^(I*(c + d*Sqrt[x]))] + (15*I)*b^2*PolyLog[4, -E^((2*I)*(c + d*Sqrt[x]))] + 240*a*b*PolyLog[5, (-I)*E^(I*(c + d*Sqrt[x]))] - 240*a*b*PolyLog[5, I*E^(I*(c + d*Sqrt[x]))] + 5*b^2*d^4*x^2*Tan[c + d*Sqrt[x]]))/(5*d^5)","A",1
57,1,247,255,0.8226703,"\int \sqrt{x} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2 \, dx","Integrate[Sqrt[x]*(a + b*Sec[c + d*Sqrt[x]])^2,x]","\frac{2 \left(a^2 d^3 x^{3/2}-12 i a b d^2 x \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)+12 i a b d \sqrt{x} \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)-12 i a b d \sqrt{x} \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)-12 a b \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)+12 a b \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)+3 b^2 d^2 x \tan \left(c+d \sqrt{x}\right)-3 i b^2 \text{Li}_2\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)+6 b^2 d \sqrt{x} \log \left(1+e^{2 i \left(c+d \sqrt{x}\right)}\right)-3 i b^2 d^2 x\right)}{3 d^3}","\frac{2}{3} a^2 x^{3/2}-\frac{8 a b \text{Li}_3\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{8 a b \text{Li}_3\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{8 i a b \sqrt{x} \text{Li}_2\left(-i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{8 i a b \sqrt{x} \text{Li}_2\left(i e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{8 i a b x \tan ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}-\frac{2 i b^2 \text{Li}_2\left(-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{4 b^2 \sqrt{x} \log \left(1+e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^2}+\frac{2 b^2 x \tan \left(c+d \sqrt{x}\right)}{d}-\frac{2 i b^2 x}{d}",1,"(2*((-3*I)*b^2*d^2*x + a^2*d^3*x^(3/2) - (12*I)*a*b*d^2*x*ArcTan[E^(I*(c + d*Sqrt[x]))] + 6*b^2*d*Sqrt[x]*Log[1 + E^((2*I)*(c + d*Sqrt[x]))] + (12*I)*a*b*d*Sqrt[x]*PolyLog[2, (-I)*E^(I*(c + d*Sqrt[x]))] - (12*I)*a*b*d*Sqrt[x]*PolyLog[2, I*E^(I*(c + d*Sqrt[x]))] - (3*I)*b^2*PolyLog[2, -E^((2*I)*(c + d*Sqrt[x]))] - 12*a*b*PolyLog[3, (-I)*E^(I*(c + d*Sqrt[x]))] + 12*a*b*PolyLog[3, I*E^(I*(c + d*Sqrt[x]))] + 3*b^2*d^2*x*Tan[c + d*Sqrt[x]]))/(3*d^3)","A",1
58,1,45,47,0.2786707,"\int \frac{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}{\sqrt{x}} \, dx","Integrate[(a + b*Sec[c + d*Sqrt[x]])^2/Sqrt[x],x]","\frac{2 \left(a^2 d \sqrt{x}+2 a b \tanh ^{-1}\left(\sin \left(c+d \sqrt{x}\right)\right)+b^2 \tan \left(c+d \sqrt{x}\right)\right)}{d}","2 a^2 \sqrt{x}+\frac{4 a b \tanh ^{-1}\left(\sin \left(c+d \sqrt{x}\right)\right)}{d}+\frac{2 b^2 \tan \left(c+d \sqrt{x}\right)}{d}",1,"(2*(a^2*d*Sqrt[x] + 2*a*b*ArcTanh[Sin[c + d*Sqrt[x]]] + b^2*Tan[c + d*Sqrt[x]]))/d","A",1
59,0,0,25,23.1260993,"\int \frac{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}{x^{3/2}} \, dx","Integrate[(a + b*Sec[c + d*Sqrt[x]])^2/x^(3/2),x]","\int \frac{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}{x^{3/2}} \, dx","\text{Int}\left(\frac{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}{x^{3/2}},x\right)",0,"Integrate[(a + b*Sec[c + d*Sqrt[x]])^2/x^(3/2), x]","A",-1
60,0,0,25,23.8109296,"\int \frac{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}{x^{5/2}} \, dx","Integrate[(a + b*Sec[c + d*Sqrt[x]])^2/x^(5/2),x]","\int \frac{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}{x^{5/2}} \, dx","\text{Int}\left(\frac{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}{x^{5/2}},x\right)",0,"Integrate[(a + b*Sec[c + d*Sqrt[x]])^2/x^(5/2), x]","A",-1
61,1,725,653,2.4922118,"\int \frac{x^{3/2}}{a+b \sec \left(c+d \sqrt{x}\right)} \, dx","Integrate[x^(3/2)/(a + b*Sec[c + d*Sqrt[x]]),x]","\frac{2 \sec \left(c+d \sqrt{x}\right) \left(a \cos \left(c+d \sqrt{x}\right)+b\right) \left(x^{5/2}+\frac{5 i b e^{i c} \left(d^4 x^2 \log \left(1+\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{e^{2 i c} \left(b^2-a^2\right)}}\right)-d^4 x^2 \log \left(1+\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{\sqrt{e^{2 i c} \left(b^2-a^2\right)}+b e^{i c}}\right)-4 i d^3 x^{3/2} \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+4 i d^3 x^{3/2} \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+12 d^2 x \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-12 d^2 x \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+24 i d \sqrt{x} \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-24 i d \sqrt{x} \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-24 \text{Li}_5\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+24 \text{Li}_5\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)\right)}{d^5 \sqrt{e^{2 i c} \left(b^2-a^2\right)}}\right)}{5 a \left(a+b \sec \left(c+d \sqrt{x}\right)\right)}","-\frac{48 i b \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^5 \sqrt{b^2-a^2}}+\frac{48 i b \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^5 \sqrt{b^2-a^2}}-\frac{48 b \sqrt{x} \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^4 \sqrt{b^2-a^2}}+\frac{48 b \sqrt{x} \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^4 \sqrt{b^2-a^2}}+\frac{24 i b x \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}-\frac{24 i b x \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}+\frac{8 b x^{3/2} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}-\frac{8 b x^{3/2} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}+\frac{2 i b x^2 \log \left(1+\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d \sqrt{b^2-a^2}}-\frac{2 i b x^2 \log \left(1+\frac{a e^{i \left(c+d \sqrt{x}\right)}}{\sqrt{b^2-a^2}+b}\right)}{a d \sqrt{b^2-a^2}}+\frac{2 x^{5/2}}{5 a}",1,"(2*(b + a*Cos[c + d*Sqrt[x]])*(x^(5/2) + ((5*I)*b*E^(I*c)*(d^4*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - d^4*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (4*I)*d^3*x^(3/2)*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (4*I)*d^3*x^(3/2)*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 12*d^2*x*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 12*d^2*x*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (24*I)*d*Sqrt[x]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (24*I)*d*Sqrt[x]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 24*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 24*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/(d^5*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))*Sec[c + d*Sqrt[x]])/(5*a*(a + b*Sec[c + d*Sqrt[x]]))","A",1
62,1,486,393,6.7001865,"\int \frac{\sqrt{x}}{a+b \sec \left(c+d \sqrt{x}\right)} \, dx","Integrate[Sqrt[x]/(a + b*Sec[c + d*Sqrt[x]]),x]","\frac{2 \left(d^3 x^{3/2} \sqrt{e^{2 i c} \left(b^2-a^2\right)}+3 i b e^{i c} d^2 x \log \left(1+\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{e^{2 i c} \left(b^2-a^2\right)}}\right)-3 i b e^{i c} d^2 x \log \left(1+\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{\sqrt{e^{2 i c} \left(b^2-a^2\right)}+b e^{i c}}\right)+6 b e^{i c} d \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-6 b e^{i c} d \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+6 i b e^{i c} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-6 i b e^{i c} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)\right)}{3 a d^3 \sqrt{e^{2 i c} \left(b^2-a^2\right)}}","\frac{4 i b \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}-\frac{4 i b \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}+\frac{4 b \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}-\frac{4 b \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}+\frac{2 i b x \log \left(1+\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d \sqrt{b^2-a^2}}-\frac{2 i b x \log \left(1+\frac{a e^{i \left(c+d \sqrt{x}\right)}}{\sqrt{b^2-a^2}+b}\right)}{a d \sqrt{b^2-a^2}}+\frac{2 x^{3/2}}{3 a}",1,"(2*(d^3*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^(3/2) + (3*I)*b*d^2*E^(I*c)*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (3*I)*b*d^2*E^(I*c)*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 6*b*d*E^(I*c)*Sqrt[x]*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 6*b*d*E^(I*c)*Sqrt[x]*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (6*I)*b*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (6*I)*b*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/(3*a*d^3*Sqrt[(-a^2 + b^2)*E^((2*I)*c)])","A",1
63,1,69,68,0.2627228,"\int \frac{1}{\sqrt{x} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)} \, dx","Integrate[1/(Sqrt[x]*(a + b*Sec[c + d*Sqrt[x]])),x]","\frac{2 \left(\frac{2 b \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)}{\sqrt{a^2-b^2}}\right)}{d \sqrt{a^2-b^2}}+\frac{c}{d}+\sqrt{x}\right)}{a}","\frac{2 \sqrt{x}}{a}-\frac{4 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)}{\sqrt{a+b}}\right)}{a d \sqrt{a-b} \sqrt{a+b}}",1,"(2*(c/d + Sqrt[x] + (2*b*ArcTanh[((-a + b)*Tan[(c + d*Sqrt[x])/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*d)))/a","A",1
64,0,0,25,4.3989842,"\int \frac{1}{x^{3/2} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)} \, dx","Integrate[1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])),x]","\int \frac{1}{x^{3/2} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)} \, dx","\text{Int}\left(\frac{1}{x^{3/2} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)},x\right)",0,"Integrate[1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])), x]","A",-1
65,0,0,25,3.9550714,"\int \frac{1}{x^{5/2} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)} \, dx","Integrate[1/(x^(5/2)*(a + b*Sec[c + d*Sqrt[x]])),x]","\int \frac{1}{x^{5/2} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)} \, dx","\text{Int}\left(\frac{1}{x^{5/2} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)},x\right)",0,"Integrate[1/(x^(5/2)*(a + b*Sec[c + d*Sqrt[x]])), x]","A",-1
66,1,2254,1925,13.8326275,"\int \frac{x^{3/2}}{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[x^(3/2)/(a + b*Sec[c + d*Sqrt[x]])^2,x]","\text{Result too large to show}","-\frac{2 i x^2 \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-\sqrt{b^2-a^2}}+1\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}+\frac{2 i x^2 \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+\sqrt{b^2-a^2}}+1\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}-\frac{8 x^{3/2} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}+\frac{8 x^{3/2} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{24 i x \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{24 i x \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{48 \sqrt{x} \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^4}-\frac{48 \sqrt{x} \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^4}+\frac{48 i \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^5}-\frac{48 i \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^5}-\frac{2 i x^2 b^2}{a^2 \left(a^2-b^2\right) d}+\frac{8 x^{3/2} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-i \sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}+\frac{8 x^{3/2} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+i \sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}-\frac{24 i x \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}-\frac{24 i x \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}+\frac{48 \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^4}+\frac{48 \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^4}+\frac{48 i \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^5}+\frac{48 i \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^5}+\frac{2 x^2 \sin \left(c+d \sqrt{x}\right) b^2}{a \left(a^2-b^2\right) d \left(b+a \cos \left(c+d \sqrt{x}\right)\right)}+\frac{4 i x^2 \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-\sqrt{b^2-a^2}}+1\right) b}{a^2 \sqrt{b^2-a^2} d}-\frac{4 i x^2 \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+\sqrt{b^2-a^2}}+1\right) b}{a^2 \sqrt{b^2-a^2} d}+\frac{16 x^{3/2} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}-\frac{16 x^{3/2} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}+\frac{48 i x \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{48 i x \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{96 \sqrt{x} \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^4}+\frac{96 \sqrt{x} \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^4}-\frac{96 i \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^5}+\frac{96 i \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^5}+\frac{2 x^{5/2}}{5 a^2}",1,"(2*x^(5/2)*(b + a*Cos[c + d*Sqrt[x]])^2*Sec[c + d*Sqrt[x]]^2)/(5*a^2*(a + b*Sec[c + d*Sqrt[x]])^2) + (2*b*E^(I*c)*(b + a*Cos[c + d*Sqrt[x]])^2*((-2*I)*b*E^(I*c)*x^2 + ((1 + E^((2*I)*c))*(4*b*d^3*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (2*I)*a^2*d^4*E^(I*c)*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - I*b^2*d^4*E^(I*c)*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 4*b*d^3*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (2*I)*a^2*d^4*E^(I*c)*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + I*b^2*d^4*E^(I*c)*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 4*d^2*((3*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*x*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 4*d^2*((-3*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*x*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 24*b*d*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (24*I)*a^2*d^2*E^(I*c)*x*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (12*I)*b^2*d^2*E^(I*c)*x*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 24*b*d*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (24*I)*a^2*d^2*E^(I*c)*x*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (12*I)*b^2*d^2*E^(I*c)*x*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (24*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 48*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 24*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (24*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 48*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - 24*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (48*I)*a^2*E^(I*c)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (24*I)*b^2*E^(I*c)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (48*I)*a^2*E^(I*c)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (24*I)*b^2*E^(I*c)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))]))/(d^4*E^(I*c)*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))*Sec[c + d*Sqrt[x]]^2)/(a^2*(a^2 - b^2)*d*(1 + E^((2*I)*c))*(a + b*Sec[c + d*Sqrt[x]])^2) + (2*(b + a*Cos[c + d*Sqrt[x]])*Sec[c + d*Sqrt[x]]^2*(b^3*x^2*Sin[c] - a*b^2*x^2*Sin[d*Sqrt[x]]))/(a^2*(-a + b)*(a + b)*d*(a + b*Sec[c + d*Sqrt[x]])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2]))","A",0
67,1,1210,1125,9.439354,"\int \frac{\sqrt{x}}{\left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[Sqrt[x]/(a + b*Sec[c + d*Sqrt[x]])^2,x]","\frac{2 \left(b+a \cos \left(c+d \sqrt{x}\right)\right) \sec ^2\left(c+d \sqrt{x}\right) \left(\frac{3 x \left(a \sin \left(d \sqrt{x}\right)-b \sin (c)\right) b^2}{(a-b) (a+b) d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right)}+\frac{3 \left(b+a \cos \left(c+d \sqrt{x}\right)\right) \left(\frac{2 i d^2 e^{i c} x \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) a^2-2 i d^2 e^{i c} x \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right) a^2+4 i e^{i c} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) a^2-4 i e^{i c} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right) a^2-i b^2 d^2 e^{i c} x \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right)+2 b d \sqrt{\left(b^2-a^2\right) e^{2 i c}} \sqrt{x} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right)+i b^2 d^2 e^{i c} x \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right)+2 b d \sqrt{\left(b^2-a^2\right) e^{2 i c}} \sqrt{x} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}+1\right)+2 \left(2 d e^{i c} \sqrt{x} a^2-i b \sqrt{\left(b^2-a^2\right) e^{2 i c}}-b^2 d e^{i c} \sqrt{x}\right) \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+2 \left(-2 d e^{i c} \sqrt{x} a^2-i b \sqrt{\left(b^2-a^2\right) e^{2 i c}}+b^2 d e^{i c} \sqrt{x}\right) \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)-2 i b^2 e^{i c} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{b e^{i c}-\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)+2 i b^2 e^{i c} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+\sqrt{\left(b^2-a^2\right) e^{2 i c}}}\right)}{\sqrt{\left(b^2-a^2\right) e^{2 i c}}}-\frac{2 i b d^2 e^{2 i c} x}{1+e^{2 i c}}\right) b}{\left(a^2-b^2\right) d^3}+x^{3/2} \left(b+a \cos \left(c+d \sqrt{x}\right)\right)\right)}{3 a^2 \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2}","-\frac{2 i x \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-\sqrt{b^2-a^2}}+1\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}+\frac{2 i x \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+\sqrt{b^2-a^2}}+1\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}-\frac{4 \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}+\frac{4 \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{4 i \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{4 i \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}-\frac{2 i x b^2}{a^2 \left(a^2-b^2\right) d}+\frac{4 \sqrt{x} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-i \sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}+\frac{4 \sqrt{x} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+i \sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}-\frac{4 i \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}-\frac{4 i \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}+\frac{2 x \sin \left(c+d \sqrt{x}\right) b^2}{a \left(a^2-b^2\right) d \left(b+a \cos \left(c+d \sqrt{x}\right)\right)}+\frac{4 i x \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b-\sqrt{b^2-a^2}}+1\right) b}{a^2 \sqrt{b^2-a^2} d}-\frac{4 i x \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{b+\sqrt{b^2-a^2}}+1\right) b}{a^2 \sqrt{b^2-a^2} d}+\frac{8 \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}-\frac{8 \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}+\frac{8 i \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{8 i \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}+\frac{2 x^{3/2}}{3 a^2}",1,"(2*(b + a*Cos[c + d*Sqrt[x]])*Sec[c + d*Sqrt[x]]^2*(x^(3/2)*(b + a*Cos[c + d*Sqrt[x]]) + (3*b*(b + a*Cos[c + d*Sqrt[x]])*(((-2*I)*b*d^2*E^((2*I)*c)*x)/(1 + E^((2*I)*c)) + (2*b*d*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*Sqrt[x]*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + (2*I)*a^2*d^2*E^(I*c)*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - I*b^2*d^2*E^(I*c)*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*b*d*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*Sqrt[x]*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (2*I)*a^2*d^2*E^(I*c)*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + I*b^2*d^2*E^(I*c)*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] + 2*((-I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] + 2*a^2*d*E^(I*c)*Sqrt[x] - b^2*d*E^(I*c)*Sqrt[x])*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 2*((-I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (4*I)*a^2*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (2*I)*b^2*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] - (4*I)*a^2*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (2*I)*b^2*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))])/Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))/((a^2 - b^2)*d^3) + (3*b^2*x*(-(b*Sin[c]) + a*Sin[d*Sqrt[x]]))/((a - b)*(a + b)*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2]))))/(3*a^2*(a + b*Sec[c + d*Sqrt[x]])^2)","A",0
68,1,163,127,0.7934883,"\int \frac{1}{\sqrt{x} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[1/(Sqrt[x]*(a + b*Sec[c + d*Sqrt[x]])^2),x]","\frac{2 \left(\frac{b \left(\left(a^2-b^2\right) \left(c+d \sqrt{x}\right)+a b \sin \left(c+d \sqrt{x}\right)\right)+a \left(a^2-b^2\right) \left(c+d \sqrt{x}\right) \cos \left(c+d \sqrt{x}\right)}{a \cos \left(c+d \sqrt{x}\right)+b}-\frac{2 b \left(b^2-2 a^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}\right)}{a^2 d (a-b) (a+b)}","-\frac{4 b \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{2 b^2 \tan \left(c+d \sqrt{x}\right)}{a d \left(a^2-b^2\right) \left(a+b \sec \left(c+d \sqrt{x}\right)\right)}+\frac{2 \sqrt{x}}{a^2}",1,"(2*((-2*b*(-2*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*Sqrt[x])/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (a*(a^2 - b^2)*(c + d*Sqrt[x])*Cos[c + d*Sqrt[x]] + b*((a^2 - b^2)*(c + d*Sqrt[x]) + a*b*Sin[c + d*Sqrt[x]]))/(b + a*Cos[c + d*Sqrt[x]])))/(a^2*(a - b)*(a + b)*d)","A",1
69,0,0,25,29.0757427,"\int \frac{1}{x^{3/2} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2),x]","\int \frac{1}{x^{3/2} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^{3/2} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2},x\right)",0,"Integrate[1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2), x]","A",-1
70,0,0,25,29.2109073,"\int \frac{1}{x^{5/2} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[1/(x^(5/2)*(a + b*Sec[c + d*Sqrt[x]])^2),x]","\int \frac{1}{x^{5/2} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^{5/2} \left(a+b \sec \left(c+d \sqrt{x}\right)\right)^2},x\right)",0,"Integrate[1/(x^(5/2)*(a + b*Sec[c + d*Sqrt[x]])^2), x]","A",-1
71,0,0,32,3.0865598,"\int (e x)^m \left(a+b \sec \left(c+d x^n\right)\right)^p \, dx","Integrate[(e*x)^m*(a + b*Sec[c + d*x^n])^p,x]","\int (e x)^m \left(a+b \sec \left(c+d x^n\right)\right)^p \, dx","x^{-m} (e x)^m \text{Int}\left(x^m \left(a+b \sec \left(c+d x^n\right)\right)^p,x\right)",0,"Integrate[(e*x)^m*(a + b*Sec[c + d*x^n])^p, x]","A",-1
72,1,38,44,0.1161566,"\int (e x)^{-1+n} \left(a+b \sec \left(c+d x^n\right)\right) \, dx","Integrate[(e*x)^(-1 + n)*(a + b*Sec[c + d*x^n]),x]","\frac{x^{-n} (e x)^n \left(a d x^n+b \tanh ^{-1}\left(\sin \left(c+d x^n\right)\right)\right)}{d e n}","\frac{a (e x)^n}{e n}+\frac{b x^{-n} (e x)^n \tanh ^{-1}\left(\sin \left(c+d x^n\right)\right)}{d e n}",1,"((e*x)^n*(a*d*x^n + b*ArcTanh[Sin[c + d*x^n]]))/(d*e*n*x^n)","A",1
73,1,188,149,0.6456684,"\int (e x)^{-1+2 n} \left(a+b \sec \left(c+d x^n\right)\right) \, dx","Integrate[(e*x)^(-1 + 2*n)*(a + b*Sec[c + d*x^n]),x]","\frac{(e x)^{2 n} \cos \left(c+d x^n\right) \left(a+b \sec \left(c+d x^n\right)\right) \left(a+\frac{b x^{-2 n} \left(2 i \left(\text{Li}_2\left(-i e^{-i \left(d x^n+c\right)}\right)-\text{Li}_2\left(i e^{-i \left(d x^n+c\right)}\right)\right)+\left(-2 c-2 d x^n+\pi \right) \left(\log \left(1-i e^{-i \left(c+d x^n\right)}\right)-\log \left(1+i e^{-i \left(c+d x^n\right)}\right)\right)-(\pi -2 c) \log \left(\cot \left(\frac{1}{4} \left(2 c+2 d x^n+\pi \right)\right)\right)\right)}{d^2}\right)}{2 e n \left(a \cos \left(c+d x^n\right)+b\right)}","\frac{a (e x)^{2 n}}{2 e n}+\frac{i b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(-i e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{i b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(i e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{2 i b x^{-n} (e x)^{2 n} \tan ^{-1}\left(e^{i \left(c+d x^n\right)}\right)}{d e n}",1,"((e*x)^(2*n)*Cos[c + d*x^n]*(a + (b*((-2*c + Pi - 2*d*x^n)*(Log[1 - I/E^(I*(c + d*x^n))] - Log[1 + I/E^(I*(c + d*x^n))]) - (-2*c + Pi)*Log[Cot[(2*c + Pi + 2*d*x^n)/4]] + (2*I)*(PolyLog[2, (-I)/E^(I*(c + d*x^n))] - PolyLog[2, I/E^(I*(c + d*x^n))])))/(d^2*x^(2*n)))*(a + b*Sec[c + d*x^n]))/(2*e*n*(b + a*Cos[c + d*x^n]))","A",1
74,0,0,235,1.3240458,"\int (e x)^{-1+3 n} \left(a+b \sec \left(c+d x^n\right)\right) \, dx","Integrate[(e*x)^(-1 + 3*n)*(a + b*Sec[c + d*x^n]),x]","\int (e x)^{-1+3 n} \left(a+b \sec \left(c+d x^n\right)\right) \, dx","\frac{a (e x)^{3 n}}{3 e n}-\frac{2 b x^{-3 n} (e x)^{3 n} \text{Li}_3\left(-i e^{i \left(d x^n+c\right)}\right)}{d^3 e n}+\frac{2 b x^{-3 n} (e x)^{3 n} \text{Li}_3\left(i e^{i \left(d x^n+c\right)}\right)}{d^3 e n}+\frac{2 i b x^{-2 n} (e x)^{3 n} \text{Li}_2\left(-i e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{2 i b x^{-2 n} (e x)^{3 n} \text{Li}_2\left(i e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{2 i b x^{-n} (e x)^{3 n} \tan ^{-1}\left(e^{i \left(c+d x^n\right)}\right)}{d e n}",1,"Integrate[(e*x)^(-1 + 3*n)*(a + b*Sec[c + d*x^n]), x]","F",-1
75,1,54,79,0.4173431,"\int (e x)^{-1+n} \left(a+b \sec \left(c+d x^n\right)\right)^2 \, dx","Integrate[(e*x)^(-1 + n)*(a + b*Sec[c + d*x^n])^2,x]","\frac{x^{-n} (e x)^n \left(a^2 d x^n+2 a b \tanh ^{-1}\left(\sin \left(c+d x^n\right)\right)+b^2 \tan \left(c+d x^n\right)\right)}{d e n}","\frac{a^2 (e x)^n}{e n}+\frac{2 a b x^{-n} (e x)^n \tanh ^{-1}\left(\sin \left(c+d x^n\right)\right)}{d e n}+\frac{b^2 x^{-n} (e x)^n \tan \left(c+d x^n\right)}{d e n}",1,"((e*x)^n*(a^2*d*x^n + 2*a*b*ArcTanh[Sin[c + d*x^n]] + b^2*Tan[c + d*x^n]))/(d*e*n*x^n)","A",1
76,1,347,221,5.1618321,"\int (e x)^{-1+2 n} \left(a+b \sec \left(c+d x^n\right)\right)^2 \, dx","Integrate[(e*x)^(-1 + 2*n)*(a + b*Sec[c + d*x^n])^2,x]","\frac{x^{-2 n} (e x)^{2 n} \left(d x^n \left(a^2 d x^n+2 b^2 \tan (c)\right)-\frac{4 a b \csc (c) \left(i \text{Li}_2\left(-e^{i \left(d x^n-\tan ^{-1}(\cot (c))\right)}\right)-i \text{Li}_2\left(e^{i \left(d x^n-\tan ^{-1}(\cot (c))\right)}\right)+\left(d x^n-\tan ^{-1}(\cot (c))\right) \left(\log \left(1-e^{i \left(d x^n-\tan ^{-1}(\cot (c))\right)}\right)-\log \left(1+e^{i \left(d x^n-\tan ^{-1}(\cot (c))\right)}\right)\right)\right)}{\sqrt{\csc ^2(c)}}+8 a b \tan ^{-1}(\cot (c)) \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x^n}{2}\right)+\sin (c)\right)-2 b^2 d \tan (c) x^n+\frac{2 b^2 d x^n \sin \left(\frac{d x^n}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} \left(c+d x^n\right)\right)-\sin \left(\frac{1}{2} \left(c+d x^n\right)\right)\right)}+\frac{2 b^2 d x^n \sin \left(\frac{d x^n}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} \left(c+d x^n\right)\right)+\cos \left(\frac{1}{2} \left(c+d x^n\right)\right)\right)}+2 b^2 \left(d \tan (c) x^n+\log \left(\cos \left(c+d x^n\right)\right)\right)\right)}{2 d^2 e n}","\frac{a^2 (e x)^{2 n}}{2 e n}+\frac{2 i a b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(-i e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{2 i a b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(i e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{4 i a b x^{-n} (e x)^{2 n} \tan ^{-1}\left(e^{i \left(c+d x^n\right)}\right)}{d e n}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left(\cos \left(c+d x^n\right)\right)}{d^2 e n}+\frac{b^2 x^{-n} (e x)^{2 n} \tan \left(c+d x^n\right)}{d e n}",1,"((e*x)^(2*n)*(8*a*b*ArcTan[Cot[c]]*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x^n)/2]] - (4*a*b*Csc[c]*((d*x^n - ArcTan[Cot[c]])*(Log[1 - E^(I*(d*x^n - ArcTan[Cot[c]]))] - Log[1 + E^(I*(d*x^n - ArcTan[Cot[c]]))]) + I*PolyLog[2, -E^(I*(d*x^n - ArcTan[Cot[c]]))] - I*PolyLog[2, E^(I*(d*x^n - ArcTan[Cot[c]]))]))/Sqrt[Csc[c]^2] + (2*b^2*d*x^n*Sin[(d*x^n)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x^n)/2] - Sin[(c + d*x^n)/2])) + (2*b^2*d*x^n*Sin[(d*x^n)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x^n)/2] + Sin[(c + d*x^n)/2])) - 2*b^2*d*x^n*Tan[c] + d*x^n*(a^2*d*x^n + 2*b^2*Tan[c]) + 2*b^2*(Log[Cos[c + d*x^n]] + d*x^n*Tan[c])))/(2*d^2*e*n*x^(2*n))","A",0
77,0,0,390,11.6616195,"\int (e x)^{-1+3 n} \left(a+b \sec \left(c+d x^n\right)\right)^2 \, dx","Integrate[(e*x)^(-1 + 3*n)*(a + b*Sec[c + d*x^n])^2,x]","\int (e x)^{-1+3 n} \left(a+b \sec \left(c+d x^n\right)\right)^2 \, dx","\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{4 a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left(-i e^{i \left(d x^n+c\right)}\right)}{d^3 e n}+\frac{4 a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left(i e^{i \left(d x^n+c\right)}\right)}{d^3 e n}+\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left(-i e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left(i e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{4 i a b x^{-n} (e x)^{3 n} \tan ^{-1}\left(e^{i \left(c+d x^n\right)}\right)}{d e n}-\frac{i b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left(-e^{2 i \left(d x^n+c\right)}\right)}{d^3 e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left(1+e^{2 i \left(c+d x^n\right)}\right)}{d^2 e n}+\frac{b^2 x^{-n} (e x)^{3 n} \tan \left(c+d x^n\right)}{d e n}-\frac{i b^2 x^{-n} (e x)^{3 n}}{d e n}",1,"Integrate[(e*x)^(-1 + 3*n)*(a + b*Sec[c + d*x^n])^2, x]","F",-1
78,1,80,87,0.2876866,"\int \frac{(e x)^{-1+n}}{a+b \sec \left(c+d x^n\right)} \, dx","Integrate[(e*x)^(-1 + n)/(a + b*Sec[c + d*x^n]),x]","\frac{(e x)^n \left(\frac{2 b x^{-n} \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} \left(c+d x^n\right)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+c x^{-n}+d\right)}{a d e n}","\frac{(e x)^n}{a e n}-\frac{2 b x^{-n} (e x)^n \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} \left(c+d x^n\right)\right)}{\sqrt{a+b}}\right)}{a d e n \sqrt{a-b} \sqrt{a+b}}",1,"((e*x)^n*(d + c/x^n + (2*b*ArcTanh[((-a + b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*x^n)))/(a*d*e*n)","A",1
79,1,861,328,1.9471102,"\int \frac{(e x)^{-1+2 n}}{a+b \sec \left(c+d x^n\right)} \, dx","Integrate[(e*x)^(-1 + 2*n)/(a + b*Sec[c + d*x^n]),x]","\frac{(e x)^{2 n} \left(b+a \cos \left(d x^n+c\right)\right) \left(1-\frac{2 b x^{-2 n} \left(2 \left(d x^n+c\right) \tanh ^{-1}\left(\frac{(a+b) \cot \left(\frac{1}{2} \left(d x^n+c\right)\right)}{\sqrt{a^2-b^2}}\right)-2 \left(c+\cos ^{-1}\left(-\frac{b}{a}\right)\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} \left(d x^n+c\right)\right)}{\sqrt{a^2-b^2}}\right)+\left(\cos ^{-1}\left(-\frac{b}{a}\right)-2 i \tanh ^{-1}\left(\frac{(a+b) \cot \left(\frac{1}{2} \left(d x^n+c\right)\right)}{\sqrt{a^2-b^2}}\right)+2 i \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} \left(d x^n+c\right)\right)}{\sqrt{a^2-b^2}}\right)\right) \log \left(\frac{\sqrt{a^2-b^2} e^{-\frac{1}{2} i \left(d x^n+c\right)}}{\sqrt{2} \sqrt{a} \sqrt{b+a \cos \left(d x^n+c\right)}}\right)+\left(\cos ^{-1}\left(-\frac{b}{a}\right)+2 i \left(\tanh ^{-1}\left(\frac{(a+b) \cot \left(\frac{1}{2} \left(d x^n+c\right)\right)}{\sqrt{a^2-b^2}}\right)-\tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} \left(d x^n+c\right)\right)}{\sqrt{a^2-b^2}}\right)\right)\right) \log \left(\frac{\sqrt{a^2-b^2} e^{\frac{1}{2} i \left(d x^n+c\right)}}{\sqrt{2} \sqrt{a} \sqrt{b+a \cos \left(d x^n+c\right)}}\right)-\left(\cos ^{-1}\left(-\frac{b}{a}\right)-2 i \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} \left(d x^n+c\right)\right)}{\sqrt{a^2-b^2}}\right)\right) \log \left(\frac{(a+b) \left(a-b-i \sqrt{a^2-b^2}\right) \left(i \tan \left(\frac{1}{2} \left(d x^n+c\right)\right)+1\right)}{a \left(a+b+\sqrt{a^2-b^2} \tan \left(\frac{1}{2} \left(d x^n+c\right)\right)\right)}\right)-\left(\cos ^{-1}\left(-\frac{b}{a}\right)+2 i \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} \left(d x^n+c\right)\right)}{\sqrt{a^2-b^2}}\right)\right) \log \left(\frac{(a+b) \left(-i a+i b+\sqrt{a^2-b^2}\right) \left(\tan \left(\frac{1}{2} \left(d x^n+c\right)\right)+i\right)}{a \left(a+b+\sqrt{a^2-b^2} \tan \left(\frac{1}{2} \left(d x^n+c\right)\right)\right)}\right)+i \left(\text{Li}_2\left(\frac{\left(b-i \sqrt{a^2-b^2}\right) \left(a+b-\sqrt{a^2-b^2} \tan \left(\frac{1}{2} \left(d x^n+c\right)\right)\right)}{a \left(a+b+\sqrt{a^2-b^2} \tan \left(\frac{1}{2} \left(d x^n+c\right)\right)\right)}\right)-\text{Li}_2\left(\frac{\left(b+i \sqrt{a^2-b^2}\right) \left(a+b-\sqrt{a^2-b^2} \tan \left(\frac{1}{2} \left(d x^n+c\right)\right)\right)}{a \left(a+b+\sqrt{a^2-b^2} \tan \left(\frac{1}{2} \left(d x^n+c\right)\right)\right)}\right)\right)\right)}{\sqrt{a^2-b^2} d^2}\right) \sec \left(d x^n+c\right)}{2 a e n \left(a+b \sec \left(d x^n+c\right)\right)}","\frac{b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^2 e n \sqrt{b^2-a^2}}-\frac{b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^2 e n \sqrt{b^2-a^2}}+\frac{i b x^{-n} (e x)^{2 n} \log \left(1+\frac{a e^{i \left(c+d x^n\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d e n \sqrt{b^2-a^2}}-\frac{i b x^{-n} (e x)^{2 n} \log \left(1+\frac{a e^{i \left(c+d x^n\right)}}{\sqrt{b^2-a^2}+b}\right)}{a d e n \sqrt{b^2-a^2}}+\frac{(e x)^{2 n}}{2 a e n}",1,"((e*x)^(2*n)*(b + a*Cos[c + d*x^n])*(1 - (2*b*(2*(c + d*x^n)*ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - 2*(c + ArcCos[-(b/a)])*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(b/a)] - (2*I)*ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[Sqrt[a^2 - b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(c + d*x^n))*Sqrt[b + a*Cos[c + d*x^n]])] + (ArcCos[-(b/a)] + (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2)*(c + d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Cos[c + d*x^n]])] - (ArcCos[-(b/a)] - (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[((a + b)*(a - b - I*Sqrt[a^2 - b^2])*(1 + I*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[((a + b)*((-I)*a + I*b + Sqrt[a^2 - b^2])*(I + Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] + I*(PolyLog[2, ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] - PolyLog[2, ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))])))/(Sqrt[a^2 - b^2]*d^2*x^(2*n)))*Sec[c + d*x^n])/(2*a*e*n*(a + b*Sec[c + d*x^n]))","B",1
80,0,0,485,1.837158,"\int \frac{(e x)^{-1+3 n}}{a+b \sec \left(c+d x^n\right)} \, dx","Integrate[(e*x)^(-1 + 3*n)/(a + b*Sec[c + d*x^n]),x]","\int \frac{(e x)^{-1+3 n}}{a+b \sec \left(c+d x^n\right)} \, dx","\frac{2 i b x^{-3 n} (e x)^{3 n} \text{Li}_3\left(-\frac{a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^3 e n \sqrt{b^2-a^2}}-\frac{2 i b x^{-3 n} (e x)^{3 n} \text{Li}_3\left(-\frac{a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^3 e n \sqrt{b^2-a^2}}+\frac{2 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^2 e n \sqrt{b^2-a^2}}-\frac{2 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^2 e n \sqrt{b^2-a^2}}+\frac{i b x^{-n} (e x)^{3 n} \log \left(1+\frac{a e^{i \left(c+d x^n\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d e n \sqrt{b^2-a^2}}-\frac{i b x^{-n} (e x)^{3 n} \log \left(1+\frac{a e^{i \left(c+d x^n\right)}}{\sqrt{b^2-a^2}+b}\right)}{a d e n \sqrt{b^2-a^2}}+\frac{(e x)^{3 n}}{3 a e n}",1,"Integrate[(e*x)^(-1 + 3*n)/(a + b*Sec[c + d*x^n]), x]","F",-1
81,1,191,157,0.9321411,"\int \frac{(e x)^{-1+n}}{\left(a+b \sec \left(c+d x^n\right)\right)^2} \, dx","Integrate[(e*x)^(-1 + n)/(a + b*Sec[c + d*x^n])^2,x]","\frac{x^{-n} (e x)^n \left(\sqrt{a^2-b^2} \left(b \left(\left(a^2-b^2\right) \left(c+d x^n\right)+a b \sin \left(c+d x^n\right)\right)+a \left(a^2-b^2\right) \left(c+d x^n\right) \cos \left(c+d x^n\right)\right)-2 b \left(b^2-2 a^2\right) \left(a \cos \left(c+d x^n\right)+b\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} \left(c+d x^n\right)\right)}{\sqrt{a^2-b^2}}\right)\right)}{a^2 d e n (a-b) (a+b) \sqrt{a^2-b^2} \left(a \cos \left(c+d x^n\right)+b\right)}","-\frac{2 b \left(2 a^2-b^2\right) x^{-n} (e x)^n \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} \left(c+d x^n\right)\right)}{\sqrt{a+b}}\right)}{a^2 d e n (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 x^{-n} (e x)^n \tan \left(c+d x^n\right)}{a d e n \left(a^2-b^2\right) \left(a+b \sec \left(c+d x^n\right)\right)}+\frac{(e x)^n}{a^2 e n}",1,"((e*x)^n*(-2*b*(-2*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x^n]) + Sqrt[a^2 - b^2]*(a*(a^2 - b^2)*(c + d*x^n)*Cos[c + d*x^n] + b*((a^2 - b^2)*(c + d*x^n) + a*b*Sin[c + d*x^n]))))/(a^2*(a - b)*(a + b)*Sqrt[a^2 - b^2]*d*e*n*x^n*(b + a*Cos[c + d*x^n]))","A",1
82,1,2450,757,10.4728924,"\int \frac{(e x)^{-1+2 n}}{\left(a+b \sec \left(c+d x^n\right)\right)^2} \, dx","Integrate[(e*x)^(-1 + 2*n)/(a + b*Sec[c + d*x^n])^2,x]","\text{Result too large to show}","\frac{2 b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 d^2 e n \sqrt{b^2-a^2}}-\frac{2 b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 d^2 e n \sqrt{b^2-a^2}}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left(a \cos \left(c+d x^n\right)+b\right)}{a^2 d^2 e n \left(a^2-b^2\right)}+\frac{2 i b x^{-n} (e x)^{2 n} \log \left(1+\frac{a e^{i \left(c+d x^n\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 d e n \sqrt{b^2-a^2}}-\frac{2 i b x^{-n} (e x)^{2 n} \log \left(1+\frac{a e^{i \left(c+d x^n\right)}}{\sqrt{b^2-a^2}+b}\right)}{a^2 d e n \sqrt{b^2-a^2}}+\frac{b^2 x^{-n} (e x)^{2 n} \sin \left(c+d x^n\right)}{a d e n \left(a^2-b^2\right) \left(a \cos \left(c+d x^n\right)+b\right)}-\frac{b^3 x^{-2 n} (e x)^{2 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 d^2 e n \left(b^2-a^2\right)^{3/2}}+\frac{b^3 x^{-2 n} (e x)^{2 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 d^2 e n \left(b^2-a^2\right)^{3/2}}-\frac{i b^3 x^{-n} (e x)^{2 n} \log \left(1+\frac{a e^{i \left(c+d x^n\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 d e n \left(b^2-a^2\right)^{3/2}}+\frac{i b^3 x^{-n} (e x)^{2 n} \log \left(1+\frac{a e^{i \left(c+d x^n\right)}}{\sqrt{b^2-a^2}+b}\right)}{a^2 d e n \left(b^2-a^2\right)^{3/2}}+\frac{(e x)^{2 n}}{2 a^2 e n}",1,"(-2*b*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^2*(2*(c + d*x^n)*ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - 2*(c + ArcCos[-(b/a)])*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(b/a)] - (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]]))*Log[Sqrt[a^2 - b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(c + d*x^n))*Sqrt[b + a*Cos[c + d*x^n]])] + (ArcCos[-(b/a)] + (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2)*(c + d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Cos[c + d*x^n]])] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] + (-ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] + I*(PolyLog[2, ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] - PolyLog[2, ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))]))*Sec[c + d*x^n]^2)/((a^2 - b^2)^(3/2)*d^2*n*(a + b*Sec[c + d*x^n])^2) + (b^3*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^2*(2*(c + d*x^n)*ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - 2*(c + ArcCos[-(b/a)])*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(b/a)] - (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]]))*Log[Sqrt[a^2 - b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(c + d*x^n))*Sqrt[b + a*Cos[c + d*x^n]])] + (ArcCos[-(b/a)] + (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2)*(c + d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Cos[c + d*x^n]])] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] + (-ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] + I*(PolyLog[2, ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] - PolyLog[2, ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))]))*Sec[c + d*x^n]^2)/(a^2*(a^2 - b^2)^(3/2)*d^2*n*(a + b*Sec[c + d*x^n])^2) + (x^(1 - n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^2*Sec[c + d*x^n]^2*(a^2*d*x^n*Cos[c] - b^2*d*x^n*Cos[c] + 2*b^2*Sin[c]))/(2*a^2*(a - b)*(a + b)*d*n*(a + b*Sec[c + d*x^n])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])) + (b^2*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^2*Sec[c]*Sec[c + d*x^n]^2*(a*Cos[c]*Log[b + a*Cos[c]*Cos[d*x^n] - a*Sin[c]*Sin[d*x^n]] + a*d*x^n*Sin[c] - ((2*I)*a*b*ArcTan[((-I)*a*Sin[c] - I*(-b + a*Cos[c])*Tan[(d*x^n)/2])/Sqrt[-b^2 + a^2*Cos[c]^2 + a^2*Sin[c]^2]]*Sin[c])/Sqrt[-b^2 + a^2*Cos[c]^2 + a^2*Sin[c]^2]))/(a*(a^2 - b^2)*d^2*n*(a + b*Sec[c + d*x^n])^2*(a^2*Cos[c]^2 + a^2*Sin[c]^2)) + (b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])*Sec[c + d*x^n]^2*(b*Sin[c] - a*Sin[d*x^n]))/(a^2*(-a + b)*(a + b)*d*n*(a + b*Sec[c + d*x^n])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])) + (b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^2*Sec[c + d*x^n]^2*Tan[c])/(a^2*(-a^2 + b^2)*d*n*(a + b*Sec[c + d*x^n])^2) - ((2*I)*b^3*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*ArcTan[(b + a*Cos[c + d*x^n] + I*a*Sin[c + d*x^n])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x^n])^2*Sec[c + d*x^n]^2*Tan[c])/(a^2*(a^2 - b^2)^(3/2)*d^2*n*(a + b*Sec[c + d*x^n])^2)","B",0
83,0,0,1384,11.748176,"\int \frac{(e x)^{-1+3 n}}{\left(a+b \sec \left(c+d x^n\right)\right)^2} \, dx","Integrate[(e*x)^(-1 + 3*n)/(a + b*Sec[c + d*x^n])^2,x]","\int \frac{(e x)^{-1+3 n}}{\left(a+b \sec \left(c+d x^n\right)\right)^2} \, dx","-\frac{2 i b^2 (e x)^{3 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{b-i \sqrt{a^2-b^2}}\right) x^{-3 n}}{a^2 \left(a^2-b^2\right) d^3 e n}-\frac{2 i b^2 (e x)^{3 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{b+i \sqrt{a^2-b^2}}\right) x^{-3 n}}{a^2 \left(a^2-b^2\right) d^3 e n}+\frac{4 i b (e x)^{3 n} \text{Li}_3\left(-\frac{a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^{-3 n}}{a^2 \sqrt{b^2-a^2} d^3 e n}-\frac{2 i b^3 (e x)^{3 n} \text{Li}_3\left(-\frac{a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^{-3 n}}{a^2 \left(b^2-a^2\right)^{3/2} d^3 e n}-\frac{4 i b (e x)^{3 n} \text{Li}_3\left(-\frac{a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^{-3 n}}{a^2 \sqrt{b^2-a^2} d^3 e n}+\frac{2 i b^3 (e x)^{3 n} \text{Li}_3\left(-\frac{a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^{-3 n}}{a^2 \left(b^2-a^2\right)^{3/2} d^3 e n}+\frac{2 b^2 (e x)^{3 n} \log \left(\frac{e^{i \left(d x^n+c\right)} a}{b-i \sqrt{a^2-b^2}}+1\right) x^{-2 n}}{a^2 \left(a^2-b^2\right) d^2 e n}+\frac{2 b^2 (e x)^{3 n} \log \left(\frac{e^{i \left(d x^n+c\right)} a}{b+i \sqrt{a^2-b^2}}+1\right) x^{-2 n}}{a^2 \left(a^2-b^2\right) d^2 e n}+\frac{4 b (e x)^{3 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^{-2 n}}{a^2 \sqrt{b^2-a^2} d^2 e n}-\frac{2 b^3 (e x)^{3 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^{-2 n}}{a^2 \left(b^2-a^2\right)^{3/2} d^2 e n}-\frac{4 b (e x)^{3 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^{-2 n}}{a^2 \sqrt{b^2-a^2} d^2 e n}+\frac{2 b^3 (e x)^{3 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^{-2 n}}{a^2 \left(b^2-a^2\right)^{3/2} d^2 e n}-\frac{i b^2 (e x)^{3 n} x^{-n}}{a^2 \left(a^2-b^2\right) d e n}+\frac{2 i b (e x)^{3 n} \log \left(\frac{e^{i \left(d x^n+c\right)} a}{b-\sqrt{b^2-a^2}}+1\right) x^{-n}}{a^2 \sqrt{b^2-a^2} d e n}-\frac{i b^3 (e x)^{3 n} \log \left(\frac{e^{i \left(d x^n+c\right)} a}{b-\sqrt{b^2-a^2}}+1\right) x^{-n}}{a^2 \left(b^2-a^2\right)^{3/2} d e n}-\frac{2 i b (e x)^{3 n} \log \left(\frac{e^{i \left(d x^n+c\right)} a}{b+\sqrt{b^2-a^2}}+1\right) x^{-n}}{a^2 \sqrt{b^2-a^2} d e n}+\frac{i b^3 (e x)^{3 n} \log \left(\frac{e^{i \left(d x^n+c\right)} a}{b+\sqrt{b^2-a^2}}+1\right) x^{-n}}{a^2 \left(b^2-a^2\right)^{3/2} d e n}+\frac{b^2 (e x)^{3 n} \sin \left(d x^n+c\right) x^{-n}}{a \left(a^2-b^2\right) d e n \left(b+a \cos \left(d x^n+c\right)\right)}+\frac{(e x)^{3 n}}{3 a^2 e n}",1,"Integrate[(e*x)^(-1 + 3*n)/(a + b*Sec[c + d*x^n])^2, x]","F",-1